black holes - What is the relative composition of Hawking radiation?

1. 
   

   For a black hole of a given starting mass or temperature, what is the relative abundance and energies of the various species of Hawking radiation?

   
   
   
   
   • Range of interest: 1 kg < mass < $10^6$ kg
   
   
   • Points of interest: mass = 1 amu, mass = 1 kg, mass = 1000 kg, end-stage (lifetime < 1 sec, mass < 229 tonnes)
   
   
   • Eg, for a 1000 kg BH, what is the abundance of gammas, neutrinos, protons, etc, and what is the distribution of their energies?
   
   
   
   


2. 
   
   

   How does the relative abundance vary with distance from the event horizon?

   
   
   
   
   • How does the decay chain affect relative abundances?
   
   
   • How do secondary reactions affect relative abundances?
   
   
   • Range of interest: 1 kg < mass < $10^6$ kg
   
   
   • Points of interest: 10 nm from the event horizon, 1 m " ", 1 km " ", 1000 km " ".
   
   
   
   

Apologies in advance if I've committed any faux pas's or if this is a common question. I've searched through ~20 papers, but honestly I'm not sure I'd recognize the answer if I saw it.

Answer

The temperature of Hawking radiation (for a Schwarzschild black hole) is $$T=\frac{\hbar c^3}{8\pi G M} .$$ For your range of interest this gives us temperatures $10\,\text{TeV}$ (in energy units) for $10^6\,\text{kg}$ black hole and higher. That is more than currently achievable in accelerators (at least per particle). At such energies most of the radiation would come from quantum chromodynamical sector of the Standard Model: quarks and gluons. Once emitted those would quickly hadronize and resulting jets would eventually decay into stable particles. The original work that covers this is:

• MacGibbon, J. H., & Webber, B. R. (1990). Quark-and gluon-jet emission from primordial black holes: The instantaneous spectra. Physical Review D, 41(10), 3052, doi.

From the paper:

a black hole emits only those particles which appear elementary on the scale of the radiated energy and the dimensions of the black hole at a given temperature. The evaporated particles then form into composite particles after emission. Thus, at temperatures above $\Lambda_\text{QH}$, we envisage the black hole emitting relativistic quark and gluon jets which subsequently fragment into the stable photons, leptons, and hadrons (i.e., neutrinos, electrons, positrons, protons, and antiprotons).

This paper has spectra of the particles for temperatures up to $100\,\text{GeV}$, which is two orders of magnitude smaller than your range, however one could hope that fractions of different species of particles would stay more or less the same, since they change very little while going from $10$ to $100\,\text{GeV}$. Here is species fractions for $T=100\,\text{GeV}$: $$ \begin{array}{l|c|c|c|c} \text{Species}&p,\bar{p} & e^\pm & \gamma & \nu,\bar{\nu} \\ \hline \dot N\,(\text{%}) &2.37 & 19.63 & 22.13 & 55.88 \end{array} $$

The scenario above did not include contributions from gravitons and Higgs boson (since Higgs field had not been observed then, and graviton is not observed to this day) although it does discuss their role. The conclusion (now that we know Higgs boson mass) is that main decay mode of Higgs into top and bottom quark/antiquarks would increase effective number of $q_b$ and $q_t$ degrees of freedom but each such additional jet would have smaller energy, while the higher spin of graviton would mean that its contribution would be less than 1% of total power and number of particles.

This paper uses Monte-Carlo jet code to simulate hadronization and subsequent fragmentation. As I understand, the descendant of this code is HERWIG event generator. So if you have experience with HEP event generators you could write your own simulation for desired range of energies while taking advantage of better understanding of Standard model incorporated in the software.

One should note, that for a time there was a theory, that for large Hawking temperatures, there would be a quasithermal 'photosphere' surrounding a black hole, so that particles emerging from it would experience bremsstrahlung and generate electron-positron pairs thus considerably lowering their effective temperature. However, it seems that this was based of some wrong assumptions, and so the vast majority of particles radiated by a black hole are noniteracting with one another or their decay products:

• MacGibbon, J. H., Carr, B. J., & Page, D. N. (2008). Do evaporating black holes form photospheres?. Physical Review D, 78(6), 064043, doi, arXiv.

What seriously could modify the presented above picture is some unknown physics. For example if supersymmetry or GUT become relevant at these energies, then there would be a much different composition of Hawking radiation.

Also, if dark matter is in fact composed of some sort of WIMP's with masses below $10\,\text{TeV}$ (or anyway below the corresponding Hawking temperature) than those would also be radiated. Depending of their exact nature and the number of degrees of freedom they carry they could seriously modify the overall power and composition of radiation.