give rise to the so-called isocurvature perturbations where density fluctuations in radiation and matter add up to zero. The isocurvature perturbations, in turn, are severely constrained by Planck's CMB measurements: their power has to be less than 4% of the adiabatic perturbations produced by the inflaton field. Hence the severe constraint on the axion: its initial displacement angle has to be small enough as to suppress the oscillation amplitude. But the same displacement angle gives rise to the production of axion dark matter. Combining these two inputs, we learn from BICEP2 that high-scale axions alone cannot account for the observed abundance of dark matter in the universe. Moreover, even giving up on axion dark matter, one has to fine-tune the initial displacement of the axion fields to a tiny value (of order 10^-8 for f = 10^16 GeV).
Of course, like for any model that crashed after the BICEP2 announcement, Microsoft-style patches are already available. In this case, one solution is simply to take f < 10^14 GeV, that is the global symmetry whose breaking gives rise to the axion particle gets broken only after the inflation ends (although this scenario has problems of its own). Another possible fix is to ensure the axion acquires a large mass (>10^14 GeV) during inflation, or that the isocurvature modes were diluted by late-time entropy production. Nevertheless, high-scale axions are clearly less motivated than they were a week ago.
A commenter pointed out that the results in this plot disagree with other literature on the
subject. The point is that, for the axion scale larger than the Hubble scale during inflation, the minimum displacement angle is of order H/f. For the relevant scales this always produces too much isocurvature perturbations. Thus, the conclusions from BICEP2 are stronger than what I wrote above: high scale axions are excluded (up to the caveats in the previous paragraph) irrespectively of any assumptions about the initial displacement angle. The plot on the right visualizes the situation for the QCD axion. The yellow region is excluded by astrophysical and CMB constraints, while the green region corresponds to the BICEP2 measurement of the Hubble scale during inflation. The QCD axion is now constrained to a narrow window of 10^9 ≤ f ≤ 10^11 GeV. At the top of this window it accounts for all dark matter in the universe.