The Pursuit of primordial non-Gaussianity in the

large scale structure of the universe

Donghui JeongTexas Cosmology Center and Astronomy Department

University of Texas at Austin

Astronomy Colloquium, Seoul National University, 27 May 2010

Contents

IntroductionPart I. Gaussian halo biasPart II. Non-Gaussian halo bias

II.1. Effect on the galaxy power spectrumII.2. Effect on the galaxy bispectrum

II.3. Effect on the galaxy-galaxy, galaxy-CMB lensing

The concordance cosmological model• Our universe is flat, and dominated by cosmological constant• From “WMAP+BAO+H0”

in Komatsu et al. (2010):

From inflation to now

Four biggest challenges in cosmology

We do not know the nature of the building blocks!

• Inflation How does our Universe begin, or what drives inflation?

• Dark Matter What is/are the dark matter(s) made of?

• Dark Energy What drives the current acceleration? Is this really the cosmological constant?

• Dark Matter, Dark Energy Is General Relativity a valid theory in the cosmological scales?

The answer will come from galaxy surveys• On-going, near future surveys

– Baryon Oscillation Spectroscopic Survey (0.2 < z < 0.7, 10,000 sq. deg.)– WiggleZ (0.2 < z < 1.0, 1,000 sq. deg.)– Hobby-Eberly Telescope Dark Energy eXperiment

(1.9 < z < 3.5, 420 sq. deg.) : Start observing from Fall 2011!!

• Surveys which may happen in the future– BigBOSS (0.2 < z < 0.7)– Euclid (0 < z < 2.0)– Cosmic Inflation Probe (1 < z < 6)– SUMIRE/LAS (?? < z < ??)– and more to be proposed

Comparing galaxy surveys

dV/dz for 1 str. (~3300 sq. deg.) survey area

~3 [Gpc/h]3

~6 [Gpc/h]3

~1.5 [Gpc/h]3

~1.3 [Gpc/h]3

420 sq. deg.1000 sq. deg.

10,000 sq. deg.

7600 sq. deg.

I. Power spectrum

• Probability of finding two galaxies at separa-tion r is given by the two-point correlation function

• P(k) is the Fourier transform of ξ(r)

• Or, in terms of density contrast, δ(k),

dV1

dV2

r

II. Bispectrum

• Probability of finding three galaxies at sepa-ration (r, s, t) is given by the two, and three-point correlation function

• B(k,k’) is the Fourier transform of ζ(r,s).

• Or, in terms of density contrast,

dV1

dV2

r s

dV3

t

Inflation sets the initial condition

Seed fluctuations predicted by most inflation models are nearly scale invariant and obey nearly Gaussian statistics, which are often parametrized as

• Initial power spectrum

• Initial bispectrum

Here, primordial curvature perturbation is defined as (local type)

Primordial cur-vature erturba-tion

Gaussian ran-dom field

Constraining inflation models

From WMAP7+BAO+H0 (Komatsu et al. 2010)

Primordial non-Gaussianity• Well-studied parameterization is “local” non-Gaussianity :

• Current best measurement of fNL

– From CMB (Komatsu et al, 2010)

– From SDSS power spectra (Slosar et al, 2009)

• Therefore, initial condition is Gaussian to ~0.04% level!Thus, I am talking about a very tiny non-Gaussianity!

Primordial curvature perturba-tion

Gaussian random field

Single-field Theorem(Consistency relation)

• For ANY single-field inflation models, where there is only one degree of freedom during inflation,Maldacena(2003);Seery&Lidsey(2005);Creminelli&Zaldarriaga(2004)

• With the current limit of ns=0.96, fNL has to be about 0.017 for single field inflation.

• Therefore, any detection of fNL would rule out all the single field models regardless of– The form of potential– The form of kinetic term (or sound speed)– The initial vacuum state

• We can detect primordial non-Gaussianity from– CMB bispectrum– High-mass cluster abundance– Scale dependent bias

• Galaxy power spectrum• Galaxy bispectrum (Jeong & Komatsu, 2009b)• Weak gravitational lensing (Jeong, Komatsu, Jain, 2009)

Implication of non-Gaussianity

Galaxies are biased tracers• Cosmological theory (or N-body simulation) tells us about the

dark matter distribution, not about the galaxy distribution.• What we observe from survey are galaxies, not dark matter.• Bias : How is galaxy distribution related to the matter distribu-

tion?

m

m

G

G

ρ

δρf

ρ

δρ

From dark matter to galaxies• In order to calculate the bias from the first principle, we need

to understand the complicated galaxy formation theory such as– Dark matter halo formation– Merger history– Chemistry and cooling– Background radiation (UV)– Feedback (SN, AGN, …)– (and even more)

• Each item is the separate research topic.

