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

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