Halos, Halo Models, and Applica0ons

Transcription

1 Halos, Halo Models, and Applica0ons Chris Hirata Ay Nov /17/09 12:30 AM Halos, Halo Models, and Applica0ons 1

2 Outline 1. The concept 2. Galaxy correla0on func0ons 3. Lensing 4. Extensions (3pcf, redshil space, ) 5. Poten0al difficul0es 6. Further Topics 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 2

3 The concept Fundamental idea: The spa0al clustering of X is determined by (i) the spa0al distribu0on of dark mawer halos; and (ii) the amount and distribu0on of X in halos as a func0on of halo mass M. Various authors have used this where X = Mass Galaxies (olen split by type/luminosity/ ) AGN (in any wave band) Hot gas (Sunyaev Zel dovich effect) Sources of UV/X ray radia0on (reioniza0on & high z 21 cm) 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 3

4 Galaxy correla0on func0on Ingredients: Halos of mass M with mass func0on n(m) and bias b(m). Each halo contains central galaxies and satellite galaxies. The probability of a halo of mass M hos0ng a central galaxy in the sample is <N c >(M). The mean number of satellite galaxies in the halo is <N s >(M). The 3D distribu0on of satellite galaxies is given by a normalized probability distribu0on p(r M). [Usually P(r M) is taken to be isotropic.] 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 4

5 Galaxy correla0on func0on Ingredients: Halos of mass M with mass func0on n(m) and bias b(m). Each halo contains central galaxies and satellite galaxies. The probability of a halo of mass M hos0ng a central galaxy in the sample is <N c >(M). The mean number of satellite galaxies in the halo is <N s >(M). The 3D distribu0on of satellite galaxies is given by a normalized probability distribu0on p(r M). [Usually P(r M) is taken to be isotropic.] FROM CDM SIMULATIONS MODEL PARAMETERS ASSUME NFW (USUALLY) 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 5

6 Compu0ng the 2PCF (I) Need to find: Mean number density of galaxies Pair probability P(x 1,x 2 ) of finding two galaxies at x 1 and x 2. Mean density is straighmorward: Simply integrate halo abundance with the mean number of galaxies per halo. n = n(m) [ N c (M) + N s (M)]dM Satellite frac0on: f sat = n(m) N s (M)dM 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 6 n

7 Compu0ng the 2PCF (II) Need to find: Mean number density of galaxies Pair probability P(x 1,x 2 ) of finding two galaxies at x 1 and x 2. Two types of pairs to be counted: 1 halo term: Galaxies in same halo 2 halo term: Galaxies in different halos 1h 2h 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 7

8 1 halo term Pairs can be either central satellite or satellitesatellite. Let r=x 2 x 1 =separa0on. Assuming independent (Poisson) satellites: [ ] 2 p(x 1 x h M) p(x 2 x h M)d 3 x h P 1h (x 1,x 2 ) = N c (M) N s (M)p(r M)n(M)dM + N s (M) [ ]n(m)dm C S S S Note C S term is simple, but in S S term must integrate over halo centroids. 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 8

9 2 halo term Now we can have C C, C S, and S S terms! P 2h (x 1,x 2 ) = N c (M) N c (M')n(M)n(M')[ 1+ b(m)b(m')ξ mm (r)]dmdm' [ ] +2 N s (M) N c (M')n(M)n(M') 1+ b(m)b(m')ξ mm (x 2 x h ) p(x 1 x h M)dMdM'd 3 x h + N s (M) N s (M')n(M)n(M') 1+ b(m)b(m')ξ mm ( x h x h ) [ ] p(x 1 x h M)p(x 2 x h M)dMdM'd 3 x h d 3 x h C C C S S S Require halo correla0on func0on, which is b(m)b(m )ξ mm. C S term has combinatorial factor of 2. Note integra0on over halo centroids. 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 9

10 Large Scale Limit Galaxy correla0on func0on is: At large r: 2 halo contribu0on dominates Halo centroid separa0ons r ξ gg (r) = P 12 (x 1,x 2 ) n 2 1 ξ gg (r) 1 [ N n 2 c (M) N c (M') + 2 N c (M) N s (M') + N s (M) N s (M')] = 1 n [1+ b(m)b(m')]n(m)n(m')dmdm' 1 [ N c (M) + N s (M)]n(M)b(M)dM ξ mm (r) Galaxies become linearly biased. 2 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 10

