Cosmology beyond BAO from the 3D distribution of the Lyman-$\alpha$ forest

We propose a new method for fitting the full-shape of the Lyman-$\alpha$ (Ly$\alpha$) forest three-dimensional (3D) correlation function in order to measure the Alcock-Paczynski (AP) effect. Our method preserves the robustness of baryon acoustic oscillations (BAO) analyses when it comes to measuring the position of the acoustic peak, while also providing extra cosmological information from a broader range of scales. We compute forecasts for the Dark Energy Spectroscopic Instrument (DESI) using the Ly$\alpha$ auto-correlation and its cross-correlation with quasars, and show how this type of analysis improves cosmological constraints. The DESI Ly$\alpha$ BAO analysis is expected to measure $H(z_\mathrm{eff})r_\mathrm{d}$ and $D_\mathrm{M}(z_\mathrm{eff})/r_\mathrm{d}$ with a precision of $\sim0.9\%$ each, where $H$ is the Hubble parameter, $r_\mathrm{d}$ is the comoving BAO scale, $D_\mathrm{M}$ is the comoving angular diameter distance and the effective redshift of the measurement is $z_\mathrm{eff}\simeq2.3$. By fitting the AP parameter from the full shape of the two correlations, we show that we can obtain a precision of $\sim0.5-0.6\%$ on each of $H(z_\mathrm{eff})r_\mathrm{d}$ and $D_\mathrm{M}(z_\mathrm{eff})/r_\mathrm{d}$. Furthermore, we show that a joint full-shape analysis of the Ly$\alpha$ auto-correlation and its cross-correlation with quasars can measure the linear growth rate times the amplitude of matter fluctuations on scales of $8\;h^{-1}$Mpc, $f\sigma_8(z_\mathrm{eff})$. Such an analysis could provide the first ever measurement of $f\sigma_8(z_\mathrm{eff})$ at redshift $z_\mathrm{eff}>2$. By combining this with the quasar auto-correlation in a joint analysis of the three high-redshift two-point correlation functions, we show that DESI will be able to measure $f\sigma_8(z_\mathrm{eff}\simeq2.3)$ with a precision of $5-12\%$, depending on the smallest scale fitted.