We use 47 gravitational wave sources from the Third LIGO-Virgo-Kamioka Gravitational Wave Detector Gravitational Wave Transient Catalog (GWTC-3) to estimate the Hubble parameter H(z), including its current value, the Hubble constant H0. Each gravitational wave (GW) signal provides the luminosity distance to the source, and we estimate the corresponding redshift using two methods: the redshifted masses and a galaxy catalog. Using the binary black hole (BBH) redshifted masses, we simultaneously infer the source mass distribution and H(z). The source mass distribution displays a peak around 34Me, followed by a drop-off. Assuming this mass scale does not evolve with the redshift results in a H(z) measurement, yielding H0 = 68+12-8 km s-1 Mpc-1 (68% credible interval) when combined with the H0 measurement from GW170817 and its electromagnetic counterpart. This represents an improvement of 17% with respect to the H0 estimate from GWTC-1. The second method associates each GW event with its probable host galaxy in the catalog GLADE+, statistically marginalizing over the redshifts of each event's potential hosts. Assuming a fixed BBH population, we estimate a value of H0 = 68+8-6 km s-1 Mpc-1 with the galaxy catalog method, an improvement of 42% with respect to our GWTC-1 result and 20% with respect to recent H0 studies using GWTC-2 events. However, we show that this result is strongly impacted by assumptions about the BBH source mass distribution; the only event which is not strongly impacted by such assumptions (and is thus informative about H0) is the well-localized event GW190814.
