Is patchy reionization an obstacle in detecting the primordial gravitational wave signal?

The large-scale CMB B-mode polarization is the direct probe to the low frequency primordial gravitational wave signal. However, unambiguous measurement of this signal requires a precise understanding of the possible contamination. One such potential contamination arises from the patchiness in the spatial distribution of free electrons during the epoch of reionization. We estimate the B-mode power spectrum due to patchy reionization using a combination of \emph{photon-conserving} semi-numerical simulation and analytical calculation, and compare its amplitude with the primordial B-mode signal. For a reionization history which is in agreement with several latest observations, we find that a stronger secondary B-mode polarization signal is produced when the reionization is driven by the sources in massive halos and its amplitude can be comparable to the recombination bump for tensor to scalar ratio $(r) \lesssim 5 \times 10^{-4}$. If contamination from patchy reionization is neglected in the analysis of B-mode polarization data, then for the models of reionization considered in this analysis, we find a maximum bias of about $30\%$ in the value of $r=\,10^{-3}$ when spatial modes between $\ell \in [50, 200]$ are used with a delensing efficiency of $50\%$. The inferred bias from patchy reionization is not a severe issue for the upcoming ground-based CMB experiment Simons Observatory, but can be a potential source of confusion for proposed CMB experiments which target to detect the value of $r< 10^{-3}$. However, this obstacle can be removed by utilizing the difference in the shape of the power spectrum from the primordial signal.