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Non-equilibrium universe and black hole evaporation
21 October 1999
Physics Letters B 465 Ž1999. 86–94
Non-equilibrium universe and black hole evaporation Iiro Vilja 1 Department of Physics, UniÕersity of Turku, FIN-20014 Turku, Finland Received 8 February 1999; received in revised form 30 August 1999; accepted 3 September 1999 Editor: P.V. Landshoff
Abstract The evaporation of the black holes during the very early universe is studied. Starting from black hole filled universe, the distributions of particle species are calculated and showed, that they differ remarkably from the corresponding equilibrium distribution. This may have great impact to the physics of the very early universe. Also the evolution of the universe during the evaporation has been studied. q 1999 Elsevier Science B.V. All rights reserved.
It is well known that in the very early universe the particles could not been equilibrated due to a too short equilibration time w1–3x. Indeed, the only possibility to have equilibrium distributions at times t ; 10y3 4 s corresponding equilibration temperature Teq ; 10 15 K is to assume, that there is some process producing particles readily to equilibrium distribution. In w2x the QCD interaction times has been explicitly studied, and the calculations show that virtually no QCD-interactions are present until the system begins to thermalise after time t ; 10y3 4 s. This means also, that no composite states of strongly interacting particles are formed before that time. ŽIt should be noted that in this case it is not really question of thermodynamic equilibrium because there are no interactions maintaining it.. Sometimes it is assumed that e.g. black hole evaporation could be such a process w3x. When the particles are originated from small black holes near Planck scale which
1
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evaporate due to Hawking process w4x, the radiation at any given time is thermal with temperature equal to Hawking temperature. Other investigated cosmological consequences of primordial black holes, as relics or source of the baryon asymmetry has been studied in w5,6x, where have been found strong constraints for the black hole spectrum. Also the dynamics of a black hole filled universe has been studied w8x. In the present paper we calculate the distribution of particles originated from black holes. We suppose that the black holes are generated at Planck time through quantum gravitation and after Planck time they begin to decay emitting particles. The production of black holes from quantum fluctuations are studied in Refs. w7x, where thermal bath of gravitons is assumed. Their analysis, which may produce very high black hole densities are, however, not applicable here, but gives a hint that remarkable black hole production is possible. In the present paper we show that the resulting particle distributions are not the equilibrium one, but something crucially different.
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 0 2 8 - X
I. Vilja r Physics Letters B 465 (1999) 86–94
This difference may play remarkable role e.g. in inflation w9x 2 , because the effective, ensemble corrected potential w3x differs now from the one calculated with an equilibrium thermal bath w11x. In the present study, for simplicity, we neglect the curvature effects, arising at Planck scale due to ambiguous concept of vacuum state w12x. By assuming isotropy, the co-moving observers have same vacuum and hence, whenever the average energy per particle is clearly smaller than Planck mass, the analysis is applicable. Consistently we omit also all direct couplings of quantum fields to curvature and treat the gravitational field classically. Let the evolution of the universe be given by the scale parameter RŽ t . and suppose that at Planck time no other matter exists in the universe than black holes. So the universe is at t i matter dominated, i.e. RŽ t . A t 2r3. One may wonder, if this is really true at initial time. However, it can be viewed as a reasonable approximation, because the radiation dilutes proportional to RŽ t .y4 whereas black holes only as RŽ t .y3 . So, unless the initial energy density of black holes were fine tuned to be extremely small, their energy density will soon after t Pl become dominant. Also, one could speculate with some more exotic objects filling the universe. In the present paper these are, however, not discussed. The ratio of the density of the universe to the critical density at that time is now given by V Ž t i . s 8p m i nŽ t i .rŽ3 M Pl2 H Ž t i . 2 ., where the Hubble parameter is H Ž t i . s 23 ty1 i . Thus we can parameterize the initial density with density ratio V i . As is well known, the dynamics of a black hole evaporation is governed by the equation m ˙ BH s y
h)
M Pl4
15 = 8 3p 2 m2BH
,
Ž 1.
where h ) is an effective number of degrees of freedom and m BH is the black hole mass. In Eq. Ž1. we have omitted the absorption of radiation back to the black holes. This is well motived whenever the radiation density Erad is small, i.e. Erad A BH < < m ˙ BH <,
Žwhere A BH is the black hole surface area.. The solution of Eq. Ž1. reads
ž
m BH Ž t . s m i 1 y
The existence of small black holes at early times would have also many other interesting consequences, see w10x.
t y ti
t
1r3
/
,
Ž 2.
where the black hole life-time t is given by
ts
5 = 8 3p 2 m3i M Pl4 h )
.
Ž 3.
We express the initial black hole mass in terms of the inverse of t mi s
M Pl 8
h)
1r3
ž /
,
5p 2d
Ž 4.
where d ' 1rŽ M Plt . has to be a small number < 1, i.e. the black hole life-time is much longer than Planck time 3. Using the expression for V i and Ž4. we can give the black hole number density n Ž ti . s
4 M Pl 3 t i2
5
ž / p h)
1r3
V i d 1r3 .
Ž 5.
We are now ready to calculate the distribution of the particles emitted by the black holes. We further simplify the situation by stating that all emitting particles are essentially massless, so that the effective particle number h ) remains constant during the evaporation 4 . Strictly speaking, this is not true, but gives a reasonable approximation of the situation, where no specific model with its particle contents is given. Also, we suppose that the black holes decay before the thermalisation starts. This means that no thermalising processes are effective during the evaporation and hence not taken into account in the calculation. Whether this is a good assumption or not is studied later in this paper. During a time w t, t q dt x Ž t ) t i . the energy emitted by a black hole and distributed to h ) particle species Ždegrees of freedom. is dE s ym ˙ BH Ž t . dt.
3
2
87
Indeed, one has to assume that t i qt 4 t Pl , because otherwise quantum gravitational effects would certainly not be omitted Žsee w8x and references therein.. 4 The massive case is as well calculable as the massless one, but leads to introducing of numerous new parameters, which is avoided here.
