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 \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 ($z\geq
0.9$).
