How to suppress exponential growth -- on the parametric resonance of
  photons in an axion background
  Axion--photon interactions can lead to an enhancement of the electromagnetic
field by parametric resonance in the presence of a cold axion background, for
modes with a frequency close to half the axion mass. In this paper, we study
the role of the axion momentum dispersion as well as the effects of a
background gravitational potential, which can detune the resonance due to
gravitational redshift. We show, by analytical as well as numerical
calculations, that the resonance leads to an exponential growth of the photon
field only if (a) the axion momentum spread is smaller than the inverse
resonance length, and (b) the gravitational detuning distance is longer than
the resonance length. For realistic parameter values, both effects strongly
suppress the resonance and prevent the exponential growth of the photon field.
In particular, the redshift due to the gravitational potential of our galaxy
prevents the resonance from developing for photons in the observable frequency
range, even assuming that all the dark matter consists of a perfectly cold
axion condensate. For axion clumps with masses below $\sim 10^{-13}\, M_\odot$,
the momentum spread condition is more restrictive, whereas, for more massive
clumps, the redshift condition dominates.
