matteo viel structure formation inaf and infn trieste sissa - 28, th february/ 3 rd march 2011
TRANSCRIPT
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MATTEO VIEL
STRUCTURE FORMATION
INAF and INFN Trieste
SISSA - 28,th February/ 3rd March 2011
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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
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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)
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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
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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
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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
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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
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LINEAR THEORY OF DENSITY FLUCTUATIONS-V
Viel et al. 2002
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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
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PS THEORY - I
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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
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PS THEORY - III
determines dependence of mass variance on volume
Synchronic U ~ l2
Diachronic U~l 1/5
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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
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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
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Excursion set approach to mass functions: random walks
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Excursion set approach to mass functions: random walks - II
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Excursion set approach to mass functions: random walks - III
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Excursion set approach to mass functions: random walks - IV
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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
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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
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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)
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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
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Mass function and its evolution -II
KEY INGREDIENT FOR HIGH REDSHIFT COSMOLOGICAL MODELS
High redshift SDSS QSOs
Reionization sources
First stars
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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
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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
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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
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Comparing a cluster at z=0 with highredshift assembly of matter
Diemand, Kuhlen, Madau (2006)
RAPID SLOW
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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
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Taken from Simon’s White talk at GGI (Florence) on February 10th 2009
Using extended Press & Schecter (EPS) for the high-z universe
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Using extended Press & Schecter (EPS) for the high-z universe-II
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Numerical N-body effects largerly affected by missing large scale power
Using extended Press & Schecter (EPS) for the high-z universe-III
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Numerical N-body effects largerly affected by missing large scale power
Using extended Press & Schecter (EPS) for the high-z universe-IV
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Numerical N-body effects largerly affected by missing large scale power
Using extended Press & Schecter (EPS) for the high-z universe-V
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Numerical N-body effects largerly affected by missing large scale power
Using extended Press & Schecter (EPS) for the high-z universe-VI
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Using extended Press & Schecter (EPS) for the high-z universe-VII
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Using extended Press & Schecter (EPS) for the high-z universeCONCLUSIONS:
Important for detection
Important for first stars
Important for diffuse HI
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FURTHER STATISTICAL TOOLS
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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
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The power spectrum P(k)
Density contrast
Corr
ela
tion f
unct
ion
Power spectral density of A
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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
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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
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A connection to particle physics and gamma rays
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The density profile convergence
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The number of sub-haloes
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Extrapolating a bit…. !!!
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DM around the sun
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-rays
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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