Evaluate fluctuations in kinetic-response regimes

Evaluate the realization-dependent flux fluctuations for the kinetic-response regimes of the stochastic scalar background, specifically cases 1, 2, and 5, by computing the full flux covariance kernel and determining the relative fluctuation with respect to the ensemble-mean flux.

Background

The paper derives an exact harmonic kernel for the connected two-time correlation function of the orbit-averaged energy flux. In kinetic-response regimes, rapid background-mode beat phases may suppress off-diagonal contributions to the covariance, but this suppression does not determine the fluctuation relative to the mean because the mean itself can involve cancellations between different velocity and harmonic sectors.

The unresolved calculation concerns cases 1 and 5, which are locally resolved and retain the environmental velocity distribution, and case 2, which is multipolar and may probe the kinetic–coherent crossover. Determining the full covariance in these regimes would establish whether the ensemble mean is representative of an individual realization.

References

An explicit determination of the relative fluctuation requires evaluating the full kernel in each kinetic regime. Cases 1 and 5 are both local and resolve the relative velocity of the environmental modes, whereas case 2 retains a multipolar source and can probe the kinetic--coherent crossover near $p\sim k_\star$. We do not perform these evaluations here. In all three cases, Eq.~eq:kernel-flux-covariance, together with the exact kernel~eq:explicit-harmonic-flux-kernel, provides the appropriate starting point for future work.

— Dynamical Friction in an Ultralight Scalar Medium across Coherent and Stochastic Regimes  (2609.28641 - Mitra et al., 23 Sep 2026) in Section 5.3, subsection “Fluctuations in kinetic-response regimes”