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

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 $w \rightarrow -1$cancorrespondtoeither$\rho_{DE} \rightarrow \rho_0$or$\rho_{DE} \rightarrow 0$\mathrm{.}Thisprovidesanaturaldivisionofmodelswith$w \rightarrow -1$intotwodistinctclasses:asymptotic$\Lambda$($\rho_{DE} \rightarrow \rho_0$)andpseudo-$\Lambda$($\rho_{DE} \rightarrow 0$). 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 $w\to -1$ are always asymptotically $\mathrm{\Lambda }$; they can never produce pseudo-$\mathrm{\Lambda }$ behavior. Quintessence models can correspond to either asymptotic $\mathrm{\Lambda }$ or pseudo-$\mathrm{\Lambda }$ evolution, but the latter is impossible when the expansion is dominated by a background barotropic fluid. We show that the distinction between asymptotic $\mathrm{\Lambda }$ and pseudo-$\mathrm{\Lambda }$ models for $w>-1$ is mathematically dual to the distinction between pseudo-rip and big/little rip models when $w<-1$.