Andrew Sornborger et al- The Structure of Cosmic String Wakes

8/3/2019 Andrew Sornborger et al- The Structure of Cosmic String Wakes

http://slidepdf.com/reader/full/andrew-sornborger-et-al-the-structure-of-cosmic-string-wakes 1/14

THE ASTROPHYSICAL JOURNAL, 482:22 È 32, 1997 June 101997. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

THE STRUCTURE OF COSMIC STRING WAKES

ANDREW SORNBORGER

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, 3 Silver Street,Cambridge CB3 9EW, England, UK

ROBERT

BRANDENBERGER

Physics Department, Brown University, Providence, RI 02912

AND

BRUCE FRYXELL AND K EVIN OLSON

Institute for Computational Science and Informatics, George Mason University, Fairfax, VA 220301

Received 1996 September 16; accepted 1997 January 2

ABSTRACT

The clustering of baryons and cold dark matter induced by a single moving string is analyzed numeri-cally, making use of the new three-dimensional Eulerian cosmological hydrocode of Sornborger et al.,which uses the piecewise parabolic method to track the baryons and the particle-in-cell method to evolvethe dark matter particles.

A long straight string moving with a speed comparable to c induces a planar overdensity (a ““ wake ÏÏ).Since the initial perturbation is a velocity kick toward the plane behind the string and there is no initial

Newtonian gravitational line source, the baryons are trapped in the center of the wake, leading to anenhanced baryon to dark matter ratio. The cold coherent Ñow leads to very low postshock temperaturesof the baryonic Ñuid.

In contrast, long strings with small-scale structure (which can be described by adding a Newtoniangravitational line source) move slowly and form Ðlamentary objects. The large central pressure due tothe gravitational potential causes the baryons to be expelled from the central regions and leads to arelative deÐcit in the baryon to dark matter ratio. In this case, the velocity of the baryons is larger,leading to high postshock temperatures.

Subject headings: cosmic strings È elementary particles È hydrodynamics Èlarge-scale structure of universe

1. INTRODUCTION

The cosmic string theory has emerged as a promisingmodel to explain the origin of structure in the universe

& Shellard & Kibble(Vilenkin 1994 ; Hindmarsh 1995 ;The primordial power spectrum pre-Brandenberger 1994).

dicted by the model is scale invariant, in reasonable agree-ment with the results from recent observations of large-scalestructure. Other predictions of the model that can be madebased on the linear theory of cosmological perturbations(e.g., the amplitude of temperature anisotropies in thecosmic microwave background on COBE scales) agree withobservations to within the observational and theoreticalerror bars Stebbins, & Bouchet(Bennett, 1992 ;Perivolaropoulos 1993).

However, nonlinear e†ects in the cosmic string model aremore important, and important at earlier times, than is thecase, e.g., in inÑation-based models. This is true because theseed perturbations are nonlinear ab initio and their dis-tribution is non-Gaussian. Hence the study of the theory ismuch more complicated than the study of inÑation-basedmodels, for which the initial perturbations are given by aGaussian random Ðeld with a small initial amplitude.

There are many uncertainties in our present understand-ing of the cosmic string theory of structure formation. Tobegin with, while it is known that the distribution of stringsobeys a scaling solution Vilenkin(Zeldovich 1980 ; 1981a,

for which the statistical properties of the string1981b)network are constant in time when all lengths are scaled tothe Hubble radius, the speciÐc properties of this scaling

1 Postal address: NASA/GSFC, Code 934, Greenbelt, MD 20771.

solution are not known. A further uncertainty concerns theso-called small-scale structure of the strings. Are stringswhich cross a Hubble volume straight or wiggly on smallerscales? The answer to this question inÑuences both themean velocity of the string and its gravitational e†ects onthe surrounding matter. Finally, very little is known aboutthe nonlinear evolution of the string-induced mass pertur-bations. SinceÈat least in a model in which the dark matteris hotÈmost of the matter moves nonlinearly in large-scalestructures, i.e., structures of the scale of the Hubble radius at

the time of equal energy in matter and radiationteq

,Brandenberger, & Stebbins(Perivolaropoulos, 1990 ;

Perivolaropoulos, & StebbinsBrandenberger, 1990 ;it is not possible to compute the pre-Brandenberger 1991),

dicted galaxy and cluster properties and quantify their non-random distribution without an understanding of thenonlinear dynamics.

There are various means of making progress toward abetter understanding of the predictions of the string model.One approach is to start from the output of high-resolutionsimulations of the cosmic string dynamics and use theresulting string networks to provide the initial conditionsfor large-scale N-body simulations (Allen et al. 1993, 1996).One difficulty with this approach is that it is not easy totake compensation correctly into account, i.e., the fact thatat the time of the string-producing phase transition, theenergy density Ñuctuations due to strings are exactly com-pensated by Ñuctuations in the radiation energy density.Possibly more importantly, the present cosmic string simu-lations & Bouchet & Shellard(Bennett 1988 ; Allen 1990 ;

& Turok do not have the resolution toAlbrecht 1989)

22



STRUCTURE OF COSMIC STRING WAKES 23

include all of the small-scale structure generated by the non-trivial evolution of the strings. It may even be that theNambu action on which the evolution of the strings is basedneglects some crucial physics Finally, it is(Carter 1990).difficult to investigate the dependence of the results of theN-body simulations on the uncertainties in the input stringdistribution.

In this work, we follow an alternative approach. We focus

on the evolution of structure induced by a single string. Wethus obtain better resolution for studying the nonlineardynamics on scales where nonlinear e†ects are crucial.More importantly, however, our approach will allow us tosystematically investigate the dependence of the results of the nonlinear evolution on the uncertainties of the inputphysics. We can study the dependence on variables such asthe small-scale structure of strings, the string velocity, andperturbations in the matter surrounding the strings, toname just a few.

