  We use data from the Parkes Half-Jansky Flat-Spectrum (PHJFS) sample
(Drinkwater et al. 1997) to constrain the cosmic evolution in the co-moving
space density of radio sources in the top decade of the flat-spectrum radio
luminosity function (RLF). A consistent picture for the high-redshift evolution
is achieved using both simple parametric models, which are the first to allow
for distributions in both radio luminosity and spectral index, and variants of
the V / V_max test, some of which incorporate the effects of radio spectral
curvature. For the most luminous flat-spectrum objects, the PHJFS sample is
extremely similar to that used by Shaver et al. (1996, 1998) to argue for an
abrupt `redshift cut-off': a decrease by a factor ~30 in space density between
a peak redshift z~2.5 and z~5. Our analysis finds that the observable co-moving
volume is too small to make definitive statements about any redshift cut-off
for the most luminous flat-spectrum sources, although both constant-space
density (no cut-off) models and models with cut-offs as abrupt as those
envisaged by Shaver et al. are outside the 90% confidence region. The inference
that the decline in space density is most likely to be gradual, by a factor ~4
between z~2.5 and z~5, is in accordance with previous work on the RLF by Dunlop
& Peacock (1990), but different to the abrupt decline favoured by studies of
optically-selected quasars. Dust obscuration provides one explanation for this
difference. We show that a significant fraction of the most radio-luminous
flat-spectrum objects are Giga-Hertz Peaked Spectrum (GPS) rather than
Doppler-Boosted (DB) sources, complicating any interpretation of the redshift
cut-off.
