$\gamma$ rays run on time

TL;DR

This paper investigates the behavior of gamma rays over cosmic distances, revealing that their optical depth is significantly larger than the expected time delay relative to gravitational waves, due to fundamental wave properties and cosmological factors.

Contribution

It demonstrates that gamma-ray absorption and refraction are related through dispersion relations, explaining the large optical depth compared to the minimal time delay.

Findings

Optical depth of gamma rays exceeds time delay by a factor of E_gamma/H_0.

Refraction index parts are related by dispersion relations.

Large factor arises from wave properties and cosmological model.

Abstract

Significant absorption of radiation is usually accompanied by refraction. This is not the case for $\gamma$ rays travelling cosmic distances. We show that the real and imaginary parts of the refraction index are indeed commensurable, as they are related by dispersion relations, but when turning to physical observables, the (finite) optical depth is way larger than the (infinitesimal) time delay of the gamma rays relative to gravitational radiation. The numerically large factor solving the apparent contradiction is $E_\gamma/H_0$ arising from basic wave properties (Bouguer-Beer-Lambert law) and the standard cosmological model, respectively.