The objective of our study is to analyze the clustering of dark matter and baryons induced by cosmic strings. Since,according to recent cosmic string evolution simulations

& Bouchet & Shellard(Bennett 1988 ; Allen 1990 ; Albrecht

& Turok most of the mass is in strings that are long1989)relative to the Hubble radius, here we focus on such strings.An extension of our study to cosmic string loops is nothard. Since we are ultimately interested in comparing thepredictions of the string model to observations from opticaland infrared large-scale redshift surveys of galaxies, it isimportant to track the evolution of dark matter andbaryons separately. Hence, cosmological hydrosimulationsare required. The questions we initially address in this paperare the determination of dark matter and baryon proÐles of cosmic-string wakes and the dependence of the results onthe amount of small-scale structure in the strings. In futurework, we intend to analyze and compare mechanisms thatinduce the fragmentation of wakes into substructures.

Understanding the nonlinear dynamics of dark matterand baryons induced by a single string will allow us, at alater stage, to combine these results with an analytical toymodel of the string scaling solution and thus obtain infor-mation about the correlation of galaxies and galaxy clustersin the string model.

The outline of this paper is as follows: in we give a° 2brief review of the features of the cosmic-string model rele-vant to our study. Next, we discuss the methods used in thiswork: the hydrodynamical equations and the basic numeri-cal techniques and their implementation. In we describe° 4our simulations and present the results. We study bothplanar collapse induced by a rapidly moving string withoutsmall-scale structure and accretion by strings that have a

substantial amount of small-scale structure and hence havea smaller translational velocity. In the Ðnal section we sum-marize the results and discuss future work.

We work in the context of an expanding, spatially ÑatFriedmann-Robertson-Walker universe with scale factora(t) [normalized to at the present time Thea(t

0) \ 1 t

0].

associated redshift is z(t). NewtonÏs constant is denotedby G.

2. COSMIC STRING REVIEW

Currently, two classes of models are receiving specialattention as possible theories for the origin of structure inthe universe. (For a recent comparative review, see

The Ðrst class considers quantumBrandenberger 1995.)

Ñuctuations produced during a hypothetical period of expo-nential expansion (inÑation) in the very early universe andgenerically (but not always) gives rise to a roughly scale-invariant spectrum of primordial adiabatic perturbationswith random phases and a Gaussian probability distribu-tion.

The second class of models considers topological defectsthat form during a phase transition of matter in the very

early universe. The cosmic string theory & Shell-(Vilenkinard & Kibble1994; Hindmarsh 1995; Brandenberger 1994)belongs to this class. Topological defect models also giverise to a roughly scale-invariant spectrum of perturbations.However, these Ñuctuations are not purely adiabatic. Moreimportantly, the models predict highly nonrandom phases.

Cosmic strings arise in a certain class of relativisticquantum Ðeld theories that are believed to describe matterat very high temperatures They are formed(Kibble 1976).during a phase transition from a high-temperature sym-metric state to a low-temperature state in which the sym-metry is broken. The formation mechanism is analogous tothe way in which defects form during crystallization of ametal and in which vortex lines form during a temperature

quench in superÑuids and superconductors.The important point is that if the microscopic theory

admits strings, then a network of string inevitably formsduring the phase transition. Strings can have no ends andare therefore either inÐnitely long or in the form of closedloops. The presence of strings simply reÑects the fact that,by causality, there can be no order over large distancescales. This causality argument in fact implies that, in atheory that admits strings, a network of strings with meanseparation of at most the Hubble radius will be present atall times t later than the phase transition. In particular,strings will be present at the time when matter can startt

eqto accrete onto seed perturbations.

The dynamics of cosmic strings is nontrivial. The action

of a single string & Olesen is the Nambu-(Nielsen 1973)Goto action. The resulting equations lead to relativisticvelocities for string motion induced by bending of thestrings. The Nambu-Goto action does not apply at thecrossing points of two strings. Numerical simulations

Myers, & Rebbi(Shellard 1987; Matzner 1988; Moriarty,& Ruback have shown that strings do1988 ; Shellard 1988)

not pass through each other but intercommute, i.e., breakand exchange ends. In a cosmological setting, this e†ectleads to a mechanism by which the long string networkloses energy to string loops formed when long strings inter-sect, and which, in turn, slowly evaporate via gravitationalradiation.

As can be inferred by analytical arguments (Zeldovich

Vilenkin and as has been veriÐed by1980 ; 1981a, 1981b)detailed numerical simulations & Bouchet(Bennet 1988 ;

& Shellard & Turok the evolu-Allen 1990 ; Albrecht 1989),tion of the string network approaches a scaling solution, adynamical Ðxed point for which the statistical properties of the string network are time-independent when all lengthsare scaled to the Hubble radius. According to the scalingsolution, on average a Ðxed number N of long strings(strings with curvature radius greater than the Hubbleradius) cross any Hubble volume. There is a remnant dis-tribution of loops that were chopped o† from the longstring network at earlier times.

The present simulations indicate that most of the energyof the string distribution is in long strings. However, there



24 SORNBORGER ET AL. Vol. 482

are still substantial uncertainties in the details of the dis-tribution. String intercommutations induce kinks on thelong strings that build up over time to give a substantialamount of small-scale structure. Numerical simulations donot have enough resolution to track the small-scale struc-ture, which leads to an inherent uncertainty in the longstring distribution, since in the numerical analysis one is, inessence, coarse-graining over the small-scale structure. This

uncertainty leads to an even greater uncertainty in the loopdistribution.

