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We study the 2nd-order scalar, vector and tensor metric perturbations inRobertson-Walker (RW) spacetime in synchronous coordinates during the radiationdominated (RD) stage. The dominant radiation is modeled by a relativistic fluiddescribed by a stress tensor $T_{\mu\nu}=(\rho+p)U_\mu U_\nu+g_{\mu\nu}p$ with$p= c^2_s \rho$, and the 1st-order velocity is assumed to be curlless. Weanalyze the solutions of 1st-order perturbations, upon which the solutions of2nd-order perturbation are based. We show that the 1st-order tensor modespropagate at the speed of light and are truly radiative, but the scalar andvector modes do not. The 2nd-order perturbed Einstein equation contains variouscouplings of 1st-order metric perturbations, and the scalar-scalar coupling isconsidered in this paper. We decompose the 2nd-order Einstein equation into theevolution equations of 2nd-order scalar, vector, and tensor perturbations, andthe energy and momentum constraints. The coupling terms and the stress tensorof the fluid together serve as the effective source for the 2nd-order metricperturbations. The equation of covariant conservation of stress tensor is alsoneeded to determine $\rho$ and $U^\mu$. By solving this set of equations up to2nd order analytically, we obtain the 2nd-order integral solutions of all themetric perturbations, density contrast and velocity. To use these solutions inapplications, one needs to carry out seven types of the numerical integrals. Weperform the residual gauge transformations between synchronous coordinates upto 2nd order, and identify the gauge-invariant modes of 2nd-order solutions.

Second-order cosmological perturbations. III. Produced by scalar-scalar coupling during radiation-dominated stage