Hidden past of dark energy cosmological models

In this paper we analyse the possibility of having homogeneous isotropic cosmological models with observers reaching $t=\mathrm{\infty }$ in finite proper time. It is shown that just observationally-suggested dark energy models with $w\in (-5\mathrm{/}3,-1)$ show this feature and that they are endowed with an exotic curvature singularity. Furthermore, it is shown that non-accelerated observers in these models may experience a duration of the universe as short as desired by increasing their linear momentum. A subdivision of phantom models in two families according to this behavior is suggested.