A redshift - observation-time relation for gamma-ray bursts: evidence of
  a distinct sub-luminous population
  We show how the redshift and peak-flux distributions of gamma-ray bursts
(GRBs) have an observation time dependence that can be used to discriminate
between different burst populations. We demonstrate how observation time
relations can be derived from the standard integral distributions and that they
can differentiate between GRB populations detected by both the BATSE and
\emph{Swift} satellites. Using \emph{Swift} data we show that a
redshift--observation-time relation (log\,$Z$\,--\,log\,$T$) is consistent with
both a peak-flux\,--\,observation time relation (log\,$P$\,--\,log\,$T$) and a
standard log\,$N$\,--\,log\,$P$ brightness distribution. As the method depends
only on rarer small-$z$ events, it is invariant to high-$z$ selection effects.
We use the log\,$Z$\,--\,log\,$T$ relation to show that sub-luminous GRBs are a
distinct population occurring at a higher rate of order $150^{+180}_{-90}
\mathrm{Gpc}^{-3}\mathrm{yr}^{-1}$. Our analysis suggests that GRB 060505 -- a
relatively nearby GRB observed without any associated supernova -- is
consistent with a sub-luminous population of bursts. Finally, we suggest that
our relations can be used as a consistency test for some of the proposed GRB
spectral energy correlations.
