THE FORMATION OF STRUCTURES AND THE GROWTH OF FLUCTUATIONS A COMPETITION BETWEEN UNIVERSAL EXPANSION AND LOCAL COLLAPSE

SOME BASIC NUMBERS

Structure Formation FUNDAMENTAL INGREDIENTS Gravitational Instability of DIFFERENTS kinds of matter (embedded in photon fluid) Primordial Fluctuations Modification of Fluctuations Linear evolution (Non-Linear Evolution) Matter distribution as a probe of Structure formation & Geometry CMB as a probe of Structure formation & Geometry The real game: A mixture of relativistic and NR (coupled) fluids: at least photons+ CDM + baryons in an expanding Universe ruled by GR Needs Use of Einstein + Hydrodynamic + Boltzmann Equations Here: a 1st order analytical and heuristic approach

Several Types of Primordial Fluctuation Adiabatic Corresponding to changes in volume in the early universe. Changes number density of photons and matter particles equally but their mass densities change differently ( predicted by inflation) Iso-curvature Start with no perturbations in the density field but with fluctuations in the matter opposed to the radiation dg = -dm Iso-thermal Radiation field unperturbed, fluctuations in matter only (ruled out by CMB observations)

Primordial Fluctuations Possibilities are quantum mechanical “Gaussian” Fluctuations which arise naturally in Inflationary theories A second possiblity is defects which might arise from phase transitions in the early Universe “Cosmic strings” 1-D “Domain walls” 2-D Or “textures” 3-D

What Inflation says about Primordial fluctuations Just after inflation, they are expected to be: Roughly scale-invariant Gaussian Consider histogram of densities Adiabatic but could also have isocurvature components Scalar, tensor and maybe even vortical

Primordial Fluctuations A common assumption is that the fluctuations have the same amplitude d ~ 10-4 when they enter the horizon This gives a scale-free or Harrison-Zeldovich spectrum Log P(k) Harrison-Zeldovich Log (k)

WHY SAME AMPLITUDE AT HORIZON IF n =1 (ANTICIPATING ON THE EQUATIONS )

A « WITH THE HANDS » FLAVOUR OF WHAT HAPPENS

GRAVITATIONAL INSTABILITY PARADIGM JEANS MASS CONCEPT (1902) : EXACT CALCULATION DONE BY LINEARIZATION OF HYDRODYNAMICAL EQUATIONS NO SCREENING EFFECT IN GRAVITATIONAL PLASMA ALWAYS ATTRACTION AND ACCRETION OF MASS , SO IN PRINCIPLE: CONTINUOUS GROWTH OF DENSITY CONTRASTS CHARATERIZED BY t collapse ~ 1/ √( G ) BUT: THERMAL PRESSION COUNTER-BALANCES (t sound ) SO: OSCILLATION REGIME (t sound < t collapse ) BELOW A GIVEN (JEANS) MASS AND COLLAPSE (t collapse < t sound) AND GROWTH ABOVE A GIVEN (JEANS) MASS AND SURDENSITY GROWS EXPONENTIALLY WITH TIME !! BUT : JEANS CALCULATION DONE WITHIN A STATIC DENSITY BACKGROUND…. UNIVERSE IN EXPANSION (t Exp ) AGAIN GROWTH BUT ONLY IN POWER LAW REMARK: SAME CONCEPT AVAILABLE FOR WIMPS USING KINETIC ENERGY AGAINST GRAVITATION

WHAT AFFECTS GROWTH OF FLUCTUATIONS… GROWTH UNEFFECTIVE IF: t SOUND < t COLLAPSE t EXP < t COLLAPSE => SO WAIT MATTER DOMINATES RADIATION (AT Z Equil) !!! (stagnation/Meszaros effect) SEVERAL PREFERED SCALES WOULD APPEAR SILK DAMPING FOR BARYONS: COMPETITION BETWEEN GRAVITATION AND RADIATION PRESSION IN A SURDENSITY SIMILARLY: FREE STREAMING FOR ~ RELATIVISTIC WIMPS, SMALL SCALES ERASED !! DIFFERENT SCALES DEPENDING ON THE « NATURE » OF MATTER ! BARYONS: NOTHING UNDER SILK MASS BUT KILLED BY CMB « HOT DARK MATTER » LIKE MASSIVE NEUTRINOS => LARGE SCALES STRUCTURES FORMED 1ST LEADING LATELY TO OTHERS BY « FRAGMENTATION » (TOP DOWN SCENARIO), KILLED BY AGE OF QSO! « COLD DARK MATTER » , (BOTTOM UP SCENARIO) SMALL SCALES FIRST , LEAD TO HIERARCHICAL MODEL

SO: THE BASIC ASSUMPTIONS OF LINEAR THEORY ARE: = Assume all fluctuations are very small Expand all equations to first order (x) ' 0 (1 + (x)) Cannot be valid today since  >>1 Expected to work at early times on large scales

LINEARIZATION OF THE HYDRODYNAMICAL EQUATIONS SOME EQUATIONS: LINEARIZATION OF THE HYDRODYNAMICAL EQUATIONS

WHICH KIND OF MATTER DOMINATES ?

