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.
