Cosmological Structure Formation I
Michael L. Norman Laboratory for Computational Astrophysics [email protected] References
• T. Padmanabhan, “Structure Formation in the Universe”, Cambridge (1994) • S. D. M. White, “The Formation and Evolution of Galaxies”, Les Houches Lectures, August 1993, astro-ph/9410043 Outline • Examples of cosmic structure • Gravity: origin of cosmic structure • Linear theory of perturbation growth • Nonlinear simulations I. dark matter – hierarchical clustering • Nonlinear theory of hierarchical clustering • Nonlinear simulations II. baryons – hydrodynamic cosmology – the Enzo code What is cosmic structure? • Inhomogeneities in the distribution of matter in the universe at any epoch δ=(ρ/<ρ>)−1 Regime Example 0 homogeneous CMB O(ε) linear CMB anisotropies O(0.1) quasi-linear galaxy LSS O(1-10) nonlinear Lyman alpha forest
O(>100) virialized galaxies, groups, clusters Cosmic Microwave Background Penzias & Wilson (1965)
back Τ=2.73 Κ Cosmic Microwave Background WMAP Year 1
-4 back ∆T/T ~ δ ~ 10 Galaxy Large Scale Structure: 2dF Galaxy Redshift Survey
back <∆ng/ng> =b<∆ρ/ρ>~0.1 Lyman Alpha Forest: HI absorption lines in quasar spectra Physical Origin of the Lyman Alpha Forest Zhang, Anninos, Norman (1995) • intergalactic medium exhibits cosmic web structure at high z • models explain observed hydrogen absorption spectra
N=1283 δ=10 5 Mpc/h
back Galaxies, Groups & δ(dynamical) ~ 103 Clusters
δ(disk) ~ 106 The Universe Exhibits a Hierarchy of Structures
dwarf galaxies
galaxy groups star clusters galaxy superclusters
100 101 102 103 104 105 106 107 108 109 Light years galaxies
galaxy clusters Structure Formation: Goals • Understand – origin and evolution of cosmic structure from the Big Bang onward across all physical length, mass & time scales – Interplay between different mass constituents (dark matter, baryons, radiation), self-gravity, and cosmic expansion – Dependence on cosmological parameters • Predict – Earliest generation of cosmic structures which have not yet been observed History of the Universe
phase transitions gravitational instability Recombination Nucleosynthesis
linear perturbation theory nonlinear simulations Our universe then and now
Recombination (~380,000 yr) δρ/<ρ> ~ 10-4
Wilkinson MAP (NASA) Present (~14x109 yr) δρ/<ρ> ~ 106 Gravitational Instability: Origin of Cosmic Structure
very small fluctuations ρ A C
<ρ> x B
gravity amplifies fluctuations ρ A C <ρ> x B Gravitational Instability in 3-D: Origin of the “Cosmic Web”
1 billion light years
Bryan & Norman (1998) N=5123 Evolution of Cosmic Structure: Key Issues • origin and character of primordial density fluctuations – Inflation: scale-free, gaussian random field • linear evolution of the power spectrum P(k) α |δ∗(k)|2 with time (redshift) for each mass component before recombination • nonlinear evolution of density fluctuations due to gravitational and internal forces after recombination • above in an expanding universe with known cosmological parameters (ΛCDM) Matter Power Spectrum
• Fourier transform of linear density field (ρ(x) − ρ ) δ (x) = ρ δ 1 = d 3 xδ (x)eik⋅x k V ∫ • Power spectrum is defined as: 2 P(k) = δk for all k between k and k + dk ΛCDM Matter Power Spectrum
http://www.hep.upenn.edu/~max Mass-Energy Budget of the Universe (WMAP)
ΩΛ
Ωb
Ωcdm
Ωcdm+ Ωb+ ΩΛ=1 Linear Theory τ Linear Perturbations
