Dynamical system approach to running $\Lambda$ cosmological models

TL;DR

This paper applies dynamical system methods to analyze various time-dependent cosmological constant models, revealing their stability, asymptotic behavior, and physical viability, including the emergence of non-covariant parametrizations from covariant ones.

Contribution

It introduces a comprehensive dynamical systems framework for analyzing different classes of running Lambda cosmologies, highlighting the physical issues with non-covariant parametrizations.

Findings

Dynamical systems methods can explore all solutions and their stability.

Non-covariant parametrizations can lead to nonphysical behaviors.

Emergent Lambda models from covariant parametrizations are physically consistent.

Abstract

We discussed the dynamics of cosmological models in which the cosmological constant term is a time dependent function through the scale factor $a(t)$, Hubble function $H(t)$, Ricci scalar $R(t)$ and scalar field $\phi(t)$. We considered five classes of models; two non-covariant parametrization of $\Lambda$: 1) $\Lambda(H)$CDM cosmologies where $H(t)$ is the Hubble parameter, 2) $\Lambda(a)$CDM cosmologies where $a(t)$ is the scale factor, and three covariant parametrization of $\Lambda$: 3) $\Lambda(R)$CDM cosmologies, where $R(t)$ is the Ricci scalar, 4) $\Lambda(\phi)$-cosmologies with diffusion, 5) $\Lambda(X)$-cosmologies, where $X=\frac{1}{2}g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}\phi$ is a kinetic part of density of the scalar field. We also considered the case of an emergent $\Lambda(a)$ relation obtained from the behavior of trajectories in a neighborhood of an invariant submanifold. In study of dynamics we use dynamical system methods for investigating how a evolutional scenario can depend on the choice of special initial conditions. We showed that methods of dynamical systems offer the possibility of investigation all admissible solutions of a running $\Lambda$ cosmology for all initial conditions, their stability, asymptotic states as well as a nature of the evolution in the early universe (singularity or bounce) and a long term behavior at the large times. We also formulated an idea of the emergent cosmological term derived directly from an approximation of exact dynamics. We show that some non-covariant parametrizations of Lambda term like $\Lambda(a)$, $\Lambda(H)$ give rise to pathological and nonphysical behaviour of trajectories in the phase space. This behaviour disappears if the term $\Lambda(a)$ is emergent from the covariant parametrization.