Axion cosmology and the energy scale of inflation: completing the joint $(E_I, f_a)$ exclusion map and deciding the fate of the anthropic window

When the Peccei-Quinn symmetry of the QCD axion is broken before or during inflation, de Sitter fluctuations of the axion field imprint uncorrelated cold-dark-matter isocurvature perturbations whose amplitude grows with both the inflationary energy scale and the fraction of dark matter in axions, tying axion physics directly to the energy scale of inflation. Hertzberg, Tegmark and Wilczek (2008) mapped the observationally allowed region of the plane spanned by the inflationary energy scale $E_I$ and the axion Peccei-Quinn scale $f_a$ using WMAP5-era data, finding a classic window and an anthropic window, and identified the error budget of that map: the relic-abundance normalization parameters ($\chi$ and $b$), the post-inflationary thermalization efficiency $\epsilon_\mathrm{eff}$, neglected cosmic-string and domain-wall production, the inflationary fluctuation mapping ($\gamma$), the prior on any additional dark-matter component, and the need for roughly an order of magnitude better isocurvature bounds ($\alpha_a \sim 0.007$). The problem is to complete this map: determine the allowed $(E_I, f_a)$ region with controlled theoretical inputs and observational bounds sufficient to decisively probe, exclude, or confirm the anthropic window. Later work has resolved parts of the budget - lattice QCD now fixes the temperature-dependent axion mass, Planck 2018 tightened the uncorrelated isocurvature bound to $\beta_\mathrm{iso} < 0.038$ (with $\beta_\mathrm{iso}$ the fraction of the total primordial perturbation power in cold-dark-matter isocurvature modes at a pivot scale), and string-network simulations yield a predicted post-inflationary axion mass - but the anthropic window remains neither detected nor excluded, and the post-inflationary normalization and thermal-history dependence are still debated, so the problem is open.

Background

The QCD axion is a hypothetical light pseudoscalar particle proposed to solve the strong CP problem, i.e. the unexplained smallness of CP violation in quantum chromodynamics (QCD), conventionally constrained by bounds on the neutron electric dipole moment. In the Peccei-Quinn (PQ) mechanism a global U(1) symmetry is spontaneously broken at a scale $f_a$ (the axion decay constant), and the axion is the pseudo-Nambu-Goldstone boson of that breaking; nonperturbative QCD effects generate an axion potential that dynamically cancels the effective CP-violating angle, giving the axion a zero-temperature mass $m_a \propto 1/f_a$ (of order micro-electronvolts for $f_a$ near $10^{12}$ GeV).

In the early universe, axion dark matter arises mainly through the vacuum-misalignment mechanism: the axion field starts at some initial angle $\theta_i$ relative to its minimum and begins coherent oscillations when its mass, which grows as the universe cools through the QCD confinement epoch, becomes comparable to the Hubble expansion rate $H$. The present axion density per photon, $\xi_a$, scales with $f_a$, $\theta_i^2$, the temperature dependence of $m_a(T)$, and anharmonic corrections. Hertzberg, Tegmark and Wilczek (2008) (the source of this problem) parameterized the remaining theory error by a constant $b$ in the high-temperature mass ($m_a(T) \approx m_a(0)\, b\, (\Lambda_{QCD}/T)^4$ with $b = 0.018$) and a dimensionless correction factor $\chi$, taking a conservative $\chi = 1/20$ and a moderate $\chi = 1$. If the PQ symmetry is broken after inflation (or restored by reheating), a network of cosmic strings (line defects of the PQ field) and domain walls (sheet-like boundaries between vacua) forms and radiates additional axions; the source deliberately treated only misalignment and noted that these additional channels carry larger theoretical uncertainties.

The connection to inflation is the heart of the problem. If PQ symmetry is already broken during inflation, the axion is light, and de Sitter fluctuations at the Gibbons-Hawking temperature $T_{GH} = H_I/2\pi$ (where $H_I$ is the Hubble scale during inflation, tied to the inflationary energy scale $E_I$ and to the tensor-to-scalar ratio $r$ that measures the amplitude of primordial gravitational waves relative to scalar density perturbations) displace the axion by $\sigma_a = \gamma H_I/2\pi$ per Hubble patch, with $\gamma$ an order-one constant the source fixed to 2. These fluctuations survive only if $f_a$ exceeds both $T_{GH}$ and the maximum post-inflationary thermalization temperature $T_{max} = \epsilon_\mathrm{eff} E_I$, set by a reheating efficiency $\epsilon_\mathrm{eff} < 1$. They then appear as uncorrelated cold-dark-matter (CDM) isocurvature perturbations - spatial fluctuations in the axion-to-entropy ratio, not accompanied by total-density fluctuations, unlike the observed adiabatic mode - with a fractional power $\alpha_a$ that grows with both $H_I$ and the axion fraction $R_a$ of the dark matter. This defines two regimes: the classic regime ($f_a < \max\{T_{GH}, T_{max}\}$, no axion isocurvature, $\theta_i$ averaged over the observable universe) and the anthropic regime (PQ broken before inflation, $\theta_i$ uniform across Hubble volumes, so large $f_a$ requires our Hubble volume to sit at an atypically small $|\theta_i|$).

