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, while also providing extra
cosmological information from a broader range of scales. We compute idealized
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 in spheres 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 could be
able to measure $f\sigma_8(z_\mathrm{eff}\simeq2.3)$ with a precision of
$5-12\%$, depending on the smallest scale fitted.
