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 O (10(-1)) and O(10(-2)) from ET and DECIGO, respectively. Possible anomalies on GW propagation are bound to vertical bar c(T)/c - 1 vertical bar <= 10(-2) (10(-2)) from ET (DECIGO), respectively. We finally discuss the consequences of our results on modified gravity phenomenology.