Probing the limits of cosmological information from the Lyman-$\alpha$ forest 2-point correlation functions

The standard cosmological analysis with the Ly$\alpha$ forest relies on a continuum fitting procedure that suppresses information on large scales and distorts the three-dimensional correlation function on all scales. In this work, we present the first cosmological forecasts without continuum fitting distortion in the Ly$\alpha$ forest, focusing on the recovery of large-scale information. Using idealized synthetic data, we compare the constraining power of the full shape of the Ly$\alpha$ forest auto-correlation and its cross-correlation with quasars using the baseline continuum fitting analysis versus the true continuum. We find that knowledge of the true continuum enables a $\sim10\%$ reduction in uncertainties on the Alcock-Paczy\'nski (AP) parameter and the matter density, $\Omega_\mathrm{m}$. We also explore the impact of large-scale information by extending the analysis up to separations of $240\,h^{-1}\mathrm{Mpc}$ along and across the line of sight. The combination of these analysis choices can recover significant large-scale information, yielding up to a $\sim15\%$ improvement in AP constraints. This improvement is analogous to extending the Ly$\alpha$ forest survey area by $\sim40\%$.