  The origin and progenitors of short-hard gamma-ray bursts remain a puzzle and
a highly debated topic. Recent Swift observations suggest that these GRBs may
be related to catastrophic explosions in degenerate compact stars, denoted as
"Type I" GRBs. The most popular models include the merger of two compact
stellar objects (NS-NS or NS-BH). We utilize a Monte Carlo approach to
determine whether a merger progenitor model can self-consistently account for
all the observations of short-hard GRBs, including a sample with redshift
measurements in the Swift era (z-known sample) and the CGRO/BATSE sample. We
apply various merger time delay distributions invoked in compact star merger
models to derive the redshift distributions of these Type I GRBs, and then
constrain the unknown luminosity function of Type I GRBs using the observed
luminosity-redshift (L - z) distributions of the z-known sample. The best
luminosity function model, together with the adopted merger delay model, are
then applied to confront the peak flux distribution (log N - log P
distribution) of the BATSE and Swift samples. We find that for all the merger
models invoking a range of merger delay time scales (including those invoking a
large fraction of "prompt mergers"), it is difficult to reconcile the models
with all the data. The data are instead statistically consistent with the
following two possible scenarios. First, that short/hard GRBs are a
superposition of compact-star-merger origin (Type I) GRBs and a population of
GRBs that track the star formation history, which are probably related to the
deaths of massive stars (Type II GRBs). Sec- ond, the entire short/hard GRB
population is consistent with a typical delay of 2 Gyr with respect to the star
formation history with modest scatter. This may point towards a different Type
I progenitor than the traditional compact star merger models.
