Inhomogeneous dusty Universes and their deceleration

Exact results stemming directly from Einstein equations imply that inhomogeneous Universes endowed with vanishing pressure density can only decelerate, unless the energy density of the Universe becomes negative. Recent proposals seem to argue that inhomogeneous (but isotropic) space-times, filled only with incoherent matter, may turn into accelerated Universes for sufficiently late times. To scrutinize these scenarios, fully inhomogeneous Einstein equations are discussed in the synchronous system. In a dust-dominated Universe, the inhomogeneous generalization of the deceleration parameter is always positive semi-definite implying that no acceleration takes place. e-mail address: [email protected] The homogeneity and isotropy of the geometry are not essential ingredients to establish a number of relevant results in relativistic cosmology. For instance, from the early sixties to the early seventies (see [1, 2, 3] and references therein), a research program on the singularity properties of general cosmological solutions has been conducted without relying on the isotropy and on the homogeneity of the geometry. The theme of the present paper is somehow opposite to the one analysed in [1, 2, 3] where the emphasis was on the rôle of the inhomogeneities (and anisotropies) in the proximity of a cosmological singularity. In the present paper we would like to understand if an inhomogeneous space-time, filled with incoherent matter, can be turned into an accelerating Universe at later times. The inhomogeneities considered in the present investigation may arise during an early inflationary stage when quantum mechanical fluctuations of the geometry and of the inflaton field are inside the Hubble radius. Depending upon the parameters of the inflationary phase, the initial quantum fluctuations will be amplified leading to a quasi-flat spectrum of curvature perturbations that accounts, through the Sachs-Wolfe effect, for the tiny temperature ripples detected in the microwave sky by several experiments. While the temperature of the Cosmic Microwave Background (CMB) is in the range of few K degrees, the inhomogeneities are in the range of the μK. This hierarchy of scales also implies that curvature fluctuations, say, after equality and before decoupling, are rather well described in the framework of linear theory. In recent months, it has been claimed [4] that since spatial inhomogeneities may have a non-trivial time dependence on length-scales larger than the Hubble radius, then, depending on the specific properties of the inhomogeneity, the deceleration parameter may well be negative, implying an effective acceleration of the Universe. While this proposal has been already criticized on various grounds [5, 6, 7, 8, 9] we find appropriate to scrutinize such a statement in the light of a set of fully inhomogeneous Einstein equations. Consider, indeed, Einstein equations R μ = 8πG [