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Publicly Accessible Penn Dissertations

Spring 2010

Dark Matter Halo Mergers and Quasars Dark Matter Halo Mergers and Quasars

Jorge Moreno University of Pennsylvania, jmoreno@haverford.edu

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Dark Matter Halo Mergers and Quasars Dark Matter Halo Mergers and Quasars

Abstract Abstract The formation and evolution of galaxies and the supermassive black holes they harbor at their nuclei depends strongly on the merger history of their surrounding dark matter haloes. First we developed a semi-analytic algorithm that describes the merger history tree of a halo. The following tests were performed: the conditional mass function, the time and mass distributions at formation, and the time distribution of the last major merger. We provide a model for the creation rate of dark matter haloes, informed by both coagulation theory and the modified excursion set approach with moving barriers. A comparison with N-body simulations shows that our square-root barrier merger rate is significantly better than the standard extended Press-Schechter rate used in the literature. The last chapter is dedicated to a simple model of quasar activation by major mergers of dark matter haloes. The model consists of two main ingredients: the halo merger rate describing triggering, and a quasar light curve, which tracks the evolution of individual quasars. In this model, the mass of the seed black hole at triggering is assumed to be a fixed fraction of its mass at the peak luminosity. The light curve has two components: an exponential ascending phase and a power-law descending phase that depends on mass of the host. We postulate a self-regulation condition between the peak luminosity of the active galactic nucleus (AGN) and the mass of the host halo at triggering. This type of model for quasar evolution is at the heart of the latest semianalytic models (SAMs) of galaxy formation and it is therefore definitely worth studying in some detail. By carefully revisiting some of the main issues linked to this approach, we were able to derive several interesting and physically meaningful constraints regarding black hole evolution. We expect simple, yet observationally-constrained models like ours to play a central role in future models of galaxy formation.

Degree Type Degree Type Dissertation

Degree Name Degree Name Doctor of Philosophy (PhD)

Graduate Group Graduate Group Physics & Astronomy

First Advisor First Advisor Ravi K. Sheth

Keywords Keywords galaxies, black holes, quasars, haloes

Subject Categories Subject Categories Cosmology, Relativity, and Gravity | External Galaxies

This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/177

DARK MATTER HALO MERGERS AND QUASARS

Jorge Moreno

A DISSERTATION

in

Physics and Astronomy

Presented to the Faculties of the University of Pennsylvania inPartial Fulfillment of the Requirements for the Degree of Doctor of

Philosophy

2010

Supervisor of Dissertation

Ravi K. Sheth, Professor of Physics and Astronomy

Graduate Group Chairman

Ravi K. Sheth, Professor of Physics and Astronomy

Dissertation Committee

Gary Bernstein, Professor of Physics and Astronomy

Adam Lidz, Assistant Professor of Physics and Astronomy

Andrea J. Liu, Professor of Physics and Astronomy

Gordon T. Richards, Assistant Professor of Physics

Dedication

This thesis is dedicated to my wife Viridiana, and our children Camila and Mateo.

You are the best thing a person can have. Thank you for your love, patience and

encouragement during these difficult years and for giving meaning to my life.

ii

Acknowledgments

To break the tradition, I wish to thank the most important people in my life first:

my wonderful wife Viridiana Moreno and our children Camila and Mateo, to whom

this thesis is dedicated. I would also like to thank my parents Martha Soto and

Juan Moreno for doing everything in their power so I had the opportunities they

could not have. Also to my brothers Hector and Juan Ernesto, for always having

my back. Likewise, I would like to thank Rafael Moreno, Irene Gomez, Vianney

Moreno, Demian Cortes, Jacqueline Ibarrola, Sandra Arvezu, little Paulo – and of

course, the rest of the family (too many to mention individually).

It is my pleasure to thank my thesis advisor, Ravi Sheth. My father always says

that the two most important decisions in a person’s life is choosing who they are

going to marry, and choosing their career. I would replace the latter with “choosing

the right Ph.D advisor”. I personally could not have chosen a better mentor. There

are not enough pages, or years, to show my gratitude to Ravi. I also want to

thank Francesco Shankar and Carlo Giocoli for all their support, encouragement,

and the wonderful opportunity of writing papers together. Also I wish to thank

iii

my other collaborators: Bepi Tormen, David Weinberg, Raul Angulo, Cheng Li,

Martin Crocce, and Federico Marulli. I thank Beth Willman for her encouragement,

guidance, and friendship during these last few difficult months.

I wish to thank Gary Bernstein, Adam Lidz, Andrea Liu, Gordon Richards, and

of course, Ravi Sheth, for agreeing to serve as committee members, and for tak-

ing the time and dedication to read this rather long thesis. Also to my professors

for imparting their knowledge and advice, including Gary Bernstein, Mariangela

Bernardi, Bhuvnesh Jain, Mark Devlin, Masao Sato, Vijay Balasubramanian, Burt

Ovrut, Mirjam Cvetic, Mark Gulian, Randy Kamien, Joe Kroll, Phil Nelson, Paul

Langacker, and Fay Ajzenberg-Selove. To CONACyT-Mexico, and Grant 2002352

from the US-Israel BSF for funding. Also to colleagues at Haverford College for their

constant support, especially Walter Smith, Steve Boughn, and Bruce Partridge for

their mentoring; and Beth Willman, Stephon Alexander, Peter Love, Anna Sajina,

Chema Diego, Adi Zolotov, and Aurelia Gomez Unamuno for their friendship. To

folks at Cinvestav for making my move to Penn possible, including my former advi-

sor Hugo Compean, and my professors Arnulfo Zepeda, Juan Jose Godina, Denjoe

O’Connor, Tonatiuh Matos, Eloy Ayon, Mauricio Carvajal, Gerardo Herrera, Piotr

Kielanowski, Masiej Pzranowski, Gabriel Lopez, Jorge Hirsch, Jose Mendez, and

Miguel Angel Perez. But mostly to Guillermo Moreno, Jerzy Plebanski and Au-

gusto Garcia, may they rest in peace. Also, my friends Ramon Castaneda, Armando

Perez, Jose Manuel Lara Bauche, Argelia Bernal, Carlos Chavez, Jorge Mercado,

iv

Juan Barranco, Idrish Huet, Eduard de la Cruz Burelo, Ildefonso Leon, Americo

Rodriguez and Pedro Podesta. To my friends at Penn, including Joey Hyde, Mike

Fischbein, Andres Plazas, Zhaoming Ma, Jacek Guzik, Matt Martino, Graziano

Rossi, Dan Swetz, Ed Chapin, Amitai Bin-Nun, Tsz Yan Lam, Laura Marian,

Reiko Nakajima, Federica Bianco, Greg Dobler, Josko Kirigin, Alex Borisov, Hite

and Willis Geffert, Dutch Ratliff, and many others for the great years together and

their friendship. I wish to thank all my teachers for encouraging me, especially Mrs.

and Mr. Blackwood, Mr. Erdman, Ms. Faridian, Mr. Guember, Ms. Kahn, Mr.

Lange, Ms. Hooper, Ms. Komatsu, Ms. Martin, Ms. O’Connell, Ms. Scharf, Mr.

Silverman, Mr. Taylor, Ms. Votto, and maestras Eva, Edith and Maria Luisa. And

a few good friends that have supported me during the years: Noah Bray-Ali, Adam

Ferrell, Alex Prieto and Rosa Elia Victorio.

Parts of these work were done with the direct help, suggestions, and inputs of

Silvia Bonoli, Marcella Brusa, Anca Constantin, Onsi Fakhouri, Joey Hyde, Avi

Loeb, Eyal Neistein, Robert Smith and Yue Shen. Many others, listed below,

contributed indirectly in many ways.

I wish to thank the following people for the insightful conversations, comments,

questions, encouragement and hospitality during my ‘talk tour’ where presenta-

tions of this work were delivered. Although it is a rather long list, the input from

every single one of you has contributed to make me a better scientist. At Cin-

vestav: Eduard de la Cruz Burelo, Miguel Garcia, Isaac Hernandez, Miguel Angel

v

Perez, Tonatiuh Matos, Arnulfo Zepeda, At UNAM: Vladimir Avila-Reese, Anto-

nio Peimbert, Octavio Valenzuela. At INAOE: Vahram Chavushyan, David Hughes,

Hector Ibarra, Omar Lopez Cruz, Divakara Mayya, Alfredo Montana. At Colgate:

Thomas Balonek, Jeff Bary. At Williams: Marek Demianski, Karen Kwitter, Steve

Souza. At Texas: Guillermo Blanc, Amy Forestell, Martin Gaskell, Karl Gebhardt,

Donghui Jeoung, Jarrett Johnson, Jun Koda, Eiichiro Komatsu, Randi Ludwig, Yi

Mao, Irina Marinova, Milos Milosavljevic, Paul Shapiro, Matashoshi Shoji, Chalence

Timer, Berverly Wills. At Harvard: Federica Bianco, Laura Blecha, Anca Con-

stantin, Claude-Andre Faucher-Guiguere, Ryan Hickox, Avi Loeb, Ryan O’Leary,

Adam Lidz, Matt Mcquinn, Ramesh Narajan, Jonathan Pritchard, Jaiyul Yoo,

Matthias Zaldarriaga. At UC Santa Barbara: Nicola Bennert, Peng Oh, Tommaso

Treu. At Stanford: Marcelo Alvarez, Michael Busha, Brian Gerke, Nelima Sehgal,

Louie Strigari, Risa Wechsler, Chen Zheng. At UC Berkeley: Kevin Bundy, Jor-

dan Carlson, Joanne Cohn, Marina Cortes, Roland De Putter, Marc Davis, Shirley

Ho, Phil Hopkins, Alexie Leauthaud, Eric Linder, Chung-Pei Ma, Reiko Nakajima,

Jonathan Pober, Linda Strubbe, Tristan Smith, George Smoot, Andrew Wetzel,

Martin White, Jun Zhang. At Caltech: Lotty Ackerman, Shin’ichiro Ando, Esfan-

diar Alizadeh, Andrew Benson, Sean Carroll, Adrienne Erickcek, Steve Furlanetto,

Chris Hirata, Elisabeth Krause, Benjamin Wandelt. At UC Irvine: Aaron Barth,

Betsy Barton, Misty Bentz, James Bullock, David Buote, Helene Flohic, Chris Pur-

cell, Devdeep Sarkar, Kyle Stewart, Erik Tollerud, Carol Thornton, Chris Trinh,

vi

Gaurang Yodh, Joanelle Walsh. At Cal Poly Pomona: Nina Abramzon, Harvey Leff,

Roger Morehouse, Alex Rudolph, Alex Small. At UC San Diego: Alison Coil, David

Kirkman, Thomas Murphy, Art Wolfe. At the Carnegie Institution of Washington:

Julio Chaname, Nick Moskowitz. At Wesleyan: Erin Arai, Tyler Desjardins, Bill

Herbs, Roy Kilgard, Amy Langford, Edward Moran, Seth Redfield, Katy Wyman.

At Heidelberg: Matthias Bartelmann, Carlo Giocoli, Matteo Maturi, Faviola Molina.

At SISSA: Michael Cook, Luigi Danese, Paolo Salucci. At MPA: Raul Angulo, Silvia

Bonoli, Michael Boylan-Kolchin, Qi Guo, Carlos Hernandez Montegudo, Guinevere

Kauffmann, Cheng Li, Andrea Merloni, Eyal Neistein, Francesco Shankar, Simon

White, Jesus Zavala. At Groningen: Muhammad Abdul Latif, Amina Helmi, J. P.

Perez Beaupuits, Laura Sales, Marco Spaans, Esra Tigrak, Rien van de Weyn-

gaert, Saleem Zaroubi. At Los Alamos National Laboratories: Suman Bhattacharya,

Stirling Colgate, Daniel Holz, Daniel Whalen. At Columbia: Greg Bryan, Zoltan

Haiman, Kristen Menou, Takamistu Tanaka. At Michigan: Eric Bell, Joel Breg-

man, Carlos Cuhna, Gus Evrard, Oleg Gnedin, Kayhan Gultekin, Elena Rasia,

Mateusz Ruszkowski, Min-Su Shin, Monica Varulli, Marta Volonteri, Marcel Zemp.

At Rutgers: Chuck Keeton, Viviana Acquaviva. At Vanderbilt: Andreas Berlind,

Kelly Holley-Bockelmann, Cameron McBride. Also to folks I have met at meetings,

including Paula Aguirre, Karla Alamo, Kaushi Bandara, Maria Beltran, Ami Choi,

David Cohen, Sanghamitra deb, Roland Dunner, Debra Elmegreen, Adiv Gonzalez,

Mandeep Gill, Lucia Guaita, Atakan Gurkan, Sarah Hansen, Luis Ho, Eric Jensen,

vii

Claudia Lagos, Felipe Marin, Kim McLeod, Joel Mendoza, Irving Morales, Fran-

cisco Mueller, Federico Pelupessy, Enrico Ramirez Ruiz, Mario Riquelme, Meghan

Roscioli, Douglas Rudd, Anil Seth, Allyson Sheffield, Greg Sivakoff, Roberto Soria

and Kendrik Smith. For answering my questions via e-mail, I wish to acknowledge

Esfandiar Alizadeh, Andrew Benson, Julie Comerford, T. J. Cox, Onsi Fakhouri,

Chris Hayward, William Keel, Eyal Neistein, Ryan O’Leary, Antonio Riotto, Marco

Spaans and Benjamin Wandelt. Lastly, to David Charbonneau, Guy Consolmagno,

Rocky Kolb and Alex Rudolph for paying me a visit in my office, despite your busy

schedule. I have been very fortunate to present my results at so many places, and

meet so many interesting people in my field, which has really shown me the human

side of this wonderful enterprise. I hope I did not leave anybody out, and I truly

look forward to bumping into you again in the future.

Doing physics and astronomy has been one of the most exciting, fascinating,

and challenging endevors in my life. I am indebted to Viridiana Moreno, the love of

my life, for her constant support. This thesis is decated to you, and to our children

Camila and Mateo for all these years together and for the years to come.

viii

ABSTRACT

DARK MATTER HALO MERGERS AND QUASARS

Jorge Moreno

Ravi Sheth, Advisor

The formation and evolution of galaxies and the supermassive black holes they

harbor at their nuclei depends strongly on the merger history of their surrounding

dark matter haloes. First we developed a semi-analytic algorithm that describes the

merger history tree of a halo. The following tests were performed: the conditional

mass function, the time and mass distributions at formation, and the time distri-

bution of the last major merger. We provide a model for the creation rate of dark

matter haloes, informed by both coagulation theory and the modified excursion

set approach with moving barriers. A comparison with N-body simulations shows

that our square-root barrier merger rate is significantly better than the standard

extended Press-Schechter rate used in the literature. The last chapter is dedicated

to a simple model of quasar activation by major mergers of dark matter haloes. The

model consists of two main ingredients: the halo merger rate describing triggering,

and a quasar light curve, which tracks the evolution of individual quasars. In this

model, the mass of the seed black hole at triggering is assumed to be a fixed fraction

of its mass at the peak luminosity. The light curve has two components: an expo-

nential ascending phase and a power-law descending phase that depends on mass of

ix

the host. We postulate a self-regulation condition between the peak luminosity of

the active galactic nucleus (AGN) and the mass of the host halo at triggering. This

type of model for quasar evolution is at the heart of the latest semianalytic models

(SAMs) of galaxy formation and it is therefore definitely worth studying in some

detail. By carefully revisiting some of the main issues linked to this approach, we

were able to derive several interesting and physically meaningful constraints regard-

ing black hole evolution. We expect simple, yet observationally-constrained models

like ours to play a central role in future models of galaxy formation.

x

Contents

1 Prologue 1

2 Introduction 7

2.1 In the beginning ... . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 The smooth expanding universe. . . . . . . . . . . . . . . . . . . . . 132.3 Perturbing the universe. . . . . . . . . . . . . . . . . . . . . . . . . 172.4 The spherical collapse model . . . . . . . . . . . . . . . . . . . . . . 212.5 The power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 The growth of structure . . . . . . . . . . . . . . . . . . . . . . . . 282.7 The excursion set theory . . . . . . . . . . . . . . . . . . . . . . . . 372.8 The unconditional mass function. . . . . . . . . . . . . . . . . . . . 442.9 The conditional mass function . . . . . . . . . . . . . . . . . . . . . 482.10 Ellipsoidal collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.11 Moving barrier models . . . . . . . . . . . . . . . . . . . . . . . . . 542.12 Comparison with N-body simulations . . . . . . . . . . . . . . . . . 60

3 Merger history trees 64

3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.2 Mass histories and merger trees . . . . . . . . . . . . . . . . . . . . 683.3 The Monte Carlo merger tree . . . . . . . . . . . . . . . . . . . . . 743.4 A conditional scaling symmetry . . . . . . . . . . . . . . . . . . . . 763.5 The progenitor mass function . . . . . . . . . . . . . . . . . . . . . 783.6 The distribution of formation redshifts . . . . . . . . . . . . . . . . 843.7 The mass distribution at formation . . . . . . . . . . . . . . . . . . 893.8 The last major merger . . . . . . . . . . . . . . . . . . . . . . . . . 92

4 Halo creation 96

4.1 Creation of dark matter haloes . . . . . . . . . . . . . . . . . . . . . 964.2 Mass history in the excursion set theory . . . . . . . . . . . . . . . 101

4.2.1 Mass history for different barriers . . . . . . . . . . . . . . . 1024.2.2 Creation times from Bayes’ rule . . . . . . . . . . . . . . . . 1064.2.3 Monte Carlo test and self-similarity . . . . . . . . . . . . . . 107

4.3 Creation time distribution . . . . . . . . . . . . . . . . . . . . . . . 110

xi

4.3.1 Distribution of creation redshifts . . . . . . . . . . . . . . . 1114.3.2 Self-similarity in halo creation . . . . . . . . . . . . . . . . . 1134.3.3 Why it doesn’t work in general . . . . . . . . . . . . . . . . 1154.3.4 Why it is a useful approximation in practice . . . . . . . . . 1194.3.5 Conditional distribution of creation redshifts . . . . . . . . . 120

4.4 Halo creation rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.4.1 The unconditional rate . . . . . . . . . . . . . . . . . . . . . 1264.4.2 The time-normalized creation rate . . . . . . . . . . . . . . . 1294.4.3 The conditional rate . . . . . . . . . . . . . . . . . . . . . . 130

4.5 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5 Merger-induced Quasars 134

5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.2.1 The halo merger rate . . . . . . . . . . . . . . . . . . . . . . 1475.2.2 The light curve . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.3 Theoretical halo merger rates . . . . . . . . . . . . . . . . . . . . . 1635.3.1 Extended Press-Schechter . . . . . . . . . . . . . . . . . . . 1635.3.2 Ellipsoidal Collapse . . . . . . . . . . . . . . . . . . . . . . . 1655.3.3 Comparison with N-body simulations . . . . . . . . . . . . . 167

5.4 Implemention of the model . . . . . . . . . . . . . . . . . . . . . . . 1725.4.1 The luminosity function . . . . . . . . . . . . . . . . . . . . 1725.4.2 The large scale bias . . . . . . . . . . . . . . . . . . . . . . . 1735.4.3 Testing the Fry formula . . . . . . . . . . . . . . . . . . . . 175

5.5 Measuring the clustering of faint AGNs and low-redshift quasars . . 1795.5.1 The clustering of faint AGNs . . . . . . . . . . . . . . . . . 1795.5.2 The clustering of low-redshift quasars . . . . . . . . . . . . . 182

5.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1865.6.1 The luminosity function . . . . . . . . . . . . . . . . . . . . 1865.6.2 Quasar clustering . . . . . . . . . . . . . . . . . . . . . . . . 1915.6.3 The predicted black hole mass function . . . . . . . . . . . . 199

5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2035.7.1 Further constraint from high-z X-ray counts. . . . . . . . . . 2035.7.2 Scaling relations . . . . . . . . . . . . . . . . . . . . . . . . . 2085.7.3 Comparison with previous models . . . . . . . . . . . . . . . 2155.7.4 Assumptions, input parameters, and degeneracies . . . . . . 2215.7.5 Black holes masses at triggering . . . . . . . . . . . . . . . . 225

5.8 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

A Diffusion theory 237

A.1 Diffusion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237A.1.1 Constant barriers . . . . . . . . . . . . . . . . . . . . . . . . 237

xii

A.1.2 Square root barriers . . . . . . . . . . . . . . . . . . . . . . 241

B Merger tree algorithm 248

B.1 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248B.1.1 The branch: from random walks to mass histories . . . . . . 248B.1.2 The tree: connecting the branches . . . . . . . . . . . . . . . 251

B.2 The main branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

C Coagulation and fragmentation 255

C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256C.2 The binary merger-fragmentation model . . . . . . . . . . . . . . . 256

C.2.1 Linear polymers . . . . . . . . . . . . . . . . . . . . . . . . . 257C.2.2 Branched polymers . . . . . . . . . . . . . . . . . . . . . . . 260

C.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265C.3.1 General linear kernel . . . . . . . . . . . . . . . . . . . . . . 269C.3.2 Multiplicative kernel . . . . . . . . . . . . . . . . . . . . . . 271

C.4 r−mer initial conditions . . . . . . . . . . . . . . . . . . . . . . . . 273C.5 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Bibliography 354

xiii

List of Tables

3.1 The conditional scaling symmetry. . . . . . . . . . . . . . . . . . . . 82

5.1 The clustering of low luminosity AGNs. . . . . . . . . . . . . . . . . 180

xiv

List of Figures

2.1 The linear power spectrum. . . . . . . . . . . . . . . . . . . . . . . 262.2 The variance in smoothed overdensities. . . . . . . . . . . . . . . . 272.3 Linearly enhanced overdensities. . . . . . . . . . . . . . . . . . . . . 292.4 The overdensity field smoothed at two scales. . . . . . . . . . . . . 312.5 Two pictures of gravitational growth. . . . . . . . . . . . . . . . . . 332.6 The spherical collapse threshold. . . . . . . . . . . . . . . . . . . . . 352.7 The smoothed overdensity field at various scales. . . . . . . . . . . . 392.8 The smoothed overdensity field with increasing variance. . . . . . . 412.9 The first crossing of a barrier by a random walk. . . . . . . . . . . . 432.10 The extended Press-Schechter argument. . . . . . . . . . . . . . . . 462.11 The two barrier problem. . . . . . . . . . . . . . . . . . . . . . . . . 492.12 Different collapse barriers. . . . . . . . . . . . . . . . . . . . . . . . 572.13 The unconditional mass function. . . . . . . . . . . . . . . . . . . . 62

3.1 The merger history tree of a halo with mass M at redshift Z. . . . . 653.2 A random walk and its associated mass history. . . . . . . . . . . . 693.3 The same as Figure 3.2, but with square-root barriers. . . . . . . . 713.4 Intersecting barriers with GIF2 time snapshots. . . . . . . . . . . . 733.5 A branch at ‘continuous’ and ‘snapshot’ redshifts. . . . . . . . . . 753.6 The progenitor mass fraction and mass function, with M/m⋆ = 6. . 793.7 Same as Figure 3.6, with M/m⋆ = 0.6. . . . . . . . . . . . . . . . . 803.8 Same as Figure 3.6, with M/m⋆ = 0.06. . . . . . . . . . . . . . . . . 813.9 The conditional scaling symmetry. . . . . . . . . . . . . . . . . . . . 833.10 Scaled distribution of formation redshifts. . . . . . . . . . . . . . . . 863.11 Distribution of formation redshifts. . . . . . . . . . . . . . . . . . . 883.12 Distribution of mass at formation. . . . . . . . . . . . . . . . . . . . 903.13 Close-up of Figure 3.12 near the peak. . . . . . . . . . . . . . . . . 913.14 The redshift distribution of the most recent major merger. . . . . . 94

4.1 The mass history associated with a random walk using a constantbarrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.2 Same as Figure 4.1, but with a square-root barrier. . . . . . . . . . 1054.3 The creation time distribution in self-similar form. . . . . . . . . . . 109

xv

4.4 Distribution of creation redshifts for a number of bins in halo mass. 1124.5 Universality of the distribution of halo creation times. . . . . . . . . 1144.6 The mass history associated with a random walk and linear barriers. 1164.7 The creation time distribution associated with linear barriers in self-

similar form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.8 Conditional distribution of creation redshifts. Symbols and style as

in Figure 4.4. We plot m = m⋆/10 (left panels) and m = m⋆/100(right panels) conditioned to end up in haloes of mass M = M⋆ (toppanels) and M = 10m⋆ (bottom panels). In all cases, T denotes thepresent time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.9 Halo creation rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.10 Conditional creation rates. . . . . . . . . . . . . . . . . . . . . . . . 132

5.1 The quasar light curve. . . . . . . . . . . . . . . . . . . . . . . . . . 1615.2 Halo merger rates divided by mass function vs. mass ratio. . . . . . 1685.3 Ellipsoidal collapse merger rates. . . . . . . . . . . . . . . . . . . . 1705.4 Ellipsoidal collapse merger rates with linear ξ. . . . . . . . . . . . . 1715.5 Testing the Fry formula. . . . . . . . . . . . . . . . . . . . . . . . . 1785.6 The bias of low-luminosity type II AGNs. . . . . . . . . . . . . . . . 1815.7 The bias of low-redshift quasars. . . . . . . . . . . . . . . . . . . . . 1845.8 Predicted bolometric luminosity function at different redshifs. . . . 1875.9 Predicted luminosity function from a model with with and without

mass dependence in the light curve. . . . . . . . . . . . . . . . . . . 1895.10 Predicted bias as a function of luminosity at different redshifts. . . . 1925.11 Comparison among models characterized by different delay times. . 1955.12 Comparison between reference and alternative models (with less mas-

sive hosts and minor mergers. . . . . . . . . . . . . . . . . . . . . . 1985.13 Predicted black hole mass function at different redshifts. . . . . . . 2005.14 Predicted black hole mass functions for the reference and alternative

models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2025.15 Predicted X-ray number counts and bolometric luminosity functions. 2045.16 Predicted MBH-σstar relation at different redshift and different values

of the ratio between circular velocity Vc and virial velocity Vvir. . . . 2105.17 Predicted MBH-Mstar relation at different redshift, as labeled, ob-

tained by matching our reference model MBH-M relation with theempirical Mstar-M relation obtained from cumulative number match-ing techniques.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

5.18 Relevant black hole masses involved in a halo merger of haloes. . . . 227

A.1 Diffusion with a constant absorbing barrier. . . . . . . . . . . . . . 239A.2 Exact square-root barrier solution. . . . . . . . . . . . . . . . . . . 242

B.1 A sample merger history tree of a halo. . . . . . . . . . . . . . . . . 249

xvi

B.2 Main branch algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 253

C.1 Haloes as branched polymers. . . . . . . . . . . . . . . . . . . . . . 262

xvii

Chapter 1

Prologue

Understanding how the universe and its constituents evolve has been one of the

longest quests in human history. The last century saw how this quest not only

enticed astronomers, but physicists as well. Our current understanding states that

slightly overdense regions in the early universe were enhanced by gravity, and

became virialized. These gravitationally-bound dark matter haloes merged with

one another, building more massive haloes in a hierarchical fashion. This process

strongly affects the assembly and nature of galaxies. Understanding galaxy forma-

tion theoretically is one of the most exciting challenges in physics today – and I

believe that it should be regarded as important as understanding critical phenom-

ena in condensed matter physics, or the unification of fundamental interactions in

high-energy physics.

Aside from observations, a great deal of knowledge can be inferred with the

1

aid of N-body simulations. From the early numerical work of Toomre & Toomre

(1972) to the Millennium Simulation of Springel & Hernquist (2005), N-body mod-

eling of gravitational growth has seen unprecedented developments. Unfortunately,

our understanding from an analytic point of view has not increased at the same

pace. One of the aims of the present thesis is to gain insight on the assembly of

gravitationally-bound dark matter haloes using analytic and semi-analytic meth-

ods. Such methods not only complement current N-body studies, but allow us to

formulate the language necessary to interpret such simulations.

The excursion set formalism, which is based on the properties of gravitational

instability, successfully describes many properties related to the abundance of dark

matter haloes. This formulation takes advantage of what is known about Brown-

ian motion random walks to gain insight. The original version describes collapse

spherically. A revised version, which incorporates ellipsoidal collapse, provides a

major improvement. In this work we advocate a model of collapse expressed in

terms of a ‘square-root’ moving barrier. This simple model correctly describes the

abundance of haloes (the mass function) and it predicts a scaling relation for those

haloes conditioned to belong to more massive haloes at a later time.

Background material, including the excursion set formalism, is introduced in

Chapter 2. Chapter 3 is dedicated to the formulation of a merger tree algorithm

based on this square-root barrier model. The assembly of haloes through mergers

is equivalent to a branching process if time is reversed. One particular feature

2

that distinguishes this algorithm from others in literature is that mass (not time) is

discretized. This property facilitates the study of many problems. Predictions of the

conditional mass function, and the distribution of redshifts and masses at formation

are tested against analytic formulas and measurements in N-body simulations. We

also test the redshift distribution of the last major merger (one where the ratio of

the masses involved is at least one third).

A description of the times at which these mergers happen – the so-called creation

time distribution – is formulated in the context of ellipsoidal collapse with moving

barriers. An interesting ‘Bayesian’ link between this quantity and the mass function

is discussed in Chapter 4. We are the first to extend this link to the conditional case.

The creation of haloes by mergers of smaller objects can be described in terms of

the Smoluchowski coagulation equation. This formalism has been applied in many

areas of physics and astronomy (see e.g., Benson, 2008, and references therein).

A formulation for halo mergers in a very special case – with white-noise initial

conditions and spherical collapse – has been available for a number of years. We

use this result as a guide to extract the creation term from the time-derivative of the

mass function with general initial conditions and ellipsoidal collapse. More details

on this formalism, its relation to polymer growth, and a discussion of fragmentation

are found in Appendix C. This is a clear example of how knowledge from other

fields can yield fruitful results. We conclude this chapter with a discussion of how

halo creation is related to the concept of halo formation described in Chapter 3.

3

Mergers of haloes (and their associated galaxies) have a huge impact on cer-

tain phenomena, including sudden bursts of star formation, and quasars. After

a merger takes place, gas is funneled towards the supermassive black hole at the

center. This phase of fast accretion is eventually quenched when the associated

luminosity is powerful enough to unbind the reservoir of gas feeding the black hole.

In Chapter 5, we develop a model that convolves the halo merger rate with a uni-

versal quasar light curve consisting of three phases: an ascending phase describing

efficient accretion, a peak luminosity controlled by a postulated self-regulation con-

dition, and a mass-dependent descending phase describing the shut down of the

quasar. The reference model, and possible extensions are discussed and constrained

by observations including the AGN luminosity function, the clustering of quasars,

the local black hole mass function, and X-ray number counts. Implications of the

model on scaling relations between the nuclear black hole and properties of the

hosts are addressed.

This thesis is the result of investigations carried during the last few years. The

presentation here does not match the chronological order in which results were ob-

tained, nor it presents all the attempts considered. This thesis is very different

from many others in astronomy. It picks and borrows ideas from random Brownian

processes and diffusion theory; polymer coagulation; and it makes use of analytic,

Monte Carlo, and N-body techniques. Its final chapter on quasars is the first step

towards many possible astrophysical applications of the ideas presented here. Our

4

quasar model is very powerful, despite its simplicity, and matches (or has the poten-

tial of matching with some modifications) several independent sets of observations.

The results presented in this thesis can be found in a number of research papers:

- An improved model for the formation times of dark matter haloes.

C. Giocoli, J. Moreno, R. K. Sheth & G. Tormen (2006).

MNRAS, 376, 977-983.

- Merger history trees of dark matter haloes in moving barrier models.

J. Moreno, C. Giocoli & R. K. Sheth (2008).

MNRAS, 391, 1729-1740

- Dark matter halo creation in moving barrier models.

J. Moreno, C. Giocoli & R. K. Sheth (2009).

MNRAS, 397, 299-310

- Models of reversible coagulation and fragmentation.

R. K. Sheth & J. Moreno (2010).

J. of Phys. A, submitted.

- Merger-induced quasars, light curves and their host haloes

F. Shankar, J. Moreno, D. H. Weinberg, R. K. Sheth, Martin Crocce,

Cheng Li, R. E. Angulo, and Federico Marulli (2010).

MNRAS, in preparation.

5

This work would not have been possible without the guidance, patience, and

support of my thesis advisor, Ravi Sheth, who is certainly the most influential

person in my career. I am indebted to Carlo Giocoli for providing the N-body

simulation data used in Chapters 3 and 4, for all his constant help, encouragement,

and for clarifying all my questions. The mentoring of Francesco Shankar on the

subject of quasars and his input in Chapter 5 was invaluable. This thesis is a

result of a collaboration with these three individuals, and it required the help of

many others, including Bepi Tormen, David Weinberg, Martin Crocce, Cheng Li,

Raul Angulo and Federico Marulli. In particular, the input and expertise of the

first two individuals in this list improved the quality of our work greatly. On the

other hand, the last four provided invaluable analyses of the passive evolution of

clustering in simulations (e.g., the ‘Fry formula’) and the clustering of faint type 2

AGNs and quasars at low redshift. Nevertheless, I take full responsibility on the

results presented here. It is my hope that you find this thesis and the problems

addressed here enjoyable and interesting.

6

Chapter 2

Introduction

The aim of this chapter is to introduce the necessary background ideas in cosmology

for this thesis. It is not our intention to present a comprehensive treatise, which

can be found in several excellent books that treat these topics in more detail. Some

of them, which were consulted at some point or another, include Peebles (1980),

Peacock (1999), Dodelson (2003) and Carroll (2004). A brief introduction to the

excursion set formalism with spherical and ellipsoidal collapse is included. An

introduction to quasars and supermassive black holes is deferred to Chapter 5.

2.1 In the beginning ...

We live in a special time in history because, for the first time, mankind has a

quantitative understanding of the universe and its evolution. The basic picture is

that the universe is homogeneous and isotropic. Observations reveal that, at large

7

enough scales, it looks pretty much the same everywhere, with no special directions.

Moreover, it is reasonable to assume that observers at distant locations would reach

similar conclusions. This view overthrows the ancient Aristotelian-Ptolemic belief

that we are at the center of the universe, with the heavens rotating around us.

Galileo’s discovery of Jupiter’s moons and the phases of Venus were among the

first pieces of evidence against that belief. The Copernican principle, which states

that the universe does not revolve around us, was put on firm ground. Perhaps

the most triumphant result in this direction is Newton’s unification of the motions

of terrestrial and heavenly bodies. Planets and apples are described by the same

natural laws.

Since that time, theories and observations have improved dramatically. In the

1700’s, Kant and Herschel realized that our sun is not the center of the universe

either, but merely a member of a group of stars called the Milky Way. It is tempting

to believe that this galaxy comprises the whole universe. But not only stars and

planets fill the sky; the presence of smudges – which Messier dubbed ‘nebulae’ –

caused great concern. Were these part of the galaxy, or faraway objects? Eventually,

Leavitt’s observations of cepheid variable stars helped measure the distance to these

objects once and for all. With this, it was established that these nebulae are in fact

their own separate galaxies. This was followed by the advent of spectroscopy, which

showed that distant stars and galaxies are made of the same elements found here on

Earth and in our Sun. The universe now seems immensely bigger than previously

8

thought, but not vastly different from our local neighborhood.

The Copernican principle came into question once more, when Hubble observed

that the spectra of distant galaxies are highly redshifted, meaning that these galaxies

are racing away from us. Did this mean that our galaxy is at the center? He also

noticed that the radial velocities of these galaxies were proportional to their distance

from us. Such behavior is allowed in a universe that is expanding everywhere – one

that in principle could reconcile Hubble’s law with the Copernican idea. If so,

neither our solar system nor our galaxy are at the center of the universe – the

universe is expanding everywhere and it has no center!

Observational discovery is often accompanied by theoretical breakthroughs. Ein-

stein replaced the Newtonian notion of absolute space and stated that gravity was

a manifestation of a dynamical space-time. Friedman and Lemaitre championed

the idea of an expanding universe as a solution to Einstein’s equations with ho-

mogeneity and isotropy. If the universe is really expanding, one could conclude

that in a distant past everything was closer together. Furthermore, that there must

have been an initial moment in which the universe came into being. Evidently,

this pleased the Church, but not those who believed in an eternally old universe.

Einstein advocated a static universe, one where space and time were homogeneous;

one that obeyed the ‘perfect’ Copernican principle. To this end, he introduced an

‘ad-hoc’ cosmological constant to his field equations, something he would later refer

to as his life’s ‘biggest blunder’. Even after Hubble reported that galaxies were

9

receding from us, Hoyle and other skeptics proposed that the universe could be

both eternal and expanding if matter was spontaneously created as space stretched.

To some extent, this contrived steady-state solution seemed reasonable, in view of

the fact that no direct evidence of a primordial ‘big bang’ was available. Big bang

theory predicts that the early universe was in a very hot and dense state, and that

a background cosmic glow must remain. As this controversy continued, Peebles

and Dicke of Princeton went forth to detect this radiation. Everything changed

when Penzias and Wilson, a few miles away, announced the discovery of a cosmic

microwave background (CMB) with primeval origins.

The universe we live in is very homogeneous and isotropic. The distribution

of galaxies is roughly the same everywhere. Even the temperature of the CMB is

nearly the same in every direction. The same cannot be said about its evolution. As

the universe expands, it cools and becomes more diffuse, experiencing several phase

transitions as a result. Our understanding of the first 10−43 seconds is somewhat

fuzzy, although a theory that describes fundamental particles as modes of extended

stringy objects seems promising. By 10−34 seconds, inflationary expansion followed,

exponentionally growing the primordial universe to great proportions. At this point,

it is believed that the strong interaction decoupled from grand unification, making

the first quarks and leptons. At 10−13 seconds, quarks bound up into protons

and neutrons. These and other particles interacted in this primordial soup, and

the lightest elements were synthesized. This plasma was so hot and dense that

10

photons scattered off electrons, preventing the formation of atoms. As the universe

expanded, this plasma cooled. The first atoms formed (recombined), light streamed

freely, and the universe became transparent. The wavelength of such light has been

stretched by the cosmic expansion, to the order of microwaves. This is the cosmic

microwave background (CMB) predicted by the Big Bang theory, and observed by

Penzias and Wilson.

Science took centuries to get to this stage, with an expanding universe that

is homogeneous and isotropic; but nature had more surprises in store. The first

is that the particles we are made of account for only about 4 percent of the con-

tents of the universe. Most of the matter consists of dark matter, which reveals

itself gravitationally in the motions of the stars and globular clusters around the

galaxy, galaxies in clusters, and gravitational lenses. The current version of the

Standard Model of particle physics does not include dark matter. Supersymme-

try could provide a viable candidate, and there is hope that the Large Hadronic

Collider at CERN or the CDMSII experiment in Minnesota could shed light on

its properties. Embarrasingly enough, matter itself only contributes to about 30

percent of the universe. Our knowledge of the other 70 percent, which causes the

universe expansion to accelerate, is even more limited. Some call it ‘dark energy’,

while others associate it with the energy of the vacuum, a cosmological constant.

Another problem is that, although the universe looks rather smooth at large scales,

it is certainly not so at smaller scales. Galaxies aggregrate in clusters, walls, and

11

filaments in the so-called cosmic web. A possible explanation is found in the theory

of inflation, which predicts that quantum fluctuations were exponentially enlarged

into density perturbations of the smooth universe – the very seeds of the large scale

structure we see today. It is reassuring to know that recent observations of the

CMB by space missions, such as the Cosmic Background Explorer (COBE) and

the Wilkinson Microwave Anisotropy Probe (WMAP), indicate that this smooth

background has tiny fluctuations of one part in 105.

Our story begins here. As the universe became matter-dominated and transpar-

ent, another transition began. The tiny perturbations were enhanced by gravity,

and became virialized objects. These objects, known as haloes, merged with other

haloes, making larger and larger objects. Their associated potential wells became

deeper and the gas within cooled and turned into stars. Aside from the primordial

elements, heavier atoms are cooked in stars. Some stars exploded into supernovae,

dumping their heavy metals into the interstellar medium. New stars formed, recy-

cled this gas into more elements, making the formation of planets and life possible.

Galaxies collided or simply became bound with others into galaxy clusters. Such

collisions funneled cool gas to the centers, triggering sudden bursts of star for-

mation. These mergers also helped supermassive black holes at the centers grow,

allowing their surrounding accretion disk to shine at immense luminosities that

can be observed at cosmological distances. Quasars, as these events are known,

are strong evidence that the universe is evolving. Understanding the interactions,

12

cannibalism, gastronomy and morphology of galaxies is a quite complex challenge.

Nevertheless, many properties can be explained by following the hierarchical assem-

bly of the dark matter haloes that contain them. This process, in turn, is strongly

determined by the properties of the initial fluctuations and the evolution of the

background universe.

But the story does not end there. If our theories are correct, our universe is about

to undergo another such phase transition, into an accelerated state dominated by a

cosmological constant (or dark energy). Cosmic acceleration will prevent the largest

regions from becoming bound, and the existing structures will become isolated in

this forever expanding universe. The structure, evolution and fate of our universe

is indeed a fascinating subject. We now introduce the basic tools necessary for

this description. For the sake of clarity, we postpone our introduction to the broad

subject of quasars and supermassive black holes for Chapter 5.

2.2 The smooth expanding universe.

In the theory of General Relativity, the metric gµν describes the geometry of space-

time. In particular, a line element is given by

ds2 = gµνdxµdxν , (2.1)

where µ, ν = 0, 1, 2, 3 and repeated indices get summed (Einstein’s convention). For

instance, for a homogeneous-isotropic universe, this is given by

13

ds2 = c2dt2 − a2(t)[ dr2

1− κr2+ r2(dθ2 + sin2 θdφ2)

]

, (2.2)

where c is the speed of light, (r, θ, φ) are the spherical spatial coordinates and a(t)

will be referred to as the scale factor. This is known as the Friedman-Robertson-

Walker (FRW) metric. Here the global geometry of the universe (and its fate) is

encapsulated in the curvature constant: κ > 0 (closed), κ = 0 (flat) or κ < 0 (open).

