  Recently, the characterisation of binary systems of neutron stars has become
central in various fields such as gravitational waves, gamma-ray bursts (GRBs),
and the chemical evolution of galaxies. In this work, we explore possible
observational proxies that can be used to infer some characteristics of the
delay time distribution (DTD) of neutron star mergers (NSMs). We construct a
sample of model galaxies that fulfils the observed galaxy stellar mass
function, star formation rate versus mass relation, and the cosmic star
formation rate density. The star formation history of galaxies is described
with a log-normal function characterised by two parameters: the position of the
maximum and the width of the distribution. We assume a theoretical DTD that
mainly depends on the lower limit and the slope of the distribution of the
separations of the binary neutron stars systems at birth. We find that the
current rate of NSMs ($\mathcal{R}=320^{+490}_{-240}$ Gpc$^{-3}$yr$^{-1}$)
requires that $\sim0.3$ per cent of neutron star progenitors lives in binary
systems with the right characteristics to lead to a NSM within a Hubble time.
We explore the expected relations between the rate of NSMs and the properties
of the host galaxy. We find that the most effective proxy for the shape of the
DTD of NSMs is the current star formation activity of the typical host. At
present, the fraction of short-GRBs observed in star-forming galaxies favours
DTDs with at least $\sim40\%$ of mergers within $100$ Myr. This conclusion will
be put on a stronger basis with larger samples of short-GRBs with host
association (e.g. $600$ events at $z \leq 1$)
