Scalar Field Cosmology -- Geometry of Dynamics

TL;DR

This paper explores the geometric phase space approach to Scalar Field Cosmology, analyzing model stability, symmetries, and potential functions to identify generic, structurally stable cosmological models.

Contribution

It introduces a geometric framework for analyzing SFC, characterizes stable models, and links symmetries to potential functions, advancing understanding of model genericity and stability.

Findings

Scaling solutions are represented by unstable separatrices of saddle points.

A class of potentials consistent with scaling symmetries was identified.

A dense subset of structurally stable models with homology symmetry was characterized.

Abstract

We study the Scalar Field Cosmology (SFC) using the geometric language of the phase space. We define and study an ensemble of dynamical systems as a Banach space with a Sobolev metric. The metric in the ensemble is used to measure a distance between different models. We point out the advantages of visualisation of dynamics in the phase space. It is investigated the genericity of some class of models in the context of fine tuning of the form of the potential function in the ensemble of SFC. We also study the symmetries of dynamical systems of SFC by searching for their exact solutions. In this context we stressed the importance of scaling solutions. It is demonstrated that scaling solutions in the phase space are represented by unstable separatrices of the saddle points. Only critical point itself located on two dimensional stable submanifold can be identified as scaling solution. We have also found a class of potentials of the scalar fields forced by the symmetry of differential equation describing the evolution of the universe. A class of potentials forced by scaling (homology) symmetries was given. We point out the role of the notion of a structural stability in the context of the problem of indetermination of the potential form of the Scalar Field Cosmology. We characterise also the class of potentials which reproduces the \Lambda CDM model, which is known to be structurally stable. We show that the structural stability issue can be effectively used is selection of the scalar field potential function. This enables us to characterise a structurally stable and therefore a generic class of SFC models. We have found a nonempty and dense subset of structurally stable models. We show that these models possess symmetry of homology.