How sound are our ultra-light axion approximations?
  Ultra-light axions (ULAs) are a promising dark-matter candidate. ULAs may
have implications for small-scale challenges to the $\Lambda$CDM model, and
arise in string scenarios. ULAs are already constrained by cosmic microwave
background (CMB) experiments and large-scale structure surveys, and will be
probed with much greater sensitivity by future efforts. It is challenging to
compute observables in ULA scenarios with sufficient speed and accuracy for
cosmological data analysis because the ULA field oscillates rapidly. In past
work, an effective fluid approximation has been used to make these computations
feasible. Here this approximation is tested against an exact solution of the
ULA equations, comparing the induced error of CMB observables with the
sensitivity of current and future experiments. In the most constrained mass
range for a ULA dark matter component ($10^{-27}~{\rm eV}\leq m_{\rm ax}\leq
10^{-25}~{\rm eV}$), the induced bias on the allowed ULA fraction of dark
matter from Planck data is less than $1\sigma$. In the cosmic-variance limit
(including temperature and polarization data), the bias is $\lesssim 2\sigma$
for primary CMB anisotropies, with more severe biases (as high as $\sim
4\sigma$) resulting for less reliable versions of the effective fluid
approximation. If all of the standard cosmological parameters are fixed by
other measurements, the expected bias rises to $4-20\sigma$ (well beyond the
validity of the Fisher approximation), though the required level of degeneracy
breaking will not be achieved by any planned surveys.
