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Galaxies form by a succession of mergers of cold dark matter halos, the baryons dissipating and forming a dense core. Isolated infall plausibly results in disk formation. Disk merging concentrates the gas into a dense spheroid. The transition from linear theory to formation of self-gravitating clouds occurs at an overdensity of about deltacrit approx 200. A simple ansatz due to Press and Schechter yields the mass function of newly nonlinear objects

Equation 1

where delta2 ident <(deltarho / rho)2 (M,t)> is the variance in the density fluctuations. The variance at 8 h-1 Mpc, delta8, is given by

Equation 2

where n approx -1 on cluster scales but n approx -2 on galaxy scales, and M = 1015 Omega h-1 (R / 8h-1 Mpc)3 Msun. Of course the luminosity function rather than the mass function is actually observed. We define sigma ident delta / deltag, where deltag is the variance in the galaxy counts. On cluster scales, one finds that sigma_8 approx 0.6 (± 0.1) yields the observed density of clusters if Omega = 1. More generally, sigma8 scales as Omega-0.6. A larger sigma is required for a given number density of objects in order to account for the reduced growth in delta as Omega is decreased below unity.

To match the observed luminosity function and predicted mass function requires specification both of sigma8 and of the mass-to-light ratio. Much of the dark mass is in objects that were the first to go nonlinear, as well as in the objects presently going nonlinear. Hence one crudely expects that M/L approx 400 h, as measured in rich clusters. The global value of M/L is M/L approx 1500 Omega h, and happens to coincide with the mass-to-luminosity ratio measured for rich clusters if Omega approx 0.4. This suggests that these clusters may provide a fair sample of the universe. Even if most dwarfs do not survive, because of subsequent merging, the relic dwarfs are expected to have high M/L. Later generations of galaxies should have undergone segregation of baryons, because of dissipation, and the resulting M/L is reduced. Many of the first dwarfs are disrupted to form the halos of massive galaxies. The predicted high M/L (of order 100) is consistent with observations, both of galaxy halos and of the lowest mass dwarfs (to within a factor of ~ 2).

However it is the detailed measurement of M/L that leads to a possible problem. One has to normalise M/L by specifying the mass-to-light ratio of luminous galaxies. The observed luminosity function can be written as

Equation 3

where alpha approx 1 - 1.5, depending on the selection criterion, and L* approx 1010 h-2 Lsun. Matching to the predicted mass function specifies M/L for L* galaxies, as well as the slope of the luminosity function. One forces a fit to alpha by invoking star formation-induced feedback and baryonic loss. This preferentially reduces the number of low mass galaxies. A typical prescription is 33 that the retained baryonic fraction is given by

Equation 4

where vc is the disk circular velocity. Dwarfs are preferentially disrupted by winds. In this way one can fit alpha. There is no longer any freedom in the luminous galaxy parameters.

Potential difficulties arise as follows. Simulations of mass loss from dwarf galaxies suggest that supernova ejecta may contribute to the wind but leave much of the interstellar gas bound to the galaxies. 34 This would be a serious problem as one relies on redistribution of the baryonic reservoir to form massive galaxies. Another problem arises with the Tully-Fisher relation. This is the measured relation, approximately L proptosim Vbetarot, between galaxy luminosity and maximum rotational velocity. In effect, the Tully-Fisher relation offers the prescription for M/L within the luminous part of the galaxy, since the virial theorem requires

Equation 5

where µL is the surface brightness of the galaxy. Since µL has a narrow dispersion for most disk galaxies, the Tully-Fisher relation, where beta approx 3 is measured in the I band and beta approx 4 is appropriate to the near infrared, effectively constrains M/L. The normalization of the Tully-Fisher relation requires M/L approx 5h for early-type spirals, as is observed directly from their rotation curves within their half-light radii. However simulations of hierarchical clustering, which incorporate baryonic cooling and star formation with a prescription designed to reproduce the luminosity function, give too high a normalization for M/L in the predicted Tully-Fisher relation: at a given luminosity the rotational velocity is too high. 35 Moreover the efficient early star formation required in order to fit the luminosity function requires the Tully-Fisher normalisation to change with redshift: galaxies are predicted to be brighter by about a magnitude at a given rotation velocity at z ~ 1, and this exceeds the observed offset. 36 Resolution of the Tully-Fisher normalization remains controversial.

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