  When the expansion of the universe is dominated by a perfect fluid with
equation of state parameter $w$ and a sound speed $c_s$ satisfying $w = c_s^2
\le 1$, the Hubble parameter $H$ and time $t$ satisfy the bound $Ht \ge 1/3$.
There has been recent interest in "ultra-slow" expansion laws with $Ht < 1/3$
(sometimes described as "fast expanding" models). We examine various models
that can produce ultra-slow expansion: scalar fields with negative potentials,
barotropic fluids, braneworld models, or a loitering phase in the early
universe. Scalar field models and barotropic models for ultra-slow expansion
are unstable to evolution toward $w = 1$ or $w \rightarrow \infty$ in the
former case and $w \rightarrow \infty$ in the latter case. Braneworld models
can yield ultra-slow expansion but require an expansion law beyond the standard
Friedman equation. Loitering early universe models can produce a quasi-static
expansion phase in the early universe but require an exotic negative-density
component. These results suggest that appeals to an ultra-slow expansion phase
in the early universe should be approached with some caution, although the
loitering early universe may be worthy of further investigation. These results
do not apply to ultra-slow contracting models.