I. Vilja r Physics Letters B 465 (1999) 86–94
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Here we emphasize, that at times studied, all particles radiated by the black holes are elementary. Also after emission they remain elementary, because there has not been enough time to form composite states. The speculative possibility, that some more more exotic states would has been emitted is not considered here. Because at a given time the radiation emitted obeys thermal spectrum with temperature TBH s M Pl2 rŽ8p m BH . Ž b BH s 1rTBH ., the energy per a degree of freedom within momentum between p and p q dp is given by dE Ž p q dp . y dE Ž p . s
k " dE v Ž p . f " Ž p, b BH Ž t . . d 3 p h)
3
Hd k v Ž k . f
"
,
Ž 6.
Ž k , b BH Ž t . .
Combining Ž7. and Ž8. we write k "m ˙ BH Ž t . dt
df f Ž b. s yn Ž t .
=
h)
3 Ž 2p . f " Ž b BH Ž t . p . 3
Hd k v Ž k . f
"
t
Ht
k "m ˙ BH Ž tX .
dtX n Ž tX .
h)
i
dE Ž p q dp . y dE Ž p .
v Ž p. s
f " Ž b BH Ž t . p . d 3 p
k " dE h)
3
Hd k v Ž k . f
"
.
Ž 7.
Ž b BH Ž t . k .
Taking into account that the black hole density reads nŽ t . s nŽ t i .Ž RŽ t i .rRŽ t .. 3, one obtains the distribution df f Ž b. of a degree of freedom: df f Ž b. Ž p,t .
d3p
Ž 2p .
3
s nŽ t .
dE Ž p q dp . y dE Ž p .
v Ž p.
.
Ž 8.
Ž 9.
Eq. Ž9. represents the distribution generated during a time-interval w t, t q dt x. The overall distribution thus if we want to end up to at a given time t is obtained by summing all contribution from t i to t, i.e. all times tX , t i F tX F t. These contribution are, however, red-shifted by a factor RŽ tX .rRŽ t ., hence the final particle distribution at time t is given by f f Ž b. Ž p;t . s y
where the fraction k " dErh ) represents the Žtotal. energy per degree of freedom Ž k " is 1 for bosons and 78 for fermions appearing due to different statistics.. The other fraction expresses the part of the energy within the given momentum range and the functions f " are the usual Fermi-Dirac Žq. and Bose-Einstein Žy. distribution functions. Here FDand BE- distribution functions are introduced to take care of different spin statistics of radiating particles, as well as h ) counts the degrees of freedom. Thus, the particle number density Žat the given momentum range. is simply given by
.
Ž b BH Ž t . k .
3 Ž 2p . f "
=
ž
RŽ t . R Ž tX .
3
Hd k v Ž k . f
"
b BH Ž tX . p
/
.
Ž b BH Ž tX . k . Ž 10 .
The formula above, Eq. Ž10., describes in principle completely the particle distributions created by black hole evaporation. It can easily be generalized to the case, where the Žinitial. black holes are not equally massive, but have some distribution. In this case one just have to sum over all contributions emerging due to different black hole masses. To be able to calculate the distributions in practise, one has to solve the evolution of the scale parameter RŽ t .. For that purpose we have to determine first the time t EQ when the universe enters the radiation dominated era after the beginning of the black hole evaporation. We have to calculate when the energy density of the black holes, EBH , equals the energy density of radiation, Erad , they have emitted: EBH Ž t EQ . s Erad Ž t EQ .. Now, the Žaverage. energy density of the black holes at a given time t is given by EBH Ž t . s m BH Ž t . nŽ t i .Ž RŽ t i .rRŽ t .. 3. During a matter dominated time RŽ t . A t 2r3 and thus EBH Ž t . s m BH Ž t . n Ž t i .
ti
ž / t
2
,
t i - t - t EQ .
Ž 11 .
I. Vilja r Physics Letters B 465 (1999) 86–94
On the other hand the radiation energy density at t - t EQ is given by Erad Ž t . s Ý g i i
sgf
d3k
H Ž 2p .
q gb
RŽ t . s
Ž k ,t . v i Ž k .
3
f f Ž k ,t . v Ž k .
d3k
H Ž 2p .
3
f b Ž k ,t . v Ž k . ,
Ž 12 .
where we have summed over all relevant bosonic Ž g b . and fermionic Ž g f . degrees of freedom 5. Eq. Ž12. can be cast in the form t
X
Ht dt m˙
Erad s y
i
X
BH
Applying this equation to the Einstein equation, one obtains 3 2
ž
1r2
8p E Ž t i .
/
3 M Pl2
6t 7
1y
mŽ t .
7r2 2r3
ž /
,
mi
Ž 15 .
d3k
H Ž 2p .
f 3 i
89
Žt .
R Ž tX .
4
R Ž ti .
ž / ž / RŽ t .
n Ž ti .
R Ž tX .
3
,
Ž 13 . where ym ˙ BH Ž tX .Ž RŽ tX .rRŽ t .. 4 corresponds the energy radiated at tX by black holes and red-shifted until time t, whereas nŽ t i .Ž RŽ t i .rRŽ tX .. 3 represents the number density of decaying black holes at the time when the radiation was produced. Evidently, Eq. Ž12. can be obtained directly without using any knowledge about particle distributions. Indeed, the connection between Eqs. Ž12. and Ž13. requires the identification h ) s g ) , where g ) is the usual effective number of degrees of freedom g ) s g b q 78 g f . In the rest of the paper we apply this identification. It should be noted here, that the correct evolution of the cosmic scale parameter at matter dominated era is not really as simple as RŽ t . A t 2r3 but more complicated. This is due to the fact that the correct energy density of matter, i.e. black holes at that time reads 3
EBH Ž t . s m Ž t . n Ž t i . Ž R Ž t i . rR Ž t . . ,
Ž 14 .
which is not as simple as just dilution of the matter.
which, however, is in quite good agreement with the usual matter dominated form. Moreover, the calculation of the scale parameter at the radiation dominated era leads to a complicated integro-differential equation. Therefore we choose here to use the conventional forms of the evolution parameter R, which anyway gives good impression of the physics involved. It is now simple matter to take the expression for EBH and Erad from Eqs. Ž11. and Ž13. Žand relation RŽ t . A t 2r3 . and equate them. One obtains a rather simply equation t EQ
2r3 m BH Ž t EQ . t EQ sy
Ht
dt t 2r3 m ˙ BH Ž t . .
