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 (F2β kernel). For tensor modes of wavenumber
kLβ, we find that effects persist for kLβΟβ«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 kLβΟβ«1 limit significantly increases the intrinsic alignment
contribution to shear B modes, especially at low redshifts zβ²2.Comment: 24 pages, 4 figures; v2: added references and corrected typos; v3:
corrected factor of 2 in Sec. VI and intrinsic alignment matching,
conclusions unchange