Axion Dark Matter Archaeology with Primordial Gravitational Waves

We investigate the complementary information to be gained from inflationary gravitational wave (IGW) signals and searches for QCD axion dark matter. We focus on post-inflationary Peccei-Quinn (PQ) breaking axion models that are cosmologically safe. Recent work has shown that a greater number of such models exist. This is because the heavy quarks required for the colour anomaly can provoke a period of heavy quark domination (HQD), which, through decay, dilutes the axion abundance. In this work we show for the first time that the axion dark matter mass can be as low as $m_a\sim10^{-8},{\rm eV}$ for models where the heavy quarks decay via dimension 6 terms. This is achieved by allowing the mass of the heavy quarks to differ from the axion decay constant, $m_Q\neq f_a$. Consequently, the observables that would distinguish between pre- and post-inflationary PQ breaking, $m_a$ and the additional relativistic degrees of freedom $ΔN_{\rm eff}$, now become indiscernible. To solve this, we propose using blue-tilted IGWs to probe HQD. In scenarios where such a blue tilt is present, the enhanced GW signal allows future interferometers to place non-trivial constraints on the parameters $m_Q$ and $f_a$, thereby complementing haloscope searches. While some degeneracies with other parameters such as $m_Q$ remain, detectors such as BBO and ET will be able to optimistically probe $f_a\gtrsim 10^{14},{\rm GeV}$.

💡 Research Summary

The paper investigates how inflationary gravitational‑wave (IGW) observations can complement searches for QCD axion dark matter, focusing on post‑inflationary Peccei‑Quinn (PQ) breaking scenarios that are cosmologically viable. In the standard KSVZ framework the heavy quark (Q) that supplies the colour anomaly is usually taken to have a mass equal to the axion decay constant, $m_Q\simeq f_a$, and to decay through low‑dimensional operators (dimension ≤5). Recent work has shown that if the heavy quark decays via higher‑dimensional operators (dimension 6 or 7) its lifetime can be long enough to dominate the energy density of the early Universe after freeze‑out, creating a period of heavy‑quark domination (HQD), analogous to an early matter‑domination (EMD) era.

During HQD the expansion rate scales as $a\propto t^{2/3}$, which delays the onset of axion field oscillations. The axion relic density is therefore diluted, and to obtain the observed dark‑matter abundance the required PQ breaking scale $f_a$ must be raised dramatically, up to $10^{14!-!15}$ GeV. Consequently the axion mass can be as low as $m_a\sim10^{-8}$ eV, far below the conventional window of $10$–$100,\mu$eV. Because the heavy quark is assumed to decay exclusively into Standard‑Model particles, the thermal axion contribution to the effective number of neutrino species, $\Delta N_{\rm eff}$, becomes negligible; the usual discriminants between pre‑ and post‑inflationary PQ breaking (the axion mass and $\Delta N_{\rm eff}$) therefore lose their power.

The authors propose to break this degeneracy by exploiting the imprint of HQD on the primordial tensor spectrum. If the inflationary tensor power has a blue tilt ($n_T>0$), the GW energy density scales as $Ω_{\rm GW}(f)\propto f^{n_T}$ during pure radiation domination but is suppressed as $f^{n_T-2}$ during any early matter‑dominated phase. When the heavy quarks finally decay, the spectrum exhibits a characteristic step‑like rise at a frequency corresponding to the decay temperature, $f_{\rm dec}\sim T_{\rm dec}$. For typical parameters ($m_Q\sim10^{11!-!12}$ GeV, $\Lambda\sim M_{\rm Pl}$) this feature lies in the 0.1–10 Hz band, precisely where next‑generation laser interferometers such as BBO, the Einstein Telescope (ET), and DECIGO will have peak sensitivity.

Using analytic estimates and numerical solutions of the coupled Boltzmann equations for the radiation, heavy‑quark, and axion fluids, the paper maps the viable $(m_Q,\Lambda,f_a)$ region that satisfies Big‑Bang Nucleosynthesis (BBN) bounds ($T_{\rm dec}>3$ MeV) and yields the correct dark‑matter density. The authors then compute the resulting GW spectra and compare them to projected detector sensitivities. They find that for $f_a\gtrsim10^{14}$ GeV the step in the GW spectrum should be detectable at the $5\sigma$ level by BBO and ET, allowing an indirect measurement of the heavy‑quark mass and decay scale.

Finally, the study emphasizes the complementarity of GW observations and traditional axion haloscopes (ADMX, HAYSTAC, DMRadio, etc.). While haloscopes can probe the axion‑photon coupling $g_{a\gamma}$ and thus constrain $f_a$, they cannot resolve whether the underlying cosmology involved HQD. Conversely, a detection of a blue‑tilted GW step would pinpoint the existence of an early matter‑dominated epoch, breaking the $m_a$–$\Delta N_{\rm eff}$ degeneracy and enabling a combined reconstruction of the full axion‑PQ sector. The work therefore opens a new avenue for probing ultra‑high PQ scales through the synergy of cosmological gravitational‑wave astronomy and laboratory axion searches.