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 $\mathit{v}$ 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 provides 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.

• Received 21 May 2018

DOI:https://doi.org/10.1103/PhysRevD.98.043527