Galaxy bias with Gaussian/non-Gaussian ini-

tial condition:a pedagogical introduction

Donghui JeongTexas Cosmology Center

The University of Texas at AustinTheory seminar, Spring 2009, May 4, 2009

What is bias?

• 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 matters.

• Bias : How does galaxy distribution related to the matter distribution?

m

m

G

G

ρ

δρf

ρ

δρ

From dark matter to Galaxy

• In order to calculate the bias from the first prin-ciple, 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. (e.g. of Jarrett, Athena, ...)

Bias : a simplest approach

• 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, , 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 whose over-density is greater than the critical over-density.”

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

c

Recipes : Finding galaxies (2D example)

x

ydm

Smooth-ing

Find peaks above thresh-old

Critical over-density sur-face

Galax-ies

I. Galaxy bias with

Gaussian initial condition

Gaussian density field

• One point statistics : The PDF of filtered density field(y) at a given point is completely deter-mined by r.m.s. of fluctuation, :

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

Note that)()()( yyxx RRR

R

]2

exp[2

1)(

2

2

2RR

yyP

)0()()(2RRRR xx

• What is the probability of finding two gal-axies separated by a distance r?– 1. Probability that a randomly chosen point has gal-

axy :

– 2. Joint Probability that two point x and x+r have gal-axy at the same time.

Where and .

Question asked by Kaiser (1984)

dyy

Pc RR

]2

exp[2

12

2

21

c c

yCydydyC

P T

]

2

1exp[

||2

1 1212

2

1

y

yy

2

2

)(

)(

RR

RR

r

rC

y

y1

y2

r

Galaxy bias is linear!

• Galaxy correlation function can be also founded as

on large scales, when .

• On large enough scale, galaxy correlation func-tion is simply proportional to the matter correlation function. That relation is called a linear bias.

)(1)(12

21

2 rP

Pr R

R

cG

Rc

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 following: For a given large scale over-dense region, what is the over-density of galaxies in that region?

m

m

G

G

ρ

δρb

ρ

δρ

Peak-background split method

• Decomposing a density field as peak (on gal-axy scale) and background (on matter fluctua-tion scale).– 1D schematic example (peaks in the over-dense re-

gion)

Peak

Background

(Gaussian Random field)

(Offset from cosmic mean)

x

m

Large scale over-den-sity

+

Galaxies with/without BG

• Positive (negative) background effectively re-duces (increases) the threshold over-density.

3 Peaks without back-ground

11 Peaks with back-ground

Threshold over-den-sity

= cosmic mean density of gal-axies(e.g. mass function)

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

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

),(

),(),(

2

22

2 2exp

ln

ln

2

2),(

R

c

R

cRc md

d

mmn

m

m

cR

c

G

G

1

2

How accurate is it?

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

Prediction from theory (Sheth-Tormen mass function)

measured from Mil-lennium simulation

II. Galaxy bias with

Non-Gaussian initial condi-tion

Primordial non-Gaussianity (nG)

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

• Current observational limit from WMAP5yr

• Therefore, initial condition is Gaussian in ~0.03% level!That is, I am talking about the very tiny non-Gaussianity!

Primordial curvature perturba-tion

Gaussian random field

fNL

= 55 ± 30 (Komatsu et al.

(2008)) fNL

= 38 ± 21 (Smith et al. (2008))

Why do we care about ~0.03%?

• Detecting of fNL larger than 1 can rule out the “plain vanilla” models of inflation– driven by a slow-rolling, single scalar field– with canonical kinetic term (meaning Ek=1/2mv2)

– originated from the Bunch-Davis vacuum

• Moreover, such a tiny non-Gaussianity generate the Huge signal!!– CMB bispectrum– High-mass cluster abundance– Galaxy bias– Galaxy bispectrum

From initial curvature to den-sity

Taking Lapla-cian

grad(φ)=0 at the potential peak

Poisson equation Laplacian(φ)∝δρ=δ<ρ>

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

N-body result IResult from Dalal et al. (2008)

“Large positive fNL accelerates the evo-lution of over-dense regions and retards the evolution of un-der-dense regions, while large negative fNL has precisely oppo-site effect”

Then, what about galaxy 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 addi-tional contribution from the primordial curvature perturbation!

Peak

BG1:density

BG2:curvature

+ +Increase once by large scale den-sity

Increase once more by large scale curva-ture

Galaxies with/without nG

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

11 Peaks with Gaussian-ity

15 Peaks with non-Gaus-sianity

Threshold over-den-sity

= mass function with additional positive offset, 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)(

N-body result II (Dalal et al. (2008))

N-body result III (Desjacques et al. 2008)

Pmh(k)

Phh(k)

Can HETDEX detect this?Galaxy power spectrum in real spaceHETDEX specification : - 1.9<z<3.5, 420 deg. sq.- .8 Million Ly-alpha emitters- “WMAP+BAO+SN” best-fit cos-mology in Komatsu et al. (2008)- ΔfNL=10.34 (68% C.L.)

Conclusion

• We discuss the theoretical tool to calculate the rela-tion between galaxy distribution and underlying matter distribution (or bias).

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

• This possibility can potentially constrain the physics of the very early universe.

• For more detailed analysis of the effect on galaxy bispectrum, see Jeong & Komatsu (2009).