  We recently improved the famous result of Parikh and Wilczek, who found a
probability of emission of Hawking radiation which is compatible with a
non-strictly thermal spectrum, showing that such a probability of emission is
really associated to two non-strictly thermal distributions for boson and
fermions. Here we finalize the model by finding the correct value of the
pre-factor of the Parikh and Wilczek probability of emission. In fact, that
expression has the "of order" sign instead of the equality. In general, in this
kind of leading order tunnelling calculations, the exponent arises indeed from
the classical action and the pre-factor is an order Planck constant correction.
But in the case of emissions of Hawking quanta, the variation of the
Bekenstein-Hawking entropy is order 1 for an emitted particle having energy of
order the Hawking temperature. As a consequence, the exponent in the Parikh and
Wilczek probability of emission is order unity and one asks what is the real
significance of that scaling if the pre-factor is unknown. Here we solve the
problem assuming the unitarity of the black hole (BH) quantum evaporation and
considering the natural correspondence between Hawking radiation and
quasi-normal modes (QNMs) of excited BHs, in a "Bohr-like model" that we
recently discussed in a series of papers. In that papers, QNMs are interpreted
as natural BH quantum levels (the "electron states" in the "Bohr-like model").
Here we find the intriguing result that, although in general it is well
approximated by 1, the pre-factor of the Parikh and Wilczek probability of
emission depends on the BH quantum level n. We also write down an elegant
expression of the probability of emission in terms of the BH quantum levels.

General Physics;High Energy Astrophysical Phenomena;General Relativity and Quantum Cosmology