Axion White-Noise Fluctuations in GR Scattering

• Axion white-noise fluctuations are characterized by a flat density power spectrum arising from post-inflationary symmetry breaking and de Broglie scale effects in ultralight dark matter.
• They accelerate early minihalo formation and induce stochastic gravitational effects, impacting disk heating, microlensing, and pulsar timing observations.
• Analytic scattering methods in general relativity provide a framework to compute density contrasts, correlation spectra, and diffusion phenomena crucial for interpreting astrophysical data.

Axion white-noise fluctuations are a foundational concept in the phenomenology of axion and axion-like particle (ALP) dark matter, arising from the combination of post-inflationary symmetry breaking, quantum/thermal misalignment, and wave properties of ultralight fields. These fluctuations underpin the cosmological initial conditions for minihalo formation, induce distinct stochastic effects in both astrophysical and laboratory environments, and place stringent constraints on the properties of axions across mass and coupling regimes.

1. Origin and Definition of Axion White-Noise Fluctuations

Axion white-noise fluctuations have two principal theoretical underpinnings, distinguished by axion mass regimes and cosmological production scenarios:

• Isocurvature white-noise: In post-inflation Peccei–Quinn (PQ) symmetry breaking, the axion field's misalignment angle is randomly and independently drawn in each Hubble patch. This yields order-unity fluctuations in local axion energy density. As a consequence, the primordial power spectrum is flat (white) for wavenumbers $k \ll k_0$, with $k_0 = a(t_0) H(t_0)/c$, where $t_0$ marks the onset of axion mass acquisition (typically QCD epoch) (Dai et al., 2019, Xiao et al., 2021). The white-noise power spectrum is truncated above $k_0$ by horizon scale causality:

$$P_{\rm iso}(k, t_0) = \Theta(k_0 - k) \frac{24\pi^2}{5\,k_0^3}$$

• Wave, or de Broglie, white-noise: For ultralight (“fuzzy”) axions, the macroscopic de Broglie wavelength, $\lambda_{\rm dB} = \hbar/(m \sigma)$, with $\sigma$ the velocity dispersion, determines the minimum coherence scale of density fluctuations (El-Zant et al., 2019). For scales $r \gg \lambda_{\rm dB}$ (i.e., $k \ll k_{\lambda} \sim 2\pi/\lambda_{\rm dB}$), the density field is spatially white-noise, but quantum interference suppresses power on smaller scales.

In both cases, “white-noise” refers to a flat density power spectrum within the allowed $k$-range, implying spatially uncorrelated density contrasts on scales below the cutoff.

2. Mathematical Structure: Density and Correlation Spectra

Formally, for a statistically homogeneous axion field $\psi(\mathbf{r}, t)$ satisfying the Schrödinger equation,

$$i\hbar \partial_t \psi = - \frac{\hbar^2}{2m} \nabla^2 \psi,$$

the density contrast $\delta(\mathbf{r}, t)$ is defined as

$$\delta(\mathbf{r}, t) = \frac{|\psi|^2}{\rho_0} - 1,\quad \rho_0 = \langle|\psi|^2\rangle,$$

with spatial power spectrum

$$P_\delta(k) = \left[\frac{2\sqrt{\pi}}{m_\hbar \sigma}\right]^3 \exp\left(-\frac{k^2}{m_\hbar^2 \sigma^2}\right),\quad m_\hbar = 2m/\hbar.$$

For $k \ll k_\lambda$, $P_\delta(k)\simeq \mathrm{const}$, so the field is white-noise; for $k \gtrsim k_\lambda$, the transfer function $T(k)$ introduces an exponential suppression:

$$T(k) = \exp\left(-\frac{k^2 \lambda_{\rm dB}^2}{4\pi^2}\right).$$

The spatial correlation function is the Fourier transform of $P_\delta(k)$:

$$\xi(r) = \langle \delta(0) \delta(r) \rangle = \exp\left(-\frac{r^2}{\lambda_{\rm dB}^2}\right) \text{ at } t=0,$$

indicating that correlations decay on the de Broglie scale.

