MATTEO VIEL STRUCTURE FORMATION INAF and INFN Trieste SISSA - 28, th February/ 3 rd March 2011

OUTLINE: LECTURES 1.Structure formation: tools and the high redshift universe 2. The dark ages and the universe at 21cm 3. IGM cosmology at z=2=6 4. IGM astrophysics at z= Low redshift: gas and galaxies 6. Cosmological probes LCDM scenario

OUTLINE: LECTURE 1 Tools for structure formation: Press & Schecther theory Power spectrum, Bispectrum Results from numerical simulations Importance of first structure for particle physics and cosmology Books: Coles & Lucchin, Peacock (chapter 15)

LINEAR THEORY OF DENSITY FLUCTUATIONS-I Newtonian equations for the evolution of density and velocity under the influence of an external gravitational potential (see also Jeans theory) We still miss Poisson equation and an equation of state relating p and  Change of variable in an expanding universe: New fluids equations: Euler equation New term Convective derivative = comoving derivative Check also Peacock’s book Sect Peculiar vel. Density contrast Conformal time Comoving position In absence of pressure and forces v ~ 1/a

LINEAR THEORY OF DENSITY FLUCTUATIONS-II Poisson’s equation 1- take divergence of Euler equation 2- eliminate gradient of v using continuity 3- use Poisson pressure-free dust universe Pressure-free dust universe + Eds Growing mode Decaying mode

LINEAR THEORY OF DENSITY FLUCTUATIONS-III Open universe  =0 Flat universe  =0 Zel’dovich approximation for structure formation Self-similar growth of density structures with time (Note that in Eds potent is const) Euler equation in linearized form Double integral which is proportional to DDouble integral which is proportional to D Check also Peebles 1980, sects EdS at high-redshift to Low  at low redshift is faster in  CDM

LINEAR THEORY OF DENSITY FLUCTUATIONS-IV Zel’dovich (1970) Formulation of linear theory Lagrangian in nature: extrapolate particles positions in the early universe, kinematic approximation Pancakes, optimized Zel’dovich approximations schemes, application to galatic spin This approximation neglects non-linear evolution of the acceleration and uses Linear theory even in the non-linear regime

LINEAR THEORY OF DENSITY FLUCTUATIONS-V Viel et al. 2002

LINEAR THEORY OF DENSITY FLUCTUATIONS: SPHERICAL COLLAPSE Simplest model for the formation of an object Birkhoff’s theorem in GR Evolution of the scale factor a First integral of evolution equation Solutions E<0 For small  values Extrapolation of linear theory describes the non-linear collapse of an object See also ellipsoidal collapse

PS THEORY - I

PS THEORY - II A method is needed for partitioning the density field at some initial time t i into a set of disjoint regions each of which will form a nonlinear object at a time t f Key-assumption:  s is a random Gaussian field  c = Time enters D Mass enters  0 and its derivative Filtering scale R

PS THEORY - III  determines dependence of mass variance on volume Synchronic U ~ l 2 Diachronic U~l 1/5

Excursion set approach to mass functions -I Variance of smoothed field Initial overdensity Low res High res Bond et al Markov Chains

Excursion set approach to mass functions-II Variance of smoothed field Initial overdensity Low res High res iii) Is the first upcrossing point! Same press & schechter derivation but with right factor 2 interpreted in a probabilistic way using Markov Chains in Fourier space

Excursion set approach to mass functions: random walks

Excursion set approach to mass functions: random walks - II

Excursion set approach to mass functions: random walks - III

Excursion set approach to mass functions: random walks - IV

PS within merger tree theory - I Conditional probability Of course important for any galaxy formation (or structure formation) model Press & Schecter theory or N-body simulations are now the inputs of any cosmological model of structure formation

PS within merger tree theory - II Distribution of formation Redshifts M/2  M Probability of having a M1 prog. Hierarchical formation but self-similarity is broken n=0 n=-2,-1,1

Sheth & Tormen mass function Sheth & Tormen 1998 PS74 ST98 Universal N-body calibrated mass function for many cosmological models (p=0.3, A=0.332,a=0.707)

Mass function and its evolution In practice it is better to compute mass variance in Fourier space: KEY INGREDIENT IS MASS VARIANCE AND DEPENDS ON P(k) Reed et al. 2003, MNRAS, 346, 565

