Measuring the speed of gravity and the cosmic expansion with time delays between gravity and light from binary neutron stars

The first observation of a gravitational wave (GW) and a short gamma-ray burst (sGRB) emitted by the same binary neutron star (BNS) merger officially opened the field of GW multimessenger astronomy. In this paper, we define and address $\textit{lagging sirens}$, a new class of multimessenger BNSs for which associated GWs and sGRBs are observed without the identification of their host galaxy. We propose a new methodology to use the observed time delay of these sources to constrain the speed of gravity that is, the propagation speed of gravitational waves, the Hubble constant and the prompt time delay distribution between GWs and sGRBs, even though a direct redshift estimation from the host galaxy is unavailable. Our method exploits the intrinsic relation between GWs and sGRBs observed and prompt time delays to obtain a statistical redshift measure for the cosmological sources. We show that this technique can be used to infer the Hubble constant at the $10\%$~level of precision with future-generation GW detectors such as the Einstein Telescope and only 100 observations of this kind. The novel procedure that we propose has systematics that differ completely from the ones of previous GW methods for cosmology. Additionally, we demonstrate for the first time that the speed of gravity and the distribution of the prompt time delays between GWs and sGRBs can be inferred conjointly with less than 10 sources even with current GW detector sensitivities.