Title: A dark standard siren measurement of the Hubble constant following LIGO/Virgo/KAGRA O4a

Abstract: We present a new constraint on the Hubble constant ($H_0$) from the standard dark siren method using a sample of $5$ well-covered gravitational wave (GW) alerts reported during the first part of the fourth LIGO/Virgo/KAGRA observing runs in combination with standard dark sirens from the first three runs. Our methodology relies on the galaxy catalog method alone. We use the full probability density estimation of photometric redshifts derived by a deep learning method using the DESI Legacy Survey and DELVE galaxy catalogs. We add the constraints from the binary black hole mergers candidates S231226av, S231206cc, S230919bj, S230627c, and S230922g to the sample of standard dark sirens analyzed in Alfradique et al. (2024). We combine the $H_0$ posterior for $5$ new standard sirens with other $10$ previous events (3 with updated posteriors), finding $H_0 = 69.9^{+13.3}_{-12.0}~{\rm km~s^{-1}~Mpc^{-1}}$ (68% Highest Density Interval) with the catalog method alone. This result represents an improvement of $\sim 23\%$ comparing the new $15$ dark siren constrain with the previous $10$ dark siren constraint, and a reduction in uncertainty of $\sim 40\%$ from the combination of $15$ dark and bright sirens compared with the GW170817 bright siren alone. The combination of dark and bright siren GW170817 with recent jet constraints yields $H_0$ of $68.0^{+4.3}_{-3.8}~{\rm km~s^{-1}~Mpc^{-1}}$, a $\sim 6\%$ precision from Standard Sirens, reducing the previous constraint uncertainty by $\sim 10\%$ .