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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).