Constraints on the cosmic expansion rate at redshift 2.3 from the Lyman-$\alpha$ forest

We determine the product of the expansion rate and angular-diameter distance at redshift $z=2.3$ from the anisotropy of Lyman-$\alpha$ (Ly$\alpha$) forest correlations measured by the Sloan Digital Sky Survey (SDSS). Our result is the most precise from large-scale structure at $z>1$. In flat $\Lambda$CDM we determine the matter density to be $\Omega_\mathrm{m}=0.36^{+0.03}_{-0.04}$ from Ly$\alpha$ alone. This is a factor of two tighter than baryon acoustic oscillation results from the same data due to our use of a wide range of scales ($25<r<180$ $h^{-1}\text{Mpc}$). Using a nucleosynthesis prior, we measure the Hubble constant to be $H_0=63.2\pm2.5$ km/s/Mpc. In combination with other SDSS tracers, we find $H_0=67.2\pm0.9$ km/s/Mpc and measure the dark energy equation-of-state parameter to be $w=-0.90\pm0.12$. Our work opens a new avenue for constraining cosmology at high redshift.