11 Bells & Whistles Is satellite distribu0on Poissonian? Yes (for sufficiently massive halos) according to simula0ons [Kravtsov et al 2004] Halo exclusion (no overlap) effect [Tinker et al 2005] Scale dependent halo bias, i.e. ξ hh (r)/ξ mm (r) constant in nonlinear regime [Tinker et al 2005] The formulas look quite complicated but the numerical integra0on poses no special difficulty. 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 11

12 The Parameterized HOD In order to constrain <N c > and <N s >, we need a parameterized halo occupa0on distribu0on (HOD). Typical example: [Zehavi et al 2004] Minimum mass M min to host a central: Minimum mass M 1 to host a satellite. Above M 1, a power law index α for the satellite abundance: N s (M) = (M / M 1 ) α 1 More parameters constrained with more observables. <N> N c (M) = Θ(M M min ) M min M 1 satellite 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 12 central M

13 Halo Model Correla0on Func0on 1h w p (r p ) = projected CF = ξ(r p,r ) dr (Insensi0ve to z space distor0ons) 2h Data pts = SDSS Line = halo model fit M 1 = (4.7±0.5) h 1 M # α = 0.89±0.05 (Formal errors) 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons Zehavi et al

14 High redshil Most detailed use of halo models has been at z~0 (e.g. SDSS) or z~1 (e.g. DEEP2) At high z, informa0on is usually so sparse that only the simplest (2 parameter) type of model can be used: Assume all observed galaxies are centrals. Then large scale bias (2 halo term) minimum or typical host mass Abundance of galaxies / abundance of halos with this mass occupa0on probability <N c >. OLen interpret as duty cycle (but there are other possibili0es). 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 14

15 Example: High z Quasars Consider occupa0on probability to be t Q /t H, where: t Q = quasar life0me t H = median halo life0me (from merger tree) Then for any t Q, can use number density to get M min b ξ(20h 1 Mpc). Clustering measurement constrains quasar life0me! Shen et al 2007 (SDSS) Solid: 2.9 z 3.5 Dashed: z /17/09 12:30 AM Halos, Halo Models, and Applica0ons 15

16 Lensing The correla0on of galaxies with lensing shear enables us to measure the galaxy mass correla0on func0on. This is amenable to study with the halo model. The modeling is actually simpler and more robust than the galaxy 2 point func0on because: Mass distribu0on in halos known from N body simula0ons (minimal gas physics except at <<r vir ) No central/satellite split for the mass No assump0on of Poissonianity for satellites, or independent satellites vs centrals (or alterna0ve assump0ons) Disadvantage of GMCF: lower S/N. 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 16

17 Galaxy Haloes via Sta0s0cs Projected 0dal field: ΔΣ(b) = Σ (< b) Σ(b) 7 bins in stellar mass separated by factors of 2; sm4 = (4 8) Msun Split into early/late types NFW + satellite + LSS fits 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons Mandelbaum et al 2005 (SDSS) 17

18 Galaxy Haloes via Sta0s0cs Rela0on of halo mass to stellar mass η = M stars /(0.17M halo ) Errors are 95% confidence Mandelbaum et al /17/09 12:30 AM Halos, Halo Models, and Applica0ons 18

19 Tes0ng the Mass Bias Rela0on σ 8 gal = b gal σ 8 Increasing L Combined galaxy clustering analysis (red points) with theore0cal predic0ons from lensing masses (black points) Different colors = values of σ 8 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons Seljak et al

20 Higher order sta0s0cs Galaxy 3 point correla0on func0on: excess probability to find triples of galaxies. ζ(r 12,r 23,r 13 ) = P(x 1,x 2,x 3 ) n 3 Depends on all 3 sides of triangle. Exists only due to nonlinear effects. Halo model has 1h (CSS+SSS), 2h (CCS+CSS+SSS), and 3h (CCC+CCS+CSS+SSS) terms. 1 1h term sensi0ve mainly to the most massive (>10 15 M ) halos, hence can break M 1 α degeneracies. 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 20

21 Example: SDSS LRGs Q(r 12,r 23,r 13 ) = r 12 = s r 23 = qs ζ (r 12,r 23,r 13 ) ξ(r 12 )ξ(r 23 ) + ξ(r 23 )ξ(r 13 ) + ξ(r 13 )ξ(r 12 ) r 13 = 1+ q 2 2qcosθ s G Kulkarni et al 2007 Parameter analysis via mock catalogs based on HOD (rather than explicit evalua\on of integrals) Suggests α ~ 1.4; larger α allowed by 2PCF but disallowed by 3PCF 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 21

