Calibrating redshift distributions at $z>2$ with Lyman-$\alpha$ forest cross-correlations

We explore the feasibility of using Lyman-$\alpha$ (Ly$\alpha$) forests to calibrate the ensemble redshift distribution of the high-redshift tail ($2<z<3$) of photometric galaxies. We use \texttt{CoLoRe} simulations to create mock DESI 5-year Ly$\alpha$ forests and Rubin Observatory LSST 10-year photometric galaxies up to $z=3$, and measure the galaxy redshift distribution via their angular cross-correlations. Due to large redshift-space distortions in the Ly$\alpha$ forest, the conventional $n(z)$ estimator for clustering redshifts does not apply, and we develope a theoretical framework to model the angular cross-correlation directly. Using the simulations, we explore effects of instrumental noise, continuum fitting, and contamination in the Ly$\alpha$ forest, cross-correlation angular scales ($\theta$), and redshift bin size ($\Delta z$) on the signal-to-noise (SNR) of the measurements. We find that continuum fitting methods strongly impact the SNR of the measurements. With our baseline continuum fitting method, \texttt{LyCAN}, at angular scales $\theta\sim10$ arcmin and $\Delta z=0.1$, we measure the cross-correlation signal at $24\sigma$. If the shape of the redshift distribution and galaxy bias evolution are known well for $z<2$, the cross-correlation can constrain the mean redshift of the galaxy sample to $\sigma_z/(1+\bar{z}) = 0.006$ at a mean redshift of $\bar{z}=2$. This demonstrates that Ly$\alpha$ cross-correlation is a reliable and promising method to calibrate the high-redshift tails of photometric Stage IV galaxy surveys.