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7.1. Gravitational collapse

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

Equation 9 (9)

for scale invariant fluctuations of the form

Equation 10 (10)

where n is index of the initial power spectrum | delta k| 2 propto 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) propto M-2. In comparison, the observed luminosity function has a faint end slope alpha approx -1 to the limits of field samples and alpha approx -1.3 in clusters, and perhaps to fainter limits in the field (see Sect. 5).