Using the fifth-year data release of the Wilkinson Microwave Anisotropy Probe (WMAP5) combined with baryon-acoustic-oscillation and supernova data (bounds $\xi_a \le 2.9$ eV on the axion density, $\alpha_a < 0.072$ on uncorrelated CDM isocurvature, and $Q_t < 9.3\times10^{-6}$ on tensor modes), the source constructed an exclusion map in the $(E_I, f_a)$ plane. Two allowed windows survive: the classic window near $f_a \sim 10^{11}$-$10^{12}$ GeV (where little can be inferred about inflation) and the inflationary anthropic window (large $f_a$ with atypically small local $\theta_i$, hence necessarily low $E_I$). For example, $f_a = 10^{16}$ GeV excludes $E_I \gtrsim 3.8\times10^{14}$ GeV (equivalently $r \gtrsim 2\times10^{-9}$) at 95% confidence. The source was explicit that the map's error budget and observational frontier constituted the open work: the $\chi$ and $b$ normalization, the unknown $\epsilon_\mathrm{eff}$, the neglected string and domain-wall production, the order-one $\gamma$ and more generally inflationary models in which $H$ varies strongly across the $\sim 55$ e-folds that set the observable modes, the lack of a well-motivated prior for a possible second dark-matter component (such as a WIMP, a weakly interacting massive particle), which controls the statistical inference on $R_a$ (the source's motivated exclusion used $R_a > 0.25\%$), and the fact that current isocurvature bounds leave most of the anthropic window unprobed - an order-of-magnitude improvement, to $\alpha_a \sim 0.007$, was identified as what is needed to push the bound deep into that window. A comprehensive modern review of this subject, including the updated conventions, is Marsh (2016), 'Axion Cosmology'.

Problem Statement

Determine the observationally allowed region of the joint parameter space of QCD-axion dark matter and inflation - the inflationary energy scale $E_I$ (equivalently $H_I$ or the tensor-to-scalar ratio $r$) versus the Peccei-Quinn scale $f_a$ (equivalently the axion mass), with the local misalignment angle $\theta_i$ and the axion fraction $R_a$ of the dark matter as environmental or statistical parameters - with controlled theoretical inputs, and in this way decide the fate of the anthropic window ($f_a \gtrsim 10^{12}$ GeV with PQ symmetry broken before or during inflation, atypically small local $\theta_i$, and correspondingly low $E_I$): decisively probed, excluded, or confirmed, or shown to be beyond the reach of realistic observation.

The requested results all bear on this single map. Specifically: (1) the axion relic-abundance normalization, including the temperature-dependent mass $m_a(T)$ (the $b$ and $\chi$ inputs of the source), anharmonic corrections, and the additional production from cosmic-string and domain-wall decay when PQ symmetry is restored after inflation; (2) the dependence of the anthropic/classic regime boundary on the post-inflationary thermalization history (the efficiency $\epsilon_\mathrm{eff}$ entering $T_{max} = \epsilon_\mathrm{eff} E_I$); (3) the mapping between inflationary dynamics and observable scales (the $\gamma$ factor in the axion fluctuation amplitude, including models where $H$ changes substantially over the $\sim 55$ observable e-folds); (4) the statistical treatment of mixed dark matter, i.e. the prior on any additional component such as a WIMP that controls the inference on $R_a$; and (5) the combination of observational bounds - CDM-isocurvature bounds pushed from the WMAP5-era $\alpha_a < 0.072$ toward the $\alpha_a \sim 0.007$ level the source identified as decisive, together with tensor-mode bounds and astrophysical and laboratory axion searches - into the joint exclusion map. Items (1)-(4) are the theoretical error budget of the same abundance normalization and regime boundary that item (5) confronts with data; combined, they determine how each input reshapes the allowed region and how much better the data must become to decide the anthropic window.

References

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