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.
