Towards classification of simple dark energy cosmological models

TL;DR

This paper characterizes simple dark energy cosmological models using a potential function framework, introduces a $C^{1}$ metric to measure distances between models, and analyzes their proximity to the $ ext{Lambda}$CDM model.

Contribution

It introduces a novel $C^{1}$ metric on the space of dark energy models and applies it to compare various models with the standard $ ext{Lambda}$CDM.

Findings

The $C^{1}$ metric quantifies distances between dark energy models.

Most models considered are close to the $ ext{Lambda}$CDM model.

The framework encompasses models with different dark energy and modified gravity theories.

Abstract

We characterize a class of simple FRW models filled by both dark energy and dark matter in notion of a single potential function of the scale factor $a(t)$; $t$ is the cosmological time. It is representing potential of fictitious particle - Universe moving in 1-dimensional well $V(a)$ which the positional variable mimics the evolution of the Universe. Then the class of all dark energy models (called a multiverse) can be regarded as a Banach space naturally equipment in the structure of the Sobolev metric. In this paper we explore notion of $C^{1}$ metric introduced in the multiverse which measure distance between any two dark energy models. If we choose cold dark matter as a reference one then we can find how so far apart are different models offering explanation of present accelerating expansion phase of the Universe. We consider both models with dark energy (models with the generalized Chaplygin gas, models with variable coefficient equation of state $w_{X}=\frac{p_{X}}{\rho_{X}}$ parameterized by redshift $z$, models with phantom matter) as well as models basing on some modification of the Friedmann equation (Cardassian models, Dvali-Gabadadze-Porati brane models). We argue that because observational data still favor the $\Lambda$CDM model all reasonable dark energy models should belong to the nearby neighborhood of this model.