Lattice QCD input for axion cosmology

Abstract

One intriguing beyond-the-Standard-Model 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 pseudo-Nambu--Goldstone boson of the conjectured Peccei--Quinn symmetry of the Standard Model. Its mass and interactions are suppressed by a heavy symmetry-breaking scale, ${f}_{a}$, the value of which is roughly greater than ${10}^{9}\text{ }\text{ }\mathrm{GeV}$ (or, conversely, the axion mass, ${m}_{a}$, is roughly less than ${10}^{4}\text{ }\text{ }\ensuremath{\mu}\mathrm{eV}$). 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 temperature-dependent QCD free energy with respect to the $CP$-violating phase, $\ensuremath{\theta}$. However, this quantity is generically nonperturbative, 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 $\ensuremath{\theta}$ 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, which is required to be less than or equal to the dark matter density. Due to a variety of lattice systematic effects at the very high temperatures required, we perform a calculation of the leading small-$\ensuremath{\theta}$ 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 ${m}_{a}\ensuremath{\ge}(14.6\ifmmode\pm\else\textpm\fi{}0.1)\text{ }\text{ }\ensuremath{\mu}\mathrm{eV}$ when Peccei--Quinn breaking occurs after inflation.