A New Constraint on the Optical Depth from the Reionization History Independent of CMB Large-Scale E-Mode Polarization

Recent studies report a mild discrepancy between baryon acoustic oscillation (BAO) and cosmic microwave background (CMB) measurements within the $\mathrm{\Lambda }$CDM framework. This discrepancy could be explained if the optical depth $\tau$ inferred from the CMB large-scale E-mode polarization is underestimated, which may be biased by foreground-subtraction or instrumental systematics. In this work, we present a determination of $\tau$ independent of the large-scale E-mode polarization, using the latest measurements of the redshift evolution of the neutral hydrogen fraction $x_{\mathrm{H}\mathrm{I}}(z)$, which is constrained by Lyman-$\alpha$ forest and damping-wing absorption measurements at $z\sim 5$-$14$, based on ground-based optical and JWST observations. Combining $x_{\mathrm{H}\mathrm{I}}(z)$ with the Planck CMB power spectra excluding the large-scale E-mode polarization, we obtain $\tau =0.0552_{-0.0026}^{+0.0019}$, a stringent constraint consistent with the previous CMB results including the large-scale E-mode. We also evaluate a potential systematic error in our method associated with absorption modeling, obtaining $\tau =0.0552_{-0.0049}^{+0.0075}$. Using this constraint on $\tau$, we resolve the degeneracy in the $\tau$-${\mathrm{\Omega }}_m$ plane and find a $2.4\sigma$ tension with the DESI DR2 BAO results, thereby confirming the claimed mild discrepancy suggestive of physics beyond $\mathrm{\Lambda }$CDM. Finally, we derive an upper limit on the sum of neutrino masses, $\mathrm{\Sigma }{m}_{\nu }<0.0550(0.0717)$ eV at the 95% (99%) confidence level. This limit favors the normal mass ordering and, when combined with the lower limits from neutrino oscillation experiments, yields a further constraint, $\mathrm{\Sigma }{m}_{\nu }=0.0594_{-0.0007}^{+0.0113}$ eV. However, the cosmological upper limit and the oscillation-based lower limit show a mild $2.2\sigma$ tension, providing an independent indication of possible physics beyond $\mathrm{\Lambda }$CDM.