Thermal corpuscular black holes

We study the corpuscular model of an evaporating black hole consisting of a specific quantum state for a large number $N$ of self-confined bosons. The single-particle spectrum contains a discrete ground state of energy $m$ (corresponding to toy gravitons forming the black hole), and a gapless continuous spectrum (to accommodate for the Hawking radiation with energy $\omega >m$). Each constituent is in a superposition of the ground state and a Planckian distribution at the expected Hawking temperature in the continuum. We first find that, assuming the Hawking radiation is the leading effect of the internal scatterings, the corresponding $N$-particle state can be collectively described by a single-particle wave function given by a superposition of a total ground state with energy $M=Nm$ and a Planckian distribution for $E>M$ at the same Hawking temperature. From this collective state, we compute the partition function and obtain an entropy which reproduces the usual area law with a logarithmic correction precisely related with the Hawking component. By means of the horizon wave function for the system, we finally show the backreaction of modes with $\omega >m$ reduces the Hawking flux. Both corrections, to the entropy and to the Hawking flux, suggest the evaporation properly stops for vanishing mass, if the black hole is in this particular quantum state.

Figure 1

Numerical estimate of some coefficients in the expansion (3.15) (black lines), and their analytical approximation (red lines), for $N=10$ (energies are in Planck units).

Figure 2

Numerical estimate of $C(E>M)$ (black line), and its analytical approximation (3.20) (red line) for $N=10$ (energies are in Planck units).