Improving cosmological distance measurements at high redshift using Lyman-Alpha forest BAO reconstruction

The Dark Energy Spectroscopic Instrument (DESI) measures baryon acoustic oscillations (BAO) in the Lyman-$α$ (Lyman-alpha) forest and provides a robust probe of the expansion history in the matter-dominated era. Yet, non-linear structure formation broadens and shifts the BAO peak. To undo these effects, we extend standard Zel'dovich reconstruction, routinely applied to galaxy surveys, to the Lyman-alpha forest. We validate our methodology on large-volume idealized Lyman-alpha forest mocks painted on top of the \textsc{AbacusSummit} $N$-body simulations, using linear control variates to suppress sample variance and thereby reduce the error bars by a factor of up to $\approx \sqrt{50}$. In fully sampled periodic boxes, reconstruction halves the radial BAO damping scale and reduces the uncertainties on the radial ($\apar$) and transverse ($\aperp$) scaling parameters by $\approx25\%$ and $\approx15\%$, respectively. It also reduces the non-linear shift from $Δ\apar=+0.3\%$ to $+0.10\pm0.07\%$, while $\aperp$ remains unbiased. To make this approach readily applicable to observational data, we demonstrate the impact of a survey mask with a DESI-like density of $50\,\mathrm{deg}^{-2}$. For this mask, we compare two approaches to estimate the displacement field: either directly from the Lyman-alpha forest or using an external tracer with a number density of $\bar n \simeq 10^{-3}\,h^3\mathrm{Mpc}^{-3}$, such as Lyman-break galaxies and Lyman-$α$ emitters---both key targets for DESI-II. Both approaches give comparable gains and reduce the uncertainties on $\apar$ and $\aperp$ by $\approx 27\%$ and $\approx 10\%$, respectively, with mild improvements for futuristic high-density surveys with $160\,\mathrm{deg}^{-2}$. Correspondingly, the figure of merit improves by factors of $1.66$ for fully sampled boxes, $1.65$ for a DESI-like, and $1.76$ for a high-density survey.