Temporal coherence is controlled by the axion field's bandwidth. In the Galactic halo, the classical field exhibits a coherence time $\tau_{\rm coh} \sim 10^6\pi/m$, over which the field amplitude and phase are approximately constant (Flambaum et al., 2023).

3. Nonlinear Collapse: Minihalo Formation and Structure

The white-noise power spectrum remarkably accelerates small-scale structure formation relative to standard $\Lambda$CDM cosmology:

• The mass scale for cut-off fluctuations is

$$M_0 = \frac{4\pi}{3} \left(\frac{\pi}{k_0}\right)^3 \bar{\rho}_{a0}.$$

For QCD axions, $M_0 \sim 5 \times 10^{-10} M_{\odot}$ (Dai et al., 2019, Xiao et al., 2021).

• Axion minihalos or “miniclusters” form from these overdensities, with the first bound objects collapsing at $z_{\rm coll}(M) \sim 10^2$ early in cosmic history.
• N-body simulations (Xiao et al., 2021) show that minihalos relax to Navarro-Frenk-White (NFW) profiles with exceptionally high concentration parameters:

$$c(M, z) \simeq 1.4 \times 10^4 (1+z) \sqrt{M/A_{\rm osc} M_0},$$

and

$$r_s(M) \approx 3.7\times10^{-3}\, h^{-1}\mathrm{pc}\; (A_{\rm osc} M_0/10^{-11} M_\odot/h)^{-1/2}(M/10^{-6} M_\odot/h)^{5/6}$$

where $A_{\rm osc}$ is the dimensionless variance.

These features differentiate axion minihalos from CDM substructure: even sub-planetary-mass clumps formed from white-noise initial conditions are ultra-compact and survive various astrophysical environments.

4. Stochastic Effects and Diffusion in Stellar Dynamics

Axion-induced white-noise fluctuations generate potential perturbations that act analogously to classical two-body relaxation in stellar systems:

• The gravitational potential power spectrum follows from Poisson’s equation:

$P_\Phi(k) = (4\pi G \rho_0)^2 k^{-4} P_\delta(k).$

The corresponding random force field produces diffusion of test-star velocities.

• The velocity variance acquired over time $T$ by a test particle is

$$\langle(\Delta v)^2\rangle = 8\pi G^2 \rho_0 m_{\rm eff} \ln\Lambda\int \frac{f_{\rm eff}(\mathbf{v})\,d^3v}{|\mathbf{v}_p - \mathbf{v}|} T,$$

where $m_{\rm eff}$ is an effective mass encoding the collective, wavelike nature of the field, and $\ln\Lambda$ is a Coulomb log reflecting the ratio of maximal to minimal wavelengths (El-Zant et al., 2019).

• The associated relaxation time,

$$t_r \simeq \frac{v^3}{8\pi G^2 \rho_0 m_{\rm eff} \ln\Lambda},$$

generalizes Chandrasekhar's classical estimate, recovering the two-body limit for $\lambda_{\rm dB} \to 0$.

This treatment is essential for evaluating disk heating and interpreting dynamical observations in galaxies and clusters.

5. Observational Signatures and Constraints

White-noise axion fluctuations manifest in diverse astrophysical and laboratory phenomena:

