Cosmological evolution of ultralight axionlike scalar fields

We examine the cosmological evolution of ultralight axionlike (ULA) scalar fields with potentials of the form $V(\ensuremath{\phi})={m}^{2}{f}^{2}[1\ensuremath{-}\mathrm{cos}(\ensuremath{\phi}/f){]}^{n}$, with particular emphasis on the deviation in their behavior from the corresponding small--$\ensuremath{\phi}$ power-law approximations to these potentials: $V(\ensuremath{\phi})\ensuremath{\propto}{\ensuremath{\phi}}^{2n}$. We show that in the slow-roll regime, when ${\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\phi}}}^{2}/2\ensuremath{\ll}V(\ensuremath{\phi})$, 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.