Although the CMBR is close to being an isotropic and thermal radiation background with simple spectral and angular distributions, it is useful to recall the formalism needed to deal with a general radiation field, since the details of the small perturbations have great physical significance. The notation used here is similar to that of Shu (1991), which may be consulted for more detailed descriptions of the quantities employed.

The state of a radiation field can be described by
distribution functions *f*_{}(**r**,**p**,*t*), such that
the number of photons in real space volume *d*^{3}*r*
about **r** and momentum space volume *d*^{3}*p*
about **p** at time *t* with
polarization (= 1 or 2) is
*f*_{}
*d*^{3}*r* *d*^{3}*p*. This
distribution function is related to the photon occupation number in
polarization state ,
*n*_{ }(**r**,**p**,*t*), by

and to the specific intensity in the radiation,
*I*_{}(**k**, **r**, *t*), by

where **k** *h* and *c*
are Planck's constant and the speed of light. The meaning of the
specific intensity is that the energy crossing area element **dS**
in time *dt* from within solid angle *d* about **k** *d* is
*I*_{} (**k** **·** **dS**) *d*
*d* *dt*.

If the occupation number is of Planck form

then the radiation field has the form of (1). The number density of photons in the Universe is then

from which can be calculated the baryon to photon number ratio,
=
*n*_{B}/*n*_{} = 2.7 x 10^{-8} _{B}
*h*_{100}^{2}. In (16) *(x)* is the Riemann zeta
function ((3) 1.202)
and the value of *T*_{rad} is taken
from a recent analysis of COBE data on the CMBR spectrum
(Fixsen *et al.*
1996).

Similarly, the energy density of the radiation field is

It is apparent that the errors on *u*_{} and *n*_{} in
(16) and (17) are so small as to have no
significant astrophysical impact, and may safely be dropped.

It is common for the specific intensity of a radiation field to be
described by radio-astronomers in units of brightness temperature,
*T*_{RJ}. This is defined as the temperature of a thermal
radiation field which in
the Rayleigh-Jeans limit of low frequency would have the same
brightness as the radiation that is being described. In the limit of
low frequency (1) reduces to
*I*_{} = 2
*K*_{B} *T*_{rad}
^{2} / *c*^{2},
so that

Thus the brightness temperature of a thermal spectrum as described by (1) is frequency-dependent, with a peak value equal to the radiation temperature at low frequencies, and tending to zero in the Wien tail.

In the presence of absorption, emission and scattering processes,
and in a flat spacetime, *I*_{} obeys a transport equation

where *j*_{} is the
emissivity along the path (the energy emitted per
unit time per unit frequency per unit volume per unit solid angle),
_{,abs} is the absorption coefficient (the fractional
loss of intensity of the radiation per unit length of propagation
because of absorption by material in the beam), _{,sca}
is the scattering coefficient (the fractional loss of intensity of the
radiation per unit length of propagation because of scattering by
material in the beam), and _{} (**k**, **k'**
**k'**
to **k**. The
absorption coefficient is regarded as containing both true absorption
and simulated emission. While this is important in astrophysical
masers, where _{,abs} is negative, this subtlety will not
affect the discussions in the present review. An important
property of *I*_{} that
follows from its definition (or equation 19) is that it is conserved in
flat spacetimes in the absence of radiation sources or absorbers.

The specific intensity of a radiation field may be changed in several ways. One is to make the photon distribution function anisotropic, for example by the Doppler effect due to the peculiar motion of the Earth relative to the sphere of last scattering, which causes the radiation temperature becomes angle-dependent

but otherwise leaves the form of (15) unchanged.
= [1 -
(*v*^{2} / *c*^{2})]^{-1/2} and
is the angle between the line
of sight and the observer's velocity vector
(Peebles & Wilkinson
1968).
The specific intensity may also be changed by redistributing photons
to different directions and frequencies (e.g., by scattering
processes), or by absorbing or emitting radiation (e.g., by thermal
bremsstrahlung). The choice of whether to describe these effects in
the photon distribution function, or in the specific intensity, is
made for reasons of convenience. Although the statistical
mechanics of photon scattering is often related to the
occupation numbers, *n*_{}, most astrophysical work is done in
the context of the specific intensity, *I*_{}.