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## The Phenomenology of Large Scale Structure

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**The Phenomenology of Large Scale Structure**• Motivation: A biased view of dark matters • Gravitational Instability • The spherical collapse model • Tri-axial (ellipsoidal) collapse • The random-walk description • The halo mass function • Halo progenitors, formation, and merger trees • Environmental effects in hierarchical models • (SDSS galaxies and their environment) • The Halo Model**Galaxy clustering depends on type**Large samples (SDSS, 2dF) now available to quantify this You can observe a lot just by watching. -Yogi Berra**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)**The Cosmic Background Radiation**Cold: 2.725 K Smooth: 10-5 Simple physics Gaussian fluctuations = seeds of subsequent structure formation = simple(r) math Logic which follows is general**N-body simulations of**gravitational clustering in an expanding universe**Cold Dark Matter**• Simulations include gravity only (no gas) • Late-time field retains memory of initial conditions • Cosmic capitalism Co-moving volume ~ 100 Mpc/h**Cold Dark Matter**• Cold: speeds are non-relativistic • To illustrate, 1000 km/s ×10Gyr ≈ 10Mpc; from z~1000 to present, nothing (except photons!) travels more than ~ 10Mpc • Dark: no idea (yet) when/where the stars light-up • Matter: gravity the dominant interaction**Models of halo abundances and clustering: Gravity in an**expanding universeGoal:Use knowledge of initial conditions (CMB) to make inferences about late-time, nonlinear structures**10.077.696.000 cpu seconds**• 10.077.696 USD (total cost!) • 10.077 GBytes of data • postdocs • EXPENSIVE!!!**Hierarchical models**Dark matter ‘haloes’ are basic building blocks of ‘nonlinear’structure Springel et al. 2005**Spherical evolution model**• Initially, Ei = – GM/Ri + (HiRi)2/2 • Shells remain concentric as object evolves; if denser than background, object pulls itself together as background expands around it • At ‘turnaround’: E = – GM/rmax = Ei • So – GM/rmax = – GM/Ri + (HiRi)2/2 • Hence (Ri/r) = 1 – Hi2Ri3/2GM = 1 – (3Hi2 /8pG) (4pRi3/3)/M = 1 – 1/(1+Di) = Di/(1+Di)≈ Di**Virialization**• Final object virializes: −W = 2K • Evir = W+K = W/2 = −GM/2rvir= −GM/rmax • so rvir = rmax/2: • Ratio of initial to final size = (density)⅓ • final density determined by initial overdensity • To form an object at present time, must have had a critical overdensity initially • Critical density ↔ Critical link-length! • To form objects at high redshift, must have been even more overdense initially**Spherical collapse**Turnaround: E = -GM/rmax size Virialize: -W=2K E = W+K = W/2 rvir = rmax/2 time Modify gravity → modify collapse model**Exact Parametric Solution (Ri/R) vs. q and (t/ti) vs.**qvery well approximated by… (Rinitial/R)3 = Mass/(rcomVolume) = 1 + d ~ (1 – dLinear/dsc)−dsc**Virial Motions**• (Ri/rvir) ~ f(Di): ratio of initial and final sizes depends on initial overdensity • Mass M ~ Ri3 (since initial overdensity « 1) • So final virial density ~ M/rvir3 ~ (Ri/rvir)3 ~ function of critical density: hence, all virialized objects have the same density, whatever their mass • V2 ~ GM/rvir ~ M2/3: massive objects have larger internal velocities/temperatures**Halos and Fingers-of-God**• Virial equilibrium: • V2 = GM/r = GM/(3M/4p200r)1/3 • Since halos have same density, massive halos have larger random internal velocities: V2 ~ M2/3 • V2 = GM/r = (G/H2) (M/r3) (Hr)2 = (8pG/3H2) (3M/4pr3) (Hr)2/2 = 200 r/rc (Hr)2/2 =W (10 Hr)2 • Halos should appear ~ ten times longer along line of sight than perpendicular to it: ‘Fingers-of-God’ • Think of V2 as Temperature; then Pressure ~ V2r**Assume a spherical cow….