Ruth Durrer Départment de physique théorique, Université de Genève

1. The Physics of the cosmic microwave background Bonn, August 31, 2005 Ruth Durrer Départment de physique théorique, Université de Genève

2. Contents • Introduction • Linear perturbation theory- perturbation varibles, gauge invariance- Einstein’s equations- conservation & matter equations- simple models, adiabatic perturbations- lightlike geodesics- polarisation • Power spectrum • Observations • Parameter estimation- parameter dependence of CMB anisotropies and LSS- reionisation- degeneracies • Conlusions

3. The cosmic micro wave background, CMB • Afterrecombination(T ~ 3000K, t~3x105 years) the photons propagate freely, simply redshifted due to the expansion of the universe • The spectrum of the CMB is a ‘perfect’ Planck spectrum: |m| < 10-4 y < 10-5

4. CMB anisotropies COBE (1992) WMAP (2003)

5. The CMB has small fluctuations, D T/T »a few£ 10-5. As we shall see they reflect roughly the amplitude of the gravitational potential. => CMB anisotropies can be treated with linear perturbationtheory. The basic idea is, that structure grew out of small initial fluctuations by gravitational instability. => At least the beginning of their evolution can be treated with linear perturbationtheory. As we shall see, the gravitational potential does not grow within linear perturbation theory. Hence initial fluctuations with an amplitude of »a few£ 10-5 are needed. During a phase of inflationary expansion of the universe such fluctuations emerge out of the quantum fluctuations of the inflation and the gravitational field.

6. metric perturbations • Decomposition into scalar, vector and tensor components Linear cosmological perturbation theory

7. Density and velocity stress tensor Perturbations of the energy momentum tensor

8. Gauge invariant metric fluctuations (the Bardeen potentials) _ In longitudinal gauge, the metric perturbations are given by h(long) = -2 d2 -2ijdxi dxj Gauge invariance Linear perturbations change under linearized coordinate transformations, but physical effects are independent of them. It is thus useful to express the equations in terms of gauge-invariant combinations. These usually also have a simple physical meaning. Y is the analog of the Newtonian potential. In simple cases F=Y.

9. The Weyl tensor The Weyl tensor of a Friedman universe vanishes. Its perturbation it therefore a gauge invariant quantity. For scalar perturbations, its ‘magnetic part’ vanishes and the electric part is given by Eij = Ciju u = ½[ij( +) -1/3(+)]

10. The anisotropic stress potential P The entropy perturbation w=p/r c2s=p’/r’ Velocity and density perturbations Gauge invariant variables for perturbations of the energy momentum tensor

11. constraints + dynamical _ • Conservation equations + • Einstein equations

12. matter radiation x=cskh Simple solutions and consequences • The D1-mode is singular, the D2-mode is the adiabatic mode • In a mixed matter/radiation model there is a second regular mode, the isocurvature mode • On super horizon scales, x<1, Y is constant • On sub horizon scales, Dg and V oscillate while Y oscillates and decays like 1/x2 in a radiation universe.

13. Simple solutions and consequences (cont.) radiation in a matter dominated background with Purely adiabatic fluctuations, Dgr = 4/3 Dm

14. + + RD ‘90 acoustic oscillations gravitat. potentiel (Sachs Wolfe) Doppler term integrated Sachs Wolfe ISW lightlike geodesics From the surface of last scattering into our antennas the CMB photons travel along geodesics. By integrating the geodesic equation, we obtain the change of energy in a given direction n: Ef/Ei = (n.u)f/(n.u)i = [Tf/Ti](1+ DTf /Tf -DTi /Ti) This corresponds to a temperature variation. In first order perturbation theory one finds for scalar perturbations

15. Polarisation • Thomson scattering depends on polarisation: a quadrupole anisotropy of the incoming wave generates linear polarisation of the outgoing wave.

16. Q§ iU are the m = § 2 spin eigenstates, which are expanded in spin 2 spherical harmonics. Their real and imaginary parts are called the ‘electric’ and ‘magnetic’ polarisations (Seljak & Zaldarriaga, 97, Kamionkowski et al. ’97, Hu & White ’97) Polarisation can be described by the Stokes parameters, but they depend on the choice of the coordinate system. The (complex) amplitude iei of the 2-component electric field defines the spin 2 intensity Aij = i*j which can be written in terms of Pauli matrices as

17. E-polarisation (generated by scalar and tensor modes) B-polarisation (generated only by the tensor mode) E is parity even while B is odd. E describes gradient fields on the sphere (generated by scalar as well as tensor modes), while B describes the rotational component of the polarisation field (generated only by tensor or vector modes). Due to their parity, T and B are not correlated while T and E are

18. An additional effect on CMB fluctuations is Silk damping: on small scales, of the order of the size of the mean free path of CMB photons, fluctuations are damped due to free streaming: photons stream out of over-densities into under-densities. Tocompute the effects of Silk damping and polarisation we have to solve the Boltzmann equation for the Stokes parameters of the CMB radiation. This is usually done with a standard, publicly available code like CMBfast (Seljak & Zaldarriaga), CAMBcode (Bridle & Lewis) or CMBeasy (Doran).