SOME BASIC DEFINITONS

Definition of P(k) Fourier Transform of 3d matter distribution Average over all directions k=(kx2 + ky2 + kz2)0.5 Present day universe P(k):

Why study P(k)? Shape depends on m total matter content b baryon content DE dark energy content Probed by galaxy surveys: 2dFGRS, SDSS Theory underpins CMB Cosmic shear Lyman- forest…

Key features Dependence of turnover position on m Baryon supression and wiggles Slope at small k ~/ k Slope at large k ~/ k-3

WHAT IS MESZAROS/STAGNATION EFFECT AND WHY A TURN OVER IN THE POWER SPECTRUM ?

STILL STAGNATION/MESZAROS EFFECT ANOTHER VIEW

WHY A TURN OVER IN P(K) ? : ANOTHER VIEW

THE Transfer Function Log Tk CDM Baryons MDM HDM Log k Large scales Small scales

AND the present Power Spectrum Large scales: primordial ! Small scales: « processed » Determined from CMB on large scales and « Astrophysics » at small scales Good concordance even in amplitude

TO SUMMARIZE A LITTLE m ~ 0.3 FOR CDM: ~ almost NOTHING Growth of density contrast: δ(x) = δρ/ρ is expected due to gravitation on all scales We start with the PRIMORDIAL Power Spectrum P(k) = <|δ(k)| 2> ~ kn (n=1) (Harrison Zeldovich spectrum from maximal ignorance principle and inflation ) BUT: Primordial P(k) is processed ( depending on HDM/CDM/baryons….) to yield P(k) today quantified by the transfer function T(k) such that: P(k) today = T 2(k) * P(k) primordal where T 2(k) is the transfer function FOR CDM: ~ almost NOTHING UNTIL Z Equil TO WHICH CORRESPONDS THE SIZE OF THE HORIZON m h2 SO INITIAL POWER SPECTRUM BEFORE THIS SCALE AND TURN OVER AFTER => m ~ 0.3

Modifications of Fluctuations Dr/r Dark matter Baryons CDM dominated Radiation dominated Post-recombination Baryons collapse into potential wells of DM NON LINEAR EVOLUTION => NUMERICAL SIMULATIONS R or t

By the way: What is 8 ? R ≡ rms mass fluctuation in spheres of radius R Mpc R=8 was chosen since gave 8 ~ 1 R2 = s W(k,R) P(k) dk Using linear theory P(k) 8 ~ amplitude of P(k) at k ~ 2  /8 Mpc-1 Often used to fix amplitude of primordial fluctuations, A in Pi(k) = A kn

Why 8 Mpc? ~ 1 at 8 Mpc Fig from Hawkins et al 2003

AND about Bias ≡ galaxy power spectrum is a constant multiple of the matter power spectrum Pg(k) = b2 P(k) 8g = b 8 Assumed for 2dFGRS and SDSS cosmological parameter analyses Could more generally have b(k) = non-linear bias eg. b=b0 + b1 k

AN INSERT: HOW COULD DARK ENERGY PLAY A ROLE? Play a role in growth since it is the global density which drives the expansion and rules t EXP No local gravitational role because smooth !!! So Newtonian dynamics valid standard  models Modification of the transfer function Décroissance plus précoce du taux de croissance

SO CDM FLUCTUATIONS SO: CDM FLUCTUATIONS GROW UNDER GRAVITATIONAL INSTABILITY LEADING TO DARK HALOES WHAT HAPPENS TO BARYONS ? BARYON STORY AND THE CMB STUFF

The Isotropy of the CMB CMB = snapshot of z~1000 universe z~1000 universe was homogeneous Leads to 'Horizon problem' Horizon size ~ c x time since Big-Bang Horizon at z~1000 is ~ 1° on sky Sky at 0° and 180° not yet 'causally connected' 'Inflation' invoked to solve Rapid expansion expands horizon scale to greater than observable universe size

A characteristic scale exists of ~ 1 degree What WMAP saw Zooming the colour scale…1 in 1000 Removing the effect of our motion through the galaxy A characteristic scale exists of ~ 1 degree

Statistical properties Spherical harmonic transform ~Fourier transform Quantifies clumpiness on different scales

What are the Clℓs? Qualitatively: ~power in each Fourier mode Quantitatively:

3 regimes of CMB power spectrum Acoustic oscillations Damping tail Large scale plateau

COUPLED PERTURBATIONS OF PHOTONS AND BARYONS:THE STORY IN SHORT

A MORE DETAILED VIEW ALTHOUGH PEDAGOGICAL SOURCES OF CMB FLUCTUATIONS

CMB- TEGMARK-SIMULATION

NB: LE «  GEL » AU DESSUS DE L’HORIZON DEPEND DE LA JAUGE, ICI ON A PRIS LA CROISSANCE