The axion-like particle (ALP), a pseudo Nambu-Goldstone boson that couples to two photons, has been studied extensively in recent years as a dark matter candidate. For initial field configurations in a minimal ALP model explaining the observed dark matter abundance, we need the potential height to exceed the ALP energy density at redshift z 5. 5 10^6 leading to: f_ϕ410^13\, GeV\, (10^-18\, eVm_ϕ), where m_ϕ and f_ϕ denote the ALP mass and decay constant, respectively. This bound is known for the ALP dark matter dominated by the homogeneous zero-momentum mode, under the requirement that coherent oscillations begin early enough to satisfy the late-forming dark matter constraint. One loop hole to evade this limit may be to introduce a large amount of the non-relativistic modes of the ALP with non-vanishing momenta. Here we show that the same limit remains valid even if nonzero-momentum modes dominate. Interestingly, when nonrelativistic gradient modes prevail, the ALP behaves relativistic radiation rather than matter, if it violates the limit. Moreover, if the typical momentum is sufficiently small, domain walls form.
Narita et al. (Thu,) studied this question.