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

Abstract

If the dark energy density asymptotically approaches a nonzero constant, ${\rho }_{DE}\to {\rho }_0$, then its equation of state parameter $w$ necessarily approaches $-1$. The converse is not true; dark energy with $w\to -1$ can correspond to either ${\rho }_{DE}\to {\rho }_0$ or ${\rho }_{DE}\to 0$. This provides a natural division of models with $w\to -1$ into two distinct classes: asymptotic $\Lambda$ (${\rho }_{DE}\to {\rho }_0$) and pseudo-$\Lambda$ (${\rho }_{DE}\to 0$). 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 $w\to -1$ 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 $w<-1$.

7 pages, no figures, references added

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