Axions, Inflation and the Anthropic Principle

The QCD axion is the leading solution to the strong-CP problem, a dark matter candidate, and a possible result of string theory compactifications. However, for axions produced before inflation, symmetry-breaking scales of $f_a \gtrsim 10^{12}$ GeV (which are favored in string-theoretic axion models) are ruled out by cosmological constraints unless both the axion misalignment angle $\theta_0$ and the inflationary Hubble scale $H_I$ are extremely fine-tuned. We show that attempting to accommodate a high-$f_a$ axion in inflationary cosmology leads to a fine-tuning problem that is worse than the strong-CP problem the axion was originally invented to solve. We also show that this problem remains unresolved by anthropic selection arguments commonly applied to the high-$f_a$ axion scenario.