  We examine the cosmological evolution of ultralight axionlike (ULA) scalar
fields with potentials of the form $V(\phi) = m^2 f^2[1 - \cos(\phi/f)]^n$,
with particular emphasis on the deviation in their behavior from the
corresponding small$-\phi$ power-law approximations to these potentials:
$V(\phi) \propto \phi^{2n}$. We show that in the slow-roll regime, when $\dot
\phi^2/2 \ll V(\phi)$, the full ULA potentials yield a more interesting range
of possibilities for quintessence than do the corresponding power law
approximations. For rapidly oscillating scalar fields, we derive the equation
of state parameter and oscillation frequency for the ULA potentials and show
how they deviate from the corresponding power-law values. We derive an analytic
expression for the equation of state parameter that better approximates the ULA
value than does the pure power-law approximation.
