  The merging of two neutron stars (MNS) is thought to be the source of short
gamma-ray bursts (SGRB) and gravitational wave transients, as well as the main
production site of r-process elements like Eu. We have derived a new delay time
distribution (DTD) for MNS systems from theoretical considerations and we have
tested it against (i) the SGRB redshift distribution and (ii) the Galactic
evolution of Eu and Fe, in particular the [Eu/Fe] vs [Fe/H] relation. For
comparison, we also tested other DTDs, as proposed in the literature. To
address the first item, we have convolved the DTD with the cosmic star
formation rate, while for the second we have employed a detailed chemical
evolution model of the Milky Way. We have also varied the DTD of Type Ia SNe
(the main Fe producers), the contribution to Eu production from core-collapse
SNe, as well as explored the effect of a dependence on the metallicity of the
occurrence probability of MNS. Our main results can be summarized as follows:
(i) the SGRB redshift distribution can be fitted using DTDs for MNS that
produce average timescales of 300-500 Myr; (ii) if the MNS are the sole
producers of the Galactic Eu and the occurrence probability of MNS is constant
the Eu production timescale must be on the order of <30 Myr; (iii) allowing for
the Eu production in core-collapse SNe, or adopting a metallicity-dependent
occurrence probability, allow us to reproduce both observational constraints,
but many uncertainties are still present in both assumptions.
