We shall now apply the formalism we have developed to understand the
temperature anisotropies in the
cosmic microwave background radiation which is probably the most useful
application of *linear*
perturbation theory. We shall begin by developing the general
formulation and the terminology
which is used to describe the temperature anisotropies.

Towards every direction in the sky, **n** =
(,
)
we can define a fractional temperature fluctuation
(**n**)
(*T* / *T*)
(,
).
Expanding this quantity in spherical harmonics on the sky plane as well
as in terms of the spatial Fourier modes, we get the two relations:

(106) |

where *L* =
_{0} -
_{LSS} is the distance to the last
scattering surface (LSS) from which we are receiving the radiation.
The last equality allows us to define the expansion coefficients
*a*_{lm} in terms of the temperature fluctuation
in the Fourier space
(**k**)
. Standard identities of mathematical physics now give

(107) |

Next, let us consider the angular correlation function of temperature anisotropy, which is given by:

(108) |

where the wedges denote an ensemble average. For a Gaussian random field
of fluctuations we can express the ensemble average as
<
*a*_{lm}
*a*_{l'm'}^{,tt>*} > =
*C*_{l}
_{ll'}
_{mm'}.
Using Eq. (107), we get a relation between *C*_{l} and
(*k*). Given
(*k*), the
*C*_{l}'s are given by:

(109) |

Further, Eq. (108) now becomes:

(110) |

Equation (110) shows that the mean-square value of temperature
fluctuations and the quadrupole anisotropy corresponding to *l* = 2
are given by

(111) |

These can be explicitly computed if we know
(*k*) from the
perturbation theory. (The motion of our local group through the CMBR
leads to a large *l* = 1 dipole contribution
in the temperature anisotropy. In the analysis of CMBR anisotropies,
this is usually
subtracted out. Hence the leading term is the quadrupole with *l*=2.)

It should be noted that, for
a given *l*, the *C*_{l} is the average over all
*m* = -*l*, ...-1, 0, 1,
... *l*. For a Gaussian random field, one can also compute the variance
around this mean value. It can be shown that this variance in
*C*_{l} is
2 *C*_{l}^{2} / (2*l* + 1). In other words,
there is an intrinsic root-mean-square fluctuation in the observed, mean
value of *C*_{l}'s which is of the order of
*C*_{l}
/ *C*_{l}
(2*l* +
1)^{-1/2}. It is not possible for any CMBR observations which
measures the *C*_{l}'s to
reduce its uncertainty below this intrinsic variance - usually called
the "cosmic variance". For large values of *l*, the cosmic variance is
usually sub-dominant to other observational errors but for low *l* this
is the dominant source of uncertainty in the measurement of
*C*_{l}'s. Current WMAP observations are indeed only
limited by cosmic variance at low-*l*.

As an illustration of the formalism developed above, let us compute the
*C*_{l}'s for low *l* which will be contributed
essentially by
fluctuations at large spatial scales. Since these fluctuations will be
dominated by gravitational effects, we can ignore the complications
arising from baryonic physics and compute these using the formalism we
have developed earlier.

We begin by noting that the
redshift law of photons in the unperturbed Friedmann universe,
_{0}
= (*a*)
/ *a*, gets modified to the form
_{0} =
(*a*) /
[*a*(1 + )] in a
perturbed FRW universe. The argument
of the Planck spectrum will thus scale as

(112) |

This is equivalent to a temperature fluctuation of the amount

(113) |

at large scales.
(Note that the observed
*T* / *T*
is not just
(_{R} / 4) as one
might have naively imagined.)
To proceed further, we recall our large scale solution (see
Eq. (78)) for the gravitational potential:

(114) |

At *y*=*y*_{dec} we can take the asymptotic solution
_{dec}
(9/10)_{i}.
Hence we get

(115) |

We thus obtain the nice result that the observed temperature
fluctuations at very large scales
is simply related to the fluctuations of the gravitational potential at
these scales. (For a discussion of the 1/3 factor, see
[19]).
Fourier transforming this result we get
(**k**) = (1/3)
(**k**,
_{LSS}) where
_{LSS} is the conformal time at the last
scattering surface. (This contribution is called Sachs-Wolfe effect.)
It follows from Eq. (109) that the contribution to *C*_{l}
from the gravitational potential is