The uncertainty in the cosmic string distribution leads touncertainties in the cosmic string structure-formation sce-nario. Long straight strings without small-scale structurehave no Newtonian gravitational potential. If their trans-verse velocity is (Vilenkin they induce av

s1981a, 1981b),

velocity perturbation of magnitude & Vilenkin(Silk 1984)

dv \ 4nGkvsc(v

s) , c(v

s) \ (1 [ v

s2)~1@2 , (2.1)

toward the plane behind the string, where k is the mass perunit length of the string. This gives rise to planar over-densities, the so-called ““ wakes.ÏÏ

Strings with a substantial amount of small-scale structure

have a coarse-grained tension T less than k and hence alsoproduce, in addition to the above velocity perturbation, aNewtonian gravitational potential (see, e.g., &VilenkinShellard 1994)

h00

(r) \ 4G(k[ T ) lnr

r0

, (2.2)

where r is the distance from the string and the stringr0

width. Since T \k leads to subrelativistic velocities for thestring, the accretion pattern of such strings is typically moreÐlamentary than planar.

In principle, the string network will induce inhomoge-neities from the time it is formed. However, for thet\ t

eqperturbations only grow logarithmically. Velocity Ñuctua-tions, in fact, decay. The large-scale structure predicted bythe string model is thus determined by strings present attimes Perturbations due to strings at are thetº t

eq. t

eqmost numerous and have the longest time to grow. Theywill thus determine the scale of the largest structures in theuniverse et al.(Vachaspati 1986; Stebbins 1987).

To study the formation of galaxies and galaxy clusters,one must understand the nonlinear evolution of the wakesand Ðlaments. There has been little work on this subject. Ina pioneering paper, discussed the evolution of Rees (1986)baryons in a cosmic string wake. Hara & Miyoshi (1987;

performed an analytical and numerical analysis of 1990)perfectly planar wake formation, including baryons. Wake

formation has been studied by means of the Zeldovichapproximation for both cold and hot dark matter

et al. et al.(Perivolaropoulos 1990; Brandenberger 1990;This method only keeps track of theBrandenberger 1991).

dark matter. The Zeldovich approximation was applied tothe formation of string Ðlaments in Vollick (1992a, 1992b)for cold and hot dark matter, respectively (see also

& Vilenkin &Vachaspati 1991; Vachaspati 1992; AguirreBrandenberger Lima, & Brandenberger1995 ; Zanchin,1996).

In this paper we present the Ðrst results of a numericalstudy of the nonlinear dynamics of both dark matter andbaryons in cosmic string wakes and Ðlaments. We use arecently developed cosmological hydrocode et(Sornborger

al. that utilizes the piecewise parabolic method (PPM)1996)to solve the hydrodynamical equations and a PIC (particle-in-cell) technique to follow the dark matter particles. One of our major goals is to determine how signiÐcantly the inter-nal structure of a wake or Ðlament depends on the amountof small-scale structure in the string.

In order to be able to compare our simulations to someanalytical results, we brieÑy review the analysis of wake

formation using the Zeldovich approximation (ZeldovichQuantities such as the overall thickness of the dark1970).

matter wake should be reproduced with reasonable accu-racy using this approximation.

The Zeldovich approximation is based on writing thephysical height h of a dark matter particle above the wakein terms of a comoving perturbation t :

h(q, t) \ a(t)[q [t(q, t)] , (2.3)

with q the initial comoving coordinate. The basic equationsused in deriving the equation of motion for t are the New-tonian gravitational force equation

h \ [L

Lh

' , (2.4)

the Poisson equation for the gravitational potential '

L2

Lh2'\ 4nGo , (2.5)

and the mass conservation equation

o(h, t)d 3h \ a3(t)o0

(t)d 3q , (2.6)

which relates the physical energy density o(h, t) to the back-ground density t). After linearizing in t, the resultingo

0(h,

equation for t becomes in the absence of any gravitationalline source on the string, et al.(Perivolaropoulos 1990;

et al.Brandenberger 1990 ; Brandenberger 1991),

t ] 2 a 5at 5 ] 3 a

at\ 0 . (2.7)

The initial conditions at the time when the string isti

passing by are given by the velocity perturbation of i.e.,equation (2.1),

t(q, ti) \ 0 , t 5 (q, t

i) \ a(t

i)~14nGkv

sc(v

s) . (2.8)

The solution to describes how the darkequation (2.7)matter particles, which are initially moving away from thewake with the Hubble Ñow, are eventually gravitationallybound to the wake. The Zeldovich approximation gives agood description of the particle motion until the point of ““turnaround,ÏÏ when It follows fromh 5 \ 0. equation (2.3)that at the time of turnaround The value of q for

t\ 1

2q.

which particles turn around at time t for perturbationsestablished at a time is denoted by t). The analysest

iq

nl(ti,

of et al. et al.Perivolaropoulos (1990), Brandenbergerand show that(1990), Brandenberger (1991)

qnl

(ti, t) \

24n5

Gkvsc(v

s)z(t

i)1@2z(t)~1t

0. (2.9)

The corresponding physical height above the center of thewake is

hnl

(ti, t) \

1

2q

nl(ti, t)z(t)~1 \

12n5

Gkvsc(v

s)z(t

i)1@2z(t)~2t

0.

(2.10)



No. 1, 1997 STRUCTURE OF COSMIC STRING WAKES 25

Once dark matter particles turn around, they virialize (see,e.g., at a distance of about t) andPeebles 1980) 1

2h

nl(t

i,

remain at that height, they have decoupled from the Hubbleexpansion.

For a simulation that starts with a velocity perturbationinduced by a string at time and ends at time t, we expectt

ithe width of the dark matter distribution to be consistentwith the value given above.

3. METHODS

In order to study the nonlinear evolution of baryons anddark matter in cosmic-string È induced wakes and Ðlaments,we have performed numerical simulations making use of anew Eulerian PPM/PIC code for cosmological hydrody-

namics et al. The construction and(Sornborger 1996).testing of the code are described in detail in etSornborgeral. Here we give a brief summary of the techniques(1996).used.

