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.