In this theory, the metric is a dynamical quantity, which obeys the Einstein field

equations:

Rµν −1

2Rgµν =

8πG

c3Tµν + Λgµν , (2.3)

where Rµν is the Riemann tensor, R is the Ricci scalar, G is Newton’s gravitational

constant. Spacetime is determined by the contents of the universe, encoded in Tµν ,

the stress-energy tensor. Notice that we have explicitly included the cosmological

constant Λ. For an ideal isotropic-homogeneous fluid with density ρ and pressure

p, from the Einstein equations one can obtain

( a

a

)2

=8πG

3ρ(a)− κc2

a2+

Λ

3. (2.4)

This is known as the Friedman equation, and it describes the evolution of the scale

factor a in time. Λ is often associated with the vacuum energy. Defining

ρΛ =Λ

8πG(2.5)

allows us to write the Friedman equation as

H2(a) =8πG

3

∑

j

ρj(a)− κc2

a2. (2.6)

14

where the Hubble parameter is defined by H(a) = a/a, which is determined by κ

and the densities of the different components. Each of these constituents evolves

differently with a. Explicitly, density is composed of relativistic matter (radiation),

ordinary matter, and vacuum energy, and it evolves as

ρ(a) =∑

i

ρi(a) = ρr(a) + ρm(a) + ρΛ(a) =ρr,0

a4+

ρm,0

a3+ ρΛ,0, (2.7)

where the quantities with the ‘0’ subscript are evaluated at the present time. It is

conventional to set the present time scale factor to unity (a0 = 1) and to express

the Hubble constant today as H0 = 100h km s−1 Mpc, with h = 0.7 as the accepted

value. The general solution is not simple. For a flat universe with matter only,

known as an Einstein de Sitter (EdS) cosmology, solving the Friedman equation is

fairly straightforward. Here, ρ(a) = ρm(a) = ρm,o/a3, a(t) = (t/t0)

2/3 and H(t) =

2/(3t). We will often make reference to this cosmology.

A flat κ = 0 is also known as a critical universe. In such a case, one can define

the critical density as

ρcrit(a) =3H2

8πG. (2.8)

Dividing equation (2.6) by ρcrit(a) and setting

Ωj(a) =ρj(a)

ρcrit(a), Ωκ(a) = − κc2

a2H2, (2.9)

the Friedman equation reads

1 =∑

j

Ωj(a) + Ωκ(a). (2.10)

15

The current values of the cosmological parameters today are (Ωm,0, ΩΛ,0, Ωκ,0, Ωr,0) ≃

(0.3, 0.7, 0, 10−4). In a flat cosmology with neglegible Ωr, the quantities Ωm(a) and

ΩΛ(a) are easily obtained. First one needs to rewrite the critical density as

ρcrit(a) = ρcrit,0

( H

H0

)2

= ρcrit,0

(Ωm,0

a3+ ΩΛ,0

)

, (2.11)

from which

Ωm(a) =Ωm,0/a

3

ΩΛ,0 + Ωm,0/a3, ΩΛ(a) =

ΩΛ,0

ΩΛ,0 + Ωm,0/a3(2.12)

immediately follow.

The Friedman equation describes the evolution of a(t). Now suppose that light

is emitted from a distant galaxy with wavelength λem. As the universe expands,

this wavelength gets stretched. If peculiar (local) motions are neglegible relative to

the bulk expansion, and we measure a wavelenght λobs on earth, then the difference

between these two is given by the redshift, defined as

z =λobs − λem

λem

=λobs

λem

− 1 =aobs

aem

− 1. (2.13)

Here, aobs and aem are the values of the scale factor at the observing and emitting

times, respectively. If one observes today light emitted at a time t < t0, then

a(t) =1

1 + z. (2.14)

Redshift, like the expansion factor, is a common proxy for cosmic time.

16

2.3 Perturbing the universe.

The homogeneous-isotropic FRW metric describes the universe well at large scales.

At smaller scales, we see that tiny inhomogeneities in the initial density field have

evolved gravitationally into the highly non-linear structures we see today: the cos-

mic web. Therefore the mean matter density in the Friedman equation, now denoted

by ρ0(t), needs to be replaced by a density field ρ(~x, t).1 If matter is treated as an

ideal fluid, then ρ obeys the following dynamical equations:

• Euler’s equation:

D~v

dt= −∇~x φ− ∇~x p

ρ. (2.15)

• The continuity equation:

Dρ

dt= −ρ∇~x · ~v. (2.16)

• Poisson’s equation:

∇2~x φ = 4πGρ. (2.17)

Here ~v(~x, t) is the velocity field, φ(~x, t) is the gravitational potential, p(~x, t) is the

pressure field,

D

dt=

∂

∂t+ ~v · ∇~x (2.18)

denotes the convective derivative and the ~x-index in ∇~x is set to emphasize that we

are working on physical (rather than comoving) coordinates.

1In this section, the ‘0’-subscript denotes ‘unperturbed’ quantities.

17

Suppose we write

ρ(~x, t) = ρ0(t) + δρ(~x, t), (2.19)

p(~x, t) = p0 + δp(~x, t), (2.20)

~v(~x, t) = ~v0 + δ~v(~x, t), (2.21)

φ(~x, t) = φ0 + δφ(~x, t). (2.22)

In the absence of δρ, δp, δ~v and δφ, the unperturbed quantities also obey the

above three fluid equations. Assuming the perturbations are small, one can linearize

the fluid equations, obtaining:

d(δ~v)

dt= −[(δ~v) · ∇~x]~v0 −∇~x (δφ)− ∇~x(δp)

ρ0

, (2.23)

d(δρ)

dt= −ρ0∇~x · ~v, (2.24)

∇2~x(δφ) = 4πG (δρ), (2.25)

where

d

dt=

∂

∂t+ ~v0 · ∇~x. (2.26)

In comoving coordinates, ∇~x = (1/a)∇, δ~v = a~u, and the fluid equations become:

d~u

dt+ 2H~u = −∇(δφ)

a2− ∇(δp)

a2ρ0

, (2.27)

dδ

dt= −∇ · ~u, (2.28)

∇2(δφ) = 4πGa2ρ0 δ, (2.29)

18

where

δ(~x, t) ≡ ρ(~x, t)− ρ0

ρ0

=(δρ)

ρ0

(2.30)

is the overdensity contrast.

Combining the divergence of the Euler equation with the time derivative of the

continuity equation, it can be shown that δ obeys

δ + 2Hδ = (4πGρ0δ +c2s∇2

a2)δ, (2.31)

where c2s = δp/δρ is the equation of state of the fluid. The Fourier transform of this

differential equation is

¨δ + 2H ˙δ = (4πGρ0 −c2sk

2

a2)δ, (2.32)

where

δ(~k, t) ≡∫

d3x δ(~x, t) ei~k·~x. (2.33)

The coefficient of ˙δ in equation (2.32) may be thought of as a friction factor, or Hub-

ble drag. The coefficient of δ can be interpreted as the competition of two effects:

gravity and gas pressure. These two effects balance out at a physical wavelength

λJ =2πa

k= cs

√

π

Gρ0

, (2.34)

called the Jeans wavelength.

Suppose the solution is separable: δ(~k, t) = D(t)δ(~k, ti), where ti is some initial

time. Inserting this into equation (2.32), one gets

D + 2HD− 4πGρ0D = −c2sk

2

a2. (2.35)

19

Notice that the left-hand side of the equation is k-independent. For this ansatz to

work, the right-hand side must also be independent of k. This is only true for scales

with λ = 2π/k ≫ λJ. Theories of structure formation favor the assumption that

dark matter is cold (pressureless), making cs neglegible. In other words, gravity

dominates for all the scales of interest. In this case, one is left with

D + 2HD− 4πGρ0D = 0. (2.36)

The solution to this second-order linear differential equation is a linear combination

of two factors: D+(t) (the growth factor) and D−(t) (the decay factor). Except for

the very earliest moments, the decay factor is neglegible and we will ignore it. From

this point on, D+ will be denoted as D.

For an EdS cosmology, recall that a ∝ t2/3. Therefore, H = 2/(3t) and 4πGρ0 =

(3/2)H2 = 2/(3t2). Equation (2.36) then becomes

D +4D

3t− 2D

3t2= 0. (2.37)

Assuming D(t) ∝ tα, then α obeys a quadratic equation, given by

3α2 + α− 2 = 0, (2.38)

The positive root, associated with the growing mode, is α = 2/3. That is, D(t) ∝

t2/3 ∝ a(t) in an Einstein de Sitter universe.

Equation (2.36) has no analytic solution for every cosmology, but Carroll et al.

(1992) provide an accurate approximation for flat cosmologies, given by

D(a) ≃ 5

2aΩm(a)

[

Ωm(a)4/7 − ΩΛ(a) +(

1 +Ωm(a)

2

)(

1 +ΩΛ(a)

70

)]−1

. (2.39)

20

For alternative expressions, we refer the reader to Heath (1977); Lightman &

Schechter (1990); Lahav et al. (1991); Navarro et al. (1997) and Percival et al.

(2000). See also Silveira & Waga (1994); Wang & Steinhardt (1998); Basilakos

(2003) and Percival (2005) for generalizations.

2.4 The spherical collapse model

Gravity tends to enhance slightly overdense regions. For sufficiently early times,

density fluctuations evolve as δ(~x, t) = δ(~x, ti)D(t)/D(ti). As soon as δ(~x, t) → 1,

linear theory breaks down, and one usually has to rely on N-body simulations to de-

scribe the evolution thereafter. Nevertheless, under certain assumptions, simplified

models that capture the essential features of nonlinear gravitational growth have

been formulated. The spherical collapse model is one of them.

Imagine two spherical regions, one unperturbed and the other perturbed, con-

taining the same mass m. The radius of the first one, denoted by Rb(t), will follow

the background expansion and evolve according to the Friedman equation. The

perturbed region (with overdensity δ) will follow the background universe early on.

Eventually, self-gravity will cause it to decouple from the overall expansion. Its

radius, denoted by Rp(t), will reach a maximum value, turn around, and collapse.

This region can be thought of as a sub-universe with a density larger than crit-

ical (i.e., a closed sub-universe). Since both regions contain the same mass, the

21

overdensity of the perturbed region can be written as

1 + δ(t) =R3

b(t)

R3p(t)

. (2.40)

Finding solutions for any cosmology is non-trivial. Gunn & Gott (1972) worked out

the case for an Einstein de Sitter (EdS) cosmology, and Lacey & Cole (1993) solved

it for an open universe (also see Navarro et al., 1997). Solutions that incorporate Λ

are found in Barrow & Stein-Schabes (1984); Lahav et al. (1991); Barrow & Saich

(1993); Eke et al. (1996); Kitayama & Suto (1996); Navarro et al. (1997); Percival

et al. (2000) and Percival (2005). For a similar treatment with underdense regions

(associated with voids), see Sheth & van de Weygaert (2004).

For the sake of illustration, we present the case with an EdS background cos-

mology. The solution for the perturbed region can be written parametrically as

Rp(θ) = A(1− cos θ), t(θ) = B(θ − sin θ), (2.41)

where 0 < θ ≤ 2π and the constants obey A3/B2 = Gm. The maximum radius

is reached when θ = π and collapse happens at θ = 2π. At turn around (θ = π),

t = πB. In principle, when θ = 2π, Rp = 0 and the region collapses to an infinitely

dense point. We will denote this time as tcoll = 2πB. In reality, dissipative physics

intervenes and random motions turn the region into a virialized object.

We will be particularly interested in the evolution at early times (the linear

regime). At early times (small θ),

Rp(θ) = A(θ2

2!− θ4

4!+ . . .

)

, t(θ) = B(θ3

3!− θ5

5!+ . . .

)

. (2.42)

22

Eliminating θ, we obtain

Rp(t) = (Gm)1/3t2/3 62/3

20

[

1− 62/3

20

(2πt

tcoll

)2/3

+ . . .]

= Rb(t)[

1− 1

20

(12πt

tcoll

)2/3

+ . . .]

.

(2.43)

The term in brackets can be interpreted as a first order correction to the background

evolution. Taking the cube of Rp/Rb and substituting into equation (2.41) we obtain

that, to leading order,

δ(t) =3(12π)2/3

20

( t

tcoll

)2/3

=3(12π)2/3

20

D(t)

D(tcoll). (2.44)

At collapse, such sphere will have an overdensity

δc ≡ δ(tcoll) =3(12π)2/3

20≃ 1.686. (2.45)

This critical threshold will be key to select virialized objects from the (linear-

extrapolated) initial overdensity field.

2.5 The power spectrum

In the cold dark matter (CDM) scenario, different Fourier modes of the perturbation

field evolve independently. In order to describe the evolution of these perturbations,

all we need is the growth factor and the initial perturbation field δ(~x) at some initial

time ti. In practice, it is impossible to know this field exactly at all points. Hope is

not lost if one acknowledges that theories of gravitational clustering are stochastic

in nature, not deterministic. That is, one does not need to follow every particle to

23

learn about the large scale structure of the universe. Instead, one may study this

process statistically, by treating δ(~x) as a Gaussian random variable. In this case,

the δ(~x)-distribution is completely specified by its two-point (correlation) function:

δ(2)(~x, ~x′) ≡< δ(~x)δ(~x′) >= δ(2)(|~x− ~x′|), (2.46)

which depends only on |~x− ~x′| if the universe is isotropic. It is common to work in

terms of the Fourier transform of this quantity:

< δ(~k)∗δ(~k′) >= (2π)3δD(|~k − ~k′|)P (k), (2.47)

where δD is the Dirac delta and P (k) = P (|~k|) is known as the power spectrum.

We are interested in the power spectrum early on in the matter-dominated universe

(which we call the initial power spectrum). This is related to the primordial power

spectrum p(k) as follows:

P (k) = T 2(k)p(k) (2.48)

where T (k) is known as the transfer function. Here we will assume a Harrison-

Zeldovich primordial power spectrum: p(k) ∝ kn (Harrison, 1970; Zeldovich, 1972),

with n = 1 being the preferred value in inflationary theories (Guth, 1981; Guth

& Pi, 1982). All the physical processes that turn the primordial into the initial

power spectrum are encapsulated in the transfer function. Several accurate fits

for this quantity exist in the literature (Bardeen et al., 1986; Efstathiou et al.,

1992; Sugiyama, 1995; Eisenstein & Hu, 1999). In this work we use the outputs

of cmbfast (Seljak & Zaldarriaga, 1996). Figure 2.1 shows the resulting power

24

spectrum (solid blue curve), along with a featureless scale-free power spectrum

P (k) ∝ k−2 (dashed red line).

So far we have only given the k-dependence of the power spectrum. We now

explain how the normalization is obtained. Consider a spherical region of radius R

at time ti. The mean overdensity in this region is

δR(~x) =

∫

d3y WR(|~x− ~y|)δ(~y), (2.49)

where WR is some spherically symmetric window (filter) function. Given this, the

variance in overdensity within spherical regions of radius R is

S(R) = σ2(R) ≡< |δR(~x)|2 > =

∫

dk3

(2π)3W 2(kR)P (k), (2.50)

where we have used equation (2.47). Here, W (kR) is the Fourier transform of WR.

The intuitive choice for the window function is

W(TH)R (r) =

(43πR3)−1 if r ≤ R;

0 if r > R,

a top-hat in real space. We will see that it is more convenient to use a so-called

sharp k-space window function, given by

WR(r) = (3/r3)(sin r − r cos r), (2.51)

whose Fourier transform is a top-hat.

To normalize the power spectrum it is common practice to use σ8, the rms of

the linearly-extrapolated density fluctuations (equation 2.50) in spherical regions

25

Figure 2.1: The linear power spectrum. The solid blue curve is the cmbfast output,

and the dashed red line is an approximation that scales like k−2.

26

Figure 2.2: The variance in smoothed overdensities. The solid blue curve is derived

from the cmbfast power spectrum. A power spectrum that scales as k−2 predicts

that S(m) scales as m−1/3 (dashed red line). Both use a sharp-k space filter function

(equations 2.51) with σ8 = 0.9 normalization.

27

of radius 8 Mpc h−1 (equation 2.50). The accepted value we use is σ8 = 0.9. The

power spectra in Figure 2.1 are normalized to this value.

For sufficiently small initial perturbations, it is safe to assign a mass m =

(4π/3)R3ρ0(1 + δR) ≃ (4π/3)R3ρ0 to regions of radius R, allowing us to write

S(m) = S(m(R)). We plot this quantity in Figure 2.2 (solid blue curve). For a

scale-free power spectrum ∝ k−2, it can be shown that S(m) ∝ m−1/3 (dashed red

line). S(m) with a sharp-k space filter will play a key role later in this chapter. For

fits to this quantity, see Kitayama & Suto (1996); Navarro et al. (1997) and Taruya

& Suto (2000). In this work we compute S(m) numerically.

2.6 The growth of structure

The pioneering work of Press & Schechter (1974), followed by the excursion set

formalism of Bond et al. (1991), provides a simple prescription for estimating the

abundance of virialized objects, the so-called dark matter haloes. The key is to ac-

knowledge that the growth of structure via gravitational instability can be described

by studying the statistics of the initial field of density perturbations.

Consider the initial density fluctuation field, filtered with a sharp-k space window

function of size R containing a mass m. This is illustrated in Figure 2.3; see the

field δm(~x, ti) in light red near the bottom. Here ~x denotes the position of the filter’s

center. As we vary this center across the universe, we get very different values of δm.

On average, δm is zero and the various fluctuations are distributed in a Gaussian

28

Figure 2.3: Linearly enhanced overdensities. The smoothed ovedensity field at an

initial time ti (light red); linearly extrapolated to t (dark red) and t0 (black). Here ~x

denotes the centers of the smoothing filters and the dotted horizontal line represents

the spherical collapse threshold δc (equation 2.45).

29

fashion about this value. Thus, the probability of drawing a particular value δm

between δ and δ + dδ is given by

Q(δ|Sm)d δ =1√

2πSm

exp(

− δ2

2Sm

)

d δ, (2.52)

where Sm = Sm(ti) is the variance of this smoothed field at t = ti.

The next step is to linearly extrapolate the initial (smoothed) overdensity field:

δm(~x, ti) → δm(~x, t) = δm(~x, ti)D(t)/D(ti). The field near the middle of Figure 2.3

(in dark red) is extrapolated to some t > ti, while the one at the top (in black) is

extrapolated to the present. These are identical to the initial field, up to a mul-

tiplicative factor. Furthermore, they are also drawn from a Gaussian distribution,

but now the variance is Sm(t) = Sm(ti)D2(t)/D2(ti). Some very strong assump-

tions are being made here. First of all, the non-linear evolution of regions beyond

δ ∼ 1 is treated linearly. Secondly, extrapolated fluctuations can be very large, and

underdensities will go well beneath −1. This is physically incorrect, strictly speak-

ing. In reality, the distribution of non-linear evolved overdensities will be highly

non-Gaussian. Nevertheless, we will see that such an oversimplified model does an

exceptional job at predicting the abundances of virialized objects.

Overdensity fluctuations will be smaller on average, if a bigger filter is used

(Figure 2.2 and equation 2.52). This is illustrated in Figure 2.4, where the thin light

green curve represents the perturbation field smoothed with a scale m, whereas that

smoothed with a larger scale M > m is shown as the thick dark green curve. In

this figure, both fields are linearly evolved to the present. The horizontal dotted

30

Figure 2.4: The overdensity field smoothed at two scales. Two smoothing scales

containing mass m (thin light green) and M > m (thick dark green) are used on

the same initial overdensity field, which is then extrapolated to the present. The

dotted horizonal line is the spherical collapse threshold, which is pierced more often

when the smaller filter is used.

31

line denotes the critical collapse threshold, δc ≃ 1.686, predicted by the spherical

collapse model (Section 2.4). Notice that the smoothed field δm(~x, t0) is more likely

to pierce this threshold at more points than δM(~x, t0), the field with the larger

filter. As we will see in the next chapter, this property is essential in determining

the abundance of haloes of a given mass.

At early times, fluctuations tend to be much more smaller than the spherical

collapse threshold. As the field (smoothed at any given scale) is linearly extrapo-

lated to later times, it will be above this threshold at more places. An equivalent

picture is to simply evolve the initial overdensity field (smoothed at different scales)

to the present time and raise the collapse barrier as

δc(t) = δc0D(t0)

D(t), (2.53)

where δc0 ≡ δc(t = t0) ≃ 1.686. This picture is illustrated in Figure 2.5 (compare

with Figure 2.3).

In the first picture, δc is fixed and the variance in the smoothed initial field is

linearly amplified in time as Sm(t) = Sm(ti)D2(t)/D2(ti). In the second picture, the

variance is fixed at Sm = Sm(ti)D2(t0)/D

2(ti) and the spherical collapse threshold

depends on time as δc(t) = δc0D(t0)/D(t). We will see that the abundance of haloes

can be conveniently described in terms of a ‘self-similar’ variable defined as

ν(m, t) ≡ δ2c

Sm

(2.54)

where the time dependence is determined by the picture chosen (in Sm in the first

picture or in δc in the second picture). Notice that in both pictures, the self-similar

32

Figure 2.5: Two pictures of gravitational growth. In Figure 2.3 we fix δc and raise

the fluctuations linearly. Here we fix the fluctuations at their linear value today

and raise the threshold back in time. The two pictures are equivalent.

33

variable evolves as

ν(m, t) =D2(ti)

D2(t)ν(m, ti). (2.55)

Hereafter, we will only use the second picture. Ultimately we will be interested

in the abundance of haloes of mass m at some redshift z. The key ingredients

are S(m) and δc(z). Figure 2.2 shows the variance of initial density fluctuations,

linearly extrapolated to the present time, as a function of m. Figure 2.6 shows the

predictions of the spherical collapse overdensity in EdS and ΛCDM cosmologies as

a function of z. For an Einstein de Sitter cosmology, the critical collapse threshold

depends on redshift as

δc(z) = δc0D(0)

D(z)=

δc0

a(z)= δc0(1 + z). (2.56)

For ΛCDM cosmology, Navarro et al. (1997) found that a good numerical approxi-

mation to this quantity is given by

δc(z) =3(12π)2/3

20Ωm(z)0.0055 D(0)

D(z). (2.57)

This is shown in Figure 2.6 for ΛCDM (solid blue curve) and EdS cosmology (dashed

red line). For more rigorous treatement of spherical collapse in the presence of Λ,

we refer the reader to the papers mentioned in Section 2.5.

We have introduced all the background results necessary for the excursion set

theory of gravitational clustering. Before we continue, a few comments are in order.

A central assumption in this work is that the initial fluctuations are Gaussian

(equation 2.52). In general, the statistics of a random field are encoded in its point

34

Figure 2.6: The spherical collapse threshold. The red dotted line corresponds to

EdS cosmology, whereas the blue solid curve represents ΛCDM.

35

functions (e.g., < δ~k >,< δ~k δ~k ′ >,< δ~k δ~k ′ δ~k ′′ >, . . . ). For a Gaussian random

field, the odd n-point functions are null, while the even n-point functions are simple

products of the 2-point function (i.e., < δ~k δ~k ′ >∝ P (k)). In other words, all the

details are contained in the power spectrum. Standard inflationary cosmologies

predict that the initial fluctuations are indeed nearly Gaussian (Maldacena, 2003;

Acquaviva et al., 2003; Creminelli, 2003; Lyth & Rodrıguez, 2005; Seery & Lidsey,

2005) – see Lo Verde et al. (2008) for a review. Recently, a variety of inflationary

models that predict non-Gaussianity have been proposed (see Bartolo et al., 2004,

and references therein). One consequence of non-Gaussianity is that the higher

n-point functions are non-zero. In other words, the different k-modes (the δ~k’s) are

correlated, and the simple description given in equation (2.50) becomes non-trivial.

Nevertheless, recent authors have pointed out that observable non-Gaussianity only

affects the largest, rarest objects (Sadeh et al., 2007; Grossi et al., 2007; Dalal et

al., 2008; Afshordi & Tolley, 2008; Pillepich et al., 2008). For this reason, in this

work we will assume that the initial field of density fluctuations are well-described

by a Gaussian random field.

Another assumption that may be brought into question is the current gravity

model: General Relativity. Several authors have attempted to modify standard

gravity, often with different motivations in mind. For instance, some seek to re-

move the need of dark matter (Milgrom, 1983; Bekenstein, 2004), the cosmological

constant (Dvali et al., 2000; Lue et al., 2004), dark energy (Mannheim, 1990; Di-

36

aferio & Ostorero, 2008) – while others are interested in unifying dark matter and

dark energy (Kamenshchik et al., 2001; Makler et al., 2003; Sen & Scherrer, 2005;

Scherrer & Sen, 2008; Bertacca et al., 2008). In modified gravity, one may no longer

assume the validity of Birkoff’s theorem. Namely, the evolution of a spherical patch

is governed by both internal and external gravitational contributions. The net ef-

fect of this is that the growth of fluctuation depends on scale – that is, the growth

factor in equation (2.36) becomes k-dependent (Shirata et al., 2005). Moreover,

while a region with zero overdensity remains that way relative to the background in

regular gravity, this is no longer true in modified gravity. In other words, whether

or not a patch in the initial field collapses at a later time does not only depend

on its density, but on its mass itself. Recently, Martino et al. (2008) proposed a

method for estimating how halo abundances are altered in modified gravity. This

formulation of spherical collapse requires one to work with the initial density field

directly, given by ‘picture one’ in Figure 2.5 (they show that using ‘picture two’

gives the wrong result). In their construction, nevertheless, only the most massive

and rarest haloes are affected. In this work, only regular gravity will be considered.

2.7 The excursion set theory

The mass function of dark matter haloes can be estimated by using the spherical

collapse model and the statistics of the (linearly-extrapolated) initial overdensity

field, smoothed at different scales with a sharp-k space window function. The larger

37

the scale, the smaller the variance of fluctuations (Figure 2.2). Intuitively one can

think of this as follows. For the very largest scales, smoothed fluctuations are very

small, and the densities of such large regions approach the background density of the

universe. For these smoothing scales, the distribution of overdensities approaches

a Dirac delta. If the same field is smoothed at smaller scales, perturbations from

the mean become more evident. With this higher resolution, a greater diversity in

overdensities results, making the width of their distribution (the variance) broader.

Figure 2.4 illustrates this, showing the same initial field as a function of ~x, times

D(t0)/D(ti), smoothed at two different scales. On average, the peaks and troughs

of the smoothed field with the smaller filter tend to be more pronounced than those

with the larger filter.

Now pick a point ~x at random early in the universe (at time t = ti). Pick a very

large sphere and smooth this field with a sharp-k space filter. Gradually shrink

the radius of this filter, and smooth the field to increasingly smaller regions. Then

linearly extrapolate the result to the present. One such process is shown in Fig-

ure 2.7. For this particular point, the smoothed overdensity enclosed in the filter’s

radius tends to increase as the region gets smaller. However, for some portions the

opposite happens, indicating that a dense region at one scale can itself be embedded

in a rather underdense region, a void.

For comparison, Figure 2.3 shows the smoothed field at only two scales and

several ~x. Figure 2.7, on the other hand, shows to only one point ~x, and it shows

38

Figure 2.7: The smoothed overdensity field at various scales. The jagged line re-

pressents the run of smoothing the initial overdensity field with a sharp-k space

filter in spheres of increasing radius R around a randomly chosen point ~x. The

dotted horizontal lines are the spherical collapse thresholds at two redshifts.

39

the value of the smoothed field at all scales. Choosing a different point ~x in the

universe would yield a different jagged trajectory. For this particular location, we

ended up with an overdensity as R → 0. In other words, this point is found in an

overdense patch of the initial density field. Had we picked a point in an underdense

patch, the jagged line would have ended at a negative height at small R. In both

situations, the jagged lines deviate more from zero (in either direction), on average,

as the radius of the smoothing filter decreases.

In Figure 2.7, we have included the present-time critical collapse threshold, and

that associated with a higher redshift (dotted horizontal lines). Take, for instance,

the lower of the two. Notice that the jagged line crosses this barrier at several

points. Identifying all the scales as collapsed objects would miscount regions that are

embedded in larger collapsed regions. This is the so-called ‘cloud-in cloud’ problem

(Epstein, 1983; Bardeen et al., 1986; Peacock & Heavens, 1990; Jedamzik, 1995;

Sheth, 1995; Avelino & Viana, 2000). The excursion set formalism of Bond et al.

(1991) solves this by selecting only the largest of such regions as the collapsed object.

In other words, the largest scale that crosses the barrier carries with it information

about the mass of the virialized halo surrounding the point ~x. Notice that the

higher the barrier (z > 0), the smaller the value of R is at which this happens. In

other words, haloes tend to be less massive at earlier times, in agreement with the

hierarchical picture of gravitational clustering.

So far we have discussed smoothed overdensities in terms of the radius of the

40

Figure 2.8: The smoothed overdensity field with increasing variance. This is the

same as Figure 2.7, but in terms of the variance S, which increases with decreasing

R. The jagged line is a Brownian motion random walk.

41

filter. A more natural choice is to use the variance S = σ2(R) itself, which truly

describes how fluctuations deviate from zero at each scale. This process is described

in Figure 2.8. Since large scales correspond to small values of S, virialized regions are

identified with the first place where the jagged line upcrosses the collapse barrier.

The excursion set approach maps the fraction of random trajectories that first

upcross a barrier of height δc(z) between S and S+dS with the fraction of mass at

redshift z in haloes with mass between m and m+dm:

f(Sm|δc(z))d Sm = f(m|z)d m. (2.58)

This situation is illustrated in Figure 2.9, where the red solid circle indicates where

the random trajectory first crosses the barrier of height δc(z). Very few random

walks will upcross this barrier immediately – most of them will wander around

before crossing. Since S is a monotonically decreasing function of mass, most of

the mass will be in low-mass haloes. At higher redshift, the height of the barrier

increases, making it more difficult for walks to cross it. This means that most haloes

will, on average, have smaller masses. This is consistent with the idea that smaller

haloes formed before massive ones.

Finding this crossing distribution is nontrivial. Nevertheless, having a sharp-k

space filter makes matters much easier. With such a filter, the action of replacing

the smoothing scale as R → R+dR amounts to changing the size of the top-hat

filter in Fourier space from some kTH to kTH+dk. The Fourier modes that contribute

to this new scale come only from a thin spherical shell of width dk. This means

42

Figure 2.9: The first crossing of a barrier by a random walk. The jagged trajectory

denotes a Brownian random walk and the horizontal dotted line is a constant barrier.

The solid red circle denotes the location where the walk first upcrosses the barrier.

43

that the new contribution carries no information on the smaller scales, making

consecutive scales uncorrelated. In other words, the steps in these trajectories

are uncorrelated, and such trajectories are true Brownian random walks. Random

walks – also known as excursion sets – have been studied in great detail for many

years in pure mathematics (where they are known as Wiener processes, Grimmett &

Stirzaker, 2001), and have been applied to subjects as diverse as genetics, gambling,

colloids, neurology, stellar dynamics, and finance. Bachelier (1900) was the first to

compute the first crossing distribution of a constant barrier. The next section is

devoted to the derivation of this result.

2.8 The unconditional mass function.

In this section we follow Chandrasekhar (1943) and Bond et al. (1991), who ex-

ploit the symmetries of the constant barrier to find the solution. A more elegant

treatment, in terms of diffusion theory, is offered in Appendix A. The formalism

presented here shows the correct way to compute the halo mass function found

originally by Press & Schechter (1974) with heuristic arguments. In this work, we

will refer to results presented in this section and the next as the ’extended Press-

Schechter’ (ePS) theory.

Let F (> m′|z) denote the cumulative fraction of mass in collapsed objects of

mass greater than m′ at redshift z. The differential fraction of m-haloes at z is

f(m|z)dm =∂

∂m′F (> m′|z)

∣

∣

∣

m′=m. (2.59)

44

In terms of random walk variables, this is given by

f(Sm|δc(z))dSm =∂

∂S ′F (< S ′|δc(z))

∣

∣

∣

S′=Sm

. (2.60)

Now we show how F (< S ′|δc(z)) is obtained in the extended Press-Schechter theory.

First consider the probability that a trajectory is located between δ and δ+dδ after

S steps (one such trajectory is shown in Figure 2.8). This is a standard stochastic

problem, whose solution has been known for a number of years (e.g., Chandrasekhar,

1943; Grimmett & Stirzaker, 2001), and it is given by

Q(δ|S)dδ =1√2πS

exp(

− δ2

2S

)

dδ, (2.61)

(compare with equation 2.52). Next, we need to compute the fraction of these

walks that cross a barrier of constant height before having performed S ′ steps. For

instance, the black and the black-gray walks in Figure 2.10 satisfy this criterion

(they cross the barrier at S < S ′), while the red one does not. All the walks that

contribute to F (< S ′|δc(z)) cross the δc(z) threshold at some S < S ′. These walks

can be classified into two groups, those with δ ≥ δc at S ′ (e.g., the black walk) and

those with δ < δc (e.g., the black-gray walk).

The cumulative fraction of walks in the first category is given by

Fup(< S ′|δc(z)) =

∫ ∞

δc(z)

dδ Q(S ′, δ). (2.62)

This is because all the walks that have δ ≥ δc at S ′, have certainly crossed the

threshold (with height δc) at some smaller S < S ′.

45

Figure 2.10: The extended Press-Schechter argument. Three random walks with

0 < S ≤ S ′ and a constant barrier of height δc(z) (horizontal dotted line) are

shown. One never crosses the barrier (in red), one crosses it and continues above it

(in black), and one that pierces it and continues beneath it (in black-gray).

46

Obtaining the contribution to F (< S ′|δc(z)) from walks belonging to the second

category is a bit tricky. Recall that, by virtue of the sharp-k space filter, consecutive

steps in a walk are independent. This means that from any point onward, a walk

has no memory of the previous steps. At the origin, a random walk is equally likely

to wander in the positive or in the negative direction. The same is true at any

other point. In particular, consider the point where the black-gray walk crosses

the barrier (the point where it changes color). The fraction of walks going up (in

black) or down (in gray) beyond S is the same: Fdown = Fup. Since the cumulative

fraction of walks in the first (second) category is Fup (Fdown), the cumulative fraction

of walks that have crossed the threshold at some S < S ′ is simply

F (< S ′|δc(z)) = Fup + Fdown = 2×∫ ∞

δc(z)

dδ Q(S ′, δ). (2.63)

This quantity includes all the walks that have crossed the barrier at some S < S ′,

without regard of whether they are above or beneath δc(z) at S ′.

Substituting Q in equation (2.61), one obtains that

F (< S ′|δc(z)) = erfc( δc(z)√

2πS ′

)

. (2.64)

Finally, substituting this expression into equation (2.60), we obtain that the fraction

of mass in m-haloes at redshift z is

f(Sm|δc(z))dSm =

√

δ2(z)

2πSm

exp(

− δ2(z)

2Sm

)dSm

Sm

. (2.65)

This is the famous Press-Schechter result, and it gives an estimate of the fraction

of mass in haloes with mass between m and m+dm at redshift z.

47

2.9 The conditional mass function

The Press-Schechter mass function, which gives the comoving number density of

haloes with mass in the (m,m+dm) at redshift z, is given by

n(m|t)d m =ρ

mf(m|z)d m =

ρ

mf(Sm|δc(z))

∣

∣

∣

dSm

dm

∣

∣

∣, (2.66)

where f(Sm|δc(z))dSm is the fraction of walks that first upcross a barrier of height

δc(z) between Sm and Sm+dSm (equation A.6 and Figure 2.9).

Suppose that instead we were only interested in the mass function of those haloes

of mass m at redshift z conditioned to end up in bound objects of mass M > m at

a more recent redshift Z < z. The conditional mass function N(m|z,M,Z)dm de-

scribes this. Some authors refer to N as the progenitor mass function, where m are

the progenitors that form a final object of mass M by merging. Just like the uncon-

ditional n(m|z)dm was obtained from f(m|z)dm; the conditional N(m|z,M,Z)dm

can be obtained from f(m|z,M,Z)dm, the fraction of mass in m-haloes at z con-

ditioned to be in M -haloes at Z.

Bower (1991) was the first to extend the Press-Schechter heuristic argument to

the conditional case (hence the word ‘extended’ in ePS). In the excursion set theory,

this can be computed by setting

f(m|z,M,Z) dm = f(Sm|δc(z), SM , δc(Z)) dSm, (2.67)

where the quantity on the right-hand side is the first crossing distribution of a

barrier of height δc(z) by random walks with origin at (SM , δc(Z)). Figure 2.11

48

Figure 2.11: The two barrier problem. The crossing distribution of a barrier at

height δc(z) by random walks originating at (SM , δc(Z)) is associated to the condi-

tional mass function of m-haloes at Z found in haloes of mass M > m at Z < z.

This becomes a one barrier problem by shifting coordinates: δ → a = δ−δc(Z) and

S → s = S − SM .

49

illustrates this. The configuration is identical to Figure 2.9, except for a simple

shift of coordinates:

S → s ≡ S − SM ,

δ → a ≡ δ − δc(Z). (2.68)

This single barrier has height ac = δc(z)− δc(Z), and its first crossing distribution

is:

f(s|ac)ds =

√

a2c

2πsexp

(

− a2c

2s

)ds

s. (2.69)

Therefore, the conditional crossing distribution is

f(Sm|δc(z), SM , δc(Z))dSm =

√

(δc(z)− δc(Z))2

2π(Sm − SM)exp

(

− (δc(z)− δc(Z))2

2(Sm − SM)

) dSm

Sm − SM

.

(2.70)

One may also write

f(m|z,M,Z) dm = f(ν) dν =

√

ν

2πe−

ν2dν

ν. (2.71)

where we have replaced

ν =δ2c (z)

Sm

→ (δc(z)− δc(Z))2

Sm − SM

=a2

c

s. (2.72)

In other words, the ePS conditional mass function is self-similar.

Lastly, N represents the mean number density of m-haloes at redshift z con-

ditioned to belong to M -haloes at a later Z. This means that we can write the

conditional f in terms of N as

f(m|z,M,Z)d m =m

MN(m|z,M,Z)d m. (2.73)

50

In other words, the conditional mass function is

N(m|z,M,Z)d m =M

mf(m|z,M,Z)d m. (2.74)

In summary, the excursion set approach extracts information from the initial

overdensity field and predicts statistical properties of collapsed haloes at later times.

One key ingredient is the spherical collapse threshold, which allows us to map halo

abundances to first crossing distributions of a constant barrier (Figure 2.9 and 2.11).

This set of properties is commonly known as the extended Press-Schechter theory

(ePS). We will see that the assumption that the spherical collapse prescription can

be generalized, thereby improving the excursion set predictions of halo abundance.

2.10 Ellipsoidal collapse

For many years, the extended Press-Schechter formalism appeared to agree well

with N-body simulations (Efstathiou & Rees, 1988; Efstathiou et al., 1988; White

& Frenk, 1991; White et al., 1993). However, as simulations improved in size and

resolutions, the limitations of this theory became evident (Lacey & Cole, 1994; Gelb

& Bertschinger, 1994). For instance, it was noted that the Press-Schechter mass

function underpredicts the abundance of haloes of high mass. The opposite trend

is true in the low-mass end. In particular, Sheth & Tormen (1999) found that

f(ν)dν = A

√

qν

2πe−

qν2 [1 + (qν)−p]

dν

ν, (2.75)

51

with (A, p, q) = (0.332, 0.3, 0.707), provides a better description of the mass func-

tion2 (in this context, see also Jenkins et al., 2001). Similar criticisms were made by

Tormen (1998) for the conditional mass function. These findings sparked a thorough

revision of the ePS theory, and its underlying assumptions.

The excursion set approach assumes that all the information regarding the abun-

dance of collapse objects is encoded in the initial overdensity field. Such objects

are selected with a spherical collapse threshold, found by following the gravitational

evolution of an isolated region. In reality, regions are affected by their surroundings,

and collapse is triaxial (Doroshkevich, 1970; Bardeen et al., 1986). It is therefore

more appropriate to assume that the growth of structures is determined by the

initial shear field (Hoffman, 1986, 1988; Dubinski, 1992; Bertschinger & Jain, 1994;

Audit & Alimi, 1996; Audit et al., 1997). The gravitational evolution of ellipsoids

has been studied by several authors (Lynden-Bell, 1964; Lin et al., 1965; Fujimoto,

1968; Zeldovich, 1970; Icke, 1973; White & Silk, 1979; Barrow & Silk, 1981; Lem-

son, 1993; Hui & Bertschinger, 1996; Bond & Myers, 1996; Jing & Suto, 2002) –

see Desjacques (2008) for a review. In such models, the deformation (shear) ten-

sor is defined as Dij ≡ ∂i∂jφ = Dji, where φ is the gravitational potential and D

has six independent components. The natural generalization of the excursion set

formalism presented earlier would be to replace the random walk the overdensity

performs with shrinking smoothing scale with six such random walks, one for each

2The constant A here is a normalization factor, not a free parameter.

52

component of D (Chiueh & Lee, 2001; Sheth & Tormen, 2002; Sandvik et al., 2007).

A much simpler prescription is proposed by Sheth, Mo & Tormen (2001), who find

that triaxial dynamics can be effectively described by a mass-dependent ellipsoidal

collapse barrier. Incorporating ellipsoidal collapse into the excursion set theory

merely requires replacing the spherical collapse barrier δc(z) for δec(m, z). We now

discuss this prescription in more detail.

For a Gaussian random field smoothed on a scale S, Doroshkevich (1970) pro-

vides a formula for the probability of having a given set of (ordered) eigenvalues

of the deformation tensor: p(λ1, λ2, λ3)dλ1dλ2dλ3 (see also Monaco, 1995, 1997a,b;

Audit et al., 1997; Lee & Shandarin, 1998). However, it is more practical to describe

an ellipsoid in terms of (e, p, δ), the initial ellipticity, prolateness and overdensity –

which are combinations of (λ1, λ2, λ3) (Bardeen et al., 1986). Particular cases are

an ellipsoid with one axis of symmetry (p = 0) and a sphere (e, p = 0). For given

e and p, the collapse threshold δec(e, p) can be found. Sheth, Mo & Tormen (2001)

propose that, to a good approximation,

δec(e, p)

δc

= 1 + β[

5(e2 ± p2)δ2ec(e, p)

δ2c

]γ

, (2.76)

where (β, γ) = (0.47, 0.615), and the plus (minus) is used for negative (positive) p.

