Page Time of Primordial Black Holes in the Standard Model and Beyond

Abstract

The Page time marks the moment when the von Neumann entropy of the emitted Hawking radiation equals the Bekenstein-Hawking entropy of an evaporating black hole, which is assumed to quantify its degrees of freedom as seen from the outside. Beyond this point, from unitarity we would expect that the entropy of the radiation begins to decrease, ensuring that information is eventually recovered. In this work, we investigate the dependence of the Page time on black hole properties and the particle content of nature. Specifically, we analyze its sensitivity to the Standard Model (SM) and potential Beyond-the-SM degrees of freedom, incorporating the effects of particle masses. We find that a Schwarzschild primordial black hole (PBH) with an initial mass of $6.23\times 10^{14}~{\rm g}$ would have a Page time equal to the age of the Universe, assuming emission of SM particles only. We further explore the impact of a non-negligible PBH angular momentum, finding that light spin-2 particles are predominantly emitted before the Page time. Specifically, for initial angular momenta values exceeding $a_\star > 0.5$, approximately $70\%$ of the total graviton emission occurs prior to the Page time for PBHs with an initial mass $M_{\rm BH} \lesssim 10^{10}~{\rm g}$. Finally, we discuss the implications for PBH phenomenology, particularly regarding potential constraints from $\Delta N_{\rm eff}$ measurements.