### 15.5. EVOLUTIONARY MODELS

The principal goals of evolutionary models are (1) to estimate the
generalized luminosity function
_{m}(*L*,
,...| *z*,
) of radio
sources, (2) to explain significant
features in the observed source counts, spectral-index distributions,
redshift distributions, etc. (e.g.,
Kellermann 1972),
and (3) to identify new observations that
will constrain future models most effectively. Existing data cannot
fully determine the evolving luminosity function, so practical models
incorporate additional
*assumptions* about source evolution. The computations must be
simplified by *approximations* as well - source spectra are taken
to be simple power laws,
spectral-index distributions may be replaced by Gaussians or even
-functions, and evolution
may be described by smooth functions with a limited number of free
parameters.

A typical model-generating procedure is:

Choose a particular world model to fix the
effective distance *D* and the volume
element *dV* (Section 15.2). The
Einstein-de Sitter (
= 1) model is most commonly picked, but changing
has only a small,
easily estimated effect
(Peacock 1985).

Guess a form for the evolving luminosity
function.

Compute the model local luminosity function,
source counts,
spectral index distributions, etc. and compare them with the data.

Revise the luminosity function.

Repeat steps (3) and (4) until satisfactory
agreement is reached.

Differences between the models produced by different authors reflect
their different data sets, assumptions about the type of evolution
occurring, computational approximations, and methods for deciding that a
model is satisfactory. Steps (2)
and (4) are not straightforward, although simplified models
(Section 15.9) can make
the relations between the input luminosity function parameters and the
output observables clearer.