Constraints on the Thompson optical depth to the CMB from the Lyman-$\alpha$ forest

We present the first constraints on the electron optical depth to reionization, $\tau_{\mathrm{e}}$, from the Lyman-$\alpha$ forest alone for physically motivated reionization models that match the reionization's end-point, $z_{\rm{end}}$, required by the same astrophysical probe, and for symmetric reionization models with fixed duration, $\Delta z$, commonly adopted in CMB reionization analyses. Compared to traditional estimates from the latter, the Lyman-$\alpha$ forest traces the ionization state of the IGM through its coupling with the thermal state. We find an explicit mapping between the two solving the chemistry and temperature evolution equations for hydrogen and helium. Our results yield $\tau_{\mathrm{e}}$=$0.042^{+0.047}_{-0.02}$ (95% C.L) and $\tau_{\mathrm{e}}$=$0.042^{+0.024}_{-0.015}$ for reionization models with $z_{\rm{end}}$ and $\Delta z$-fixed, respectively, disfavoring a high $\tau_{\mathrm{e}}$=0.09 by 2.57$\sigma$ and 4.31$\sigma$. With mock Lyman-$\alpha$ forest data that mimics the precision of future larger quasar sample datasets, we would potentially obtain tighter $\tau_{\mathrm{e}}$ constraints and exclude such a high $\tau_{\mathrm{e}}$ with a higher significance, paving the way for novel constraints on the epoch of reionization from a large-scale structure probe independent of the CMB.