Annu. Rev. Astron. Astrophys. 1992. 30:
653-703
Copyright © 1993 by . All rights reserved |

There are two very different statistical questions which arise in studies of the microwave background radiation - 1. what is the distribution of fluctuations on the sky? and 2. what is the ``best'' statistical approach in analyzing microwave background radiation observations? These are discussed separately below.

**2.1 The Amplitude Distribution and Spatial Spectrum of
Fluctuations**

The question of the amplitude distribution and spatial spectrum of the
fluctuations is of fundamental physical interest since this must be
related to the origin of the fluctuations for which there is as yet no
accepted theory. It should be noted that the *observed*
distribution on large angular scales might be difficult to interpret
since we have only one realization of the distribution to observe.
Thus, even if the fluctuations have a random Gaussian density field it
may be difficult to determine this from the observations because of
the dominance of low-order multipoles of the temperature fluctuations
(Scaramella & Vittorio
1990,
1991).
In the present case we are
concerned with angular scales < 20° on which this is less of a
problem but it cannot be completely ignored.

Gaussian approximations for *W(k)* are often used by both theorists
and observers. As discussed above, this permits easy comparisons of
observations to models and we remind the reader that it is rather a
matter of convenience than of sound practice. It is important to
remember that most models can at best only be approximated by Gaussian
*W(k)* over a restricted range of angular scales, and many models
produce spectra which are non-Gaussian on all scales. The cold dark
matter models of
Bond and Efstathiou
(1987),
for example, are roughly
approximated by a Gaussian *W(k)* on angular scales less than a few
degrees, but not on larger scales, while string
(Bouchet et al 1988)
and texture
(Turok & Spergel 1990)
models predict non-Gaussian
microwave background radiation fluctuations, as is also possibly true
of some inflationary models
(Matarrese et al 1990,
Kofman et al 1990).

We will use the Gaussian approximation because of the convenience that
it provides, but we will not combine observations over angular scales
on which different models yield very different distributions, in order
to emphasize the fact that these data are *not* directly
comparable in terms of such a simplistic model.