The code follows the evolution of baryons and noninter-acting dark matter in an expanding universe. The e†ects of strings are put in as external velocity or gravitational poten-tial Ñuctuations. The code does not take radiation intoaccount and hence cannot be used to evolve the baryonsbefore the time of recombination. Cooling of baryons ist

rec,

also not included in this code. The baryonic Ñuid is treatedas a single Ñuid of hydrogen. This approximation is a rea-sonable one if we are interested in understanding shockbehavior and the initial stages of nonlinear evolution.However, it does not allow us to study the evolution andhydrodynamically induced fragmentation of high-densitypeaks.

There are two classes of simulation methods used tostudy cosmological hydrodynamics numerically. The Ðrstapproach is grid-based (Eulerian). The hydrodynamicalequations are discretized and solved on a Ðxed comovinggrid. The second approach is smoothed particle hydrody-

namics (SPH), in which the Ñuid is treated as a set of par-ticles statistically representing the Ñuid with interactionsthat follow from the hydrodynamic equations. The Eulerianapproach has the advantage of good shock resolution,whereas the advantage of SPH is its better resolution of high-density regions. We chose the grid-based Eulerianapproach, since we expect shocks to be very important forcosmic-string È induced structure formation and since weare, at the moment not very interested in resolving thehighest density peaks.

3.1. Equations

The baryonic Ñuid equations are obtained by setting the

covariant divergence of the energy-momentum tensor equalto zero. In the limit of nonrelativistic velocities and pressuremuch less than the rest-mass density of the Ñuid, the equa-tions in comoving coordinates are

o 5 ]$ Æ (o¿) \ 0 , (3.1)

(ovi)~ ]+ Æ (ov

i¿ ] e

i p) \ [2

a 5aov

i[

oa3

Li/ , (3.2)

(oE)~ ]$ Æ (oE ] p)¿ \ [4a 5aoE [

oa3

vkLk/ . (3.3)

These are the equations that describe an(Peebles 1980)inviscid Ñuid with no shear or stress terms. In the above, o is

the matter density, p is the pressure, E is the total energywhich can be written as

E \ 12

v2 ] u , (3.4)

with u denoting the internal energy; v is the comoving pecu-liar velocity, and / is the gravitational potential. The vari-ables have been chosen such that the di†erential operatorson the left-hand side of equations and have(3.1), (3.2), (3.3)

the same form as in the Euler equations in nonexpandingspace (see, e.g., The variables are related toPeebles 1980).physical variables (subscripted by p) via

o\ a3op

, p \ app

, a2u \ up

, a2T \ T p

,

a2E \ Ep

, /\ a/p

]a2

2a x

p. (3.5)

We also assume that the Ñuid is adiabatic,

p \ coT , (3.6)

where c is the gas constant, and obeys an ideal gas equationof state

ou(c[ 1) \ p . (3.7)

Noninteracting dark matter constitutes the second com-ponent of our system. Although present analyses indicatethat the cosmic string model is in better agreement with theobserved power spectrum of structures in the universe if thedark matter is hot, here we, for simplicity, consider colddark matter, collisionless particles with negligible thermalvelocity In this case, the equation of motion fordispersion.2dark matter particles is the collisionless Boltzmann equa-tion (Vlasov equation). For particles in Lagrangian coordi-nates, the equation is

¿ 0 ] 2a 5

a

¿ \1

a3+/ ,

x 5 \ ¿ . (3.8)

The only forces are due to gradients in the gravitationalpotential.

In the Newtonian approximation to the Einstein equa-tions, the gravitational potential / is determined via thePoisson equation from the matter density o of both bary-

onic and dark matter:

+2/\ 4nG(o[o 6) , (3.9)

where is the spatial average of o.o 6The e†ects of strings enter as initial conditions in our

analysis. At the beginning of the simulation, we take the

distribution of baryons and dark matter to be unperturbed.The baryonic matter density is taken to be 5% of the totalmatter density. In order to justify neglecting the radiationpressure, we take our initial time to be after i.e., at at

rec,

redshift The expansion rate of the universe canzin\ 1200.

then be taken to be

a(t) \A t

t0

B2@3. (3.10)

2 Since we will primarily study wakes which are formed at late times,hot dark matter will already be cold at the relevant times, and hence willbehave as in our treatment.



26 SORNBORGER ET AL. Vol. 482

The strings produce velocity and gravitational potentialÑuctuations. We consider a straight string at z \ 0 withtangent vector moving with velocity in the y direction.e

xvs

Such a string will produce a velocity perturbation of magni-tude given by toward the x-y plane. Theequation (2.1)velocity perturbation for a Ðxed value of y sets in once thestring passes that value of y and then propagates with thespeed of light in the z-direction.

For strings without small-scale structure, there is noinitial gravitational potential perturbation. The e†ects of strings with small-scale structure can be modeled by givingthe string an e†ective Newtonian mass per unit length k

effof magnitude

keff

\k[ T (3.11)

(see which, via the Poisson equation, induces aeq. [2.2]),Newtonian gravitational potential Ñuctuation.

3.2. Numerical Techniques and Implementation

The key issue in selecting a numerical technique forsolving the hydrodynamical equations is accurateresolution of nonlinear e†ects. Of particular importance isgood shock resolution, since shocks are expected to play acrucial role in cosmological Ñows, in particular in the caseof cosmic string wakes.

PPM & Woodward is a technique that has(Colella 1984)been well tested as an accurate method for treating hydro-dynamical Ñows with discontinuities. Hence, we havechosen the PPM algorithm to evolve the baryonic Ñuid.Since PPM is grid-based, it is most natural to use a grid-based method to evolve the dark matter distribution. Weuse the PIC method & Eastwood an exten-(Hockney 1988),sively tested scheme that combines particle and gridmethods, to evolve the dark matter in our system. (Thismethod is more commonly referred to as the particle-mesh

[PM] method, but we use the acronym PIC to avoid confu-sion of PM with PPM.) We have combined the PIC andPPM codes to form a cosmological hydrocode that simulta-neously evolves both the collisionless dark matter and thecollisional baryons to simulate our two-Ñuid system.

