Explain the discrepancy between spectral-shape dependence of growth rates and coherence-time scaling of absolute thresholds

Determine the physical reason for the observed difference whereby, in a homogeneous plasma, the peak temporal growth rates of the two-plasmon decay instability driven by a temporally incoherent laser depend on the laser power spectrum’s shape at the carrier wavenumber G(k0), while in inhomogeneous plasmas near the quarter-critical density the absolute thresholds for Raman scattering and two-plasmon decay have been found to scale universally with laser coherence time (equivalently with ∫|G(k)|^2 dk).

Background

Within the paper’s homogeneous-plasma analysis, the broadband dispersion relation predicts that the peak growth rate depends on the laser spectral shape G(k), with different quantitative reductions for Lorentzian versus top‑hat spectra. This appears to conflict with prior results (e.g., Follett 2019) for absolute instabilities in inhomogeneous plasmas, where thresholds were reported to scale with coherence time independent of spectral shape.

The authors note this mismatch explicitly and highlight that the absolute instability problem near quarter‑critical density in an inhomogeneous plasma is substantially more complicated than homogeneous temporal growth, suggesting a distinct bandwidth dependence. Understanding the underlying mechanism would reconcile these regimes and clarify bandwidth’s role in instability control.