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

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The physics of the cosmic microwave background bonn august 31 2005

The Physics of the cosmic microwave background Bonn, August 31, 2005

Ruth Durrer

Départment de physique théorique, Université de Genève


  • 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

The cosmic micro wave background cmb
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

Cmb anisotropies
CMB anisotropies

COBE (1992)

WMAP (2003)

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.

Linear cosmological perturbation theory

  • Decomposition into scalar, vector and tensor components

Linear cosmological perturbation theory

Perturbations of the energy momentum tensor

Density and velocity

stress tensor

Perturbations of the energy momentum tensor

Gauge invariance

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.

The weyl tensor
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(+)]

Gauge invariant variables for perturbations of the energy momentum tensor

The anisotropic stress potential


The entropy perturbation



Velocity and density perturbations

Gauge invariant variables for perturbations of the energy momentum tensor





  • Conservation equations


  • Einstein equations

Simple solutions and consequences




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.

Simple solutions and consequences cont
Simple solutions and consequences (cont.)

radiation in a matter dominated background with

Purely adiabatic fluctuations, Dgr = 4/3 Dm

Lightlike geodesics



RD ‘90

acoustic oscillations

gravitat. potentiel

(Sachs Wolfe)

Doppler term

integrated Sachs Wolfe


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


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

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


(generated by scalar and tensor modes)


(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

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).


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 .

Gunn peterson trough

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)

The power spectrum of cmb anisotropies

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

The physics of cmb fluctuations

  • Large scales :The gravitational potential on the surface of last scattering, time dependence of the gravitational potentialY ~ 10-5 .

q > 1o


6’< q < 1o


  • 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

Wmap data
WMAP data

Temperature (TT = Cl)

Polarisation (ET)

Spergel et al (2003)

Newer data i
Newer data I


From Readhead et al. 2004

Newer data ii
Newer data II

The present knowledge of the

EE spectrum.

(From T. Montroyet al. 2005)

Acoustic oscillations
Acoustic oscillations

Determine the angular distance to the last scattering surface, z1

Dependence on cosmological parameters

more baryons

Most cosmological parameters

have complicated effects on

the CMB spectrum

larger L

Dependence on cosmological parameters

Geometrical degeneracy


Flat Universe:


Geometrical degeneracy

Flat Universe (ligne of constant curvature WK=0 )

degeneracy lines:

=  h2

Primordial parameters

blue, nS > 1

nS = 1 : scale invariant spectrum


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

Mesured cosmological parameters

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

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

Forecast2 planck 2 year data

Forecast2: Planck 2 year data

(Rocha et al. 2003)

Forecast2: Planck 2 year data

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

Evidence for a cosmological constant

Sn1a, Riess et al. 2004


CMB + Hubble


Bi-spectrum , Verde 2003


Evidence for a cosmologicalconstant

(from Verde, 2004)


  • 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!