19. Reionization The absence of the so called Gunn-Peterson trough in quasar spectra tells us that the universe is reionised since, at least, z» 6. Reionisation leads to a certain degree of re-scattering of CMB photons. This induces additional damping of anisotropies and additional polarisation on large scales (up to the horizon scale at reionisation). It enters the CMB spectrum mainly through one parameter, the optical depth t to the last scattering surface or the redshift of reionisation zre .

20. normal emission no emission Gunn Peterson trough In quasars with z<6.1 the photons with wavelength shorter that Ly-a are not absorbed. (from Becker et al. 2001)

21. consequence of statistical isotropy observed mean cosmic variance (if the alm ’s are Gaussian) The power spectrum of CMB anisotropies DT(n) is a function on the sphere, we can expand it in spherical harmonics

22. Large scales :The gravitational potential on the surface of last scattering, time dependence of the gravitational potentialY ~ 10-5 . q > 1o l<100 6’< q < 1o 100<l<800 • Intermediate scales : Acoustic oscillations of the baryon/photon fluid before recombination. • Small scales : Damping of fluctuations due to the imperfect coupling of photons and electrons during recombination • (Silk damping). q < 6’ 800 > l The physics of CMB fluctuations

23. Power spectra of scalar fluctuations l

24. WMAP data Temperature (TT = Cl) Polarisation (ET) Spergel et al (2003)

25. Newer data I CBI From Readhead et al. 2004

26. Newer data II The present knowledge of the EE spectrum. (From T. Montroyet al. 2005)

27. Observed spectrum of anisotropies Tegmark et al. 03

28. Acoustic oscillations Determine the angular distance to the last scattering surface, z1

29. more baryons Most cosmological parameters have complicated effects on the CMB spectrum larger L Dependence on cosmological parameters

30. Degeneracy: Flat Universe: shift Geometrical degeneracy Flat Universe (ligne of constant curvature WK=0 ) degeneracy lines: =  h2

31. blue, nS > 1 nS = 1 : scale invariant spectrum (Harrison-Zel’dovich) red, nS < 1 The ‘smoking gun’ of inflation, has not yet been detected: B modes of the polarisation (QUEST, 2006). nT > 0 Tensor spectum:(gravity waves) nT > 0 Primordial parameters Scalar spectum: scalar spectral index nS and amplitude A

32. On the other hand: Wtot = 1.02 +/- 0.02 with the HST prior on h... a rigid constraint which is in slight tension with nucleosynthesis? wbar = 0.02 + 0.002 WL =0.73§0.11 zreion ~ 17 unexpectedly early reionisation Attention: FLATNESS imposed!!! Mesured cosmological parameters (With CMB + flatness or CMB + Hubble) Spergel et al. ‘03

33. Forecast1: WMAP 2 year data (Rocha et al. 2003) wb = Wbh2 wm = Wmh2 wL = WLh2 ns spectral index Q quad. amplit. R angular diam. t optical depth

34. Forecast2: Planck 2 year data (Rocha et al. 2003) Forecast2: Planck 2 year data

35. Forecast3: Cosmic variance limited data (Rocha et al. 2003)

36. Sn1a, Riess et al. 2004 (green) CMB + Hubble (orange) Bi-spectrum , Verde 2003 (blue) Evidence for a cosmologicalconstant (from Verde, 2004)

37. Conclusions • The CMB is a superb, physically simple observational tool to learn more about our Universe. • We know the cosmological parameters with impressive precision which will still improve considerably during the next years. • We don’t understand at all the bizarre ‘mix’ of cosmic components:Wbh2 ~ 0.02, Wmh2 ~ 0.16, WL~ 0.7 • The simplest model of inflation (scale invariant spectrum of scalar perturbations, vanishing curvature) is a good fit to the data. • What is dark matter? • What is dark energy? • What is the inflaton? ! We have not run out of problems in cosmology!