Probing a QCD String Axion with Precision Cosmological Measurements

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.