In general, one may only solve equation (2.76) for δec numerically. To obtain the

desired δec(m, z), Sheth, Mo & Tormen (2001) set p to its mean value and e to its

most probable value. From this, one must compute p(e, p|δ)de dp, the distribution

of e and p for a given δ (Bardeen et al., 1986). The average prolateness is p = 0.

53

The ellipticity distribution peaks at e = (√

S/δ2)/√

5 when p = 0. Thus, the

ellipsoidal collapse threshold at redshift z is δec(e, p, z). Setting p = p = 0 and

e = e = (1/√

S/δ2ec)/√

5 in equation (2.76), they obtain

δec(m, z) = δc(z)

1 + β[ S

δ2c (z)

]γ

. (2.77)

A few comments are in order. First note that all the redshift dependence is

contained in δc(z), the spherical collapse threshold. For large masses, S → 0 and

δec → δc; massive regions are barely affected by their surroundings and their collapse

is spherical. For small masses, S is large and δec(m, z) increases. Physically, this

mean that small regions are more affected by their surroundings and therefore must

be much denser to keep themselves together and be able to collapse. The following

section will show how this simplified version of ellipsoidal collapse can be readily

incorporated into to the excursion set approach (Sheth, Mo & Tormen, 2001).

2.11 Moving barrier models

Recall that in the excursion set approach, the problem of estimating halo abun-

dances is mapped to one of estimating the distribution of the number of steps a

Brownian-motion random walk must take before it first crosses a barrier of specified

height. For instance, the Press-Schechter mass function is associated with barriers of

constant height δc – such model assumes that haloes form from a spherical collapse.

In constrast, the mass function associated with the simplified version of ellipsoidal

54

collapse (Sheth, Mo & Tormen, 2001) is actually related to barriers whose height

increase monotonically with the number of steps:

B(S, δc) =√

qδc

1 + β

[

S

qδ2c

]γ

, (2.78)

This barrier is identical to δec(S, δc) in equation (2.77), except for the additional

constant q (Sheth & Tormen, 2002) (see below). In the context of diffusion theory,

this is called a ‘moving’ barrier (see Section A.1.2).

Having a barrier that increases with S makes it more difficult for random walks

to cross it. The net effect of this is to decrease the abundance of low-mass objects.

Replacing δc → √qδc with q < 1 decreases the height of the barrier for all values

of S. The effect of this is that it is easier for random walks to cross the barriers at

low S, associated with high masses. By choosing this ‘ad-hoc’ constant correctly,

the abundance of massive objects can be such that the number of small ones is

not increased significantly. This solves the discrepancy between ePS and N-body

simulations. We will see that the value of q in the Sheth & Tormen (1999) formula

(equation 2.75) works well.

This free parameter can be considered a weakness of the theory. There are

two possible explanations. First, Sheth & Tormen (2002) point out that N-body

simulations need to rely on one free parameter to identify dark matter haloes. For

instance, the ‘spherical overdensity’ method groups particles in spherical regions

into collapsed objects if such regions are denser than some critical density (typically

55

ρSO ≃ 200ρ). There is some freedom in how ρSO is chosen.3 Others use a ‘friends-of-

friends’ algorithm to connect particles so that each is at least at some fixed distance

from another in the group (typically this distance bFOF ≃ 0.2× the mean inter-

particle distance). In this method too, the choice of ‘FOF’ distance is somewhat

arbitrary. The second explanation, also discussed by Sheth, Mo & Tormen (2001),

is somewhat more technical. The excursion set theory predicts the abundance

of collapse objects from the statistics of random walks associated with random

positions in the initial density field. However, it is more realistic to assume that

haloes collapse around the peaks in the initial density field, and not just around

any random position (see also Bond & Myers, 1996). Calculating quantities like the

mass function from the more realistic approach is a rather difficult task. Sheth, Mo

& Tormen (2001) argue that the net effect of using the first approach is that the

predicted halo masses are underestimated. This effect is, in principle, encoded in

the q parameter. For our present purposes, q will be treated as a free parameter.

Figure 2.12 shows the different barriers for different collapse models (see below)

at two redshifts. The solid dots are included for easy comparison with Figure 2.11.

The red (dotted) horizonal lines represent the constant barriers associated with ePS

at the two redshifts. Here, (q, β, γ) = (1, 0, 0). The green (long-dashed) curves show

the ellipsoidal collapse barrier in equation (2.78), which has (0.707, 0.47, 0.615). A

few comments regarding the nature of this barrier are in order. Because γ > 1/2,

3We must emphasize that this is not the same as the ePS approach, where linear-extrapolations

of the initial overdensity field are used.

56

Figure 2.12: Different collapse barriers. Barriers associated with spherical (constant

– red dotted lines), and ellipsoidal (γ > 1/2 – green long-dashed curves) at different

redshifts are shown along with the square-root barriers (γ = 1/2 – blue dashed

curves). The filled circles are included for easy comparison with Figure 2.11; these

show that the mass functions (unconditional and conditional) will change with

different choice of barriers for a given random walk.

57

not all walks are guaranteed to cross it, because the rms of the walks scales as

√S (Sheth & Tormen, 2002), which increases slower than S0.615. Another problem

is that the ellipsoidal collapse barriers associated with two different times may

intersect; of course, this never happens for the spherical collapse barriers. Sheth &

Tormen (2002) suggest that this intersection of barriers may represent the possibility

that haloes can fragment. We will be interested in models which assume that

fragmentation never occurs. This is one of the reasons why we study the limiting

case of ‘square-root’ barriers for which γ = 1/2:

B(S, δc) =√

qδc + β√

S. (2.79)

Such barriers are shown in blue (short-dashed), and their predicted halo abundances

are very similar to those in N-body simulations if one sets (q, β, γ) = (0.55, 0.5, 0.5)

(see Figure 2.13 in Section 2.12). Aside for the value of q, this barrier is very similar

to the γ = 0.615 barrier. Notice that this barrier is the sum of two factors, a redshift-

dependent one and a mass-dependent one. All the redshift information is encoded

in the barrier’s ‘δ-intercept’; and the barriers at different redshifts are identical,

up to a vertical shift. These properties are also true for the constant barrier, but

not for the Sheth & Tormen (2002) barrier with γ > 1/2. Also, this ‘square-root’

model is particularly interesting, in view of the fact that an analytic solution to

the first crossing distribution of square-root barriers is available (Breiman, 1967).

(See Section A.1.2 for expressions and a derivation of the exact solution to the

diffusion equation with square-root barriers.) Analytic expressions for the first

58

crossing distribution of barriers with γ = 1 and γ = 2 are also known (Schroedinger,

1915; Groeneboem, 1989). Unfortunately, an exact solution for general γ does not

exist – but see Zhang & Hui (2006) for an integral-equation solution.

For general barriers, Sheth & Tormen (2002) provide a series approximation

that works reasonably well for a large range of regimes, and it is given by

f(S|δc)dS =|T (S)|√

2πSexp

[

− B(S, δc)2

2S

]

dS

S, (2.80)

where T (S) includes the first few terms of the Taylor expansion of B(S, δc):

T (S) =5∑

n=0

(−S)n

n!

∂nB(S, δc)

∂Sn.

They point out that, when γ = 0.615, setting n = 5 is sufficient to reach reasonable

accuracy. We will see that this is true when γ = 0.5 as well. Notice that this

approximation is exact for the constant and linear barrier (Sheth, 1998).

The barrier in equation (2.78) can be written as

B(S, δc)√qδc

= 1 + β(qν)−γ. (2.81)

where ν = δ2c/S is the self-similar variable. The mass functions arising from such

barriers are also self-similar. The series approximation (equation 2.80) becomes

f(ν)dν =

√

qν

2πe−

qν2

[1+ β(qν)γ

]2 [1 +βα

(qν)γ]dν

ν, (2.82)

where α is the result of the Taylor sum; and it is given by α ≃ (0.094, 0.2461) for

the models with γ = (0.615, 0.5), respectively.

59

Similarly, Sheth & Tormen (2002) provide an approximate formula for the con-

ditional crossing distribution. This is given by

f(Sm|δc(z), SM , δc(Z))dSm =|T (Sm|SM)|

√

2π(Sm − SM)exp

[

− (Bm(z)−BM(Z))2

2(Sm − SM)

]

dSm

Sm − SM

,

(2.83)

where the Taylor-series factor is

T (Sm|SM) =5∑

n=0

(SM − Sm)n

n!

∂n

∂Smn

[

Bm(z)−BM(Z)]

. (2.84)

and we have denoted Bm(z) = B(Sm, δc(z)), etc., (compare to equation 2.70 with

constant barriers). We will now compare these findings with measurements in N-

body simulations.

2.12 Comparison with N-body simulations

In this section, we compare the unconditional mass function for the various mod-

els presented above, against N-body simulations. A similar comparison with the

conditional mass function will be postponed until Chapter 3.

In this thesis we use the GIF2 N-body simulation, which follows the evolution

of 4003 particles in a periodic cubic box 110h−1Mpc on a side in a flat ΛCDM back-

ground cosmology with parameters (Ωm,0, σ8, h, Ωb,0h2, n) = (0.3, 0.9, 0.7, 0.0196, 1)

(Gao et al., 2004b). We use these cosmological parameters everywhere, unless stated

otherwise. Fifty simulation snapshots were output, equally spaced in log(1+ z). At

each snapshot, haloes were identified using the spherical overdensity (SO) criterion,

60

adopting for virial mass the definition of Eke et al. (1996) (i.e., with virial density at

∼ 324ρ at redshift zero). The particle mass is mp = 1.73× 109h−1M⊙ and only ob-

jects with at least ten particles are considered. m⋆(z), defined by δ2c (z) = S(m⋆(z)),

is the typical mass scale at redshift z (that for which ν = 1). It is common prac-

tice to express halo masses in terms of m⋆ = m⋆(z = 0) (the z-dependence is

suppressed for the present time). For this cosmology and initial power spectrum,

m⋆ = 8.7×1012h−1M⊙ ≃ 5030mp. The simulation data and halo catalogs are avail-

able at http://www.mpa-garching.mpg.de/Virgo. The N-body data used in this

work was acquired by the Giocoli-Tormen pipeline in Padova, Italy. See Giocoli et

al. (2008a) for more details regarding the post-processing of the simulation.

In Figure 2.13 we plot the Sheth & Tormen (1999) mass function (long-dashed

black curve) and the different barrier predictions against the N-body simulation

data (filled black circles). Notice the improvement over ePS (red dotted curve) when

γ = 0.615 is used (dashed dark-green curve). The square-root barrier also gives an

excellent description in this range of masses. Moreover, the Sheth & Tormen (2002)

series (equation 2.80 – short-dashed blue curves) agrees well with the exact square-

root solution (solid magenta curve), given in Section A.1.2 below. The bottom panel

shows the ratio with respect to the Sheth & Tormen (1999) formula.

For the sake of simplicity, we have only presented the mass function at the

present time. This is valid because, as shown by Sheth & Tormen (1999), the mass

function is self-similar and it can be rescaled for other redshifts. These authors

61

Figure 2.13: The unconditional mass function. Top panel: N-body data are the

filled black circles and the long-dashed black curve is the Sheth & Tormen (1999)

prediction. The green (dashed) and blue (short-dashed) are the series approxi-

mations to the γ = (0.615, 0.5) barriers, respectively. The ePS mass function is

presented in red (dotted) and the exact square-root barrier solution in magenta

(solid). Bottom panel: All data and curves divided by the Sheth & Tormen (1999)

formula.

62

acknowledge that the halo mass function is not expected to be self-similar exactly.

This is confirmed by recent precision investigations (e.g., Warren et al., 2006; Reed

et al., 2007; Lukic et al., 2007; Tinker et al., 2008), but the deviations are small

enough for our purposes to be ignored.

63

Chapter 3

Merger history trees

In the early matter-dominated universe, overdensities get enhanced by gravity,

which later become virialized dark matter haloes. Small haloes form first and merge

with others, creating more massive haloes as a result. Looking backwards in time,

this merger picture is equivalent to a branching process, and the history of a final

halo is contained in a merger tree (Figure 3.1). Our main objective in this chapter

is to show that the excursion-set formalism can be used to reconstruct the merger

history tree of a halo. We provide an algorithm based on the ellipsoidal collapse

model with square-root barriers, and several tests are performed. Some technical

details on the implementation of this algorithm can found in Appendix B.

64

Figure 3.1: The merger history tree of a halo with mass M at a final redshift Z.

Redshift z increases in the vertical direction, and the masses of the merging pieces

are proportional to the width of the branches. The horizontal lines represent sample

redshift snapshots.

65

3.1 Motivation

Most models of galaxy formation in a hierarchical universe assume that the merger

history of the surrounding dark matter halo plays an important role in determining

the properties of a galaxy (White & Rees, 1978; Blumenthal et al., 1984; White

& Frenk, 1991) – (see Baugh, 2006; Avila-Reese, 2006, for reviews). Although

halo merger histories can be measured using N-body simulations, these can be

time consuming and computationally intensive (Springel & Hernquist, 2005). This

has fueled considerable study of the formation and merger histories of dark matter

haloes from a excursion-set Monte Carlo perspective. Monte Carlo merger trees have

the advantage of being fast and one may easily probe mass regimes inaccessible to

current N-body simulations. Moreover, unlike N-body experiments, the cosmology

and initial conditions may be easily modified.

Fast algorithms for generating halo merger trees, in which haloes were assumed

to form from a spherical collapse, were developed in the 1990s (Kauffmann & White,

1993; Somerville & Kolatt, 1999; Sheth & Lemson, 1999; Cole et al., 2000) – see

Zhang et al. (2008b) for a comparison. However, recall that spherical collapse over-

predicts (underpredicts) the abundance of haloes in the low (high) mass regime.

To address these issues, Sheth & Tormen (1999) extended the excursion set frame-

work to include ellipsoidal collapse (Sheth, Mo & Tormen, 2001; Sheth & Tormen,

2002). This clearly showed that merger-trees which assume spherical collapse are

inadequate.

66

Hiotelis & Del Popolo (2006) and Zhang et al. (2008b) describe merger tree

algorithms which extend some of the older algorithms to incorporate aspects of

the ellipsoidal collapse results. In addition, a number of new algorithms have re-

cently been published (Parkinson et al., 2008; Neistein & Dekel, 2008a); although

efficient and accurate, such methods side-step the idea of ellipsoidal collapse alto-

gether. Moreover, these methods are calibrated to match N-body simulations, and

are therefore limited by the accuracy and scope of these simulations.

The most significant difference between the algorithm we derive here and all the

others described above (full N-body simulations included) is that it takes discrete

steps in mass rather than time. The horizontal lines in Figure 3.1 show how tradi-

tional merger trees are built. Each of these discrete ‘snapshots’ contains the pro-

genitors of the final halo at the corresponding redshifts. If one takes the limit where

the separation between these snapshots is small, the full merger history tree can

be recovered. Then one typically uses the progenitor mass function to break haloes

into their progenitors at each step. In the present approach, on the other hand,

redshift snapshots do not enter the picture. Instead, we generate random walks

directly, and extract ‘mass history’ information from them. This way a branch is

built and subsequent branches can be connected. In this approach, the number of

steps S of the random walk – or equivalently the mass – is discretized. This feature

allows us to study a number of problems which are more difficult to address with

the other methods. We now proceed to introduce the notion of ‘mass history’ in

67

the spherical and ellipsoidal collapse contexts.

3.2 Mass histories and merger trees

Figure 3.2 illustrates how the mass growth history of an object is encoded in the

excursion set approach if objects form from a spherical collapse (also see Figure 1 of

Lacey & Cole, 1993). The jagged line shows a random walk which starts from the

origin: (S, δ) = (0, 0). Imagine drawing a horizontal line with height δc0 = 1.686 and

marking the smallest value of S at which the walk intersects this barrier of constant

height (recall that δc0 corresponds to the present time and δc > δc0 corresponds to

higher redshifts). The dotted horizontal line denotes such a barrier. Then increase

the height of this barrier, and record how this value of S changes as δ increases. Such

mass history points are depicted as dark red filled circles in Figure 3.2. The dashed

lines show that S will occasionally jump from a small value to a larger one. Since

S is a proxy for mass, and δc for time, such a jump is a proxy for an instantaneous

change in mass: a merger. Note that the random walk steps under such jumps are

not part of the mass history (e.g., the gray portion with Sm < S < Sm′).

The key to our merger tree algorithm is to recognize that these jumps mean

that there are a set of other walks which one might associate with this one – one

for each jump. One such walk is illustrated by the second jagged curve, which

starts at about the middle of the panel. If the jump from Sm to Sm′ occurred when

the barrier height was δc(z), then this other walk starts from (Sm−m′ , δc(z)). The

68

Figure 3.2: A random walk and its associated mass history. The dark-red filled

circles represent the history of a halo of mass M at redshift z = 0. A merger

(m′,m − m′) → m at redshift z is depicted by the Sm → Sm′ jump (in gray) at

height δc(z). A new branch associated with (m−m′) is connected at (Sm−m′ , δc(z)).

The light-red filled circles denote the mass history of this object.

69

‘merger history’ associated with this new branch is represented by the light-shade

red filled circles in Figure 3.2. For every such jump, a new random walk must be

drawn. For each jump within each of those new walks, the same process applies

– more walks must be drawn. In summary, the bundle of such walks encodes the

entire merger history of a present-day object. Notice that jumps can occur at any

z – there is no constraint that they happen at discrete times. However, if one is

interested in the mass function of progenitors at some fixed z, one simply reads-off

the list of values of S at which this bundle of walks first cross δc(z).

So far, we have discussed how to generate trees in the spherical collapse model.

Figure 3.3 shows the same walk as before, but now the mass growth history associ-

ated with the walk is given by its intersection with square-root barriers of gradually

increasing height. This shows clearly that the jumps in mass, and the times at

which they occur, are modified. But the overall logic remains the same. Each

jump gives rise to a new walk from (Sm−m′ , B(Sm−m′ , δc(z))), where B is given by

equation (2.79).

The natural generalization would be to incorporate the original γ > 1/2 barrier

of Sheth, Mo & Tormen (2001). One feature of this model is that barriers with

different values of δc intersect (Sheth & Tormen, 2002). This is because the barrier

in equation (2.78) increases faster with S when δc is small. Consider two barriers

with δ-intercepts given by δc1 and δc2 (with δc2 > δc1). These two barriers cross at

70

Figure 3.3: The same random walk as in Figure 3.2, but now with square-root

rather than constant barriers, illustrating that the mass accretion history depends

on the barrier shape. In our algorithm, the new object with mass (m−m′) is now

connected at (Sm−m′ ,√

qδc(z) + β√

Sm−m′).

71

some S = S× that satisfies

β(S×/q)γ =

√q(δc2 − δc1)

1/δ2γ−1c1 − 1/δ2γ−1

c2

, where 2γ − 1 > 0. (3.1)

The value S× depends on δc1 and δc2. For instance, if δc2 & δc1, the denominator in

the above expression will be much smaller than the numerator, and S× will be large.

Likewise, if δc1 >> δc1, the denominator will be much larger than the numerator,

and S× will be small.

For merger tree algorithms that rely on time snapshots, having barriers that

intersect can be problematic (e.g., Hiotelis & Del Popolo, 2006; Zhang et al., 2008b).

For instance, consider two barriers B1 and B2 with δc1 and δc2 respectively (with

δc2 > δc1 and z2 > z1). Also, consider a point (SM , B2), with SM > S×, and

the first crossing distribution of the B1 barrier by random walks originating from

that point. This distribution is associated to the conditional mass function of the

progenitors of the M -halo. Notice that the redshift of this progenitors is lower than

that of the parent halo. This can be interpreted as a halo of mass M at redshift z2,

which fragments into its progenitors at z1. There is nothing particularly problematic

about fragmenting haloes – and this phenomenon is certainly seen in simulations

(Tormen, 1998; Fakhouri & Ma, 2008a). However, it is not clear that the intersection

of moving barriers correctly describes fragmentation in N-body simulations. This

problem can be avoided altogether by choosing the time snapshots close enough so

that S× > Sdust. For example, consider the time snapshots in the GIF2 simulation.

For every two consecutive snapshots, there is a value of S× associate to each pair.

72

Figure 3.4: Intersecting barriers with GIF2 time snapshots. The black dots are

located at each (S×, δc(z)), where δc(z) refers to the redshift snapshots in the GIF2

simulation and S× are the points where the barriers (with γ > 1/2) associated with

a pair of consecutive snapshots intersect (see equation 3.1). The vertical dotted line

is located at S = Sdust and the horizontal dotted line is located at δc0.

73

Figure 3.4 illustrates this. Notice that all the values of S× > Sdust (where the

vertical dotted line is located at Sdust). In fact, Figure 2.2, shows that the masses

associated with the crossing points in Figure 3.4 are all ≪ 107M⊙h−1. Thus, for

the problems studied in this thesis, the mass regime where intersection of barriers

occurs is never probed.1

Nevertheless, avoiding the issue of intersecting barriers simplifies matters sig-

nificantly. For instance, recall that the algorithm considered here does not rely on

time snapshots. Instead, for each point along each random walk generated, one

must be able to associate a value of m and z. If barriers with γ > 1/2 are used,

this mapping is not unique. This is one of the main reasons we choose the limiting

case with γ = 1/2. Moreover, we will see that non-intersecting moving barriers will

play a key role in our formulation of halo creation (Chapter 4).

3.3 The Monte Carlo merger tree

To compare our merger histories with those in the GIF2 simulation, we generated

2000 realizations of our tree for each final halo mass bin M of interest. In all cases,

the minimum mass considered was mdust = M/1000, and the merger histories of

haloes with mass below this were not followed (we call this minimum mass the

‘branching-mass resolution’). We used random walks with 105 steps in between SM

1See, however, Angulo & White (2009). These authors probe the S ≃ 700 regime, corresponding

to M ≃ 10−3M⊙, where the physics of free-streaming of neutralinos matters.

74

Figure 3.5: A branch at ‘continuous’ and ‘snapshot’ redshifts. The blue filled cirles

show the mass history retrieved from a random walk (‘Cont’). The black circles

show the result of constraining this history to the redshift snapshots of the GIF2

simulation (‘Snap’).

75

and Sdust to ensure that the mass change between the steps was less that mdust.

Having a small step size is essential to faithfully reproduce random walks in a

computer. Moreover, if the step size is too large, we run the risk of missing branches.

See Appendix B for more details on the implementation of our Monte Carlo tree.

Recall that our tree does not take discrete steps in time. Nevertheless, for fair

comparison with the measurements from the GIF2 simulation, the tree data were

stored in the same discrete redshift bins as were output from the simulation. We use

the word ‘Cont’ to denote the original tree data and the word ‘Snap’ for the data

stored in redshift snapshots. Figure 3.5 shows the result of storing a given branch

using these two prescriptions. For instance, near z = 7, the ‘snapshot’ version shows

a jump in mass, whereas the ‘continuous’ one sees two. Sections 3.7 and 3.8 study

some merger-related quantities which are sensitive to the differences between these

two ways of storing trees (Figures 3.12, 3.13 and 3.14). Before testing our tree’s

prediction of the conditional mass function, we will discuss a scaling symmetry

inherent to the square-root barrier model.

3.4 A conditional scaling symmetry

Recall that the unconditional mass function is associated to the first crossing dis-

tribution associated with walks which start from the origin: (S, δ) = (0, 0). When

constant barriers are used, this can be expressed self-similarly as

f(m|z)dm = f(S|δc)dS = f(ν)dν, (3.2)

76

where ν ≡ δ2c/S. The conditional mass function of m-haloes at z that end up

in bound objects of mass M > m at Z < z is given by f(m|z,M,Z)dm =

f(Sm|δ1, SM , δ0)dSm, where where δ1 = δc(z), δ0 = δc(Z). In other words, the

conditional mass function is associated with the first crossing distribution of a bar-

rier of height δ1 by random walks with origin at (SM , δ0). Because a straight-line

is straight whatever the origin of the coordinate system, the conditional mass func-

tion, in the spherical collapse model has the same functional form as that of the

unconditional mass function, provided one sets ν = (δ1 − δ0)2/(Sm − SM).

However, for the square-root barrier, a walk which starts from (√

q δ0+β√

SM , SM)

must cross a barrier of shape B =√

q(δ1 − δ0) + β√

Sm − SM + SM (see Sec-

tion A.1.2). This is not quite of the same form as equation (2.79). As a result,

the conditional mass function is not simply a rescaled version of the unconditional

one. Rather, in this model,

f(m|z,M, z0) dm = f(Sm/SM |ηc) d(Sm/SM), (3.3)

where

ηc ≡δ1 − δ0√

SM

= ηβ − β. (3.4)

Thus, final haloes of different masses will have similar progenitor mass functions

when expressed in terms of Sm/SM , provided they have similar values of ηc. While

this scaling is like that for the constant barrier model, in the square-root barrier

model, the progenitor mass function is not a function of the combination ν2 =

η2c/(Sm/SM − 1). This is interesting, because Sheth & Tormen (2002) have shown

77

that the conditional mass function in simulations is not well-fit by a function of

ν. In what follows, we will present evidence that it is, however, a function of ηc

and Sm/SM separately, so the qualitatively different scaling associated with the

square-root barrier is indeed seen in simulations.

3.5 The progenitor mass function

Figures 3.6-3.8 show the progenitor mass fractions and mass functions at five dif-

ferent redshifts (z = 0.5, 1, 2, 3, 5), for haloes identified at Z = 0 with final masses

given by M/m⋆ = 0.06, 0.6 and 6 (recall that m⋆ ≃ 8.7 × 1012M⊙h−1 in this cos-

mology). The corresponding values of ηc (equation 3.4) are shown in each panel. In

all three figures, filled circles show measurements in the GIF2 simulation, and open

circles show results from our square-root trees. We probe the m < mp regime with

our trees to verify consistency with analytic excursion-set predictions (the smooth

curves in all the panels). The short-dashed red curve shows the constant barrier

(β, q) = (0, 1) prediction associated with spherical collapse. The solid magenta

curves show the exact square-root barrier solution with (β, q, γ) = (0.5, 0.55, 0.5).

Dashed blue curves show the considerably simpler approximation to the solution

which is due to Sheth & Tormen (2002); this approximation is excellent over the

entire range of interest. The long-dashed dark-green curves show this same ap-

proximation for the ellipsoidal collapse barrier: (β, q, γ) = (0.707, 0.47, 0.615). The

square-root barrier prediction agrees well with the γ = 0.615 curve, except in the

78

Figure 3.6: The progenitor mass fraction (left) and mass function (right) at redshifts

z = (0.5, 1, 2, 3, 5), for haloes of mass M/m⋆ = 0.06 at z = 0. Filled circles show

measurements in the GIF2 simulation, and open circles are from our square-root

trees. The smooth solid (magenta) and dashed curves (blue) show the exact square-

root barrier solution, and the series approximation, respectively. The long-dashed

curve (dark green) shows the ellipsoidal collapse model with γ > 1/2, and the short-

dashed curve is the constant barrier prediction. Values of the scaling parameter ηc

(equation 3.4) are also shown (see Figure 3.9).

79

Figure 3.7: Same as Figure 3.6, but with M/m⋆ = 0.6.

80

Figure 3.8: Same as Figures 3.6, but with M/m⋆ = 6.

81

M/m⋆ z ηc — M/m⋆ z ηc

0.06 1 0.3 — 6 0.5 0.31

0.06 2 0.65 — 6 1 0.66

0.6 3 1.45 — 6 2 1.44

Table 3.1: The conditional scaling symmetry. Pairs in with similar ηc (equation 3.4)

in Figures 3.6-3.8 are listed here and plotted in Figure 3.9.

high-mass regime. This discrepancy becomes evident when ηc > 1 and it is amplified

with increasing ηc.

Before we ask how our merger tree algorithm compares with simulations, we note

that it produces progenitor mass functions that are well-described by the theory

curves over a wide range of masses and redshifts. At high redshifts, our tree data

lie slightly below the theory curves at both high and low m/M , and slightly above

in between, although where the cross-over points occur depends on z and M . In

all other regimes, our Monte Carlo trees match the square-root barrier predictions.

Any additional disagreement with the GIF 2 simulation measurements (compare

open and filled circles) is due to limitations of the γ = 1/2 model.

Finally, recall that the square-root and constant barrier models make specific

predictions for how the conditional mass functions should scale with final halo mass

and time. Table 3.5 lists pairs with similar ηc, yet quite distinct values of M

and z. Figure 3.9 compares the associated conditional mass functions. The black

82

Figure 3.9: The conditional scaling symmetry. Different combinations of M and z

with similar ηc (equation 3.4 and Table 3.5). N-body simulation measurements and

the Sheth & Tormen (2002) result with γ > 1/2 are shown.

83

squares and long-dashed black lines refer to the left-hand side of Table 3.5 (low

final masses), whereas the gray triangles and short-dashed gray lines refer to the

right-hand side (high final masses). Notice that the curves are remarkably similar

to one another, as are the symbols. This is true despite the fact that the values of ηc

are not perfectly identical, and that f(m, z|M,Z)dm = f(Sm/SM |ηc)d(Sm/SM) ≃

f(m/M |ηc)d(m/M). The results for low-mass haloes (black squares) are truncated

at higher m/M than they are for larger M (gray triangles), simply because only

haloes with at least ten particles are considered. Evidently, the conditional mass

functions are indeed functions of ηc and Sm/SM separately, rather than of the

combination ν.

3.6 The distribution of formation redshifts

Following Lacey & Cole (1993), a halo is said to have ‘formed’ when it first acquires

half of its final mass. For a given halo mass M (at a final redshift Z), there is a dis-

tribution of formation redshifts, denoted by p(zF |M,Z)dzF . We now show that this

distribution is intimately related to the progenitor mass function N(m, z|M,Z)dm.

Consider the merger history tree of an M -halo (e.g., Figure 3.1). At different red-

shifts, this halo has a set of progenitors. However, by conservation of mass, if one of

the progenitors has mass m > M/2, there may only be one such halo. Computing

the cumulative formation-redshift distribution up to zF is equivalent to counting

the number progenitors of that have been formed by that epoch (with m > M/2).

84

Namely,

P (> zF ) ≡∫ ∞

zF

dz p(z|M,Z) =

∫ M

M/2

dm N(m, zF |M,Z). (3.5)

Lacey & Cole (1993) realized that for spherical collapse, it is more convenient to

cast this equation in terms of

ω ≡ δc(z)− δc(Z)√

S(M/2)− S(M), and S =

S(m)− S(M)

S(M/2)− S(M). (3.6)

In these variables, they find that for white-noise initial conditions,

P (> ω) =

√

2

πωe−ω2/2 + (1− ω2)erfc

( ω√2

)

. (3.7)

Therefore, the distribution of formation redshifts is

p(zF |Z,M)dzF = p(ωF )dωF = −dωFd

dωP (> ω)

∣

∣

∣

ω=ωF

= 2ωF erfc(ωF√

2

)

dωF . (3.8)

For general Gaussian initial conditions, this can only be computed numerically.

Nevertheless, as pointed out by Lacey & Cole (1993), the dependence on the initial

power spectrum is very weak. We now compare these predictions with our merger

trees and with N-body simulations.

The filled circles in Figure 3.10 show the scaled formation redshift distributions

for haloes with masses in the range 0.9M ≤ M ≤ 1.1M with log10(M/m∗) =

1, . . . ,−1.5 in steps of −0.5 in the GIF2 simulation. We use the first snapshot

when at least half the mass is in a single progenitor at the formation time, and

make no attempt to interpolate our simulation formation redshifts between these

discretely spaced output times (Harker et al., 2006; Giocoli et al., 2007). Recall that

85

Figure 3.10: Scaled distribution of formation redshifts. Filled black circles show

simulation data, open blue circles and red triangles show results from the square-

root and constant barrier trees. Smooth curves show equation (3.8) with q = 1,

0.707 and 0.55 (short-dashed red, long-dashed green, and dashed blue), correspond-

ing to the predicted distribution for constant barriers (spherical collapse), moving

(ellipsoidal collapse) and square-root barriers.

86

we do not discretize redshift in our tree, so the question of interpolation does not

arise. The open circles and triangles show the corresponding formation time distri-

butions from our square-root and constant barrier trees. The same distribution, but

now in terms redshift is presented in Figure 3.11. Notice that for all models, Monte

Carlos and the GIF2 data, the formation time distribution peaks at higher red-

shifts (and are broader) when the final masses are smaller. In other words, massive

objects formed recently, whereas less massive ones are typically older.

For the ellipsoidal collapse model, Giocoli et al. (2007) showed that the formation

redshift is well-described by equation (3.8) if one replaces ωF → √qωF . The smooth

curves show this with q = 1, 0.707 and 0.55 (short-dashed red, long-dashed green,

and dashed blue), which represent the (constant, γ = 0.615, and square-root) barrier

predictions. For higher values of q, the peaks are located at lower redshifts and the

widths of the curves decrease. Recall that, strictly speaking, equation (3.8) only

holds for white-noise initial conditions. Nevertheless, as pointed out Lacey & Cole

(1993), it remains a reasonable approximation to the CDM case. Furthermore, note

that it provides an excellent description of the formation times generated by our

trees. However, no choice of q provides particularly good agreement with the GIF2

simulation, a discrepancy noted by previous authors (Lin et al., 2003; Hiotelis &

Del Popolo, 2006; Giocoli et al., 2007). This is likely a consequence of the excursion

set assumption that different steps in the walk are uncorrelated (Sheth & Tormen,

2002). Brownian motion random walks with this property are said to be Markovian.

87

Figure 3.11: Distribution of formation redshifts. Same as Figure 3.10, but in terms

of redshift.

88

To induce non-Markovianity, it is necessary to replace the sharp-k space filter, which

is a rather non-trivial matter. See Pan et al. (2008) and references therein for how

one might improve on this.

3.7 The mass distribution at formation

The previous section studied halo formation, where formation was defined as the

first time that the mass of one of the progenitors exceeds half the total. Therefore,

this mass can have any value between 1/2 and 1 times the final mass, and one can

study the distribution of masses at, and just prior to, formation. The excursion set

constant barrier model makes a prediction for this distribution (Nusser & Sheth,

1999). The mass distribution at formation is expected to be

p(µ) dµ =2

π

√

1− µ

2µ− 1

dµ

µ2, where 1/2 ≤ µ ≤ 1, (3.9)

and µ ≡ m/M , and the distribution just before formation is

q(µ) dµ =1

π(1− µ)

(

√

µ

1− 2µ−√

1− 2µ) dµ

µ2, (3.10)

where 1/4 ≤ µ ≤ 1/2. We have found that, to a very good approximation,

µ p(µ) → µ q(µ) if one replaces µ → 1/4µ (solid and dashed curves in Figure 3.12,

respectively).

Although these expressions were derived assuming a white-noise power spec-

trum, they are expected to be relatively independent of P (k). Sheth & Tormen

(2004) showed that they did indeed match numerical simulations well for different

89

Figure 3.12: Distribution of the mass at formation for several final masses. The left

half of each panel shows the mass just prior to formation, whereas the right half

shows the mass just after formation. Filled circles show simulation data, open circles

and triangles are from the square-root and constant barrier trees. The solid curve

shows µq(µ) (right half) and µp(µ) (left half) (equations 3.10 and 3.9 respectively).

The dashed curve shows these same expressions with µ→ 1/4µ.

90

Figure 3.13: Same as Figure 3.12, but showing only the region around m/M = 1/2.

The peak in the simulations (filled symbols) is less pronounced than in the merger

trees (jagged lines). Open circles show the result of sampling the merger trees at

the same redshifts as the simulation snapshots: this makes a dramatic difference

around 0.49 ≤ µ ≤ 0.51, suggesting that the sharp cusp predicted by the theory will

also be present in simulations with sufficiently closely spaced outputs. The smaller

discrepancies further from the peak remain.

91

cosmologies and initial power spectra. Figure 3.12 shows that they also work well

for square-root barriers.

The agreement between the theory curves (smooth curves) and our Monte Carlo

trees (jagged lines, labeled ‘Cont’) is excellent. All panels show that agreement with

simulation data is also quite good. However, there is a systematic discrepancy: the

cusp at µ = 1/2 appears to be less pronounced in the simulation, with correspond-

ingly lower tails. A similar discrepancy was seen by Sheth & Tormen (2004), who

suggested that the fact that the simulations only provide discrete snapshots in time

may be smoothing out the peak. By sampling our trees at the simulation snapshots

(open symbols, labeled ‘Snap’), we have attempted to model the magnitude of this

effect. Figure 3.13 illustrates that the cusp has indeed been smoothed, but this is

a dramatic effect only around 0.49 ≤ µ ≤ 0.51. The discrepancies further from the

peak remain (Figure 3.12).

3.8 The last major merger

Mergers of galaxies with similar masses are expected to produce strong short-lived

periods of star formation (i.e., starbursts) and quasars (see Chapter 5). Moreover,

recent numerical studies suggest that the mass ratio of the galaxies involved plays

an important role: merger-induced bursts occur when the galaxies have similar

masses (Gao et al., 2004a; Springel & Hernquist, 2005; Cox et al., 2008). Moreover,

it has been suggested that a galaxy’s Hubble type is strongly correlated with its last

92

major merger (e.g., Toomre & Toomre, 1972; Barnes, 1988; Hernquist, 1992, 1993).

(See Maller et al., 2006, for a discussion of the redshifts associated with the last

major mergers of galaxies). Understanding such mergers requires understanding

the mergers their host haloes undergo. Consider a merger (m′,m′ −m)→ m, with

m′ < m−m′. For ease of comparison with Parkinson et al. (2008), we will define a

‘major’ merger as one with mass ratio µ ≡ m′/(m−m′) ≥ 1/3. The filled circles in

Figure 3.14 show the redshift distribution of the last major merger onto the main

branch. (The last major merger does not necessarily happen on the main branch.

However, Figure 3 of Parkinson et al. (2008) suggests that, in most cases, it does.

Presumably, this is because the assembly of haloes in recent times is dominated by

mergers.) Curves show measurements in our full trees (‘Cont’), and open symbols

show the result of only sampling the trees at the GIF2 simulation outputs (‘Snap’).

For the discretely-sampled data, only mergers involving haloes with at least ten

particles are considered. Note that the anomalously low data point that is second

from the left in all panels appears to be an effect of seeing the tree at discrete

snapshots – the smooth curves show no such dip. This feature is also present in the

simulation analysed by Parkinson et al. (2008); we expect it to disappear if more

finely spaced snapshots are analysed.

For high masses, the data from the square-root trees peak at about the same

redshifts as the simulations; the constant barrier, spherical collapse trees peak at

lower redshifts. This improvement relative to the spherical collapse case is similar

93

Figure 3.14: The redshift distribution of the most recent major merger, for several

final masses. A major merger is defined as one in which the minor component

has at least 1/3 of the mass of the major component. Filled circles show the

GIF2 simulation data, open symbols show the corresponding measurements in the

same ‘snapshot’ versions of our trees, and smooth curves show the ‘continuous’

distributions which would be seen with arbitrarily closely spaced output times.

94

to that in the modified galform trees of Parkinson et al. (2008). However, our

square-root trees tend to lie above the simulation at low redshifts, and below at

higher redshifts. The discrepancy with simulation becomes increasingly worse at

small masses, although it is possible that the GIF2 results for our two smallest mass

bins are not reliable – the high redshift mergers involve haloes with few particles.

Because we require haloes to have more than 10 particles, we are likely to miss

major mergers once the typical mass becomes of this order.

95

Chapter 4

Halo creation

The previous chapter concentrated on describing the assembly of a dark matter halo

as a product of mergers. A numerical algorithm was introduced and its predictions

were evaluated. The aim of this chapter is to estimate analytically the rate at which

haloes are ‘created’ by mergers of smaller haloes.

4.1 Creation of dark matter haloes

The aim of this section we introduce the notion of halo ‘creation’. Let n(m|t) dm

denote the number density of haloes with mass in the range dm about m at time

t. As a result of mergers, dn/dt is the sum of two competing effects - the number

of objects in a given mass bin increases if smaller mass objects merge to form an

object of precisely this mass - an event we call creation - or the number decreases

as objects of this mass merge with others, thus depleting the number in the bin -

96

an event we call destruction. Thus, the time derivative of the halo mass function is

the difference of the creation and destruction rates:

dn/dt = C(m, t)−D(m, t). (4.1)

Given n(m|t), it is easy enough to take the time derivative; the problem is to

separate dn/dt into its two contributions. Roughly speaking, low mass objects

may have undergone significant mergers in the past but they are not being created

in merging events any more - their evolution is expected to be dominated by the

destruction term. In contrast, extremely massive objects are undergoing substantial

merging activity at the current time, and the time derivative of the halo mass

function should be a good estimator of the creation rate of these objects. But

quantifying the general case requires a richer model.

Early work used the Press & Schechter (1974) form for n(m|t), and advocated

equating the ‘positive’ term in dn/dt with the creation rate, and the ‘negative’

term with the destruction rate (e.g., Haehnelt et al., 1998). But this is clearly

not a solution at all, since it provides no rule for how to determine what one

should correctly equate with ‘positive’. For example, if dn/dt = P − N , there

is no particular reason why one could not have written the right hand side as

(P − ǫ) − (N − ǫ). The second part of this chapter is devoted to extracting the

creation rate C from dn/dt. See Blain & Longair (1993a,b), Sasaki (1994) and

Kitayama & Suto (1996) for other attempts to solve this problem.

Before addressing the creation term C (the number density of mergers per Gyr),

97

in the first part of this chapter we discuss the distribution of times c(t|m) when

these creation events take place (i.e., c ≡ C/∫

C dt is the rate normalized by the

total number of creation events that will ever occur). Although c and C differ

only by normalization constant, it turns out that c is somewhat easier to model.

This is because the excursion set formalism from which the Press-Schechter mass

function can be derived (Bond et al., 1991; Lacey & Cole, 1993), carries with it a

prescription for computing c(t |m), the distribution of creation times (Percival &

Miller, 1999). In this case, the functional form of c(t |m) is very similar to that of

f(m|t) ≡ m n(m|t)/ρ, where ρ is the mean comoving background density.

