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It is known that the Newtonian cosmology (both for the Friedmann background and its linear perturbations) is not purely based on Newton’s gravity, but a guided one by Einstein’s theory. The action‐at‐a‐distance nature of Newton’s gravity requires further verification from Einstein’s theory for its use in the large‐scale nonlinear simulation. We provide its domain of validity in cosmology by investigating the weakly and fully nonlinear regimes in relativistic cosmology based on Einstein’s gravity. In the weakly nonlinear approach the Einstein guided Newtonian cosmology is exactly valid to the second order in perturbation and pure relativistic correction terms start appearing from the third order, and are small in the large‐scale near horizon. We also provide fully nonlinear relativistic correction terms in the post‐Newtonian approach in cosmological situation. Perturbation approach is valid in weakly nonlinear regime but is applicable in strong gravity (including near and beyond horizon), whereas post‐Newtonian approach is valid in fully nonlinear regime but is applicable in the weak gravity (thus far inside horizon). These two approaches provide complementary relativistic correction terms, thus showing the limit and validity of the Newtonian approach in cosmology.

We present the power spectra of the scalar- and tensor-type structures generated in an inflation model based on the nonminimally coupled scalar field with a self coupling. By comparing the contributions of these structures to the anisotropy of the cosmic microwave background radiation with the four year COBE DMR data we derive strong constraints on model parameters and the inflation model.

The evolutions of linear structures in a spatially homogeneous and isotropic world model are characterized by some conserved quantities. The amplitude of gravitational wave is conserved in the super-horizon scale, the perturbed three-space curvature in the comoving gauge is conserved in the super-sound-horizon scale, and the angular momentum of rotational perturbation is generally conserved.