Non-Linear Theory of Gravitational Instability in the Expanding Universe. II

The general relativistic non-linear theory of gravitational instability is reformulated in order to get a hydrodynamical system of equations which are applicable at any early stage of cosmic expansion. Non-linearity is taken into account to the second-order smallness. On this basis, the second-order density perturbations associated with the first-order rotational and gravitational waves are derived, and the amplitudes of the density perturbations are shown to become comparable with the square of the amplitudes of the first-order waves. Moreover the second-order density perturbations corresponding to the first-order ones are derived and examined.