Age of the universe: Influence of the inhomogeneities on the global expansion factor

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 second-order Zel’dovich-approximation scheme for irrotational dust we use observational results in the form of the normalization constant fixed by the Cosmic Background Explorer results and we check different power spectra, namely, for adiabatic cold dark matter (CDM), isocurvature CDM, hot dark matter, warm dark matter, 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 constrained to comparing globally homogeneous and averaged inhomogeneous matter distributions. In addition we relate our work to previous treatments.