**4.4. Constraints on
from the cosmic
microwave background**

On large angular scales
1° photons
of the cosmic microwave background traveling to us from the last
scattering surface probe scales that were causally unconnected at the
time of recombination. ^{(8)}
As a result observations of the CMB anisotropy on large scales provide us
with a very clean probe of the primordial matter fluctuation spectrum
before its distortion by astrophysical processes. On such large scales
the main contribution to the CMB anisotropy comes from the Sachs-Wolfe
effect

(39) |

which relates temperature fluctuations to the integral of the variation of the metric evaluated along the line of sight [167]. The evaluation of (39) in a flat matter dominated universe is simplified by the fact that linearized the gravitational potential does not evolve with time, with the result that the above expression reduces to

(40) |

which relates fluctuations in the CMB to those in the gravitational potential at the surface of last scattering. Equation (40) can therefore be successfully used to determine the amplitude of primordial metric fluctuations with the help of COBE data. The presence of a cosmological constant however causes the linearized gravitational potential to evolve with time, the full Sachs-Wolfe integral (39) must therefore be used both to determine and normalize the primordial fluctuation spectrum [113].

The CMB temperature distribution can be written as

(41) |

where *T*_{0} is the blackbody temperature
*T*_{0} = 2.728 ± 0.004°*K*
[65].
*T* / *T* can
be written in terms of a multipole expansion
on the celestial sphere:

(42) |

Information pertaining to a particular theoretical model is contained
in the coefficients *a*_{lm} which are usually assumed to be
statistically independent and distributed in the manner of a Gaussian
random field with zero mean and variance

(43) |

where the angle brackets indicate an ensemble average over possible universes.

The quantity that is directly measured by observations is the angular correlation of the temperature anisotropy

(44) |

where cos =
_{1}^{ . }
_{2},
*P*_{l} are Legendre polynomials
and *W*_{l} is the filter
function of the experiment used to measure the CMB;
<> denote an ensemble average in the case of theoretical
predictions and angular average in the context of observations.
(The relationship between
*C*() and the
angular power spectrum *C*_{l} is analogous to that between
the two point correlation function
and the matter
power spectrum *P*(*k*).)

At low multipoles
*l* 60
the contribution to *C*_{l} is mainly from the Sachs-Wolfe
effect due
to scalar density perturbations and (in some models) tensorial gravity
waves.
(The value of the tenth multipole provides a convenient choice for
normalization of the perturbation spectrum
[22].)
At large *l* > 60 however, the main contribution to
*C*_{l} is due to oscillations
in the photon-baryon plasma before decoupling, which leave their
imprint in the CMB at the time of last scattering.
These oscillations give rise
to Doppler peaks in *C*_{l} the location of the peak being
determined by the angle subtended by the sound horizon at the time of
recombination (see figure 7).
The sound horizon depends upon
_{baryon}
& _{m}
whereas the angular diameter distance
to the last scattering surface depends upon
_{},
_{m} and the
spatial curvature of the universe. (Both
_{} and the spatial
curvature are extremely small
at the time of last scatter and therefore do not contribute to the
sound horizon. On the other hand, the location of the doppler peak is not
very sensitive to
_{baryon}
provided
_{baryon}
<< _{m}
+ _{}.)

Since the angular scale corresponding to the first Doppler peak
is sensitive to both the curvature of the
universe and its matter content, its location
can be used to place strong constraints on cosmological models.
There are some indications that the first Doppler peak has been measured
near *l* 260
Einstein is quoted as saying
[86].
(The height of the peak is related to the baryon fraction in the
universe and also to the scalar/tensor ratio S/T,
the larger the baryon density the higher the peak, a small value of S/T
reduces the peak height. The peak height also depends on the rate of
expansion of the universe and hence on *H*_{0}
[97]; for low values
_{baryon}
0.05 the peak
height decreases if *H*_{0} increases,
whereas the reverse is true for a larger baryon fraction.)
In figure 7 we show the angular power spectrum
of the cosmic microwave background for the flat
CDM model with
_{} = 0.7 (dotted
line), for comparison we also
show spatially flat (solid line) and open (dashed line)
matter dominated models with
_{m} = 1 and
_{m} = 0.3
respectively.

It should however be pointed out that the CMB alone cannot uniquely
differentiate between two models having identical matter content,
perturbation
spectra and with the same angular diameter distance to the last scattering
surface. Such models will be degenerate in the sense that they will
produce very similar CMB anisotropies
[56,
57].
A degeneracy in parameter space happens to be a common feature of most
cosmological tests. Fortunately different tests often have complementary
degeneracies.
(A degeneracy arises when a result remains unaffected by a specific
combination of parameter changes.) For instance the degeneracy in the
_{m} -
_{} plane
from high redshift supernovae tests is almost orthogonal to that
in a CMB analysis. Thus combining Type 1a supernovae measurements
with the results from CMB experiments can serve to substantially
decrease the errors on expected values of
_{m} and
_{} as
illustrated in figure 8 and
figure 9
[209,
190,
56,
57].
Since the location of the Doppler peak near
*l* 260 supports
a spatially flat universe
[86],
a combined likelihood analysis of CMB anisotropy
and Type 1a Supernovae data gives the best fit values
[57]

(45) |

which strongly favour a flat universe with
_{m} +
_{}
1 (also see
[130,
191]).

^{8} In matter dominated models
the horizon at last scattering subtends an angle
1.8°
_{m}^{1/2}(1000 /
*z*_{rec})^{1/2}
1.8° for
_{m}
1 and
*z*_{rec}
1000. In flat
dominated models the dependence of
on
_{m} is much
weaker, consequently
1.8° provides a
good approximation for most values of
_{m}.
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