(116) |

with

(117) |

For a scale invariant spectrum, *k*^{3}
|_{k}|^{2}
is a constant independent of *k*. (Earlier on, in Eq. (103) we said
that scale invariant spectrum has *k*^{3}
|_{k}|^{2}
= constant. These statements are equivalent since
-2 at the large scales
because of Eq. (85) with
the extra correction term in Eq. (85) being about 3× 10^{-4}
for *k*
*L*^{-1},*y* = *y*_{dec}.) As we shall
see later, inflation generates such a perturbation. In this case, it is
conventional to introduce a constant amplitude *A* and write:

(118) |

Substituting this form into Eq. (116) and evaluating the integral, we find that

(119) |

As an application of this result, let us consider the observations of
COBE which measured the temperature fluctuations for the first time in
1992. This satellite obtained the RMS fluctuations and the quadrupole
after smoothing over an angular scale of
about _{c}
10°. Hence the
observed values are slightly different from those in Eq. (111). We have,
instead,

(120) |

Using Eqs. (118), (119) we find that

(121) |

Given these two measurements, one can verify that the fluctuations are
consistent with the
scale invariant spectrum by checking their ratio. Further, the numerical
value of the observed
(*T* /
*T*) can be used to determine the amplitude *A*. One finds
that *A* 3
× 10^{-5} which sets the scale of fluctuations in the
gravitational potential at the time when the perturbation enters the
Hubble radius.

Incidentally, note that the solution
_{R} =
4 -
6_{i}
corresponds to
_{m} =
(3/4)_{R} =
3 -
(9/2)_{i}. At
*y* = *y*_{dec}, taking
_{dec} =
(9/10)_{i}, we get
_{m} =
3_{dec} -
(9/2)(10/9)_{dec}
= -2_{dec}.
Since (*T* /
*T*)_{obs} =
(1/3)_{dec}
we get _{m} =
-6(*T*
/ *T*)_{obs}. This shows that the amplitude of matter
perturbations is a factor six larger that the amplitude of temperature
anisotropy for our adiabatic initial conditions. In several other
models, one gets
_{m} =
(1)(*T* /
*T*)_{obs}. So, to reach a given level of nonlinearity in
the matter distribution at later times, these models will require higher
values of
(*T* /
*T*)_{obs} at decoupling. This is one reason for such
models to be observationally ruled out.

There is another useful result
which we can obtain from Eq. (109) along the same lines as we derived
the Sachs-Wolfe effect. Whenever
*k*^{3}|(*k*)|^{2} is a slowly varying function of
*k*, we can pull out this factor
out of the integral and evaluate the integral over
*j*_{l}^{2}. This will give the result for any
slowly varying *k*^{3}
|(*k*)|^{2}

(122) |

This is applicable even when different processes contribute to temperature anisotropies as long as they add in quadrature. While far from accurate, it allows one to estimate the effects rapidly.

**6.1. CMBR Temperature Anisotropy: More detailed
theory**

We shall now work out a more detailed theory of temperature anisotropies of CMBR so that one can understand the effects at small scales as well. A convenient starting point is the distribution function for photons with perturbed Planckian distribution, which we can write as:

(123) |

The
*f*_{p}()
is the standard Planck spectrum for energy
and we take
= *a* *E*
(1 + )^{-1}
to take care
of the perturbations. In the absence of collisions, the distribution
function is conserved along the trajectories of photons so that
*df* /
*d* =
0. So, in the presence of collisions, we can write the time
evolution of the distribution function as

(124) |

where the right hand side gives the contribution due to collisional terms. Equivalently, in terms of , the same equation takes the form:

(125) |

To proceed further, we need the expressions for the two terms on the left hand side. First term, on using the standard expansion for total derivative, gives:

(126) |

(Note that we are assuming
/
*E* = 0 so
that the perturbations do not depend on the frequency of the photon.)
To determine the second term, we note that it vanishes in the
unperturbed Friedmann universe and arises essentially due to the
variation of . Both the
intrinsic time variation of
as well
as its variation along the photon path will contribute, giving:

(127) |

(The minus sign arises from the fact that the we have (1 +
2) in
*g*_{00} but (1 -
2) in the spatial
perturbations.) Putting all these together, we can bring the evolution
equation Eq. (125) to the form:

(128) |

Let us next consider the collision term, which can be expressed in the form:

(129) |

Each of the terms in the right hand side of the first line has a simple
interpretation. The first term describes the removal
of photons from the beam due to Thomson scattering with the electrons
while the second term gives the scattering contribution into the
beam. In a static universe, we expect these two terms to cancel if
=
(1/4)_{R} which
fixes the relative coefficients of these
two terms. The third term is a correction due to the fact that the
electrons which are scattering the photons are not at rest relative to
the cosmic frame. This leads to a Doppler shift which is accounted for
by the third term. (We denote electron number *density* by
*N*_{e}
rather than *n*_{e} to avoid notational conflict with
*n*^{}.)

Formally, Eq. (128) is a first order linear differential equation for
. To eliminate the
-*N*_{e}
_{T}
term which is linear in
in the right hand
side, we use the standard integrating factor
exp(-) where

(130) |

We can then formally integrate Eq. (128) to get:

(131) |

We can write

(132) |

On integration, the first term gives zero at the lower limit and an
unimportant constant (which does not depend on **n**). Using the
rest of the terms, we can write Eq. (131) in the form:

(133) |

The first term gives the contribution due to the intrinsic time
variation of the gravitational potential along the path of the photon
and is called the integrated Sachs-Wolfe effect. In the second term one
can make further simplifications. Note that
*e*^{-} is
essentially unity (optically thin) for *z* <
*z*_{rec}
and zero (optically thick) for *z* > *z*_{rec}; on
the other hand, *N*_{e}
_{T} is zero for
*z* < *z*_{rec} (all the free electrons have
disappeared) and is large
for *z* > *z*_{rec}. Hence the product (*a*
*N*_{e}
*e*^{-} ) is
sharply peaked at =
_{rec} (i.e. at
*z* 10^{3}
with *z*
80 ).
Treating this sharply peaked quantity as essentially a Dirac delta
function (usually called the
instantaneous recombination approximation) we can approximate the second
term in Eq. (133) as a contribution occurring just on the LSS:

(134) |

In the second term we have put
=
for
<
_{LSS}
and = 0 for
>
_{LSS}.

Once we know _{R},
and **v** on the LSS
from perturbation theory, we can take
a Fourier transform of this result to obtain
(*k*) and use
Eq. (109) to compute *C*_{l}.
At very large scales the velocity term is sub-dominant and we get back
the Sachs-Wolfe effect
derived earlier in Eq. (118). For understanding the small scale
effects, we need to introduce baryons into the picture which is our next
task.

** 6.2. Description of photon-baryon fluid**

To study the interaction of photons and baryons in the fluid limit, we need to again start from the continuity equation and Euler equation. In Fourier space, the continuity equation is same as the one we had before (see Eq. (52)):

(135) |

The Euler equations, however, gets modified; for photons, it becomes:

(136) |

The first two terms in the right hand side are exactly the same as the ones in Eq. (55). The last term is analogous to a viscous drag force between the photons and baryons which arises because of the non zero relative velocity between the two fluids. The coupling is essentially due to Thomson scattering which leads to the factor . (The notation, and the physics, is the same as in Eq. (130)). The corresponding Euler equation for the baryons is:

(137) |

where

(138) |

Again, the first two terms in the right hand side of Eq. (137) are the
same as what we had before in Eq. (55). The last term has the
same interpretation as in the case of Euler equation Eq. (136)
for photons, except for the factor *R*. This quantity essentially takes
care of the inertia of baryons relative to photons. Note that the
the conserved momentum density of photon-baryon fluid has the form

(139) |

which accounts for the extra factor *R* in Eq. (137).