• Disk heating limits: In an isothermal halo, radial and vertical stellar velocity dispersions accrue from FDM fluctuations. For the Milky Way disk, the requirement that FDM-induced velocity dispersions not exceed observed errors imposes $m_a \gtrsim 2 \times 10^{-22}\ \mathrm{eV}$ (El-Zant et al., 2019).
• Dense star clusters: Application to the Eridanus II central cluster yields $m_a \gtrsim 8.8\times10^{-20}\ \mathrm{eV}$, but for such large masses, the cluster lies within the soliton core and diffusion treatments break down, suggesting that the constraint must be interpreted with caution.
• Lensing in galaxy clusters: The post-inflation scenario predicts a white-noise surface convergence fluctuation $\Delta_\kappa \sim 10^{-4} - 10^{-3}$ on $10$–$10^4$ AU scales (Dai et al., 2019, Xiao et al., 2021). For highly magnified stars ($\mu \sim 10^3 - 10^4$), these fluctuations generate detectable, $O(1)$ modulations in microlensing light curves.
• Pulsar timing arrays: Shapiro and Doppler delays due to stochastic minihalo substructure may be detectable if future arrays achieve the required sensitivity band (Xiao et al., 2021).

6. Laboratory Searches and Stochastic Frequency Fluctuations

Axion field fluctuations induce stochastic variations in atomic and nuclear transition frequencies, operating on the field's coherence time scale and accessible to high-precision clock experiments (Flambaum et al., 2023):

• Classical field regime: For $m$ in the $10^{-17}$–$10^{-13}$ eV range, $v\sim10^{-3}c$, and $\rho_{\rm DM}\approx0.4$ GeV/cm$^3$, the axion quotient per de Broglie volume is large and the field is treated classically.
• Energy shifts: Quadratic coupling leads to first-order shifts $\Delta E^{(1)}(t) = C\left[a(t)/f_a\right]^2$, while pseudoscalar (derivative) interactions produce smaller, second-order shifts.
• Statistical properties: For averaging intervals short compared to $\tau_{\rm coh}$, clock measurements sample a Rayleigh distribution in amplitude $a_0$, with time-averaged mean and standard deviation that coincide.
• Power spectral density: For frequencies $\omega \ll 1/\tau_{\rm coh}$, measurement bandwidth sees an effectively white power spectrum.
• Experimental limits: Rb/Cs and H–Si cavity–hyperfine comparisons set $f_a \gtrsim 1.8\times10^9$ and $5.8\times10^9$ GeV$\cdot(10^{-15}$ eV$/m)$ over $m$-ranges covered by integration durations and total time spans (Flambaum et al., 2023). The constraints fill parameter space not reached by direct detection, albeit at levels above the QCD axion band.

A dedicated protocol for extracting $\tau_{\rm coh}$ and thus $m$ from clock data exploits the exponential nature of the distribution induced by axion fluctuations, yielding model-independent mass determination strategies.

7. Extensions, Uncertainties, and Alternative Scenarios

White-noise fluctuation phenomenology extends to other contexts and carries intrinsic model uncertainties:

• Axion string decay: Emission from cosmic axion strings enhances the initial small-scale power by $O(1)$, affecting the effective variance $A_{\rm osc}$ and softening the $k$ cutoff (Xiao et al., 2021). Non-Gaussianity in the most nonlinear patches remains subdominant for the excursion-set mass function at late times.
• Other ALPs and blue spectra: The gravitational evolution under white-noise or blue initial $P(k)$ is scale-invariant, so results generalize to ALPs with different $k_c$ and $A_{\rm osc}$, or to primordial black hole (PBH) and early matter-dominated scenarios by suitable $P(k)$ substitution.
• Breakdown of diffusion approach: For structures smaller than the de Broglie wavelength or for systems embedded in soliton cores (e.g., dense clusters for large $m_a$), the assumptions of uncorrelated “white-noise” fluctuations and conventional relaxation theory no longer hold, requiring explicit modeling of coherent core oscillations and locality criteria (El-Zant et al., 2019).

The direct implication is that while white-noise fluctuations are a robust signature of post-inflation axion models and fuzzy DM, care must be taken in translating their effects into observable or exclusionary constraints, particularly near theoretical and physical cutoffs. Further, ongoing and upcoming observational initiatives, from caustic microlensing to PTAs and clock networks, provide complementary avenues to test these predictions.