Detecting the tensor-to-scalar ratio with the pure pseudospectrum reconstruction of $B$-mode

In this work we employ the pure-pseudo formalism devised to minimise the effects of the leakage on the variance of power spectrum estimates and discuss the limits on the tensor-to-scalar ratio, $r$, that could be realistically set by current and forthcoming measurements of the $B$-mode angular power spectrum. We compare those with the results obtained using other approaches: na\"{i}ve mode-counting, minimum-variance quadratic estimators, and re-visit the question of optimizing the sky coverage of small-scale, suborbital experiments in order to maximize the statistical significance of the detection of $r$. We show that the optimized sky coverage is largely insensitive to the adopted approach at least for reasonably compact sky patches. We find, however, that the mode-counting overestimates the detection significance by a factor $\sim1.17$ as compared to the lossless maximum variance approach and by a factor $\sim1.25$ as compared to the lossy pure pseudo-spectrum estimator. In a second time, we consider more realistic experimental configurations. With a pure pseudospectrum reconstruction of $B$-modes and considering only statistical uncertainties, we find that a detection of $r\sim0.11$, $r\sim0.0051$ and $r\sim0.0026$ at 99$\%$ of confidence level is within the reach of current sub-orbital experiments, future arrays of ground-based telescopes and a satellite mission, respectively. This means that an array of telescopes could be sufficient to discriminate between large- and small-field models of inflation, even if the $E$-to-$B$ leakage is consistently included but accounted for in the analysis. However, a satellite mission will be required to distinguish between different small-field models depending on the number of e-folds.