We can now combine the Eqs. (135), (136),
(137) to obtain, to lowest order in (*k* /
) the equation:

(140) |

with

(141) |

An exact solution to this equation is difficult to obtain.
However, we can try to understand several features by an approximate
method in which
we treat the time variation of *R* to be small. In that case, we can
drop the
terms on both sides of the equation. Since we know that the physically
relevant temperature fluctuation is
=
(1/4)_{R} +
, we can recast the
above equation for
as:

(142) |

Let us further ignore the time variation of all terms (especially
on the right hand
side). Then, the solution is just
=
-*R* + *A*
cos(*kc*_{s}
_{LSS}) +
*B*sin(*kc*_{s}
_{LSS}). To fix the initial conditions
which determine *A* and *B*, we recall that early on
(
0), we have
/3 (see Eq. (115)) and
corresponding velocity should vanish. This gives the solution:

(143) |

(One can do a little better by using WKB approximation in which
(*kc*_{s}
_{LSS}) can be replaced by the integral
of *kc*_{s} over
but it is not
very important.) Given this solution, one can proceed as before and
compute the *C*_{l}'s.
Adding the effects of
[ +
(1/4)_{R}] and
that of [**v · n**] in quadrature
and noticing that the angular average of <(**v ·
n**)^{2}> = (1/3) *v*^{2} we can estimate the
*C*_{l} for scale invariant
( *k*^{3}|_{k}|^{2} =
2^{2}
*A*^{2} ) spectrum to be:

(144) |

with *k*^{*}*L*
*l* with *L* =
_{0} -
_{LSS}
_{0}.
The key feature is, of course, the maxima and minima which arises from
the trigonometric functions. The peaks of *C*_{l} are
determined by the
condition *k*^{*} *c*_{s}
_{LSS} = *lc*_{s}
_{LSS} /
_{0}
= *n* ; that is

(145) |

More precise work gives the first peak at *l*_{peak}
200.
It is also clear that because of non zero *R* the peaks are larger
when the cosine term is negative; that is, the odd peaks corresponding
to *n*= 1, 3, ... have
larger amplitudes than the even peaks with *n* = 2, 4, ....

Incidentally, the above approximation is not very good for modes which
enter the Hubble radius during the radiation dominated phase since
does evolve with time
(and decays) in the radiation dominated phase. We saw that
-3_{i}
(*ly*)^{-2} cos(*ly*) asymptotically
in this phase (see Eq. (80)). From Eq. (80) we find that during this
phase, for modes which are inside the Hubble radius, we can take
_{R}
6_{i}
cos(*ly*), so that
_{R}/4
(3/2)_{i}
cos(*ly*). On the other hand, at very large scale, the amplitude was
=
/3 =
(1/3)(9/10)_{i}
= (3/10)_{i}.
Hence the amplitude of the modes that enter the
horizon during the radiation dominated phase
is enhanced by a factor (3/2)(10/9) = 5, relative to the large scale
amplitude contributed by modes which enter during matter dominated
phase. This is essentially due to the driving term
which is nonzero in the
radiation dominated phase but zero in the matter dominated phase.
(In reality, the enhancement is smaller because the relevant modes have
*k*
*k*_{eq} rather than *k* >>
*k*_{eq}; see Figs. 3
and 4.)

If this were the whole story, we will see a series of peaks and troughs
in the temperature anisotropies as a function of angular scale. In
reality, however, there are processes which damp out the anisotropies at
small angular scales (large -*l*) so that only the first few peaks and
troughs are really relevant. We will now discuss two key damping
mechanisms which are responsible for this.

The first one is the finite width of the last scattering surface which
makes it uncertain from which event we are receiving the photons.
In general, if (*z*)
is the probability that the photon was last
scattered at redshift *z*, then we can write:

(146) |

From Eq. (41) we know that
(*z*) is a Gaussian
with width
*z* = 80. This
corresponds to a length scale

(147) |

over which the temperature fluctuations will be smoothed out.

It turns out that there is another effect, which is slightly more important. This arises from the fact that the photon-baryon fluid is not tightly coupled and the photons can diffuse through the fluid. This diffusion can be modeled as a random walk and the root mean square distance traveled by the photon during this diffusion process will smear the temperature anisotropies over that length scale. This photon diffusion length scale can be estimated as follows:

(148) |

Integrating, we find the mean square distance traveled by the photon to be

(149) |

The corresponding proper length scale below which photon diffusion will wipe out temperature anisotropies is:

(150) |

It turns out that this is the dominant sources of damping of temperature
anisotropies at large *l*
10^{3}.