  We investigate the tensor clustering fossils as a possible probe to constrain
the theory of gravity, in particular the deviation of the sound speed of
gravitational waves from the speed of light at high redshifts. We develop the
formalism of the effective Poisson equation to include the novel
phenomenological model of the scalar-tensor tidal interactions that are
expected to be induced by the modification of the theory of gravity. We show
that the tensor clustering fossils can arise from the propagation of
gravitational waves, the growth of the large-scale structures, and the
second-order contributions from the effective Poisson equation. We construct
the small-scale effective Lagrangian from the Horndeski scalar-tensor theory
and derive the formula applicable to the tensor clustering fossils in the
language of the effective field theory of dark energy. As a demonstration, we
apply the formalism to the constraint on the sound speed of gravitational waves
in the futuristic survey.
