Second-order cosmological perturbations. IV. Produced by scalar-tensor and tensor-tensor couplings during the radiation dominated stage

We continue to study the second-order cosmological perturbations in synchronous coordinates in the framework of general relativity (GR) during the radiation dominated (RD) stage and to focus on the scalar-tensor and tensor-tensor couplings. The first-order curl velocity and the associated first-order vector metric perturbations are assumed to be vanishing. By analytically solving the second-order Einstein equation and the energy-momentum conservation equations, we obtain the second-order formal solutions (in the integral form) of all the metric perturbations, density contrast and velocity; perform the transformation between the synchronous coordinates; and identify the residual gauge modes in the second-order solutions. In addition, we present the second-order gauge transformations of the solutions from synchronous to Poisson coordinates. To apply these formal solutions to concrete cosmological study, one needs to choose proper initial conditions and carry out several numerical integrations.