1. Photon spectra from final stages of primordial black hole evaporation in different theoretical models
Edgar Bugaev, Peter Klimai, Valery Petkov
INR, Moscow, Russia

2. Primordial Black Holes
Main assumptions (B.Carr, 1975)
Initial overdensity in region of initial mass M is normally distributed about zero with standard deviation given by ,
where Mi is horizon mass at the moment of PBH formation, and is constant corresponding to the assumption of scale invariance of the primordial density fluctuations.
In this case, one has an extended mass spectrum of PBHs,
, ,
In radiation era, =4/3, =5/2.
In this case the mass spectrum has no exponential cut-off at large PBH masses, nevertheless, the density of PBHs larger than M falls of as M-1/2, and most of the PBH density is contained in the smallest PBHs.

3. ii) Experimental limits on
Approximate comoving number density of PBHs of mass M* is, in Carr’s scenario, .
M* is the mass of PBH having the age equal to the age of the Universe,
If t0=16 Gyr and = g3s-1, then M* = ( ) 1014g.
The fraction of energy density contained in PBHs is
If ~ 104 pc-3 (Page-Hawking limit, see below), then
Bolometric intensity (the energy received by detector per unit time per unit area, from all PBHs which have been shining since PBH formation time) is

4. It is known that the actual bolometric intensity of background radiation in EdS Universe is of order of
QEdS ~ erg s-1 cm-2.
So, for = 2.5 one has
< h0-1.
Spectral intensity of radiation from PBHs is given by formula ,
where is the PBH comoving density,
is the SED (spectral energy distribution) of Hawking radiation.
From comparison with the extragalactic photon background data one obtains (Page and Hawking, 1976)
~ 10-8 h0-1 , < (2-3) 104 h0 pc-3 .

5. Finally,
if Mi << M* , < 10-8 ;
if Mi > M* , < (cosmological limit) .
- fraction of energy density in PBHs in the moment of their formation (it depends on Mi).
For Mi < 1011g , ;
For Mi ~ M* , ;
For Mi > M* , (cosmological limit) .

6. It is evident that PBHs in the simplest formation scenario cannot be significant contributors to the dark matter.The limits obtained above can be weakened or evaded if PBH formation occurs in such a way as to produce fewer low-mass objects. A narrow spectrum of masses might be expected if PBH formed during spontaneous phase transition rather than arising from primordial fluctuations (the quark-hadron transition, grand unified symmetry breaking transition, Weinberg-Salam phase transition etc).
iii) Clustering
The rate of bursts from PBHs is, approximately, (F.Halzen et al, 1991) .
If ~ 104 h0 pc-3 , t0= yr, then
If an enhancement factor due to clustering is about 108 , then

7. It is stronger than the experimental limit from direct search using ground-based detectors (Alexandreas et al, 1993; Amenomori et al, 1995; Funk et al, 1995; Connaughton et al, 1998; Linton et al, 2006) which is
According to recent work (J.Chisholm, 2005) a local clustering enhancement can be as large as ,
where is the “threshold to sigma ratio”
is the threshold value of the density contrast needed for PBH formation (analytically, ) is the standard deviation at horizon crossing (which is, in the simplest scenario, constant, and is equal to in formulas above.
If =10 (e.g., if =1/3, ~ 0.03) , then

8. 2. Final stages of PBH evolution
i) Model of MacGibbon and Webber (MacGibbon & Webber, 1990)
In first calculations of photon spectra from BH evaporations it had been shown that this spectrum is far from being thermal, simply because emitted particles such as quarks fragment into hadrons, photons, neutrinos, etc, and just this fragmentation and decay of unstable hadrons produces the final photon spectrum. It was assumed that the radiation evaporated from the BH interacts too weakly and all emitted quarks propagate freely and fragment independently of each other.
The resulting time-integrated spectrum can be parameterized in the form (E0=105 GeV)
the average energy of time-integrated spectrum is
If Emin = 100 GeV,
The connection between initial Hawking temperature (in GeVs) and PBH lifetime (in seconds) is

9. ii) Heckler model (Heckler, 1997)
According to Heckler’s idea, once a black hole reaches some critical temperature, the emitted Hawking radiation begins to interact with itself and forms nearly thermal chromosphere. This is based on a simple argument: at relativistic energies, the cross section for gluon bremsstrahlung in a qq collision in approximately constant, while the density of emitted particles around BH grows with its temperature. The evaporation of BH creates a radiation shell which propagates outward at the speed of light. The quark and gluon spectrum in the observer rest frame is obtained by boosting a thermal spectrum at (and ) with the Lorenz factor of the outer surface of the thermal chromosphere : ,
where the integration over the surface of the chromosphere of radius has been carried out. The observed photon spectrum of the chromosphere is a convolution of the quark-gluon spectrum with the quark-pion fragmentation function and the Lorentz-boosted spectrum from neutral pion decay.