2 δ define H ( ) ≡ a& / a, Ω ≡ 3H /8 G π ρ δthen, pressure - free perturbation obeys δ 2 a 3 a &&+ & & − Ω & = 0 δ a 2 a δ for EdSτ universe (Ω =1), above becomes δ τ &&+ 2 & / − 62 / 2 = 0 δ 2 2/3 growing mode : = Dτ( δ) 0 ∝ ∝ t ∝ a δ τ decaying mode : ∝τ −3 ∝1/ t ∝ a−3/ 2 Saturated Growth in Low Ωm Universe −Ωis the driving term in growth of δ - Makes a sudden transition in Λ model
- Growth like EdS for a
ac = value of expansion factor when Ω = 0.5 Mass Fluctuations in Spheres
• Approximate δP(k) locally with power-law 2 2 n k ∝ D (τ )k
δ • Mean square mass fluctuation (variance) ( M / M )2 = d 3k W 2 (kR) 2 ∝ D2 R−(3+n) ∝ D2M −(3+n)/3 ∫ T k
WT (kR) is Fourier transformδ of top - hat window function 3/ 4πR3 , x < R W(x) = 0, x ≥ R 2 →WT (kR) = 3[]sin(kR) / kR − cos(kR) /(kR) CDM Power Spectrum
LSS n=0 super horizon n=-1 galaxy clusters n=1
P(k)
n=-2 galaxies
n=-3 first objects
k Mass Fluctuations, cont’d
• RMS mass fluctuations σ 1/ 2 (M ) ≡ (δM / M )2 ∝ DM −(3+n)/ 6
For n>-3, RMS fluctuations a decreasing function of M • Nonlinear mass scale setting σ (M ) =1 τ nl 6/(3+n) −6/(3+n) get M nl ( ) ∝ D(τ ) (∝ (1+ z) for EdS)
For n>-3, smallest mass scales become nonlinear first Evolution of nonlinear mass scale with redshift in ΛCDM
Log(1+znl) Structure formation is a multiscale problem 100
0.01 Gyr first structures Rare fluctuations 10 galaxies Ly α forest 1 Gyr Typical fluctuations 1 galaxy clusters 2 4 6 8 10 12 14 16 Log(Mnl) Numerical Cosmology Goals
• Simulate the processes governing the formation and evolution of the observable structures in the universe – galaxies, quasars, clusters, superclusters • Find the “best fit” cosmological model through detailed observational comparison • Make quantitative predictions for new era of high redshift observations Nonlinear Simulations I: Cold Dark Matter Cold Dark Matter
• Dominant mass constituent: Ωcdm~0.23 •Only interacts gravitationally with ordinary matter (baryons) • Collisionlessυ dynamics governed by Vlasov- Poisson equationυ φ r r r ∂t f (x, π,t) + ⋅∇ x f − ∇φ ⋅∇υ f = 0 υ ∇2 = 4 G∫ d 3 r f (xr,υr,t) • Solved numerically using N-body methods The Universe is an IVP suitable for computation
• Globally, the universe evolves according to the Friedmann equation 2 π a 8 G k Λ H (t)2 ≡ & = ρ − + a 3 a2 3
cosmological constant Hubble parameter mass-energy spacetime scale factor a(t) density curvature Friedmann Models: The Omega Factor
Ω=<ρ>/ρcrit
Ω+Λ=1 Ω<1 “Flat” universe “Open” universe Ω=1 which accelerates expands forever
a(t) “Flat” universe expands forever
Ω>1 The Universe is an IVP...
• Locally, its contents obey: – Newton’s laws of gravitational N-body dynamics for stars and collisionless dark matter (CDM) – Euler or MHD equations for baryonic gas/plasma – Atomic, molecular, and radiative processes important for the condensation of stars and galaxies from diffuse gas Gridding the Universe
• Transformation to • Triply-periodic comoving boundary conditions coordinates x=r/a(t)
a(t1) a(t2) a(t3) Dark Matter Dynamics in an Expanding Universe dx dm = v dt dm Newton’s laws dv a 1 φdm = −2 & v − ∇ dt π a dm a 2 φ 2 2 ρ ∇ = 4 Ga ( − ρ ) Poisson equation 1/ 2 da 1 2 = H 0 Ω m − 1 + Ω Λ (a − 1) + 1 dt a Friedmann equation Hierarchical Clustering of Cold Dark Matter Hierarchical Structure Formation: Like a River
• large galaxies form from mergers of sub-galactic units • Galaxy groups form from merger of galaxies • Galaxy clusters form from merger of groups • Where did it begin, and where will it end?
