  Weak lensing distortion of the background cosmic microwave background (CMB)
temperature and polarization patterns by the foreground density fluctuations is
well studied in the literature. We discuss the gravitational lensing
modification to CMB anisotropies and polarization by a stochastic background of
primordial gravitational waves between us and the last scattering surface.
While density fluctuations perturb CMB photons via gradient-type deflections
only, foreground gravitational waves distort CMB anisotropies via both
gradient- and curl-type displacements. The latter is a rotation of background
images, while the former is related to the lensing convergence. For a
primordial background of inflationary gravitational waves, with an amplitude
corresponding to a tensor-to-scalar ratio below the current upper limit of
$\sim$ 0.3, the resulting modifications to the angular power spectra of CMB
temperature anisotropy and polarization are below the cosmic variance limit. At
tens of arcminute angular scales and below, these corrections, however, are
above the level at which systematics must be controlled in all-sky anisotropy
and polarization maps with no instrumental noise and other secondary and
foreground signals.

Astrophysics