**(Gunn & Gott 1972; Press & Schechter 1974; Bond et al. 1991; Fosalba & Gaztanaga 1998)**The Random Walk Model**Higher Redshift Critical over- density smaller mass patch within more massive region This patch forms halo of mass M MASS**From Walks to Halos: Ansätze**• f(dc,s)ds= fraction of walks which first cross dc(z) at s ≈ fraction of initial volume in patches of comoving volume V(s) which were just dense enough to collapse at z ≈ fraction of initial mass in regions which each initially contained m =rV(1+dc) ≈rV(s) and which were just dense enough to collapse at z (r is comoving density of background) ≈ dm m n(m,dc)/r**The Random Walk Model**Higher Redshift Critical over- density Typical mass smaller at early times: hierarchical clustering MASS**Scaling laws**• Recall characteristic scale V(s) defined by dc2(z) ~ s ~ s2(R) ~∫dk/k k3 P(k) W2(kR) • If P(k) ~ knwith n>−3, then s ~ R–3–n • Since M~R3, characteristic mass scale at z is M*(z)~ [dc(z)] –6/(3+n) • Since dc(z) decreases with time, characteristic mass increases with time → Hierarchical Clustering**Random walk with absorbing barrier**• f(first cross d1 at s) = 0∫sdSf(first cross d0 at S) × f(first cross d1 at s | first cross d0 at S) • (where d1 >d0 and s>S ) • But second term is function of d1 −d0 and s − S • (because subsequent steps independent of previous ones, so statistics of subsequent steps are simply a shift of origin –– a key assumption we will return to later) • f(d1,s) = 0∫sdSf(d0,S) f(d1−d0|s−S) • To solve….**…take Laplace Transform of both sides:**• L(d1,t) = 0∫∞dsf(d1,s) exp(–ts) = 0∫∞ds exp(–ts) 0∫sdSf(d0,S)f(d1–d0,s–S) = 0∫∞dSf(d0,S) e-tSs-S∫∞dsf(d1–d0,s–S) e-t(s-S) = L(d0,t) L(d1–d0,t) • Solution must have form:L(d1,t) = exp(–Cd1) • After some algebra (see notes): L(d1,t) = exp(–d1√2t) • Inverting this transform yields: • f(d1,s) ds = (d12/2πs)½exp(–d12/2s) ds/s • Notice: few walks cross before d12=2s**The Mass Function**• f(dc,s) ds = (dc2/2πs)½exp(–dc2/2s) ds/s • For power-law P(k):dc2/s = (M/M*)(n+3)/3 • n(m,dc) dm = (r/m)/√2p (n+3)/3 dm/m (M/M*)(n+3)/6exp[–(M/M*)(n+3)/3/2] • (Press & Schechter 1974; Bond et al. 1991)**Simplification because…**• Everything local • Evolution determined by cosmology (competition between gravity and expansion) • Statistics determined by initial fluctuation field: since Gaussian, statistics specified by initial power-spectrum P(k) • Fact that only very fat cows are spherical is a detail (crucial for precision cosmology)**Only very fat cows are spherical….**(Sheth, Mo & Tormen 2001; Rossi, Sheth & Tormen 2007)**(Reed et al. 2003)**The Halo 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)**Theory predicts…**• Can rescale halo abundances to ‘universal’ form, independent of P(k), z, cosmology • Greatly simplifies likelihood analyses • Intimate connection between abundance and clustering of dark halos • Can use cluster clustering as check that cluster mass-observable relation correctly calibrated • Important to test if these fortunate simplifications also hold at 1% precision (Sheth & Tormen 1999)**Non-Maxwellian Velocities?**• v = vvir + vhalo • Maxwellian/Gaussian velocity within halo (dispersion depends on parent halo mass) + Gaussian velocity of parent halo (from linear theory ≈ independent of m) • Hence, at fixed m, distribution of vis convolution of two Gaussians, i.e., p(v|m)is Gaussian, with dispersion svir2(m) + sLin2 = (m/m*)2/3svir2(m*)+ sLin2**Two contributions to velocities**~ mass1/3 • Virial motions (i.e., nonlinear theory terms) dominate for particles in massive halos • Halo motions (linear theory) dominate for particles in low mass halos Growth rate of halo motions ~ consistent with linear theory**Exponential tails are generic**• p(v) = ∫dm mn(m) G(v|m) F(t) = ∫dv eivt p(v) = ∫dm n(m)m e-t2svir2(m)/2e-t2sLin2/2 • For P(k) ~ k−1, mass function n(m) ~ power-law times exp[−(m/m*)2/3/2], so integral is: F(t) = e-t2sLin2/2 [1 +t2svir2(m*)]−1/2 • Fourier transform is product of Gaussian and FT of K0 Bessel function, so p(v) is convolution of G(v) with K0(v) • Sincesvir(m*)~ sLin, p(v) ~ Gaussian at |v|<sLin but exponential-like tails extend to large v(Sheth 1996)**Comparison with simulations**Sheth & Diaferio 2001 • Gaussian core with exponential tails as expected!