Axion Cosmology Revisited

TL;DR

This work revisits axion cosmology using a temperature-dependent axion mass calculated from the interacting instanton liquid model, bridging high-temperature instanton physics to low-temperature chiral dynamics. By solving the misalignment dynamics exactly and incorporating precise effective degrees of freedom, it finds that analytic estimates typically underestimate the relic density by a factor of 2–3, and it places updated bounds on the decay constant $f_a$ and mass $m_a$ from the misalignment mechanism. The analysis shows that axion-string radiation imposes a stricter bound than misalignment in the classic window, pushing $f_a$ to below about $3.2\times10^{10}$ GeV (corresponding to $m_a \gtrsim 0.20$ meV). In the anthropic/anthropic-isocurvature sector, anharmonic effects near $\theta_a \to \pi$ open a viable inflationary dark matter window with axion masses up to $\sim$ 1 meV and inflation scales up to $H_I \sim 10^9$ GeV, while quantum fluctuations constrain heavier dominant DM axions. Overall, the results reinforce the robustness of the axion as a dark matter candidate and provide refined guidance for experimental searches in the thermal and inflationary regimes.

Abstract

The misalignment mechanism for axion production depends on the temperature-dependent axion mass. The latter has recently been determined within the interacting instanton liquid model (IILM), and provides for the first time a well-motivated axion mass for all temperatures. We reexamine the constraints placed on the axion parameter space in the light of this new mass function. We find an accurate and updated constraint $ f_a \le 2.8(\pm2)\times 10^{11}\units{GeV}$ or $m_a \ge 21(\pm2) \units{μeV}$ from the misalignment mechanism in the classic axion window (thermal scenario). However, this is superseded by axion string radiation which leads to $ f_a \lesssim 3.2^{+4}_{-2} \times 10^{10} \units{GeV}$ or $m_a \gtrsim 0.20 ^{+0.2}_{-0.1} \units{meV}$. In this analysis, we take care to precisely compute the effective degrees of freedom and, to fill a gap in the literature, we present accurate fitting formulas. We solve the evolution equations exactly, and find that analytic results used to date generally underestimate the full numerical solution by a factor 2-3. In the inflationary scenario, axions induce isocurvature fluctuations and constrain the allowed inflationary scale $H_I$. Taking anharmonic effects into account, we show that these bounds are actually weaker than previously computed. Considering the fine-tuning issue of the misalignment angle in the whole of the anthropic window, we derive new bounds which open up the inflationary window near $θ_a \to π$. In particular, we find that inflationary dark matter axions can have masses as high as 0.01--1$\units{meV}$, covering the whole thermal axion range, with values of $H_I$ up to $10^9$GeV. Quantum fluctuations during inflation exclude dominant dark matter axions with masses above $m_a\lesssim 1$meV.

Figures (10)

• Figure 1: The mass for the QCD axion follows from the topological susceptibility, $m^2_a f^2_a = \chi$. The fit goes over to the dilute gas approximation for moderately high temperatures $T\approx 400\, \mathrm{MeV}$, in accordance with the IILM data. Note that the large errors are mostly due to the large uncertainties in the determination of $\Lambda$, used to set dimensions.
• Figure 2: Shown are the mass for the QCD axion from IILM simulations (\ref{['eq:mass:iilm']}), from a lattice inspired fit that uses the IILM mass shifted towards higher temperatures to mimic the phase transition at $T^{lat}_c\approx 160 \, \mathrm{MeV}$, from the classic dilute gas approximation (DGA) by Turner turner:axion:cosmology and its update by Bae et al. bae:huh:kim:axion, and from the DGA derived in this paper (\ref{['eq:mass:dga']}). The simple power-law DGA axion masses are cut off by hand once they exceed $m_a(T=0)$ and give a surprisingly good approximation to the full IILM result; we believe this is a coincidence. The differences that persist to high temperatures, between the update and our DGA model, arise from the slightly different quark masses. Our choice has the merit that the masses were determined self-consistently within the IILM at $T=0$wantz:iilm:1.
• Figure 3: In an adiabatically evolving universe the scale factor and the temperature are related through the condition of constant entropy. Given the knowledge of the effective degrees of freedom $g_{*,S}$, it amounts to solving an implicit equation. The QCD phase transition occurs at around $T_\mathrm{QCD}\approx 180 \, \mathrm{MeV}$, when the number of hadronic excitations rises very sharply, and $g_{*,S}$ is almost discontinuous; the would-be latent heat 'reheats' the universe, which is clearly seen in the graph.
• Figure 4: As long as the axion Compton wavelength is well outside the horizon, the axion zero mode is frozen; this corresponds to the late-time solution of (\ref{['eq:axion:evolution']}) with $m_a$ neglected. The axion starts to feel the pull of its mass at $m_a \approx 3H$, and evolves to its minimum at $\theta_a=0$, i.e. the PQ mechanism to solve the strong CP problem. After a few oscillations the axion number per comoving volume stays constant as long as the axion mass and the scale factor change slowly (adiabatic approximation). This is then used to extrapolate the result to today.
• Figure 5: The effective degrees of freedom $g_{*,R}$ and $g_{*,S}$ are given for the temperature range up to $T\approx 100\, \mathrm{GeV}$. The decoupling of the neutrinos is included and manifests itself in the differences between $g_{*,R}$ and $g_{*,S}$ after $e^\pm$ annihilation, when $T_\nu \neq T_\gamma$. We followed closely coleman:roos:geff, but included some minor changes to take into account a better understanding of the QCD phase transition from recent lattice studies (see main text). We have determined fits by using a sequence of smoothed step functions, see appendix \ref{['app:geff:fit']}. As seen from the graph, the fits are good, generally with an accuracy below $1\%$.