22 RedshiL space (I) So far we have considered correla0ons in real space But galaxy clustering is done in redshil space: Radial direc0on is affected by peculiar veloci0es. Hence correla0on func0on is a func0on of transverse separa0on (r p ) and radial separa0on (π). Key differences: Halo clustering anisotropic, depends on rela0ve velocity PDF A new parameter α v = velocity bias is required to fit the satellite velocity dispersion: α v = σ sat σ vir 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 22

23 RedshiL space (II) Fits to simula\ons Redshib/real space ξ 0 / R (r) = π / 2 0 ξ(r,θ)sinθdθ ξ R (r) ξ lin 0 / R (r) =1+ 2 β + 1 β Quadrupole asymmetry Q(r) = 5 Q lin (r) = π / 2 ξ(r,θ)p 2 (θ)sinθdθ 0 π / 2 ξ(r,θ) 3 r [ ξ(r',θ)r' 2 dr ' r 3 0 ] sinθdθ β β + 21β 2 β Tinker /17/09 12:30 AM Halos, Halo Models, and Applica0ons 23

24 The parameters: What Can Go Wrong? How many parameters in an HOD are physically relevant? Can be usefully constrained? What can be done by combining observables? Usually >3 (or 4) parameters renders the HOD underconstrained. The physics: Central vs satellite split: who is central in Local Group? In Coma cluster? What to do about merging systems? Do halos of different merger histories (hence different galaxy popula0ons) have the same large scale clustering? Or, for clustering purposes, is P(N M) appropriate or do we need P(N M,environment)? 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 24

25 Press Schechter Revisited PS theory is based on the no0on that collapse to form a halo of mass M occurs when the smoothed (on mass scale M) density field δ M δ c. Smoothed field has variance S(M)=σ 2 (M). The smoothed field starts at δ =0 and random walks as we move to smaller M by picking up contribu0ons from higher k ~ (ρ/m) ⅓. First crossing of δ c threshold halo mass M containing that par0cle. PDF of M. Extended PS (e.g. Lacey & Cole): as δ M grows with growth func0on, trace merger tree containing DM par0cle. 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 25

26 Random Walks In EPS theory, for a par0cular halo mass, the halo merger history is independent of large scale environment. This follows from two assump0ons: Halo merger tree δ M δ M is a Markovian random walk (exact if cutoff in Fourier space, for Gaussian fields) Consequence: two types of halo δ M today of mass M have the same bias b(m), i.e. no b(m,c), b(m,t form ) But no reason for these to be exact. z=1 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 26 δ c M

27 Bias Revisited You learned in Ay 127 that galaxies are related to mawer via linear biasing. More precisely: If galaxy formation is a local process (in the sense of depending only on densities and velocities within some scale length R), if the underlying density field is in the linear regime (σ(r)<<1), the initial perturbations were Gaussian, General Relativity is valid, and sound waves cannot travel distances R in the lifetime of the Universe, then the galaxy bias [ξ gg (r)/ξ mm (r)] ½ approaches a constant at separations r>>r. This is (almost) the most general exact statement (could extend to allow for bulk flows) EPS result on all halos of mass M having the same bias is NOT exact! 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 27

28 Assembly Bias Gao et al. (2005) examined b(m,t form ) where t form is the earliest 0me at which a halo progenitor had mass ½M. 10% oldest 20% oldest 10,20% youngest M * = h 1 M Gao et al /17/09 12:30 AM Halos, Halo Models, and Applica0ons 28

29 Assembly Bias (II) This effect represents a poten0al systema0c error in both HOD fi ng (e.g. using b to obtain M) and in awempts to simultaneously fit HOD + cosmological parameters. Gao, Springel, White /17/09 12:30 AM Halos, Halo Models, and Applica0ons 29

30 Topics 1. Assembly Bias and Its Implica0ons for Halo Models Theory: Gao et al (2005) MNRAS 363, L66; Wechsler et al (2006) ApJ 652, 71; Dalal et al (2008) ApJ 687, 12 How convincing are the observa0onal claims for assembly bias of galaxy groups by e.g. Yang et al (2006) ApJ 638, L55 (and more recent works). 2. Can HOD Modeling Constrain Cosmology by Breaking the σ 8 b gal Degeneracy? Seljak et al (2005) PRD 71, Zheng & Weinberg (2007) ApJ 659, 1 11/17/09 12:30 AM Halos, Halo Models, and Applica0ons 30