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

Understand

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 $a(t)$.