Since that time, interest has shifted to functional forms for n(m|t) which more

closely approximate the abundances measured in numerical simulations (e.g., Sheth

& Tormen, 1999; Warren et al., 2006; Reed et al., 2007; Lukic et al., 2007; Tinker

et al., 2008) – see Chapter 2 for a discussion. So it is interesting to ask how

the creation time distributions are modified. Percival et al. (2000) argue that the

relation between the functional forms of c(t|m) and f(m|t) should survive these

modifications, and show that this does indeed provide a good description of halo

creation in simulations. However, although they use intuition from the excursion set

approach to motivate their arguments, their method side-steps the generalization of

the excursion set approach from which the modified mass functions may be derived

– this is the ellipsoidal collapse ‘moving’ barrier approach (Sheth, Mo & Tormen,

2001; Sheth & Tormen, 2002). The first goal of this chapter is to calculate the

98

creation time distribution self-consistently within the moving barrier excursion set

approach. We do find that c and f are simply related, but that this is actually

extremely fortuitous – Section 4.3.3 demonstrates that this scaling does not hold

generally.

Getting the normalization constant which relates c to the creation rate C(m, t) is

a more challenging problem. Percival et al. (2000) obtained this quantity by match-

ing the creation time distribution to the rate measured in N-body simulations in

the low redshift regime (see also Percival & Miller, 1999), but they acknowledge

that they have no theory for the normalization factor. One possible solution to this

problem is to explore the evolution of the halo population in terms of coagulation

theory, where the creation and destruction terms are estimated separately (Smolu-

chowski, 1916, 1917). Early applications to galaxy formation and dark-matter halo

interactions include Silk & White (1978), Cavaliere et al. (1991a), Cavaliere et al.

(1991b), Cavaliere et al. (1992), Cavaliere & Menci (1993), Sheth & Pitman (1997)

and Menci et al. (2002). For a more recent treatment, see Benson et al. (2005) and

Benson (2008). For white-noise initial conditions, both the Smoluchowski and the

Press-Schechter excursion-set expressions for n(m|t) agree, so, for this case, the cre-

ation and destruction rates are known (Sheth & Pitman, 1997). However, obtaining

the rates for more general initial conditions, or for the modified mass functions that

are of more current interest, remains unsolved. The second goal of the present

chapter is to provide a model for the creation rate of dark matter haloes that is

99

informed by both coagulation theory and the modified excursion set approach with

moving barriers.

We also study the problem of how halo creation is modified if it is known that

the merging haloes are bound up in objects of mass M at some later time T .

The excursion set theory provides a way to compute the conditional mass func-

tion N(m|t,M, T ). We show how the conditional distribution of creation times

c(t|m,M, T ) is related to f(m|t,M, T ) = (m/M)N(m|t,M, T ). For the conditional

rate, the problem is to separate dN/dt into creation and destruction components.

Sheth (2003) argues that this conditional distribution may be the basis for under-

standing the phenomenon known as down-sizing (also see Neistein et al., 2006).

A few final remarks regarding the different uses of the term ‘halo creation’ are

in order. The first part of the chapter focuses on the creation time distribution,

c(t|m), which can be derived within the excursion-set formalism. The second part

focuses on the creation rate, C(m, t), the first term in the coagulation equation

(equation 4.1). The former is a normalized time distribution, while the latter is not.1

Another source of confusion is that halo ‘creation’ is distinct from halo ‘formation’;

following Lacey & Cole (1993) (and Chapter 3), the latter is typically defined as

the time that an object first reaches half its current mass. See Giocoli et al. (2007)

for an explicit calculation showing how creation and formation are related.

1The normalized distribution is denoted with lower case c, while the un-normalized rate is

denoted with capital C.

100

4.2 Mass history in the excursion set theory

Recall that in the excursion set approach, the problem of estimating the halo abun-

dances is mapped to one of estimating the distribution of the number of steps a

Brownian-motion random walk must take before it first crosses a barrier of speci-

fied height (Bond et al., 1991). In this approach, the height of the barrier plays a

crucial role. The Press-Schechter mass function is associated with barriers of con-

stant height - such barriers arise naturally in models in which haloes form from a

spherical collapse model. In constrast, the more accurate mass functions may be

related to ellipsoidal-collapse moving-barriers (equation 2.78).

Recall too that for the γ > 1/2 model of Sheth & Tormen (2002), not all

walks are guaranteed to cross the barrier, and two distinct barriers intersect. To

avoid these two problems, we introduced the ‘square-root’ barriers with γ = 1/2

(equation 2.79).

The dependence on S of the square root barrier means that it is more like the

ellipsoidal than spherical collapse barrier (which has constant height, independent

of S). However, there is one important respect in which the square root model

is very like the constant one. Consider the barriers associated with two different

times. For square-root barriers, the difference between the barrier heights is

B(S, δc2)−B(S, δc1) = (δc2 + β√

S)− (δc1 + β√

S)

= δc2 − δc1. (4.2)

101

Notice that this difference is independent of S. This is also (trivially) true for

constant barriers, but it is not true for any other values of γ. In this respect, the

excursion set model based on square-root barriers is extremely special. This will be

important later.

4.2.1 Mass history for different barriers

Figure 4.1 illustrates the relation between Brownian motion random walks and

the mass growth history of an object. The jagged line shows an example of a

random walk — this walk represents the run of smoothed overdensity around a

randomly chosen position in the initial fluctuation field, as the region over which

the overdensity is smoothed changes from large (left) to small (right). (The plot

actually shows the initial overdensity evolved to the present time using linear theory

— it differs from the initial overdensity by a multiplicative constant.) The initial

overdensities are all small compared to unity, so one may associate a mass with

each smoothing scale: this mass is larger for the larger smoothing scales.

Consider a horizontal line, and consider the places where it first intersects the

random walk, as the height of this line, this barrier, is raised. Clearly, this position

shifts to the right as the barrier is raised (red filled circles) — mass decreases as

redshift increases. Whereas the halo abundance problem corresponds to fixing the

barrier height δc (to illustrate, the height of dotted line corresponds to δc0 = 1.686)

and asking for the distribution of S values at which the barrier is first crossed,

102

Figure 4.1: The mass history associated with a random walk (jagged line) using a

constant barrier. Time increases as δ decreases and mass decreases as S increases.

The filled circles on the random walk denote the history, and the horizonal jumps

denote mergers. For reference, the horizontal dotted line denotes the barrier asso-

ciated with the present.

103

the halo creation problem corresponds to asking for the distribution of δc values

for a fixed S. We will use f(S|δc) dS to denote the first crossing distribution, and

c(δc|S) dδc to denote the distribution of creation times, where, for haloes of a given

mass m, c(δc|S) dδc = c(t |m) dt.

Notice that the mass increases relatively smoothly sometimes, and rather abruptly

at others. For instance, in the interval 4.2 . S . 4.8 in Figure 4.1, mass decreases

smoothly. Compare this situation to the sudden jump from S ≃ 4.8 → S ≃ 8.7

(red long-dashed line). Thus, this walk does not contribute to the creation time

distribution for any values of S between 4.8 and 8.3. But it does contribute in the

calculation of halo abundances for every δc.

It is interesting to compare this mass accretion history with that shown in Fig-

ure 4.2. The same walk is shown in both figures. However, now the horizontal dotted

line at δc0 has been replaced by a curve that has δ-intercept at√

qδc0, and increases

with increasing S — this is the square-root barrier (equation 2.79) associated with

the same epoch as the constant one. The comparison clearly shows that the mass

accretion history (blue solid circles) depends on the barrier shape. For instance, the

values with S . 4.3 and S ≃ 8.8, 9.7 and S . 10.4 are no longer included in the

mass accretion history (red solid circles). Moreover, even if a point on the random

walk happens to be part of the mass history in both cases, its associated time is

different under the two barrier prescriptions. To illustrate, the point at S ≃ 4.4 is at

the present for the square-root barrier case and in the past when constant barriers

104

Figure 4.2: Same as Figure 4.1, but with a square-root barrier. The blue soid circles

depict the history associated with the square-root barrier, while the red circles (for

spherical collapse) were kept to emphasize that different barriers predict different

mass histories for a given random walk. For reference, the horizontal dotted curve

denotes the barrier associated with the present.

105

are used. As a result, the halo mass function and the creation-times distribution

are modified. One of the goals in this thesis is to quantify these changes.

4.2.2 Creation times from Bayes’ rule

Consider the joint probability that a random walk first upcrosses a constant barrier

(with δ-intercept between δc and δc + dδc) between S and S + dS. Using Bayes’

Theorem, this can be written as

P (S, δc)dSdδc = f(S|δc)f(δc)dSdδc = c(δc|S)c(S)dδcdS,

For constant barriers, Percival & Miller (1999) argue that the δc-prior must be

uniform. First they note that all walks must have a creation event for any barrier,

regardless of its height (see Figure 4.1, left panel). Moreover, for any two equal-

sized intervals dδc1 and dδc2, the probability that such a creation event exists must

be equal. This is because the steps in the walk are uncorrelated, implying that any

point along the walk can be regarded as the starting point of a new walk. Therefore,

the walk is not altered at different values of δc and the probability of crossing two

different barriers at some point is the same. In other words, the δc-prior is given by

f(δc)dδc =dδc

∆δc

, (4.3)

where the constant ∆δc is infinite, since δc ∈ [0,∞).

Given f(δc)dδc and f(S|δc)dS, one can marginalize the joint distribution in δc.

106

This yields

c(S)dS = dS

∫ ∞

0

dδc[P (S, δc)] =dS

2A√

S∆δc

, (4.4)

where A = (√

π/2, 2, 2.08, 1.893) respectively for each of the constant, exact square-

root, square-root series approximation and the Sheth & Tormen (1999) result (see

Chapter 2 for a review). Inserting (4.3) and (4.4) in Bayes’ formula gives

c(δc|S)dδc = 2A√

Sf(S|δc)dδc. (4.5)

Let us write this expressions in terms of δ2c/S. In this work we will denote this

variable as ν if δc is fixed and as νc if S is fixed. Thus, f(S|δc) = (νc/S)f(νc) (S is

fixed) and c(δc|S) = (2νc/δc)c(νc), from which

c(νc)dνc = A√νcf(νc)dνc. (4.6)

Thus, for constant barriers, there is a remarkably simple relation between the cre-

ation distribution c and the first crossing distribution f .

4.2.3 Monte Carlo test and self-similarity

Barriers of the form given in equation (2.78) are self-similar, in the sense that

if δc is increased by a factor κ, then so is√

S. As a result, the first crossing

distribution f(S|δc) and the distribution of halo creation times c(δc|S) are both

simply functions of δ2c/S. In other words, if equation (2.78) describes ellipsoidal

collapse, then f(S|δc)dS = f(ν)dν and c(δc|S)dδc = c(νc)dνc. In the previous

107

section we showed that

c(νc) dνc = A√νc f(νc) dνc (4.7)

for the constant (A =√

π/2) and square root (A ≃ 2) barriers. This simple

relationship is one of the central results of this chapter, as is the warning that it

does not hold in general. To illustrate, Section 4.3.3 uses a system of linear barriers

in where this simple result does not apply.

To test equation (4.7) for constant and square-root barriers, we generated 105

random walks with 104 steps between S = 0 and S = S(mp) ≃ 28. Then we

stored the corresponding mass histories for the constant and square-root barriers

(e.g., solid circles in Figures 4.1-4.2). Every (S, δc) point along the history has a

corresponding νc. To study how the creation time distribution depends on S, we

could have chosen the subset of walks which have the correct value of S, and then

found the distribution of νc = δ2c/S values for those walks. However, the self-similar

scaling above means that c(νc) should be the same for all S. As a result, there is

no need to select a subset in S before binning in νc. If we simply bin up all the νc

values, whatever the associated values of S, then we can compare the result with

the predicted c(νc).

The symbols in Figure 4.3 show the creation time distributions for the constant

(triangles) and square-root (circles) barriers: the distribution associated with the

square-root barrier is broader and peaks at slightly higher νc. The curves show the

predicted creation time distributions (equation 4.7); they are in excellent agreement

108

Figure 4.3: The creation time distribution in self-similar form. The variable νc de-

notes δ2c/S at fixed mass. Triangles and circles show the distribution measured from

an ensemble of random walks with constant and square-root barriers respectively.

Dotted and solid (dashed) lines show the associated predictions. The long dashed

curve shows the result of inserting the Sheth & Tormen (1999) form for f(ν)dν into

equation (4.7).

109

with the measurements. Equation (4.7) shows that these creation time distributions

depend on the functional form of the first crossing distribution f(ν). For the case of

square-root barriers, we show the Breiman (1967) exact but complicated expression

for f(ν), and a much simpler approximation for it from Sheth & Tormen (2002).

The two curves are almost indistinguishable.

The use of barriers which scale self-similarly (equation 2.78) was motivated

by the observation that, when expressed as a function of ν, halo abundances in

simulations could be scaled to a universal form (Sheth & Tormen, 1999). We have

added the long-dashed curve in the Figure, which shows the result of using the

Sheth & Tormen (1999) functional form for f(νc) in equation (4.7); it is almost

indistinguishable from the curves associated with the square-root barrier.

4.3 Creation time distribution

In Section 4.2 we discussed the creation time distribution from the excursion set

point of view. Having shown that the analytic expressions accurately reproduce

our Monte Carlo measurements, we now study if they provide a good description of

halo creation in N-body simulations. Simulated haloes are labeled as having been

‘created’ if at least half of their particles were not observed in a more massive halo

at an earlier time.

110

4.3.1 Distribution of creation redshifts

The creation time distributions we measure in simulations and shown as filled circles

in Figure 4.4 are normalized to unity. However, the simulations only sample δc at

epochs before the present time, whereas the theory curves assume that 0 ≤ δc ≤ ∞.

Therefore, for haloes of mass m, we set

c(z|m) dz =c(δc|Sm)

∫∞

δc0dδ′c c(δ′c|Sm)

∣

∣

∣

∣

dδc

dz

∣

∣

∣

∣

dz, (δc ≥ δc0). (4.8)

These are the curves in Figure 4.4. The open triangles and circles show the creation

times measured in constant and square-root random walk ensembles sampled at the

same redshifts as the simulations. We studied bins of size dlog10 m = 0.2 in mass

centered at log10(m/m⋆) = 0.5 to −3 in steps of −0.5.

A couple of remarks are in order. Only the haloes with the highest redshift

in each mass bin were treated as newly-created. These measurement were tested

with different bin sizes (not shown), yielding similar results. One limitation is that if

dlog10 m is too small, most bins are empty, and the data does not follow a continuous

curve. One should not take dlog10 m to be too large – in particular, the mass bins

for the different values of m should be disjoint. Our choice of dlog10 m satisfied

both criteria. Lastly, we found that our choice of redshift bin dlog10(1 + z) = 0.05

was sufficiently large to capture enough creation events and sufficiently small for

comparison with the different theory curves.

The solid circles in the Figure denote the N-body measurement, where only

haloes with m > 10mp are considered (notice that the lowest-mass panel has no

111

Constant Exact Sqrt Approx Sqrt

ST-99

MC (Const) MC (Sqrt) N-Body

Figure 4.4: Distribution of creation redshifts for a number of bins in halo mass.

Filled circles show measurements in the simulations, and open triangles and circles

show analogous measurements made from sampling our constant and square-root

barrier random walk ensembles similarly to the simulations. Dotted curve shows the

prediction associated with a constant barrier; the exact square-root solution and its

series approximation are the solid and dashed curves; long-dashed curve shows the

result of inserting the Sheth & Tormen (1999) form into equation (4.7).

112

black filled circles). In all cases, improvement over the location of the peaks is seen

when the square-root barrier is used. However, these curves are slightly broader

than those traced by the simulation data, making the height of these normalized

distributions lower.

4.3.2 Self-similarity in halo creation

The excursion set model suggests that if the halo mass function f(ν)dν can be scaled

to a self-similar form, then the creation time distribution c(νc)dνc is also self-similar.

To test this we have scaled the values of δc(z) associated with each mass bin in the

simulations to νc = δc(z)2/S(m), and measured the resulting distribution of νc.

However, because all mass bins sample the same range in δc, they sample different

ranges in νc. We account for this by dividing the measured distribution of νc by

a normalization factor given by∫∞

νc0dν ′

c c(ν ′c), where νc0 = δ2

c0/S(m) and c(νc) is

associated with the Sheth & Tormen (1999) formula.

Figure 4.5 shows the result: different symbols show the rescaled distributions

associated with the various masses. Note that they do indeed appear to trace out a

universal curve. The various smooth curves show the constant barrier, square-root

barrier, and Sheth & Tormen (1999) based predictions. The symbols approximately

split the difference between the constant and square-root barrier models.

113

Figure 4.5: Universality of the distribution of halo creation times. The same theory

curves as in Figure 4.3 are shown. Different symbols show results for different halo

masses in the GIF2 simulation data, as indicated.

114

4.3.3 Why it doesn’t work in general

One of the key results of this work is equation (4.7), which relates the creation times

distribution to the mass function in a simple way. It is tempting to assume that

this holds for every mass function (e.g., Percival et al., 2000). The purpose of this

section and the next is to offer a counter-example, and to show why the general

case is more complicated. The next section discusses why this result works well for

constant and square-root barriers.

Consider the linear barrier

B(S, δc) = δc

[

1− β( S

δ2c

)]

. (4.9)

This is special case of the barrier in equation (2.78) with γ = 1 (we have suppressed

the parameter q). Notice that we have deliberately replaced β → −β in the above

expression. In this discussion we will only consider the β > 0 case to avoid barriers

that intersect. Moreover, if β > 0, all walks are guaranteed to cross. The exact

solution is known, and it is given by

f(ν)dν =

√

ν

2πe−

ν2(1−β

ν)2 dν

ν, (4.10)

where ν = δ2c/S (fixed S) (Schroedinger, 1915; Sheth, 1998). If equation (4.3) holds,

then the creation time distribution should be given by

c(νc)dνc = A√νcf(νc)dνc, where A =

√

π

2e−

β2 , (4.11)

and νc = δ2c/S (fixed δc).

115

Figure 4.6: The mass history associated with a random walk (jagged line) and linear

barriers. The brown filled circles on the random walk denote the creation events.

Notice that the barriers become steeper as δc increases, and the separation between

any two barriers increases with increasing S. For reference, the dotted line in the

lower left represents the linear barrier associated with δc = δc0.

116

To test this, we performed a Monte Carlo simulation of the mass histories as-

sociated with random walks where linear barriers (with β = 1) are used to select

creation events. Figure 4.6 shows one sample mass history of this ensemble. Com-

pare this with Figure 4.1, which shows the same process, but with constant and

square-root barriers. The jagged line is a random walk, and the brown solid circles

are the associated history. The long dashed lines denote jumps in the history. For

completeness we have included the linear barrier with δc = δc0, depicted as a dotted

line in the lower left. Figure 4.7 shows our Monte Carlo data (brown open squares).

The solid black curve is the prediction in equation (4.11). This disagreement inval-

idates the claim that c(νc)dνc is always proportional to√

νcf(νc)dνc.

Before moving on, note that the Monte Carlo data in Figure 4.7 follow a smooth

curve that is quite different from that associated with constant or square-root bar-

riers (Figure 4.3). The main difference is that the latter peak at some intermediate

value of νc, whereas the former decreases monotonically with νc. This is because it

is unlikely for a random walk to upcross a constant barrier (or a square-root barrier)

after a few steps. In other words, creation events with S ≪ δ2c (i.e., νc ≪ 1) are very

unlikely. Similarly, it is unlikely that a walk survives for many steps without being

absorbed by a constant barrier. That is, creation events with S ≫ δ2c (i.e., νc ≫ 1)

are unlikely. For the linear barrier, the ensemble of creation events is dominated

by points with small δc (i.e., small νc). This is because the linear barrier becomes

steeper as δc decreases. Since the height of a linear barrier decreases with S, it

117

Figure 4.7: The creation time distribution associated with linear barriers in self-

similar form. The variable νc denotes δ2c/S at fixed mass. The squares show the

distribution measured from an ensemble of random walks with linear barriers. The

solid curves show the associated predictions – assuming that equation (4.3) applies

(the ‘Bayesian’ result). The discrepancy between the theory prediction and the

data indicates that care must be taken when using the result in equation (4.7).

118

becomes easier for walks to upcross a barrier as δc decreases. This even allows for

cases where creation events are selected from points along a walk with δ < 0. Such

cases would be impossible for the constant and the square-root barrier models.

In principle, the creation time distribution with linear barriers can be computed

analytically, without using the Bayesian approach presented here (e.g., Karlin &

Taylor, 1975). Since the mass function associated with linear barriers does not

resemble halo abundances in N-body simulations, we do not pursue this any further

(but see Sheth, 1998, for interesting applications of this barrier).

4.3.4 Why it is a useful approximation in practice

If equation (4.6) is incorrect in general, then why then did it work so well in for con-

stant and square-root barriers (see e.g., Figure 4.3)? The first step is to recognize

that, because of the property highlighted by equation (4.2), a uniform distribution

in δc is appropriate for the family of square-root barriers of interest to us (equa-

tion 2.79), and so, for square-root barriers, equation (4.6) is exact. Our Monte

Carlo simulations shown in the main text confirm that this is indeed the case.

The second step is to note that, for barriers with general γ, the difference be-

tween two barriers carries additional factors of δc (e.g., equation 2.78). As a result,

assumption (4.3) is no longer valid. E.g., for linear barriers,

B(S, δc2)−B(S, δc1) = δc2 − δc1 − β( S

δc2

− S

δc1

)

6= δc2 − δc1. (4.12)

This property makes the linear barrier considerably different from the constant and

119

square-root barriers. Figure 4.7 shows that the distance between any two linear

barriers increases with increasing S (compare with Figure 4.1), so the δc-prior is

not uniform.

It is important to emphasize that the central conclusion of this and the previous

subsection – that the assumptions behind equation (4.7) do not hold in general –

do not depend on the fact that the linear barrier decreases in height with S. E.g.,

barriers of the form given by equation (2.78) with β > 0 and 0 < γ < 1/2 increase

with S. However,

B(S, δc2)−B(S, δc1) 6= δc2 − δc1; (4.13)

the separation between any two barriers increases with S, so the δc-prior is not

uniform in this case either. For these barriers too, equation (4.6) is incorrect.

Nevertheless, for γ close to 0 or 1/2, equation (4.6) should provide a reasonable

approximation. This is the fundamental reason why barriers with γ = 0.6, or of the

form required to give the Sheth & Tormen (1999) formula as the first crossing dis-

tribution, are likely to have creation time distributions which are well approximated

by equation (4.6).

4.3.5 Conditional distribution of creation redshifts

So far we have discussed the unconditional creation-time distribution c(t |m) and

its relation to f(m|t). In this section we study the creation time distribution,

c(t |m,T,M), of m-haloes at time t conditioned to be bound up in M -haloes at

120

a later time T . We also discuss how this quantity is related to f(m|t,M, T ), the

fraction of mass in m-progenitors at time t of a final halo of mass M at time T .

The latter is derived from the excursion-set theory by setting

f(m|t,M, T )dm = f(S|δ1, S0, δ0)dS, (4.14)

where S = S(m), S0 = S(M), δ1 = δc(t) and δ0 = δc(T ). The right-hand term

is the crossing distribution of a barrier B(S, δ1) by random walks with origin at

(S0, B(S0, δ0)). In Figure 4.1, this amounts to shifting the origin from (0, 0) to

(S0, δ0) (left panel) or to (S0,√

qδ0 + β√

S0) (right panel). In this work, the condi-

tional mass function is denoted by N(m|t,M, T )dm = (M/m)f(m|t,M, T )dm.

In essence, conditioning is equivalent to finding the (unconditional) crossing

distribution of a barrier

B(s, δ1, δ0) = B(s + S0, δ1)−B(S0, δ0), s = S − S0. (4.15)

In the constant barrier problem, B = δ1 − δ0. This means one simply replaces all

the unconditional expressions given previously with δc(z) → δ1 − δ0 and S → s =

S − S0. As a result, the only change occurs in the self-similar variable ν (and νc):

δ2c/S → (δ1 − δ0)

2/(S − S0) (Lacey & Cole, 1993; Percival & Miller, 1999; Sheth,

2003).

The square-root barrier is slightly more complicated:

B(s, ac) = ac + β√

s + S0, ac = δ1 − δ0 − β√

S0. (4.16)

Because equation (4.16) is not quite the same functional form as equation (2.79), the

121

first crossing distribution is not simply a suitably rescaled version of the uncondi-

tional distribution. Rather, it is a function of ηβ ≡ ac/√

S0 and s/S0. Nevertheless,

the logic one follows to arrive at the creation time distribution is the same. In

particular, the conditional versions of c and f are simply related:

c(ηβ|S/S0)dηβ = A(S/S0)f(S/S0|ηβ)d(S/S0), (4.17)

where now the A factor is a function of S/S0. This is another central result of this

chapter (compare to equation 4.7).

Let us derive this result for constant and square-root barriers. First notice that

the conditional crossing distribution can be written as

f(s/S0|ηβ)d(s/S0) = g(s/S0, ηβ)d(s/S0)

s/S0 + 1. (4.18)

where two forms of g are given next (Breiman, 1967; Sheth & Tormen, 2002). First,

for the exact square-root barrier solution:

g(s/S0, ηβ) =∑

λ

eηβ2/4Dλ(ηβ)lλ(−β)

(s/S0 + 1)λ/2. (4.19)

Likewise, for the Sheth & Tormen (2002) series approximation:

g(s/S0, ηβ) =

∣

∣

∣ηβ + β

√

s/S0 + 1[

1 + α(s/S0)

]

∣

∣

∣

√

2πs/S0

× exp

− (ηβ + β√

s/S0 + 1)2

2s/S0

s/S0 + 1

s/S0

, (4.20)

where

α(s/S0) =5∑

n=1

αn

s/S0 + 1, α 1 = −1

2, and αn = (1− 3

2n)αn−1.

122

The joint distribution of s/S0 and ηβ is given by

P (s/S0, ηβ)d(s/S0)dηβ = f(s/S0|ηβ)f(ηβ)d(s/S0)dηβ

= c(ηβ|s/S0)c(s/S0)d(s/S0)dηβ. (4.21)

Following equation (4.3) for δ1 and δ0 and using the fact that S0 is fixed, it can be

shown that the ηβ-prior is uniform:

f(ηβ)dηβ =dηβ

∆η(4.22)

where ∆η is an infinite constant. Marginalizing over the joint distribution in ηβ we

obtain

c(s/S0)d(s/S0)G

s/S0 + 1

d(s/S0)

∆η, (4.23)

where

G(s/S0) =

∫ ∞

0

dηβ g(s/S0, ηβ). (4.24)

Inserting equations (4.18), (4.22) and (4.23) in Bayes’ rule, we obtain

c(ηβ|s/S0)dηβ =g

Gdηβ. (4.25)

Comparing (4.18) to (4.25), we find that

c(ηβ|s/S0)dηβ = A(s/S0)f(s/S0|ηβ)dηβ, (4.26)

where

A(s/S0) = (s/S0 + 1)/G. (4.27)

123

Constant Exact Sqrt Approx Sqrt

Constant Exact Sqrt Approx Sqrt

MC (const)MC (Sqrt)N-Body

MC (const)MC (Sqrt)N-Body

Figure 4.8: Conditional distribution of creation redshifts. Symbols and style as

in Figure 4.4. We plot m = m⋆/10 (left panels) and m = m⋆/100 (right panels)

conditioned to end up in haloes of mass M = M⋆ (top panels) and M = 10m⋆

(bottom panels). In all cases, T denotes the present time.

124

Figure 4.8 shows the conditional distribution of creation redshifts for m/m⋆ =

(0.1, 0.01) that end up in haloes of mass M/m⋆ = (1, 10) today. Filled symbols show

the GIF2 measurements, open circles show our Monte Carlos with square root bar-

rier, and open triangles show Monte Carlos with a constant barrier. Smooth curves

show the corresponding predictions – notice that they are in excellent agreement

with the Monte Carlos.

The simulation bin sizes dlog10 m and dlog10(1+z) were used as in the uncondi-

tional case. Selecting haloes to be conditioned to belong to a final M -halo reduces

the number of creation events significantly. We selected haloes bound to end-up

in clumps with mass in a bin of size dlog10 M = 0.5. As in the unconditional case

(Figure 4.4), the distributions peak at higher redshifts and are slightly broader for

lower m (compare left and right panels). The same trends are seen as one increases

the final M (compare top and bottom panels).

In general, the moving barrier based curves provide a much better description

of the simulations, although the agreement is by no means perfect. For example,

the square root barrier tends to produce distributions which are slightly broader

than those in the N-body simulation. This effect is more pronounced in the right-

hand side panels. A similar effect was found in Chapter 3 for the formation time

distribution.We speculate that the discrepancy there was due to non-Markovian

effects. We refer the interested reader to Pan et al. (2008) for a discussion of this

topic.

125

4.4 Halo creation rates

Extracting the creation rate from dn/dt (or dN/dt) is a non-trivial problem. In

this section we make use of halo coagulation theory to estimate this quantity.

4.4.1 The unconditional rate

In the coagulation formalism (Smoluchowski, 1917), the halo mass function n(m|t)

obeys

dn(m|t)dt

= C(m, t)−D(m, t), (4.28)

where

C(m, t) =

∫ m

0

K(m′,m−m′; t)

2n(m′|t)n(m−m′|t)dm′, (4.29)

is our creation term, and the destruction term is

D(m, t) =

∫ ∞

0

K(m,m′; t)n(m|t)n(m′|t)dm′. (4.30)

In these expressions, the coagulation kernel K(m,m′; t) is symmetric in m and

m′.

Few analytic solutions to Smoluchowski’s equation exist. However, when the

kernel is additive in mass, then the associated mass function is given by the Press-

Schechter formula for white-noise initial conditions (Silk & White, 1978; Sheth

& Pitman, 1997). Of course, white-noise is a bad approximation to the initial

conditions in the CDM models of current interest. Moreover, we have shown that

ellipsoidal collapse gives a better description of halo abundances and creation times

126

than Press-Schechter (spherical collapse). Nevertheless, let us discuss the expression

obtained for the creation term in that special case.

White-noise initial conditions can be regarded as the continuum limit (large m

and small δc) of a discrete system initially with Poisson initial conditions. Therefore,

the creation rate in the discrete Smoluchowski equation is

C(m, t) =m−1∑

m′=1

K(m′,m−m′; t)

2n(m′|t)n(m−m′|t). (4.31)

Sheth & Pitman (1997) showed that the above equation can be written as

C(m, t) = n n(m|t)m− 1

1 + δc

∣

∣

∣

dδc

dt

∣

∣

∣

m−1∑

m′=1

pm′|m−m′ , (4.32)

where n is the mean number-density of particles. If we picture haloes as a collection

of particles held together by (m − 1) non-intersecting bonds (branched polymers),

pm′|m−m′ gives the probability of obtaining an m′-halo and and (m − m′)-halo by

deleting one random bond in the m-halo. The sum over a normalized probability is

trivial, so the creation term is simply

C(m, t) = n n(m|t)m− 1

1 + δc

∣

∣

∣

dδc

dt

∣

∣

∣. (4.33)

In the continuum limit, this becomes

C(m, t) = ρ m n(m|t)∣

∣

∣

dδc

dt

∣

∣

∣(4.34)

Figure 4.9 compares this assumption with the measured creation rates in the

simulations. The simulation measurements in Figures 4.9 and 4.4 are the same,

127

Constant Exact Sqrt Approx Sqrt

ST-99 N-Body

Figure 4.9: Halo creation rates. Symbols, line styles and choices of mass as in

Figure 4.4. As in Figure 4.4, the lowest panel on the right shows not data because

this case is beneath the resolution of the N-body simulation.

128

except that in the latter, data is normalized in time. Notice that the heights of the

curves increase with decreasing mass. This reflects that fact that, in hierarchical

models, more small haloes are created during the history of the Universe than are

massive halos (which are only created at later times). The constant barrier model

works well for massive haloes, but it overpredicts the creation rate of less massive

haloes – showing a similar discrepancy as in the Press-Schechter mass function in

that mass regime. In all cases, the square-root barrier and the Sheth & Tormen

(1999) creation rates match N-body results reasonably well.

4.4.2 The time-normalized creation rate

In this section, we show that creation rate C(m, t) (equation 4.34, Section 4) is

related to the creation time distribution c(t|m) (equation 4.7, Section 3) in a simple

way. First, notice that c(t|m) can be written in terms of νc = δ2c/S. That is,

c(t|m) = c(δc|S)∣

∣

∣

dδc

dt

∣

∣

∣= c(νc)

∣

∣

∣

dνc

dδc

∣

∣

∣

∣

∣

∣

dδc

dt

∣

∣

∣=

2νc

δc

c(νc)∣

∣

∣

dδc

dt

∣

∣

∣, (4.35)

where we have used the fact that |d ln νc/d ln δc| = 2.

Now, following equation (4.7), the above expression can be written as

c(t|m) =2νc

δc

A√νcf(νc)∣

∣

∣

dδc

dt

∣

∣

∣. (4.36)

On the other hand,

mn(m|t)ρ

= f(m|t) = f(S|δc)∣

∣

∣

dS

dm

∣

∣

∣=

ν

Sf(ν)

∣

∣

∣

dS

dm

∣

∣

∣, (4.37)

where we have used the fact that |d ln ν/d ln S| = 1.

129

Our prescription for the creation rate (equation 4.34) is

C(m, t) = ρ2mn(m|t)ρ

∣

∣

∣

dδc

dt

∣

∣

∣= ρ2 ν

Sf(ν)

∣

∣

∣

dS

dm

∣

∣

∣

∣

∣

∣

dδc

dt

∣

∣

∣. (4.38)

Taking the ratio of (4.38) and (4.36), we obtain

C(m, t)

c(t|m)=

ρ2 νSf(ν)

∣

∣

∣

dSdm

∣

∣

∣

∣

∣

∣

dδcdt

∣

∣

∣

2νc

δcA√νcf(νc)

∣

∣

∣

dδcdt

∣

∣

∣

=ρ2

2A√S

∣

∣

∣

dS

dm

∣

∣

∣≡ g(m), (4.39)

where we have dropped the distinction in notation between ν and νc. Simply put,

C(m, t) = g(m)c(t|m). (4.40)

Integrating both sides with respect to time,

∫

C(m, t)dt =

∫

g(m)c(t|m)dt = g(m)

∫

c(t|m)dt = g(m), (4.41)

since c(t|m) is, by definition, a time-normalized distribution. Thus,

c(t|m) =C(m, t)

g(m)=

C(m, t)∫

C(m, t)dt. (4.42)

In words, c(t|m) can be obtained from C(m, t) by normalizing the latter in time.

4.4.3 The conditional rate

We assume that the same prescription can be applied for the conditional case – i.e.,

the rate of creation of haloes of mass m at time t conditioned to belong to M haloes

at a later time T > t. In the discrete Poissonian case, the conditional rate is given

by

C(m, t|M,T ) = n N(m|t,M, T )m− 1

1 + δc

∣

∣

∣

dδc

dt

∣

∣

∣, (4.43)

130

(Sheth, 2003). In the continuum limit, this becomes

C(m, t |M,T ) = ρ m N(m, t |M,T )∣

∣

∣

dδc

dt

∣

∣

∣. (4.44)

Figure 4.10 shows our results for the same set of m and M as in Figure 4.8. As

in the unconditional case, the simulation measurements in Figures 4.10 and 4.8 are

the same, except that in the latter, data is normalized in time.

For both choices of m, the height of the curves decreases with increasing final

mass M (compare top and bottom panels). This effect is less pronounced for the

smaller m (compare left and right panels). In all panels, curves based on the square-

root barrier provide a more accurate description of the measurements.

Lastly, one interesting fact is that the two rates (equations 4.34 and 4.44) are

related by the following consistency condition:

∫ ∞

m

dM C(m, t |M,T ) n(M |T ) = C(m, t). (4.45)

In other words, the unconditional rate is recovered from the conditional rate by

multiplying by the number density of M -haloes at T and integrating over all possible

M > m. This consistency relation is true in general, not just when the initial

conditions are white-noise.

4.5 Final remarks

An alternative derivation for the creation rate can be obtained by using R(m,m′|t),

the rate of mergers of m-haloes with m′-haloes, creating (m+m′)-haloes as a result.

131

Constant Exact Sqrt

Approx SqrtN-Body

Figure 4.10: Conditional creation rates. Symbols, line styles and choice of masses

as in Figure 4.8.

132

This merger rate has been measured recently in cosmological simulations (Fakhouri

& Ma, 2008a,b) and studied in the spherical (Lacey & Cole, 1993) and ellipsoidal

(Zhang et al., 2008a) collapse versions of the excursion set approach. The problem

in this case is that, with the exception of white-noise initial conditions, the excursion

set theory predicts that R(m,m′|t) 6= R(m′,m|t) (Sheth & Pitman, 1997) (but see

Neistein & Dekel, 2008b, for a possible solution). Even if the merger rate were

defined unambiguously within the excursion set approach with moving barriers,

care must be taken when the integrals in equations (4.29) and (4.30) are computed.

Our prescription of the creation rate circumvents these problems altogether. A third

possible method is to use the merger rate developed by Benson et al. (2005) (see also

Benson, 2008). This method avoids the complications in the excursion set theory by

estimating the coagulation kernel K numerically. Unfortuntely, the answer in this

case depends on the choice of regularization technique. The formulation a complete

model for halo mergers with general initial conditions and with the advantages of

ellipsoidal collapse remains an open, and quite interesting, problem.

133

Chapter 5

Merger-induced Quasars

So far, the focus of this work has been on dark matter haloes and their assembly.

In this chapter, we present a model where quasars are triggered by major mergers

of dark matter haloes. In particular, a detailed comparison among theoretical halo

merger rates is provided. This project is the culmination of this thesis, and it

marks the beginning of the many astrophysics projects that can be studied with

our formalism.

5.1 Background

Our understanding of quasars has increased dramatically since their discovery over

four decades ago (Schmidt, 1963; Hazard et al., 1963). Evidence has shown that

these objects are associated with intense activity at the nuclei of galaxies at cosmo-

logical distances, produced by the intense accretion of gas onto supermassive black

134

holes (SMBHs or BHs) (Salpeter, 1964; Zeldovich & Novikov, 1964; Lynden-Bell,

1969; Bardeen, 1970). Likewise, our understanding of cosmology and structure for-

mation has increased at a similar pace. Our universe is well-described by a ΛCDM

expanding cosmology, where structure assembles hierarchically via mergers of dark

matter haloes. These haloes provide the gravitational potential wells where gas can

cool and fragment, allowing the formation of stars and galaxies (White & Rees,

1978; Blumenthal et al., 1984; White & Frenk, 1991). It is natural to expect that

violent halo mergers are followed by the mergers of the galaxies they harbor. More-

over, it has been suggested that major mergers of gas-rich galaxies are an efficient

mechanism to feed SMBHs. This suggestion is supported by hydrodynamic simula-

tions, where mergers produce gravitational torques, which induce strong gas inflows

into the center (e.g., Hernquist, 1989; Barnes & Hernquist, 1991, 1996). In this sce-

nario, the hierarchical assembly of haloes serves as the stage, or back bone of the

coevolution of galaxies and their nuclear black holes. The problem with this picture

is that while the halo hierarchy continues today, observations show that the peak

of quasar activity (and galaxy formation) has already passed (Osmer, 1982; Warren

et al., 1994; Schmidt et al., 1995; Shaver et al., 1996; Steffen et al., 2003; Ueda et

al., 2003; Barger et al., 2005; Brusa et al., 2009). Reconciling these two seemingly

contradictory pictures requires a deeper understanding of the evolution of BHs and

their host galaxies.

In recent years, dynamical observation have revealed that SMBHs are ubiquitous

135

at the centers of most, if not all local massive galaxies (Richstone et al., 1998).

Moreover, these observations show that tight correlations exist between the mass

of the nuclear black hole and properties of the spheroidal component of the host

galaxy. These include scalings between the mass of the black hole and the bulge’s

stellar mass (Kormendy & Richstone, 1995; Magorrian et al., 1998; Marconi &

Hunt, 2003; Haring & Rix, 2004), surface brightness (early-type) (van der Marel,

1999), light concentration (Graham et al., 2001; Graham & Driver, 2007), luminosity

(Bettoni et al., 2003; McLure & Dunlop, 2002, 2004; Graham, 2007; Bernardi et

al., 2007; Gultekin et al., 2009; Bentz et al., 2009), kinetic energy (Feoli & Mele,

2005, 2007), and velocity dispersion (Ferrarese & Merritt, 2000; Gebhardt et al.,

2000; Tremaine et al., 2002; Shields et al., 2003; Bettoni et al., 2003; McLure &

Dunlop, 2004; Bernardi et al., 2007; Tundo et al., 2007; Gultekin et al., 2009) –

this last correlation is commonly referred to as the ‘MBH-σstar relation’. Beyond

the spheroidal component, scaling relations between the black hole mass and either

the host’s circular velocities (Baes et al., 2003; Ferrarese, 2002) or the mass of

the surrounding dark matter halo (Ferrarese, 2002; Bandara et al., 2009) have also

been proposed. See Novak et al. (2006) for a comparison amongst some of these

scaling relations. These observations suggest that the formation of galaxies and

their nuclear black holes are intimately linked; where the dormant SMBHs in the

local universe are the present-day remnants of distant luminous quasars.