Bias : a simple approach(works only on large scales)

• Galaxies are formed inside of dark matter halos.• Halos form at the peak of density field!• But, not every peak forms a halo.

– The peak has to have sufficient over-density.– Let’s say there is a threshold over-density, δc, above which dark matter

clumps to form a halo.– A halo of mass M is

“a region in the space around the peak of smoothed density field (depends on M) whose over-density is greater than the critical over-density.”

• Simplest assumption : – Every dark matter halo hosts a galaxy, or galaxies.

Recipes : Finding galaxies (2D example)

x

ydm

Smoothing(or picking up a certain mass M)

Find peaks above threshold

Critical over-den-sity surface

Galaxies

Contents

IntroductionPart I. Gaussian halo biasPart II. Non-Gaussian halo bias

II.1. Effect on the galaxy power spectrumII.2. Effect on the galaxy bispectrum

II.3. Effect on the galaxy-galaxy, galaxy-CMB lensing

Gaussian density field• One point statistics : The PDF of filtered density field(y) at a

given point is completely determined by r.m.s. of fluctuation, :

• Two point statistics : The covariance of filtered density field is given by the two point correlation function.

Note that

)()()( yyxx RRR

R

]2

exp[2

1)(

2

2

2RR

yyP

)0()()(2RRRR xx

• Fraction of collapsed objects with scale larger than R

Cumulative mass function

dyy

Pc RR

]2

exp[2

12

2

21

• What is the probability of finding two galaxies separated by a distance r?– Joint Probability that two points x and x+r have galaxies at the same

time.

where and .

Question asked by Kaiser (1984)

c c

yCydydyC

P T

]

2

1exp[

||2

1 1212

2

1

y

yy

2

2

)(

)(

RR

RR

r

rC

y1

y2

r

Galaxy bias is linear on large scales!• Galaxy correlation function can be calculated as

on large scales, when, δc σ≫ R (high peak approximation).

• On large enough scale, galaxy correlation function is simply proportional to the matter correlation function.

• This relation is called a linear bias.

)(1)(12

221

2 rP

Pr R

R

cG

Galaxy bias : 2nd method• Linear bias means the linear relation between matter density

contrast and the galaxy density contrast.

• One can change the question to the following: For a given large scale over-dense region, what is the over-density (excess number) of galaxies in that region?

m

m

G

G

ρ

δρb

ρ

δρ

Peak-background split method• Decomposing a density field as peak (on galaxy scale) and

background (on matter fluctuation scale).– 1D schematic example (peaks in the over-dense region)

Peak Background

(Gaussian Random field) (Offset from cosmic mean)

x

m

Large scale over-density+

Galaxies with/without BG• Positive (negative) background effectively reduces (increases)

the threshold over-density.

3 Peaks without background 11 Peaks with background

Threshold over-density

= cosmic mean density of galaxies(e.g. mass function)

= mass function with a positive offset, or reduced threshold!!(see, dashed line in the left figure)

All we need is a Mass function• Therefore, mass function determines the bias!

– Example: for Press-Schetchter mass function,

bias is given by

mc

c

c

cdmc

G

G mn

mn

mnmn

),(ln

),(

),(),(

How accurate is it?