PPM is a higher order Godunov method for integratingpartial di†erential equations. Our code is based on a codeoriginally developed for nonlinear astrophysical problems

Mueller, & Arnett(Fryxell, 1991).The Godunov method is a Ðnite-volume scheme. The

equations of motion are considered in their integral form.Thus, the problem of calculating spatial gradient termsbecomes a problem of determining Ñuxes. The advantage of this procedure is that mass, energy, and momentum are

exactly conserved in the absence of source terms, e.g., theexpansion of the universe.The simulation volume is divided into a set of cells; for

each cell the code keeps track of the average values of theÑuid variables. To Ðnd an approximate solution to the inte-grated Euler equations, one needs to determine the Ñuxesacross the cell boundaries. For instance, the integrated con-tinuity equation

P d 3x L

to]

P d 3x$ Æ (o¿) \ 0 , (3.12)

(integration over one cell) becomes

Lto 6 ] ;

sides

o¿ Æ S \ 0 , (3.13)

where is the total density in the cell and S is the vectoro 6normal to the side of the cell. Obviously, a prescription isneeded in order to compute the Ñuxes knowing only theaverage values of the Ñuid variables in the cells.

In the Godunov method, the Ñuxes are computed in twosteps. First, proÐles of the Ñuid variables in each cell areconstructed based on the average values in the cell and itsneighbors. In the second step, the Riemann shock tube

problem is solved at the cell interfaces, giving a set of non-linear discontinuities in the Ñuid variables propagatingaway from the interface with characteristic velocities. Thesepropagating discontinuities give the Ñuxes from and to eachcell, which are used in the Ðnal step of the algorithm toupdate the Ñuid variables. For a discussion of the Riemannshock tube problem (which is in essence a way of solving theintegrated form of the Euler equations across adiscontinuity) see et al. andSornborger (1996) Hirsch(1988).

The advantage of the Godunov method is that the non-linearities in the evolution equations are incorporateddirectly in the di†erencing scheme via the solution of theRiemann shock tube problem. Linear schemes for calcu-

lating Ñuxes are unable to simultaneously reproduce boththe width of the discontinuity and the amplitude of thewaves traveling away from it. Linear schemes may also spu-riously allow sound waves to propagate supersonically.Both of these problems are avoided by using the Godunovmethod.

PPM introduces a number of changes to the Godunovmethod & Woodward to achieve higher(Colella 1984)order resolution. Most importantly, instead of using proÐlesof the Ñuid variables that are constant across any cell, inter-polating parabolas are employed. This method gives betterspatial resolution and allows a more precise determinationof the initial data for the Riemann shock tube problemsolving routine. In order to damp spurious oscillations at

shocks, the parabolas are Ñattened out near shocks. As aconsequence, a much smaller artiÐcial numerical viscosity isrequired in order to damp the oscillations.

The initial PPM code et al. was written for(Fryxell, 1991)hydrodynamics in nonexpanding space. The inclusion of dynamical gravity necessitates two main changes. First, thegravitational potential is introduced. It is computed at eachtime step and at each grid point by solving the Poisson

(with o being the sum of baryon and darkequation (3.9)matter mass density) by means of a standard FFT scheme.The second major change is, as can be seen from equations

and the appearance of additional source terms in(3.2) (3.3),the Ñuid equations. These terms are local and hence do nota†ect the computation of gradients. They are incorporated

into our code using standard operator-splitting methods,while keeping the code accurate to second order.

Since baryonic and dark matter interact with each otheronly gravitationally, it is straightforward to combine thePPM and PIC codes. Some nontrivial issues concerningthis combination of codes are discussed in et al.Sornborger(1996).

The code was implemented on a MasPar MP-2 at theGoddard Space Flight Center. It was tested extensively withand without including the expansion of the universe. Forsmall sinusoidal Ñuctuations, the numerical results weretested against the analytical predictions of linear theory(Jeans test). Over a time interval of 40 Hubble expansiontimes, the relative error etween the numerical and analytical



22 24 26 28 30 32 34 36 38 40 42−5

0

5

10

15

dark matter density

o v e r d e n s i t y

22 24 26 28 30 32 34 36 38 40 42−5

0

5

10

15

20

baryonic matter density


z = 19.68

22 24 26 28 30 32 34 36 38 40 42−5

0

5

10

15

dark matter density


22 24 26 28 30 32 34 36 38 40 42−5

0

5

10

15

20



z = 11.89

22 24 26 28 30 32 34 36 38 40 42−5

0

5

10

15

dark matter density


22 24 26 28 30 32 34 36 38 40 42−5

0

5

10

15

20



z = 9.705

22 24 26 28 30 32 34 36 38 40 42−5

0

5

10

15

dark matter density


22 24 26 28 30 32 34 36 38 40 42−5

0

5

10

15

20



z = 8.545

22 24 26 28 30 32 34 36 38 40 42−5

0

5

10

15

dark matter density


22 24 26 28 30 32 34 36 38 40 42−5

0

5

10

15

20



z = 7.773

22 24 26 28 30 32 34 36 38 40 42−5

0

5

10

15

dark matter density


22 24 26 28 30 32 34 36 38 40 42−5

0

5

10

15

20



z = 7.207

FIG. 1a FIG. 1b

FIG. 1c FIG. 1d

FIG. 1e FIG. 1 f

FIG. 1.È(aÈf ) Evolution in time of the matter overdensity in dark matter and baryons of a planar symmetric cosmic string wake formed at Thezi

\ 100.grid-zone size is 0.0016 Mpc (the Ðgures are still at a relatively early stage in the evolution of the wake, so the wake is still relatively thin in comovingcoordinates).




FIG. 2.ÈPhase space distribution of dark matter particles accreting in awake formed at Position is in cm and velocity in cm s~1.z

i\ 100.

larger than the corresponding factor for the central regionof an individual wake.