Mass function and its evolution -II KEY INGREDIENT FOR HIGH REDSHIFT COSMOLOGICAL MODELS High redshift SDSS QSOs Reionization sources First stars

Summary of theory Linear theory simple and powerful: modes scale as scale factor Press & Schecter is a relatively good fit to the data Support for a hierarchical scenario of structure formation for the dominant dark matter component (baryons are a separate issue at this stage) Springel, Frenk, White, Nature 2006

Formation of structures in the high redshift universe - I Main results found recently: Typical first generation haloes are similar in mass to the free-streaming mass limit (Earth mass or below) They form at high redshift (universe is denser) and are thus dense and resistant to later tidal disruption The mass is primarily in small haloes at z>20 Structure builds up from small mass (Earth like) to large (e.g. MW) by a subsequence of mergers

Formation of structures in the high redshift universe - II Primordial CDM inhomogeneities are smeared out by collisional damping and free-streaming Damping scale depends on the actual dark matter model but tipically is sub-parsec Green, Hofmann, Schwarz 2004, MNRAS, 353, L23 Sharp cutoff generation of haloes form abruptly. Mass variance independent of mass and many masses collapse

Comparing a cluster at z=0 with high redshift assembly of matter Diemand, Kuhlen, Madau (2006) RAPID SLOW

Subhaloes population at z=0 Kuhlen, Diemand, Madau, Zemp, 2008, Subhaloes are self-similar and cuspy Tidally truncated in the outer regions Subhaloes Main haloProxy for halo mass

Taken from Simon’s White talk at GGI (Florence) on February 10 th 2009 Using extended Press & Schecter (EPS) for the high-z universe

Using extended Press & Schecter (EPS) for the high-z universe-II

Numerical N-body effects largerly affected by missing large scale power Using extended Press & Schecter (EPS) for the high-z universe-III

Numerical N-body effects largerly affected by missing large scale power Using extended Press & Schecter (EPS) for the high-z universe-IV

Numerical N-body effects largerly affected by missing large scale power Using extended Press & Schecter (EPS) for the high-z universe-V

Numerical N-body effects largerly affected by missing large scale power Using extended Press & Schecter (EPS) for the high-z universe-VI

Using extended Press & Schecter (EPS) for the high-z universe-VII

Using extended Press & Schecter (EPS) for the high-z universe CONCLUSIONS: Important for detection Important for first stars Important for diffuse HI

FURTHER STATISTICAL TOOLS

0-pt, 1-pt, 2-pt, 3-pt,……. n-pt statistics of the density field Ideally one would like to deal with  DARK MATTER in practice  ASTROPHYSICAL OBJECTS (galaxies,HI, etc…) 0-pt: calculate the mean density 1-pt: calculate probability distribution function (pdf) 2-pt: calculate correlations between pixels at different distances (powerspectrum) 3-pt: calculate correlations in triangles (bispectrum) STATISTICS OF DENSITY FIELDS Viel, Colberg, Kim 2008

The power spectrum P(k) Density contrast Correlation function Power spectral density of A

Nichol arXiv: k eq ~  m h 2 Cutoff in the P(k) sets transition matter-radiation: fluctuations below this scale cannot collapse in the radiation era z eq ~  m h 2 The power spectrum P(k): an example of its importance

Matarrese, Verde, Heavens 1997 – Fry 1994 The bispectrum Use Gaussian part -- NonGaussian part Note that in the pure gaussian case The statistics is fully determined by the Power spectrum Applied by Verde et al. (2002) on 2dF galaxies To measure b 1 =1

A connection to particle physics and gamma rays

The density profile convergence

The number of sub-haloes

Extrapolating a bit…. !!!

DM around the sun

 -rays

SUMMARY 1 – Linear theory + Press & Schechter: simple tool to get abundance of collapsed haloes at any redshift 2- Sheth & Tormen and other fitting N-body based formulae Importance of describing the number of haloes at high redshift as a potentially fundamental cosmological tool 3- Numerical simulations and EPS in the high redshift universe (neutralino dark matter) 4- Further statistical tools (power spectrum, bispectrum mainly) 5- The link to the z~0 universe. Perspectives for indirect DM detection