• Only qualitatively useful. (e.g. Jeong & Komatsu (2009))

Prediction from theory (Sheth-Tormen mass function)

measured from Millen-nium simulation Higher peaks

Lower peaks

Higher peaks

Lower peaks

Contents

IntroductionPart I. Gaussian halo biasPart II. Non-Gaussian halo bias

II.1. Effect on the galaxy power spectrumII.2. Effect on the galaxy bispectrum

II.3. Effect on the galaxy-galaxy, galaxy-CMB lensing

From initial curvature to density

Taking Laplacian

grad(φ)=0 at the potential peak

Poisson equation Laplacian(φ)∝δρ=δ<ρ>

Dalal et al.(2008); Matarrese&Verde(2008); Carmelita et al.(2008); Afshordi&Tolly(2008); Slosar et al.(2008);

Dalal et al. (2008)

N-body result : effect of nG

Result from Dalal et al. (2008)

“Large positive fNL acceler-ates the evolution of over-dense regions and retards the evolution of under-dense regions, while large negative fNL has precisely the opposite effect”

Then, what about gal-axy bias?

fNL =-5000

fNL =-500

fNL =0

fNL =500

fNL =5000

375 Mpc/h

80 M

pc/h

Again, peak-background split• Non-Gaussian density field at the peak has additional contri-

bution from the primordial curvature perturbation!

Peak BG1:densityBG2:curvature

+ +Increase once by large scale density

Increase once more by large scale curvature

Galaxies with/without nG

• Positive (negative) fNL effectively reduces (increases) the threshold over-density further more.– Remember, δΦ is always positive!

11 Peaks with Gaussianity 17 Peaks with non-Gaussianity

Threshold over-density

= mass function with additional positive off-set, or reduced threshold!!

Galaxy bias with nG• The primordial non-Gaussianity changes the galaxy power

spectrum by

where change of linear bias is given by

Linear bias depends on the scale!!

snkkP 4

1)(

~1/k2

N-body test of P(k) I (Dalal et al. 2008)H

alo-

matt

er c

orre

latio

n

N-body test of P(k) II (Desjacques et al. 2009)

Pmh(k)Phh(k), Halo-Halo correlation

Can we detect this? (e.g. HETDEX)

Galaxy power spectrum in real spaceΔfNL=21 (68% C.L.) Comparable to WMAP7!!

The galaxy P(k) in redshift space • Peculiar velocity, which further shift the redshift of the galaxy

induces yet another change in power spectrum (Kaiser effect)• As a result, power spectrum becomes anisotropic:

increase in clustering along line of sight direction

μ

Prediction for galaxy surveys

• Predicted 1-sigma marginalized error of non-linearity pa-rameter (fNL): 10000 sq. deg. galaxy survey (CIP)

z V[Gpc/h]3

ng

10-3[h/Mpc]3B1 ΔfNL

(Kmax=0.1 [h/Mpc])ΔfNL

(Kmax=0.2 [h/Mpc])

1.25 17.43 93.120 1.37 6.11 4.991.75 20.91 52.738 1.61 5.04 4.022.25 22.20 28.501 1.96 4.46 3.492.75 22.26 13.217 2.44 4.12 3.163.25 21.69 5.067 3.10 3.90 2.943.75 20.83 3.744 3.57 3.77 2.794.25 19.84 2.586 4.09 3.68 2.694.75 18.82 1.537 4.71 3.64 2.635.25 17.82 0.7726 5.46 3.65 2.625.75 16.87 0.3898 6.23 3.71 2.666.25 15.97 0.2058 7.00 3.85 2.76

Contents

IntroductionPart I. Gaussian halo biasPart II. Non-Gaussian halo bias

II.1. Effect on the galaxy power spectrumII.2. Effect on the galaxy bispectrum

II.3. Effect on the galaxy-galaxy, galaxy-CMB lensing

Bispectrum and non-Gaussianity• Bispectrum is the Fourier space counter part of three point

correlation function:

• CMB (z~1100) bispectrum is a traditional tool to test the non-Gaussianity, because it should vanish when density field is Gaussian.– The latest limit on fNL is (Smith et al. 2009) fNL= 38±21 (68% C.L.)

– Predicted 68% C.L. range of Planck satellite is ΔfNL= 5.

The galaxy bispectrum: theory• The galaxy bispectrum consists of four pieces

I. Matter bispectrum due to primordial non-GaussianityII. Matter bispectrum due to non-linear gravitational evolutionIII. Non-linear galaxy biasIV. Non-Gaussianity term from peak correlation

I

II

III

IV

Jeong & Komatsu (2009b)

Triangular configurations

Bispectrum of Gaussian Universe• We can measure bias from Equilateral and Folded triangles:

Δb1/b1= 0.01, Δb2/b2= 0.05 for HETDEXBispectrum from non-linear gravitational evolution

Bispectrum from non-linear galaxy bias

Jeong & Komatsu (2009b)

Linearly evolved primordial bispectrum

• Notice the factor of k2 in the denominator.• Sharply peaks at the squeezed configuration!

