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We present a general framework for analyzing spatially inhomogeneous cosmological dynamics. It employs Hubble-normalized scale-invariant variables which are defined within the orthonormal frame formalism, and leads to the formulation of Einstein's field equations with a perfect fluid matter source as an autonomous system of evolution equations and constraints. This framework incorporates spatially homogeneous dynamics in a natural way as a special case, thereby placing earlier work on spatially homogeneous cosmology in a broader context, and allows us to draw on experience gained in that field using dynamical systems methods. One of our goals is to provide a precise formulation of the approach to the spacelike initial singularity in cosmological models, described heuristically by Belinskiǐ, Khalatnikov and Lifshitz. Specifically, we construct an invariant set which we conjecture forms the local past attractor for the evolution equations. We anticipate that this new formulation will provide the basis for proving rigorous theorems concerning the asymptotic behavior of spatially inhomogeneous cosmological models.

The quantum field theoretic prediction for the vacuum energy density leads to a value for the effective cosmological constant that is incorrect by between 60 to 120 orders of magnitude. We review an old proposal of replacing Einstein's Field Equations by their trace-free part (the Trace-Free Einstein Equations), together with an independent assumption of energy-momentum conservation by matter fields. While this does not solve the fundamental issue of why the cosmological constant has the value that is observed cosmologically, it is indeed a viable theory that resolves the problem of the discrepancy between the vacuum energy density and the observed value of the cosmological constant. However, one has to check that, as well as preserving the standard cosmological equations, this does not destroy other predictions, such as the junction conditions that underlie the use of standard stellar models. We confirm that no problems arise here: hence, the Trace-Free Einstein Equations are indeed viable for cosmological and astrophysical applications. *

In this letter we investigate the nature of generic cosmological singularities using the framework developed by Uggla et al. We do so by studying the past asymptotic dynamics of general vacuum G2 cosmologies, models that are expected to capture the singular behavior of generic cosmologies with no symmetries at all. In particular, our results indicate that asymptotic silence holds, i.e., that particle horizons along all timelines shrink to zero for generic solutions. Moreover, we provide evidence that spatial derivatives become dynamically insignificant along generic timelines, and that the evolution into the past along such timelines is governed by an asymptotic dynamical system which is associated with an invariant set -the silent boundary. We also identify an attracting subset on the silent boundary that organizes the oscillatory dynamics of generic timelines in the singular regime. In addition, we discuss the dynamics associated with recurring spike formation.PACS numbers: 98.80.Jk, 04.20.Dw, 04.25.Dm, 04.20.Ha The singularity theorems of Penrose and Hawking [1] state that generic cosmological models contain an initial singularity, but do not give any information on the nature of this singularity. Heuristic investigations of this issue led Belinskiǐ, Khalatnikov and Lifshitz [2] (BKL) to propose that a generic cosmological initial singularity is spacelike, local and oscillatory. Uggla et al.[3] (UEWE) reformulated Einstein's field equations (EFEs) by introducing scale-invariant variables which have the property that all individual terms in EFEs become asymptotically bounded, for generic solutions. This made it possible to characterize a generic cosmological initial singularity in terms of specific limits. The numerical study of the picture proposed by UEWE was initiated in Ref.[4], specializing to Gowdy vacuum spacetimes which have a nonoscillatory singularity. This letter presents the results of the first detailed study of the oscillatory asymptotic dynamics of inhomogeneous cosmologies from the dynamical systems point of view introduced in UEWE.Here we focus on vacuum cosmologies with an Abelian symmetry group G 2 with two commuting spacelike Killing vector fields, and the spatial topology of a 3-torus. This is arguably the simplest class of inhomogeneous models that is expected to capture the properties of a generic oscillatory singularity. Numerical investigations of G 2 spacetimes supporting the BKL proposal were carried out by Weaver et al. in Ref. [5,6].UEWE used an orthonormal frame formalism and factored out the expansion of a timelike reference congruence e 0 by normalizing the dynamical variables with the isotropic Hubble expansion rate H of e 0 . This yielded a dimensionless state vector X = (E α i )⊕S, where E α i are the Hubble-normalized components of the spatial frame vectors orthogonal to e 0 ; e α = e α i ∂ i , E α i = e α i /H.The approach to an initial singularity will be said to be asymptotically silent for timelines along which E α i → 0, and asymptotically silent and local for timel...