### 1. DEFINITION AND USES OF THE LUMINOSITY FUNCTION

The luminosity function is a distribution function, specifically the distribution of luminosities of objects in a sample. Luminosity itself is difficult to measure, since the total or bolometric luminosity requires both accurate distance measurements and integration of each object's spectrum over all frequencies. Generally we measure specific luminosity, L, over a given band, or range of frequency, so the units of L are erg sec-1 Hz-1. Then the luminosity function is n(L), where n(L) dL is the number of galaxies with luminosity in the range L to L + dL; n(L) has units Mpc-3 (watt Hz-1)-1. The integral of n(L) dL over all luminosities is just the density of galaxies, n.

Note that if n(L) is described by a power law, for L to be finite requires that the power law index be less than -1 at high luminosities, and greater than -1 at low luminosities, where L is the total luminosity of all galaxies in a unit volume, i.e.,

A simple combination of power laws is often a useful approximate form to assume for a luminosity function, i.e.,

1.

What we know about luminosity functions generally comes from the distribution of fluxes in a survey which is complete in some way. The best is a volume-limited sample (meaning every galaxy in the volume has been measured). This is rarely available, since all surveys have some minimum detectable flux, Smin, which translates into a cutoff distance rcut = sqrt(L / 4 Smin) which is a function of luminosity. So the sample size is a function of luminosity. This bias can be corrected if we know that the distribution of galaxies is homogeneous, i.e., that the total density

is independent of position. In general n is a strong function of position; it varies by several orders of magnitude between rich clusters and voids; indeed, the study of this function is the main subject of this course.

If the distribution of objects is homogeneous, so that n is a constant independent of position, which presumably olds when we average over very large scales, then we can easily evaluate some simple integrals of the luminosity function which apply to a flux limited sample. The number of objects brighter than the minimum flux Smin is just given by

2.

where we have simply interchanged the integration over volume and over luminosity. (Note: the distance r, which on small scales is simply cz / H0, generalizes in a Friedman universe with = 0, = 1 to

see Condon (1984a). In the simplest case of ``standard candles'' the luminosity function is

so that

In magnitude notation N>(Smin) Smin-3/2 becomes N<(m) m0.6. The ``differential source count`` function, n(Smin) = -dN> (Smin) / dSmin depends on Smin to the minus 5/2 power, and the total flux from all sources is proportional to S n(S) dS which diverges. This is a statement of Olber's paradox, which can only be resolved in a Euclidian universe if the luminosity function evolves, i.e., changes with time. The only tracers for which we have complete flux limited samples reaching to large redshifts (z > 1) are QSO's and radio sources, both of which apparently have luminosity functions which evolve strongly on cosmological time scales.

In a flux limited sample the contribution to n(s) of sources of different luminosities is most easily seen in von Hoerner's (1973) ``visibility function'':

3.

whose dimensions are watt3/2 Hz-3/2 pc-3, which is usually converted to Jy3/2. A plot of log (L) vs. log L immediately shows what range of luminosities contribute most to a flux limited sample, since

4.

(see Condon 1984a, b).

A useful integral of the luminosity function gives the median distance to objects in a flux limited sample, r1/2, given by

5.

where again we can interchange integration to get

6.

where L1/2 = 4 Smin r1/22. This can easily be evaluated for n(L) having the simple form of equation 1. For example, if n2 = 0 and 1 = 0 [i.e., a step function n(L)], we find simply that r1/2 is 0.64 sqrt(L0 / 4 Smin). Using (L) equation 6 becomes simply

where x = log L1/2, which is the obvious median value of (L) when plotted vs. log L.