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.