General K=-1 Friedman-Lema\^itre models and the averaging problem in cosmology

TL;DR

This paper introduces generalized K=-1 Friedmann-Lemaître cosmological models, analyzes their evolution, and discusses the averaging problem, gravitational energy effects, and stability properties in these models.

Contribution

It defines and studies the properties of general K=-1 Friedmann-Lemaître cosmologies, including averaging, gravitational energy roles, and stability analysis, advancing understanding of open universe models.

Findings

Long-time stability in H^{3} x H^{2} implies smoothing in higher Sobolev spaces.

No mathematical restriction on initial gravitational energy at the big bang.

Detailed estimations of gravitational energy evolution and its impact on cosmic deceleration.

Abstract

We introduce the notion of general K=-1 Friedman-Lema\^itre (compact) cosmologies and the notion of averaged evolution by means of an averaging map. We then analyze the Friedman-Lema\^itre equations and the role of gravitational energy on the universe evolution. We distinguish two asymptotic behaviors: radiative and mass gap. We discuss the averaging problem in cosmology for them through precise definitions. We then describe in quantitative detail the radiative case, stressing on precise estimations on the evolution of the gravitational energy and its effect in the universe's deceleration. Also in the radiative case we present a smoothing property which tells that the long time H^{3} x H^{2} stability of the flat K=-1 FL models implies H^{i+1} x H^{i} stability independently of how big the initial state was in H^{i+1} x H^{i}, i.e. there is long time smoothing of the space-time. Finally we discuss the existence of initial "big-bang" states of large gravitational energy, showing that there is no mathematical restriction to assume it to be low at the beginning of time.