An accurate linear model for redshift space distortions in the void-galaxy correlation function

Redshift space distortions within voids provide a unique method to test for environmental dependence of the growth rate of structures in low density regions, where effects of modified gravity theories might be important. We derive a linear theory model for the redshift space void-galaxy correlation that is valid at all pair separations, including deep within the void, and use this to obtain expressions for the monopole $\xi^s_0$ and quadrupole $\xi^s_2$ contributions. Our derivation highlights terms that have previously been neglected but are important within the void interior. As a result our model differs from previous works and predicts new physical effects, including a change in the sign of the quadrupole term within the void radius. We show how the model can be generalised to include a velocity dispersion. We compare our model predictions to measurements of the correlation function using mock void and galaxy catalogues modelled after the BOSS CMASS galaxy sample using the Big MultiDark $N$-body simulation, and show that the linear model with dispersion provides an excellent fit to the data at all scales, $0\leq s\leq120\;h^{-1}$Mpc. While the RSD model matches simulations, the linear bias approximation does not hold within voids, and care is needed in fitting for the growth rate. We show that fits to the redshift space correlation recover the growth rate $f(z=0.52)$ to a precision of $2.7\%$ using the simulation volume of $(2.5\;h^{-1}\mathrm{Gpc})^3$.