10. The high energy behavior of time-integrated spectrum is
The average energy of time-integrated spectrum is
iii) Kapusta & Daghigh model (Kapusta & Daghigh, 2001)
This approach differs from those of Heckler mainly in two respects. It is assumed that the hadronization of quarks occurs before the thermal freeze-out and the onset of free-streaming. It is assumed, further, that it happens suddenly at a temperatute Tf in the range 100−140 MeV (it is the parameter of the model), and to this moment all particles whose mass is greater than Tf have annihilated leaving only photons, electrons, muons and pions.

11. Photons emitted by BH come from two sources
Photons emitted by BH come from two sources. Either they are emitted directly from a (boosted) black body spectrum or they arise from the neutral pion decay. The Lorenz factor of the outer edge of the chromosphere and its radius are
The formula for particle spectrum from the chromosphere is analogous to Heckler’s. The high energy behavior of time-integrated photon flux is also similar, but the flux is approximately 1.5 orders of magnitude higher:
The average energy is

12. Instantaneous photon spectra from a black hole with Hawking temperature 10 TeV.
Time-integrated photon spectra from a black hole with initial Hawking temperature 10 TeV.

13. The average energy of photons emitted by a black hole, as a function of its lifetime shown on the figure is very sensitive to the evaporation model.
For model of MacGibbon and Webber, the minimum energy was taken to be 100 GeV.
It is seen that in chromosphere model of Kapusta the average photon energy is much higher than in Heckler’s model.

14. Kapusta
Kapusta
Photon flux from a black hole with different Hawking temperatures in model of Daghigh and Kapusta.
Left figure: instantaneous flux.
Right figure: time-integrated flux, curves are labeled by a BH initial temperature.

15. 3. Experimental search of PBH bursts with ground based detectors
One can see from the figures above that the average photon energy as a function of PBH lifetime is highly model dependent. Therefore, various methods should be used for the search. Up to now the search of high energy gamma radiation from PBH with ground – based arrays had been carried out using the predictions of the model of MacGibbon and Webber, and, specifically, data on distribution of events (extensive air showers (EAS)) in time and space were used. The results were reported in (Alexandreas et al, 1993; Amenomori et al, 1995; Funk et al, 1995; Connaughton et al, 1998; Linton et al, 2006), time intervals used were from 1 to 10 seconds.
If the chromosphere models of black hole evaporation are close to reality the average photon energy at last instants of PBH life is rather low. In the model of hadronic chomosphere (Daghigh & Kapusta, 2001) this energy is about 30 GeV for A search of PBH bursts in this case is possible using the method suggested in works of EAS-TOP collaboration (Aglietta et al, 1996) and Baksan group (Petkov et al, 2003; Smirnov et al, 2005). Shower particles generated in atmosphere by photons with energy ~ GeV are strongly absorbed before reaching the detector level, so, the average number of signals in the detector module is smaller than 1.

16. Therefore, in this energy range PBH bursts can be sought for by operating with the modules in single particle mode, that is, by measuring the single particle counting rate of the individual modules. The primary arrival directions of photons are not measured, and bursts can be detected only as transients (short-time increases) of the cosmic ray counting rate.
Using the photon spectra from PBH bursts shown above and registration efficiences of photons (as functions of zenith angle, the array altitude and features of concrete modules) one can calculate the average energies of PBH bursts for a given array and corresponding time durations of these bursts.

17. On upper figure the dependencies of average photon energy on zenith angle in model of Kapusta for two arrays of Baksan observatory are shown. Andyrchy and Carpet arrays are situated practically on the same place (horizontal distance between them is about 1 km), but their altitudes above sea level are different: Andyrchy m, Carpet m .
On lower figure, time duration of bursts as a function of zenith angle is given for the same arrays. One can see that time duration of burst strongly depends on the zenith angle.

18. 4. Conclusions
It is shown that predictions for high energy photon spectra from PBH burst depend rather strongly on the model used for a description of final stage of PBH evaporation. In particular, chromosphere models predict too steep spectra in TeV region.
It is argued that, if such models are correct, the practical search of PBH bursts can be carried out using the technique developed by EAS-TOP and Baksan groups for a search for gamma-bursts at energies E>10 GeV.
It is shown that in such search the average energy and time duration of the signal depend on the zenith angle of its arrival and on the altitude of the array above the sea level.