A simple way to constrain the growth of these SMBHs is to link their local

136

abundances to the luminosity evolution of quasars throughout the history of the

universe. By comparing the local black hole mass function (BHMF) to the integral

of the luminosity function (LF) of active galactic nuclei (AGN) at all epochs, it

has been established that SMBHs acquire most of their mass through fast-accreting

‘quasar’ phases (see e.g., Soltan, 1982; Salucci et al., 1999; Yu & Tremaine, 2002;

Aller & Richstone, 2002; Yu & Lu, 2004a,b; Marconi et al., 2004; Merloni, 2004;

Shankar et al., 2004; Barger et al., 2005; Yu et al., 2005; Hopkins et al., 2006a;

Yu & Lu, 2008; Shankar et al., 2009).1 One way to describe the evolution of black

holes is with semi-empirical models, which numerically follows the ‘flow’ of BH mass

induced by accretion by means of the continuity equation (Cavaliere et al., 1971,

1983; Cavaliere & Szalay, 1986; Cavaliere et al., 1988; Cavaliere & Padovani, 1989;

Caditz & Petrosian, 1990; Caditz et al., 1991; Small & Blandford, 1992; Cavaliere &

Vittorini, 2000; Yu & Tremaine, 2002; Yu & Lu, 2004a; Shankar et al., 2009). While

it is true that these models provide very valuable constrains on BH evolution (e.g.,

on the radiative efficiency and the mean Eddington ratio), they do not address the

coevolution of SMBHs and their hosts directly.

In order to connect the evolution of BHs to their hosts, early works use the

Press & Schechter (1974) model, or the more accurate Sheth & Tormen (1999)

model. This is typically done by using the halo mass function (Monaco et al.,

2000; Haiman & Menou, 2000; Hatziminaoglou et al., 2001), its time derivative

1Throughout this work, the term ‘quasar’ refers to AGN in the brightest, fast-accretion phases.

137

(Efstathiou & Rees, 1988; Haehnelt & Rees, 1993; Haiman & Loeb, 1998; Haiman,

2004), or ad-hoc versions of the halo creation rate (Cavaliere & Vittorini, 1998;

Haehnelt et al., 1998; Percival & Miller, 1999; Cavaliere & Vittorini, 2002; Vittorini

et al., 2005; Lapi et al., 2006; Miller et al., 2006). Aside from inconsistencies in the

formalism of the creation rate (see Chapter 4), it is unclear whether the merger of

two haloes with any mass ratio can trigger nuclear activity. In particular, numerical

simulations show that minor mergers of galaxies only affect the outskirts, and are

therefore rather inefficient at transporting fuel towards the center (Cox, 2004; Cox

et al., 2008). This raises one of the central questions addressed in this thesis: do

mergers trigger quasars?

For several years, it has been suspected that mergers of galaxies trigger quasar

activity (Stockton, 1982; Sanders et al., 1988a). Many quasar hosts undergoing

mergers and interactions have been observed at low redshifts (e.g., Heckman et al.,

1984; Hutchings et al., 1988; Canalizo & Stockton, 2001; Kauffmann et al., 2003;

Hutchings, 2003a; Hutchings et al., 2003, 2006; Zakamska et al., 2006; Letawe et al.,

2007; Urrutia et al., 2008). Also, strong Ca II/Mg II absorption lines in quasars are

associated with a disturbed interstellar medium, possibly due to a recent merger

(Wang et al., 2005). Further evidence comes from ultraluminous infrared galaxies

(ULIRGs) (see e.g., Sanders & Mirabel, 1996; Jogee, 2006). Almost all of these

objects appear to be in the last stages of merging (Allen et al., 1985; Joseph &

Wright, 1985; Armus et al., 1987; Kleinmann et al., 1988; Borne et al., 1999; Veilleux

138

et al., 2002; Dasyra et al., 2006a,b), and CO observations show that many of these

galaxies contain huge amounts of gas in their nuclei (Scoville et al., 1986; Sargent

et al., 1987, 1989). Moreover, the presence of ‘warm’ IR spectra (Sanders et al.,

1988b) suggests that ULIRGs are likely to be a dust-enshrouded ‘buried’ quasars,

triggered by galaxy mergers (Sanders et al., 1988a; Genzel et al., 1998; Tran et

al., 1999; Sajina et al., 2007). One would expect, therefore, that all quasar hosts

should have visible imprints of a recent merger. Recently, high-resolution imaging

studies have revealed the existence of spectacular shells in quasar hosts (Canalizo

et al., 2007; Bennert et al., 2008), many of which had been previously catalogued

as ‘bona-fide undisturbed galaxies’ (Dunlop et al., 2003).

An attractive feature of major mergers is that during these events, stars can be

easily scrambled from circular to random orbits, producing a change in morphology

as a result. Indeed, simulations show that the only way to reproduce the kine-

matic properties of elliptical galaxies and bulges is via mergers (Toomre & Toomre,

1972; Toomre, 1977; Barnes, 1988; Hernquist, 1989; Barnes, 1992; Hernquist, 1992;

Schweizer, 1992; Hernquist, 1993; Naab et al., 1999; Naab & Burkert, 2003; Naab

et al., 2006a; Naab & Trujillo, 2006b; Naab et al., 2006c; Bournaud et al., 2005;

Cox et al., 2006; Jesseit et al., 2007). If the spheroidal component of galaxies and

SMBHs are formed by the same mechanism as quasar activity, then it is plausible

that this ‘merger hypothesis’ helps explain the aforementioned tight correlations

between the mass of the black hole and the properties of the host.

139

Aside from feeding the BH, the new reservoir of available cold gas provided by

the merger is capable of producing sudden bursts of star formation. A connection

between AGNs and starburst in merging galaxies has been suggested by several

authors (Canalizo & Stockton, 1997; Stockton et al., 1998; Brotherton et al., 1999)

– see Ho (2005) for a review. For instance, spectroscopic signatures, which are

indicative of post-starburst stellar populations, are present in the spectral energy

distributions (SEDs) of quasars (Brotherton et al., 1999; Canalizo & Stockton, 2001;

Kauffmann et al., 2003; Yip et al., 2004; Jahnke et al., 2004a; Sanchez et al., 2004;

Vanden Berk et al., 2006; Barthel, 2006; Zakamska et al., 2006).

One interesting observation is that since redshift z ∼ 2, the most massive galax-

ies appear to shut down star formation earlier than their less-massive counterparts

(Juneau et al., 2005; Treu et al., 2005; Bundy et al., 2006; Borch et al., 2006;

Cimatti et al., 2006; Bell et al., 2007; Bundy et al., 2008; Cowie & Barger, 2008).

This phenomenon, known as ‘downsizing’ (Cowie et al., 1996), coincides with the

drop in number of quasars at recent epochs – pointing to the possibility that black

hole accretion and star formation are quenched by the same mechanism (Shaver

et al., 1996; Dunlop, 1997; Boyle & Terlevich, 1998). Several authors advocate

AGN feedback as the primary quenching mechanism (e.g., Ciotti & Ostriker, 1997;

Silk & Rees, 1998; Fabian, 1999; Ciotti & Ostriker, 2001; King, 2003; Begelman,

2004). In this scenario, when the nuclear luminosity becomes comparable to the

binding energy of the host, it prevents gas from cooling and turning into stars,

140

while accretion by the black hole is terminated. Invoking supernova feedback as

the primary quenching mode of star formation is another possibility (Dekel & Silk,

1986; Kauffmann & Charlot, 1998). However, this method predicts that the most

massive galaxies should harbor the youngest stellar populations, contrary to ob-

servations. It seems, therefore, that supernovae feedback is insufficient to inhibit

star formation in massive galaxies (Benson et al., 2003; Schawinski et al., 2007),

further enforcing the idea that AGN feedback regulates the coevolution of SMBHs

and their spheroidal hosts.

In particular, the most massive elliptical galaxies tend to harbor old, metal-rich,

quiescent stellar populations (Sandage & Visvanathan, 1978; Bernardi et al., 1998;

Jørgensen et al., 1999; Trager et al., 2000; Terlevich & Forbes, 2002; Caldwell et

al., 2003; Nelan et al., 2005; Thomas et al., 2005) – and so do their precursors,

the hosts of the most luminous quasars (Jahnke et al., 2004b; Vanden Berk et al.,

2006). This is corroborated by observations of the [Mg/Fe] ratio (Faber et al.,

1992; Carollo et al., 1993), which an indicator that the spheroid forms before Type

Ia supernovae explode (Matteucci, 2001). Interestingly, the [Mg/Fe] ratio increases

with the mass of the host (Worthey et al., 1992; Weiss et al., 1995; Kuntschner,

2000; Kuntschner et al., 2001; Thomas et al., 2005; Nelan et al., 2005). Lastly,

using AEGIS X-ray observations, Bundy et al. (2008) recently found that not only

is star formation quenched first in massive galaxies, but also faster! These last few

points will motivate our modeling of quasars in the last stages of activity.

141

Gunn (1979) summarized the quasar phenomenon in terms of three key ingre-

dients: an engine, a fuel supply, and a transport mechanism connecting the two.

Observational evidence supports a picture where the nuclear SMBH is the engine,

gas-rich galaxies provide the fuel, and major mergers are the mechanism. More-

over, a fourth ingredient may be added to Gunn’s list: AGN feedback quenches both

BH accretion and star formation, regulating the coevolution of SMBHs and their

spheroidal hosts as a result. In this picture, supernova feedback plays a secondary

role (Benson et al., 2003; Schawinski et al., 2007). Putting the pieces together into

a coherent picture has taken many years. In parallel, we have witnessed the emer-

gence of several theoretical scenarios of merger-induced quasars. These methods

come in many flavors, and can be classified as follows:

• Analytic models (Wyithe & Loeb, 2002; Hatziminaoglou et al., 2003; Wyithe

& Loeb, 2003; Scannapieco & Oh, 2004; Mahmood et al., 2005; Rhook &

Haehnelt, 2006; Marulli et al., 2006; Shen, 2009b, and the present work).2

With the exception of Shen (2009b), these models typically involve some ver-

sion of the extended Press-Schechter (ePS) merger rate (Lacey & Cole, 1993).

Also, they often invoke simple prescriptions linking the BH mass and the lu-

minosity of the associated quasar – the latter is modeled by means of light

curves with universal shapes.

2See also Roos (1981), Roos (1985a), Roos (1985b), De Robertis (1985) and Carlberg (1990)

for much earlier theoretical attempts to link quasars to mergers.

142

• Semi-analytic models (SAMs) (Cattaneo et al., 1999; Cattaneo, 2001; Kauff-

mann & Haehnelt, 2000; Volonteri et al., 2003; Enoki et al., 2003; Koushiappas

et al., 2004; Yoo & Miralda-Escude, 2004; Bromley et al., 2004; Volonteri &

Rees, 2005; Volonteri et al., 2006; Marulli et al., 2006, 2007; Malbon et al.,

2007; Yoo et al., 2007; Somerville et al., 2008; Tanaka & Haiman, 2008). These

models rely on Monte Carlo realizations of the merger history of dark matter

haloes, along with galaxy formation recipes. Monte Carlo merger trees repro-

duce the merger history to great accuracy, and are flexible enough to allow a

great number of realizations with greater resolution and scope (both in mass

and time) than N-body simulations (see Chapter 3 for a discussion). Alterna-

tively, the MORGANA method (Monaco et al., 2007; Fontanot et al., 2007)

models the merger history of haloes with the Zel’dovich approximation-based

PINOCCHIO code (Monaco et al., 2002a,b; Taffoni et al., 2002), which has

similar advantages as the Monte Carlo merger tree methods.

• Hybrid models (Cattaneo et al., 2005; Bower et al., 2006; Hopkins et. al.,

2006; Li et al., 2007; Marulli et al., 2008; Lagos et al., 2008; Bonoli et al.,

2008). These are very similar to the SAMs, but the galaxy-formation recipes

are incorporated in the post-processing of cosmological N-body simulations

(Springel et al., 2005). These simulations capture all the nonlinear features of

halo assembly – something often missed in the Monte Carlo approach (Gao

et al., 2005; Wechsler et al., 2006; Gao & White, 2007; Wetzel et al., 2007;

143

Angulo et al., 2008).

• Hydrodynamic models (Springel et al., 2005; Di Matteo et al., 2005; Hopkins

et al., 2006a; Robertson et al., 2006; Hopkins et al., 2008a,b; Younger et al.,

2008; Khalatyan et al., 2008; Johansson et al., 2009). These models follow the

merger of individual pairs of gas-rich galaxies in great detail. One limitation

is that these galaxy mergers are not put in a cosmological context, and thus

the nature of the selected participating galaxies and their type of encounters

are not always necessarily representative.

• Direct models (Sijacki et al., 2007; Di Matteo et al., 2008; Colberg & di Matteo,

2008; Booth & Schaye, 2009). These new methods incorporate hydrodynamic

methods into cosmological simulations. One drawback is that testing the

physical inputs requires performing a new run of the full simulation, severely

limiting a thorough study of the impact of the different sub-grid recipes em-

ployed.

The advent of supercomputers in recent years has been key to many of these

approaches. However, although the size and resolution of simulations has increased

dramatically, a full multi-scale approach is not yet available. Even the so-called ‘self-

consistent’ hydrodynamic and direct methods have to rely on separate analytical

prescriptions (e.g., Hoyle & Lyttleton, 1939; Bondi & Hoyle, 1944; Bondi, 1952) to

describe the unresolved circumnuclear region near the SMBH. And while detailed

modeling of accreting BHs has been performed (Gammie, 2001; Lodato & Rice,

144

2004; Shapiro, 2005; Lodato & Rice, 2005; Rice et al., 2005), it is often disconnected

with the potential impact its feedback may have on the stellar evolution of its host

(but see Escala, 2007; Mayer et al., 2008; Levine et al., 2008; Kawakatu & Wada,

2008, for recent attempts in this direction).

In adding computational complexity to a model, flexibility and scope are often

sacrificed. Moreover, our poor knowledge of BH evolution and accretion physics still

prevents us to widely adopt the above techniques in extreme detail. Contrary to this

trend, the spirit of this work is to follow the footsteps of the early analytic methods,

and revisit some of their underlying assumptions. Another issue is that different

SAMs often reach opposite conclusions on BH evolution, even though they match

similar sets of observables (e.g., Lapi et al., 2006; Malbon et al., 2007; Marulli et

al., 2008, see section § 5.7.3). For this reason it is essential to step back a little and

attempt to probe some of the central mechanisms regulating BH evolution with more

transparent and basic recipes. In this work we propose a simple and concise model,

based on very general and widely adopted (and accepted) physical assumptions. We

will show that by matching the predictions of such a basic model with a variety of

different observations, we can identify quite general and important characteristics

of BH evolution, which must be seriously considered by more advanced SAMs.

We assume that quasars are triggered by major mergers of dark matter haloes.

A universal light curve describes the evolution of individual quasars. This light

curve consists of an exponential ascending phase and a mass-dependent power-law

145

descending phase. Physically, right after the halo merger, a torque is exerted in the

host and gas is quickly funnelled into the center of the newly-formed gravitational

potential. In this phase, the BH accretes the fuel very efficiently. However, accretion

cannot last indefinitely, and we assume that it is quenched when the luminosity is

powerful enough to expel this gas or at least to balance its infall, thus inhibiting

further growth. It is precisely the depth of the potential (Vvir) that controls the

peak luminosity of the quasar. We postulate a self-regulating condition relating the

peak luminosity the AGN can reach, and the mass of the host halo at triggering.

This peak luminosity dictates the transition from the ascending to the descending

phase. This is a very general condition adopted by all SAMs and verified, to some

extent, in most numerical simulations (e.g., Di Matteo et al., 2005; Ciotti et al.,

2009; DeBuhr et al., 2009), although several aspects of the actual underlying physics

still remain obscure. In our model, this simple scaling condition encapsulates the

physics responsible of halting nuclear activity. While it is true that the host halo

may grow further through accretion of satellites and diffuse material, this has little

effect on the core of the potential where the spheroid forms (e.g., Zhao et al.,

2003; Diemand et al., 2007; Vass et al., 2009). One key feature in our modeling is

that we assume that quasars inhabiting more massive haloes shut down faster. As

mentioned before, this last ingredient is in line with recent observations indicating

that star formation in massive galaxies is quenched first and faster. Motivated by

this empirical evidence, we will show that shortening the light curves for quasars in

146

massive haloes will help to reproduce the bright-end of the AGN luminosity function

at low redshifts.

5.2 The model

Our model consists of two ingredients: (1) the halo merger rate, which describes

major mergers as the trigger of AGN activity, and (2) the light curve, which de-

scribes the evolution of individual quasars. These two ingredients are described

below.

5.2.1 The halo merger rate

There are a number of theoretical models for the dark matter halo merger rate in the

literature (Lacey & Cole, 1993; Parkinson et al., 2008; Zhang et al., 2008a; Neistein

& Dekel, 2008b; Benson, 2008). These are typically based on either the excursion

set formalism (Bond et al., 1991; Sheth, Mo & Tormen, 2001) or obtained by tuning

to numerical simulations. Section 5.3 includes a detailed comparison among various

excursion-set predictions.

More recently, direct fits to the merger rate in numerical simulations have been

performed (Fakhouri & Ma, 2008a; Genel et al., 2009). It is common practice to

express the merger rate in terms of the sum M = m + m′ and the ratio3 ξ = m/m′

3Note that in previous chapters we have used the notation µ for this quantity. For clarity, we

opted to stick to the original notation used by Fakhouri & Ma (2008a) in this discussion.

147

(or m′/m), where m < m′ (m′ < m):

R(m,m′, z) = R(ξ,M, z)∣

∣

∣

∂(ξ,M)

∂(m,m′)

∣

∣

∣= R(ξ,M, z)(1 + ξ)2/M. (5.1)

This particular form facilitates the selection of major mergers, which may be re-

sponsible for triggering quasars. Moreover, Fakhouri & Ma (2008a) found that the

evolution of R(ξ,M, z) is strongly determined by the mass function (their Figure

6). For this reason, one often focuses instead in the merger rate per unit halo,

R(ξ,M, z)/n(M, z). These authors provide a fit given by

R(ξ,M, z)

n(M, z)= κ0

(M

M

)κ1(ξ

ξ

)−κ2

exp[(ξ

ξ

)κ3](dδc

dz

)κ4

. (5.2)

where δc is the spherical overdensity required for collapse, M = 1.2 × 1012M⊙,

ξ = 0.098, and (κ0, κ1, κ2, κ3, κ4) = (3.0798, 0.083, 2.01, 0.409, 0.371). This fit is a

simple product of functions of M , ξ and z, and is accurate to the 10-20 percent level,

a negligible uncertainty with respect to the error bars in the observables we will

be comparing with. For our purposes, the Sheth & Tormen (1999) mass function

is accurate enough too. As shown in Chapter 2, this is essentially identical to our

square-root barrier model. (In this context, see also the fits by Warren et al., 2006;

Reed et al., 2007; Lukic et al., 2007; Tinker et al., 2008).

5.2.2 The light curve

The light curve consists of three components: an ancending phase of fast accre-

tion, the peak luminosity, and a descending phase where the AGN is gradually

extinguished by its own feedback.

148

The ascending phase

First, we assume that once the major merger takes place (at some triggering time

ttri) the black hole begins to accrete at some constant multiple of the Eddington

rate. The associated bolometric luminosity is

L = λ0LEdd, (5.3)

where the parameter λ0 is the Eddington ratio. The Eddington luminosity is defined

as

LEdd ≡4πGmp

σecMBHc2 ≡ MBHc2

τS

, (5.4)

where G is Newton’s gravitational constant, mp is the mass of the proton, σe is

the electron-photon Thompson scattering cross section, c is the speed of light, and

τS ≃ 0.43 Gyrs. is known as the Salpeter time.4 It is also common to write to above

expression as

LEdd = 1.26× 1038 erg s−1

(

MBH

M⊙

)

≡ l

(

MBH

M⊙

)

. (5.5)

Notice that in this phase, the luminosity scales linearly with the black hole mass:

L =

(

λ0l

M⊙

)

MBH =

(

λ0c2

τS

)

MBH. (5.6)

4This order-of-magnitude result comes from taking an idealized sphere in hydrostatic equilib-

rium, where the pressure force Frad = LEdd/4πr2c associated with the radiation being emitted

balances the gravitational force Fgrav = −GMBHmp/r2 exerted on the material being accreted –

see e.g., Peterson (1997) and Martini (2004) for a review.

149

Black hole growth results from the mass that falls into the black hole that is not

radiated away. If a fraction ǫ of the gravitational potential energy of the infalling

material is converted into radiation, the associated luminosity is

L = ǫMinfallc2, (5.7)

where Minfall is the rate of accretion of infalling material and the parameter ǫ is

known as the radiative efficiency. While a fraction ǫ of the infalling matter is

radiated away, the rest (i.e., 1 − ǫ) feeds the black hole (see e.g., Yu & Tremaine,

2002). In other words, MBH = (1 − ǫ)Minfall, and the luminosity can be expressed

in terms of the black hole’s mass growth rate:

L =ǫ

1− ǫMBHc2 ≡ fMBHc2. (5.8)

Notice that in this phase, the luminosity scales linearly with the black hole mass

growth rate. Combining equations (5.6) and (5.8), one obtains that MBH/MBH =

λ0/fτS, and the black hole mass increases exponentially:

MBH(t) = MBH(ttri) exp(t− ttri

tef

)

, t ≥ ttri, (5.9)

where

tef ≡f

λ0

τS ≃ 0.43( f

λ0

)

Gyr., (5.10)

is the e-folding time. By virtue of equation (5.6), the ascending phase of the light

curve evolves as

Lasc(t) = L(ttri) exp(t− ttri

tef

)

, (5.11)

150

with L(ttri) = λ0lMBH(ttri)/M⊙ (we will discuss the value of MBH(ttri) shortly).

Physically, this exponential growth cannot be sustained indefinitely. Eventually,

this phase is halted at some time tpeak after triggering, where the maximum lumi-

nosity Lpeak = L(tpeak) is reached. At that point, we denote the black hole mass by

MBH, peak = MBH(tpeak).

The peak luminosity

The ascending phase reaches its peak luminosity when the feedback is powerful

enough to unbind gas from the potential well. Subsequently, accretion by the black

hole becomes substantially inefficient, and is eventually terminated. In our model,

the physics of AGN feedback is encapsulated in a self-regulation condition between

the peak luminosity and the mass of the host halo at triggering. Following Wyithe

& Loeb (2002) and Wyithe & Loeb (2003), we postulate this condition to have the

form

Lpeak

1044ergs−1= 7

(

M

1012M⊙h−1

)5/3

(1 + z)5/2 , (5.12)

where M is the mass of the host dark matter halo at the time of triggering.

The role of this condition is to control the peak luminosity at the end of the

ascending phase. Physically, when the peak luminosity is attained, AGN feedback

kicks in, and the light curve switches from the ascending to the descending phase (see

below). Wyithe & Loeb (2002) provide a simple explanation for this self-regulation

condition (see also Silk & Rees, 1998). First, consider the luminosity capable of

151

unbinding the entire energy of the gravitating system, which is proportional to

the ratio of the binding energy (∝ MV 2vir) and the dynamical time (∝ rvir/Vvir),

where rvir is the virial radius and Vvir is the virial velocity. This ratio is given

by Lpeak ∝ MV 3vir/rvir. From the virial relation, GM/rvir ∝ V 2

vir, we can write

Lpeak ∝ V 5vir/G ∝ (M/rvir)

5/2. But the volume r3vir ∝ M/ρ ∝ M/(1 + z)3, and

M/rvir ∝M2/3(1 + z) results.5 Thus, with this order of magnitude calculation, and

assuming that the halo mass at triggering scales with the gravitating mass that

regulates AGN feedback, one obtains

Lpeak ∝M5/3(1 + z)5/2, (5.13)

which is precisely our postulated self-regulation condition.

Recall that we assume that the BH ends its rapid-accretion phase when the

feedback energy is sufficient to unbind, or just energetically balance the infall of,

the surrounding gas reservoir. The potential well Vvir refers to the core of the

halo’s gravitational potential, and it is dominated by the inner regions of the halo

and the newly-formed spheroid (the gravitating system). Note that our feedback-

constrained condition refers to the virial velocity Vvir at triggering. While it is true

that the halo mass may increase between ttri and tpeak – though a second major

merger is unlikely within that (short) time – it has been numerically proven that the

5Strictly speaking, M/rvir ∝M2/3(1+z)ξ(z)1/3, where ξ(z) = (Ωm,0/0.25)(1/Ωm(z))(∆/18π2),

∆ = 18π2 + 82d = 39d2 is the mean density contrast relative to the critical density, and d ≡

Ωm(z) − 1 (Bryan & Norman, 1998; Barkana & Loeb, 2001). Since ξ(z) varies very weakly with

redshift, it is safe to ignore it.

152

central potential well remains rather constant for a long time since the virialization

epoch (e.g., Zhao et al., 2003; Vass et al., 2009). Moreover, minor mergers and

accretion only affect the outskirts (Cox, 2004; Cox et al., 2008), leaving the core of

the potential intact.

Alternatively, some authors argue that feedback momentum (rather than feed-

back energy) may dominate self-regulation (King, 2003; Di Matteo et al., 2005;

Lidz et al., 2006). In that case, Lpeak ∝ v4vir/G ∝ M4/3(1 + z)2, which depends

more weakly on halo mass and redshift than our energy-driven condition. We do

not discuss this model any further below, although we have checked that the main

conclusions of this work do not significantly change when switching to the latter

feedback condition, provided that the appropriate normalization in the Lpeak-M re-

lation (a factor of∼ 2−3 higher) is considered (and some rather small readjustments

in other parameters).

Lastly, in § 5.7.2 we discuss how this self-regulation condition has an impact on

the tight correlations between the nuclear BH and the host spheroid in the local

universe. These correlations have scatter. In our model, we assume that the our

Lpeak −M relation has a scatter of Σ dex (see equation 5.30 below).

The descending phase

This portion of the light curve describes the gradual shutdown of the quasar after

it reaches its peak luminosity. For instance, in hydrodynamic simulations (e.g.,

153

Hopkins et al., 2006b), accretion is halted as the feedback energy couples to the gas

reservoir, leading to a nearly self-similar power-law decay, L(t) ∝ t−α. Motivated

by this, Shen (2009b) recently adopted decaying power-law light curves, where the

parameter α is mass-independent.

Throughout this work, the evolution of SMBHs and their host spheroids has

been emphasized. In particular, the most luminous massive ellipticals tend to be

redder. The color suggests older stellar population, while higher luminosity suggests

that these populations are more metal rich. Moreover, high metallicity is indicative

of efficient periods of star formation, taking place during short periods of time (see

e.g., Lapi et al., 2006, for a discussion). This is further corroborated by Bundy et

al. (2008), who conclude that star formation in massive systems is quenched faster

than in less massive ones. Lastly, even hydrodynamic simulations model a shutdown

phase that depends on mass (Hopkins et al., 2006b; Hopkins & Hernquist, 2009).

Motivated by this, we model our descending phase in terms of a mass-dependent

power-law:

Ldsc(t) = L(tpeak)

(

t

tpeak

)−α(M)

. (5.14)

In our model, the shutdown parameter α(M) is described by the following em-

pirical law:

log α(M) = log α0 + α1 log

(

M

1012.5M⊙

)

ΘH(M − 1012.8M⊙), (5.15)

154

where α0, α1 are free parameters, and ΘH(x) is the Heavyside step function.6 In

our reference model, we use (α0, α1) = (3.5, 0.9). Other parametrizations of this

effect are allowed, provided that the yield the desired behavior that quasars in more

massive host shut down faster.

Major mergers and host masses

Simulations show that minor mergers (those with ξ < 1/10, where ξ ≡ m′/m, m

and m′ are the masses of the merging haloes, and m′ < m) only affect the outskirts

(Cox, 2004; Cox et al., 2008). Likewise, mild mergers (with 1/4 > ξ > 1/10) may

trigger some AGN activity (Hernquist, 1989; Hernquist & Mihos, 1995; Bournaud et

al., 2005), but these are limited to an specific set of orbital geometries (Younger et

al., 2008; Johansson et al., 2009). In this work, we assume that major mergers (with

ξ > 1/4) are the triggers of AGN activity.7 This assumption is further supported by

observations (Dasyra et al., 2006a; Woods et al., 2006), which show that starbursts

and strong inflows of gas are only seen in major mergers.

In the major-merger regime, it is natural to assume that a merger of the haloes

6Formally, the Heavyside step function is defined as

ΘH(x) =

1, if x ≥ 0;

0, if x < 0.

In practice, we use a smooth version, given by (1− e−2x)−1.7In Chapter 3 we used a more strict version of ‘major mergers’ with ξ > 1/3. The operational

difference between the two is largely irrelevant.

155

is followed by a merger of the galaxies within, after some delay time necessary for

the galaxies to sink to the bottom of the potential. However, as also discussed by,

e.g., Cavaliere & Vittorini (2000) and Shen (2009), the time for quasar triggering is

not necessarily associated to the time of the actual merging of the host galaxies, and

actually observations and numerical simulations do indeed favor scenarios where the

tidal forces start acting on the gas reservoirs already at the very initial stages of the

merger (references above). In the following we will therefore assume that quasar

activity is triggered at the moment of the halo merging8, and we will discuss the

consequences of loosening such an assumption in § 5.7.3. Also note that for mild

mergers, the delay times are much longer, and the satellites also suffer from tidal

stripping, which diminishes their ability to trigger any AGN activity (e.g., Colpi et

al., 1999).

Another issue is that not all mergers are gas-rich; and gas-poor ‘dry’ mergers

are certainly not expected to trigger quasars. Theoretical models predict that dry

mergers between ellipticals are seldom, except perhaps at low redshifts (Khochfar

& Burkert, 2003; Ciotti et al., 2007; Stewart et al., 2009; Hopkins et al., 2009). This

assertion is supported by observations (Lin et al., 2008; Bundy et al., 2009), and

will not be addressed in this work.

Lastly, very massive haloes, common at low redshifts, often harbor more than

one galaxy. Significant research on AGNs in galaxy clusters has been done obser-

8This condition is even more valid at higher redshifts, when hosts are smaller at fixed mass,

rendering tidal torques even more effective.

156

vationally (e.g., Houghton et al., 2006; Gebhardt et al., 2007; Georgakakis et al.,

2008a,b; Martini et al., 2009; Heinz et al., 2009) and theoretically (e.g., Thacker et

al., 2006; Scannapieco & Bruggen, 2008; Brueggen & Scannapieco, 2009). In this

work, we do not explore this regime. One reason is that tight correlations between

the SMBHs in central galaxies and other properties of the clusters have not been

found, and therefore our self-regulation condition (equation 5.12) is unlikely to hold

for very massive haloes. For instance, Gebhardt et al. (2007) found that for NGC

1399 (the central galaxy in the Fornax cluster), the mass of of the SMBHs is well

outside the MBH-σstar relation. Haloes more massive that 1013M⊙h−1, tend to be

groups and clusters of galaxies (Cavaliere & Vittorini, 2000; Menci et al., 2005; Vit-

torini et al., 2005). In such cases, major mergers are not likely to trigger significanly

bright AGN activity (Georgakakis et al., 2008b; Croton et al., 2006; Cattaneo et

al., 2007; Okamoto et al., 2008). For these reasons, in our model we constrain the

masses of haloes to be ‘galactic’ (e.g., 1011.5M⊙h−1 ≤ M ≤ 1013M⊙h−1), and one

galaxy per halo is assumed. For a potential way to extend our model, see Hopkins

et al. (2008b), who use halo occupation models (Yang et al., 2003; Yan et al., 2003;

Zheng et al., 2005; Conroy et al., 2006; Vale & Ostriker, 2006) and subhalo mass

functions (Kravtsov et al., 2004; Zentner et al., 2005; van den Bosch et al., 2005)

to populate haloes with more than one galaxy.

157

Universal light curves with two phases

Our theoretical light curve describes how quasars evolve ‘on average’; reality need

not be that simple. Nevertheless, using a universal prescription like ours is actually

quite powerful. Traditional models often adopt a light curve with only an ascending

phase (Salpeter, 1964; Rees, 1984; Small & Blandford, 1992; Haehnelt & Rees, 1993;

Salucci et al., 1999; Cavaliere & Vittorini, 2000; Marconi et al., 2004; Shankar et

al., 2004), only a descending phase (Haehnelt et al., 1998; Haiman & Loeb, 1998;

Kauffmann & Haehnelt, 2000), or assume that quasars shine at a fixed luminosity

for a fixed period of time – the so-called ‘light bulb’ models (Wyithe & Loeb, 2002,

2003). The problem with these prescriptions is that they are fail to match the quasar

bias (Marulli et al., 2006), the local BHMF (Yu & Lu, 2008) or the Eddington-ratio

distribution (Hopkins & Hernquist, 2009). In light of this, Shen (2009b) proposed a

model similar to ours, but with a mass-independent α. The simulations of Hopkins

et al. (2006b) – and the subsequent model by Marulli et al. (2008) – also invoke

a universal two-phase light curve with a descending phase (albeit with a different

mass dependence than in our model).

Let us now briefly discuss our choice of parameters describing the light curve.

First, notice that tef increases with increasing radiative efficienty (equation 5.10).

Physically, allowing more infalling material to turn into radiation slows down the

growth of the black hole. Soltan-like arguments comparing the amount of light

received from past quasars to the mass in local BHMF favors a radiative efficiency

158

of ǫ & 0.1 (Yu & Tremaine, 2002; Aller & Richstone, 2002; Marconi et al., 2004;

Shankar et al., 2004; Yu & Lu, 2004a, 2008; Shankar et al., 2009b). This value is

compatible with standard models of BH accretion (Shakura & Sunyaev, 1973; Rees,

1984; Narayan & Quataert, 2005), and even with more exotic process involving

electromagnetic extraction of energy from spinning BHs (Blandford & Znajek, 1977;

Phinney, 1983; Gammie, 2001; Shapiro, 2005).

The Eddington ratio, on the other hand, allows us to translate the observed

bolometric luminosity into the actual BH mass for a given quasar (equation 5.6).

The higher λ0, the more luminous the quasar is for a given black hole mass. In

this case, the e-folding time is shorter (equation 5.10), and the black hole accretes

faster. In this work we assume that the accretion in the ascending phase is mildly

super-Eddington, which is allowed by theoretical models (Begelman, 1978, 2002;

King, 2002; Ohsuga et al., 2005; Ohsuga & Mineshige, 2007), and below we will

discuss that this choice is actually favored by high-redshift observations.

The black hole in the earliest stages

As we just discussed, more realistic light curves can be more complicated. For

instance, some hydrodynamic simulations show that before the two galaxies sink to

the center of the remnant halo, a mild preliminary accreting phase can triggered by

the ‘first passage’ of the two interacting galaxies (Hopkins et al., 2005). Moreover,

accretion may occur in the two black holes associated with the merging galaxies

159

before BH coalescence. Indeed, sub-Mpc dual AGN have been observed at various

separations, indicating various stages of merging. For instance, the dual AGN in 3C

75 are 7 kpc apart, those in Mrk 463 are ∼ 3.8 kpc apart, while NGC 640 exhibits

active nuclei with a separation of ∼ 700 pc, and those in Arp 299 are separated

by . 50 pc (see Colpi & Dotti, 2009, and references therein). Also, Comerford et

al. (2009) report a dual AGN with separation of ∼ 1.75 kpc, and Chornock et al.

(2009) found another pair at ∼ 5 kpc. Likewise, Foreman et al. (2009) found 85

quasar pairs whose separation distribution peaks at ∼ 30 kpc. Sub-parsec binaries

are harder to observe. Ongoing SDSS surveys have found an increasing fraction of

observed quasar binaries (Hennawi et al., 2006, 2009; Shen et al., 2009c). Binary

quasars are expected to further constrain triggering mechanisms.

In our model, the initial quasar phase is modeled by using a single seed black

hole undergoing extremely fast accretion, which grows from some triggering mass

MBH, tri to some large mass MBH, peak before switching into the descending phase.

Since the bulk of the black hole mass is acquired during the last e-folding time

prior to the peak, we expect activity during this period to be insensitive to the

detailed physics of binary BH coalescence. Choosing the mass at triggering is not a

trivial matter. For instance, while stellar theory predicts seed BH mass ≃ 102−4M⊙

arising from population-III stars (Omukai & Nishi, 1998; Madau & Rees, 2001;

Heger & Woosley, 2002; Ripamonti et al., 2002; Bromm & Loeb, 2003; Pelupessy

et al., 2007; Begelman et al., 2008), the problem of seeding galaxies with BHs at

160

Figure 5.1: The quasar light curve. After triggering, the quasar grows exponentially

for some time tasc = tef ln(µBH). The peak is controlled by the MBH −M relation

(or Lpeak − M relation), where M is the mass of the host dark matter halo at

virialization. The descending phase decays as a power law, with a decaying exponent

that increases monotonically with halo mass.

161

early epochs remains largely unanswered (see e.g, Haiman, 2004). The truth of the

matter is that, at the present moment, our observational window into the early

stages of BH accretion is still very limited. For this reason, we choose the BH mass

at triggering to be a fixed fraction of the BH mass at the peak luminous stages:

MBH, seed ≡ MBH, peak/µBH. A direct consequence of this is that the time spent in

the ascending phase is fixed,

tasc ≡ tpeak − ttri = tef ln(µBH), (5.16)

and so is the number of e-foldings. Notice that in this scenario, tasc ∝ log(µBH)/λ0.

Namely, choosing large BH masses at triggering and high Eddington ratios has the

same effect on tasc. In § 5.6.2 we explore different combinations of log µBH and λ0

and their impact on the LF and quasar clustering. In this context, see also Shen

(2009b), who simply sets tasc at 6.9τS. In principle, a model with mass-dependent

tasc can be formulated. We made various attempts, which introduce breaks and

noise into our outputs of the luminosity function. Given that our knowledge of the

actual distribution of tpeak values is very limited, we adopted a universal duration of

the ascending phase and introduced mass-dependence only in the descending phase

(longer durations for less massive hosts).

In summary, the light curve is given by

L(t) =

L(ttri) exp(

t−ttritef

)

, ttri ≤ t ≤ tpeak;

L(tpeak)(

ttpeak

)−α(M)

, t > tpeak.

where α(M) increases monotonically with halo mass, and the duration of the ascend-

162

ing phase is set by the ratio of black hole mass at the time of maximum luminosity

and the triggering time (tpeak and ttri). Figure 5.1 illustrates the shape of the light

curve for three given values of MBH, peak. Before implementing the model, we will

discuss the different theoretical halo merger rate scenarios.

5.3 Theoretical halo merger rates

Consider the merger of an m-halo with an m′-halo at redshift z, producing a halo

of mass M = m + m′ as a result. The excursion-set theory with constant barriers

(ePS) provides a formula for this quantity. In this section, we show how this rate

is modified when moving barriers (such as our square-root barrier) are considered.

These are then compared against results from N-body simulations.

5.3.1 Extended Press-Schechter

In the excursion-set framework with constant barriers (spherical collapse), the

merger rate is given by

R(m,m′, z) = n(M, z) lim∆z→0

1

∆zN(m, z + ∆z|M, z), (5.17)

where n(M, z) is the (unconditional) mass function of haloes of mass M at redshift

z. The second factor, N(m, z+∆z|M, z), is the mass function of m-haloes at z+∆z

conditioned to be in bound objects of mass M at z. It is common to work in terms

of the ratio R/n because the evolution of R is strongly determined by n (Fakhouri

163

& Ma, 2008a). This ratio is given by

R(m,m′, z)

n(M, z)=

1√2π

M/m

(Sm − SM)3/2

∣

∣

∣

dSm

dm

∣

∣

∣

∣

∣

∣

dδc

dz

∣

∣

∣, (5.18)

where δc(z) is the spherical collapse threshold and S is the variance of the initial

density-perturbation field (linearly-extrapolated to the present), with a (sharp-k

space) window function containing a given mass (see Chapter 2 for a review).

Unfortunately, this prescription is formally self-inconsistent because, for the ini-

tial conditions considered here (encoded in the initial power spectrum), N(m|M) 6=

N(M −m|M). Sheth & Pitman (1997) show that for white-noise initial conditions,

the equality holds (see Chapter 4). In general, however, equation (5.18) predicts

two distinct merger rates, one for m < m′ and one for m > m′. As pointed out by

Benson (2008), one can still use a symmetrized version, such as

Rsym(m,m′, z) =

R(m,m′, z) if m < M/2;

R(m′,m, z) if m ≥M/2,

where R is given by equation (5.18). Alternatively, one could use

Rsym(m,m′, z) =

R(m′,m, z) if m < M/2;

R(m,m′, z) if m ≥M/2.

Zhang et al. (2008a) refer to these as ‘Option A’ and ‘Option B’, respectively. De-

spite this conceptual problem, we find that for all the models considered, ‘Option

A’ does a more reasonable job at reproducing the merger rates in N-body simula-

tions. We refer the reader to Neistein & Dekel (2008b) for a possible solution to

this inconsistency.

164

5.3.2 Ellipsoidal Collapse

In order to generalize the ePS rate in equation (5.18) for ellipsoidal collapse, the

limit of N(m, z + ∆z|M, z) as ∆z → 0 must be computed in that model. Recently,

Zhang et al. (2008a) showed that, while the Sheth & Tormen (2002) approximation

series to general barriers works well in general, it fails in this regime. Likewise, our

recipes for the exact square-root solution have numerical problems in that limit.

Nevertheless, expressions can be obtained by analyzing the barriers directly. (See

Zhang et al., 2008a, whose method we follow for square-root barriers).

Recall that the conditional mass function is obtained by solving the uncondi-

tional crossing distribution of a barrier

B(ac, s) = ac +√

qδ1

(s− S0

qδ21

)γ

(5.19)

where

s = S − S0, and ac ≡√

q(δ1 − δ0)−√

qδ0

( S0

qδ20

)γ

. (5.20)

Here, S = S(m), S0 = S(M), δ1 = δc(z + ∆z), δ0 = δc(z) and (q, β, γ) =

(0.707, 0.45, 0.615). By expanding equation (5.19) in s/S0 and (δ1 − δ0)/δ0, we

find that the leading order terms are given by

B(ac, s) = b0 + b1s + b2s2, (5.21)

After some algebra, one can show that the coefficients can be written as

b0 = G0(δ1 − δ0), b1 =G1√S0

, b2 = − 4G2√2πS

3/20

, (5.22)

165

where we have conveniently defined:

G0 =√

q[1 + (1/2− γ)(2β)(qν)−γ], (5.23)

G1 = γβ(qν)1/2−γ, (5.24)

G2 =

√2π

8γ(1− γ)β(qν)1/2−γ. (5.25)

The crossing distribution of this quadratic barrier is well-approximated by

f(s)ds ≃ b0√2πs

[

exp(

− (b0 + b1s)2

2s

)

−√

2πs3/2

4S0

b2 exp(

− b21s

2

)

]

ds

s. (5.26)

(For details, see Zhang & Hui, 2006; Zhang et al., 2008a). Notice that when b1 = 0

and b2 = 0, this reduces to the Press-Schechter result (equation 2.65). Similarly,

when b2 = 0, this reduces to the exact linear-barrier solution (equation 4.10).

