Relativistic Axion with Nonrelativistic Momenta: A Robust Bound on Minimal ALP Dark Matter

TL;DR

The paper demonstrates that the robust lower bound on the ALP decay constant, $f_ \gtrsim 4\times 10^{13}\, {\rm GeV}\left(\frac{10^{-18}\, {\rm eV}}{m_}\right)$, derived for dark matter dominated by a homogeneous mode, persists even when nonrelativistic, nonzero-momentum ALP modes dominate the energy density. Through lattice simulations of a cosine potential, it shows that configurations with gradient/kinetic energy can temporarily exhibit radiation-like behavior, while small typical momenta can give rise to Baumkuchen-like domain walls that eventually collapse, allowing the system to transition toward a matter-like regime and preserving the bound. The results imply that large low-momentum fluctuations do not threaten the minimal ALP dark matter scenario and highlight rich nonlinear dynamics, including domain-wall formation and potential gravitational-wave signals. The findings constrain the ALP-photon coupling in the minimal model and inform relic abundance calculations, with potential observational consequences depending on the domain-wall dynamics.

Abstract

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 $z\approx 5.5\times 10^{6}$ leading to: $$ f_φ\gtrsim4\times10^{13}\,{GeV}\,\biggl(\frac{10^{-18}\,eV}{m_φ}\biggr), $$ 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, Baumkuchen-like domain walls form, which play an important role in understanding the transition.

Figures (9)

• Figure 1: Time evolution of the energy density normalized by $a^3$. The red dot-dashed, blue dashed, green dotted, and black solid lines represent the kinetic, gradient, potential, and total energy densities, respectively. In all panels we set $H_i = 0.5\,m_\star$, $q = 2$, $K_{\rm IR} = H_i$, and a comoving box size $L \simeq 63/m_\star$ with $a_i = 1$. The top-left, top-middle, top-right, bottom-middle, and bottom-right panels correspond to $(\tilde{\phi}, K_{\rm UV}) = (f_\phi, m_\phi)$, $(10 f_\phi, m_\phi)$, $(40 f_\phi, m_\phi)$, $(40 f_\phi, 0.5 m_\phi)$, and $(40 f_\phi, \approx 0.84 m_\phi)$, respectively. In all panels we take $m_\phi^2 = 5 m_\star^2$, except for the bottom-right one, where $m_\phi^2 = 10 m_\star^2$. The bold black segments are added for visual reference to check the scaling behavior. The purple star in the bottom-middle panel indicates the point corresponding to the plots shown in Figs. \ref{['fig:2']} and \ref{['fig:5']}.
• Figure 2: Distribution of $\phi/f_\phi$ for the bottom-middle panel of Fig. \ref{['fig:1']}. The lower panel focuses on the short time interval until $a= 3$ (c.f. the transition at $a\sim 2.5$).
• Figure 3: Snapshots of $\phi/f_\phi$ on an $x$-$y$ slice (we choose the 513th slice along the $z$ direction) at $a = 1, 3, 5, 7, 9,$ and $11$. The black contours in the top panels mark the locations of the domain walls.
• Figure 4: The generic limits derived from the minimal ALP dark matter model. The solid, dashed, and dotted lines correspond to the case for $c_\gamma =3, \, 1, \, 1/3$, respectively, from top to bottom. The green region indicates astrophysical constraints which are taken from the data compiled in Ref AxionLimits.
• Figure 5: Same as Fig. \ref{['fig:5']}, but for the bottom-right panel of Fig. \ref{['fig:1']}.