Large-Scale Structure and Gravitational Waves III: Tidal Effects
  The leading locally observable effect of a long-wavelength metric
perturbation corresponds to a tidal field. We derive the tidal field induced by
scalar, vector, and tensor perturbations, and use second order perturbation
theory to calculate the effect on the locally measured small-scale density
fluctuations. For sub-horizon scalar perturbations, we recover the standard
perturbation theory result ($F_2$ kernel). For tensor modes of wavenumber
$k_L$, we find that effects persist for $k_L\tau \gg 1$, i.e. even long after
the gravitational wave has entered the horizon and redshifted away, i.e. it is
a "fossil" effect. We then use these results, combined with the "ruler
perturbations" of arXiv:1204.3625, to predict the observed distortion of the
small-scale matter correlation function induced by a long-wavelength tensor
mode. We also estimate the observed signal in the B mode of the cosmic shear
from a gravitational wave background, including both tidal (intrinsic
alignment) and projection (lensing) effects. The non-vanishing tidal effect in
the $k_L\tau \gg 1$ limit significantly increases the intrinsic alignment
contribution to shear B modes, especially at low redshifts $z \lesssim 2$.