We also have detailed information about the temperatureproÐle through a wake. shows the temperature of Figure 5the baryonic gas along a cross section of the wake at di†er-ent times. The temperature peaks are at the locations of thestrong shocks, where the infalling stream of baryons hits thedistribution of baryons that fell in previously and lost theirkinetic energy to shocks. The temperature at the shock posi-tions is determined by the conversion of kinetic to thermalenergy. For an ideal cosmic string wake studied here, thepreshock Ñow of baryons is a cool and coherent Ñow.Hence, the resulting postshock temperature is low:T

s

T s

DmH

(d

v)2DmH

vs

2(4n

Gk

)2D 102 K , (4.2)

where is the mass of a hydrogen atom.mH

Note that for wakes formed earlier than zD 800, theinitial thermal velocities of the baryons dominate over the

FIG. 3.ÈRadius of secondary turnaround (Ðrst caustic) vs. time. Alsoplotted is the theoretical turnaround. Error bars are the width of 1 gridzone. This data is taken from a high-resolution simulation with 2048 gridzones and z

i\ 10000.

FIG. 4.ÈRatio of baryonic to dark matter at zD8 in a planar-symmetric cosmic string wake formed at The grid size is 0.0016z

i\ 100.

Mpc.

string-induced velocity perturbation. This leads to an initialdi†usion of the baryons over a distance larger than thewidth of the dark matter density enhancement (see Fig. 6).

4.2. Moving Newtonian L ine Source

There are two main motivations for studying clusteringof dark and baryonic matter induced by a slow movingstring with a substantial amount of small-scale structure,modeled as a moving Newtonian line source. First, there areindications that over time a substantial amount of small-scale structure builds up on the long string network, andthat hence this situation might well be realized for cosmicstrings resulting from grand uniÐed phase transitions. Asecond motivation for performing these simulations is the

attempt to identify distinctive predictions of the stringmodel in the regime of nonlinear gravitational clustering. Amoving Newtonian line source would give rise to planesand Ðlaments that would have clustering properties similarto those of the objects that emerge in N-body simulations of large-scale structure formation in models based on adia-batic perturbations with a scale-invariant spectrum. It isinteresting to use the same numerical code to analyze andcompare the baryon and dark matter distribution resultingfrom nonlinear clustering for string wakes (no initial New-tonian potential, only deÐcit angle) and for initial pertur-bations with a Newtonian source. In particular, it is of interest to study any relative baryon enhancement in thesemodels.

and (Plates 1 È 3) show results from simu-Figures 7, 8, 9lations of clustering induced by a slow-moving cosmicstring. The string tension was with the non-T \ 0.95k

0, k

0renormalized mass per unit length. The renormalized massper unit length k is related to and T viak

0(Carter 1990;

Vilenkin 1990)

T k\k02 . (4.3)

The initial string velocity was taken to be Withvs

\ 0.0005.thise parameter values, the Newtonian gravity of the stringhas a much larger e†ect than the deÐcit angle. This fact canbe seen by computing the relative velocity U in the z-direction developed by two particles after a string haspassed between them (Vollick &1992a, 1992b; Vachaspati



20 25 30 35 40 450

20

40

60

80

100

120

140

160

180

200

grid number (grid size = 0.0016 Mpc)

b a r y o n

i c t e m p e r a t u r e ( K )

z = 22.05

20 25 30 35 40 450

20

40

60

80

100

120

140

160

180

200


b a r y o n

i c t e m p e r a t u r e ( K )

z = 12.16

20 25 30 35 40 450

20

40

60

80

100

120

140

160

180

200


b a r y o n i c t e m p e r a t u r e ( K )

z = 9.822

20 25 30 35 40 450

20

40

60

80

100

120

140

160

180

200



z = 8.616

20 25 30 35 40 450

20

40

60

80

100

120

140

160

180

200



z = 7.824

20 25 30 35 40 450

20

40

60

80

100

120

140

160

180

200



z = 7.245

FIG. 5a FIG. 5b

FIG. 5c FIG. 5d

FIG. 5e FIG. 5 f

FIG. 5.È(aÈf ) Evolution in time of the temperature of the baryonic matter in a planar-symmetric cosmic string wake formed at zi

\ 100




FIG. 6.ÈDark and baryonic matter distributions for a wake formed atFor wakes formed so early the initial sound speed is high and az

i\ 1200.

weak shock forms, spreading the baryonic overdensity beyond the width of the dark matter. Later, as the gravity of the dark matter begins to domi-

nate, the baryonic matter clumps at the core of the wake.

Vilenkin 1991):

k\ 8nGkvsc(v

s) ]

4nG(k[ T )

vsc(v

s)

, (4.4)

where the Ðrst term is due to the deÐcit angle and thesecond due to the Newtonian force. For the valueGk\ 10~6 used in the simulations, the term due to Newto-nian gravity dominates by a factor of 2] 105.

As shown in Figures and at zD 8, from a simulation7 8with the distribution of baryons and dark matterz

i\ 100,

perpendicular to the plane spanned by the tangent vector of the string and its velocity vector is completely di†erent from

the corresponding distribution for strings without small-scale structure (see Now the baryons no longerFig. 1).remain conÐned to the central region of the wake. There is abuildup of pressure in the vicinity of this plane whichimparts an outward velocity to the baryons: the baryonsare shock-heated and expelled from the central regions.Instead of a baryon overdensity, there is a relative baryondeÐcit at zD 80.

From it is also manifest that the accretionFigure 7pattern is no longer planar but rather Ðlamentary. Thereare substantial velocities in the x-direction toward theinstantaneous location of the string.

