Download SISSA lect 1 28/02/11 and 03/03/11 - INAF

STRUCTURE FORMATION
MATTEO VIEL
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
Books: Coles & Lucchin, Peacock (chapter 15)
Results from numerical simulations
Importance of first structure for particle physics and cosmology
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)
Convective derivative = comoving derivative
We still miss Poisson equation and an equation of state relating p and r
Change of variable in an expanding universe:
Comoving position
Peculiar vel.
Density contrast
Conformal time
New fluids equations:
New term
Euler equation
In absence of pressure and forces v ~ 1/a
Check also Peacock’s book Sect. 15.2
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
Check also Peebles 1980, sects.10-13
Open universe L=0
Flat universe k=0
EdS at high-redshift to Low W at low redshift is faster in LCDM
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
D
o
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 h 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 ti into a set
of disjoint regions each of which will form a nonlinear object at a time tf
Filtering scale R
Key-assumption: ds is a random Gaussian field
d c = 1.686
Time enters D
Mass enters D0 and
its derivative
PS THEORY - III
Diachronic U~l 1/5
Synchronic U ~ l2
a determines dependence of mass variance on volume
Bond et al. 1991
Initial overdensity
Excursion set approach to mass functions -I
Variance of smoothed field
Low res
High res
Markov Chains
Initial overdensity
Excursion set approach to mass functions-II
iii) Is the first upcrossing point!
Variance of smoothed field
Low res
High res
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
Probability of having a M1 prog
Hierarchical formation but self-similarity is broken
Distribution of formation
Redshifts M/2  M
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
Reed et al. 2003, MNRAS, 346, 565
In practice it is better to
compute mass variance in
Fourier space:
KEY INGREDIENT IS MASS VARIANCE AND DEPENDS ON P(k)
Mass function and its evolution -II
High redshift SDSS QSOs
Reionization sources
First stars
KEY INGREDIENT FOR HIGH REDSHIFT COSMOLOGICAL MODELS
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
RAPID
SLOW
Comparing a cluster at z=0 with high
redshift assembly of matter
Diemand, Kuhlen, Madau (2006)
Subhaloes population at z=0
Kuhlen, Diemand, Madau, Zemp, 2008,
Subhaloes are self-similar and cuspy
Tidally truncated in the outer regions
Main halo
Subhaloes
Proxy for halo mass
Using extended Press & Schecter (EPS) for the high-z universe
Taken from Simon’s White talk at GGI (Florence) on February 10th 2009
Using extended Press & Schecter (EPS) for the high-z universe-II
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
Numerical N-body effects largerly affected by missing large scale power
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
STATISTICS OF DENSITY FIELDS
0-pt, 1-pt, 2-pt, 3-pt,……. n-pt statistics of the density field
Ideally one would like to deal with d DARK MATTER
in practice d 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)
Viel, Colberg, Kim 2008
The power spectrum P(k)
Correlation function
Density contrast
Power spectral density of A
The power spectrum P(k): an example of its importance
Cutoff in the P(k) sets transition matter-radiation: fluctuations below
this scale cannot collapse in the radiation era
k eq ~ 0.075 Wm h2
Nichol arXiv: 0708.2824
z eq ~ 25000 Wm h2
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 b1=1
Matarrese, Verde, Heavens 1997 – Fry 1994
A connection to particle physics and gamma rays
The density profile convergence
The number of sub-haloes
Extrapolating a bit…. !!!
DM around the sun
g-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