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

2. SOME BASIC NUMBERS

6. 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

7. 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)

8. 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” D
Or “textures” 3-D

9. 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

10. 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)

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

12. A « WITH THE HANDS » FLAVOUR OF WHAT HAPPENS

13. 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

14. 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

15. 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

16. LINEARIZATION OF THE HYDRODYNAMICAL EQUATIONS
SOME EQUATIONS:
LINEARIZATION OF THE HYDRODYNAMICAL EQUATIONS

17. WHICH KIND OF MATTER DOMINATES ?

19. SOME BASIC DEFINITONS

20. 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):

22. 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…

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

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

25. STILL STAGNATION/MESZAROS EFFECT ANOTHER VIEW

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

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

28. 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

29. 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

30. 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

31. 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

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

33. 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

34. 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

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

36. 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

37. 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

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

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

40. 3 regimes of CMB power spectrum
Acoustic oscillations
Damping tail
Large scale plateau

41. COUPLED PERTURBATIONS OF PHOTONS AND BARYONS:THE STORY IN SHORT

42. A MORE DETAILED VIEW ALTHOUGH PEDAGOGICAL SOURCES OF CMB FLUCTUATIONS

48. CMB- TEGMARK-SIMULATION

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