Lagrangian theory of structure formation in relativistic cosmology. II. Average properties of a generic evolution model

Kinematical and dynamical properties of a generic inhomogeneous cosmological model, spatially averaged with respect to free-falling (generalized fundamental) observers, are investigated for the matter model "irrotational dust.'' Paraphrasing a previous Newtonian investigation, we present a relativistic generalization of a backreaction model based on volume-averaging the "relativistic Zel'dovich approximation.'' In this model we investigate the effect of "kinematical backreaction'' on the evolution of cosmological parameters as they are defined in an averaged inhomogeneous cosmology, and we show that the backreaction model interpolates between orthogonal symmetry properties by covering subcases of the plane-symmetric solution, the Lemaitre-Tolman-Bondi solution and the Szekeres solution. We so obtain a powerful model that lays the foundations for quantitatively addressing curvature inhomogeneities as they would be interpreted as "dark energy'' or "dark matter'' in a quasi-Newtonian cosmology. The present model, having a limited architecture due to an assumed Friedmann-Lemaitre-Robertson-Walker background, is nevertheless capable of replacing 1/4 of the needed amount for dark energy on domains of 200 Mpc in diameter for typical (one-sigma) fluctuations in a cold dark matter initial power spectrum. However, the model is far from explaining dark energy on larger scales (spatially), where a 6% effect on 400 Mpc domains is identified that can be traced back to an on average negative intrinsic curvature today. One drawback of the quantitative results presented is the fact that the epoch when backreaction is effective on large scales and leads to volume acceleration lies in the future. We discuss this issue in relation to the initial spectrum, the dark matter problem, the coincidence problem, and the fact that large-scale dark energy is an effect on the past light cone (not spatial), and we pinpoint key elements of future research.