Using Gray Sirens to Resolve the Hubble-Lema\^{i}tre Tension

The measurement of the Hubble-Lema\^{i}tre constant $(H_0)$ from the cosmic microwave background and the Type IA supernovae are at odds with each other. One way to resolve this tension is to use an independent way to measure $H_0$. This can be accomplished by using gravitational-wave (GW) observations. Previous works have shown that with the onset of the next-generation of GW detector networks, it will be possible to constrain $H_0$ better than $2\%$ (which is enough to resolve the tension) with binary black hole systems, also called dark sirens. Bright sirens like binary neutron star systems can also help resolve the tension if both the GW and the following electromagnetic counterpart are detected. In this work, we assess the potential of using neutron star-black hole (NSBH) mergers to measure the Hubble-Lema\^{i}tre constant, both as dark sirens as well as bright sirens, thus, assigning them the term gray sirens. We find that the Voyager network might be able to resolve the tension using NSBH mergers in an observation span of 5 years, whereas next-generation networks which include the Cosmic Explorer detectors and the Einstein Telescope will be able to measure the $H_0$ to sub-percent level.