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Weak lensing distortion of the background cosmic microwave background (CMB)temperature and polarization patterns by the foreground density fluctuations iswell studied in the literature. We discuss the gravitational lensingmodification to CMB anisotropies and polarization by a stochastic background ofprimordial gravitational waves between us and the last scattering surface.While density fluctuations perturb CMB photons via gradient-type deflectionsonly, foreground gravitational waves distort CMB anisotropies via bothgradient- and curl-type displacements. The latter is a rotation of backgroundimages, while the former is related to the lensing convergence. For aprimordial background of inflationary gravitational waves, with an amplitudecorresponding to a tensor-to-scalar ratio below the current upper limit of$\sim$ 0.3, the resulting modifications to the angular power spectra of CMBtemperature anisotropy and polarization are below the cosmic variance limit. Attens of arcminute angular scales and below, these corrections, however, areabove the level at which systematics must be controlled in all-sky anisotropyand polarization maps with no instrumental noise and other secondary andforeground signals.Comment: 11 pages, 4 figures; Revised version updates the numerical  calculation for several corrections to the analytical formulation of lensing  by foreground gravitational waves. Main conclusions unchanged. Version  accepted for publication in Phys. Rev.

Weak lensing of the cosmic microwave background by foreground gravitational waves