Lattice QCD input for axion cosmology

TL;DR

The paper develops a first-principles, lattice-based pipeline to constrain axion cosmology by computing the temperature dependence of the QCD topological susceptibility $\chi(T)$ at small $\theta$ in SU(3) Yang–Mills and constraining the axion mass via the misalignment mechanism. By fitting $\chi(T)$ to DIGM- and IILM-inspired forms, it translates high-temperature QCD dynamics into a bound on the present axion mass for the post-inflation PQ-breaking scenario, finding $m_a \ge (14.6 \pm 0.1)\ \mu\text{eV}$ with $f_a$ around $4\times 10^{11}$ GeV for pure-glue input. The study also carefully analyzes lattice discretization and finite-volume systematics, establishes a robust methodology to propagate lattice uncertainties into cosmological predictions, and provides a concrete roadmap toward full QCD calculations to yield sharper axion constraints for experiments like ADMX. This work thus bridges nonperturbative QCD topology with early-Universe cosmology and experimental searches in a controlled, first-principles framework.

Abstract

One intriguing BSM particle is the QCD axion, which could simultaneously provide a solution to the Strong CP problem and account for some, if not all, of the dark matter density in the universe. This particle is a pNGB of the conjectured Peccei-Quinn (PQ) symmetry of the Standard Model. Its mass and interactions are suppressed by a heavy symmetry breaking scale, $f_a$, whose value is roughly greater than $10^{9}$ GeV (or, conversely, the axion mass, $m_a$, is roughly less than $10^4\ μ\text{eV}$). The density of axions in the universe, which cannot exceed the relic dark matter density and is a quantity of great interest in axion experiments like ADMX, is a result of the early-universe interplay between cosmological evolution and the axion mass as a function of temperature. The latter quantity is proportional to the second derivative of the QCD free energy with respect to the CP-violating phase, $θ$. However, this quantity is generically non-perturbative and previous calculations have only employed instanton models at the high temperatures of interest (roughly 1 GeV). In this and future works, we aim to calculate the temperature-dependent axion mass at small $θ$ from first-principle lattice calculations, with controlled statistical and systematic errors. Once calculated, this temperature-dependent axion mass is input for the classical evolution equations of the axion density of the universe. Due to a variety of lattice systematic effects at the very high temperatures required, we perform a calculation of the leading small-$θ$ cumulant of the theta vacua on large volume lattices for SU(3) Yang-Mills with high statistics as a first proof of concept, before attempting a full QCD calculation in the future. From these pure glue results, the misalignment mechanism yields the axion mass bound $m_a \geq (14.6\pm0.1) \ μ\text{eV}$ when PQ-breaking occurs after inflation.

Figures (9)

• Figure 1: Temperature as a function of $\beta$, given $\beta_c = 6.338$ and ${N_{\tau}} = 12$.
• Figure 2: The topological charge as a function of Monte Carlo time for the 80$^3\times$6 $\beta=6.301$ ($T/T_c = 1.90$) ensemble. In the left panel, the yellow points are the measurements of $Q_\mathbb{R}$ and the purple points are the globally-fit values $Q_f$. In both panels, the brown dotted lines indicate the local maxima for the distribution of $Q_\mathbb{R}$. The right panel shows a histogram of those points with a bin width of 0.05. The dashed purple line is a gaussian distribution with a standard deviation given by the second moment of the $Q_f$ distribution.
• Figure 3: A study of finite-volume effects at three different temperatures. All lattices have a temporal extent ${N_{\tau}}=6$. Each ensemble is slightly offset from its temperature for ease of visibility. The statistical errors are smaller than the markers shown.
• Figure 4: A study of lattice spacing effects at two different temperatures. At $T/T_c$=1.31 the lattices shown are $48^3\times6$ and $64^3\times8$ which have exactly the same physical volume, while at $T/T_c$=1.8 the lattice shown are $64^3\times6$ and $96^3\times8$, which have physical spacial extents that differ by only 13%. Each ensemble is slightly offset from its temperature for ease of visibility. The statistical errors are smaller than the markers shown.
• Figure 5: The largest volumes from Gattringer et al.Gattringer:2002mr and our best ${N_{\tau}}=6$ data points at each temperature. The statistical errors on our points are smaller than the markers shown.