Lacey & Cole (1993) Billion Particle Simulation of Large Scale Structure
P. Bode & J. Ostriker Nonlinear Theory Dark Matter Halo Mass Function
• Principal quantity of interest is the globally averaged number density of collapsed objects of mass M as a function of redshift=z < n(M , z) > • Dark matter halos define the gravitational potential wells galaxies, galaxy groups, and clusters of galaxies form in • Sensitive function of cosmological parameters Spherical Top-Hat Model
• Simplest analytic model of nonlinear evolution of a discrete perturbation consider spherical region with uniform overdensity and radius R by Birkhoff's theorem (GR), EOM is : d 2 R GM π 4 G = − = − (1+ )R R dt 2 R2 ρ3 δ δ whereas the Friedmann equation is : π δ d 2a 4 G = − ρa dt 2 3 ∴ perturbation evolves like a universe of different density, but the same initial time and expansion rate Spherical Top-Hat, cont’d
E>0
E=0 R Rm first integral of motion : t + m 2 1 dR GM (Ri, ti) E<0 − = E 2 dt R t if E < 0, perturbation is bound, and obeys 1 R = (1− cosΘ), t = (Θ − sin Θ) /π R m 2 t m where Rm and tm are radius and time at "turnaround" δ Turnaround and Collapse π δ mean linear overdensity with respect to EdS universe: δ 3 2/3 lin = (6 t / tm ) ∝ aEdS ( <<1) δ 20
lin (tm ) =1.063 nonlinear overdensity at turnaround :
nlδ(tm ) = 4.6 δ Collapse: R → 0 as t → 2t m 3 ∴ = (2t ) = (12π )2/3 =1.686 collapse lin m 20 Virialization collapse halted when virial equilibrium established : U = 2K assume total energy E = K-U is conserved 3 GM 2 at turnaround : K = 0, E = -U = − (homogeneous sphere) 5 Rm at virialization : E = K-2K = -K 1 1 3 GM 2 ∴K = −E = U = ⋅ 2 2 5 Rvir equating 1 3 GM 2 3 GM 2 1 ⋅ = → Rvir = Rm 2 5 Rvir 5 Rm 2 Evolution of Top-hat Perturbation
Padmanabhan (1994) Statistics of Hierarchical Clustering
• Press & Schechter (1974) derived a simple, yet accurate formula for estimating the number of virialized objects of mass M as a function of τ (or equivalently, z) • Basic idea is to smooth density field on a scale R at such that mass scale of interest satisfies
4π M = ρ (τ )R3 3 Press-Schechter Theory
• Because the density field (both smoothed and unsmoothed) is a Gaussian random field, probabilityτ that mean overdensity in spheres of δ radius R exceeds a critical value δc is πσ τ δ ∞ 1 2 p(R, ) = d exp− σ c ∫ 2 (R, ) 2 2 (R,τ ) δc where σ 2(R,τ) = d 3k W 2(kR)P(k,τ) ∫ T P-S Theory, cont’d
• P-S suggested this fraction be identified with the fraction of particles which are part of a nonlinear lump with mass exceeding M if we
take δc=1.686 ρ ∞ dn i.e. n( > M,τ) = 2 p(R,τ) =δ∫ dM ′ M π M σ dM ′ σ dn ρ ∂p 2 ρ d δ 2 ∴ = −2 = − c exp− c 2 2 dM M ∂M M dM 2σ n.b. factor of 2 is a fudge factor so that every particle belongs to a halo of some mass P-S theory, cont’d σ τ n+3 n+3 τ − − σ 2 6 (R, ) = D( ) 0 (R) ∝ DR ∝ DM explicitly, we find
n−3 3+n 1/ 2 dn 2 n M 6 M 3 dM = ρ 1+ exp− / 2dM π 2 dM M 3 M nl ( ) M nl ( )
σwhere now M nl satisfies τ τ (M nl , ) = δ c τ Mass function is a power-law for M
• PS under-predicts most massive objects and over- predicts “typical” objects
Jenkins et al. (1998)