The cosmic evolution of the neutron star merger (NSM) rate can be deduced from the observed cosmic star formation rate. This allows to estimate the rate expected in the horizon of the gravitational wave detectors advanced Virgo and ad LIGO and to compare those rates with independent predictions. In this context, the rapid neutron capture process, or r process, can be used as a constraint assuming NSM is the main astrophysical site for this nucleosynthetic process. We compute the early cosmic evolution of a typical r process element, Europium. Eu yields from NSM are taken from recent nucleosynthesis calculations. The same approach allows to compute the cosmic rate of Core Collapse SuperNovae (CCSN) and the associated evolution of Eu. We find that the bulk of Eu observations at high iron abundance can be rather well fitted by either CCSN or NSM scenarios. However, at lower metallicity, the early Eu cosmic evolution favors NSM as the main astrophysical site for the r process. A comparison between our calculations and spectroscopic observations at very low metallicities allows to constrain the coalescence timescale in the NSM scenario to about 0.1 to 0.2 Gyr. These values are in agreement with the coalescence timescales of some observed binary pulsars. Finally, the cosmic evolution of Eu is used to put constraints on the NSM rate, the merger rate in the horizon of the gravitational wave detectors advanced Virgo/ad LIGO, as well as the expected rate of electromagnetic counterparts to mergers (kilonovae) in large near-infrared surveys.
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