Large scale structure: Phenomenology The halo model: Theory Halo abundances, clustering, profiles In practice: HOD, CLF, SHAM (Assembly bias)
Luminous Not luminous Zehavi et al. 2010 (SDSS)
blue red Zehavi et al. 2010 (SDSS)
Complication: Light is a biased tracer Not all galaxies are fair tracers of dark matter; To use galaxies as probes of underlying dark matter distribution, must understand ‘bias’
How to describe different point processes which are all built from the same underlying density field? THE HALO MODEL Review in Physics Reports (Cooray & Sheth 2002)
Center-satellite process requires knowledge of how 1) halo abundance; 2) halo clustering; 3) halo profiles; 4) number of galaxies per halo; all depend on halo mass (+ ...) (Revived, then discarded in 1970s by Peebles, McClelland & Silk)
• Late time field is a collection of ‘halos’ – Halos have a range of masses – Spatial distribution depends on halo mass; e.g. more massive halos are more strongly clustered – Average density within a halo is approximately 200x background, independent of halo mass • Galaxies form in halos – Galaxy ~ halo substructure is good approximation, so halo formation affects star formation and assembly histories
The Halo (Reed et al. 2003) Mass Function •Small halos collapse/virialize first •Can also model halo spatial distribution •Massive halos more strongly clustered (current parametrizations by Sheth & Tormen 1999; Jenkins etal. 2001)
• Can also measure/model halo spatial distribution (and its evolution) • On large scales, linear bias ξ hm (r) = b ξ mm (r) is good approximation • At any given time, massive halos are more strongly clustered
Universal Navarro, Frenk & White (1996) Halo Profiles ρ (r) = 4 ρ s /(r/r s )/(1+ r/r s ) 2 •Not quite isothermal •Scale radius r s depend on halo mass, formation time •Massive halos less concentrated (partially built-in from GRF initial conditions) • Distribution of shapes (axis-ratios) known (Jing & Suto 2001)
• Late time field is a collection of ‘halos’ – Halos have a range of masses – Spatial distribution depends on halo mass; e.g. more massive halos are more strongly clustered – Average density within a halo is approximately 200x background, independent of halo mass • Galaxies form in halos – Galaxy ~ halo substructure is good approximation, so halo formation affects star formation and assembly histories
Baryonic effects on the profile: Adiabatic contraction
Adiabatic contraction … r [M g (<r) + M dm (<r)] = r i M g+dm (<r i ) • Dark matter initially within r i and now within r is M dm (<r) = (1 - f g ) M tot (<r i ) • Circular velocity from 2 (r) = GM(<r)/r = (r i /r) 2 GM g+dm (<r i )/r i V circ V circ (r) = (r i /r) V circ (r i ) • In general, solve numerically. But, for (realistic) Hernquist galaxy M g (<r) = M g (r/s g ) 2 /(1+r/s g ) 2 result is analytic: f g r 3 + (r+s g ) 2 [(1-f g ) r - r i ] m g+dm (r i ) = 0 where f g = M g /M tot . Get r by solving the cubic ( Keeton 2001 ). … increases circular velocities.
Inclusion of star formation feedback related effects can heat (expand) the gas, thus the dark matter as well: remove the cusp Binding energy is M(GM/R) ~ M 5/3 so removal of cusp Fewer stars easier at low mass formed , so less feedback, What remains has so cusp smaller V circ , thus remains resolving the too- big-to-fail problem with no new physics Madau et al. 2014
The halo-model of clustering • Two types of pairs: both particles in same halo, or particles in different halos • 1+ ξ (r) = 1+ ξ 1h (r) + 1+ ξ 2h (r) • All physics can be decomposed similarly: ‘nonlinear’ effects from within halo, ‘linear’ from outside
The dark-matter correlation function ξ dm (r) = 1+ ξ 1h (r) + ξ 2h (r) • 1+ ξ 1h (r) ~ ∫ dm n(m) m 2 ξ dm (r|m)/ ρ 2 • n(m): comoving number density of m-halos • Comoving mass density: ρ = ∫ dm n(m) m • ξ dm (r|m): fraction of total pairs, m 2 , in an m - halo which have separation r; depends on (convolution of) density profile within m -halos • This term only matters on scales smaller than the virial radius of a typical M * halo (~ Mpc) – Need not know spatial distribution of halos!
