A Halo Model Approach to the $\rm{21\, cm}$ and $\rm{Ly}\alpha$ Cross-correlation

We present a halo-model-based approach to calculate the cross-correlation between $\rm{21\,cm}$ HI intensity fluctuations and $\rm{Ly}\alpha$ emitters (LAE) during the epoch of reionization (EoR). Ionizing radiation around dark matter halos are modeled as bubbles with the size and growth determined based on the reionization photon production, among other physical parameters. The cross-correlation shows a clear negative-to-positive transition, associated with transition from ionized to neutral hydrogen in the intergalactic medium during EoR. The cross-correlation is subject to several foreground contaminants, including foreground radio point sources important for $\rm{21\,cm}$ experiments and low-$z$ interloper emission lines, such as $\rm{H}\alpha$, OIII, and OII, for $\rm{Ly}\alpha$ experiments. Our calculations show that by masking out high fluxes in the $\rm{Ly}\alpha$ measurement, the correlated foreground contamination on the $\rm{21\,cm}$-$\rm{Ly}\alpha$ cross-correlation can be dramatically reduced. We forecast the detectability of $\rm{21\,cm}$-$\rm{Ly}\alpha$ cross-correlation at different redshifts and adopt a Fisher matrix approach to estimate uncertainties on the key EoR parameters that have not been well constrained by other observations of reionization. This halo-model-based approach enables us to explore the EoR parameter space rapidly for different $\rm{21\,cm}$ and $\rm{Ly}\alpha$ experiments.