The history of the universe after the recombination probably involves a reionization epoch, as the Gunn-Peterson test seems to suggest: if this is the case, the consequences of such a phenomenon should be relevant, both for the induced enhancement of the cosmic microwave background (CMB) polarization and for the possible damping of the CMB fluctuations on small angular scales (θ~1º). In this paper, I will study a model of reionization at redshifts z≤60 caused by the evaporation of primordial black holes; photon emission only from nonrotating black holes is considered. A system of coupled differential equations, giving the time evolution of the ionization degree x, of the plasma temperature Te and of the photon number density nγ, is solved in an analytical way: the results obtained show that such a kind of reionization is possible, being able to increase the ionization degree of the universe from a value x=0.002 (just after the recombination) to values near 1 (when the black holes evaporation ends). In particular, taking the evaporation redshift equal to the reionization redshift zR, one obtains total reionization (i.e. x=1) for 15≤zR≤30, while only a partial effect (x~0.75÷0.90) is present for higher values of zR (40≤zR≤60). The fast increase of x seems to agree with the predictions of an exponential reionization model discussed in a previous study of the CMB polarization induced by gravitational waves. The evolution of the plasma temperature Te is also estimated: it is affected in a less important way by the primordial black holes evaporation process, as we expect from the experimental FIRAS upper limit on the comptonization parameter yc (yc<2.5×10−5). The photoionization process here studied seems generally able to maintain the plasma in a ionized state without heating it up at very high temperatures; however, an improvement in the numerical calculation of Te is necessary in order to take into account in a more satisfactory way the collisional and excitation cooling, that can limit the increase of the plasma temperature. In this model, the density of primordial black holes (PBH’s) necessary to give a nonnegligible reionization is an important parameter: here I will consider various birth times tin and various initial density for the PBH’s, showing that the most effective reionization is obtained for zR≤30 and for PBH’s formed at tin~10−28 secafter the big bang. An estimate of their present density for this formation time gives a value ρ0=2.44×10−38g cm−3, corresponding to a present density parameter ΩPBH equal to 5.20×10−9. This result agrees with the experimental upper limit ΩPBH≤(7.6±2.6)×10−9 h(−1.95±0.15) A future improvement of this work will consider also massive particle emission from both rotating and nonrotating black holes and a spectrum taking also into account quarks and gluons jets emission.
Primordial black holes from the QCD epoch: linking dark matter, baryogenesis, and anthropic selection