ξ dm (r) = 1+ ξ 1h (r) + ξ 2h (r) • ξ 2h (r) ≈ ∫dm 1 m 1 n(m 1 ) ∫dm 2 m 2 n(m 2 ) ξ 2h (r|m 1 ,m 2 ) ρ ρ • Two-halo term dominates on large scales, where peak-background split estimate of halo clustering should be accurate: δ h ~ b(m) δ dm • ξ 2h (r|m 1 ,m 2 ) ~ ‹ δ h 2 › ~ b(m 1 )b(m 2 ) ‹ δ dm 2 › • ξ 2h (r) ≈ [∫dm mn(m) b(m)/ ρ ] 2 ξ dm (r) • On large scales, linear theory is accurate: 2 ξ Lin (r) ξ dm (r) ≈ ξ Lin (r) so ξ 2h (r) ≈ b eff
Dark matter power spectrum • Convolutions in real space are products in k-space, so P(k) is easier than ξ 1h (r) P(k) = P 1h (k) + P 2h (k) • P 1h (k) = ∫ dm n(m) m 2 |u dm (k|m)| 2 / ρ 2 • P 2h (k) ≈ [ ∫ dm n(m) b(m) m u dm (k|m)/ ρ] 2 P dm (k)
The halo-model of galaxy clustering • Two types of particles: central + ‘satellite’ • Two types of pairs: both particles in same halo, or particles in different halos • 1+ ξ obs (r) = 1+ ξ 1h (r) + 1+ ξ 2h (r) n t (n t -1)[1+ ξ 1h (r)] = 2n c n s [1+ ξ cs (r)] + n s (n s -1)[1+ ξ ss (r)]
The halo-model of galaxy clustering • Write as sum of two components: – 1+ ξ 1gal (r) = ∫ dm n(m) g 2 (m) ξ dm (m|r) / ρ gal 2 – ξ 2gal (r) ≈ [∫dm n(m) g 1 (m) b(m)/ ρ gal ] 2 ξ dm (r) ρ gal = ∫ dm n(m) g 1 (m): number density of galaxies – – ξ dm (m|r): fraction of pairs in m-halos at separation r • Think of mean number of galaxies, g 1 (m) = <N|m> , as a weight applied to each dark matter halo • And g 2 (m) = <N(N-1)|m> is mean number of distinct pairs – Galaxies ‘biased’ if g 1 (m) not proportional to m, …, g n (m) not proportional to m n ( Jing, Mo & Boerner 1998; Benson et al. 2000; Peacock & Smith 2000; Seljak 2000; Scoccimarro et al. 2001 ) – Central + Poisson satellites model (see later) works well • Similarly, Y SZ or T X are just a weight applied to halos, so same formalism can model cluster clustering
Power spectrum • Convolutions in real space are products in k-space, so P(k) is easier than ξ (r): P(k) = P 1h (k) + P 2h (k) • P 1h (k) = ∫ dm n(m) g 2 (m) |u dm (k|m)| 2 / ρ 2 • P 2h (k) ≈ [ ∫ dm n(m) b(m) g 1 (m) u dm (k|m)/ ρ] 2 P dm (k) • Galaxies ‘biased’ if g n (m) not proportional to m n
Type-dependent clustering: Why? populate massive halos populate lower mass halos Spatial distribution within halos second order effect (on >100 kpc)
Comparison with simulations: OK! • Halo model calculation of ξ (r) • Red galaxies • Dark matter ←ξ 1h › ξ 2h • Blue galaxies ξ 1h ‹ ξ 2h → • Note inflection at scale of transition from 1halo term to 2- halo term (~ virial radius) • Bias constant at large r
Galaxy-lensing power spectrum P(k) = P 1h (k) + P 2h (k) • P 1h (k) = ∫ dm n(m) mu(k|m) g 1 (m)u g (k|m)/n g ρ • P 2h (k) ≈ [ ∫ dm n(m) b(m) m u(k|m)/ ρ] x [ ∫ dm n(m) b(m) g 1 (m) u g (k|m)/n g ] P dm (k)
Redshift space distortions
Two redshift space distortions: Linear + nonlinear
Redshift space distortions On large scales, use Gaussian statistics to compute (Fisher 1995)
Linear redshift space distortions • The same velocities which lead to Zeldovich displacements make redshift space position different from real space position. • x s = x + [v(x).d los /|d los |]/H = q + v(q)/(afH) + [v(q)∙d los /|d los |]/H • Hence, same Jacobian which gives δ now gives δ s = (1 + f µ 2 ) δ where µ is angle wrt los, so P s (k, µ ) = (1 + f µ 2 ) 2 P(k) and averaging over µ → (1 + 2f/3 + f 2 /5) P(k) (Kaiser 1987)
Nonlinear Fingers-of-God • Virial equilibrium: V 2 = GM/r = GM/(3M/4 π200ρ ) 1/3 • Since halos have same density, massive halos have larger random internal velocities: V 2 ~ M 2/3 V 2 = GM/r = (G/H 2 ) (M/r 3 ) (Hr) 2 = (8 π G/3H 2 ) (3M/4 π r 3 ) (Hr) 2 /2 = 200 ρ / ρ c (Hr) 2 /2 = Ω (10 Hr) 2 • Halos should appear ~ten times longer along line of sight than perpendicular to it: ‘Fingers-of-God’ • Think of V 2 as Temperature; then Pressure ~ V 2 ρ
Redshift space power spectrum P s (k) = P 1h (k) + P 2h (k) Nonlinear – fingers of god u s (k|m) = u(k|m) e -k2 µ 2 σ 2vir(m)/2 Linear ~ Zeldovich • P 1h (k) = (1 + f µ 2 ) 2 ∫ dm n(m) g 2 (m) |u s (k|m)| 2 /n g 2 • P 2h (k) ≈ [ ∫ dm n(m) (b(m) + f µ 2 ) g 1 (m) u s (k|m)/n g ] 2 bias + linear × P dm (k)
Halo Model: HOD, CLF, SHAM • Goal is to infer p(N|m) from measurements of abundance and clustering – Abundance constrains <N|m> = g 1 (m) – 1-halo term of n-pt clustering constrains g n (m) • HOD uses abundance and 2pt statistics to constrain p(N|m) from different samples (Zehavi et al. 2011; Skibba et al. 2014) • CLF now does too, to constrain φ (L|m) (Lu et al. 2014) • Since <N(>L)|m> = φ (>L|m), HOD~CLF but with different systematics • SHAM (Klypin+ 1999; Sheth-Jain 2003; Conroy+ 2006) uses abundance only, but gets 2pt stats quite well anyway (Moster et al. 2013) – Problematic for samples where relation to halo mass is not monotonic (e.g., color selected samples)
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