Ž 16 .
i
It is noteworthy, that Eq. Ž16. does not depend on the initial black hole number density nŽ t i . but only on initial time and black hole mass. According to the approximated analysis, the radiation dominance begins quite late compared to the black hole life-time. When t irt varies from 0 to 0.5, the combination Ž t EQ y t i .rt decreases from 0.928 to 0.908. In any case over 90% of the black hole life-time belongs to the matter dominated era. This is a consequence of the fact that the soft radiation produced at the early phase of the black hole life-time further red-shifts during the expansion of the universe. Using the expression Ž15. for cosmic scale parameter, one obtains Ž t EQ y t i .rt s 0.924 which is in good agreement with the approximated result. Therefore, it is reasonable to use the simpler equations. The scale factor is now simply given by equation R Ž t . s u Ž t y t EQ .
1r2
t
ž / ž /
qu Ž t EQ y t .
t EQ
t
t EQ
2r3
R Ž ti .
t EQ
ž / ti
2r3
,
Ž 17 . 5
Here we remind that, for sake of simplicity, we have assumed that all relevant particles are essentially massless, thus v Ž k . s k.
where u is the usual Heaviside-function. Eq. Ž17. can be inserted to formulas like Ž10. and Ž13.. In
I. Vilja r Physics Letters B 465 (1999) 86–94
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particular we are now interested in the energy density of the universe at the end of the black hole evaporation era: the energy density at that time determines the later evolution of the universe. We obtain the energy density of radiation at a given time t from Eq. Ž13. R Ž ti .
3
t
R Ž tX .
ž / H ž /ž / ž / H ž/ ž / ž /
Erad Ž t . s y
RŽ tf .
2
ti
sy
t EQ
=
t
ti
n Ž ti .
t EQ
n Ž ti .
X
X
dt m ˙ BH Ž t .
3r2
tf
X
ti
X
RŽ tf .
X
t EQ tf
1r2
4
RŽ t .
dt m ˙ BH Ž t . u Ž t y t EQ .
qu Ž t EQ y tX .
RŽ tf .
tX
t EQ
tX
1r2
tf
2r3
,
Ž 18 . where t i - t - t f s f i q t . For times t ) t f the energy density is only red-shifted as usual until it will thermalise. We define a pseudo-temperature T˜ to describe the energy density and particle distribution by setting simply
p2 30
4
g ) T˜ Ž t . s E Ž t . .
Ž 19 .
Now, T˜ is by no means a real temperature, simply because there exists no thermal equilibrium. However, it describes the energy density in familiar way and, moreover, when the thermal equilibrium is finally maintained, T˜ coincides with the true temperature. Therefore, at the time t f it is enlightening to compare the actual particle distribution to the ther˜ In Fig. 1a we have mal one at pseudo-temperature T. displayed the pseudo-temperature T˜ at t f as a function of t irt Žusing t i s t Pl .. An interesting feature emerges in the behaviour of T˜ Ži.e. energy density.. For t irt - z c ' 0.334 the
pseudo-temperature has a minimum during the black hole evaporation era. In Fig. 1b we have plotted the evolution of two pseudo-temperatures, one with t irt s 0.1 - z c Žlower curve. and another with t irt s 0.35 ) z c Župper curve.. We have scaled the time so, that T˜ is presented as function of Ž t y t i .rt . Note, that the large t irt values correspond small black holes with evaporation times comparable to t i . Thus, if t , t i , the approximations are not very good and. Practically in all relevant cases the pseudo-temperature has a minimum. In this stage we have to make a notion about the homogeneity of the matter and radiation in the universe. All calculations done as far, and all calculations to be done, assume that the universe is homogeneous at large scales. We can consider the matter to be homogeneously distributed if there are several black holes within a single horizon volume. On the other hand the radiation can be viewed to be roughly homogeneous, if the distance between the radiation sources, i.e. black holes is shorter than the length propagated by radiation. These two conditions are essentially equivalent leading to n Ž t . s nŽ t i .1r3 RŽ t i .rRŽ t . ) ty1 . In particular, this requirement at transition time t EQ means, that d 0.14Ž V i . 3r2 Ž160rh ) .1r2 where we have used t EQ s 0.924t . So, if V i s 1 we obtain d - 0.14, which in terms of black hole mass reads m B H ) 0.36Ž h ) r160.1r2Vy1r2 M Pl . i One can also ask, when the last emitted radiation Žat t s t f . is homogenized. Spatially homogeneous distribution is reached at time t h Žassuming the radiation domination after t f . if nŽ t f .y1 r3 RŽ t h .r RŽ t f . - t h y t f . Performing a numerical calculation it shows up Žwith V i s 1., that Ž t h y t f .rt varies from 0 to 3 as d increases from 0 to 0.5. Thus the homogenization time is of the same order of magnitude as t f itself or even shorter. Keeping in mind all above, we are now finally ready to calculate the distributions themselves. Applying Eq. Ž1. and formulae for black hole density and temperature to the Eq. Ž10., one obtains the
˜ Pl at t s t f as a function of the ratio t irt . Ž t i s t Pl , V i s 1.. Žb. Evolution of the pseudo-temperaFig. 1. Ža. The pseudo-temperature TrM ˜ Pl as a function of Ž t y t i .rt . The upper curve corresponds t irt s 0.35 whereas the lower one t irt s 0.1. Ž V i s 1.. ture TrM
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I. Vilja r Physics Letters B 465 (1999) 86–94
92
particle distributions. The distributions can be written as f f Ž b. Ž p;t . s
16 n i M Pl4
t
Ht
dtX m BH Ž tX .
i
=f "
RŽ t .
