Future detectability of gravitational-wave induced lensing from
  high-sensitivity CMB experiments
  We discuss the future detectability of gravitational-wave induced lensing
from high-sensitivity cosmic microwave background (CMB) experiments.
Gravitational waves can induce a rotational component of the weak-lensing
deflection angle, usually referred to as the curl mode, which would be
imprinted on the CMB maps. Using the technique of reconstructing lensing
signals involved in CMB maps, this curl mode can be measured in an unbiased
manner, offering an independent confirmation of the gravitational waves
complementary to the B-mode polarization experiments. Based on the Fisher
matrix analysis, we first show that with the noise levels necessary to confirm
the consistency relation for the primordial gravitational waves, the future CMB
experiments will be able to detect the gravitational-wave induced lensing
signals. For a tensor-to-scalar ratio of $r < 0.1$, even if the consistency
relation is difficult to confirm with a high significance, the
gravitational-wave induced lensing would be detected at more than $3\,\sigma$
significance level. Further, we point out that high-sensitivity experiments
will be also powerful to constrain the gravitational waves generated after the
recombination epoch. Compared to the B-mode polarization, the curl mode is
particularly sensitive to gravitational waves generated at low redshifts ($z <
10$) with a low frequency ($k < 10^{-3}$ Mpc$^{-1}$), and it could give a much
tighter constraint on their energy density $\Omega_{\rm GW}$ by more than three
orders of magnitude.
