High accuracy on $H_0$ constraints from gravitational wave lensing
  events
  In light of the newly opened and rapidly growing gravitational waves window
in multi-messenger astronomy, in order to fully take advantage of the new
opportunities we are provided with, new ideas are required for a better and
deeper employ of the state-of-the-art probes we handle. Following this goal,
here we suggest a method to constrain the cosmological background, and the
Hubble constant in particular, by future observations of gravitationally lensed
radiation emitted by a single source in both the gravitational wave and the
electromagnetic regimes. The lensing of the gravitational wave radiation, in
fact, can leave a clear imprinting in the corresponding waveform, and we want
to analyze if such kind of measurements can be successfully employed to better
constrain the cosmological background. Thus, by making use of wave optics for
the gravitational wave lensed signal, and of standard geometrical optics
approximation for the electromagnetic one, we study the impact of different
cosmological parameters on the value of the arrival time delay due to
gravitational lensing, given specific gravitational wave frequencies, mass
models of the lens, and redshifts and positions (with respect to the lens) of
the source. Although the rate of lensing of gravitational waves is expected to
be low, we show that even one single lensing event, combined with a prior on
$\Omega_m$ from \textit{Planck}, could provide us with an uncertainty on $H_0$
comparable with present independent probes in a "pessimistic" scenario (with a
pulsar population similar to present Pulsar Timing Array state), and of two
orders smaller in an optimistic one (with a number of observed pulsars as large
as that expected from the Square Kilometer Array). Thus, its role in the
solution of the Hubble tension could be decisive.