ž
R Ž tX .
2
R Ž ti .
3
ž / / R Ž tX .
b BH Ž tX . p .
Ž 20 .
Here we have used Eq. Ž1. as well integrated out the momentum from Eq. Ž10.. This integration produces 4 a factor Ty4 ˙ BH BH A m BH to the integrand, whereas m 2 A m BH , and hence the integrand is proportional to m2BH . In Fig. 2a we have displayed the distribution at t f for bosons multiplied by p 2 as a function of the momentum p together with a reference distribution. The reference distribution is the equilibrium distribution with T s T˜Ž t f .. All are given in Plank units with V i s 1. The shape of distribution given in Fig. 2a is generic and it is noteworthy that the black hole originated distribution has more power in large momenta. On the other hand, the IR end of the distribution, p ; 0, there is less power. Indeed, the IR behaviour of reads f b ; krp like for equilibrium Žwhere fy; Trp ., but
ks
2 n i R Ž ti .
p M Pl2 R Ž t f .
tf
Ht
i
dt m BH Ž t .
R Ž ti .
ž / RŽ t .
2
- T˜ Ž t f .
Ž 21 . for all relevant d : the ratio T˜Ž t f .rk is for all d - 0.5 larger than ; 5.6. This may be important, because the IR effects are essential in many physical considerations, in particular when some kind of effective potential andror interactions are used. The fermionic distribution function is presented in Fig. 2b. As for bosonic case we have presented for comparison the equilibrium distribution at T˜Ž t f ., too. The shape of the distribution in Fig. 2b is generic, and again there is more power at large p in f f than in equilibrium distribution. In general, the distribu-
tions arising from black hole evaporation are flatter and lower than corresponding equilibrium distributions. This can be understood, because at late times t ; t f the black hole radiates particles with large momenta. The particle density, n bŽ f . , is thus always lower than at equilibrium with same energy density, EQ n EQ bŽ f . . The ratio n bŽ f .rn bŽ f . - 0.36 for all d - 0.5 1r6 and is proportional to d when d ™ 0. Finally the question, how small the parameter d can be, i.e. how large the black holes can be. If one supposes that the black holes evaporate before the thermalisation takes place due to GUT processes w1x, strong interactions w2x or electroweak interactions w3x, one has to require that t f < t th s 1rG , where G is the Žaverage. thermalisation rate of the particles in˜ where inspired by GUT’s a ; 3 volved. If G ; a T, = 10y2 , we conclude that drT˜4 a 2 or d 4 3 = 10y8 . This estimate has many uncertainties e.g. because as dimensional parameter the pseudo-temperature T˜ has been used. Nevertheless, this approximation gives a general idea about the time-scales involved. On the other hand, one should address some interest to the question, when the masses are negligible. Concerning the tree-level masses, general remarks are impossible without specifying a particular model. However, in the case of the ensemble masses Ž‘‘thermal masses’’. some remarks can be ˜ where g is a done. The ensemble mass m e ; gT, generic coupling, should be small compared to the black hole temperature T BH , i.e. m e < TBH . This can be rewritten as d 1r3rT˜ 4 gp Ž h ) r5p 2 .1r3 My1 Pl , where T BH s TBH Ž t i . has been used because it corresponds the lowest black hole temperature giving a conservative limit. Now the left hand side of this equation is always larger than 7.9 Žat d s 0.5.. Thus Žfor h ) s 160. the ensemble mass can be ignored, if the coupling g < 1.7! In the present, paper a general study of the particle distributions inspired by the black hole evaporation in the Õery early uniÕerse is presented. It is found that the distribution produced differ clearly from the equilibrium distributions. The distribution
Fig. 2. Ža. Boson distribution function p 2 f b Ž p;t f .rM Pl2 as a function of momentum p Žsolid line.. For reference, also the equilibrium distribution at the temperature T˜Ž t f . has been presented Ždashed line.. d s 0.1. Žb. Corresponding fermion distribution function p 2 f f Ž p;t f .rM Pl2 Žsolid line. with equilibrium reference curve at T˜Ž t f . Ždashed line.. d s 0.1.
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I. Vilja r Physics Letters B 465 (1999) 86–94
has much more power at large momenta and the boson distribution has slower increase when p ™ 0 q . The conditions of validity of the analysis have been studied and found that under wide range of parameters calculation can be performed with good reliability. Also the evolution of the universe has been studied during the black hole evaporation era. In particular, the time when it makes transition from matter domination to radiation domination and the generic behaviour of radiation density are calculated. It should be noted, however, that the precise characterization of the physics, including particle distributions, would require information of the model of elementary particles at the energies in hand. This information is, unfortunately, not available and therefore in this paper we have used a typical GUT degrees of freedom to illustrate the possible physics. These considerations may have great impact to the calculations of the very early universe. In particular an early inflation may depend crucially on the particle distributions at the non-thermalised era of the early universe due to the ensemble corrected quantities. However, more detailed studies are definitely needed, when any particular model is considered.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x
w11x w12x
J. Ellis, G. Steigman, Phys. Lett. B 89 Ž1980. 186. K. Enqvist, J. Sirkka, Phys. Lett. B 314 Ž1993. 298. K. Enqvist, P. Elmfors, I. Vilja, Phys. Lett. B 326 Ž1994. 37. S. Hawking, Nature 248 Ž1974. 30; Commun. Math. Phys. 43 Ž1975. 199. J.D. Barrow, Mon. Not. R. Astr. Soc. 192 Ž1980. 427; D 46 Ž1992. 645. J.D. Barrow, E.J. Copeland, E.W. Kolb, A.R. Liddle, Phys. Rev. D 43 Ž1991. 984. Gross, Perry, Yaffe, Phys. Rev. D 25 Ž1982. 330; J. Kapusta, Phys. Rev. D 30 Ž1984. 831. J.D. Barrow, E.J. Copeland, A.R. Liddle, Mon. Not. R. Astr. Soc. 253 Ž1991. 675. For review, see: E. Kolb, M. Turner, The Early Universe, Addison-Wesley, 1990. B.J. Carr, in: J.L. Sanz, L.J. Giocoechea ŽEds.., Observatonal and theoretical aspects of relativistic astrophysics and cosmology, World Scientific, Singapore, 1985; Ap. J. 201 Ž1975. 1; 206 Ž1976. 8. A. Riotto, I. Vilja, Mod. Phys. Lett. A 11 Ž1996. 1151. See, e.g.: N.D. Birrel, P.C.W. Davies, Quantum Fields in Curved space, Cambridge University Press, 1982.
