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Current cosmological data exhibit discordance between indirect and somedirect inferences of the present-day expansion rate, $H_0$. Early dark energy(EDE), which briefly increases the cosmic expansion rate prior torecombination, is a leading scenario for resolving this "Hubble tension" whilepreserving a good fit to cosmic microwave background (CMB) data. However, thiscomes at the cost of changes in parameters that affect structure formation inthe late-time universe, including the spectral index of scalar perturbations,$n_s$. Here, we present the first constraints on axion-like EDE using data fromthe Lyman-$\alpha$ forest, i.e., absorption lines imprinted in backgroundquasar spectra by neutral hydrogen gas along the line of sight. We consider twoindependent measurements of the one-dimensional Ly$\alpha$ forest flux powerspectrum, from the Sloan Digital Sky Survey (SDSS eBOSS) and from theMIKE/HIRES and X-Shooter spectrographs. We combine these with a baselinedataset comprised of Planck CMB data and baryon acoustic oscillation (BAO)measurements. Combining the eBOSS Ly$\alpha$ data with the CMB and BAO datasetreduces the 95% confidence level (CL) upper bound on the maximum fractionalcontribution of EDE to the cosmic energy budget, $f_{\rm EDE}$, from 0.07 to0.03 and constrains $H_0=67.9_{-0.4}^{+0.4}$ km/s/Mpc (68% CL), with maximum aposteriori value $H_0=67.9$ km/s/Mpc. Similar results are obtained for theMIKE/HIRES and X-Shooter Ly$\alpha$ data. Our Ly$\alpha$-based EDE constraintsyield $H_0$ values that are in $>4\sigma$ tension with the SH0ESdistance-ladder measurement and are driven by the preference of the Ly$\alpha$forest data for $n_s$ values lower than those required by EDE cosmologies thatfit Planck CMB data. Taken at face value, the Ly$\alpha$ forest severelyconstrains canonical EDE models that could resolve the Hubble tension.

Canonical Hubble-Tension-Resolving Early Dark Energy Cosmologies are Inconsistent with the Lyman-$\alpha$ Forest