2 d F Galaxy Redshift Survey M galaxies

2 d. F Galaxy Redshift Survey ¼ M galaxies 2003 n o iz 4 1/ CFA Survey 1980 of th e r o h

The Initial Fluctuations At Inflation: Gaussian, adiabatic a realization of an ensemble ensembe average ~ volume average fluctuation field Fourier Power Spectrum rms δ Pk k M

Scale-Invariant Spectrum (Harrison-Zel’dovich) mass M time

Cosmological Scales mass teq zeq~104 t 0 time

CDM Power Spectrum mass growth when matter is self-gravitating teq CDM t time Pk HDM free streeming Meq mass CDM kpeak m h k

Formation of Large-Scale Structure Fluctuation growth in the linear regime: rms fluctuation at mass scale M: Typical objects forming at t: example HDM: top-down CDM: bottom-up 1 1 free streaming 0 M

Power Spectrum

ΛCDM Power Spectrum normalization:

Lecture Non-linear Growth of Structure Spherical Collapse, Virial Theorem, Zel’dovich Approximation, N-body Simulations

Formation of Large-Scale Structure: comoving

Filamentary Structure: Zel’dovich Approximation Approximate the displacement from initial position Velocity & acceleration along displacement → trajectories straight lines as in linear central force → potential flow In physical coordinates Density (Lagrangian): Jacobian continuity → caustics filament 42% cluster 8% pancake 42% deformation tensor eigenvalues

Zel’dovich Approximation cont’d linear → D is the growing mode of GI obeying Error: plug density in Poisson eq. → error is 2 nd +3 rd terms error small in linear regime or pancakes error big in spherical collapse

Non-dissipative Pancakes: why flat? R oscilation time << expansion time adiabatic invariant pancake becomes flatter in time

N-body simulation CDM

N-body simulation

N-body simulation

N-body simulation spherical collapse or mergers

N-body simulation spherical collapse or mergers

N-body simulation spherical collapse or mergers

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N-body simulation of Halo Formation

N-body simulation of Halo Formation

Top-Hat Model (Λ=0, matter era) a bound sphere (k=1) in Ed. S universe (k=0) conformal time universe overdensity: 5. 55 linear perturbation Taylor turnaround linear equivalent to collapse 0 tmax tvir

universe 5. 55 4 200 8 perturbation 0 tmax tvir t

Spherical Collapse universe radius perturbation virial equilibrium time virial equilibrium:

Virial Scaling Relations Virial equilibrium: Spherical collapse: Weak dependence on time of formation: Practical formulae:

Lecture 6 Hierarchical Clustering Press Schechter Formalism

Press Schechter Formalism halo mass function n(M, a) Gaussian random field nonlinear random spheres of mass M linear-extrapolated δrms at a: σ linear fraction of spheres with δ>δc =1. 68: a(t) a 0 =1 δ δc x PS ansaz: F is the mass fraction in halos >M (at a) derivative of F with respect to M: Mo & White 2002

Press Schechter Formalism cont. log n(M) Example: M*(a) self-similar evolution, scaled with M* approximate log M time Pk in a flat universe Top Hat k. R=π R x k

Press Schechter cont. Better fit using ellipsoidal collapse (Sheth & Tormen 2002) Comparison of PS to N-body simulations log M 2 n(M) factor 2 log M

Press-Schechter in ΛCDM 14 2σ 13 12 1σ log M/Mʘ 11 10 9 Mo & White 2002

Press-Schechter Mo & White 2002

Merger Tree t

Mass versus Light Distribution halo mass galaxy stellar mass 40% of baryons