Can inflationary axion fluctuations dominate over misalignment?

Technical Summary: Inflationary Axion Fluctuations vs. Misalignment

Problem Statement
The relic abundance of light stable axion dark matter is conventionally determined by the misalignment mechanism, where a homogeneous field displacement is frozen by Hubble friction until the Hubble rate drops below the axion mass, triggering oscillations. However, if the Peccei-Quinn (PQ) symmetry is broken before observable scales exit the horizon during inflation, the axion acquires quantum fluctuations of order $H_I/(2\pi)$. While these fluctuations are known to generate isocurvature perturbations constrained by the Cosmic Microwave Background (CMB), their contribution to the total dark matter abundance is often subdominant. This paper investigates whether inflationary fluctuations can, under specific conditions, dominate the axion abundance over the misalignment mechanism. A critical challenge is that the axion energy density is quadratic in the field; thus, even if the mean displacement vanishes, the fluctuations generate a non-zero energy density and a residual isocurvature perturbation (the quadratic term) that evades standard linear constraints.

Methodology
The authors analyze the production of axion-like particles (ALPs) during an instantaneous transition from de Sitter expansion to radiation domination. They employ a spectator axion model with a canonical inflaton and study two specific couplings that modify the axion spectrum:

1. Non-minimal kinetic coupling: A field-dependent kinetic function $K(\phi)$, realized via linear ($e^{\lambda \phi}$) or quadratic ($e^{\gamma \phi^2}$) exponentials.
2. Curvature coupling: A non-minimal coupling to the Ricci scalar, $\xi R \eta^2$.

The study follows the modes analytically through the transition, neglecting the axion mass during inflation but retaining it afterward. The authors derive the late-time spectrum fully analytically by matching Bunch-Davies modes in de Sitter space to parabolic cylinder functions (Weber equation solutions) in radiation domination. This approach allows for a direct comparison between the fluctuation-induced abundance and the misalignment abundance without relying on arbitrary onset prescriptions.

Key to the analysis is the calculation of the isocurvature power spectrum, $P_S(k)$, which includes both a linear term (proportional to the mean displacement $\eta_c$) and a quadratic term (proportional to the variance of fluctuations). The authors confront the fluctuation-dominated abundance with the Planck isocurvature limit at the pivot scale, specifically accounting for the quadratic term that survives when the mean vanishes.

Key Contributions and Results

• Dominance of Fluctuations: The paper demonstrates that inflationary fluctuations can dominate the axion abundance, but this requires a "blue" spectral tilt ($n_\eta - 1 > 0$) to suppress the isocurvature power at CMB scales while maintaining sufficient power at smaller scales to generate the required relic density.
• Canonical Axion Constraints: For a canonical, minimally coupled axion, the spectrum is scale-invariant ($n_\eta - 1 = 0$). In this case, the fluctuation-dominated abundance exceeds the Planck isocurvature limit by approximately eight orders of magnitude. Thus, a canonical axion cannot be fluctuation-dominated dark matter.
• Constant Kinetic Slope: For a kinetic function with a constant slope $p$ (linear exponential), satisfying the pivot isocurvature limit requires a significant blue tilt.
  • At $H_I = 10^{10}$ GeV: $n_\eta - 1 \gtrsim 0.27$ and $m_\eta \gtrsim 200$ GeV.
  • At $H_I = 10^{13}$ GeV: $n_\eta - 1 \gtrsim 0.34$ and $m_\eta \gtrsim 4.5$ keV.
  • In these scenarios, the mean displacement must be extremely small (e.g., $|\theta_c| \lesssim 4 \times 10^{-6}$ for $H_I = 10^{10}$ GeV) to avoid violating the linear isocurvature constraint.
• Running Slope (Quadratic Exponential): A kinetic function with a running slope (quadratic exponential) significantly relaxes these constraints. The spectrum is very blue at the CMB pivot (suppressing isocurvature) but less blue at the smaller scales that dominate the abundance.
  • This lowers the mass threshold to $m_\eta \approx 1.4$ GeV at $H_I = 10^{10}$ GeV and $m_\eta \approx 0.7$ eV at $H_I = 10^{13}$ GeV.
  • At the tensor bound ($H_I \approx 4.7 \times 10^{13}$ GeV), the mass can be as low as $0.01$ eV with an $\mathcal{O}(1)$ coupling.
• QCD Axion: For the QCD axion, where oscillations begin late ($T \sim 1$ GeV), the authors estimate that any single power-law spectrum blue enough to satisfy the pivot limit yields insufficient fluctuation abundance. Consequently, misalignment dominates for the QCD axion in this framework. However, a running slope brings the fluctuation abundance closer to the misalignment value, though thermal restoration constraints on the PQ scale ($f_\eta$) may still close this window.
• Isocurvature Structure: The analysis highlights that for fluctuation-dominated dark matter, the isocurvature spectrum is blue with a tilt of $2(n_\eta - 1)$. While the CMB pivot is safe, the isocurvature power grows on small scales. The authors note that for $n_\eta - 1 = 0.3$, the power exceeds the adiabatic power at $k \sim 10^3$ Mpc$^{-1}$, which is currently allowed by Lyman-$\alpha$ forest and spectral distortion data but warrants further investigation.

Significance and Claims
The paper claims to resolve the question of whether inflationary fluctuations can dominate axion dark matter production, concluding that the answer is "yes," provided the CMB isocurvature constraints are satisfied via a blue tilt. The primary significance lies in:

1. Analytic Precision: Providing a fully analytic derivation of the late-time spectrum and abundance, including the transition effects and the specific normalization of the misalignment component, removing order-one ambiguities found in previous estimates.
2. Quadratic Isocurvature: Explicitly demonstrating that the quadratic term in the isocurvature perturbation is the dominant constraint when the mean vanishes, invalidating previous analyses that relied solely on the linear term.
3. Mass Thresholds: Establishing that fluctuation-dominated dark matter requires heavy ALPs (GeV to TeV scale) for constant slopes, but running slopes can lower this to the eV scale, potentially allowing lighter axions to be produced entirely from inflation.
4. UV Realizations: Identifying that realizing these kinetic functions at low inflationary scales requires large couplings or strongly curved field spaces, while high-scale inflation ($H_I \sim 10^{13}$ GeV) offers a more natural realization with potentially detectable primordial gravitational waves ($r \sim 1.6 \times 10^{-3}$).

The authors maintain a modest stance regarding the QCD axion, noting that their estimate suggests the fluctuation mechanism is insufficient for it under standard assumptions, and they defer the full analysis of non-Gaussian isocurvature and small-scale structure formation to future work.