In the standard Friedmann model of the universe, neutral atomic systems
form at a redshift of about
*z*
10^{3} and the photons decouple from the matter at this
redshift. These photons, propagating freely in spacetime since then,
constitute the CMBR observed around us today. In an ideal Friedmann
universe, for a comoving observer, this radiation will appear to be
isotropic. But if physical process has led to inhomogeneities in the
*z* = 10^{3} spatial surface, then these inhomogeneities
will appear as angular anisotropies in the CMBR in the sky today. A
physical process operating at a proper length scale *L* on the
*z* = 10^{3} surface will lead to an effect at an angle
=
*L* / *d*_{A}(*z*). Numerically,

(95) |

To relate the theoretical predictions to observations, it is usual to
expand the temperature anisotropies in the sky in terms of the
spherical harmonics. The temperature anisotropy in the sky will provide
=
*T* / *T*
as a function of two angles
and
.
If we expand the temperature
anisotropy distribution on the sky in spherical harmonics:

(96) |

all the information is now contained in the angular coefficients
*a*_{lm}.

If **n** and **m** are two directions in the sky with an angle
between them, the
two-point correlation function of the temperature fluctuations in the
sky can be defined as

(97) |

Since the sources of temperature fluctuations are related linearly to
the density inhomogeneities, the coefficients *a*_{lm}
will be random fields with some power spectrum. In that case
< *a*_{lm}*a*^{*}_{l'm'}
> will be nonzero only if *l* = *l'* and *m* =
*m'*. Writing

(98) |

and using the addition theorem of spherical harmonics, we find that

(99) |

with *C*_{l} = < | *a*_{lm}|^{2} >.
In this approach, the pattern of anisotropy is contained in the variation
of *C*_{l} with *l*. Roughly speaking,
*l*
^{-1} and we can
think of the
(, *l* ) pair as
analogue of (**x**, **k**) variables
in 3-D. The *C*_{l} is similar to the power spectrum
*P*(**k**).

In the simplest scenario,
the primary anisotropies of the CMBR arise from three different sources.
(i) The first is the gravitational potential fluctuations at the last
scattering surface (LSS) which will contribute an anisotropy
(*T* /
*T*)_{}^{2}
*k*^{3}
*P*_{}(*k*) where
*P*_{}(*k*)
*P*(*k*) /
*k*^{4} is the power spectrum of gravitational potential
. This
anisotropy arises because photons climbing out of deeper gravitational
wells lose more energy on the average.
(ii) The second source is the Doppler shift of the frequency of the photons
when they are last scattered by moving electrons on the LSS.
This is proportional to
( *T* /
*T*)_{D}^{2}
*k*^{3}
*P*_{v} where *P*_{v}(*k*)
*P* /
*k*^{2} is the power spectrum of the velocity field.
(iii) Finally, we also need to take into account the intrinsic
fluctuations of the radiation field on the LSS. In the case of adiabatic
fluctuations, these will be proportional to the density fluctuations of
matter on the LSS and hence will vary as
( *T* /
*T*)_{int}^{2}
*k*^{3}
*P*(*k*).
Of these, the velocity field and the density field (leading to the
Doppler anisotropy and intrinsic anisotropy described in (ii) and (iii)
above) will oscillate at scales smaller than the Hubble radius at
the time of decoupling since pressure support will be effective at these
scales. At large scales, if
*P*(*k*)
*k*, then

(100) |

where
*k*^{-1}
is the angular scale
over which the anisotropy is measured. Obviously, the fluctuations due
to gravitational potential dominate at large scales while
the sum of intrinsic and Doppler anisotropies
will dominate at small scales. Since the latter two
are oscillatory, we sill expect an oscillatory behaviour in the
temperature anisotropies at small angular scales.

There is, however,
one more feature which we need to take into account. The above analysis
is valid if recombination was instantaneous; but in reality the thickness
of the recombination epoch is about
*z*
80
([220];
[44], chapter 3).
This implies that the anisotropies will
be damped at scales smaller than the length scale corresponding to a
redshift interval of
*z* = 80. The
typical value
for the peaks of the oscillation are at about 0.3 to 0.5 degrees depending
on the details of the model. At angular scales smaller than about
0.1 degree, the anisotropies are heavily damped by the thickness of the
LSS.

The fact that several different processes contribute to the
structure of angular anisotropies make CMBR a valuable
tool for extracting cosmological information. To begin with, the
anisotropy at very large scales directly probe modes which are
bigger than the Hubble radius at the time of decoupling
and allows us to directly determine the primordial spectrum.
Thus, in general, if the angular dependence of the spectrum at very large
scales is known, one can work backwards and determine
the initial power spectrum.
If the initial power spectrum is assumed to be *P*(*k*) =
*Ak*, then the observations
of large angle anisotropy allows us to fix the amplitude *A* of the
power spectrum
[207,
208].
Based on the results of COBE satellite
[221],
one finds that the amount of initial power per logarithmic band in
*k* space is given by

(101) |

This corresponds to
*A*
(28.6*h*^{-1} Mpc)^{4} and an initial fluctuation
in the gravitational potential of
3.1 × 10^{-5}.
This result is powerful enough to rule out matter dominated,
= 1 models
when combined with the data on the
abundance of large clusters which determines the
amplitude of the power spectrum at
*R*
8*h*^{-1} Mpc. For example the parameter values
*h* = 0.5,
_{0}
_{DM} = 1,
_{} =
0, are ruled out by this observation when combined with COBE observations
[207,
208].

