Cosmological Problems with Multiple Axion-like Fields
  Incorporating the QCD axion and simultaneously satisfying current constraints
on the dark matter density and isocurvature fluctuations requires non-minimal
fine-tuning of inflationary parameters or the axion misalignment angle (or
both) for Peccei-Quinn symmetry-breaking scales $f_a > 10^{12}$ GeV. To gauge
the degree of tuning in models with many axion-like fields at similar
symmetry-breaking scales and masses, as may occur in string theoretic models
that include a QCD axion, we introduce a figure of merit ${\cal F}$ that
measures the fractional volume of allowed parameter space: the product of the
slow roll parameter $\epsilon$ and each of the axion misalignment angles,
$\theta_0$. For a single axion, $\mathcal{F} \lesssim 10^{-11}$ is needed to
avoid conflict with observations. We show that the fine tuning of $\mathcal{F}$
becomes exponentially more extreme in the case of numerous axion-like fields.
Anthropic arguments are insufficient to explain the fine tuning because the
bulk of the anthropically allowed parameter space is observationally ruled out
by limits on the cosmic microwave background isocurvature modes. Therefore,
this tuning presents a challenge to the compatibility of string-theoretic
models with light axions and inflationary cosmology.
