Second-order cosmological perturbations. I. Produced by scalar-scalar coupling in synchronous gauge

We present a systematic study of the 2nd-order scalar, vector, and tensor metric perturbations in the Einstein-de Sitter Universe in synchronous coordinates. For the scalar-scalar coupling between 1st-order perturbations, we decompose the 2nd-order perturbed Einstein equation into the respective field equations of 2nd-order scalar, vector, and tensor perturbations, and obtain their solutions with general initial conditions. In particular, the decaying modes of solution are included, the 2nd-order vector is generated even if the 1st-order vector is absent, and the solution of the 2nd-order tensor corrects that in literature. We perform general synchronous-to-synchronous gauge transformations up to 2nd order generated by a 1st-order vector field ${\xi }^{(1)\mu }$ and a 2nd-order ${\xi }^{(2)\mu }$. All the residual gauge modes of 2nd-order metric perturbations and density contrast are found, and their number is substantially reduced when the transformed 3-velocity of dust is set to zero. Moreover, we show that only ${\xi }^{(2)\mu }$ is effective in carrying out 2nd-order transformations that we consider, because ${\xi }^{(1)\mu }$ has been used in obtaining the 1st-order perturbations. Holding the 1st-order perturbations fixed, the transformations by ${\xi }^{(2)\mu }$ on the 2nd-order perturbations have the same structure as those by ${\xi }^{(1)\mu }$ on the 1st-order perturbations.