Scalar Corrections to the Cosmological Equations

A consistent approach to Cosmology requires an explicit averaging of the Einstein equations, to describe a homogeneous and isotropic geometry. Such an averaging will in general modify the Einstein equations. The averaging procedure due to Buchert has attracted considerable attention recently since it offers the tantalizing hope of explaining the phenomenon of dark energy through such corrections. This approach has been criticized, however, on the grounds that its effects may be gauge artifacts. We apply the fully covariant formalism of Zalaletdinov’s Macroscopic Gravity and show that, after making some essential gauge choices, the Cosmological equations receive spacetime scalar corrections which are therefore observable in principle, and further, that the broad structure of these corrections is identical to those derived by Buchert. e-mail address: [email protected] 1 Cosmology crucially assumes that the matter distribution in the Universe, when averaged on large enough scales, is homogeneous and isotropic. Assuming that the metric can also be “averaged” on these scales, one then models this averaged metric of spacetime as having the Friedmann-Lemâıtre-Robertson-Walker (FLRW) form with homogeneous and isotropic spatial sections, and applies Einstein’s General Relativity to determine the geometry of the Universe. The averaging operation is usually assumed implicitly and rather vaguely. It has long been known however [1], that any explicit averaging scheme for the metric g of spacetime and energy-momentum tensor T of matter, must necessarily yield corrections to the Einstein equations (which should ideally be imposed on length scales comparable to, say, the Solar System). This is a consequence of the Einstein tensor E[g] being a nonlinear functional of the metric g : given some averaging operator 〈·〉, in general one has 〈E[g]〉 6 = E[〈g〉], and by using E[〈g〉] = 〈T 〉 as the field equations, one is ignoring these corrections. Clearly, one needs a systematic averaging scheme within which to ask how large these corrections are. Among the averaging schemes available in the literature, only two – Buchert’s spatial averaging of scalars [2] and Zalaletdinov’s Macroscopic Gravity (MG) [3] – are capable of addressing the issue of averaging in General Relativity in a nonperturbative manner. (This is important since one expects the effects of averaging, if any, to show up only through nonlinear inhomogeneities.) Buchert’s approach has attracted considerable attention recently due partly to its tractability, but mainly because it offers the tantalizing hope of solving the dark energy problem completely within the framework of classical General Relativity. However, this approach has been criticized on the grounds that an averaging operation such as Buchert’s which is defined only for a particular 3+1 splitting of spacetime, is likely to lead to observationally irrelevant gauge artifacts which could wrongly be interpreted as solving the dark energy problem. While it is difficult to refute such criticism from within Buchert’s noncovariant ap2 proach, it might be possible to address the issue beginning with a covariant averaging scheme. Zalaletdinov’s MG is precisely such a scheme – fully covariant and mathematically elegant – which unfortunately comes with the somewhat steep price of extreme technical complexity. Nevertheless, the following question can be posed : Starting with the fully covariant structure of MG, and then making some appropriate gauge choices, is it possible to derive equations resembling Buchert’s modified FLRW equations? We answer this question in the affirmative [4], and address some of the issues that emerge in this construction. We begin by describing the broad structure of MG. Zalaletdinov’s approach considers a general differentiable manifoldM with metric gab, and defines a spacetime averaging operation for tensors which is then used to construct an “averaged” differentiable manifold M̄. The average of the affine connection on M is shown to itself behave like a connection, and is used as the affine connection for the abstract manifold M̄. The averaging operation is very sophisticated – it is defined using a Lie dragging of averaging regions along chosen vector fields, and ensures that the average of some tensor field pij(x), say, is itself a local tensor field 〈 pij 〉 (x) on the manifold M. One defines the (tensorial) connection correlation terms,