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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
• After recombination (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
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 perturbation theory.
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
perturbation theory.
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
• metric perturbations
•Decomposition into scalar, vector and tensor components
Perturbations of the energy momentum
Density and velocity
stress tensor
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.
Gauge invariant metric fluctuations (the Bardeen potentials)
Y is the analog of the Newtonian potential. In simple cases F=Y.
In longitudinal gauge, the metric perturbations are given by
hm(long) = -2Y d2 -2Fijdxi dxj
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 = Cmijum u = ½[i j(F +Y) -1/3D(F+Y)]
Gauge invariant variables for perturbations of the
energy momentum tensor
The anisotropic stress potential
The entropy perturbation
Velocity and density perturbations
•Einstein equations
• Conservation equations
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.)
radiation in a matter dominated background with
Purely adiabatic fluctuations, Dgr = 4/3 Dm
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
RD ‘90
acoustic oscillations
Doppler term
gravitat. potentiel
(Sachs Wolfe)
integrated Sachs Wolfe
• Thomson scattering depends on polarisation: a
quadrupole anisotropy of the incoming wave
generates linear polarisation of the outgoing wave.
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
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)
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).
(generated by scalar and tensor modes)
(generated only by the tensor mode)
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
To compute 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
In quasars with z<6.1 the photons
with wavelength shorter that Ly-a
are not absorbed.
normal emission
(from Becker et al. 2001)
no emission
The power spectrum of CMB anisotropies
DT(n) is a function on the sphere, we can expand it
in spherical harmonics
consequence of
statistical isotropy
observed mean
cosmic variance
(if the alm ’s are Gaussian)
The physics of CMB fluctuations
• Large scales : The gravitational potential
on the surface of last scattering, time
dependence of the gravitational potential
Y ~ 10-5 .
q > 1o
• Intermediate scales : Acoustic oscillations of
the baryon/photon fluid before
6’< q < 1o
• Small scales : Damping of fluctuations due to
the imperfect coupling of photons and
electrons during recombination
q < 6’
800 > l
(Silk damping).
Power spectra of scalar fluctuations
WMAP data
Temperature (TT = Cl)
Polarisation (ET)
Spergel et al (2003)
Newer data I
From Readhead et al. 2004
Newer data II
The present knowledge of the
EE spectrum.
(From T. Montroy et al. 2005)
Observed spectrum of anisotropies
Acoustic oscillations
Determine the angular distance to the last scattering surface, z1
Dependence on cosmological parameters
larger L
Most cosmological parameters
have complicated effects on
the CMB spectrum
Geometrical degeneracy
Flat Universe (ligne of
constant curvature WK=0 )
degeneracy lines:
 = W h2
Flat Universe:
Primordial parameters
Scalar spectum:
blue, nS > 1
scalar spectral index nS and
amplitude A
nS = 1 : scale invariant spectrum
red, nS < 1
The ‘smoking gun’ of
Tensor spectum: inflation, has not yet been
(gravity waves) detected: B modes of the
polarisation (QUEST,
nT > 0
nT > 0
Mesured cosmological parameters
(With CMB + flatness or CMB + Hubble)
a rigid constraint which is in slight
tension with nucleosynthesis?
bar = 0.02 + 0.002
WL =0.73§0.11
zreion ~ 17
unexpectedly early reionisation
Attention: FLATNESS imposed!!!
Spergel et al. ‘03
On the other hand: Wtot = 1.02 +/- 0.02 with the HST prior on h...
Forecast1: WMAP 2 year data
(Rocha et al. 2003)
b = Wbh2
m = Wmh2
L = WLh2
ns spectral index
Q quad. amplit.
R angular diam.
t optical depth
Forecast2: Planck 2 year data
(Rocha et al. 2003)
Forecast2: Planck 2 year data
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
(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:
The simplest model of inflation (scale invariant spectrum of scalar
perturbations, vanishing curvature) is a good fit to the data.
Wbh2 ~ 0.02, Wmh2 ~ 0.16, WL~ 0.7
• What is dark matter?
• What is dark energy?
• What is the inflaton?