Precise cosmological overclosure bound on the invisible-axion decay constant: QCD-crossover microphysics, inflation and thermal history, and the string–domain-wall axion yield

In 1983 Preskill, Wise and Wilczek showed that vacuum misalignment of the invisible axion after the QCD chiral transition overcloses the universe unless the Peccei–Quinn scale obeys $f_a\lesssim10^{12}\,$GeV, and identified what makes the precise numerical bound provisional: the temperature dependence of the axion mass across the QCD transition, imperfectly known cosmological parameters, the initial misalignment angle set by the timing of inflation relative to Peccei–Quinn breaking, and the unquantified axion energy budget of cosmic strings and domain walls. Four decades of literature have resolved the first two dependencies — lattice QCD now fixes the topological susceptibility and hence $m_a(T)$, the QCD transition is a smooth crossover, and the cosmological parameters are measured — and have classified the initial-angle issue into pre- and post-inflationary scenarios, with an anthropic window extending to $f_a\sim10^{17}\,$GeV in the former. The surviving open core is the string–domain-wall contribution: independent simulation groups still obtain post-inflationary axion-dark-matter masses spanning roughly $25$--$500\,\mu$eV. The problem is to determine the total relic axion abundance as a function of $f_a$ and the cosmological history with controlled systematics, fixing the precise overclosure bound.

Background

The strong $CP$ problem of quantum chromodynamics (QCD) is the unexplained smallness of the effective $CP$-violating angle $\bar\theta$, bounded by null searches for a neutron electric dipole moment to $|\bar\theta|\lesssim10^{-10}$. The Peccei–Quinn (PQ) solution postulates a global $U(1)_{\rm PQ}$ symmetry, spontaneously broken at a scale $f_a$ (the axion decay constant); the resulting light scalar — the axion, a pseudo-Nambu–Goldstone boson of the broken symmetry, where a pseudo-Nambu–Goldstone boson is the nearly massless excitation left by breaking a symmetry that is only approximate — dynamically relaxes $\bar\theta$ to zero. Because the original, electroweak-scale axion was ruled out, invisible-axion models decouple the PQ scale from the electroweak scale, allowing $f_a$ to be very large: stellar-cooling arguments require roughly $f_a\gtrsim10^{9}\,$GeV, while the scale could naturally sit near grand-unification scales of $\sim10^{15}$--$10^{16}\,$GeV.

Preskill, Wise and Wilczek (1983), together with Abbott and Sikivie and with Dine and Fischler in the same issue of Physics Letters B, computed the cosmological consequence. As the universe cools through the QCD epoch, nonperturbative QCD effects turn on and tilt the axion potential; a spatially homogeneous, initially misaligned axion field of amplitude $A_i\sim f_a$ begins coherent oscillations once the axion mass $m_a(T)$ (which grows as the temperature $T$ drops through the QCD chiral transition) exceeds the Hubble expansion rate $H$. The oscillations behave as a gas of nonrelativistic, nearly zero-momentum axions whose number per comoving volume is conserved, so their energy density survives to the present as cold, pressureless matter (this is the vacuum misalignment mechanism). Requiring that this energy density not exceed the observed dark-matter density — the overclosure requirement, where overclosure means an axion energy density $\Omega_a$ (relative to the critical density) so large that the universe is too young and too matter-dominated — yields the bound $f_a\lesssim4\times10^{12}\,$GeV (times a factor $h^2\lambda^{3/4}$, where $h$ is the Hubble constant in units of 100 km/s/Mpc and $\Lambda_{\rm QCD}=200\,{\rm MeV}\times\lambda$) for an adiabatic QCD transition, quoted more conservatively as $f_a\lesssim10^{12}\,$GeV if the transition is sudden. The same computation made axions a cold-dark-matter candidate.

