Measuring the reionization optical depth without large-scale CMB polarization

We study the possibility of measuring the optical depth at reionization $\tau$ without relying on large-scale cosmic microwave background (CMB) polarization. Our analysis is driven by the need to obtain competitive measurements that can validate the state-of-the-art constraints on this parameter, widely based on $E$-mode polarization measurements at $\ell \le 30$. This need is partially motivated by the typical concerns regarding anomalies observed in the Planck large-scale CMB data as well as by the remarkable fact that, excluding these latter, $\tau$ consistently exhibits correlations with anomalous parameters, such as $A_{\mathrm{lens}}$ and ${\Omega }_k$, suggesting that slightly higher values of the optical depth at reionization could significantly alleviate or even eliminate anomalies. Within the $\Lambda$ cold dark matter model, our most constraining result is $\tau =0.080\pm 0.012$, obtained by combining Planck temperature and polarization data at $\ell >30$, the Atacama Cosmology Telescope (ACT) and Planck measurements of the lensing potential, baryon acoustic oscillations (BAOs), and type-Ia supernova data from the $\mathrm{Pantheon}+\text{catalog}$. Notably, using only ACT temperature, polarization, and lensing data in combination with BAOs and supernovae, we obtain $\tau =0.076\pm 0.015$, which is entirely independent of Planck. The relative precision of these results is approaching the constraints based on large-scale CMB polarization ($\tau =0.054\pm 0.008$). Despite the overall agreement, we report a slight $1.8\sigma$ shift toward larger values of $\tau$. We also test how these results change by extending the cosmological model. While in many extensions they remain robust, in general, obtaining precise measurements of $\tau$ may become significantly more challenging.