Annu. Rev. Astron. Astrophys. 1992. 30: 653-703
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2. STATISTICS AND ASSUMPTIONS

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.

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