Isocurvature forecast in the anthropic axion window

We explore the cosmological sensitivity to the amplitude of isocurvature fluctuations that would be caused by axions in the ``anthropic window'' where the axion decay constant f a >> 10 12 GeV and the initial misalignment angle Θ i << 1. In a minimal ΛCDM cosmology extended with subdominant scale-invariant isocurvature fluctuations, existing data constrain the isocurvature fraction to α < 0.09 at 95% C.L. If no signal shows up, Planck can improve this constraint to 0.042 while an ultimate CMB probe limited only by cosmic variance in both temperature and E -polarisation can reach 0.017, about a factor of five better than the current limit. In the parameter space of f a and H I (Hubble parameter during inflation) we identify a small region where axion detection remains within the reach of realistic cosmological probes.