  The number of galaxies at a given flux as a function of the redshift, $z$, is
derived when the $z$-distance relation is non-standard. In order to compare
different models, the same formalism is also applied to the standard cosmology.
The observed luminosity function for galaxies of the zCOSMOS catalog at
different redshifts is modelled by a new luminosity function for galaxies,
which is derived by the truncated beta probability density function. Three
astronomical tests, which are the photometric maximum as a function of the
redshift for a fixed flux, the mean value of the redshift for a fixed flux, and
the luminosity function for galaxies as a function of the redshift, compare the
theoretical values of the standard and non-standard model with the observed
value. The tests are performed on the FORS Deep Field (FDF) catalog up to
redshift $z=1.5$ and on the zCOSMOS catalog extending beyond $z=4$. These three
tests show minimal differences between the standard and the non-standard
models.