N-body test of matter bispectrum

fNL

Matt

er b

ispe

ctru

m

Nishimichi et al. (in preparation)20 runs of (8 Gpc/h)3 simulations

large scale

squeezed

Non-Gaussian peak correlation terms

• The galaxy bispectrum also depends on trispectrum (four point function) of underlying mass distribution!!

Jeong & Komatsu (2009)

Matter trispectrum I. TΦ

• For local type non-Gaussianity,

• Primordial trispectrum is given by

• For more general multi-field inflation, trispectrum is

Shape of TΦ terms

• Both of TΦ terms peak at squeezed configurations.

• fNL2 term peaks more sharply than gNL term!!

Matter trispectrum II. T1112

• Trispectrum generated by non-linearly evolved primordial non-Gaussianity.

Shape of T1112 terms

• T1112 terms also peak at squeezed configurations.

• T1112 terms peak almost as sharp as gNL term.

Are new terms important? (z=0)Jeong & Komatsu (2009)

N-body galaxy bispectrum

fNL

Gal

axy

bisp

ectr

um

Nishimichi et al. (in preparation)20 runs of (8 Gpc/h)3 simulations

large scale

squeezed

Even more important at high-z!! (z=3)Jeong & Komatsu (2009)

Prediction for galaxy surveys• Predicted 1-sigma marginalized error of non-linearity parame-

ter (fNL) from the galaxy bispectrum alone

z V[Gpc/h]3

ng

10-5[h/Mpc]3b1 ΔfNL

(SK07)ΔfNL

(JK09)

SDSS-LRG0.315 1.48 136 2.17 60.38 5.43

BOSS0.35 5.66 26.6 1.97 31.96 3.13

HETDEX2.7 2.96 27 4.10 20.39 2.35

CIP2.25 6.54 500 2.44 8.96 0.99

ADEPT1.5 107.3 93.7 2.48 5.65 0.92

EUCLID1.0 102.9 156 1.93 5.56 0.77

Jeong (2010)

Contents

IntroductionPart I. Gaussian halo biasPart II. Non-Gaussian halo bias

II.1. Effect on the galaxy power spectrumII.2. Effect on the galaxy bispectrum

II.3. Effect on the galaxy-galaxy, galaxy-CMB lensing

fNL from Weak gravitational lensing

Picture from M. Takada (IPMU)

Jeong, Komatsu, Jain (2009)

• Mean tangential shear is given by

It is often written as

where, Σc is the “critical surface density”

Mean tangential shear

G

R

Mean tangential shear, status

Mean tangential shear from SDSS Sheldon et al. (2009)

What about larger scales?

BAO in mean tangential shear

RBAO=106.9 Mpc/h

Δχ2 = 0.85

Errors are dominated by cosmic variance.

(SDSS LRG)

Jeong, Komatsu, Jain (2009)

BAO in mean tangential shear

Even larger scale?

Result from the thin (delta function) lens plane

RBAO=106.9 Mpc/hΔχ2 = 1.34

(LSST cluster)

Jeong, Komatsu, Jain (2009)

fNL in mean tangential shear (LRG)Jeong, Komatsu, Jain (2009)

fNL in mean tangential shear (LSST)Jeong, Komatsu, Jain (2009)

CMB anisotropy as a backlight

Picture from Hu & Okamoto (2001)

Unlensed

Lensed

Galaxy-CMB lensing, z=0.3Jeong, Komatsu, Jain (2009)

Galaxy-CMB lensing, z=0.8Jeong, Komatsu, Jain (2009)

Cluster-CMB lensing, z=5High-z population provide a better chance of finding fNL.

Jeong, Komatsu, Jain (2009)

Conclusion• Detecting primordial non-Gaussianity will rule out all single

field inflation models. We can estimate the primordial non-Gaussianigy from large scale structure of the universe.

• For Gaussian case, large scale bias is a constant, but for non-Gaussian case, it depends on the wave-number sharply.

• The galaxy power spectrum and the galaxy bispectrum are very promising tools to prove primordial non-Gaussianities!!

• Weak gravitational lensing on large scales can provides inde-pendent cross-checks of non-Gaussianity.