  String and M-theory compactifications generically have compact moduli which
can potentially act as the QCD axion. However, as demonstrated here, such a
compact modulus can not play the role of a QCD axion and solve the strong CP
problem if gravitational waves interpreted as arising from inflation with
Hubble constant $H_inf \gsim 10^{13}$ GeV are observed by the PLANCK
polarimetry experiment. In this case axion fluctuations generated during
inflation would leave a measurable isocurvature and/or non-Gaussian imprint in
the spectrum of primordial temperature fluctuations. This conclusion is
independent of any assumptions about the initial axion misalignment angle, how
much of the dark matter is relic axions, or possible entropy release by a late
decaying particle such as the saxion; it relies only on the mild assumption
that the Peccei-Quinn symmetry remains unbroken in the early universe.
