  We present a method for modelling the cluster-galaxy correlation function in
redshift-space, down to ~ Mpc scales. The method builds upon the so-called
Galaxy Infall Kinematics (GIK) model, a parametric model for the pairwise
velocities of galaxies with respect to nearby galaxy clusters. We fit the
parameters of the GIK model to a suite of simulations run with different
cosmologies, and use Gaussian Processes to emulate how the GIK parameters
depend upon cosmology. This emulator can then be combined with knowledge of the
real-space clustering of clusters and galaxies, to predict the cluster-galaxy
correlation function in redshift space. Fitting this model to an observed
correlation function enables the extraction of cosmological parameter
constraints, and we present forecasts for a DESI-like survey. We also perform
tests of the robustness of our constraints from fitting to mock data extracted
from N-body simulations, finding that fitting to scales < 3 Mpc/h leads to a
biased inference on cosmology, due to model misspecification on these scales.
Finally, we discuss what steps will need to be taken in order to apply our
method to real data.
