Comparison of CMB lensing efficiency of gravitational waves and large scale structure

We provide a detailed treatment and comparison of the weak lensing effects due to large scale structure (LSS), or scalar density perturbations and those due to gravitational waves (GWs) or tensor perturbations, on the temperature and polarization power spectra of the cosmic microwave background (CMB). We carry out the analysis both in real space by using the correlation function method, as well as in the spherical harmonic space. We find an intriguing similarity between the lensing kernels associated with LSS lensing and GW lensing. It is found that the lensing kernels only differ in relative negative signs and their form is very reminiscent of even and odd parity bipolar spherical harmonic coefficients. Through a numerical study of these lensing kernels, we establish that lensing due to GW is more efficient at distorting the CMB spectra as compared to LSS lensing, particularly for the polarization power spectra. Finally we argue that the CMB B-mode power spectra measurements can be used to place interesting constraints on GW energy densities.