As we move to smaller scales we are probing the behaviour of baryonic
gas coupled to the
photons. The pressure support of the gas leads to modulated acoustic
oscillations
with a characteristic wavelength at the *z* = 10^{3}
surface. Regions of high and low
baryonic density contrast will lead to anisotropies in the temperature
with the same characteristic wavelength.
The physics of these oscillations has been studied in several papers in
detail
[222,
223,
224,
225,
226,
227,
228,
229,
230].
The angle subtended by the wavelength of these
acoustic oscillations will lead to a series of peaks in the temperature
anisotropy which has been detected
[231,
232].
The structure of acoustic peaks at small scales
provides a reliable procedure for estimating the cosmological parameters.

To illustrate this point let us consider the location of the
first acoustic peak.
Since all the Fourier components of the growing density perturbation
start with zero amplitude at high redshift, the condition for a mode
with a given wave vector *k* to reach an extremum amplitude at
*t* = *t*_{dec} is given by

(102) |

where *c*_{s} =
(*P* /
)^{1/2}
(1 / 3) is the speed of
sound in the baryon-photon fluid. At high redshifts,
*t*(*z*)
_{NR}^{-1/2}(1 + *z*)^{-3/2}
and the proper wavelength of the first acoustic peak scales as
_{peak} ~
*t*_{dec}
*h*^{-1}
_{NR}^{-1/2}. The angle subtended by this
scale in the sky depends on *d*_{A}. If
_{NR} +
_{} = 1
then the angular diameter distance varies as
_{NR}^{-0.4} while if
_{} = 0, it varies
as _{NR}^{-1}.
It follows that the angular size of the acoustic peak varies with the
matter density as

(103) |

Therefore, the angle subtended by acoustic peak is quite sensitive to
_{NR} if
= 0 but not if
_{NR} +
_{} = 1.
More detailed computations show that the multipole index corresponding
to the acoustic peak scales as
*l*_{p}
220 _{NR}^{-1/2} if
= 0 and
*l*_{p}
220 if _{NR} +
_{} = 1 and
0.1
_{NR}
1.
This is illustrated in figure 17
which shows the variation in the structure of
acoustic peaks when
is changed keeping
_{} = 0.
The four curves are for
=
_{NR} =
0.25, 0.45, 1.0, 1.15
with the first acoustic peak moving from right to left.
The data points on the figures are from the first results of MAXIMA and
BOOMERANG experiments and are included
to give a feel for the error bars in current observations.
It is obvious that the overall geometry of the
universe can be easily fixed by the study of CMBR anisotropy.

The heights of acoustic peaks also contain important information.
In particular, the height of the first acoustic peak
relative to the second one depends sensitively on
_{B}. This is
because the damping of the anisotropies arise from the finite
thickness of the surface of recombination which, in turn, depends on the
strength of the coupling between photons and baryons. Increasing the
amount of baryons increases this coupling and thus increases
the effect of damping on the second peak compared to first.

However, not all cosmological parameters
can be measured *independently*
using CMBR data alone. For example, different models with the same
values for
(_{DM} +
_{}) and
_{B}
*h*^{2} will
give anisotropies which are fairly indistinguishable. The structure
of the peaks will be almost identical in these models. This shows
that while CMBR anisotropies can, for example, determine the total
energy density
(_{DM} +
_{}), we will need
some other independent cosmological observations to determine the
individual components.

At present there exists several observations of the small scale anisotropies in the CMBR from the balloon flights, BOOMERANG [231], MAXIMA [232], and from radio telescopes CBI [233], VSA [234], and DASI [235, 236]. These CMBR data has been extensively analyzed in isolation as well as in combination with other results [59, 60, 221, 233, 234, 237, 238, 239, 240, 241, 242]. (The information about structure formation arises mainly from galaxy surveys like SSRS2, CfA2 [243], LCRS [244], Abell-ACO cluster survey [245], IRAS-PSC z [240] and the 2-D survey [246, 239].) While there is some amount of variations in the results, by and large, they support the following conclusions.

- The data strongly supports a
*k*= 0 model of the universe [242] with_{tot}= 1.00 ±^{0.03}_{0.02}. - Combined with large scale structure data, the results suggest
_{NR}= 0.29 ± 0.05 ± 0.04 [59, 60, 242, 247]. The initial power spectrum is consistent with being scale invariant and*n*= 1.02 ± 0.06 ± 0.05 [59, 60, 242]. In fact, combining 2dF survey results with CMBR suggest [248]_{}0.7 independent of the supernova results. - A similar analysis based on BOOMERANG data leads to
_{tot}= 1.02 ± 0.06(see for example, [238]). Combining this result with the HST constraint [49] on the Hubble constant*h*= 0.72 ± 0.08, galaxy clustering data as well SN observations one gets_{}= 0.62^{0.10}_{-0.18},_{}= 0.55^{0.09}_{-0.09}and_{}= 0.73^{0.10}_{-0.07}respectively [249]. - The analysis also gives an independent handle on baryonic
density in the universe which is consistent with the BBN value:
_{B}*h*^{2}= 0.022 ± 0.003 [59, 60]. This is gratifying since the initial data had an error and gave too high a value [250].

There has been some amount of work on the effect of dark energy on the
CMBR anisotropy
[251,
252,
253,
254,
255,
256,
257].
The shape of the CMB spectrum is relatively insensitive to the dark energy
and the main effect is to alter the angular diameter distance to the
last scattering surface and thus the position of the first acoustic peak.
Several studies have attempted to put
a bound on *w* using the CMB observations. Depending on the
assumptions which were invoked, they all lead to a bound broadly in the
range of *w*
- 0.6. At present it is not clear whether CMBR anisotropies
can be of significant help in discriminating between different dark energy
models.