  We estimate the rate and the luminosity function of short (hard) Gamma-Ray
Bursts (sGRBs) that are non-Collapsars, using the peak fluxes and redshifts of
BATSE, Swift and Fermi GRBs. Following Bromberg2013 we select a sub-sample of
Swift bursts which are most likely non-Collapsars. We find that these sGRBs are
delayed relative to the global star formation rate (SFR) with a typical delay
time of a 3-4 Gyr (depending on the SFR model). However, if two or three sGRB
at high redshifts have been missed because of selection effects, a distribution
of delay times of ~1/t would be also compatible. The current event rate of
these non-Collapsar sGRBs with L_iso > 5*10^49 erg/s is 4.1(-1.9,+2.3)Gpc^-3
yr^-1. The rate was significantly larger around z ~ 1 and it declines since
that time. The luminosity function we find is a broken power law with a break
at 2.0(-0.4,+1.4) * 10^52~erg/s and power-law indices 0.95(-0.12,+0.12) and
2.0(-0.8,+1.0). When considering the whole Swift sGRB sample we find that it is
composed of two populations: One group (~ 60%-80% of Swift sGRBs) with the
above rate and time delay and a second group (~ 20%-40% of Swift sGRBs) of
potential "impostors" that follow the SFR with no delay. These two populations
are in very good agreement with the division of sGRBs to non-Collapsars and
Collapsars suggested recently by Bromberg2013. If non-Collapsar sGRBs arise
from neutron star merger this rate suggest a detection rate of 3-100 yr^-1 by a
future gravitational wave detectors (e.g. Advanced Ligo/Virgo with detection
horizon on 300 Mpc), and a co-detection with Fermi (Swift) rate of 0.1-1 yr^-1
(0.02-0.14 yr^-1). We estimate that about 4 * 10^5 (f_b^-1 / 30) mergers took
place in the Milky Way. If $0.025 m_\odot$ were ejected in each event this
would have been sufficient to produce all the heavy r-process material in the
Galaxy.
