  In this work, we compute rates of merging neutron stars (MNS) in galaxies of
different morphological type, as well as the cosmic MNS rate in a unitary
volume of the Universe adopting different cosmological scenarios. Our aim is to
provide predictions of kilonova rates for future observations both at low and
high redshift. In the adopted galaxy models, we take into account the
production of r-process elements either by MNS or core-collapse supernovae. In
computing the MNS rates we adopt either a constant total time delay for merging
(10 Myr) or a distribution function of such delays. Our main conclusions are:
i) the observed present time MNS rate in our Galaxy is well reproduced either
with a constant time delay or a distribution function $\propto t^{-1}$. The
[Eu/Fe] vs. [Fe/H] relation in the Milky Way can be well reproduced with only
MNS, if the time delay is short and constant. If the distribution function of
delays is adopted, core-collapse supernovae as are also required. ii) The
present time cosmic MNS rate can be well reproduced in any cosmological
scenario, either pure luminosity evolution or a typical hierarchical one, and
spirals are the main contributors to it. iii) The spirals are the major
contributors to the cosmic MNS at all redshifts in hierarchical scenarios. In
the pure luminosity evolution scenario, the spirals are the major contributors
locally, whereas at high redshift ellipticals dominate. iv) The predicted
cosmic MNS rate well agrees with the cosmic rate of short Gamma Ray Bursts if
the distribution function of delays is adopted, in a cosmological hierarchical
scenario observationally derived. v) Future observations of Kilonovae in
ellipticals will allow to disentangle among constant or a distribution of time
delays as well as among different cosmological scenarios.
