Tensor clustering fossils in modified gravity and high-redshift gravitational-wave sound speed

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.