Mapping the cosmic expansion history from LIGO-Virgo-KAGRA in synergy with DESI and SPHEREx

The measurement of the expansion history of the Universe from the redshift unknown gravitational wave (GW) sources (dark GW sources) detectable from the network of LIGO-Virgo-KAGRA (LVK) detectors depends on the synergy with the galaxy surveys having accurate redshift measurements over a broad redshift range, large sky coverage, and detectability of fainter galaxies. In this work, we explore the possible synergy of the LVK with the spectroscopic galaxy surveys such as DESI and SPHEREx to measure the cosmological parameters which are related to the cosmic expansion history and the GW bias parameters. We show that by using the three-dimensional spatial cross-correlation between the dark GW sources and the spectroscopic galaxy samples, we can measure the value of Hubble constant with about $2%$ and $1.5%$ precision from LVK+DESI and LVK+SPHEREx respectively from the five years of observation with $50%$ duty-cycle for the GW merger rates driven by the star formation history. Similarly, the dark energy equation of state can be measured with about $10%$ and $8%$ precision from LVK+DESI and LVK+SPHEREx respectively. We find that due to the larger sky coverage of SPHEREx than DESI, the performance in constraining the cosmological parameters is better from the former than the latter. By combining Euclid along with DESI, and SPHEREx a marginal gain in the measurability of the cosmological parameters is possible from the sources at high redshift ($zgeq 0.9$).