Golden and Silver Dark Sirens for precise H0 measurements with the Hobby-Eberly telescope

Gravitational waves (GWs) provide an independent measurement of the Hubble constant (H0) through the standard siren method. While bright sirens remain rare, dark sirens are more common but typically suffer from weak constraining power due to the large number of possible host galaxies. We focus on two promising subclasses of dark sirens: golden and silver dark sirens, where the H0 posterior is dominated by one or a few host galaxies because of their small sky areas, respectively. Our forecasts indicate that such events will be present under LIGO A+ and A# sensitivities. In this regime, precise redshifts and high catalog completeness are essential. To address this, we incorporate the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX), a spectroscopic survey offering both high completeness and precision, into our forecasts. These results highlight the importance of dedicated spectroscopic follow-up surveys for realizing the full potential of dark sirens in measuring the Hubble constant.