**Spherical evolution model**• ‘Collapse’ depends on initial over-density Di; same for all initial sizes • Critical density depends on cosmology • Final objects all have same density, whatever their initial sizes • Collapsed objects called halos; • ~ 200× denser than background, whatever their mass (Figure shows particles at z~2 which, at z~0, are in a cluster)**Initial spatial distribution within patch (at z~1000)...**…stochastic (initial conditions Gaussian random field); study `forest’ of merger history ‘trees’. …encodes information about subsequent ‘merger history’ of object (Mo & White 1996; Sheth 1996)**Random Walk = Accretion history**High-z Low-z over- density Major merger small mass at high-z larger mass at low z MASS**Other features of the model**• Quantify forest of merger histories as function of halo mass (formation times, mass accretion, etc.) • Model spatial distribution of halos: (halo clustering/biasing) • Abundance + clustering calibrates Mass • Halos and their environment: • Nature vs. nurture—key to simplifying models of galaxy formation**Merger trees**• (Bond et al. 1991; Lacey & Cole 1993) • Fraction of M (halo which virialized at T) which was in m<M at t<T:f[s,dc(t)|S,dc(T)] ds= f[s−S,dc(t)−dc(T)] ds= f(m,t|M,T) dm =(m/M) N(m,t|M,T) dm • N(m|M) is mean number of smaller halos at earlier time • (see Sheth 1996 and Sheth & Lemson 1999 for higher order moments) (from Wechsler et al. 2002)**Correlations with environment**Critical over- density over-dense Easier to get here from over-dense environment This patch forms halo of mass M ‘Top-heavy’ mass function in dense regions under-dense MASS**The Peak-Background Split**• Consider random walks centered on cells which have overdensity d when smoothed on some large scale V: M=rV(1+d)» M* • On large scales (M» M*, so S(M)«1), fluctuations are small (i.e., d «1), so walks start from close to origin: • f(m,t|M,T) dm = f[s−S,dc(t)−d] ds ≈ f[s,dc−d] ds ≈ f[s,dc] ds −d (df/ddc) ≈ f(s,dc) ds [1 −(d/dc) (dlnf/dlndc)] ≈f(s,dc) ds [1 −(d/dc) (1 – dc2/s)]**Halo Bias on Large Scales**• Ratio of mean number density in dense regions to mean number density in Universe: N(m,t|M,T)/n(m,t)V = (M/m) f(m,t|M,T)/(rV/m)f(m,t) [recall dense region had mass M = rV(1+d)] • But from peak-background split: f(m,t|M,T) ≈ f(m,dc) [1 −(d/dc)(1 – dc2/s)] • N(m,t|M,T)/n(m,t)V ≈ (1+d) [1 −(d/dc) (1 – dc2/s)] ≈ 1 − (d/dc) (1 – dc2/s) + d = 1 + b(m)d • Large-scale bias factor: b(m) ≡ 1 + (dc2/s – 1)/dc • Increases rapidly with m at m»m* (Cole & Kaiser 1989; Mo & White 1996; Sheth & Tormen 1999)**Halos and their environment**• Easier to get to here from here than from here • Dense regions host more massive halos n(m,t|d) = [1 + b(m,t)d] n(m,t) b(m,t) increases with m, so n(m,t|d) ≠ (1+d) n(m,t) Fundamental basis for models of halo bias (and hence of galaxy bias)**Most massive halos populate densest regions**over-dense under-dense Key to understand galaxy biasing (Mo & White 1996; Sheth & Tormen 2002) n(m|d) = [1 + b(m)d] n(m)**Correlations with environment**PAST Critical over- density over-dense FUTURE This patch forms halo of mass M At fixed mass, formation history independent of future/environment under-dense MASS