The temperature distribution of the gas along the z-axis isshown in (also for redshift 8). Since the baryonFigure 9

velocity at the shock location is much larger than in the caseof the wakes studied in the previous section, the postshocktemperature is signiÐcantly higher. From itequation (4.4)follows that, for our values of T / k and the velocity Ñow kv

s,

induced by the Ðlament is about a factor of 102 larger thanthe velocity dv obtained for wakes in the previous simula-tions. Hence, by the postshock temperatureequation (4.2),is expected to be higher than 106 K, as is conÐrmed inFigure 9.

5. DISCUSSION

Making use of a new three-dimensional cosmologicalhydrocode, we have simulated the clustering of baryons andcold dark matter induced by long straight strings with and

without small-scale structure. We have studied the clus-tering induced by a single string.

Strings with no small-scale structure give rise to planarwakes. We have shown that the dark matter distribution isin excellent agreement with the known self-similar solutionsfor planar accretion. In particular, the Zeldovich approx-imation yields a good estimate for the thickness of the darkmatter wake. For Gk\ 10~6 and the thickness isv

s\ 0.5,

about 0.7 Mpc for perturbations generated at the timeteq,of equal energy in matter and radiation.

We have shown that the baryon overdensity in wakes forstrings without small-scale structure is thinner than theoverdensity of the dark matter, leading to a relativeenhancement of the baryon density in the center of thewake. In our simulations with the above parameters, theenhancement factor was about 2.4, with a thickness of baryon wakes of only about 0.3 Mpc.

For strings with a substantial amount of small-scalestructure, the baryon distribution is completely di†erent.Instead of a baryon-density enhancement, there is a deÐcitin the center of the structures. The di†erence is due to thefact that small-scale structure on a string induces a Newto-

nian gravitational line source on the string. The velocity of sound is then larger than the impulse due to the Newtonianpotential of the string. Hence baryons can more easily ther-malize and a high pressure builds up, creating a rapidlyoutward-moving shock.

The di†erence between string wakes and string Ðlamentsis very pronounced when considering the postshock baryontemperatures. For wakes, the cold coherent Ñow leads totemperatures of only about 102 K, whereas the large veloci-ties induced by the Newtonian gravitational line source onthe string Ðlaments lead to very high temperatures. For thevalues of T / k\ 0.95 and used in our simulations,v

s\ 0.005

the postshock temperature was about 106 K, comparable tothe temperature of clusters in inÑation-based CDM models

(see, e.g., et al. and references therein :Kang 1994, PenThe di†erence in baryon temperatures between the1996a).

string wake and string Ðlament models may lead to verydi†erent ionization Based on our simulations, wehistories4.conclude that the temperatures in string wakes is too low tolead to ionization at high redshifts since no atomic coolingwill occur, whereas in string Ðlaments the baryons are suffi-ciently hot to lead to substantial cooling, emission of ener-getic photons, and subsequent ionization. These issues willbe explored in future work.

We can compare our results on the segregation of baryons and cold dark matter with recent results of Pen

who studied local properties of gas in rich clusters(1996a),of galaxies in a theory with Gaussian adiabatic pertur-

bations with a scale-invariant spectrum by means of numerical simulations using a new adaptive-mesh hydro-code This codeÈapart from the adaptive(Pen 1996b).moving meshÈis based on numerical techniques similar tothose of our code. The results of these simulations show amarked baryon deÐcit in the cluster centers. Note, however,that neither our code nor the code of Pen include cooling,which might change the results.

However, by comparing our results for strings with andwithout small-scale structure, we have identiÐed a furtherdistinguishing characteristic of the cosmic string wakemodel: there will be a baryon enhancement in the central

4 We thank D. Scott for a useful discussion on this subject.



32 SORNBORGER ET AL.

region of the wakes. If, as proposed by & MiyoshiHaraand et al. rich clusters of galaxies are(1993) Hara (1994),

identiÐed with the crossing sites of three wakes, the baryonenhancement factor in clusters should be about three timesthat in a wake, i.e., about 7. Thus, a cosmic string modelmay be able to explain in a natural way the observed highbaryon fraction in clusters such as the Coma cluster (Whiteet al. in the context of a spatially Ñat universe.1993)

A further distinguishing feature of the string model thathas been conÐrmed by our simulations concerns thegeometry of the nonlinear density perturbations produced.The most numerous and thickest string wakes were gener-ated by strings at Their comoving length l and width wt

eq.

are given by

wP l P mteq

z(teq

)vsc(v

s)Pm] 40 h~1 Mpc , (5.1)

where m is a constant of order unity that sets the curvatureradius of the long strings relative to the Hubble radius. The

thickness of these planar structures is about 0.7 Mpc. Thesepredictions are in encouraging agreement with recentobservations et al. which indicate that(Doroshkevich 1997)the super È large-scale structure of the universe is dominatedby planar structures of dimension comparable to 50 h~1Mpc, and that these walls are indeed very thin Lap-(deparent 1991).

The work of A. S. was supported in part by a NASAGraduate Student Researcher award NGT-51011, and byUK Particle Physics and Astronomy Research Councilgrant GR/K29272. The work of R. B. was supported inpart by the US Department of Energy under grantDEFG0291ER40688, Task A. The work of B. F. and K. O.was supported under NASA grant NAG 5-2652. One of us(R. B.) thanks Douglas Scott and Ue-Li Pen for fruitfuldiscussions, and the Physics and Astronomy Department of the University of British Columbia for hospitality.

