Gravitational redshift in the void-galaxy cross-correlation function in redshift space

We construct an analytic model for the void-galaxy cross-correlation function that enables theoretical predictions of the dipole signal produced dominantly by the gravitational redshift within voids for the first time. By extending a theoretical formulation for the redshift-space distortion of galaxies to include the second order terms of the galaxy peculiar velocity v and the gravitational potential, we formulate the void-galaxy cross-correlation function multipoles in the redshift space, the monopole ξ0(s), dipole ξ1(s), and quadrupole ξ2(s). We find that the dipole ξ1(s) is dominated by the gravitational redshift, which provides a unique opportunity to detect the gravitational potential of voids. Thus, for the dipole ξ1(s)(s), the gravitational redshift is crucial. Although the higher order effect is almost negligible on the monopole ξ0(s), it has an influence on the quadrupole ξ2(s). The effects from the random velocity of galaxies and the definition of the void center on the dipole signal are also discussed. Our model offers a new theoretical probe for the detection of gravitational redshift with voids and further tests on cosmology and gravity.