Constraining Warm Dark Matter with high-$z$ supernova lensing
  We propose a new method to constrain the warm dark matter (WDM) particle
mass, $m_\chi$, based on the counts of multiply imaged, distant supernovae (SN)
produced by strong lensing by intervening cosmological matter fluctuations. The
counts are very sensitive to the WDM particle mass, assumed here to be
$m_\chi=1, 1.5, 2$ keV. We use the analytic approach developed by Das &
Ostriker to compute the probability density function of the cold dark matter
(CDM) convergence ($\kappa$) on the lens plane; such method has been
extensively tested against numerical simulations. We have extended this method
generalizing it to the WDM case, after testing it against WDM $N$-body
simulations. Using the observed cosmic star formation history we compute the
probability for a distant SN to undergo a strong lensing event in different
cosmologies. A minimum observing time of 2 yr (5 yr) is required for a future
100 square degrees survey reaching $z \approx 4$ ($z \approx 3$) to disentangle
at 2$\sigma$ a WDM ($m_\chi=1$ keV) model from the standard CDM scenario. Our
method is not affected by any astrophysical uncertainty (such as baryonic
physics effects), and, in principle, it does not require any particular
dedicated survey strategy, as it may come as a byproduct of a future SN survey.
