Galaxy bias with Gaussian/non- Gaussian initial condition: a pedagogical introduction Donghui Jeong Texas Cosmology Center The University of Texas at Austin Theory 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?

From dark matter to Galaxy 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. (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.

Recipes : Finding galaxies (2D example) x y Smoothing Find peaks above threshold Critical over- density surface Galaxies

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 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

What is the probability of finding two galaxies separated by a distance r? –1. Probability that a randomly chosen point has galaxy : –2. Joint Probability that two point x and x+r have galaxy at the same time. Where and. Question asked by Kaiser (1984) y y1y1 y2y2 r

Galaxy bias is linear! Galaxy correlation function can be also founded as on large scales, when. On large enough scale, galaxy correlation function is simply proportional to the matter correlation function. That relation is called a linear bias.

Galaxy bias : 2 nd 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?

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) Pea k Background (Gaussian Random field)(Offset from cosmic mean) x 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

How accurate is it? Only qualitatively useful. ( e.g. Jeong & Komatsu (2009)) Prediction from theory (Sheth-Tormen mass function) measured from Millennium simulation

II. Galaxy bias with Non-Gaussian initial condition

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 perturbation Gaussian random field f NL = 55 ± 30 (Komatsu et al. (2008)) f NL = 38 ± 21 (Smith et al. (2008))

Why do we care about ~0.03%? Detecting of f NL 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 E k =1/2mv 2 ) –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 density Taking Laplacian grad(φ)=0 at the potential peak Poisson equation Laplacian(φ) ∝ δρ=δ Dadal et al.(2008); Matarrese&Verde(2008); Carmelita et al.(2008); Afshordi&Tolly(2008); Slosar et al.(2008);

N-body result I Result from Dalal et al. (2008) “Large positive f NL accelerates the evolution of over-dense regions and retards the evolution of under-dense regions, while large negative f NL has precisely opposite effect” Then, what about galaxy bias? f NL =-5000 f NL =-500 f NL =0 f NL =500 f NL = Mpc/h 80 Mpc/h

Again, peak-background split Non-Gaussian density field at the peak has additional contribution from the primordial curvature perturbation! Pea k BG1:density BG2:curvature ++ Increase once by large scale density Increase once more by large scale curvature

Galaxies with/without nG Positive (negative) f NL effectively reduces (increase) the threshold over-density further more. –Remember, δΦ is always positive! 11 Peaks with Gaussianity 15 Peaks with non-Gaussianity Threshold over-density = 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!!

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

N-body result III (Desjacques et al. 2008) P mh (k) P hh (k)

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

Conclusion We discuss the theoretical tool to calculate the relation 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).