Modified gravitational-wave propagation and standard sirens
  Studies of dark energy at advanced gravitational-wave (GW) interferometers
normally focus on the dark energy equation of state $w_{\rm DE}(z)$. However,
modified gravity theories that predict a non-trivial dark energy equation of
state generically also predict deviations from general relativity in the
propagation of GWs across cosmological distances, even in theories where the
speed of gravity is equal to $c$. We find that, in generic modified gravity
models, the effect of modified GW propagation dominates over that of $w_{\rm
DE}(z)$, making modified GW propagation a crucial observable for dark energy
studies with standard sirens. We present a convenient parametrization of the
effect in terms of two parameters $(\Xi_0,n)$, analogue to the $(w_0,w_a)$
parametrization of the dark energy equation of state, and we give a limit from
the LIGO/Virgo measurement of $H_0$ with the neutron star binary GW170817. We
then perform a Markov Chain Monte Carlo analysis to estimate the sensitivity of
the Einstein Telescope (ET) to the cosmological parameters, including
$(\Xi_0,n)$, both using only standard sirens, and combining them with other
cosmological datasets. In particular, the Hubble parameter can be measured with
an accuracy better than $1\%$ already using only standard sirens while, when
combining ET with current CMB+BAO+SNe data, $\Xi_0$ can be measured to $0.8\%$
. We discuss the predictions for modified GW propagation of a specific nonlocal
modification of gravity, recently developed by our group, and we show that they
are within the reach of ET. Modified GW propagation also affects the GW
transfer function, and therefore the tensor contribution to the ISW effect.
