Forecasts on the speed of gravitational waves at high $z$
  The observation of GW170817 binary neutron star (BNS) merger event has
imposed strong bounds on the speed of gravitational waves (GWs) locally,
inferring that the speed of GWs propagation is equal to the speed of light.
Current GW detectors in operation will not be able to observe BNS merger to
long cosmological distance, where possible cosmological corrections on the
cosmic expansion history are expected to play an important role, specially for
investigating possible deviations from general relativity. Future GW detectors
designer projects will be able to detect many coalescences of BNS at high $z$,
such as the third generation of the ground GW detector called Einstein
Telescope (ET) and the space-based detector deci-hertz interferometer
gravitational wave observatory (DECIGO). In this paper, we relax the condition
$c_T/c = 1$ to investigate modified GW propagation where the speed of GWs
propagation is not necessarily equal to the speed of light. Also, we consider
the possibility for the running of the Planck mass corrections on modified GW
propagation. We parametrize both corrections in terms of an effective GW
luminosity distance and we perform a forecast analysis using standard siren
events from BNS mergers, within the sensitivity predicted for the ET and
DECIGO. We find at high $z$ very strong forecast bounds on the running of the
Planck mass, namely $\mathcal{O}(10^{-1})$ and $\mathcal{O}(10^{-2})$ from ET
and DECIGO, respectively. Possible anomalies on GW propagation are bound to
$|c_T/c - 1| \leq 10^{-2} \,\,\, (10^{-2})$ from ET (DECIGO), respectively. We
finally discuss the consequences of our results on modified gravity
phenomenology.
