On the impact of large angle CMB polarization data on cosmological
  parameters
  (abridged) We study the impact of the large-angle CMB polarization datasets
publicly released by the WMAP and Planck satellites on the estimation of
cosmological parameters of the $\Lambda$CDM model. To complement large-angle
polarization, we consider the high-resolution CMB datasets from either WMAP or
Planck, as well as CMB lensing as traced by Planck. In the case of WMAP, we
compute the large-angle polarization likelihood starting over from
low-resolution frequency maps and their covariance matrices, and perform our
own foreground mitigation technique, which includes as a possible alternative
Planck 353 GHz data to trace polarized dust. We find that the latter choice
induces a downward shift in the optical depth $\tau$, of order ~$2\sigma$,
robust to the choice of the complementary high-l dataset. When the Planck 353
GHz is consistently used to minimize polarized dust emission, WMAP and Planck
70 GHz large-angle polarization data are in remarkable agreement: by combining
them we find $\tau = 0.066 ^{+0.012}_{-0.013}$, again very stable against the
particular choice for high-$\ell$ data. We find that the amplitude of
primordial fluctuations $A_s$, notoriously degenerate with $\tau$, is the
parameter second most affected by the assumptions on polarized dust removal,
but the other parameters are also affected, typically between $0.5$ and
$1\sigma$. In particular, cleaning dust with \planck's 353 GHz data imposes a
$1\sigma$ downward shift in the value of the Hubble constant $H_0$,
significantly contributing to the tension reported between CMB based and direct
measurements of $H_0$. On the other hand, we find that the appearance of the
so-called low $\ell$ anomaly, a well-known tension between the high- and
low-resolution CMB anisotropy amplitude, is not significantly affected by the
details of large-angle polarization, or by the particular high-$\ell$ dataset
employed.
