The present expansion stage of the universe is believed to be mainly governed by the cosmological constant, collisionless dark matter and baryonic matter. The latter two components are often modeled as zero-pressure fluids. In our previous work we have shown that to second order in the cosmological perturbations, the relativistic equations for the density and velocity perturbations of the zero-pressure, irrotational, multi-component fluids in a spatially near flat background without gravitational waves effectively coincide with the Newtonian equations. As the Newtonian equations only have quadratic order non-linearity, it is of practical interest to derive the third-order perturbation terms in a general relativistic treatment, which correspond to pure general relativistic corrections. In our previous work we have shown that even in a single-component fluid there exist a substantial number of pure relativistic third-order correction terms. We have, however, shown that those correction terms are independent of the horizon scale, and are quite small (similar to 5 x 10(-5) smaller compared with the relativistic/Newtonian second-order terms) near the horizon scale due to the weak level anisotropy of the cosmic microwave background radiation. Here, we present pure general relativistic correction terms appearing in the third-order perturbations of the multi-component zero-pressure fluids. The forms of the pure general relativistic correction terms are quite similar to the ones in a single-component situation. The third-order correction terms involve only the 'linear order spatial curvature perturbation in the comoving gauge' phi(v) which has the order of the 'perturbed Newtonian gravitational potential divided by c(2,), and thus is small on nearly all scales. Consequently, we show that, as in a single-component situation, the third-order correction terms are quite small (similar to 5 x 10(-5) smaller) near the horizon, and independent of the horizon scale. We emphasize that these results are based on our proper choice of perturbation variables and gauge conditions for describing the relativistic perturbations. Still, there do exist a substantial number of pure general relativistic correction terms in third-order perturbations which could potentially become important in future development of precision cosmology. Although phi(v) is small on nearly all scales, our third-order corrections are applicable only in weakly non-linear regimes where perturbation analysis is viable. We include the cosmological constant in all our analyses.
