  Recently, many works have tried to realize cosmological accelerated expansion
in string theory models in the asymptotic regions of field space, with a
typical scalar potential $V(\varphi)$ having an exponential fall-off
$e^{-\gamma\, \varphi}$. Those attempts have been plagued by the fact that $V$
is too steep, namely $\gamma \geq 2/\sqrt{d-2}$ in a $d$-dimensional spacetime.
We revisit the corresponding dynamical system for arbitrary $d$ and $\gamma$,
and show that for an open universe ($k=-1$), there exists a new stable fixed
point $P_1$ precisely if $\gamma > 2/\sqrt{d-2}$. Building on the recent work
arXiv:2210.10813, we show in addition that cosmological solutions asymptoting
to $P_1$ exhibit accelerated expansion in various fashions (semi-eternal,
eternal, transient with parametrically controlled number of e-folds, or
rollercoaster). We finally present realizations in string theory of these
cosmological models with asymptotically accelerating solutions, for $d=4$ or
$d=10$. We also show that these solutions do not admit a cosmological event
horizon, and discuss the possibility of this being a generic feature of quantum
gravity.
