Dark energy with $w \rightarrow -1$: Asymptotic $\Lambda$ versus pseudo-$\Lambda$

TL;DR

This paper classifies dark energy models with $w ightarrow -1$ into asymptotic $ ho_{DE} ightarrow ho_0$ and pseudo-$ ho_{DE} ightarrow 0$ types, analyzing their properties and realizations in barotropic and quintessence frameworks.

Contribution

It provides a clear division and boundary criteria between asymptotic $ ho_{DE}$ and pseudo-$ ho_{DE}$ models, including their realizations and mathematical dualities.

Findings

Barotropic models with positive sound speed and $w ightarrow -1$ are always asymptotic $ ho_{DE} ightarrow ho_0$.

Quintessence models can be either class, but pseudo-$ ho_{DE}$ is impossible with background fluid domination.

The distinction between the two classes is dual to rip models for $w < -1$.

Abstract

If the dark energy density asymptotically approaches a nonzero constant, $\rho_{DE} \rightarrow \rho_0$, then its equation of state parameter $w$ necessarily approaches $-1$. The converse is not true; dark energy with $w \rightarrow -1$ can correspond to either $\rho_{DE} \rightarrow \rho_0$ or $\rho_{DE} \rightarrow 0$. This provides a natural division of models with $w \rightarrow -1$ into two distinct classes: asymptotic $\Lambda$ ($\rho_{DE} \rightarrow \rho_0$) and pseudo-$\Lambda$ ($\rho_{DE} \rightarrow 0$). We delineate the boundary between these two classes of models in terms of the behavior of $w(a)$, $\rho_{DE}(a)$, and $a(t)$. We examine barotropic and quintessence realizations of both types of models. Barotropic models with positive squared sound speed and $w \rightarrow -1$ are always asymptotically $\Lambda$; they can never produce pseudo-$\Lambda$ behavior. Quintessence models can correspond to either asymptotic $\Lambda$ or pseudo-$\Lambda$ evolution, but the latter is impossible when the expansion is dominated by a background barotropic fluid. We show that the distinction between asymptotic $\Lambda$ and pseudo-$\Lambda$ models for $w> -1$ is mathematically dual to the distinction between pseudo-rip and big/little rip models when $w < -1$.