Taking the limit as (δ1 − δ0) → 0, the halo merger rate (modulo the mass

function) is

R(m,m′, z)

n(M, z)=

R(m,m′, z)

n(M, z)

∣

∣

∣

ePS×G(s/S, ν). (5.27)

where ν = δ2c (z)/S(M), and

G(Sm/SM , ν) = G0

1+G2

(Sm

SM

−1)

32[

1+2G1√

π

(Sm

SM

−1)]

exp

− G21

2

(Sm

SM

− 1)

.

(5.28)

The factor of G can be thought of as a ‘perturbation’ to the ePS prediction (equa-

tion 5.18). Notice that all the dependence on redshift is contained only in |dδc/dz|

and in the (G0, G1, G2) parameters (which are functions of ν).

When (q, β, γ) = (1, 0, 0), B reduces to the ePS constant barrier. In this case,

(G0, G1, G2) = (1, 0, 0) and G = 1, as expected. The square-root barrier has

166

(q, β, γ) = (0.55, 0.5, 0.5), and in this case (G0, G1, G2) = (√

q, β/2,√

2πβ/32). No-

tice that in this case the perturbation G = G(s/S) is independent of redshift. See

Parkinson et al. (2008) and Benson (2008) for other expressions of G, which are not

derived from the excursion set theory. We now compare our expressions against a

fit to measurements in N-body simulations.

5.3.3 Comparison with N-body simulations

Figure 5.2 shows the halo merger rate modulo the mass function, (R/n) vs. the mass

ratio (ξ) for various masses, redshifts and models. The different panels show R/n

at redshifts z = (0, 1.2, 2.4, 3.6, 4.8, 6). Each panel shows three bundles of curves,

each bundle corresponding to each of the following halo masses: M = 1013,12,11M⊙

(top-to-bottom). For clarity, the curves with mass M = 1013M⊙ (M = 1011M⊙)

are shifted up (down) by an order of magnitude. The dotted, dashed and long-

dashed curves (in red, blue and green) denote the ePS, the square-root barrier,

and γ > 1/2 prediction of Zhang et al. (2008a). We will maintain these styles

throughout this chapter. The solid black line represents the Fakhouri & Ma (2008a)

fit to the Millennium Simulation (equation 5.2) – and it is truncated at low ξ to

indicate the scope of the simulation. Notice that the cutoff occurs at a higher

value of ξ for lower values of M . This is because, for each M , the minimum value

of ξ is ξmin ≡ Mmin/M . In the Millennium Simulation, only haloes with mass

≥ Mmin = 20Mp are considered, where the particle mass is Mp = 1.2 × 109M⊙.

167

Figure 5.2: Halo merger rates divided by mass function vs. mass ratio. Different

panels show redshifts (z = 0, 1.2, 2.4, 3.6, 4.8, 6) at three masses (bottom-to-top):

M = 1011,12,13M⊙. At each panel, each bundle of curves corresponds to a given

mass M . For clarity, curves with M = 1013M⊙ (1011M⊙) are shifted up (down) by

an order of magnitude. Please see key for each style corresponding to each model.

168

Thus, for smaller halo masses M , the value of ξmin is larger.

Before comparing the different models (Figures 5.3 and 5.4), some remarks are

in order. First of all, notice that in every case, R/n does not evolve strongly with

mass or redshift. As pointed out by Fakhouri & Ma (2008a), this quantity is indeed

‘nearly universal’. In other words, any evolution in the merger rate itself comes

from the mass function. Also, notice that R/n drops strongly with increasing ξ,

indicating that minor mergers occur more frequently than major ones. Let us now

compare the different theoretical predictions.

The curves in Figure 5.3 show R/n, modulo the fit of Fakhouri & Ma (2008a) to

the Millennium Simulation (equation 5.2). The choices of redshift, mass, and styles

are the same as in Figure 5.2, except that here we don’t shift curves corresponding

to different masses. In this case, the theoretical curves are truncated at low ξ

to indicate the scope of the N-body fit. Notice that for each model, the curve

shifts up with increasing mass and increasing redshift (although this trend is hardly

noticeable at high redshifts). Also, in all cases, curves shift up with increasing ξ. For

the most part, theory curves are above (below) the Fakhouri & Ma (2008a) at high

(low) ξ. Exceptions to this trend are found at redshift z = 0, where some ellipsoidal

collapse prediction are below the fit for all values of the mass ratio. Overall, the ePS

curves are the steepest. The Zhang et al. (2008a) prediction follows in steepness,

and our square-root barrier curves are the flattest. The Zhang et al. (2008a) curves

change (shift up) very strongly with increasing mass. This dependence is weaker in

169

Figure 5.3: Ellipsoidal collapse merger rates. Different panels show redshifts

(z = 0, 1.2, 2.4, 3.6, 4.8, 6) at three masses (bottom-to-top): M = 1011,12,13M⊙. All

models are divided by the Fakhouri & Ma (2008a) fit to the Millennium Simulation.

In this case we don’t shift the curves for different masses. Please see key for each

style corresponding to each model.

170

Figure 5.4: The same as Figure 5.3, but in terms of linear (as opposed to logarith-

mic) ξ. This allows us to see the large-ξ regime (associated with major mergers) in

more detail.

171

the ePS curves, and the weakest is our square-root barrier model. In other words,

the square-root barrier curves are closer to one another, the Zhang et al. (2008a)

curves are farther apart, and the separation of ePS curves is somewhere in between.

Ranking which model best resembles the fit to the Millennium simulation de-

pends on what one is actually interested in. In this work, we are interested in major

mergers quite above ξ > 1/10. To this end, we include Figure 5.4, which contains

the same information as Figure 5.3, except that we have plotted ξ linearly, as op-

posed to logarithmically. It is gratifying to see that in all cases our square-root

barrier model (depicted in blue) provides the best match to N-body simulations.

5.4 Implemention of the model

We now show how our model can be implemented to predict the statistics of AGNs

and black holes.

5.4.1 The luminosity function

The luminosity function is defined as the comoving number density of quasars shin-

ing with bolometric luminosity between L and L+dL at redshift zshi. We model

this as

Φ(L, zshi)dL = dL

∫ zmax

zzhi

dztri

∫ M/2

M/5

dm×R(m,M−m, ztri)δD(L−L(zshi)) WΣ[Lpeak,M, ztri].

(5.29)

172

Let us explain this expression physically – which can be interpreted as two

selection effects. First consider all mergers producing haloes of mass M at some

redshift ztri > zshi. The first selection is imposed by the Dirac delta, which only

chooses those M -haloes whose associated AGN light curve L shines in the (L,L+dL)

bin precisely at our redshift zshi of interest. For some, this luminosity will be in

the ascending phase while for others it will be in the descending phase. The second

selection is on the type of mergers involved. The integral from M/4 to M/2 only

chooses those mergers where the ratio of the constituents is m/(M − m) > 1/3.

(Notice that in this notation, m is the smaller of the two.) Lastly, just like the

Magorrian relations have scatter, it is natural to model our self-regulation condition

with some scatter. For this purpose, we assume that the Lpeak −M relation9 has a

log-normal distribution of width Σ = 0.28 dex, given by

WΣ[Lpeak,M, ztri] =1√

2πΣ2exp

[

−(log Lpeak − log Lpeak(M, ztri))2

2Σ2

]

. (5.30)

In this context, see Lapi et al. (2006) and § 5.7.2.

5.4.2 The large scale bias

To further test our models we compute the large scale clustering as a function of

luminosity and redshift via the equation

b(L, zshi) =

∫∞

0dM

∫ zshi

zmaxdz′WΣ[Lpeak,M, z′]R(M, z′)bh(M, zshi; z

′)∫∞

0dM

∫ zshi

zmaxdz′WΣ[Lpeak,M, z′]R(M, z′)

, (5.31)

9Unless it is discussed in the context of WΣ, we will denote the self-regulation condition as

Lpeak −M , as opposed to Lpeak −M .

173

where R(M, z) denotes the integral of R(m,M−m, z) in m from M/5 to M/2. The

evolution of the bias between z′ and zshi is

bh(M, zshi; z′)− 1 =

(

D(z′)

D(zshi)

)

[bh(M, zshi)− 1] (5.32)

as in Fry (1996). Hereafter we refer to this expression as the ‘Fry formula’ (see

the next section for a more thorough discussion). Following Sheth, Mo & Tormen

(2001), the halo bias is given by

bh(M, z) = 1 +1√qδc

[√q(qν) +

√qβ(qν)1−γ − (qν)γ

(qν)γ + β(1− γ)(1− γ/2)

]

, (5.33)

where (q, β, γ) = (0.75, 0.47, 0.615). Here ν(M, z) = δc(z)2/σ2(M) is the square of

the ratio of the critical overdensity required for collapse to the rms density fluctua-

tion on scales containing a mass M . Alternatively, many authors (e.g, Shen, 2009b)

use the Jing (1999) formula. However, the simulations of Shankar et al. (2009c)

show that the Sheth & Tormen (1999) result is more accurate.

We will discuss in § 5.6.2 that although the large systematic and statistical un-

certainties still prevent very accurate quasar clustering measurements, the present

available data are already good enough to set interesting constraints on BH evo-

lution, especially at low and high redshifts. More specifically, we will show that if

we allow the bias to evolve passively (e.g., Fry, 1996) since the triggering epoch ztri

until the time of shining zsh, the high quasar clustering signal measured at z > 3

constraints the delay time tdelay = t(zshi)− t(ztri) to be rather short, thus implying

an initial fast, super-Eddington BH growth. On the other hand, at low redshifts

174

z < 1 and low luminosities L . 1045erg s−1, the flat bias as a function of luminosity,

measured from SDSS, suggests that low-luminosity AGNs are hosted by rather mas-

sive haloes. This constraints models of AGN evolution and implies a lower number

of BHs with MBH< 3× 107M⊙ in the local universe (see § 5.6.3).

5.4.3 Testing the Fry formula

We discuss here the model for the evolution of bias that we used in our model in

more detail, and its validation against N-body simulations. The model, introduced

initially by Fry (1996), results from assuming that objects form by some arbitrary

local process at redshift zf and then move with the streaming velocity caused by

the gravitational potential. If one allows for a biased relation between fluctuations

in the number density of objects and that in the matter density at the time of

formation zf , then the subsequent evolution of this relation to linear order is given

by

b(z)− 1 = D(zf)/D(z)[b(zf)− 1] (5.34)

where D(z) is the linear growth factor and z ≤ zf .

The same result has also been derived in the context of the Press-Schechter for-

malism Mo & White (1996) (provided that the impact of halo mergers is negligible)

or using the Peak-Background split argument of Sheth & Tormen (1999). Moreover

it has been already used in this context in the analysis of QSO clustering (e.g., see

the early work of Croom & Shanks, 1996).

175

To test this model against N-body measurements, we used the Millennium sim-

ulation Springel et al. (2005), that tracked the joint gravitational evolution of a

billion particles within a cubic box-size of Lbox = 500 h−1 Mpc, from the initial con-

ditions at z = 127 until today. The cosmological parameters employed were slightly

different from the ones assumed in this thesis, namely Ωm = 0.26, ΩΛ = 0.74,

Ωb = 0.044, σ8 = 0.78 and h = 0.73 .

As discussed by Tegmark & Peebles (1998), the basic assumption underlying

equation (5.34) is that, at the time of formation, objects are assigned a near perma-

nent ‘observational tag’ and then move satisfying the matter continuity equation.

We thus selected haloes formed at some initial time and tracked their evolution

through the merger history tree of the Millennium run until today, and computed

their clustering amplitude along the way. We chose all FoF groups at z = 3.87

and z = 4.89 in two mass-bins, M ≥ 1012 h−1 M⊙ and M ≥ 3.16× 1012 h−1 M⊙ (at

z = 3.87 we found 40790 and 3477 haloes in each mass bin, while at z = 4.89 there

were 10152 and 471 haloes respectively). We then found their descendants at several

lower redshifts (a “descendant” is defined as the halo that contains most of the 10%

most bound particles of the ‘parent’ halo), and measured their bias by computing

their power spectrum in ratio with the corresponding linear power spectrum, i.e.

b =√

Phh/Plin, at large scales k ≤ 0.2 h Mpc−1.

The mass distribution of the descendants turned to be roughly Gaussian in

log(Mfinal/Minitial), with mean ∼ 1.35 and variance ∼ 0.52 at z = 0 for formation

176

time zf = 4. This implies that these haloes grew by a factor of ∼ 20 in mass

since they were formed until today (for zf = 5 we found a factor of ∼ 40 in mass

increment).

The result for the bias evolution is shown in Fig. 5.5, with different symbols

(circles and squares) corresponding to the measured bias in each mass-bin, as la-

beled. Different line-types (solid and dashed) correspond to the prediction in equa-

tion (5.34) for each formation time. The bottom panel displays the relative differ-

ence with respect to the prediction, showing a remarkably good agreement for both

mass-bins at all z ≤ zf .

Nonetheless, due to the low number density of haloes our measurements are

slightly sensitive to the shot-noise subtraction, and its impact is redshift dependent.

Also these results are a bit sensitive to the range of scales used to compute the bias

(±5% at z = 0 using k ≤ 0.1 h Mpc−1 or k ≤ 0.3 h Mpc−1).

Finally, notice that the comparison performed in this section differs from measur-

ing the bias of all haloes more massive than a given mass threshold, as a function

of redshift. In that case, the evolution of bias evolves much more strongly (e.g,

Matarrese et al., 1997).

177

Figure 5.5: The evolution of bias for haloes formed at redshifts z = 3.87 and

z = 4.89 and their descendants as a function time, for two different halo mass-cuts

(as labeled). The ‘parent’ haloes increase their mass by a factor of ∼ 20 (∼ 40)

from zf ∼ 4 (zf ∼ 5) until today. The bias, defined as b =√

Phh/Plin at large scales,

decreases with evolving time as the inverse of the growth factor, as predicted by

equation (5.34).178

5.5 Measuring the clustering of faint AGNs and

low-redshift quasars

Before presenting how various version of our model are constrained by observed

data, we take a brief detour into some technical aspects related to data we analyzed

and used. In particular, we focus on the clustering of faint AGNs and quasars at

low redshift.

5.5.1 The clustering of faint AGNs

In this section we describe in more detail how we compute the bias associated with

the measurements of low-luminosity AGN by Constantin & Vogeley (2006). The

redshift-space correlation function is commonly quantified in terms of a best-fitting

power law

ξ(s) =

(

s

s0

)−κ

, (5.35)

where s is the comoving separation between the pairs. The parameter s0 repre-

sents the amplitude of clustering, while κ quantifies the ratio of small-to-large-scale

clustering. Table 5.5.1 and Figure 5.6 report the best-fitting values for different

measured samples.

Another common estimator of clustering is the integrated correlation function

(Croom et al., 2005), defined as

ξ(s20) ≡3

s320

∫ s20

s5

ξ(s)s2d s =3

3− κ

[

ξ(s20)−ξ(s5)

64

]

, (5.36)

179

AGN type s0 κ ξ(s20) b

[OI], [OI]/[H α] criteria included:

Emission line 6.33 ± 0.94 1.30 ± 0.16 0.40 ± 0.13 1.20 ± 0.23

Seyferts 5.67 ± 0.62 1.56 ± 0.17 0.29 ± 0.13 1.00 ± 0.27

LINERs 7.82 ± 0.64 1.39 ± 0.09 0.51 ± 0.08 1.38 ± 0.13

Pure LINERs 7.39 ± 0.67 1.33 ± 0.11 0.48 ± 0.10 1.33 ± 0.15

Transition 5.38 ± 0.71 1.35 ± 0.23 0.31 ± 0.17 1.03 ± 0.34

H II galaxies 5.81 ± 0.53 1.28 ± 0.11 0.36 ± 0.09 1.13 ± 0.16

[OI], [OI]/[H α] criteria excluded:

Emission line 6.45 ± 0.24 1.24 ± 0.04 0.42 ± 0.03 1.23 ± 0.06

Seyferts 6.00 ± 0.64 1.41 ± 0.15 0.35 ± 0.12 1.10 ± 0.22

LINERs 7.26 ± 0.61 1.29 ± 0.08 0.47 ± 0.07 1.33 ± 0.11

H II galaxies 5.73 ± 0.24 1.22 ± 0.06 0.37 ± 0.05 1.14 ± 0.11

Table 5.1: The clustering of low luminosity AGNs.

where the right-hand-side follows from equation (5.35). Following Shen et al. (2007),

we integrate from s5 = 5 Mpc h−1 (to minimize non-linear effects) up to s20 = 20

Mpc h−1 (which is large enough for linear theory to apply)10. Table 5.5.1 reports

ξ(s20) for various types of low-luminosity AGN.

10Unlike us, Croom et al. (2005) integrate from 0 Mpc h−1.

180

Figure 5.6: The bias of low-luminosity type II AGNs.

181

The bias is computed as in da Angela et al. (2008):

b(z) =

√

ξ(s20)

ξρ(r20)− 4Ω1.2

m (z)

45− Ω0.6

m (z)

3. (5.37)

Here, ξρ is the correlation function in real space, and

ξρ(r20) ≡3

r320

∫ r20

r5

ξρ(r)r2d r = 0.206, (5.38)

where r5 = 5 Mpc h−1 and r20 = 20 Mpc h−1. Table 5.5.1 and Figure 5.6 show the

bias for these samples, with median redshift z = 0.1.

The corresponding error bars are obtained by using standard propagation of

uncertainty. One can show that

∆ξ(s)

ξ(s)=

√

(

κ

s0

)2

(∆s0)2 +

(

ln

(

s

s0

))2

(∆κ)2,

∆ξ(s20) =3

3− κ

√

(∆ξ(s20))2 +

(

∆ξ(s5)

64

)2

,

and

∆b =1

2

∆ξ(s20)

ξρ(r20)

[

b +Ω0.6

m (z)

3

]−1

. (5.39)

5.5.2 The clustering of low-redshift quasars

We use a sample of 88178 type 2 AGNs, which is publicly available from the

MPA/JHU SDSS DR4 database11, constructed by Kauffmann et al. (2003) using

the SDSS DR4 spectroscopic data. We divide all the AGNs into 10 subsamples

11http://www.mpa-garching.mpg.de/SDSS/DR4/

182

with equal sample size according to their extinction-corrected [O III] luminosity.

We then compute the projected two-point cross-correlation function wp(rp) of each

subsample with respect to a reference sample of galaxies. The reference sample

is composed of ∼ 0.5 million galaxies selected from the Sample dr72 of the New

York University Value Added Galaxy Catalogue (NYU-VAGC)12 (Blanton et al.,

2005), which is based on the SDSS Data Release 7 (DR7; Abazajian et al., 2009).

A random sample is constructed so as to have the same selection effects as the

reference sample. The reference and random samples are cross-correlated with each

of the AGN subsamples, and wp(rp) as a function of the projected separation rp is

defined by the ratio of the two pair counts minus one. Errors of the wp(rp) mea-

surements are estimated using the Bootstrap resampling technique. Details about

our methodology for computing the correlation functions and for constructing the

reference and random samples can be found in Li et al. (2006a) and in Li et al.

(2006b). Finally we convert our wp(rp) measurements to the real-space correlation

function ξ(r), using the Abel transform (see for example Hawkins et al., 2003).

The quasar-galaxy cross-correlation function is

ξQg(r, z) = bQ(r, z)bg(r, z)D2(z)ξρ(r). (5.40)

Similary, the galaxy-galaxy auto-correlation function is

ξgg(r, z) = b2g(r, z)D2(z)ξρ(r). (5.41)

12http://wassup.physics.nyu.edu/vagc/

183

Figure 5.7: The bias of low-redshift quasars. The solid line represents our average

sample bias and the dotted lines represent the estimated error (equation 5.50)

184

One can the solve for the galaxy bias:

bg(r, z) =

√

ξgg(r, z)

D2(z)ξρ(r), (5.42)

and the quasar bias

bQ(r, z) =

(

ξQg(r, z)

ξgg

)

bg(r, z). (5.43)

The associated uncertainties are

∆bg

bg

=∆ξgg

2ξgg

, (5.44)

and

∆bQ

bQ

=

√

(

∆ξQg

ξQg

)2

+

(

∆ξgg

2ξgg

)2

, (5.45)

Our SDSS sample has mean redshift z = 0.1, with growth factor D(z) = 0.819.

Defining

ξgg(r20) ≡3

r320

∫ r20

r5

ξgg(r)r2d r, (5.46)

the mean galaxy bias (Peebles, 1980) can be estimated as

bg =ξgg(r20)

D2(z)ξρ(r20), (5.47)

From our sample, we report

bg(z = 0.1) = 1.207± 1.716. (5.48)

Estimating the mean quasar bias as

bQ =

(

ξQg(r20)

D2(z)ξgg(r20)

)

bg. (5.49)

From our sample, we report

bQ(z = 0.1) = 1.282± 0.192. (5.50)

185

5.6 Results

5.6.1 The luminosity function

Figure 5.8 shows our reference model bolometric luminosity function at different

redshifts, as labeled, compared with the compilation of data from (Shankar et al.,

2009, and references therein). Our model reproduces the bright-end of the AGN

luminosity function at all redshifts. In the alternative framework in which all BHs

of any mass are characterized by the same (mass-independent) light curve, the low-

z bright-end AGN luminosity function is inevitably overproduced (e.g, Wyithe &

Loeb, 2003). As an example, Figure 5.9 compares our reference model (solid lines)

to a model in which no mass dependence in the light curve is allowed (long-dashed

lines). The latter overproduces the bright-end by a factor of & 2 − 3. Also note

that fine-tuning the redshift dependence of the Lpeak-M relation to find a better fit

to the z < 1 bright end of the AGN luminosity function does note overcome the

latter problem.

The success of the model discussed here is actually quite noticeable. As dis-

cussed by several authors (e.g., Cavaliere & Vittorini, 2002; Wyithe & Loeb, 2003;

Scannapieco & Oh, 2004; Vittorini et al., 2005; Marulli et al., 2008; Bonoli et al.,

2008; Shen, 2009b), simultaneously matching the low and high-redshift AGN num-

ber counts through AGN feedback-type Lpeak-M relations, often requires additional

input physics (and parameters) for the models to be successful (such as progressive

186

Figure 5.8: Predicted bolometric luminosity function at different redshifts, as la-

beled. At low redshifts the major-merger model fails to reproduce the faint-end of

the luminosity function, as expected. All the data are from Shankar et al. (2009)

and references therein.

187

gas exhaustion, radio feedback, etc.). Improving on Wyithe & Loeb (2003), Scanna-

pieco & Oh (2004) invoked, for the first time, an additional ‘radio mode’ feedback to

limit the growth of very massive black holes at low redshifts, thus lowering the num-

ber of luminous quasars, and improving the match to the luminosity function. The

more detailed works by Croton et al. (2006) and Marulli et al. (2008) have essen-

tially adopted similar recipes to cope with the low-redshift, high-luminosity portion

of the AGN luminosity function. Similarly, other groups such as Mahmood et al.

(2005) and Shen (2009b) assumed that the maximum halo mass hosting quasars

decreases by a factor of a few at lower redshifts, mimicking a progressive starvation

of massive black holes in massive haloes.

Another interesting feature of our model is that the faint end of the predicted

luminosity function flattens out with increasing redshift (and even turns over for

extremely dim luminosities). If confirmed by future surveys, this behavior has

significant implications for the quasar population. The break claimed by some

groups in the AGN luminosity function at high redshifts (e.g., Small & Blandford,

1992; Willott et al., 2004; Fontanot et al., 2007; Shankar & Mathur, 2007) would be

real, and present at all wavelengths. Also, the obscured fraction of sources should

have a different behavior with respect to luminosity, being constant, or even maybe

decreasing with increasing luminosity (see discussion in Shankar & Mathur, 2007).

If planned deep future surveys, such as PanSTARRS, instead unveil a non-negligible

population of faint AGNs at high redshifts; this will be a direct proof that major

188

Figure 5.9: Predicted luminosity function from a model with no mass dependence

in the light curve (long-dashed lines), and the reference model (solid lines), at two

redshifts, as labeled. Ignoring the mass-dependence in the light curve (see text)

implies growing more numerous massive black holes and, therefore, more luminous

quasars at late epochs.

189

mergers alone cannot be the only trigger of quasar activity, not even at very high

redshifts. Other in-situ and still not well-understood, AGN triggering mechanisms

must be already efficient in fueling BHs at high redshifts.

At z > 2, the model tends to somewhat overpredict the number of faint sources

below the knee of the AGN luminosity function. Allowing for the minimum halo

mass that can host a quasar Mmin to slightly increase with redshift will considerably

decrease the number of faint quasars and also increasing their bias (we will further

discuss this point in § 5.7.1).

We also find that a model like this one characterized by a more short-lived post-

peak phase, yields Eddington ratio distributions at fixed bolometric luminosity well

described by a Gaussian with dispersion around 0.3-0.4 dex, much lower than what

expected when assuming a long descending phase (see discussion in Yu & Lu 2008

and Shen 2009). Taken at face value, the outputs of our model would match well the

observations by AGES (Kollmeier et al. 2006), SDSS (e.g., Netzer & Trakhtenbrot

2007), and 2dF (e.g., Fine et al. 2008), that actually show comparable Eddington

ratio dispersions, although flux limited biases in the observations (e.g., Shen et al.

2008, Yu & Lu 2008) prevent us to draw any firmer conclusion.

A shortcomming of this model – a common feature of any major-merger model

– is instead the underestimation of the number of faint, low-redshift AGNs, as also

recently re-addressed by Shen (2009b). This is principally because the major-merger

events involving massive haloes get too rare at low redshifts to accommodate for

190

the numerous faint, low-z AGNs. We will further discuss below ways to constrain

extensions of our model tuned to reproduce this portion of the AGN luminosity

function.

5.6.2 Quasar clustering

The large scale clustering

Figure 5.10 compares our reference model to the bias measurements (listed in the

caption) from several large and small quasar surveys at different redshifts and lumi-

nosities. The grey stripes encompass a systematic 10% uncertainty in the analytical

fits of the bias (e.g., Shankar et al., 2009c). Overall, our model provides a reasonable

good match to the large scale clustering, although the systematic and statistical er-

rors in the measurements are still large. It is also fair to mention that not all the

clustering measurements have the same statistical weight. For example, the data

based on large scale surveys such as SDSS and 2dF (e.g., Myers et al., 2007; da

Angela et al., 2008; Shen et al., 2009a) should be less affected by cosmic variance

with respect to those obtained from small fields, such as the compilation of data of

Plionis et al. (2008) from X-ray surveys (see also discussions in Gilli et al., 2007;

Marulli et al., 2009). Also, the SDSS and 2dF clustering measurements are based

on thousands of quasars and therefore should be more reliable than data obtained

from much smaller samples (see, e.g., discussion in Coil et al., 2007, 2009, and

references therein).

191

Figure 5.10: Predicted bias as a function of luminosity at different redshifts, as

labeled. Our merger model is consistent with all the available data. Data are

from Padmanabhan et al. (2008) (inverse solid triangles), Mountrichas et al. (2009)

(filled circles), da Angela et al. (2008) (open triangles), Francke et al. (2008) (large

triangle), Shen et al. (2009a) (double circles), Myers et al. (2007) (open squares),

Porciani & Norberg (2006) (filled triangles), Shen et al. (2007) (diamonds and

inverse open triangles), Plionis et al. (2008) (open circles), Coil et al. (2007) (filled

squares). In particular, measurements from small samples (e.g., Plionis et al., 2008)

suffer from cosmic variance and are not as reliable as those from broad surveys like

SDSS and 2dF.

192

Nevertheless, several features are worth mentioning. At high redshifts (z > 3),

the model is only marginally consistent with the data. This is a feature common

to all models based on ‘low’ radiative efficiencies of ǫ . 0.1 (see, e.g., Shen, 2009b).

To boost the predicted bias some modification of the input parameters must be

adopted. As discussed in detail by Shankar et al. (2009c) (see also White et al.,

2008), boosting the bias implies placing the quasars in more massive, but less nu-

merous, haloes. This in turn implies increasing the median radiative efficiency to

at least ǫ & 0.15 (at fixed Eddington ratio λ & 0.15), to allow for the less numerous

hosts to produce the same amount of emissivity to match observations.

Alternatively, some authors (e.g., Wyithe & Loeb, 2008, and references therein)

have put forward the hypothesis that recently merged haloes are characterized by

some ‘excess’ bias with respect to haloes of similar mass. Adopting this extreme

bias would then ease the tension between model predictions and the high clustering

signal measured at z > 3. Detailed studies with cosmological simulations will be

able to test such interesting proposals (e.g., Gao & White, 2007; Bonoli et al., 2009).

Another important point concerns the clustering of low-luminosity sources at

z . 1. Taken at face value, the model predicts a flat bias below log(L/ergs−1) . 45

over several orders of magnitude in luminosity. This is due to the lower limit

in massive haloes which can host quasars, that in our reference model is set to

Mmin = 3 × 1011M⊙. We will further discuss this point, and modifications to this

assumption, in § 5.6.2.

193

A signature of rapid black hole growth

As anticipated at the end of § 5.4.2, the high redshift clustering measured from

SDSS at z > 3 is also powerful in constraining the delay time experienced by

quasars from their triggering epoch until their shining time. The high redshift

quasars are on average very luminous, and therefore they must be produced at

times close to the peak of the light curve, when the luminosity emitted by the BH is

around the maximum. Therefore the delay time tdelay = t(zshi)− t(ztri) is primarily

composed of the time spent by the BH in the ascending phase. In other words,

tdelay ≃ tasc ∝ log(µBH)/λ0 (see equation 5.16).

Figure 5.11 compares the predicted z = 4 bias as a function of luminosity for

models characterized by different delay times, as labeled. Models with longer delays

between the virialization epoch and the shining of the quasar, i.e., with lower λ0

and/or higher µBH = MBH,peak/MBH,seed, tend to generate a lower bias at fixed

luminosity, due to the passive evolution of the bias, especially at high redshifts (see

Section 5.4.2). The high-z quasar clustering measured by Shen et al. (2007) favors

models characterized by massive seeds and high Eddington ratios, conditions which

minimize the delay. We conclude from our analysis that a delay time of . 108 yr is

favored by the high-z data on quasar clustering (and number counts, see § 5.7.1).

This delay time is hard to reconcile with models that adopt small BH seeds of

. 103M⊙ and, especially, Eddington or sub-Eddington rates in the initial growth

phases of the BH (a 109M⊙ BH in these conditions would in fact require a growth

194

Figure 5.11: Comparison among models characterized by different delay times at

two different redshifts, as labeled. Models with longer delays between the virial-

ization epoch and the shining of the quasar, i.e., with lower λ0 and/or higher ratio

between peak and seed masses, tend to generate a lower bias at fixed luminosity,

due to the passive evolution of the bias, especially at high redshifts. The high-z

quasar clustering measured by Shen et al. (2007) favors models characterized by

massive seeds and high Eddington ratios, conditions which minimize the delay.

195

time in the ascending phase of & 5× 108 years).

Observationally, our understanding of BH evolution is richer in their final, op-

tically visible stages than in the earlier (possibly obscured) ones. Therefore, con-

straints like the one presented here derived from clustering, are invaluable for mod-

els that attempt to describe the full evolution of BHs. For example, Granato et

al. (2006) and Lapi et al. (2006) discuss the importance of the relative duration

of the obscured and visible phases within their quasar evolutionary models tuned

to simultaneously reproduced both the dust-enshrouded number counts of SCUBA

galaxies and ULIRGs, and the quasar optical luminosity functions. They conclude

that successful models must be characterized by a rather short tdelay, a condition

they are able to meet by assuming an initial fast, super-Eddington accretion phase

(see also Di Matteo et al. 2008).

Making faint AGNs at z < 1

As anticipated in § 5.6.1, the merging model fails at reproducing the faint-end of

the AGN luminosity function at z < 1 and L . 1045erg s−1. This is an old problem

several times discussed in the literature (e.g, Vittorini et al., 2005; Hopkins et al.,

2006b; Shen, 2009b, and references therein), although the solution to this issue has

not yet found a definite answer.

Faint AGNs at these epochs are predominantly Seyfert-like (see e.g., Hopkins

& Hernquist, 2009), usually associated to disky galaxies (e.g., Kim et al., 2008).

196

However, it is still not clear what the exact masses of the haloes that host them

actually are. In Figure 5.12 we compare our reference model (solid lines), at z = 0.5

(upper panels) and z = 0.1 (lower panels), with the outputs of two extension of our

model. The first is characterized by (1) faint AGNs predominantly hosted by low-

mass haloes with log(Mmin/M⊙) = 10 undergoing major merger events (dot-dashed

lines); while the other one has (2) faint AGNs hosted by the same range of halo

masses considered in the reference model (i.e., log(Mmin/M⊙) = 11.5), but allowing

minor mergers as triggering agents. It is clear that both alternative models can

reproduce the number counts of faint AGNs although, as expected, their clustering

predictions are significantly different.

In the lower right panel of Figure 5.12 we compare with the data on clustering

extracted by us for faint AGNs from SDSS (see Section 5.5). Given the large

number of sources in each luminosity bin (about 8,000 AGNs), we are able to derive

very small error bars which can rule out the model with low-mass haloes at a high

significance level. We conclude that the faint-end of the AGN luminosity function at

low redshifts is mainly produced in haloes more massive than log(Mmin/M⊙h−1) ∼

11.5−12 (see also Lidz et al., 2006). Obviously, our simple model cannot isolate the

true triggering mechanisms causing the shining of the faint, low-z AGNs; although

these are probably in-situ processes, uncorrelated from large scale events as in the

more luminous quasars. As discussed below, having constrained the haloes which

are hosting the faint AGNs also implies a different local BH mass distribution.

197

Figure 5.12: Comparison among the reference model (solid lines) with other two

models, one specified by having fainter AGNs produced in less massive haloes, with

log(Mmin/M⊙) = 10 (dot-dashed lines), and another one where the faint-end of

the AGN luminosity function is produced by black holes in more massive haloes

triggered by minor events (long-dashed lines). These models can be readily tested

by current and future measurements.

198

5.6.3 The predicted black hole mass function

Following § 5.4, we can compute the expected BH mass function at any redshift with

our formalism. The result for our reference model is shown in Figure 5.13, plotted at

different redshifts, as labeled. It is clear that a natural, mild downsizing is generated

during the evolution in the model, with the most massive BHs completing their

growth faster than less massive ones. This general feature is a natural outcome

of all models tied to the observed AGN luminosity (see discussion in SWM and

references therein).

The long-dashed line is the local BH mass function found by convolving the

Sheth (2003) velocity dispersion function with the MBH-σstar relation for early-type

galaxies recently calibrated by Tundo et al. (2007) and (Shankar & Ferrarese, 2009,

and references therein). The grey band is the statistical 1-σ error bar estimated

derived from Monte Carlo simulations (see Shankar & Ferrarese, 2009, for further

details). We find that our merger model predicts a cumulative BH mass function

at z = 0 consistent with the local distribution of BHs in early-type galaxies, when

adopting ǫ . 0.07 − 0.1, consistent with Soltan-type arguments (SWM and ref-

erences therein), with the systematic uncertainty of ∼ 30% caused by the actual

bolometric corrections adopted (see discussion in SWM). Our model therefore nat-

urally predicts that all BHs in early-type galaxies were mainly triggered in major

merging events, as long as ǫ . 0.07− 0.1. Also note that our model grows massive

BHs less than the model adopted by Shen (2009b) because our input light curves

199

Figure 5.13: Predicted black hole mass function at different redshifts, as labeled.

There is sign of a mild downsizing, with the cumulative number the more massive

black holes being in place at higher redshifts. The grey band is the local black

mass function computed from the velocity dispersion function of early-type galaxies,

while the dotted lines are from Shankar et al. (2009c) and bracket the systematic

uncertainties in estimating the total local black hole mass function.

200

for massive BHs have negligible descending phase. This allows a better match with

the local BH mass function at the high-mass end.

Figure 5.14 shows the predicted local BH mass function derived from the three

models discussed in § 5.6.2 in reference to Figure 5.12. The models characterized

by quasars shining only in more massive / less abundant haloes with major and

minor mergers (solid and long-dashed lines) produce, as expected, less numerous

BHs with mass log MBH < 107M⊙ by a factor & 2−3, relative to the model in which

BHs are allowed to be triggered by major mergers even in less massive haloes (dot-

dashed lines). SDSS clustering measurements presented in the lower right panel

of Figure 5.12 and discussed in § 5.6.2, however, support the former massive-host

models. This has profound implications regarding our knowledge of the local BH

mass function. Many of the previous attempts (e.g., Salucci et al., 1999; Shankar

et al., 2004; Marconi et al., 2004) might have overestimated the actual contribution

of late-type galaxies to the local BH mass function. We here find that models

that match the SDSS bias values are consistent with a much flatter local BH mass

function. This in turn implies that the Eddington ratio distribution characterizing

the local population of AGNs is peaked at significantly lower λ with respect to the

more luminous high-z quasars. This is simply due to fact that the characteristic

BH mass is always growing in time while the L∗ in the AGN luminosity function is

decreasing (e.g., Hopkins et al., 2006b; Shankar et al., 2009; Shankar, 2009).

This behavior in the Eddington ratio distribution with time is also confirmed

201

Figure 5.14: Predicted black hole mass functions for the same models discussed in

§ 5.6.2 and also labeled here. The models characterized by quasars shining only in

massive haloes produce less numerous black holes with mass log MBH < 107M⊙ by

a factor & 2 − 3 (and a flatter bias as shown in Figure 5.12). The grey band and

dotted lines are as in Figure 5.13.

202

by continuity equation arguments. Shankar et al. (2004) and Shankar et al. (2009)

(as also reviewed in Shankar 2009), have shown that a continuous decrease of the

characteristic λ with decreasing redshift is the only way to reduce the number

of low-mass BHs predicted by direct integration of the observed AGN luminosity

function.

5.7 Discussion

5.7.1 Further constraint from high-z X-ray counts.

As discussed by several authors (e.g., Richards et al. 2006, Fontanot et al. 2007,

Hopkins et al. 2007, Shankar et al. 2009), the shape of the bolometric z > 3

AGN luminosity function is still quite uncertain, as also directly evident from the

data in Figure 5.8. Brusa et al. (2009) have recently measured the space density

of X-ray selected quasars in the deep and uniform multiwavelength coverage of the

COSMOS survey in the redshift range 3 . z . 4.5. In agreement with previous

findings on optically selected quasars by Richards et al. (2006), Brusa et al. (2009)

confirm an exponential drop in the space density of quasars at z & 3. Recent re-

calibrations of the global hard X-ray luminosity function based on 2 Ms Chandra

Deep Fields and the AEGIS-X 200 ks survey (Aird et al., 2009, open squares in the

upper right panel of Figure 5.15) also point towards a lower number density of faint

quasars at z > 3. These findings set further constraints on the actual shape of the

203

Figure 5.15: Upper panels : Predicted X-ray number counts for different models, as

labeled, compared with the data by Brusa et al. (2009) at z = 3 (left) and z = 4

(right). Lower panels : corresponding AGN luminosity functions predicted by the

same models at the same redshifts. Data at z & 3 favor models with a higher Mmin

and minimal descending phase.

204

overall, high-z bolometric AGN luminosity function at luminosities L & (0.5−1)L∗,

favoring a flat faint-end for the AGN luminosity function at L ≤ L∗. Note in fact

that the minimum luminosity probed by the COSMOS survey is L ∼ 3×1045erg s−1

(LX ∼ 1044), a factor of a few lower than the knee of the AGN luminosity function,

occurring around L ∼ 1046erg s−1 at z & 2 (Figure 5.8).

As already partially discussed above, our reference model showed in Figure 5.8

tends to predict too many faint quasars with respect to the data, although the

mismatch is somewhat dependent on the exact bolometric corrections, completeness

corrections, and several other systematics affecting the data (see, e.g., Fontanot

et al. 2007). The thick dotted lines in Figure 5.15 show the corresponding X-ray

number counts at z = 3 and z = 4 in our ‘reference’ model discussed above. As note

before, the model overproduces the number of faint AGNs at z > 2 and therefore,

as expected, it also overpredicts the X-ray number counts.

Here we plot, in agreement with the data, the number counts of all X-ray sources

with column densities in the range 21 < log NH/cm−2 < 24. The number counts

are computed following, e.g., Gilli et al. (2001, and references therein). We first

determine the relative contribution of X-ray sources of a given column density, red-

shift and luminosity to the overall number counts by first correcting the bolometric

AGN luminosity function by the appropriate redshift- and luminosity-dependent

fraction of sources f(NH , z, LX) (taken from Ueda et al. 2003). We then compute

the integral N(> S) ∝∫ Lmax

Lmin(NH ,z,LX)Φ(LX , z, NH)dzdLX with Lmin(NH , z, LX) dif-

205

ferently K-corrected depending on the type of spectrum considered (each spectrum

is computed from the PEXRAV code of Magdziarz & Zdziarski (1995), and de-

pends on the amount of column density NH considered). We note that our results

do not depend significantly on the exact choice of column density distribution or

X-ray spectrum adopted, but more on the shape of the underlying AGN bolometric

luminosity function.

We find that to decrease the number of sources shining at luminosities L . L∗,

thus simultaneously improving the match to the AGN luminosity function and X-

ray number counts, we could either increase the minimum halo mass Mmin host-

ing quasars, and/or minimize the time spent by BHs in their descending, low-

Eddington ratios, post-peak phases. Figure 5.15 shows examples of such mod-

els characterized by higher minimum halo masses of Mmin= 1012M⊙ (long-dashed

lines), of Mmin= 1012.3M⊙ (dot-dashed lines), and a model with higher minimum

mass Mmin= 1012M⊙ and without descending phase (solid lines). All these models

characterized by a higher Mmin provide better matches to the data, further sup-

porting the need for an evolving minimum halo mass hosting quasars.

