Impact of next-to-leading order contributions to CMB lensing
  In this Letter we study the impact on cosmological parameter estimation, from
present and future surveys, due to lensing corrections on CMB temperature and
polarization anisotropies beyond leading order. In particular, we show how
post-Born corrections, LSS effects and the correction due to the change in the
polarization direction between the emission at the source and the detection at
the observer, are non-negligible in the determination of the polarization
spectra. They have to be taken into account for an accurate estimation of
cosmological parameters sensitive to or even based on these spectra. We study
in detail the impact of higher order lensing on the determination of the
tensor-to-scalar ratio $r$ and on the estimation of the effective number of
relativistic species $N_\text{eff}$. We find that neglecting higher order
lensing terms can lead to misinterpreting these corrections as a primordial
tensor-to-scalar ratio of about $\mathcal{O}(10^{-3})$. Furthermore, it leads
to a shift of the parameter $N_\text{eff}$ by nearly two sigma considering the
level of accuracy aimed by future S4 surveys.
