  Inflationary models can correlate small-scale density perturbations with the
long-wavelength gravitational waves (GW) in the form of the
Tensor-Scalar-Scalar (TSS) bispectrum. This correlation affects the
mass-distribution in the Universe and leads to the off-diagonal correlations of
the density field modes in the form of the quadrupole anisotropy.
Interestingly, this effect survives even after the tensor mode decays when it
re-enters the horizon, known as the fossil effect. As a result, the
off-diagonal correlation function between different Fourier modes of the
density fluctuations can be thought as a way to probe the large-scale GW and
the mechanism of inflation behind the fossil effect. Models of single field
slow roll inflation generically predict a very small quadrupole anisotropy in
TSS while in models of multiple fields inflation this effect can be observable.
Therefore this large scale quadrupole anisotropy can be thought as a
spectroscopy for different inflationary models. In addition, in models of
anisotropic inflation there exists quadrupole anisotropy in curvature
perturbation power spectrum. Here we consider TSS in models of anisotropic
inflation and show that the shape of quadrupole anisotropy is different than in
single field models. In addition in these models the quadrupole anisotropy is
projected into the preferred direction and its amplitude is proportional to
$g_* N_e$ where $N_e$ is the number of e-folds and $g_*$ is the amplitude of
quadrupole anisotropy in curvature perturbation power spectrum. We use this
correlation function to estimate the large scale GW as well as the preferred
direction and discuss the detectability of the signal in the galaxy surveys
like Euclid and 21 cm surveys.