REFERENCES

A., & Brandenberger, R. 1995, Int. J. Mod. Phys. D, 4, 711Aguirre,A., & Turok, N. 1989, Phys. Rev. D, 40,Albrecht, 973

B., Caldwell, R., Shellard, E. P. S., Stebbins, A., & Veeraraghavan, S.Allen,

1993, in CMB Anisotropies Two Years after COBE, ed. L. Krauss (NewYork: World ScientiÐc)

1996, Phys. Rev. Lett., ÈÈÈ. submittedB., & Shellard, E. P. S. 1990, Phys. Rev. Lett., 64,Allen, 119D., & Bouchet, F. 1988, Phys. Rev. Lett., 60,Allen, 257

D., Stebbins, A., & Bouchet, F. 1992, ApJ, 399,Bennett, L5R. 1991, Phys. Scr., T36,Brandenberger, 114

1994, Int. J. Mod. Phys. A, 9, ÈÈÈ. 21171995, ÈÈÈ. preprint

R., Perivolaropoulos, L., & Stebbins, A. 1990, Int. J. Mod.Brandenberger,Phys. A, 5, 1633

B. 1990, Phys. Rev. D, 41,Carter, 3869P., & Woodward, P. 1984, J. Comput. Phys., 54,Collela, 174

Lapparent, V., Geller, M., & Huchra, J. 1991, ApJ, 369,de 273A., Tucker, D., Oemler, A., Kirshner, R., Lin, H., Shecht-Doroshkevich,

man, S., & Landy, S. 1997, MNRAS, submittedJ., & Goldreich, P. 1984, ApJ, 281,Filmore, 1

B., Mueller, E., & Arnett, D. 1991, ApJ, 367,Fryxell, 619T., & Miyoshi, S. 1987, Prog. Theor. Phys., 77,Hara, 1152

1990, Prog. Theor. Phys., 84, ÈÈÈ. 8671993, ApJ, 412, ÈÈÈ. 22

T., Yamamoto, H., Maho nen, P., & Miyoshi, S. 1994, ApJ, 432,Hara, 31M., & Kibble, T. W. B. 1995, Rep. Prog. Phys., 58,Hindmarsh, 477

C. 1988, Numerical Computation of Internal and External FlowsHirsch,(New York: Wiley)

R., & Eastwood, J. 1988, Computer Simulations using ParticlesHockney,(New York: Hilger)

H., Ostriker, J., Cen, R., Ryu, D., Hernquist, L., Evrard, A., Bryan,Kang,G., & Norman, M. 1994, ApJ, 430, 83

T. W. B. 1976, J. Phys. A, 9,Kibble, 1387J. 1992, Phys. Rev. D, 46,Magueijo, 1368

R. 1988, Computers in Physics, 1,Matzner, 51K., Myers, E., & Rebbi, C. 1988, Phys. Lett. B, 207,Moriarty, 411

H., & Olesen, P. 1973, Nucl. Phys. B, 61,Nielsen, 45

P. J. E. 1980, The Large-Scale Structure of the UniversePeebles,(Princeton: Princeton Univ. Press)

U.-L. 1996a,Pen. preprint1996b, ÈÈÈ. preprint

L. 1993, Phys. Lett. B, 298,Perivolaropoulos, 305L., Brandenberger, R., & Stebbins, A. 1990, Phys. Rev.Perivolaropoulos,

D, 41, 1764M. 1986, MNRAS, 222,Rees, 27

E. P. S. 1987, Nucl. Phys. B, 283,Shellard, 624E. P. S., & Ruback, P. 1988, Phys. Lett., 209,Shellard, 262

J., & Vilenkin, A. 1984, Phys. Rev. Lett., 53,Silk, 1700A., Fryxell, B., Olson, K., & MacNeice, P. (1996),Sornborger, astro-

ph/9608019A., Veeraraghavan, S., Brandenberger, R., Silk, J., & Turok, N.Stebbins,

1987, ApJ, 322, 1T. 1986, Phys. Rev. Lett., 57,Vachaspati, 1655

1992, Phys. Rev. D, 45, ÈÈÈ. 3487T., & Vilenkin, A. 1991, Phys. Rev. Lett., 67,Vachaspati, 1057

A. 1981a, Phys. Rev. Lett., 46,Vilenkin, 1169

1981b, Phys. Rev. D, 23, ÈÈÈ. 8521990, Phys. Rev. D, 41, ÈÈÈ. 3038

A., & Shellard, E. P. S. 1994, Cosmic Strings and Other Topo-Vilenkin,logical Defects (Cambridge: Cambridge Univ. Press)

D. 1992a, Phys. Rev. D, 45,Vollick, 18841992b, ApJ, 397, ÈÈÈ. 14

S., Navarro, J., Evrard, A., & Frenk, C. 1993, Nature, 366,White, 429V., Lima, J. A. S., & Brandenberger, R. 1996, Phys. Rev. D, 54,Zanchin,

7129Ya. B. 1970, A&A, 5,Zeldovich, 84

1980, MNRAS, 192, ÈÈÈ. 663



0

2

4

6

8

10

12

14

16

18

20

10 20 30 40 50 60

10

20

30

40

50

60

dark matter overdensity

FIG

. 7.ÈDark matter overdensity atzD

8 for a cosmic string Ðlament formed at The white square indicates where pixels containing the highz

i\

100.density from thestring were removed. The area of the simulation is 1 Mpc]1 Mpc.

SORNBORGER et al. (see 482, 29)

PLATE 1



0

0.5

1

1.5

2

2.5

10 20 30 40 50 60

10

20

30

40

50

60

baryonic matter overdensity

z = 26.32

FIG. 8.ÈBaryonic matter overdensity at zD 8 for a cosmic string Ðlament formed at Notice the low density in the core and the shell of matterz

i\100.

thathas been pushed out of the core bythe high pressure. The volumeof the simulation is 1 Mpc]1 Mpc.


PLATE 2



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

10 20 30 40 50 60

10

20

30

40

50

60

baryonic temperature (K)

FIG. 9.ÈBaryonic temperature at zD 8 for a cosmic string Ðlament formed at Very high temperatures are concentrated at the Ðlament core. Thez

i\ 100.

volume of the simulation is 1 Mpc]1 Mpc. The temperature is in units of 1]106K.


Andrew Sornborger et al- The Structure of Cosmic String Wakes

Documents

Transcript of Andrew Sornborger et al- The Structure of Cosmic String Wakes