Light axion-like dark matter must be present during inflation
  Axion-like particles (ALPs) might constitute the totality of the cold dark
matter (CDM) observed. The parameter space of ALPs depends on the mass of the
particle $m$ and on the energy scale of inflation $H_I$ , the latter being
bound by the non-detection of primordial gravitational waves. We show that the
bound on HI implies the existence of a mass scale $m_\chi = 10 {\rm \,neV}
{\div} 0.5 {\rm \,peV}$, depending on the ALP susceptibility $\chi$, such that
the energy density of ALPs of mass smaller than $m_\chi$ is too low to explain
the present CDM budget, if the ALP field has originated after the end of
inflation. This bound affects Ultra-Light Axions (ULAs), which have recently
regained popularity as CDM candidates. Light ($m < m_\chi$) ALPs can then be
CDM candidates only if the ALP field has already originated during the
inflationary period, in which case the parameter space is constrained by the
non-detection of axion isocurvature fluctuations. We comment on the effects on
these bounds from additional physics beyond the Standard Model, besides ALPs.
