Numerical study of Hawking radiation photosphere formation around microscopic black holes

Heckler has recently argued that the Hawking radiation emitted from microscopic black holes has sufficiently strong interactions above a certain critical temperature that it forms a photosphere, analogous to that of the Sun. In this case, the visible radiation is much cooler than the central temperature at the Schwarzschild radius, in contrast with the naive expectation for the observable spectrum. We investigate these ideas more quantitatively by solving the Boltzmann equation using the test particle method. We confirm that at least two kinds of photospheres may form: a quark-gluon plasma for black holes of mass $M_{\mathrm{BH}}\lesssim 5\times {10}^{14}\mathrm{g}$ and an electron-positron-photon plasma for $M_{\mathrm{BH}}\lesssim 2\times {10}^{12}\mathrm{g}.$ The QCD photosphere extends from the black hole horizon to a distance of 0.2–4.0 fm for ${10}^9\mathrm{g}\lesssim M_{\mathrm{BH}}\lesssim 5\times {10}^{14}\mathrm{g},$ at which point quarks and gluons with average energy of order ${\Lambda }_{\mathrm{QCD}}$ hadronize. The QED photosphere starts at a distance of approximately 700 black hole radii and dissipates at about 400 fm, where the average energy of the emitted electrons, positrons and photons is inversely proportional to the black hole temperature, and significantly higher than was found by Heckler. The consequences of these photospheres for the cosmic diffuse gamma ray and antiproton backgrounds are discussed: bounds on the black hole contribution to the density of the universe are slightly weakened.