Gauge-invariant formulation of second-order cosmological perturbations

Gauge-invariant treatments of the second-order cosmological perturbation in a four dimensional homogeneous isotropic universe filled with the perfect fluid are completely formulated without any gauge fixing. We derive all components of the Einstein equations in the case where the first order vector and tensor modes are negligible. These equations imply that the tensor and the vector mode of the second-order metric perturbations may be generated by the scalar-scalar mode coupling of the linear order perturbations as the result of the nonlinear effects of the Einstein equations