The luminosity function (LF) of galaxies reflects both the initial conditions (e.g. the power spectrum of density fluctuations in the gravitational instability scenario) and the complicated physics of collapse, cooling, star formation, and feedback that govern how mass is converted into light. Based on arguments for hierarchical structure formation (Press and Schechter 1974), Schechter (1976) proposed the following functional form for the luminosity function:
where is
While the Schechter function is a good first-order approximation to the
overall LF, there is information as well in the type specific
luminosity functions, which are typically not well-fit by Schechter
functions.
Binggeli et
al. (1988)
reviewed the type-specific LF in
clusters and the field and hypothesized that the LF's of the individual
morphological types are constant, independent of environment. The
observed environmental variation in the total LF (summed over all
types) could then be explained as a simple change in the morphological
mix with environment, via the morphology-density relation (see also
Thompson and
Gregory 1980;
Jerjen et al. 1992).
The theoretical implication is
that the physics that governs the LF within a given Hubble class is (to
first order at least) independent of environment, while the
creation/destruction processes that govern the mix of types are not.
The faint end of the LF is dominated by dE galaxies in clusters, while
Sd and Im galaxies dominate in lower-density environments.
A working hypothesis
(Binggeli et
al. 1988)
is that variations in the faint end slope of the LF can be explained
entirely by variations in the relative proportions of dE
and Sd-Im galaxies. However, before assessing this hypothesis, we must
highlight the uncertainties in current LF estimates and the
selection effects that could bias the faint-end slope.
(L) is the number
of galaxies per unit volume per unit luminosity, L* is a
``characteristic'' luminosity, and 
is a ``characteristic'' faint-end
slope. The arguments behind the
Press and
Schechter (1974)
formalism have recently been made more rigorous
(Bond et al. 1991;
Bower 1991),
but these apply to the collapse of galaxy halos, and do
not directly translate into a luminosity function.
The various attempts to incorporate star-formation physics
into hierarchical models (Sect. 7)
produce a break at L*, due to the requirement that galaxies
radiatively cool while they are collapsing. The
power-law form at faint magnitudes is imposed by the initial
power spectrum but modified by a mass-dependent star-formation
efficiency due to galactic winds and by galaxy mergers.