Physics Letters B 465 Ž1999. 86–94
Non-equilibrium universe and black hole evaporation Iiro Vilja 1 Department of Physics, UniÕersity of Turku, FIN-20014 Turku, Finland Received 8 February 1999; received in revised form 30 August 1999; accepted 3 September 1999 Editor: P.V. Landshoff
Abstract The evaporation of the black holes during the very early universe is studied. Starting from black hole filled universe, the distributions of particle species are calculated and showed, that they differ remarkably from the corresponding equilibrium distribution. This may have great impact to the physics of the very early universe. Also the evolution of the universe during the evaporation has been studied. q 1999 Elsevier Science B.V. All rights reserved.
It is well known that in the very early universe the particles could not been equilibrated due to a too short equilibration time w1–3x. Indeed, the only possibility to have equilibrium distributions at times t ; 10y3 4 s corresponding equilibration temperature Teq ; 10 15 K is to assume, that there is some process producing particles readily to equilibrium distribution. In w2x the QCD interaction times has been explicitly studied, and the calculations show that virtually no QCD-interactions are present until the system begins to thermalise after time t ; 10y3 4 s. This means also, that no composite states of strongly interacting particles are formed before that time. ŽIt should be noted that in this case it is not really question of thermodynamic equilibrium because there are no interactions maintaining it.. Sometimes it is assumed that e.g. black hole evaporation could be such a process w3x. When the particles are originated from small black holes near Planck scale which
1
E-mail: [email protected]
evaporate due to Hawking process w4x, the radiation at any given time is thermal with temperature equal to Hawking temperature. Other investigated cosmological consequences of primordial black holes, as relics or source of the baryon asymmetry has been studied in w5,6x, where have been found strong constraints for the black hole spectrum. Also the dynamics of a black hole filled universe has been studied w8x. In the present paper we calculate the distribution of particles originated from black holes. We suppose that the black holes are generated at Planck time through quantum gravitation and after Planck time they begin to decay emitting particles. The production of black holes from quantum fluctuations are studied in Refs. w7x, where thermal bath of gravitons is assumed. Their analysis, which may produce very high black hole densities are, however, not applicable here, but gives a hint that remarkable black hole production is possible. In the present paper we show that the resulting particle distributions are not the equilibrium one, but something crucially different.
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 0 2 8 - X
I. Vilja r Physics Letters B 465 (1999) 86–94
This difference may play remarkable role e.g. in inflation w9x 2 , because the effective, ensemble corrected potential w3x differs now from the one calculated with an equilibrium thermal bath w11x. In the present study, for simplicity, we neglect the curvature effects, arising at Planck scale due to ambiguous concept of vacuum state w12x. By assuming isotropy, the co-moving observers have same vacuum and hence, whenever the average energy per particle is clearly smaller than Planck mass, the analysis is applicable. Consistently we omit also all direct couplings of quantum fields to curvature and treat the gravitational field classically. Let the evolution of the universe be given by the scale parameter RŽ t . and suppose that at Planck time no other matter exists in the universe than black holes. So the universe is at t i matter dominated, i.e. RŽ t . A t 2r3. One may wonder, if this is really true at initial time. However, it can be viewed as a reasonable approximation, because the radiation dilutes proportional to RŽ t .y4 whereas black holes only as RŽ t .y3 . So, unless the initial energy density of black holes were fine tuned to be extremely small, their energy density will soon after t Pl become dominant. Also, one could speculate with some more exotic objects filling the universe. In the present paper these are, however, not discussed. The ratio of the density of the universe to the critical density at that time is now given by V Ž t i . s 8p m i nŽ t i .rŽ3 M Pl2 H Ž t i . 2 ., where the Hubble parameter is H Ž t i . s 23 ty1 i . Thus we can parameterize the initial density with density ratio V i . As is well known, the dynamics of a black hole evaporation is governed by the equation m ˙ BH s y
h)
M Pl4
15 = 8 3p 2 m2BH
,
Ž 1.
where h ) is an effective number of degrees of freedom and m BH is the black hole mass. In Eq. Ž1. we have omitted the absorption of radiation back to the black holes. This is well motived whenever the radiation density Erad is small, i.e. Erad A BH < < m ˙ BH <,
Žwhere A BH is the black hole surface area.. The solution of Eq. Ž1. reads
ž
m BH Ž t . s m i 1 y
The existence of small black holes at early times would have also many other interesting consequences, see w10x.
t y ti
t
1r3
/
,
Ž 2.
where the black hole life-time t is given by
ts
5 = 8 3p 2 m3i M Pl4 h )
.
Ž 3.
We express the initial black hole mass in terms of the inverse of t mi s
M Pl 8
h)
1r3
ž /
,
5p 2d
Ž 4.
where d ' 1rŽ M Plt . has to be a small number < 1, i.e. the black hole life-time is much longer than Planck time 3. Using the expression for V i and Ž4. we can give the black hole number density n Ž ti . s
4 M Pl 3 t i2
5
ž / p h)
1r3
V i d 1r3 .
Ž 5.
