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 $\bm v$ and the gravitational potential, we formulate the void-galaxy cross-correlation function multipoles in the redshift space, the monopole $\xi_0^{(s)}$, dipole $\xi_1^{(s)}$ and quadrupole $\xi_2^{(s)}$. We find that the dipole $\xi_1^{(s)}$ is dominated by the gravitational redshift, which provide a unique opportunity to detect the gravitational potential of voids. Thus, for the dipole $\xi_1^{(s)}(s)$, the gravitational redshift is crucial. Although the higher order effect is almost negligible on the monopole $\xi_0^{(s)}$, it has an influence on the quadrupole $\xi_2^{(s)}$. 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.