Detectability of the gravitational redshift effect from the asymmetric
  galaxy clustering
  It has been recently recognized that the observational relativistic effects,
mainly arising from the light propagation in an inhomogeneous universe, induce
the dipole asymmetry in the cross-correlation function of galaxies. In
particular, the dipole asymmetry at small scales is shown to be dominated by
the gravitational redshift effects. In this paper, we exploit a simple
analytical description for the dipole asymmetry in the cross-correlation
function valid at quasi-linear regime. In contrast to the previous model, a new
prescription involves only one dimensional integrals, providing a faster way to
reproduce the results obtained by Saga et al. (2020). Using the analytical
model, we discuss the detectability of the dipole signal induced by the
gravitational redshift effect from upcoming galaxy surveys. The gravitational
redshift effect at small scales enhances the signal-to-noise ratio (S/N) of the
dipole, and in most of the cases considered, the S/N is found to reach a
maximum at $z\approx0.5$. We show that current and future surveys such as DESI
and SKA provide an idealistic data set, giving a large S/N of $10\sim 20$. Two
potential systematics arising from off-centered galaxies are also discussed
(transverse Doppler effect and diminution of the gravitational redshift
effect), and their impacts are found to be mitigated by a partial cancellation
between two competitive effects. Thus, the detection of the dipole signal at
small scales is directly linked to the gravitational redshift effect, and
should provide an alternative route to test gravity.
