Constraints on Tensor-to-scalar Ratio for Planck Mission

One of the main goals of modern CMB missions is to measure the tensor-to-scalar ratio $r$ accurately to constrain inflation models. Due to the ignorance of the reionization history $X_{e}(z)$, this analysis is usually done by assuming an instantaneous reionization $X_{e}(z)$ occurring at $z\sim10$. Moreover, due to strong mixing of B-mode and E-mode polarizations in cut-sky measurements, applying the simple $f_{sky}$ modification on full-sky forecast would not give satisfactory results. In this work, we make a forecast of the constraints on $r$ for the Planck mission taking into account the general reionization scenario and cut-sky. We also study the degeneracy caused by $r$ and $X_{e}(z)$ on B-mode polarization. Our results show that by applying N-point interpolation analysis on reionization history, it can correct the bias induced by instantaneous reionization assumption and the best fit of $r$ is constrained within $5\%$ error level by including both temperature and polarization correlation in likelihood, if the true value of $r$ is 0.1.