We are now ready to calculate the distribution of the particles emitted by the black holes. We further simplify the situation by stating that all emitting particles are essentially massless, so that the effective particle number h ) remains constant during the evaporation 4 . Strictly speaking, this is not true, but gives a reasonable approximation of the situation, where no specific model with its particle contents is given. Also, we suppose that the black holes decay before the thermalisation starts. This means that no thermalising processes are effective during the evaporation and hence not taken into account in the calculation. Whether this is a good assumption or not is studied later in this paper. During a time w t, t q dt x Ž t ) t i . the energy emitted by a black hole and distributed to h ) particle species Ždegrees of freedom. is dE s ym ˙ BH Ž t . dt.
3
2
87
Indeed, one has to assume that t i qt 4 t Pl , because otherwise quantum gravitational effects would certainly not be omitted Žsee w8x and references therein.. 4 The massive case is as well calculable as the massless one, but leads to introducing of numerous new parameters, which is avoided here.
I. Vilja r Physics Letters B 465 (1999) 86–94
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Here we emphasize, that at times studied, all particles radiated by the black holes are elementary. Also after emission they remain elementary, because there has not been enough time to form composite states. The speculative possibility, that some more more exotic states would has been emitted is not considered here. Because at a given time the radiation emitted obeys thermal spectrum with temperature TBH s M Pl2 rŽ8p m BH . Ž b BH s 1rTBH ., the energy per a degree of freedom within momentum between p and p q dp is given by dE Ž p q dp . y dE Ž p . s
k " dE v Ž p . f " Ž p, b BH Ž t . . d 3 p h)
3
Hd k v Ž k . f
"
,
Ž 6.
Ž k , b BH Ž t . .
Combining Ž7. and Ž8. we write k "m ˙ BH Ž t . dt
df f Ž b. s yn Ž t .
=
h)
3 Ž 2p . f " Ž b BH Ž t . p . 3
Hd k v Ž k . f
"
t
Ht
k "m ˙ BH Ž tX .
dtX n Ž tX .
h)
i
dE Ž p q dp . y dE Ž p .
v Ž p. s
f " Ž b BH Ž t . p . d 3 p
k " dE h)
3
Hd k v Ž k . f
"
.
Ž 7.
Ž b BH Ž t . k .
Taking into account that the black hole density reads nŽ t . s nŽ t i .Ž RŽ t i .rRŽ t .. 3, one obtains the distribution df f Ž b. of a degree of freedom: df f Ž b. Ž p,t .
d3p
Ž 2p .
3
s nŽ t .
dE Ž p q dp . y dE Ž p .
v Ž p.
.
Ž 8.
Ž 9.
Eq. Ž9. represents the distribution generated during a time-interval w t, t q dt x. The overall distribution thus if we want to end up to at a given time t is obtained by summing all contribution from t i to t, i.e. all times tX , t i F tX F t. These contribution are, however, red-shifted by a factor RŽ tX .rRŽ t ., hence the final particle distribution at time t is given by f f Ž b. Ž p;t . s y
where the fraction k " dErh ) represents the Žtotal. energy per degree of freedom Ž k " is 1 for bosons and 78 for fermions appearing due to different statistics.. The other fraction expresses the part of the energy within the given momentum range and the functions f " are the usual Fermi-Dirac Žq. and Bose-Einstein Žy. distribution functions. Here FDand BE- distribution functions are introduced to take care of different spin statistics of radiating particles, as well as h ) counts the degrees of freedom. Thus, the particle number density Žat the given momentum range. is simply given by
.
Ž b BH Ž t . k .
3 Ž 2p . f "
=
ž
RŽ t . R Ž tX .
3
Hd k v Ž k . f
"
b BH Ž tX . p
/
.
Ž b BH Ž tX . k . Ž 10 .
The formula above, Eq. Ž10., describes in principle completely the particle distributions created by black hole evaporation. It can easily be generalized to the case, where the Žinitial. black holes are not equally massive, but have some distribution. In this case one just have to sum over all contributions emerging due to different black hole masses. To be able to calculate the distributions in practise, one has to solve the evolution of the scale parameter RŽ t .. For that purpose we have to determine first the time t EQ when the universe enters the radiation dominated era after the beginning of the black hole evaporation. We have to calculate when the energy density of the black holes, EBH , equals the energy density of radiation, Erad , they have emitted: EBH Ž t EQ . s Erad Ž t EQ .. Now, the Žaverage. energy density of the black holes at a given time t is given by EBH Ž t . s m BH Ž t . nŽ t i .Ž RŽ t i .rRŽ t .. 3. During a matter dominated time RŽ t . A t 2r3 and thus EBH Ž t . s m BH Ž t . n Ž t i .
ti
ž / t
2
,
t i - t - t EQ .
Ž 11 .
I. Vilja r Physics Letters B 465 (1999) 86–94
On the other hand the radiation energy density at t - t EQ is given by Erad Ž t . s Ý g i i
sgf
d3k
H Ž 2p .
q gb
RŽ t . s
Ž k ,t . v i Ž k .
3
f f Ž k ,t . v Ž k .
d3k
H Ž 2p .
3
f b Ž k ,t . v Ž k . ,
Ž 12 .
where we have summed over all relevant bosonic Ž g b . and fermionic Ž g f . degrees of freedom 5. Eq. Ž12. can be cast in the form t
X
Ht dt m˙
Erad s y
i
X
BH
Applying this equation to the Einstein equation, one obtains 3 2
ž
1r2
8p E Ž t i .
/
3 M Pl2
6t 7
1y
mŽ t .
7r2 2r3
ž /
,
mi
Ž 15 .
d3k
H Ž 2p .
f 3 i
89
Žt .
R Ž tX .
4
R Ž ti .
ž / ž / RŽ t .
n Ž ti .
R Ž tX .
