On Integral Upper Limits Assuming Power Law Spectra and the Sensitivity in High-Energy Astronomy

The high-energy non-thermal universe is dominated by power law-like spectra. Therefore results in high-energy astronomy are often reported as parameters of power law fits, or, in the case of a non-detection, as an upper limit assuming the underlying unseen spectrum behaves as a power law. In this paper I demonstrate a simple and powerful one-to-one relation of the integral upper limit in the two dimensional power law parameter space into the spectrum parameter space and use this method to unravel the so far convoluted question of the sensitivity of astroparticle telescopes.