Testing chirality of primordial gravitational waves with Planck and
  future CMB data: no hope from angular power spectra
  We use the 2015 Planck likelihood in combination with the Bicep2/Keck
likelihood (BKP and BK14) to constrain the chirality, $\chi$, of primordial
gravitational waves in a scale-invariant scenario. In this framework, the
parameter $\chi$ enters theory always coupled to the tensor-to-scalar ratio,
$r$, e.g. in combination of the form $\chi \cdot r$. Thus, the capability to
detect $\chi$ critically depends on the value of $r$. We find that with present
data set $\chi$ is \textit{de facto}unconstrained. We also provide forecasts
for $\chi$ from future CMB experiments, including COrE+, exploring several
fiducial values of $r$. We find that the current limit on $r$ is tight enough
to disfavor a neat detection of $\chi$. For example, in the unlikely case in
which $r\sim0.1(0.05)$, the maximal chirality case, i.e. $\chi = \pm1$, could
be detected with a significance of $\sim2.5(1.5)\sigma$ at best. We conclude
that the two-point statistics at the basis of CMB likelihood functions is
currently unable to constrain chirality and may only provide weak limits on
$\chi$ in the most optimistic scenarios. Hence, it is crucial to investigate
the use of other observables, e.g. provided by higher order statistics, to
constrain these kind of parity violating theories with the CMB.