3
,
Ž 13 . where ym ˙ BH Ž tX .Ž RŽ tX .rRŽ t .. 4 corresponds the energy radiated at tX by black holes and red-shifted until time t, whereas nŽ t i .Ž RŽ t i .rRŽ tX .. 3 represents the number density of decaying black holes at the time when the radiation was produced. Evidently, Eq. Ž12. can be obtained directly without using any knowledge about particle distributions. Indeed, the connection between Eqs. Ž12. and Ž13. requires the identification h ) s g ) , where g ) is the usual effective number of degrees of freedom g ) s g b q 78 g f . In the rest of the paper we apply this identification. It should be noted here, that the correct evolution of the cosmic scale parameter at matter dominated era is not really as simple as RŽ t . A t 2r3 but more complicated. This is due to the fact that the correct energy density of matter, i.e. black holes at that time reads 3
EBH Ž t . s m Ž t . n Ž t i . Ž R Ž t i . rR Ž t . . ,
Ž 14 .
which is not as simple as just dilution of the matter.
which, however, is in quite good agreement with the usual matter dominated form. Moreover, the calculation of the scale parameter at the radiation dominated era leads to a complicated integro-differential equation. Therefore we choose here to use the conventional forms of the evolution parameter R, which anyway gives good impression of the physics involved. It is now simple matter to take the expression for EBH and Erad from Eqs. Ž11. and Ž13. Žand relation RŽ t . A t 2r3 . and equate them. One obtains a rather simply equation t EQ
2r3 m BH Ž t EQ . t EQ sy
Ht
dt t 2r3 m ˙ BH Ž t . .
Ž 16 .
i
It is noteworthy, that Eq. Ž16. does not depend on the initial black hole number density nŽ t i . but only on initial time and black hole mass. According to the approximated analysis, the radiation dominance begins quite late compared to the black hole life-time. When t irt varies from 0 to 0.5, the combination Ž t EQ y t i .rt decreases from 0.928 to 0.908. In any case over 90% of the black hole life-time belongs to the matter dominated era. This is a consequence of the fact that the soft radiation produced at the early phase of the black hole life-time further red-shifts during the expansion of the universe. Using the expression Ž15. for cosmic scale parameter, one obtains Ž t EQ y t i .rt s 0.924 which is in good agreement with the approximated result. Therefore, it is reasonable to use the simpler equations. The scale factor is now simply given by equation R Ž t . s u Ž t y t EQ .
1r2
t
ž / ž /
qu Ž t EQ y t .
t EQ
t
t EQ
2r3
R Ž ti .
t EQ
ž / ti
2r3
,
Ž 17 . 5
Here we remind that, for sake of simplicity, we have assumed that all relevant particles are essentially massless, thus v Ž k . s k.
where u is the usual Heaviside-function. Eq. Ž17. can be inserted to formulas like Ž10. and Ž13.. In
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particular we are now interested in the energy density of the universe at the end of the black hole evaporation era: the energy density at that time determines the later evolution of the universe. We obtain the energy density of radiation at a given time t from Eq. Ž13. R Ž ti .
3
t
R Ž tX .
ž / H ž /ž / ž / H ž/ ž / ž /
Erad Ž t . s y
RŽ tf .
2
ti
sy
t EQ
=
t
ti
n Ž ti .
t EQ
n Ž ti .
X
X
dt m ˙ BH Ž t .
3r2
tf
X
ti
X
RŽ tf .
X
t EQ tf
1r2
4
RŽ t .
dt m ˙ BH Ž t . u Ž t y t EQ .
qu Ž t EQ y tX .
RŽ tf .
tX
t EQ
tX
1r2
tf
2r3
,
Ž 18 . where t i - t - t f s f i q t . For times t ) t f the energy density is only red-shifted as usual until it will thermalise. We define a pseudo-temperature T˜ to describe the energy density and particle distribution by setting simply
p2 30
4
g ) T˜ Ž t . s E Ž t . .
Ž 19 .
Now, T˜ is by no means a real temperature, simply because there exists no thermal equilibrium. However, it describes the energy density in familiar way and, moreover, when the thermal equilibrium is finally maintained, T˜ coincides with the true temperature. Therefore, at the time t f it is enlightening to compare the actual particle distribution to the ther˜ In Fig. 1a we have mal one at pseudo-temperature T. displayed the pseudo-temperature T˜ at t f as a function of t irt Žusing t i s t Pl .. An interesting feature emerges in the behaviour of T˜ Ži.e. energy density.. For t irt - z c ' 0.334 the
pseudo-temperature has a minimum during the black hole evaporation era. In Fig. 1b we have plotted the evolution of two pseudo-temperatures, one with t irt s 0.1 - z c Žlower curve. and another with t irt s 0.35 ) z c Župper curve.. We have scaled the time so, that T˜ is presented as function of Ž t y t i .rt . Note, that the large t irt values correspond small black holes with evaporation times comparable to t i . Thus, if t , t i , the approximations are not very good and. Practically in all relevant cases the pseudo-temperature has a minimum. In this stage we have to make a notion about the homogeneity of the matter and radiation in the universe. All calculations done as far, and all calculations to be done, assume that the universe is homogeneous at large scales. We can consider the matter to be homogeneously distributed if there are several black holes within a single horizon volume. On the other hand the radiation can be viewed to be roughly homogeneous, if the distance between the radiation sources, i.e. black holes is shorter than the length propagated by radiation. These two conditions are essentially equivalent leading to n Ž t . s nŽ t i .1r3 RŽ t i .rRŽ t . ) ty1 . In particular, this requirement at transition time t EQ means, that d 0.14Ž V i . 3r2 Ž160rh ) .1r2 where we have used t EQ s 0.924t . So, if V i s 1 we obtain d - 0.14, which in terms of black hole mass reads m B H ) 0.36Ž h ) r160.1r2Vy1r2 M Pl . i One can also ask, when the last emitted radiation Žat t s t f . is homogenized. Spatially homogeneous distribution is reached at time t h Žassuming the radiation domination after t f . if nŽ t f .y1 r3 RŽ t h .r RŽ t f . - t h y t f . Performing a numerical calculation it shows up Žwith V i s 1., that Ž t h y t f .rt varies from 0 to 3 as d increases from 0 to 0.5. Thus the homogenization time is of the same order of magnitude as t f itself or even shorter. Keeping in mind all above, we are now finally ready to calculate the distributions themselves. Applying Eq. Ž1. and formulae for black hole density and temperature to the Eq. Ž10., one obtains the
˜ Pl at t s t f as a function of the ratio t irt . Ž t i s t Pl , V i s 1.. Žb. Evolution of the pseudo-temperaFig. 1. Ža. The pseudo-temperature TrM ˜ Pl as a function of Ž t y t i .rt . The upper curve corresponds t irt s 0.35 whereas the lower one t irt s 0.1. Ž V i s 1.. ture TrM
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particle distributions. The distributions can be written as f f Ž b. Ž p;t . s
16 n i M Pl4
t
Ht
dtX m BH Ž tX .
