How accurately does Hawking emission survive modified short-distance dispersion?
In plain words
In a fluid, sound waves stop behaving like ordinary sound at very short wavelengths, just as light might at the tiny Planck length. The question is to put an exact bound on how much this changes the thermal glow from a sonic horizon.
Precise statement
Take a $1+1\mathrm{D}$ field with dispersion $(\omega - v(x) k)^2 = c^2 k^2 (1 + s k^2/k_d^2)$, $s = +1$ (superluminal) or $-1$ (subluminal), on a stationary flow $v(x)$ with a sonic horizon of surface gravity $\kappa = \left|d(v - c)/dx\right|$ at the horizon. Compute the emitted spectrum $n(\omega)$ and its deviation from the Planck law at $T_H = \hbar \kappa/(2 \pi)$, including the cutoff frequency $\omega_{\mathrm{max}}$ and the greybody factor. An answer is a proven bound on $\left|n(\omega) - n_{\mathrm{Planck}}(\omega)\right|$ as a function of $\kappa/(c k_d)$ and the width $D$ of the near-horizon region, valid also for steep profiles with $\kappa \sim c k_d$.
What would settle it
An analytic $S$-matrix error bound, checked against exact numerical mode solutions over the full range of $\kappa/(c k_d)$ and $D k_d$.
Status in the literature
Unverified note
Analytic and numerical work since Unruh (1995) and Corley and Jacobson (1996) establishes near-thermal emission when $c k_{d} >> \kappa$ and the horizon region is broad; Finazzi and Parentani (PRD 85, 124027, 2012) separated the broad-horizon and steep-horizon regimes, but the steep-horizon result is numerical and no general proven bound exists as of 2026.