For the first time we calculate quantitatively the influence of inhomogeneities on the global expansion factor by averaging the Friedmann equation. In the framework of the relativistic secondorder Zel'dovich-approximation scheme for irrotational dust we use observational results in form of the normalisation constant fixed by the COBE results and we check different power spectra, namely for adiabatic CDM, isocurvature CDM, HDM, WDM, Strings and Textures. We find that the influence of the inhomogeneities on the global expansion factor is very small. So the error in determining the age of the universe using the Hubble constant in the usual way is negligible. This does not imply that the effect is negligible for local astronomical measurements of the Hubble constant. Locally the determination of the redshift-distance relation can be strongly influenced by the peculiar velocity fields due to inhomogeneities. Our calculation does not consider such effects, but is contrained to comparing globally homogeneous and averaged inhomogeneous matter distributions. In addition we relate our work to previous treatments.

The gravitational instability of inhomogeneities in the expanding universe is studied by the relativistic second-order approximation. Using the tetrad formalism we consider irrotational dust universes and get equations very similar to those given in the Lagrangian perturbation theory in Newtonian cosmology.Neglecting the cosmological constant and assuming a flat background model we give the solutions of the nonlinear dynamics of cosmological perturbations.We present the complete second-order solutions, which extend and improve earlier works.

We derive a formula relating the large-scale temperature anisotropy of the cosmic microwave background radiation with the cosmological perturbations responsible for them using the local covariant and gauge-invariant formalism developed by Ellis and Bruni. Comparisons of our covariant expression with previously derived Sachs-Wolfe formulas are given. Expanding our covariant variables in terms of Bardeen quantities, we derive a generalization of a result due to Panek. PACS number(s): 98.70.Vc, 98.80.H~