The rate of Type-Ia supernovae in galaxy clusters and the delay-time
  distribution out to redshift 1.75
  The observed delay-time distribution (DTD) of Type-Ia supernovae (SNe Ia) is
a valuable probe of SN Ia progenitors and physics, and of the role of SNe Ia in
cosmic metal enrichment. The SN Ia rate in galaxy clusters as a function of
cluster redshift is an almost-direct measure of the DTD, but current estimates
have been limited out to a mean redshift z=1.1, corresponding to time delays,
after cluster star-formation, of over 3.2 Gyr. We analyze data from a Hubble
Space Telescope monitoring project of 12 galaxy clusters at z=1.13-1.75, where
we discover 29 SNe, and present their multi-band light curves. Based on the SN
photometry and the apparent host galaxies, we assess cluster membership and SN
type, finding 11 cases that are likely SNe Ia in cluster galaxies and 4 more
cases which are possible but not certain cluster SNe Ia. We conduct simulations
to estimate the SN detection efficiency, the experiment's completeness, and the
photometric errors, and perform photometry of the cluster galaxies to derive
the cluster stellar masses. Separating the cluster sample into high-z and low-z
bins, we obtain rest-frame SN Ia rates per unit formed stellar mass of $2.2
^{+2.6}_{-1.3}\times 10^{-13}{\rm yr}^{-1}{\rm M}_\odot^{-1}$ at a mean
redshift z=1.25, and $3.5^{+6.6}_{-2.8} \times 10^{-13}{\rm yr}^{-1}{\rm
M}_\odot^{-1}$ at z=1.58. Combining our results with previous cluster SN Ia
rates, we fit the DTD, now down to delays of 1.5 Gyr, with a power-law
dependence, $t^\alpha$, with $\alpha=-1.30^{+0.23}_{-0.16}$. We confirm
previous indications for a Hubble-time-integrated SN Ia production efficiency
that is several times higher in galaxy clusters than in the field, perhaps
caused by a peculiar stellar initial mass function in clusters, or by a higher
incidence of binaries that will evolve into SNe Ia.
