The leading paradigm for galaxy formation involves gravitational collapse of primordial density fluctuations, followed by cooling of the gas and star formation. The sucess of the COBE satellite in detecting fluctuations in the microwave background both supports gravitational instability as the mechanism for forming structure, at least on large scales, and rules out competing models such as explosions (Wright et al. 1992; Efstathiou et al. 1992). Blanchard et al. (1992) provide a detailed discussion of the theoretical construct, which leads to a mass function

for scale invariant fluctuations of the form

where *n* is index of the initial power spectrum
| ^{k}| ^{2} *k ^{n}*
(Peebles 1980).

The *mass* function of dwarf galaxies is in principle a
sensitive measure of the shape of the primoridial fluctuation
spectrum, as it is not very sensitive to the nature of the density
fluctuations (Gaussian or non-Gaussian) or the details
of nonlinear collapse
(Blanchard et
al. 1992).
However, the analytical theory is difficult to test at the resolution of
current numerical codes and the mass function is difficult
to derive from observations. The CDM power spectrum has
-3 < *n* < -2 on the scale of dwarf galaxies, leading to a
mass function *N(M)*
*M*^{-2}. In comparison, the observed
*luminosity* function has a faint end slope -1
to the limits of field samples and -1.3 in
clusters, and perhaps to fainter limits in the field
(see Sect. 5).