Quantum Gravity

240 QUANTIZATION OF BLACK HOLESlogλ physH −1 (t)a iλ k (t) ≡ a(t)kaf a eqt k,exit t k,enterlog a(t)Fig. 7.4. Time development of a physical scale λ(t) ≡ a(t)/k, wherea(t) isthescale factor of a Friedmann universe, and the Hubble scale H −1 (t). Duringan inflationary phase, H −1 (t) remains approximately constant. After the endof inflation (a f ) the Hubble scale H −1 (t) increases faster than any scale.Therefore the scale described by λ k , which has left the Hubble scale at timet k,exit , enters the Hubble scale again at t k,enter in the radiation- (or matter-)dominated phase.since, according to (7.14), black holes with masses smaller than M PBH ≈ 5×10 14 ghave by now evaporated due to Hawking radiation. PBHs with bigger mass arestill present today. At t ≈ 10 −5 s, one can create a solar-mass black hole and att ≈ 10 s (the time of nucleosynthesis) one could form a PBH with the mass ofthe galactic black hole. The initial mass can increase by means of accretion, butit turns out that this is negligible under most circumstances.In the presence of an inflationary phase in the early universe, all PBHs producedbefore the end of inflation are diluted away. This gives the boundM PBH >M H (T RH ) ≈m 3 P10.88T 2 RH∼ 1g, (7.151)if for the reheating temperature T RH a value of 10 16 GeV is chosen.According to the numerical calculations by Niemeyer and Jedamzik (1999),there exists a whole spectrum of initial masses,