The past attractor in inhomogeneous cosmology

TL;DR

This paper introduces a new dynamical systems framework for analyzing the behavior of inhomogeneous cosmological models near the initial singularity, extending previous homogeneous models and aiming to rigorously characterize their asymptotic dynamics.

Contribution

It develops a general, scale-invariant formulation of Einstein's equations for inhomogeneous cosmologies, connecting to homogeneous cases and proposing a conjectured attractor for the past evolution.

Findings

Constructed an invariant set as a candidate past attractor.

Unified inhomogeneous and homogeneous cosmological dynamics.

Provides a foundation for future rigorous theorems on asymptotic behavior.

Abstract

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 Belinski\v{\i}, 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.