The 1983 authors were explicit that the numerical bound is provisional, for four reasons that define the present problem. First, $m_a(T)$ across the QCD transition was computed only in the dilute-instanton-gas approximation (an approximation treating the nonperturbative QCD configurations that generate the axion mass as a sparse gas), and a rapid, nonadiabatic transition would raise the relic density by a factor of a few. Second, the numerical prefactors carry then-unmeasured cosmological parameters ($h$, $\Lambda_{\rm QCD}$, and the effective number of relativistic degrees of freedom $g_*$ governing entropy reheating). Third, the initial misalignment amplitude depends on the cosmological history: if PQ symmetry breaking precedes or coincides with inflation, the misalignment angle in our observable universe is an uncomputable accident that can be atypically small and relax the bound, while if PQ symmetry is broken after inflation the angle is a super-horizon-random field. Fourth, if PQ symmetry is broken after inflation, topological defects form — a network of global cosmic strings at PQ breaking, and below the QCD crossover domain walls (with a degeneracy of disconnected vacua set by the domain-wall number $N_{\rm DW}$, fixed by the color anomaly of $U(1)_{\rm PQ}$) — and axions radiated by the decaying string–wall system, including wall pieces bounded by strings and damped by gravitational radiation — the dissipative channel proposed by Vilenkin and Everett (1982), which PWW argued can remove only a small fraction of the axion energy — add to the axion energy budget in a way the 1983 computation did not quantify.

Four decades of work reorganized this list. The QCD input is now first-principles: lattice QCD at physical quark masses determines the topological susceptibility $\chi(T)$ — the free-energy curvature that sets $m_a(T)\simeq\sqrt{\chi(T)}/f_a$ — and shows that the QCD transition at physical masses is a smooth crossover rather than a phase transition (Aoki et al. 2006), so the sudden-transition branch does not arise in the Standard Model; the misalignment-only prediction is the standard power law $\Omega_a h^2\simeq0.12\,\theta_i^2(f_a/2\times10^{11}\,{\rm GeV})^{7/6}$ modulo known anharmonic corrections (reviews: Marsh 2016; Di Luzio et al. 2020). The timing of inflation relative to PQ breaking defines two standard scenarios: pre-inflationary PQ breaking, with a single accidental initial angle, an anthropic window extending $f_a$ to $\sim10^{17}\,$GeV, and constraints from CMB isocurvature perturbations (dark-matter fluctuations not accompanied by radiation fluctuations); and post-inflationary PQ breaking, with an average over the random initial angle plus strings and walls. What remains unsettled is the size of the defect contribution: the axion yield of the string network, which must be extrapolated over the enormous logarithmic range between the string-core scale and the Hubble scale, and its fate through the nonlinear QCD-crossover annihilation of walls bounded by strings. Published predictions of the axion mass for which axions constitute all of the observed cold dark matter in the post-inflationary scenario still range over roughly $25$--$500\,\mu$eV, so the numerical overclosure bound on $f_a$ still carries a factor-of-few to order-of-magnitude systematic uncertainty.

Problem Statement

Determine, with controlled and quantified systematics, the total present-day axion relic abundance $\Omega_a$ as a function of the axion decay constant $f_a$ and of the cosmological history — and therefore the precise numerical overclosure bound on $f_a$, equivalently the axion mass for which axions constitute all of the observed cold dark matter. The total abundance is the sum of: (a) the vacuum-misalignment contribution, computed with the lattice-QCD input for the temperature dependence of the axion mass $m_a(T)$ across the QCD crossover; (b) the contribution of the initial-condition statistics of the misalignment angle, set by the timing of inflation relative to Peccei–Quinn symmetry breaking (including the inflationary-accident/anthropic branch and the isocurvature constraints that bound it); and (c) the axion yield of the cosmic string–domain-wall network — the axions radiated during the string scaling regime (the attractor evolution in which the network maintains of order one long string per Hubble volume), the spectrum with which they are emitted, and the products of the final annihilation of walls bounded by strings at the QCD crossover, including the energy budget of the competing decay and gravitational-radiation channels. The determination must be given per cosmological scenario (pre- versus post-inflationary PQ breaking, with PQ symmetry thermally restored or not; standard radiation domination versus non-standard pre-BBN histories such as early matter domination or late string networks; domain-wall number $N_{\rm DW}=1$ versus $N_{\rm DW}>1$ with explicit PQ breaking), and must identify and propagate the dominant systematics of each: the extrapolation from the simulated to the physical string tension (the string energy per unit length, set logarithmically by the ratio of the Hubble scale to the string-core scale), the shape of the axion emission spectrum, initial-condition and discretization artifacts, and the nonlinear evolution at the QCD crossover.

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