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CMB anisotropies are modified by the weak lensing effect of intervening largescale structures on the photon path from the last scattering surface to theobserver. This has to be accounted for when observational data of sensitiveexperiments are used to constrain cosmological models. A common approximationto analyze the CMB angular power spectra is to include only the Gaussian partof the lensing correction and to ignore the non-gaussian terms in the errorcovariance matrix of the spectra. In order to investigate the validity of thisapproximation, we computed these non-Gaussian terms by using a perturbativeexpansion method. We present a graphical method to write down any N-pointcorrelation functions at any order in lensing. We use a pedagogical approach todemonstrate that neglecting non-gaussian terms is an accurate approximation forall polarizations but B, and it will remain so even for the analysis of verysensitive post-Planck experiments. For the B polarization, non-gaussiancontributions up to order 4 must be taken into account.Comment: 29 pages, 13 figures, JCAP style. Improved to match published version  : annex A expanded, references added, few minor correction

Probing inflation with cosmic microwave background polarization: the weak lensing effect on the covariance of cosmic microwave background spectra