MATTEO VIEL

STRUCTURE FORMATION

INAF and INFN Trieste

SISSA - 28,th February/ 3rd 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=2-6

5. 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 densityand velocity under the influence of an externalgravitational 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. 15.2

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 equation2- eliminate gradient of v using continuity3- 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 D

Check also Peebles 1980, sects.10-13

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 positionsin the early universe, kinematic approximation

Pancakes, optimized Zel’dovich approximations schemes, application to galaticspin

This approximation neglects non-linear evolution of the acceleration and usesLinear 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 describesthe 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 ti into a setof disjoint regions each of which will form a nonlinear object at a time tf

Key-assumption: s is a random Gaussian field

c = 1.686

Time enters DMass enters 0 and its derivative

Filtering scale R

PS THEORY - III

determines dependence of mass variance on volume

Synchronic U ~ l2

Diachronic U~l 1/5

Excursion set approach to mass functions -I

Variance of smoothed field

Init

ial overd

ensi

ty

Low res High res

Bond et al. 1991

Markov Chains

Excursion set approach to mass functions-II

Variance of smoothed field

Init

ial overd

ensi

ty

Low res High res

iii) Is the first upcrossing point!

Same press & schechter derivation but with right factor 2 interpretedin 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 importantfor any galaxy formation (or structure formation) model

Press & Schecter theory or N-body simulations are now the inputsof any cosmological model of structure formation

PS within merger tree theory - II

Distribution of formationRedshifts 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 functionSheth & Tormen 1998

PS74

ST98

Universal N-body calibrated mass functionfor many cosmological models (p=0.3,A=0.332,a=0.707)

Mass function and its evolution

In practice it is better tocompute mass variance inFourier 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 theoryLinear 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 dominantdark 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 masslimit (Earth mass or below)

They form at high redshift (universe is denser) and are thus dense and resistantto 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 subsequenceof 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 formabruptly. Mass variance independent of mass and many masses collapse

Comparing a cluster at z=0 with highredshift assembly of matter

Diemand, Kuhlen, Madau (2006)

RAPID SLOW

Subhaloes population at z=0Kuhlen, Diemand, Madau, Zemp, 2008,

Subhaloes are self-similar and cuspyTidally truncated in the outer regions

Subhaloes

Main halo Proxy for halo mass

Taken from Simon’s White talk at GGI (Florence) on February 10th 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 universeCONCLUSIONS:

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 fieldIdeally one would like to deal with DARK MATTER

in practice ASTROPHYSICAL OBJECTS (galaxies,HI, etc…)

0-pt: calculate the mean density1-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

Corr

ela

tion f

unct

ion

Power spectral density of A

Nichol arXiv: 0708.2824

k eq ~ 0.075 m h2

Cutoff in the P(k) sets transition matter-radiation: fluctuations belowthis scale cannot collapse in the radiation era

z eq ~ 25000 m h2

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 caseThe statistics is fully determined by thePower spectrum

Applied by Verde et al.(2002) on 2dF galaxiesTo measure b1=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