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 \texttt{ LIGO-Virgo-KAGRA} (\texttt{ 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 \texttt{ DESI} and \texttt{ 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 \texttt{ LVK+DESI} and \texttt{ LVK+SPHEREx} respectively within the five years of observation time with $50\%$ duty-cycle. Similarly, the dark energy equation of state can be measured with about $10\%$ and $8\%$ precision from \texttt{ LVK+DESI} and \texttt{ LVK+SPHEREx} respectively. We find that due to the large sky coverage of \texttt{ SPHEREx} than \texttt{ DESI}, performance in constraining the cosmological parameters is better from the former than the latter. By combining \texttt{ Euclid} along with \texttt{ DESI} and \texttt{ SPHEREx}, a marginal gain in the measurability of the cosmological parameters is possible from the sources at high redshift ($z\geq 0.9$).
Categories: astro-ph.CO astro-ph.HE gr-qc