i
=f "
RŽ t .
ž
R Ž tX .
2
R Ž ti .
3
ž / / R Ž tX .
b BH Ž tX . p .
Ž 20 .
Here we have used Eq. Ž1. as well integrated out the momentum from Eq. Ž10.. This integration produces 4 a factor Ty4 ˙ BH BH A m BH to the integrand, whereas m 2 A m BH , and hence the integrand is proportional to m2BH . In Fig. 2a we have displayed the distribution at t f for bosons multiplied by p 2 as a function of the momentum p together with a reference distribution. The reference distribution is the equilibrium distribution with T s T˜Ž t f .. All are given in Plank units with V i s 1. The shape of distribution given in Fig. 2a is generic and it is noteworthy that the black hole originated distribution has more power in large momenta. On the other hand, the IR end of the distribution, p ; 0, there is less power. Indeed, the IR behaviour of reads f b ; krp like for equilibrium Žwhere fy; Trp ., but
ks
2 n i R Ž ti .
p M Pl2 R Ž t f .
tf
Ht
i
dt m BH Ž t .
R Ž ti .
ž / RŽ t .
2
- T˜ Ž t f .
Ž 21 . for all relevant d : the ratio T˜Ž t f .rk is for all d - 0.5 larger than ; 5.6. This may be important, because the IR effects are essential in many physical considerations, in particular when some kind of effective potential andror interactions are used. The fermionic distribution function is presented in Fig. 2b. As for bosonic case we have presented for comparison the equilibrium distribution at T˜Ž t f ., too. The shape of the distribution in Fig. 2b is generic, and again there is more power at large p in f f than in equilibrium distribution. In general, the distribu-
tions arising from black hole evaporation are flatter and lower than corresponding equilibrium distributions. This can be understood, because at late times t ; t f the black hole radiates particles with large momenta. The particle density, n bŽ f . , is thus always lower than at equilibrium with same energy density, EQ n EQ bŽ f . . The ratio n bŽ f .rn bŽ f . - 0.36 for all d - 0.5 1r6 and is proportional to d when d ™ 0. Finally the question, how small the parameter d can be, i.e. how large the black holes can be. If one supposes that the black holes evaporate before the thermalisation takes place due to GUT processes w1x, strong interactions w2x or electroweak interactions w3x, one has to require that t f < t th s 1rG , where G is the Žaverage. thermalisation rate of the particles in˜ where inspired by GUT’s a ; 3 volved. If G ; a T, = 10y2 , we conclude that drT˜4 a 2 or d 4 3 = 10y8 . This estimate has many uncertainties e.g. because as dimensional parameter the pseudo-temperature T˜ has been used. Nevertheless, this approximation gives a general idea about the time-scales involved. On the other hand, one should address some interest to the question, when the masses are negligible. Concerning the tree-level masses, general remarks are impossible without specifying a particular model. However, in the case of the ensemble masses Ž‘‘thermal masses’’. some remarks can be ˜ where g is a done. The ensemble mass m e ; gT, generic coupling, should be small compared to the black hole temperature T BH , i.e. m e < TBH . This can be rewritten as d 1r3rT˜ 4 gp Ž h ) r5p 2 .1r3 My1 Pl , where T BH s TBH Ž t i . has been used because it corresponds the lowest black hole temperature giving a conservative limit. Now the left hand side of this equation is always larger than 7.9 Žat d s 0.5.. Thus Žfor h ) s 160. the ensemble mass can be ignored, if the coupling g < 1.7! In the present, paper a general study of the particle distributions inspired by the black hole evaporation in the Õery early uniÕerse is presented. It is found that the distribution produced differ clearly from the equilibrium distributions. The distribution
Fig. 2. Ža. Boson distribution function p 2 f b Ž p;t f .rM Pl2 as a function of momentum p Žsolid line.. For reference, also the equilibrium distribution at the temperature T˜Ž t f . has been presented Ždashed line.. d s 0.1. Žb. Corresponding fermion distribution function p 2 f f Ž p;t f .rM Pl2 Žsolid line. with equilibrium reference curve at T˜Ž t f . Ždashed line.. d s 0.1.
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I. Vilja r Physics Letters B 465 (1999) 86–94
has much more power at large momenta and the boson distribution has slower increase when p ™ 0 q . The conditions of validity of the analysis have been studied and found that under wide range of parameters calculation can be performed with good reliability. Also the evolution of the universe has been studied during the black hole evaporation era. In particular, the time when it makes transition from matter domination to radiation domination and the generic behaviour of radiation density are calculated. It should be noted, however, that the precise characterization of the physics, including particle distributions, would require information of the model of elementary particles at the energies in hand. This information is, unfortunately, not available and therefore in this paper we have used a typical GUT degrees of freedom to illustrate the possible physics. These considerations may have great impact to the calculations of the very early universe. In particular an early inflation may depend crucially on the particle distributions at the non-thermalised era of the early universe due to the ensemble corrected quantities. However, more detailed studies are definitely needed, when any particular model is considered.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x
w11x w12x
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