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.