Primordial Black Hole Evaporation in Modified-Entropy Horizons

Non-additive black-hole entropies, such as the Barrow and Tsallis-Cirto forms, change the thermodynamic temperature [Formula: see text] obtained from the horizon first law and therefore modify the evaporation rate, lifetime and particle-emission history of primordial black holes (PBHs). Since many PBH constraints depend on the epoch at which a PBH evaporates, these entropy deformations can shift them substantially. We derive the modified temperature laws from the first law, solve the resulting evaporation equation in closed form, and use the solution to (i) obtain analytic expressions for the evaporating-today threshold mass [Formula: see text] and [Formula: see text], (ii) show that the PBH mass window probed by CMB [Formula: see text] spectral distortions shifts and narrows as a function of the deformation parameter, with a width fixed by the lifetime exponent, and (iii) identify which part of the standard “already evaporated” mass range is no longer excluded by the evaporation argument alone. For a fixed greybody/particle-content normalization, a modest Barrow deformation [Formula: see text] lowers the threshold from [Formula: see text] g to [Formula: see text] g and moves the CMB-distortion-sensitive window from [Formula: see text] g to [Formula: see text] g. These values correspond to the case in which the emission temperature follows the generalised first law, the dependence on this assumption is analysed explicitly. The existing evaporation-based and CMB-distortion PBH constraints must therefore be re-derived at the shifted masses before the surrounding dark-matter parameter space can be assessed. We also find that sufficiently non-extensive Tsallis horizons ([Formula: see text]) have no finite evaporation time in this treatment.