Back reaction of inhomogeneities on the expansion: The evolution of cosmological parameters

Averaging and evolving inhomogeneities are noncommuting operations. This implies the existence of deviations of an averaged model from the standard Friedmann-Lemaître cosmologies. We quantify these deviations, encoded in a back reaction parameter, in the framework of Newtonian cosmology. We employ the linear theory of gravitational instability in the Eulerian and Lagrangian approaches, as well as the spherically and plane-symmetric solutions as standards of reference. We propose a model for the evolution of the average characteristics of a spatial domain for generic initial conditions that contains the spherical top-hat model and the planar collapse model as exact subcases. A central result is that the back reaction term itself, calculated on sufficiently large domains, is small but, still, its presence can drive the cosmological parameters on the averaging domain far away from their global values of the standard model. We quantify the variations of these parameters in terms of the fluctuations in the initial data as derived from the power spectrum of initial cold dark matter density fluctuations. For example, in a domain with a radius of 100 Mpc today and initially one-σ fluctuations, the density parameters deviate from their homogeneous values by 15%; three-σ fluctuations lead to deviations larger than 100%.