The%20Physics%20of%20the%20cosmic%20microwave%20background%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20Bonn,%20August%2031,%202005

1
The Physics of the cosmic microwave background
Bonn,
August 31, 2005

• Ruth Durrer
• Départment de physique théorique, Université de
  Genève

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Contents

• Introduction
• Linear perturbation theory- perturbation
  varibles, gauge invariance- Einsteins
  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

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The cosmic micro wave background, CMB

• After recombination (T 3000K, t3x105 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 lt 10-4 y lt 10-5
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CMB anisotropies

• COBE (1992)

WMAP (2003)

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• 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.
• gt CMB anisotropies can be treated with linear
  perturbation theory.
• The basic idea is, that structure grew out of
  small initial
• fluctuations by gravitational instability.
• gt 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.

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Linear cosmological perturbation theory
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Perturbations of the energy momentum tensor
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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 FY.
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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?(??)

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Gauge invariant variables for perturbations of
the energy momentum tensor
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• Einstein equations

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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, xlt1, Y is constant
• On sub horizon scales, Dg and V oscillate while Y
  oscillates and decays like 1/x2 in a radiation
  universe.

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

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Polarisation

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

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• 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

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• 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
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• 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.
• 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).

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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 .
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Gunn Peterson trough
In quasars with zlt6.1 the photons with wavelength
shorter that Ly-a are not absorbed.
(from Becker et al. 2001)
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The power spectrum of CMB anisotropies
DT(n) is a function on the sphere, we can
expand it in spherical harmonics
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The physics of CMB fluctuations
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Power spectra of scalar fluctuations
l
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WMAP data
Temperature (TT Cl)
Polarisation (ET)
Spergel et al (2003)
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Newer data I
CBI
From Readhead et al. 2004
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Newer data II
The present knowledge of the EE spectrum.
(From T. Montroy et al. 2005)
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Observed spectrum of anisotropies
Tegmark et al. 03
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Acoustic oscillations
Determine the angular distance to the last
scattering surface, z1
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Dependence on cosmological parameters
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Geometrical degeneracy
degeneracy lines
Flat Universe (ligne of constant curvature WK0 )
? ? h2
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Primordial parameters
Scalar spectum
scalar spectral index nS and amplitude A
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Mesured cosmological parameters
(With CMB flatness or CMB Hubble)
Spergel et al. 03
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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
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Forecast2 Planck 2 year data
Forecast2 Planck 2 year data (Rocha et al. 2003)
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Forecast3 Cosmic variance limited data (Rocha et
al. 2003)
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Evidence for a cosmological constant
(from Verde, 2004)
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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 dont 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!