Second-order cosmological perturbations. II. Produced by scalar-tensor and tensor-tensor couplings

We study the second-order perturbations in the Einstein-de Sitter Universe in synchronous coordinates. We solve the second-order perturbed Einstein equation with scalar-tensor and tensor-tensor couplings between 1st-order perturbations, and obtain, for each coupling, the solutions of scalar, vector, and tensor metric perturbations, including both the growing and decaying modes for general initial conditions. We perform general synchronous-to-synchronous gauge transformations up to 2nd order, which are generated by a 1st-order vector field and a 2nd-order vector field, and obtain all the residual gauge modes of the 2nd-order metric perturbations in synchronous coordinates. We show that only the 2nd-order vector field is effective for the 2nd-order transformations that we consider because the 1st-order vector field was already fixed in obtaining the 1st-order perturbations. In particular, the 2nd-order tensor is invariant under 2nd-order gauge transformations using ${\xi }^{(2)\mu }$ only, just like the 1st-order tensor is invariant under 1st-order transformations.