  Asymptotic (late-time) cosmology depends on the asymptotic
(infinite-distance) limits of scalar field space in string theory. Such limits
feature an exponentially decaying potential $V \sim \exp(- c \phi)$ with
corresponding Hubble scale $H \sim \sqrt{\dot \phi^2 + 2 V} \sim \exp(-
\lambda_H \phi)$, and at least one tower of particles whose masses scale as $m
\sim \exp( - \lambda \phi)$, as required by the Distance Conjecture. In this
paper, we provide evidence that these coefficients satisfy the inequalities
$\sqrt{(d-1)/(d-2)} \geq \lambda_H \geq \lambda_{\text{lightest}} \geq
1/\sqrt{d-2}$ in $d$ spacetime dimensions, where $\lambda_{\text{lightest}}$ is
the $\lambda$ coefficient of the lightest tower. This means that at late times,
as the scalar field rolls to $\phi \rightarrow \infty$, the low-energy theory
remains a $d$-dimensional FRW cosmology with decelerated expansion, the light
towers of particles predicted by the Distance Conjecture remain at or above the
Hubble scale, and both the strong energy condition and the dominant energy
condition are satisfied.