Also, these models predict a strong break and a flat faint-end slope in the high-z

AGN luminosity function. In principle, all these improved models are quite degener-

ate with respect to the data (although the model with no descending phase provides

a better match to the bright end of the AGN luminosity function). In contrast with

what has been discussed in reference to the low redshift universe, however, the

206

still scarce data on the faint-end, high-z AGN luminosity function and the still too

loose clustering measurements in the corresponding region in the (L, z)-plane (e.g.,

Francke et al., 2008) prevent us to draw any firmer conclusions.

Nevertheless, we note that those models characterized by BH light curves sharply

peaked with a negligible post-peak phase (e.g., Lapi et al., 2006), better reproduce

the obscured fraction as a function of luminosity within evolutionary models (e.g.,

Lapi et al., 2006; Shankar et al., 2008). In fact, we have checked that a long

post-peak luminous, unobscured phase requires a non-trivial, physically challenging,

fine tuning of the parameters to reproduce an obscured fraction decreasing with

increasing luminosity. We reserve a full study of the obscuration fraction as a

function of luminosity and redshift and its connection with evolutionary/orientation

models and input light curves for future work.

One key result in several studies that attempted to empirically constrain the

average quasar light curve (e.g., Yu & Lu, 2008; Hopkins & Hernquist, 2009; Shen,

2009b), is that most BHs should possess a significant post-peak luminous phase,

most of the times characterized by a more or less pronounced descending phase.

However, most of these studies are sensitive to data at z . 1.5 − 2, and cannot

really rule out alternative models characterized by only an ascending phase at higher

redshifts. In fact, most of the models predict that the bright high-z quasar emissivity

is produced around the peak of the light curve (e.g., Haehnelt et al., 1998; Wyithe &

Loeb, 2003; Lapi et al., 2006). However, we have also verified that adopting a light

207

curve with no descending phase even at very low redshifts implies that only a really

tiny portion of the AGN luminosity function at z . 1.5 is produced by the model,

specifically only the number density of AGNs shining with above L & 1047erg s−1.

We have seen that in this case the natural extensions of our models, as the ones

discussed in § 5.6.2 characterized by either less massive haloes hosting AGNs and/or

lower fraction of the minimum triggering ratio ξmin, do not provide any match

to the data, and some non-trivial, heavy fine-tuning of the models (such as an

evolutionary, mass and redshift-dependent normalization in the Lpeak−M relation)

must be invoked to get better agreement with the data. We therefore feel that a

much cleaner, simpler, and possibly more physical model is to just allow lower-mass

BHs to have more extended light curves (at least at low redshifts).

5.7.2 Scaling relations

It is now of wide interest the understanding of the redshift and, possibly, the mass

dependence of the basic scaling relations between BHs and their host galaxies and

dark matter haloes (e.g., Peng et al., 2006; Shankar et al., 2009b). Although flux-

limited biases might still affect in part some of the observational results (Lauer

et al., 2007), an increasing number of groups are finding a non-negligible increase,

within a factor of ∼ 2 − 3 in the overall normalization of the MBH-Mstar relation

at 0.5 < z < 2 (e.g., Peng et al., 2006; Jahnke et al., 2009; Somerville, 2009). The

actual evolution in the MBH-σstar relation seems instead to be milder, at least for

208

the bulk of the BH population, as concluded both from cumulative, Soltan-type

arguments (e.g., Shankar et al., 2009b) and detailed, direct observations of high-z

AGNs (e.g., Shields et al., 2006; Woo et al., 2008; Gaskell, 2009).

Understanding the true evolution of these scaling relations is, of course, invalu-

able for setting constraints on the actual co-evolution of BHs and galaxies. In this

section we address this issue by making use of our output MBH-M relation. The

feedback-constrained model developed in this work allows us in fact to trace the

underlying relation between BH mass and host halo mass/virial velocity at the peak

of the quasar activity. We can then work out the MBH-σstar relation predicted by

this model by first assuming a mapping between σstar and circular velocity Vc, as

observed in the local universe (e.g., Ferrarese, 2002), and mapping circular velocities

to virial velocities Vvir, which are linked to halo masses by the virial theorem (e.g.,

Barkana & Loeb, 2001).

We perform this exercise in Figure 5.16, where we show the predicted MBH-σstar

relations in four different redshift bins, as labeled. The grey band represents one

of the latest determinations of the local MBH-σstar relation by Tundo et al. (2007).

In each panel we show with a solid line the predicted MBH-σstar relation at the

moment of the shining of the quasar, derived by imposing the circular velocity Vc

equal to Vvir. This proves that if the velocity dispersion σstar is a good tracer of the

potential well, then a tight relation between virial velocity and BH mass, such as the

feedback-type relation used here, naturally translates into a non-evolving MBH-σstar

209

Figure 5.16: Predicted MBH-σstar relation at different redshift and different values

of the ratio between circular velocity Vc and virial velocity Vvir, as labeled. The

reference model is consistent with a nearly constant MBH-σstar relation with time,

as also found from independent empirical studies. Coupled with the information

of a smaller sizes at fixed stellar mass, the non-evolution in the MBH-σstar relation

implies a significant evolution in the Magorrian relation (see text).

210

relation (e.g., Wyithe & Loeb, 2003). Such findings, in turn, are in agreement with

the above mentioned empirical results concerning the MBH-σstar relation observed

in distant quasars, which seem to be consistent with the local one (e.g., Gaskell,

2009) – (see also Shankar et al., 2009b). For completeness, we also show with filled

circles the predicted remnant MBH-σstar relation at z = 0, obtained by assuming the

velocity dispersions do not evolve since the quasar shining epoch. It is clear that

our models predict final BH masses still within the range of the observed scatter in

the local MBH-σstar relation. Models characterized by much more prolonged light-

curves than the ones adopted here, would obviously produce BHs more massive

than allowed by the scatter in the z = 0 relation.

Direct observations of high-z massive, early-type galaxies have shown that, at

fixed stellar mass, galaxies are more compact (e.g., Trujillo et al., 2006, 2007;

Cimatti et al., 2008; Buitrago et al., 2008; Franx et al., 2008; Tacconi et al., 2008;

van Dokkum et al., 2008; Younger et al., 2008; Saracco et al., 2009; van der Wel et

al., 2009; Damjanov et al., 2009; Williams et al., 2009), and have higher velocity

dispersions (e.g., Cenarro & Trujillo, 2009; Cappellari et al., 2009; van Dokkum et

al., 2009). In particular, van Dokkum et al. (2009) measured the velocity dispersion

of a z ∼ 2 massive galaxy to be about around 510 km/s, a factor of ∼ 1.4 higher

than the most rapidly rotating early type galaxies in the local universe. We show

with dot-dashed lines in Figure 5.16 the results of assuming such a scaling relation

between stellar and dark matter velocities. It is clear that if the stellar circular

211

velocity or velocity dispersion is significantly higher than that of the underlying

dark matter, the resulting MBH-σstar relation gets shifted to higher σstar at fixed

BH mass, and just the opposite is true assuming Vc to be lower than Vvir, as in the

long-dashed lines. However, such trends contradict empirical results (e.g., Woo et

al., 2008; Gaskell, 2009; Shankar et al., 2009b).

The only way to reconcile higher velocity dispersions in high-z quasar hosts and,

at the same time, minimizing any (negative) evolution in the MBH-σstar relation,

would be to allow for a lower host stellar bulge hosting the BH at the moment of the

shining. This would automatically decrease the velocity dispersion associated with

a given BH mass, without affecting the underlying Mstar-σstar relation. A direct

one-to-one matching between the stellar mass and halo mass functions at different

redshifts, allows to determine the mean Mstar-M relation at all times (e.g., Vale &

Ostriker, 2004; Shankar et al., 2006; Conroy & Wechsler, 2009). The most recent

estimate of this relation has been performed by Moster et al. (2009), who also

show that the resulting stellar mass-halo relation is consistent with a compilation

of galaxy large-scale clustering at different masses and redshifts. Associating their

Mstar-M relation to our empirical MBH-M relation yields a redshift-dependent MBH-

Mstar relation which we show in Figure 5.17. Although it is difficult to draw firmer

conclusions due to our ignorance in how to exactly parameterize the true stellar

mass fractions associated to a given bin of BH mass at different redshift, there is

nevertheless tentative evidence for a positively evolving MBH-Mstar relation, in a

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Figure 5.17: Predicted MBH-Mstar relation at different redshift, as labeled, obtained

by matching our reference model MBH-M relation with the empirical Mstar-M rela-

tion obtained from cumulative number matching techniques. The reference model

is consistent with a relation with normalization increasing with increasing redshift,

i.e., higher BH masses at fixed stellar bulge. Coupled with the information of a

smaller sizes at fixed stellar mass, the evolution in the MBH-Mstar relation can be

translated into a non-evolving MBH-σstar relation, as observed (see text).

213

way that higher BH masses correspond to the same stellar bulge at higher redshifts.

We also note that the above discussion relies on the basic assumption that our

adopted MBH-M relation actually holds at the shining epoch. However, as also

discussed in § 5.2 and § 5.7.3, although supported by simulations and observations,

dropping the assumption of no dynamical friction delay between halo mergers and

quasar triggering, significantly reduces the number of mergers per halo at z & 3 (by a

factor of ∼ 2−4; see, e.g. Shen, 2009b). Recovering the match to the observed AGN

luminosity function without worsening the predicted bias then implies increasing

the normalization of the MBH-M relation by a proportionally higher factor. While

this has a minor impact on the overall local BH mass function (as long as the

condition of a higher normalization is limited to z & 3), it has a major impact on

the resulting MBH-σstar relation. In fact, a higher BH mass compensates the lower

MBH-σstar normalization induced by the higher σstar at the quasar shining due to

a more compact host, thus yielding a resulting MBH-σstar at z > 0 closer to the

local one, and in better agreement with direct observations. A drawback of this

model is, however, that due to the larger BH mass for a given host, the resulting

MBH-Mstar relation is much more offset with respect to the local one than the one

showed in Figure 5.17, requiring a major stellar mass growth at late times, difficult

to reconcile with observations and most SAMs.

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5.7.3 Comparison with previous models

The approach described in this work is significantly different from previous ones. Al-

though previous models aimed at describing the AGN luminosity function and clus-

tering properties using similar techniques of convolution between halo merger/creation

rates and light curves, the physical assumptions were, sometimes, substantially dif-

ferent.

As anticipated in § 5.6, the seminal paper by Wyithe & Loeb (2003) showed

that reproducing the AGN luminosity function at all relevant redshifts through

feedback-constrained relations, is not trivial. Possibly the main concern with these

kind of models is the overproduction of bright quasars at z . 2. In order to solve

such a problem, Scannapieco & Oh (2004) proposed that due to quasar preheating,

massive haloes at late times with an entropy floor of ∼ 100 keV cm−2 cannot cool

within a Hubble time inhibiting further star formation and BH fueling (see also

Marulli et al., 2007).

Our model differs from theirs in two ways. First, we here just assume that

quasar activity occurs only within haloes with mass below Mmax, in line with a

number of other groups (e.g., Granato et al., 2004; Mahmood et al., 2005; Lapi et

al., 2006). Although such a cut has a minor effect on the predictions at z > 2, due

to the scarce number density of very massive, cluster-size haloes at high redshifts,

it does have a significant impact on the AGN luminosity function at lower redshifts,

as discussed above. However, cutting out the mergers of very massive haloes as

215

triggers of quasar activity is a reasonable approximation. In fact, as discussed in

detail by, among others, Cavaliere & Vittorini (2000), Menci et al. (2005), and

Vittorini et al. (2005), most of the haloes more massive than Mmax at late times

are massive groups and clusters of galaxies with much longer cooling time-scales

and in which mergers among galaxies are replaced by milder dynamical events such

interactions, rarely capable of triggering very luminous quasars. In other words, the

massive BHs at the center of massive groups and clusters, such as their galaxies,

do not experience many major mergers, especially at late times (e.g., Bouwens et

al., 2009), and thus cannot be accounted for in quasar statistics. The SAMs by

Marulli et al. (2008) and Bonoli et al. (2009) also find, for similar reasons, that

AGN activity is negligible in haloes more massive than ∼ (1 − 2) × 1013M⊙. The

large hydro-simulations of Colberg & di Matteo (2008), which also include recipes

for the build-up of BHs, also show a similar drop in the star formation rate and BH

accretion rate above Mmax, at least at z & 3.

However, simply cutting out haloes more massive than Mmax from quasar num-

ber counts is not enough. As clearly re-discussed recently by Shen (2009b), allowing

for quasars to evolve with the same constant light curve at all times and luminosities

still produces too many bright sources at z . 2 (despite the upper cut at a fixed

Mmax). A further, progressive inhibition of BH gas accretion at late times still needs

to be inserted into the models to make them successful. Mahmood et al. (2005)

and Shen (2009b) mimicked this behavior by inserting a redshift dependence Mmax,

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and/or a decreased BH growth efficiency at late times. While this behavior in the

BH accretion mode might at least in part be true, here we have shown that one

may alternatively allow for mass-dependent light curves (with no redshift variation

in halo hosts or BH efficiency), and still reproduce the observables. Moreover, the

latter choice can be physically coonnected with the notion of galaxy downsizing, as

discussed above, in the hypothesis that AGN and star formation in the galaxy are

coupled at some level and proceed on similar timescales.

The faint end of the AGN luminosity function at low redshifts seems to be better

fitted by models that allow for a significantly long post-peak phase, as discussed in

§ 5.6. These conclusions agree with the results of more advanced SAMs, such as

those by Marulli et al. (2009) and Bonoli et al. (2009), and also with the statistics of

AGNs as extracted from large hydro-simulations of Di Matteo et al. (2008), Colberg

& di Matteo (2008), and DeGraf et al. (2009). We also note here that the Marulli

et al. (2008), Bonoli et al. (2009), and Di Matteo et al. (2008) models have similar

mass-dependent light curves to the ones used here, with more massive BHs shutting

down faster than low mass ones. In particular, Marulli et al. (2008) and Bonoli et

al. (2009) showed that, in agreement with our findings, a significant post-peak phase

is required to fit the faint-end of the AGN luminosity function at z . 0.5, although

it is not necessary, and possibly disfavored, at higher redshifts (z & 2).

Moreover, we also find evidence from the z < 0.3 SDSS AGN clustering, that

faint sources predominantly live in haloes more massive than 5× 1011M⊙h−1. This

217

information breaks the degeneracies discussed by several groups (e.g., Vittorini et

al., 2005; Marulli et al., 2008; Bonoli et al., 2009), favoring models where the faint

end is dominated by the post-peak phases of BHs shining in haloes more massive

than 5× 1011M⊙h−1(in this context, see also Lidz et al., 2006).13

A key difference, however, is that the model in our work consistent with the

SDSS z < 0.3 clustering predicts a local BH mass function flatter than the one by

Marulli et al. (2008) and Bonoli et al. (2009). The discrepancy can be ascribed by

the fact that still a large fraction of sources in their models still lives in low mass

haloes (see, e.g., Figure 14 of Bonoli et al., 2009). A more precise measurement

of the local BH mass function and AGN duty cycles as a function of redshift and

luminosity will further break these degeneracies.

The fact that the SDSS clustering measurements point to high mass hosts can

favor AGN models which grow their BHs through in situ processes, such as those

by Lagos et al. (2008), where most of the BH growth is triggered by disk instability,

especially at late times. The models put forward by Granato et al. (2004), Cat-

taneo et al. (2005) and Fontanot et al. (2006) link the star formation rate to the

accretion rate on to the central BH, all of them finding good agreement with the

AGN luminosity function at different epochs. In particular, Fontanot et al. (2006)

push their model to very low redshifts, predicting an Eddington ratio distribution

13We cannot directly compare with the results from hydro-simulations here as no clustering

information has still been discussed by Di Matteo et al. (2008) and Colberg & di Matteo (2008)

at z < 1.

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peaked at very low values, in agreement with observational data (e.g., Ballo et. al.,

2004; Kauffmann & Heckman, 2009). Also, they predict a flatter BH mass function,

consistent with our model where AGNs are preferentially hosted by massive haloes

(long-dashed line in Figure 5.14), although a closer comparison cannot be made

given that no clustering predictions have been discussed by these groups.

Particularly interesting is the Malbon et al. (2007) model that predicts, at vari-

ance with several other SAMs, a late evolution for massive BHs governed by BH

coalescence with negligible gas accretion. They predict all local BHs to be hosted

by haloes more massive than 1011M⊙h−1, only marginally consistent with our SDSS

data, and with a cumulative BH number density higher than the models discussed

above. Their match to the optical luminosity function is good, at least at z . 2,

although a comparison with the full bolometric AGN luminosity function prevents

us to further test and compare with their model.

We showed that the z & 3 SDSS clustering measurements of luminous quasars

allows us to set stringent constraints on the delay time (< 5× 108 yr) between the

triggering and shining of the quasar, mostly due to the rapid decline in time of

the clustering strength of the high-σ peaks hosting quasars. Such findings are in

line with several other groups. Marulli et al. (2008) and Bonoli et al. (2009) use

light curves with a delay time < 1 Gyr. (see also Li et al., 2007; Sijacki et al.,

2009, for comparable time-scales from simulations), and similarly Lapi et al. (2006)

adopt super-Eddington accretion at z & 3 to enhance the initial growth of the most

219

massive BHs. Our values of the median bias and scatter in the Lpeak-M relation

are also consistent with these studies.

Moreover, all these studies agree in showing that a negligible post-peak phase is

better suited to reproduce the faint-end of the high-z AGN luminosity function and,

in the case of Lapi et al. (2006), to reproduce the fraction of obscured sources as

a function of luminosity. We also showed that the exponential drop in the number

density of deep X-ray counts further supports the need for a flatter faint-end slope

of the AGN z > 3 luminosity function, thus further limiting the actual need for

a long post-peak phase. Integrating over the pre-peak obscured phases of quasars,

Granato et al. (2006) also showed consistency with the cumulative SCUBA number

counts at 850µm.

A noticeable difference between the model described here and the one discussed

by Marulli et al. (2008) and Bonoli et al. (2009) is the fact that the latter falls short

in reproducing the bright end of the z & 2 AGN luminosity function. We believe one

of the main reasons for this discrepancy lies on our assumption to neglect any delay

time for the progenitors galaxies to actually merge. Such an assumption naturally

enhances the merger triggering rates of quasars at z & 2 while it has a much more

negligible impact at lower redshifts (e.g., Shen 2009), irrespective of the differences

in dynamical friction delay times (Chandrasekhar, 1943; Zentner & Bullock, 2003;

Taylor & Babul, 2004; Zentner et al., 2005; Boylan-Kolchin et al., 2008; Hopkins et

al., 2008b; Wetzel et al., 2009). As discussed in § 5.2 and § 5.7.2, neglecting any

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delay is a good approximation given that the central BH can be triggered at any

time during the interaction, as suggested by numerical experiments (e.g., Hopkins

et al., 2006a) and observational evidence (e.g, Ballo et. al., 2004; Guainazzi et. al.,

2005).

Overall, we believe that the basic approach undertaken here has revealed several

interesting, and quite general, aspects of BH / AGN evolution that should be taken

seriously into account by more advanced SAMs that attempt to connect BH and

galaxy evolution.

5.7.4 Assumptions, input parameters, and degeneracies

In line with the conclusions of the previous paragraph, we stress here that the aim

of this paper is not to provide the reader with a specific model for BH evolution, but

rather to show how the simultaneous match to a number of independently derived

observables is effective in constraining some of the main ‘characteristics’ that any

model must satisfy. In other words, the model presented here is by no means unique,

in fact several of the parameters are degenerate, and this is the main reason why we

do not present a full χ2 fitting. However, our study shows that some basic ‘trends’

must be satisfied for a basic AGN evolution model to actually work. In this section

we attempt to discuss these issues in some detail.

First of all, we stress that this work is based on a set of very general and widely

accepted model assumptions that cannot be considered as true free parameters. Our

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first assumption is that quasar activity is triggered in major merger events (with

ξmin & 0.25 − 0.3), close to the epoch of virialization of the halo and build up of

the central potential. The number density of these events as a function of mass and

redshift, is well described by the analytical formalism presented in § 5.2.1, which

was tuned to fit the results of large, high-resolution numerical simulations.

The redshift and halo dependence of the Lpeak-M relation have also been fixed

to the values predicted by theory (see § 5.2), and the normalization chosen in way to

match the z = 0 MBH-σstar relation (adopting the observed σstar-Vvir relation). The

model described here therefore, by definition, satisfies the local relations between

BH and its host masses.

Our prediction of the delay time between triggering and quasar shining is not

very well constrained by the AGN luminosity function, at least at z ≤ 2. There-

fore, also the associated parameters are not very well defined, such as the initial

Eddington ratio λ0 which can take any value in the range ∼ 0.1 − 10, or the ra-

diative efficiency, which is degenerate with λ0. Nevertheless, additional constraints

for these parameters come from other observables. For instance, the strong z > 3

clustering signal forces the model to have short delays (see § 5.6.2), which imply

super-Eddington values for λ0, at least at high-z. From arguments linking the the

integrated BH mass function from luminous quasars events matches the local early-

type BH mass function (e.g., Salucci et al., 1999), we get ǫ . 0.1 (see § 5.6.3),

independent of the systematic uncertainties linked to the AGN luminosity function

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and / or bolometric corrections (e.g., Shankar et al. 2009).

The parameter α0 (in equation 5.15) that controls the overall ‘strength’ of the

light curve descending phase, is by no means unique, and it is in fact degenerate

with the normalization of the Lpeak-M relation. Lower values of α0, such as in Shen

(2009b) and Yu & Lu (2008), imply longer decreasing phases and, overall, more

AGN counts at approximately all luminosities, and larger final BH masses. For

example, setting α0 = 1.5, requires a reduction of ∼ 40% in the normalization of

the Lpeak-M relation to recover the match to the AGN luminosity function, which

anyhow provides a MBH-σstar relation still well within the observational constraints.

However, the major variation in this kind of model is that the final BH mass in-

creases by a factor of ∼ 10, with respect to the its mass at the shining, i.e. a much

higher BH mass at fixed σstar. While this effect might help in reconciling high-z

observations of a higher σstar and a non-negative MBH-σstar redshift evolution (see

§ 5.7.2), it is for sure at variance with the very low z > 3 X-ray number counts

(§ 5.7.1). Also, allowing for too much post-peak BH growth produces far too many

massive BHs in the local universe, in clear contradiction with measurements of the

local BHMF. In conclusion, the multiple match to different data sets favors higher

values of α0.

The mass-dependence of the α parameter presented in Eq. 5.15, is obviously not

unique. For example, lowering (increasing) its mass-dependence can be counteracted

by a decrease (increase) in Mmax. Nevertheless, models characterized by a sharp

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cut-off for BH masses above a few times 108M⊙ are better-suited to reproduce the

bright end of the AGN luminosity if no redshift dependence in Mmax is invoked.

As discussed above, the maximum and minimum masses hosting quasars are

input parameters in our model. A natural upper mass cut-off Mmax is expected

because multiple quasars will be hosted by too massive haloes with longer cooling

timescales (e.g., Cavaliere & Vittorini, 2000). However, the exact value of this

parameter is degenerate, again, with the normalization of the Lpeak-M relation.

For example, increasing (decreasing) Mmax by a factor of ∼ 2, implies decreasing

(increasing) the normalization by∼ 40%, which has a minimal effect on the resulting

MBH-σstar relation or other observables. Similar uncertainties affect the minimum

halo mass Mmin, although, as discussed above, larger variations of this parameter

are ruled at high significance by clustering and number counts measurements.

Finally, Mmax and the normalization of the Lpeak-M relation, are degenerate

with the value of the intrinsic scatter Σ. While Σ = 0.28 dex was chosen to be

in agreement with the average value of the scatters in the local relations, its exact

value does not have much impact in the overall results, provided that (rather small)

appropriate changes in Mmax and/or Lpeak-M normalizations are taken into account.

The parameter Σ has a larger impact on the results at z > 3, where smaller values

of Σ are preferred because the boost the clustering associated to a given luminosity

bin (see § 5.6.2 and references therein), although the main conclusions on delay,

number counts, etc.. discussed above, again, do not depend much on Σ.

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5.7.5 Black holes masses at triggering

The purpose of this section is to discuss, for the first time, a conceptual limitation

present in most theoretical models of quasar activity (ours included). Recall that

in our framework, we assume that circumnuclear material is initially accreted at

a mildly super-Eddington rate by the central BH, which in turn grows from some

triggering mass MBH, tri to some mass MBH, peak during some time tasc. One usually

argues that choosing the BH mass at triggering is not trivial, due to our lack of

knowledge in the early obscured phases. For this reason, we simply assume that

the BH mass at triggering is some fixed fraction µBH of its mass at the peak.

Nevertheless, this issue is not merely an observational limitation, but a theoretical

one as well.

In general, models based on halo merger / formation rates (the so-called ‘analytic

models’) and models involving detailed simulations of individual galactic mergers

(‘hydrodynamic simulations’) often ignore the past history of the black holes prior

to the triggering event in question. An interesting question to pose is, how massive

are the incoming BHs in the merging haloes? In particular, did they follow the

same MBH, peak −M relation? Are the BH masses in the incoming haloes compa-

rable with the initial mass MBH, tri of the final halo? Below we perform a simple

analytic estimate to address this problem. As a first step, recall that picking the

BH mass at triggering to be a fixed fraction of the BH mass at the peak of the light

curve is equivalent to fixing the time tasc spend by the BH in the ascending phase.

225

Conversely, one could pick a value of tasc and infer µBH (and MBH, tri). The result

of this exercise is shown in Figure 5.18.

Each column in Figure 5.18 corresponds to a given host halo mass at triggering

(as labeled, note the different vertical mass scales used in each column) and each

row corresponds to different triggering redshifts. In each panel, the lower dotted

line represents MBH, tri, while the upper dotted line represents MBH, peak. Their mass

ratio is given by MBH, peak/MBH, tri = µBH = exp (tasc/tef), where the e-folding time

is tef = 0.43(f/λ0) Gyr. Notice that our values of MBH, tri are compatible or above

those used in hydrodynamical simulations (e.g, Di Matteo et al., 2005; Robertson

et al., 2006; Hopkins et al., 2008a,b). In the Figure we assume that tasc = 0.11 Gyr

and λ0 = 3, which corresponds to an e-folding time of 15.9 Myr, and a BH mass

ratio of log µBH = 3. This is our original reference model, broadly consistent with

the bias at high redshift (see Figure 5.11).

Consider a merger of two haloes with masses m and m′ respectively (with m′ <

m), producing a halo of mass M = m + m′ at redshift z = Ztri as a result. In our

model, we would invoke the MBH, peak −M relation at Ztri to find MBH, peak. Then,

with our assumed values of tasc and λ0, the black hole mass MBH, tri at triggering

could be inferred (the upper and lower dotted lines in Figure 5.18, respectively).

Estimating the masses of the incoming BHs (at the centers of the merging haloes)

is a more difficult task. First of all, recall that only major mergers (with ξ =

m′/m > 1/4) are considered as triggering agents. Denote the mass of the BH in

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Figure 5.18: Relevant black hole masses involved in a halo merger of haloes with

masses m and m′ vs. the mass ratio of the merging haloes (ξ = m/m′). We consider

final halo masses M = 1012M⊙ (left panel) and 1013M⊙ (right panel). The rows refer

to the merger redshifts z = 0.5, 2, 4, 6 (top-to-bottom). See text for a discussion of

the different line styles.

227

the m-halo as mBH, and that in the m′-halo as m′BH. Without loss of generality,

we will focus our discussion in terms of ‘unprimed’ quantities – similar expressions

hold for ‘primed’ quantities. It is safe to assume that the bulk of their masses was

accreted during their respective ascending phases, especially at these early epochs

(see § 5.7.1). In other words, let us assume that mBH ≃ mBH, peak. To estimate the

masses of the incoming BHs at their peak luminosity, we must know their respective

triggering epochs and the masses of their hosts at those epochs. Once these pieces

of information are established, the mBH −m at ztri can be invoked for each of the

merging haloes. A possible way to do this is to use a merger history tree algorithm

(e.g., Moreno et. al. , 2008). In the present work we will use a simpler, but

still reasonable estimate instead. A more rigorous approach with our merger tree

algorithm will be the subject of future work.

Recall that in Chapter 3, we introduced the term ‘halo formation’ to describe

when a dark matter halo is formed (see also, Lacey & Cole, 1993; Harker et al.,

2006; Sheth & Tormen, 2004). Formally, this is defined as the first time a halo had

at least one half its final mass. In our estimate of the incoming BH masses, we will

assume that each host halo (i.e., with ‘final’ masses m at Ztri, etc.) was triggered

at redshifts ztri = zF. At triggering, we denote the host halo mass by mtri = mF.

One caveat is that for a halo at the final epoch Ztri, one does not expect a single

value of formation redshift or formation mass, but a distribution of these formation

values with a certain width. Given that we here interested in average trends, we

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will simply estimate the formation redshift as zF = zF, where zF is the median

formation redshift. For similar reasons we will also assume that mF = m/2, since

one-half the final mass is the most typical formation mass (Sheth & Tormen, 2004).

Following Giocoli et al. (2007), the median formation redshift (of haloes identified

at final redshift Ztri) is given by

δc(zF) = δc(Ztri) + 1.26√

σ2(m/2)− σ2(m). (5.51)

Here, δc(z) is the overdensity threshold required for spherical collapse at redshift z

and σ(m) is the rms in the initial linear fluctuation field, smoothed with a top-hat

filter of size R = (3m/4πρ)13 (with comoving background density ρ). With this,

we can invoke the mBH −mtri self-regulation condition (Eq. 5.12 with λ0 = 3) with

mtri = mF = m/2 and ztri = zF (and similarly so for the ‘primed’ quantities).

For each halo of mass M and triggering redshift z = Ztri, Figure 5.18 shows

the values of mBH and m′BH versus the mass ratio of their merging hosts (ξ =

m′/m). These are denoted by the short-dash-dotted and long-dash-dotted lines

respectively. Notice that m′BH/mBH approaches unity as ξ → 1, and become greater

than unity for smaller values of ξ. In all panels, the incoming BH masses are

orders of magnitude larger than our BH mass at triggering. One might think that

this is good news, since growing massive BHs takes less time. From the point of

view of clustering, shorter accretion phases implies that the bias (as described by

the Fry formula) does not drop so much, improving the match of our model with

observations. However, this feature is spoiled by a more important issue. Notice

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that in many cases, the incoming BHs are more massive than the MBH, peak (at

z < Ztri) predicted by our self-regulation condition. This is especially true for

massive hosts and recent epochs. In other words, if either of the two incoming BHs

gets triggered, then the correspoding AGN would extinguish itself immediately,

completely inhibiting the very existence of quasars, which is obviously contrary to

observations.

Several models of BH growth assume that the two incoming BHs coalesce imme-

diately after the merger of their hosts (Kauffmann & Haehnelt, 2000; Menou et al.,

2001; Erickcek et al., 2006; Malbon et al., 2007). In Figure 5.18, the solid line rep-

resents the sum mBH + m′BH. Except for low-mass hosts at high redshifts, the mass

of the BH resulting from the coalescence of the binary is larger than that allowed

by the MBH−M relation. Conserving the total entropy of the system instead (e.g.,

Ciotti & Ostriker, 2001) leads to a final mass given by√

m2BH + m′2

BH < mBH+m′BH,

while the rest leaves the system in the form of gravitational waves. This final mass

is depicted as dashed lines. While this is certainly smaller than the sum of the

incoming BH masses, the effect is small.

From this exercise we conclude that merging models like ours cannot, by con-

struction, be strictly self-consistent. This means that such models require that at

each merging event a “seed” BH must exist at the center of the remnant halo that is

not necessarily related to any previous, self-regulated quasar phase. Indeed, these

findings seem to support the choice of some of the current SAMs that actually take

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the BH seed as an external free parameter not related to the previous merging his-

tory of the haloes (e.g., Granato et al. 2004, Marulli et al. 2008). For these reasons,

we here caution against the use of predicting, at least in merger-driven models of

this kind, the BH mass evolution at each time during the hierarchy, by simple wild

extrapolations of some self-regulated condition. Nevertheless, despite this concep-

tual complication, simple models like ours still provide very powerful descriptions of

quasar activity. Our formalism remains an excellent approximation. This is because

galactic haloes are not expected to experience more than one major merger during

the epochs studied here (e.g., Angulo & White, 2009). In other words, in our regime

of interest, it is for the most part safe to assume that there is only one triggering

event, allowing us to ignore any previous history of the supermassive black hole.

It is true that a halo might had experience a series of major mergers prior to its

final major merger. Nevertheless, this would only occur at extremely high redshifts,

when the first generation of black holes first assembled. This regime is beyond the

scope of models of quasar activation by major mergers like ours.

5.8 Final remarks

In this work we discuss the predictions of a simple model of quasar activation by

major mergers of dark matter ‘galactic’ haloes in the range 1011.5 − 1013M⊙h−1.

The model consists of two main ingredients: (1) the halo merger rate as the main

triggering agent, and (2) the quasar light curve, which describes the evolution of

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individual quasars. After a phase of exponential growth, the BH reaches a peak

luminosity set by a feedback-constrained relation of the type Lpeak ∝ V 5vir, after

which the AGN luminosity progressively decreases. This model has already been

widely used in the literature and, in its essence, it is the core of all SAMs that tried

to explain the evolution of BHs and their hosts. By carefully revisiting some of

the main issues linked to this approach, we were able to derive several interesting

and physically meaningful constraints regarding BH evolution that will be useful

for updating more advanced SAMs.

Our results can be summarized as follows:

• We find that allowing less massive BHs to possess more extended light curves,

i.e., to be active for a longer time after their peak luminosity, can well repro-

duce the bright end of the AGN luminosity function at all epochs, without the

need to invoke a not very well defined declining AGN efficiency at low red-

shifts. In turn, this recipe is in line with models that postulate AGN feedback

to be the ‘clock’ regulating the star-formation timescales in the host galax-

ies. Chemical evolution models of early-type galaxies in fact require rapid

shut-downs after a burst of intense star formation, the duration of which is

inversely proportional to the host potential well and final stellar mass.

• The integrated BH mass function from the major-merger model is consistent

with the local BH mass function, derived from the early-type galaxy pop-

ulation, if the radiative efficiency is ǫ ≤ 0.1. This result is robust against

232

systematics affecting the bolometric corrections or the exact shape of the

quasar luminosity function.

• We measure the average bias of Type II AGNs in SDSS to be, on average, b =

1.282 ± 0.192, independent of luminosity in the range 42.5 ≤ log L/ergs−1 ≤

45.5. Such a high value of the bias implies that faint AGNs at z < 0.3 are

mainly hosted by haloes more massive than M ∼ 1011.5−12M⊙/h, thus favoring

models in which AGN activity in the local universe is triggered by minor,

low-Eddington ratio events, such as galaxy interactions, minor mergers, and

in-situ processes (e.g., bar instability). We find that if AGNs are preferentially

hosted by more massive haloes, then the model predicts at z = 0 much less

numerous low-mass BHs, i.e., a flat BH mass function at the low-mass end,

which needs to be confirmed by independent measurements of the local BH

mass function. As discussed in § 5.7.3, an independent, secure estimate of the

shape of the low-mass end of the local BH mass function will be invaluable to

break some degeneracies now present in current, different SAMs.

• The high clustering signal measured at z > 3 in SDSS forces successful models

to be characterized by rather short delay times of tdelay . 5× 108 yr. between

the triggering and the shining epochs. If the delay time is too long in fact, the

bias characterizing the high-σ peaks of quasar host haloes will rapidly drop as

b ∝ (1 + z)1.5 for the masses of interest here. Such conditions of short delays

are more easily met if the ‘seed’ BH mass is & 105M⊙, and the initial growth

233

is super-Eddington.

• The exponential drop characterizing the number counts of X-ray AGNs – as

measured in recent deep surveys and the observed obscured AGN fraction

decreasing with increasing luminosity – are better reproduced by models with

a minimal post-peak phase, as expected from a more efficient AGN feedback

at high redshifts and a higher minimum halo mass (of & 1012M⊙h−1) hosting

quasars.

• We are the first to notice, intriguingly, that merger-driven models (including

the one presented here) are, by construction, conceptually limited. In fact,

by applying the self-regulation condition to the incoming, merging haloes at

their respective triggering epochs, would force them to host BHs much more

massive than the final BH in the descendant halo. This indicates that the

BH mass at the triggering in the descendant halo must be close to the true

seed BH mass, or at least it should not have experienced any previous major,

self-regulated growth. Scenarios with a single major merger remain a good

approximation because galactic haloes are not expected to experience several

major mergers in the epochs and range of halo masses of interest.

• Our feedback-constrained relation MBH-Vvir, consistent with the clustering and

AGN luminosity function at all redshifts, implies a non-evolving MBH-σstar re-

lation consistent with the local one, assuming σstar and Vvir scale as observed in

234

the local universe. Observations suggest, however, that the host galaxy might

be more compact and with a higher σstar at the moment of the quasar shining,

implying a lower BH mass at fixed velocity dispersion (i.e., a lower normal-

ization in the MBH-σstar relation), if the BH mass is mostly formed during the

quasar phase. On the other hand, however, direct observations and cumulative

arguments support, if anything, a mild positive evolution in the normalization

of the MBH-σstar relation. We speculate that such apparently contradictory

results might be resolved either by a non-trivial, redshift-dependent mapping

between σstar and Vvir, or either by assuming that the BH host had a smaller

bulge stellar mass associated to the BH at the moment of the shining of the

quasar.

• Cross-correlating the feedback-constrained MBH-M relation, with the redshift-

dependent Mstar-M relation obtained from the cumulative number matching

of the stellar and halo mass functions, we find tentative evidence for a factor

of ∼ 2 larger BH-to-stellar mass ratio at high redshifts, which might sup-

port the above argument and also some direct observations and clustering

measurements.

The wealth of observational information now available is capable of constraining

the main effects describing BH growth in AGNs. Simple models with very few

realistic assumptions, like the one presented here, are flexible enough to easily probe

and study the impact of these different effects (e.g., host halo masses, delay times

235

in the early stages, shut-down in the late stages, etc.). Such models have already

been widely used in the literature and, in essence, they are the core of all SAMs

that tried to explain the evolution of BHs and their hosts. By carefully revisiting

some of the main issues linked to this approach, we were able to derive several

interesting and physically meaningful constraints regarding BH evolution. It is our

hope and intention that simple, yet powerful models like ours, will help guide the

next generation of semi-analytic models of galaxy formation.

236

Appendix A

Diffusion theory

A.1 Diffusion theory

We now present the solutions to the constant and square-root barrier problems

from the point of view of the diffusion equation. This Appendix is particularly

technical, and may be skipped. Nevertheless, we point out the most important

formulas. These are equation (A.38) and equation (A.33) for the exact square-root

unconditional and conditional mass functions, respectively.

A.1.1 Constant barriers

Here we present an alternative derivation of the Press-Schechter formula. The

statistics of random walks have been studied in a completely different context.

Imagine an ensemble of weakly-interacting dust particles in one-dimensional random

237

motion. A random walk then can be assigned to each particle. δ can be interpreted

as the distance walked, and S as a time variable. The question is, what is the

distribution of times these particles take before they are deposited on a barrier

at a distance δc. In our cosmological context, the meaning of these variables is

completely different, but the underlying mathematics is the same. The key idea

here is to acknowledge that Q(S, δ) obeys the one-dimensional diffusion equation:

∂Q(S, δ)

∂S=

1

2

∂2Q(S, δ)

∂δ2. (A.1)

In the absence of a barrier, the solution is simply

Q0(S, δ) =1√2πS

exp(

− δ2

2S

)

. (A.2)

(Notice the ‘0’ index denoting the barrier-free situation.)

Including a barrier amounts to imposing the constraint

Q(S, δ = δc) = 0 (A.3)

at the barrier, known as a Dirichlet boundary condition. Such barriers are called

‘absorbing’ barriers (Chandrasekhar, 1943). Notice that the solution is only valid

for −∞ < δ ≤ δc. An additional condition, which allows Q to be normalizable, is

that Q and its first partial derivatives vanish as S →∞ and δ → −∞. The points

along the red walk in Figure A.1 satisfy equation (A.1) with the boundary condition

given in (A.3). To obtain this solution, we must take the set of all walks beneath

the barrier at S ′, and substract off those forbidden ones that have been absorbed

238

Figure A.1: Diffusion with a constant absorbing barrier. The solution Q (in red)

of the diffusion equation (A.1) with boundary condition (A.3) is obtained by sub-

stracting off forbidden walks (black-gray). One can get the latter by exploting the

symmetry Q0(S, δ)→ Q0(S, 2 δc − δ) about the barrier of height δc.

239

already by the barrier (e.g., the black-gray walk). By symmetry, a walk is equally

likely to go above (black) or beneath (gray) the barrier after the crossing point. To

get the latter from the former, one just needs to replace δ → 2 δc − δ. Thus, to get

the set of walks that have never touched the barrier, we must take all and substract

off the forbidden ones. Explicitly,

Q(S, δ) = Q0(S, δ)−Q0(S, 2 δ − δc) (A.4)

=1√2πS

[

exp(

− δ2

2S

)

− exp(

− (2δc − δ)2

2S

)]

.

(Notice the absence of the ‘0’ index on the left, indicating the presence of a barrier).

This is the solution to the diffusion equation in the presence of an absorbing-barrier

boundary condition. See Zentner (2007) for an alternative method of solving the

diffusion equation, which uses a Fourier transform.

