Emulating galaxy and peculiar velocity clustering on non-linear scales

We explore the potential of cross-correlating galaxies and peculiar velocities on non-linear scales to enhance cosmological constraints. Leveraging the \textsc{AbacusSummit} simulation suite and the halo occupation distribution (HOD) formalism, we train emulator models to describe the non-linear clustering of galaxies and velocities in redshift space. Our analysis demonstrates that combining galaxy and peculiar velocity clustering, provides tighter constraints on both HOD and cosmological parameters, particularly on $\sigma_8$ and $w_0$. We further apply our models to realistic mock catalogues, reproducing the expected density and peculiar velocity errors of type-Ia supernovae and Tully-Fisher/fundamental plane measurements for the combined ZTF and DESI measurements. While systematic biases arise in the HOD parameters, the cosmological constraints remain unbiased, yielding $3.8\%$ precision measurement on $f\sigma_8$ compared to $4.7\%$ using galaxy clustering alone. We demonstrate that, while combining tracers with realistic velocity measurements still yields improvement, the gains are diminished, highlighting the need for further efforts to reduce velocity measurement uncertainties and correct observational systematics on small scales.