Disentangling patchy reionization signatures from primordial gravitational waves using CMB $E$-mode and $B$-mode polarization

The detection of large angular scale $B$-mode in the Cosmic Microwave Background (CMB) polarization signal will open a direct window into not only the primary CMB anisotropies caused by the primordial gravitational waves (PGW) originating in the epoch of inflation, but also the secondary anisotropies imprinted during the epoch of cosmic reionization. The existence of patchiness in the electron density during reionization produces a unique distortion in the CMB $B$-mode polarization, which can be distinguished from the PGW signal with the aid of spatial frequency modes. In this work, we employ an $EB$ estimator by combining $E$-mode and $B$-mode polarization for the $\tau$ power spectrum signal generated in a photon-conserving semi-numerical reionization model called SCRIPT. We developed a Bayesian framework for the joint detection of the PGW and reionization signal from CMB observations and show the efficacy of this technique for upcoming CMB experiments. We find that, for our model, the $\tau$ power spectrum signal effectively tracks the inhomogeneous electron density field, allowing for robust constraints on the patchy $B$-mode signal. Further, our results indicate that employing the $EB$ estimator for the $\tau$ signal will facilitate ground-based CMB-S4 to detect the patchy $B$-mode signal at approximately $\geq 2\sigma$ confidence level while observations with space-based PICO will improve this detection to $\geq 3\sigma$ going as high as $\geq 7\sigma$ for extreme reionization models. These findings not only highlight the future potential of these experiments to provide an improved picture of the reionization process but also have important implications towards an unbiased measurement of $r$.