The cumulative fraction of walks that have crossed the barrier at some S < S ′

includes all the walks minus those that have never crossed the barrier (the red

walks), and it is given by

F (< S ′|δc) = 1−∫ δc

−∞

dδ Q(S ′, δ), (A.5)

From this, one can compute

f(S|δc)dS =∂

∂S ′F (< S ′|δc)

∣

∣

∣

S′=SdS = −

∫ δc

−∞

dδ∂Q(S ′, δ)

∂S ′

∣

∣

∣

S′=SdS

= −1

2

∫ δc

−∞

dδ∂2Q(S, δ)

∂δ2dS =

√

δ2c

2πSexp

(

− δ2c

2S

)dS

S, (A.6)

which is precisely the extended Press-Schechter result. Notice that in the second

240

line we have used the diffusion equation and the vanishing condition as δ → −∞.

We now show a derivation of the exact solution to the square-root barrier.

A.1.2 Square root barriers

In this section we solve the diffusion equation in the presence of square-root barrier

(Breiman, 1967). In terms of the ensemble of random-motion dust particles, the

distance to the absorbing barrier is increasing with S (interpreted as a time variable)

– that is, the barrier is moving. Another interpretation is that the barrier is fixed,

and the particles ‘drift’ away because they experience a constant force. Such random

walk models with drift have been studied extensively in the literature (see Grimmett

& Stirzaker, 2001, and references therein).

In this presentation we follow Mahmood & Rajesh (2005). We will work out the

conditional solution because, and the unconditional one will be obtained as a special

case. We are interested in the first crossing distribution of a barrier B(Sm, δc(z))

(equation 2.79) by random walks with origin at (SM , B(Sm, δc(Z)). Close inspection

of Figure A.2 shows that this problem is simplifed by a change of coordinates:

s = SM − S,

a = δ − δc(Z)− β√

SM . (A.7)

Compare with the change of variables with constant barrier (equation 2.68). With

this, the above two-barrier problem reduces to the crossing distribution of one com-

241

Figure A.2: Exact square-root barrier solution. The first crossing distribution can

be obtained from the first crossing distribution of a barrier B(s, ac) (see equa-

tion A.8), shown in magenta. For this it is necessary to change coordinates

δ → a = δ − δc(Z)− β√

SM and S → s = S − SM . Compare to Figure 2.11.

242

plicated barrier,

B(s, ac) = ac + β√

s + SM ,

ac ≡ δc(z)− δc(Z)− β√

SM . (A.8)

We suppress the parameter q here for simplicity. The shape of the effective barrier

B depends explicitly on SM . This barrier does not belong to the family of barriers

in equation (2.78), and cannot be written in self-similar fashion (equation 2.81).

For this reason, the moving barrier prediction of the conditional mass function is

not expected to be self-similar.

The diffusion equation in these coordinates is given by

∂Q(s, a)

∂s=

1

2

∂2Q(s, a)

∂a2. (A.9)

In the presence of an absorbing square-root barrier (equation A.8), the Dirichlet

boundary condition at the barrier is

Q(s, a = B(s, ac)) = 0. (A.10)

The solution and its derivatives vanish as s→∞ and a→ −∞. Finally, at s = 0,

it is certain that a = 0. Mathematically,

Q(s = 0, a) = δD(a = 0). (A.11)

Our calculations are facilitated by using the distance to the barrier as one of the

coordinates: a→ ξ = B − a. In this case, the diffusion equation becomes

∂Q(s, ξ)

∂s+

β

2√

s + SM

∂Q(s, ξ)

∂ξ=

1

2

∂2Q(s, ξ)

∂ξ2. (A.12)

243

This is seemingly more complicated than equation (A.9), but the boundary condi-

tions are simplified considerably. Explicitly, these become

Q(s, ξ = 0) = 0 ,

Q(s = 0, ξ) = δD(ξ − B) . (A.13)

We will make another change of variables, which the will allow the solution to

be separable. By setting ξ → y = ξ/√

s + SM , the diffusion equation becomes

∂Q(s, y)

∂s+

1

2

β − y

s + SM

∂Q(s, y)

∂y− 1

2(s + SM)

∂2Q(s, y)

∂y2= 0 . (A.14)

Setting

Q(s, y) = Θ(s)Y (y) , (A.15)

and substituting in equation (A.14) yields

s + SM

Θ(s)

dΘ(s)

ds=

1

2Y (y)

[

(y − β)dY (y)

dy+

d2Y (y)

dy2

]

= −ζ . (A.16)

The left-hand side only depends on s, while the right-hand side depends only on y.

In reality, both sides are just a constant −ζ. Let’s solve for Θ(s) first. The integral

of the left-hand size of (A.16) is:

Θζ(s) =1

(s + SM)ζ, (A.17)

where the constant of integration is zero because Q = 0 as s→∞.

For the y-factor, let x = y − β, which turns the right-hand side of (A.16) into

Y ′′(x) + xY ′(x) + 2µY (x) = 0 . (A.18)

244

Now define

D(x) = e−x2/4Y (x) . (A.19)

Substituting back into (A.18) and defining λ = 2ζ − 1 we are left with

D′′(x) + (λ +1

2− x2

4)D(x) = 0 . (A.20)

This is known as the Webber differential equation, whose solutions are the Parabolic

Cylinder Functions Dλ(x) (Abramowitz & Stegun, 1972). The general solution is a

linear combination

Q(s, x) =∑

λ

AλΘλ(s)Dλ(x)e−x2/4 . (A.21)

where the Θλs can be thought of as eigenfunctions. The set λ of eigenvalues and

the Aλ coefficients will be fixed by the boundary conditions.

In these variables, the first (Dirichlet) boundary condition is

Q(s, x = −β). (A.22)

This fixes the eigenvalues, which are defined by this condition:

Dλ(−β) = 0 . (A.23)

The second boundary condition (at the origin) is

Q(s = 0, x) =δD(x = ηβ)√

s + SM

, and ηβ ≡ac√SM

. (A.24)

245

Here we have used the fact that δD(κx) = δD(x)/|κ|. With equation (A.23), this

becomes

∑

λ

AλΘλ(0)Dλ(x)e−x2/4√

SM = δD(x = ηβ) . (A.25)

In order to solve for the Aλ coefficients, we use the fact that the Parabolic cylinder

functions are orthogonal:

∫ +∞

−β

Dλ(x)Dλ′(x)dx ≡ δλλ′Iλ(−β) , (A.26)

where δλλ′ is the Kronecker delta. Multiplying equation (A.25) by Dλ′(x)ex2/2 and

integrating, we get

Aλ =S

λ/2M eη2

β/4Dλ(ηβ)

Iλ(−β)(A.27)

and the solution to the diffusion equation is

Q(s, η) =∑

λ

( SM

s + SM

)λ2 eη2

β/4e−(ηβ−η)2/4Dλ(ηβ)Dλ(ηβ − η)

Iλ(−β)√

s + SM

, (A.28)

where we have set

x = ηβ − η, and η ≡ a√s + SM

. (A.29)

The first crossing distribution is

f(s|ac)ds = − ∂

∂s′

∫ B(s,ac)

−∞

daQ(s′, a)∣

∣

∣

s′=sds = −1

2

∂Q(s, a)

∂a

∣

∣

∣

B(s,ac)

−∞

= − 1

2√

s + SM

∂Q(s, η)

∂η

∣

∣

∣

ηβ+β

−∞=(ηβ − η

2+

D′λ(ηβ − η)

Dλ(−β)

) Q(s, η)

2√

s + SM

∣

∣

∣

ηβ+β

−∞. (A.30)

Using the fact that Q and its derivatives vanish as η → −∞, we obtain

f(s|ac)ds =∑

λ

( SM

s + SM

)λ2eη2

β/4Dλ(ηβ)lλ(−β)

ds

s + Sm

, (A.31)

246

where

lλ(−β) ≡ e−β2/4

2

D′λ(−β)

Iλ(−β). (A.32)

This is the exact first-crossing distribution to the square-root barrier we were after.

The conditional crossing distribution is

f(Sm|δc(z), SM , δc(Z))dSm =∑

λ

(SM

Sm

)λ2eη2

β/4Dλ(ηβ)lλ(−β)

dSm

Sm

, (A.33)

where

ηβ =δc(z)− δc(Z)√

SM

− β. (A.34)

For the unconditional case, we need to set (SM , δc(Z)) = (0, 0). That is, we must

take the limit as ηβ →∞. We can use the fact that as x→∞,

Dλ(x)→ xλe−x2/4 . (A.35)

With this

(SM

Sm

)λ2eη2

β/4Dλ(ηβ)→

(SM

Sm

)λ2ηλ

β =(δ2

c (z)

Sm

)λ2, (A.36)

and the unconditional mass function becomes

f(Sm|δc(z))dSm =∑

λ

(δ2c (z)

Sm

)λ2lλ(−β)

dSm

Sm

. (A.37)

As expected, this solution is self-similar

f(ν)dν =∑

λ

νλ2 lλ(−β)

dν

ν. (A.38)

Our square-root barrier limit of ellipsoidal collapse allows us to compare against

a exact solution. Obtaining exact solutions for the crossing distribution of a bar-

rier of general shape remains an open problem (but see Zhang & Hui, 2006, for a

numerical solution).

247

Appendix B

Merger tree algorithm

B.1 The algorithm

The aim of this section is to discuss the implementation of the merger history tree

algorithm (presented in Chapter 3) in more detail. Consider a dark matter halo

of mass M0 at redshift Z0 (Figure B.1). Realizations of its merger history may be

constructed with our tree algorithm. Below we explain how each branch is built

and how these branches are connected.

B.1.1 The branch: from random walks to mass histories

The mass growth history of a halo is contained in a random walk (Section 3.2). First

we discuss the constant-barrier (spherical collapse) model and then the square-root

barrier (ellipsoidal collapse) case.

248

Figure B.1: A sample merger history tree of a halo with final mass M0 at Z0. The

mass history (black branch) is constructed with a random walk. Mass jumps larger

than mdust are identified, and branches are connected there (medium-gray). This

process is repeated for each branch, and new branches are connected (light-gray)

until the tree is complete.

249

A random walk is essentially a collection of steps and heights: s0, . . . , si, . . .

and h0, . . . , hi, . . . (e.g., the jagged line in Figure 3.2). Consider a horizontal

barrier of height δc(z). This line may intersect the walk at several values of S. The

smallest of such of values, Sm, indicates what mass the halo had at redshift z. As

z increases, so does δc – and m decreases accordingly, as expected.

As we increase the height of the barrier, a subset (S0, H0), . . . , (Sj, Hj), . . .

of the walk is chosen. These points contain the mass history (e.g., the dark-filled

circles in Figure 3.2). Every point in this subset has the following property: they

are higher than all their predecessors. This is illustrated by the point at (Sm, δc(z))

Figure 3.2: it has the maximum height in the range SM ≤ S ≤ Sm. In other words,

for any point (si, hi) on the walk to be selected as part of the history, it must satisfy:

hi > hk, ∀ k < i. (B.1)

Now we discuss what modifications are necessary when the square-root barrier

model of ellipsoidal collapse is used (Figure 3.3). Consider a barrier (equation 2.79)

with δ-intercept√

qδc(z). This curve may intersect the random walk at several

places. We pick the smallest of these to find the mass at that redshift. As z

increases, the δ-intercept increases, but the overall shape of the barrier remains

unchanged. This is how the mass history (S0, H0), . . . , (Sj, Hj), . . . is selected.

250

From this subset we can construct a string √qD0, . . . ,√

qDi, . . . of ‘δ-intercepts’:

Di =1√q(Hi − β

√

Si)1 (B.2)

Every point on the history has the following property: its δ-intercept is higher than

that of its predecessors. In other words, for a point (si, hi) on the walk to belong

to the mass history, condition (B.1) is replaced by

di > dk, ∀ k < i, where di =1√q(hi − β

√si). (B.3)

For a given cosmology and initial power spectrum one may map excursion set vari-

ables into physical ones: (S(m), δc(z)) ↔ (m, z). With this prescription, the mass

accretion history is obtained: (M0, Z0), . . . , (Mj, Zj) . . . . The masses in the his-

tory are such that S(Mj) = Sj and the redshifts are given by δc(Zj) = Dj. When

barriers are constant, Dj = Hj.

B.1.2 The tree: connecting the branches

Consider a halo of mass M0 at redshift Z0. We are interested in constructing its

merger history tree. The first step is to draw a random walk with origin at

s0 = S(M0), h0 = B(S(M0), δc(Z0)), (B.4)

where B(S, δc) is given by equation (2.79). The mass history is then collected:

(M0, Z0), . . . , (Mj, Zj), . . . (see section B.1.1). In Figure B.1, this is represented

by the black branch.

1Note that this mapping is ill-defined when γ > 1/2, because barriers cross in that case.

251

The mass history can also be seen as a series of jumps in mass: (M0 ←

M ′0, Z0), . . . , (Mj ← M ′

j, Zj), . . . , with M ′j = Mj+1. Such jumps are interpreted

as binary mergers: M ′j + (Mj −M ′

j) → Mj. In practice, we only care about the

mass jumps above the branching-mass resolution. Denote this subset of jumps as

. . . , (mJ ← m′J , zJ), . . . , where

mJ −m′J > mdust. (B.5)

These jumps are shown in Figure B.1. The next step is to construct the history of

each halo with mass (mJ −m′J) that falls on to the mass history of the M0-halo.

This is done by generating random walks originating from

s0 = S(mJ −m′J), h0 = B(S(mJ −m′

J), δc(zJ)), (B.6)

Such mass histories correspond to the medium-gray branches in Figure B.1. The

above process is repeated for each of these branches: retrieve their mass history,

identify large enough mass jumps, and attach new branches there (light-gray in

Figure B.1). Eventually, all the new branches will only involve mass jumps that do

not satisfy condition (B.5). At this point the tree is complete.

B.2 The main branch

In a merger history tree, the ‘main’ branch of a halo is obtained by following the

most massive progenitor at each mass split. In this section, we show that the mass

history algorithm described in section B.1.1 can be easily modified to construct

252

Figure B.2: Main branch algorithm. The symbols and styles are the same as in

Figure 3.3. The dotted vertical line denotes S = Sm/2. The main branch is built by

following the most massive piece at each split: Sm′ > Sm/2 indicates that m′ < m/2,

implying that m−m′ > m′, whose mass history we must follow.

253

the main branch. This process requires continuous monitoring of the walk and its

associated history, and is illustrated in Figure C.7 (compare to Figure 3.3).

Recall that mass decreases with increasing S. For the portions in the mass

history consisting of jumps where the mass loss is less than half, the algorithm is

unchanged (e.g., the portion with SM ≤ S ≤ Sm in Figure C.7). Occasionally,

there are jumps where more than half the mass is lost. Such is the case for the

Sm → Sm′ jump illustrated in Figure C.7, with Sm′ > Sm/2 (i.e., m′ < m/2 and

(m − m′) > m/2). To construct the main branch in that situation, one must

simply follows (m−m′), not m′. In other words, instead of continuing the walk at

(Sm′ ,√

qδc(z)+β√

Sm′), one must continue from (Sm−m′ ,√

qδc(z)+β√

Sm−m′) (i.e.,

the dark filled circles in Figure C.7).

254

Appendix C

Coagulation and fragmentation

In Chapter 4, we used information from the Smoluchowski coagulation formalism

to infer the functional form of the halo creation rate (equation 4.34). In this Ap-

pendix, we discuss coagulation theory (the discrete Smoluchowski equation) in more

detail. In particular we address varios kernels, their associated solutions, and their

interpretation in the context of polymers. Coagulation theory assumes that clumps

never fragment, in contrast to what is actually observed in N-body simulations of

dark matter haloes (Tormen, 1998; Fakhouri & Ma, 2008a). Fragmentation can

change the shape of the mass function significantly. In this Appendix, we present a

special case known as ‘reversible’ fragmentation, in which the mass function is left

intact.

255

C.1 Introduction

Whereas coagulation models (Smoluchowski, 1916, 1917; Ziff, 1980) and fragmenta-

tion models (Montroll & Simha, 1940; Ziff & McGrady, 1985) have been reasonably

well-studied, models which combine the two processes are less well-developed.

Barrow & Silk (1981) studied perhaps the simplest merger-fragmentation model:

one in which the merger and fragmentation rates were independent of both time

and of the masses of the objects which were merging and fragmenting. That model

admitted a solution in which the mass spectrum did not evolve, and could therefore

be interpreted as the equilibrium state. They noted that it would be interesting if

equilibrium states also existed when the merger and fragmentation rules were more

complex. Below we show that this is indeed the case.

C.2 The binary merger-fragmentation model

The binary merger-fragmentation model is

dn(m, t)

dt=

1

2

m−1∑

m′=1

K(m′,m−m′)n(m′, t)n(m−m′, t)−∞∑

m′=1

K(m,m′)n(m, t)n(m′, t)

+∞∑

M=m+1

F (M |m)n(M, t)−m−1∑

m′=1

F (m|m′)n(m, t),(C.1)

where K and F denote the coagulation and fragmentation rates which are assumed

to be independent of time t. In what follows, we will derive expressions for n(m, t)

which solve this model for specific choices of K and F . We will be particularly

256

interested in solutions which are independent of time, which we will interpret as the

equilibrium state.

C.2.1 Linear polymers

If, in addition to being independent of time, K and F are both also independent of

m, then

dn(m, t)

dt=

K0

2

m−1∑

m′=1

n(m′, t)n(m−m′, t)−K0n(m, t)∞∑

m′=1

n(m′, t)

−F0(m− 1)n(m, t) + 2F0

∞∑

M=m+1

n(M, t), (C.2)

where we have used K0 and F0 to denote the constant coagulation and fragmentation

rates, respectively. The coagulation terms are the same as those which describe the

growth of linear polymers. The fragmentation terms can be understood as follows.

In the model, all bonds in the system are equally likely to be broken. Since an

m−mer contains (m − 1) bonds which may be broken, the rate at which m−mers

fragment is the fragmentation rate, times the number of bonds associated with

an m−mer, times the number density of m−mers: F0 (m − 1) n(m, t). Each such

fragmentation decreases the number of m−mers. The number of m−mers increases

if an M−mer with M > m fragments into an m−mer and an (M − m)−mer.

For linear polymers, of the (M − 1) bonds in a massive M−mer, only a fraction

p = 2/(M − 1) will lead to the creation of an m−mer (recall m < M). The rate at

257

which this happens is F (M |m) = F0 (M − 1)p = 2F0. If

n(m, t) = (1− b)2 bm−1 for some b(t), (C.3)

then (C.2) requires that

dn(m, t)

dt= n(m, t)

(

m−1b− 2

1−b

)

dbdt

= n(m, t)(

m−1b− 2

1−b

)

[

K0(1−b)2

2− F0b

]

, (C.4)

where the first equality comes from evaluating dn/db and the second from inserting

(C.3) in the right hand side of (C.2). Thus, the term in square brackets shows what

is required of db/dt if (C.3) is to solve (C.2). Notice that this solution corresponds

to monomer initial conditions. We must point out that in this Appendix, the term

‘initial conditions’ refers to the size-distribution of clumps, not the distribution of

particles (the power spectrum). In the continuum limit at late times, the solution

has no memory of the initial mass function. Mergers erase this information, making

the mass spectrum approach a universal shape.

Before solving for b(t), notice that when

F0

K0

=(1− b)2

2b, (C.5)

then dn/dt = 0 for all m. This shows that the merger-fragmentation process leads

to an equilibrium mass spectrum given by (C.3), with the parameter b related to

the ratio of the coagulation and fragmentation rates. Notice that an equilibrium

state exists even if the coagulation and fragmentation rates are vastly different! If

258

fragmentation is large, F0/K0 ≫ 1, then b≪ 1 and there are very few objects with

large m (in the limit, almost all polymers are monomers). If mergers dominate,

then b→ 1 and the mass spectrum tends to

(1− b)2

bbm → 2F0

K0

exp(

−m(1− b))

≈ 2F0

K0

exp

(

−m

√

2F0

K0

)

.

Our expression for the equilibrium mass spectrum represents the discrete version of

the calculation presented in Barrow & Silk (1981). As Barrow notes, the interesting

thing about this equilibrium mass spectrum is that an exponential form is also the

solution to the case in which there is no fragmentation. This remains true in the

discrete calculation here.

However, our calculation demonstrates that it is not just the equilibrium mass

spectrum which has the same form as when fragmentation is absent. In particular, in

our merger-fragmentation model, dn/db does not depend on F and K individually,

so the functional form of the mass spectrum is the same as when fragmentation

is absent—the net result of allowing for fragmentation is to rescale the relation

between b and t. For instance, if the initial distribution at t = t0 is given by (C.3)

with b0, then (C.3) with

b(t) = b0 +K0(t− t0)(1− b0)

2

K0(1− b0)(t− t0) + 2

solves (C.2) when F0 = 0.

When both coagulation and fragmentation are present

b(t) =F0

K0

r+ + κ(t)r−1− κ(t)

,

259

where

r± =√

1 + 2K0/F0 ± (1 + K0/F0), κ(t) = κ0 exp[−√

1 + 2K0/F0(t− t0)],

and

κ0 =b0K0/F0 − r+

b0K0/F0 + r−.

Notice that b(t) depends on the ratio K0/F0 and not on K0 or F0 individually. The

pure coagulation solution is recovered in the limit as F0 → 0. The number density

n(m, t) depends on K0/F0 implicitly through b(t). Thus, fragmentation modifies

the rate at which n(m, t) evolves, but not the functional form of the mass spectrum

itself. sequence of states it takes. The system reaches equilibrium when

b→(

1 +F0

K0

)

[

1−√

1− 1

(1 + F0/K0)2

]

; (C.6)

equilibrium is reached earlier as F0/K0 is increased.

We are not aware of any astrophysical applications of this kernel (or any others

besides the additive one discussed below). But, in the presence of a constant kernel,

the growth of clumps is associated with percolation – similar to that behind the

‘friends-of-friends’ algorithm for indentifying haloes in N-body simulations.

C.2.2 Branched polymers

A simple model of the evolution of branched polymers follows from setting K(m,m′) =

K0(m + m′) (see Chapter 4). In this case, an m-mer is best visualized not as being

linear, but as being a ‘tree’: a singly connected planar graph containing (m − 1)

260

bonds which connect m vertices with no loops. We must emphasize that on this

context, the word ‘tree’ refers to spatial structure of the clumps, not to their as-

sembly history, as in Chapter 3. Nevertheless, the two are related by following

the creation of the individual bonds as subclumps merge (Sheth & Pitman, 1997).

If fragmentation follows from the same simple process as before, i.e., the deletion

of a randomly chosen bond, then F (M |m) 6= 2F0 as it was for the linear graphs,

since more than two possible choices of the bond to cut may give rise to the pair of

subtrees (m,M −m). Rather, the probability that deletion of a random edge of an

M−tree leads to an m−tree is

pm|M =1

2(M − 1)

(

M

m

)

(m

M

)m−1 (

1− m

M

)M−m−1

(C.7)

where 1 ≤ m ≤M − 1 (see equation 10 in Sheth & Pitman, 1997) – see Figure C.1.

Note that although this fragmentation term is considerably more complicated than

those usually studied in the literature, it is, in fact, the natural choice.

With this choice, F (M |m) = F0 (M − 1) pm|M , so (C.1) becomes

dn(m, t)

dt=

K0m

2

m−1∑

m′=1

n(m′, t)n(m−m′, t)−K0n(m, t)∞∑

m′=1

(m + m′) n(m′, t)

−F0(m− 1)n(m, t) + F0

∞∑

M=m+1

n(M, t) (M − 1)(

pm|M + pM−m|M

)

.(C.8)

A little algebra shows that if

n(m, t) = (1− b)(mb)m−1

m!e−mb, (C.9)

261

Figure C.1: Haloes as branched polymers. An m-clump and and m′-clump are

obtain by deleting a random bond in an (m + m′) clump.

262

then

dn(m, t)

dt= n(m, t)

(

m− 1

b−m− 1

1− b

)

db

dt

db

dt= K0(1− b)− F0b. (C.10)

Note that when

F0

K0

=1− b

b(C.11)

then dn/dt = 0 for all m. If fragmentation dominates, then b → 0, and almost all

clusters are monomers. If mergers dominate, b→ 1, at t =∞.

As in the linear polymer (constant kernel) model, the equilibrium mass spectrum

(Borel, 1942) has the same form as the solution to (C.8) when F0 = 0. The only

difference is that here, if b = b0 at t0, then

b(t)− b0 =(

1− exp[

−K0(t− t0)(1 + F0/K0)])

(

1

1 + F0/K0

− b0

)

. (C.12)

From (C.11), equilibrium is reached when

b→(

1 + F0/K0

)−1

; (C.13)

as before, larger values of F0/K0 lead to earlier equilibrium.

Ziff & McGrady (1985) discuss the pure fragmentation model (K = 0) for linear

polymers with random bonds, so, for completeness, we now discuss the correspond-

ing solution for branched polymers. Let N0(t) ≡∑

n(m, t) denote the total number

density of polymers present at time t. If all polymers initially have the same num-

ber of particles, say M (note that branched polymers are not indistinguishable from

263

each other, as there are MM−2 different tree configurations; in contrast, linear poly-

mers of mass M are indistinguishable), and the system then fragments according

to (C.8) with K0 = 0 and F0 = 1, then the number density of polymers increases

with time: N0(t) = M + (1−M) e−t. In this case, the mass spectrum evolves as

n(m, t|M) = (1−B)

(

M

m

)(

mB

M

)m−1(

1− mB

M

)M−m−1

, (C.14)

where B(t) = exp(−t) and 1 ≤ m ≤M . This can be verified by direct substitution

(tedious), or by noting that this expression for n(m, t) is the same as that in Sheth

(1996) and Sheth & Pitman (1997) for the average number of m−subtrees within

an M−tree, given that a fraction (1 − B), chosen at random from among the

available M−1 branches, has been cut. If there is a distribution of initial polymers,

then the solution is got by simple superposition of the one derived here. If the

initial distribution is given by (C.9) with parameter b0, then this superposition

can be solved analytically: the distribution at some later time is given by (C.9)

with parameter 0 ≤ b1 ≤ b0. To see this, simply set B = b1/b0 and compute

∑

n(M, b0)n(m, t|M).

Barrow & Silk (1981) asked if a coagulation–fragmentation process could give

rise to a mass spectrum of the form n(m) ∝ m−3/2 exp(−m/m∗(t)). If K0 ≫ F0,

then b→ 1−F0/K0, and the mass spectrum in our (C.9) tends to this form. Thus,

our model shows that the answer to his question is ‘yes’, provided coagulation is

proportional to the sum of the masses, and fragmentation is described by (C.7). Al-

though our fragmentation term is considerably more complicated than those usually

264

studied in the literature, we argued that it is, in fact, a natural choice: it corre-

sponds to a model in which a system of branched (rather than linear) polymers

evolves by the addition or deletion of randomly chosen bonds.

C.3 Generalization

Our model of reversible coagulation and fragmentation can be generalized by setting

F (M |m) = F0 (M − 1)n(m, t)n(M −m, t) K(m,M −m)

∑M−1m=1 n(m, t)n(M −m, t) K(m,M −m)/2

and

F (m|m′) = F0 (C.15)

in the binary merger-fragmentation model of (C.1). Notice that the denominator in

F (M |m) is simply the rate of creation of objects of mass M from pure coagulation.

So F (M |m), the rate at which objects of mass M fragment to produce objects of

mass m is the product of a fragmentation rate F0 times the number of bonds (M−1)

times twice the probability that coagulation produced an object with M − 1 bonds

by the merger of objects with mass m and M −m. [Twice, because fragmentation

can produce m from M either by reversing the merger of m with M − m, or of

M −m with m, and, by symmetry F (M |m) = F (M |M −m).]

If N0(t) ≡∑∞

m=1 n(m, t) and N1(t) ≡∑∞

m=1 mn(m, t). then mass conservation

265

means that N1 is constant. This can be checked explicitly, since

dN1(t)

dt=

1

2

∑

m>0

m∑

m′<m

K(m,m′)n(m′, t)n(m−m′, t)

−∑

m>0

m∑

M>0

K(m,M)n(M, t)n(m, t)

+∑

m>0

m∑

M>m

F (M |m)n(M, t)−∑

m>0

m∑

m′<m

F (m|m′)n(m, t)

=1

2

∑

m′>0

∑

m>m′

(m′ + m−m′)K(m,m′)n(m′, t)n(m−m′, t)

−∑

m>0

m∑

M>0

K(m,M)n(M, t)n(m, t)

+∑

M>0

n(M, t)∑

m<M

mF (M |m)−∑

m>0

m(m− 1)F0n(m, t). (C.16)

Notice that the first two terms, which arise from the coagulation part of the process,

cancel each other. The second two terms, arising from the fragmentation process,

also cancel, as can be seen by inserting (C.15) for F (M |m) and using the fact that

M−1∑

m=1

m n(m, t)n(M −m, t) K(m,M −m)

=M−1∑

m=1

(m + M −m)

2n(m, t)n(M −m, t) K(m,M −m)

=M

2

M−1∑

m=1

n(m, t)n(M −m, t) K(m,M −m). (C.17)

Thus, dN1(t)/dt = 0, and it is usual to set this constant to unity: N1 = 1. Similarly,

(C.1) and (C.15) imply that

dN0(t)

dt= −

∞∑

m=1

n(m, t)∞∑

m′=1

n(m′, t)K(m,m′)

2− F0

[

N0(t)−N1

]

. (C.18)

A pure coagulation model does not have the extra term involving F0. Equilibrium

is reached if dN0(t)/dt = 0.

266

Writing out dn(m, t)/dt explicitly yields

dn(m, t)

dt=

1

2

m−1∑

m′=1

K(m′,m−m′)n(m′, t)n(m−m′, t)−∞∑

m′=1

K(m,m′)n(m, t)n(m′, t)

+∞∑

M=m+1

F (M |m)n(M, t)−m−1∑

m′=1

F (m|m′)n(m, t),

=1

2

m−1∑

m′=1

K(m′,m−m′)n(m′, t)n(m−m′, t)−∞∑

m′=1

K(m,m′)n(m, t)n(m′, t)

+2F0

∞∑

M=m+1

(M − 1)n(M, t)n(m, t)n(M −m, t)K(m,M −m)

∑M−1m=1 n(m, t)n(M −m, t)K(m,M −m)

−F0

m−1∑

m′=1

n(m, t),

=1

2

m−1∑

m′=1

K(m′,m−m′)n(m′, t)n(m−m′, t)−∞∑

m′=1

K(m,m′)n(m, t)n(m′, t)

+2F0n(m, t)∞∑

M=m+1

(M − 1)n(M, t)n(M −m, t)K(m,M −m)∑M−1

m=1 n(m, t)n(M −m, t)K(m,M −m)

−F0(m− 1)n(m, t),

=1

2

m−1∑

m′=1

K(m′,m−m′)n(m′, t)n(m−m′, t)−∞∑

m′=1

K(m,m′)n(m, t)n(m′, t)

+2F0n(m, t)∞∑

m′=1

(m′ + m− 1)n(m′ + m, t)n(m′, t)K(m,m′)∑m′+m−1

m′′=1 n(m′′, t)n(m′ + m−m′′, t)K(m′′,m′ + m−m′′)

−F0(m− 1)n(m, t),

=1

2

m−1∑

m′=1

n(m′, t)n(m−m′, t)[

K(m′,m−m′)−F(m′,m−m′, t)]

−n(m, t)∞∑

m′=1

n(m′, t)[

K(m,m′)−F(m,m′, t)]

(C.19)

267

where we have defined

F(m,m′, t) ≡ 2F0 (m′ + m− 1)n(m′ + m, t)K(m,m′)∑m′+m−1

m′′=1 n(m′′, t)n(m′ + m−m′′, t) K(m′′,m′ + m−m′′), (C.20)

and used the fact that

1

2

m−1∑

m′=1

F(m′,m−m′, t) n(m′, t) n(m−m′, t)

= F0 (m− 1) n(m, t)m−1∑

m′=1

n(m′, t)n(m−m′, t) K(m′,m−m′)∑m−1

m′′=1 n(m′′, t)n(m−m′′, t) K(m′′,m−m′′)

= F0 (m− 1) n(m, t). (C.21)

Thus,

dn(m, t)

dt=

1

2

m−1∑

m′=1

n(m′, t)n(m−m′, t)[

K(m′,m−m′)−F(m′,m−m′, t)]

−n(m, t)∞∑

m′=1

n(m′, t)[

K(m,m′)−F(m,m′, t)]

(C.22)

(Note that F is symmetric in m and m′.) This illustrates that the system with co-

agulation and reversible fragmentation has the form of a pure coagulation problem,

but with time-dependent coagulation kernel

K(m,m′, t) ≡ K(m,m′)−F(m,m′, t). (C.23)

Because of this time dependence, it is possible for the system to evolve to an equi-

librium state. This corresponds to the time when K = F(t).

If F(m,m′, t) separates into the product of a mass-dependent part and a time-

dependent part, and the mass-dependent part is similar to the mass-dependence of

K(m,m′), then the mass spectrum associated with the coagulation-fragmentation

268

process is the same as that associated with pure coagulation, except for a rescaling of

the time variable. Equation (C.20) shows that this will happen for all systems where

the creation term associated with coagulation (notice that it is the denominator

in the expression which defines F) can be written as (m + m′ − 1)n(m + m′, t)

times a piece which depends on time but not mass. (This is not an unreasonable

requirement, since this simply says the creation rate can be written as the product

of the number of bonds in a system with M = m + m′ particles, times the number

density of such systems, times a function of time.) When this is true, the ensemble

averaged coagulation-fragmentation system evolves through the same sequence of

states as when fragmentation is absent, only at a different rate. Of course, the

system associated with both coagulation and fragmentation evolves only until the

equilibrium state is reached.

C.3.1 General linear kernel

To illustrate, consider the general linear kernel

K(m,m′) = A + B(m + m′). (C.24)

In the pure coagulation case and monomer initial conditions,

n(m, t) = (1− b) (1− A∗b)m/A∗−m+1 (A∗b)

m−1

m!

Γ(m/A∗ + 1)

Γ(m/A∗ −m + 2)(C.25)

with b =1− exp(−Bt)

1− A∗ exp(−Bt)and A∗ =

A

A + 2B(C.26)

269

(Lu , 1987; Treat, 1990), so

dn(m, t)

dt= n(m, t)

[

m− 1

b− 1

1− b−(

m

A∗

−m + 1

)

A∗

1− A∗b

]

db

dt, (C.27)

where

db

dt= B (1− b)

1− A∗b

1− A∗

. (C.28)

Computing F(m,m′, t) involves finding the creation rate in the pure coagulation

equation (C.27). However, the required sum involves Pochhammer symbol iden-

tities, making the calculation cumbersome. To circumvent this, one can instead

compute dn(m, t)/dt and then subtract-off the destruction rate (which only involves

moments of n(m, t)). This shows that the creation rate is

m−1∑

m′=1

n(m′, t) n(m−m′, t)K(m′,m−m′)

2= (m− 1) n(m, b)

d ln b

dt

∣

∣

∣

F0=0, (C.29)

so

F(m,m′, t) = F0 K(m,m′)

(

d ln b

dt

∣

∣

∣

F0=0

)−1

, (C.30)

and the net coagulation kernel is

K(m,m′, t) = K(m,m′)

[

1− F0

(

d ln b

dt

∣

∣

∣

F0=0

)−1]

. (C.31)

This is equivalent to replacing

db

dt→ db

dt

∣

∣

∣

F0=0− F0b. (C.32)

(e.g. C.22). These four equations hold for all the kernels considered so far; only the

evolution in b(t) differs.

270

In particular, for the general linear kernel, (C.27) still holds in the presence of

fragmentation, but now

db

dt=

(1− b)(1− A∗b)

1− A∗

B − F0b. (C.33)

Solving for b(t) here is analogous to the constant kernel case, so we do not show it

here. The condition for equilibrium is

F0

A + 2B= (1− b)

1− A∗b

2b. (C.34)

It is a simple matter to check that when (A,B) = (1, 0), then equilibrium is given

by (C.5). Likewise, when (A,B) = (0, 1), equilibrium is given by (C.11). The

larger the value of F0, the sooner the equilibrium state is reached. In the limit

when F0/A≫ 1 and F0/B ≫ 1, then b→ 0 and one finds mostly monomers.

Moreover, because the creation rate has the form given by (C.29), the net coag-

ulation rate K(m,m′, t) separates into the product of a mass-dependent part and a

time-dependent part. Therefore, the mass-dependence of K is the same as that of

the original coagulation piece K.

C.3.2 Multiplicative kernel

There is another exactly solvable case for which our merger-fragmentation model

shows similar behavior: the multiplicative kernel

K(m,m′) = K0 mm′. (C.35)

271

In contrast to the generalized linear case (and its two special instances), this model

evolves towards gelation—an infinite-size cluster is created in finite time—if there

is no fragmentation. Prior to gelation, the mass spectrum is

n(m, t) =(2mb)m−1

mm!e−2mb where 0 ≤ b(t) = t/2 < 1. (C.36)

The explicit factor of 2 is chosen so that N0(t) ≡∑

m n(m, t) = 1− b(t), as it is for

the constant, additive and general linear kernels, and the upper limit on b derives

from requiring that N0 be positive. In fact, one could argue that N0(t) is a more

natural choice for the time parameter than is b(t).

In the pure coagulation case, this kernel results in

dn(m, t)

dt= n(m, t)

(

m−1b− 2m

)

dbdt

= n(m, t)(

m−1b− 2

1−b

)

K0

2. (C.37)

Here

b(t) = b0 + (K0/2) (t− t0) (C.38)

and the system evolves towards gelation in finite time:

tgel ≡ t0 + (2/K0) (1− b0). (C.39)

Can fragmentation prevent the onset of gelation? The creation term in this case

has the form given by (C.29), so the three equations which follow it also apply.

Therefore, in the presence of fragmentation, b(t) obeys

db

dt=

C0

2− F0b, (C.40)

272

and the condition for equilibrium is simply

F0

K0

=1

2b. (C.41)

The evolution is given by

b(t) =K0

2F0

−(

K0

2F0

− b0

)

exp[

− F0(t− t0)]

. (C.42)

Notice that equilibrium is only reached if F0 > 2K0; if F0 is smaller, then gelation

will still occur.

As in all the other cases studied in this Appendix, if fragmentation dominates

over coagulation, b→ 0 and one finds mostly monomers. In this case too, because

F(m,m′, t) is the product of a time dependent piece and a mass dependent piece

which scales as K(m,m′), the evolution to equilibrium is through the same sequence

of mass spectra as when fragmentation was absent, only at a different rate.

C.4 r−mer initial conditions

The above examples assumed that the initial conditions were monodisperse. For

arbitrary initial conditions, the solution to the coagulation equation with kernel

K(m,m′) = A + B(m + m′) is

n(m, t) = (1− b) (1− A ∗ b)(1+2Bm/A)

m−1∑

n=0

(A∗b)n

(n + 1)!(2 + 2Bm/A)nc(n+1)

m (C.43)

(Treat, 1990). Here

b =1− exp(−Bµt)

1− A∗ exp(−Bµt), A∗ =

A

A + 2µB, (C.44)

273

µ ≡ N1(0), (a)n ≡ Γ(a + n)/Γ(a) denotes the Pochhammer symbol, and

∞∑

m=1

c(n+1)m sm =

(

∞∑

i=1

n(i, 0)si)n+1

. (C.45)

The case of monomers considered before corresponds to n(m, 0) = δm,1 and

therefore c(n+1)m = δm,n+1. It is straightforward to generalize to r-mer initial condi-

tions:

n(m, 0) = δm,r, where c(n+1)m = δm,r(n+1).

In this case

dn(m, t)

dt= n(m, t)

(

m/r − 1

b− 1

1− b− (

m/r

A∗

−m/r + 1)A∗

1− A∗b

)

db

dt(C.46)

if there is no fragmentation. This differs from the solution (C.25) for monomers

only by the replacements m → m/r and µ = 1 → r, reflecting the fact that, with

pure coagulation, only polymers of size being a multiple of r will ever form. The

evolution of b is given by

db

dt= rB

(1− b)(1− A∗b)

(1− A∗). (C.47)

Our prescription for fragmentation yields

F(m,m′, t) = rF0

(

m + m′ − 1

m + m′ − r

)

K(m,m′)

d ln b/dt|F0=0

; (C.48)

this does not have the same mass dependence as K(m,m′) unless r = 1. Therefore,

in this case, fragmentation not only changes the rate with which the system evolves,

but also the mass spectrum at any give time. This is easily understood: absent

274

fragmentation, the only allowed states were multiples of r−mers; our fragmentation

treats all bonds equally—the primordial bonds between the initial r-mers as well

as those formed by coagulation—so fragmentation results in a system in which

polymers whose sizes are no longer integer multiples of r. Therefore, in the presence

of fragmentation, (C.43) for the mass spectrum no longer applies. The only way

to preserve the same states as in the pure coagulation case is to have a different

fragmentation rule in which the bonds in the initial r−mers where indestructible.

C.5 Final remarks

We have studied a number of systems in which fragmentation is the time-reverse

process of coagulation. The particular examples presented here for which exact

analytic solutions were obtained suggest a general conclusion: When the fragmen-

tation is the time-reverse process of coagulation, then the mass spectrum evolves

to an equilibrium state provided that the pure-coagulation process would not have

formed an infinite cluster. In this case, the shape of the equilibrium spectrum is

described by the same functional form as when fragmentation was absent. More-

over, until it reaches this equilibrium state, the mass spectrum evolves similarly to

how it does when there is no fragmentation; the only difference is that the rate of

evolution is different. The constant, additive and generalized linear polymers are

examples of such systems.

On the other hand, if the pure-coagulation process would have led to forma-

275

tion of an infinite cluster, then an equilibrium state exists only if fragmentation is

sufficiently efficient. The results presented here allow one to estimate this critical

efficiency. They also show that the evolution to equilibrium is through a similar

series of mass spectra as when fragmentation was absent, only the rate of evolution

is slower, and the mass spectrum reaches a steady state before infinite clusters form.

The multiplicative polymer is a particular example of this case, in which F0 > 2K0

results in equilibrium before gelation.

These conclusions depend on the initial spectrum of clusters in the system. In

general, if the initial conditions contain r-mers, rather than monomers, the spectrum

of states is changed in the presence of fragmentation. This is because in pure

coagulation, the sizes of polymers are always multiples of r; this is no longer true if

fragmentation is able to break the primordial bonds between the initial r-mers.

276

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