arXiv:0905.3243v1 [gr-qc] 20 May 2009
ACCRETION, PRIMORDIAL BLACK HOLES AND STANDARD COSMOLOGY B. Nayak1 and L. P. Singh2 Department of Physics Utkal University Bhubaneswar-751004 India 1
[email protected] 2 lambodar
[email protected] Abstract Primordial Black Holes evaporate due to Hawking radiation. We find that the evaporation time of primordial black holes increase when accretion of radiation is included. Thus depending on accretion efficiency more and more number of primordial black holes are existing today, which strengthens the idea that the primordial black holes are the proper candidate for dark matter.
PACS numbers : 98.80.-k, 97.60.Lf Key words : Primordial Black Hole, accretionn, accretion efficiency .
1
1
INTRODUCTION
Black holes which are formed in the early Universe are known as Primordial Black Holes (PBHs). These black holes are formed as a result of initial inhomogeneities [1,2], inflation [3,4], phase transitions [5], bubble collisions [6,7] or the decay of cosmic loops [8]. In 1974 Hawking discovered that the black holes emit thermal radiation due to quantum effects [9]. So the black holes get evaporated depending upon their masses. Smaller the masses of the PBHs, quicker they evaporate. But the density of a black hole varies as inversely with it’s mass. So high density is needed for forming lighter black holes. And such high densities is available only in the early Universe. Thus Primordial Black Holes are the only black holes whose masses could be small enough to have evaporated by present time. Further, PBHs could act as seeds for structure formation[10] and could also form a significant component of dark matter[11,12,13]. Using standard cosmology Barrow and Carr [14] have studied the evaporation of PBHs. They have, however, not included the effect of accretion of radiation which seems to play an important role. Majumdar, Das Gupta and Saxena [15] have solved the baryon asymmetry problem including accretion. In the present work, we include accretion of radiation while studying the evaporation of PBHs and have shown that how evaporation times of PBHs change wih accretion efficiency.
2
PBH EVAPORATION IN STANDARD COSMOLOGY
For a spatially flat(k=0) FRW Universe with scale factor a, the Einstein equation is [16] a˙ 2
8πG ρ 3
(1)
(1 + γ)ρ = 0
(2)
=
a
where ρ is the density of the Universe. The energy conservation equation is ρ˙ + 3
a˙
a
on assuming that the universe is filled with perfect fluid describrd by equation of state p = γρ .The parameter γ is 13 for radiation dominated era(t < t1 ) and is 0 for matter dominated era(t > t1 ), where time t1 marks the end of the radiation
2
dominated era ≈ 1011 sec. Now equation(2) gives ρ∝
(
a−4 (t < t1 ) a−3 (t > t1 )
Using this solution in equation(1), one gets the wellknown temporal behaviour of the scale factor a(t) as a(t) ∝
(
1
t 2 (t < t1 ) 2 t 3 (t > t1 )
(3)
Due to Hawking evaporation, the rate at which the PBH mass (M) decreases is given by 2 4 M˙ evap = −4πrBH aH TBH
(4)
where rBH ∼ black hole radius=2GM with G as Newton’s gravitational constant. aH ∼ black body constant 1 . and TBH ∼ Hawking Temperature= 8πGM Now equation (4) becomes aH 1 M˙ evap = − 3 2 256π G M 2
(5)
Integrating the above equation, we get h
M = Mi3 + 3α(ti − t) where α =
3
aH 1 256π 3 G2
i1
3
(6)
.
ACCRETION
When a PBH passes through radiation dominated era, the accretion of radiation leads to increase of its mass with the rate given by 2 M˙ acc = 4πf rBH ρr
(7) 2
3 a˙ where ρr is the radiation energy density of the sorrounding of the black hole= 8πG a and f is the accretion efficiency. The value of the accretion efficiency f depends upon complex physical processes such as the mean free paths of the particles comprising
3
the radiation sorrounding the PBHs. Any peculiar velocity of the PBH with respect to the cosmic frame could increase the value of f [15,17]. Since the precise value of f is unknown, it is customary [18] to take the accretion rate to be proportional to the product of the surface area of the PBH and the energy density of radiation with f ∼ O(1). After substituting the expressions for rBH and ρR equation(7) becomes a˙ 2 M˙ acc = 6f G M2 a
(8)
M2 3 M˙ acc = f G 2 2 t
(9)
Using equation(3), we get
On integration, the above eqution gives h 3 1 1 i−1 M(t) = Mi−1 + f G − 2 t ti
(10)
The variation of accreting mass with time for different f is shown in figure-1.
1.4
MMi
1.3
1.2
1.1
1.0 1.0
1.2
1.4
1.6
1.8
2.0
tti
Figure 1: Variation of accreting mass for f=0.1,0.2,0.3,0.4 From figure-1, it is cleared that the mass of the PBH increases with accretion efficiency. 4
Now equation(10) gives 1 3 G 1 1 + f = − M(t) Mi 2 t Mi
(11)
But Horizon mass varies with t as MH (t) = G−1 t . So 1 1 3 1 1 − = 1− f × − M(t) MH (t) 2 Mi MH (t)
(12)
But for accretion to become effective, Horizon mass MH (t) must grow faster than 1 1 black hole mass M(t) which implies M (t) − MH (t) is a positive quantity.
Hence equation(12) demands 1 − 32 f is a positive quantity which gives f
t1 ). CASE-I (t < t1 ) Black hole evaporation equation (6) implies h
M = Mi 1 +
i1 3α 3 (t − t) i Mi3
(14)
If we consider both evaporation and accretion simultaneously, then the rate at which primordial black hole mass changes is given by 3 M2 1 M˙ P BH = f G 2 − α 2 2 t M
(15)
This equation can n’t be solved analytically. So we have solved it by using numerical methods. 3 (For our calculation purpose, we have used α ≈ G−2 = 1028 ( gs ) and G = 10−38 ( gs ).) For a given Mi the equation(14) and the solution of the equation(15) are shown in figure-3. 8
Mi =10 g 1.4 1.2
MMi
1.0 0.8 0.6 0.4 0.2 0.0
0
25
2. ´ 10
25
25
4. ´ 10
6. ´ 10
25
8. ´ 10
26
1. ´ 10
tti
Figure 3: Variation of PBH mass for f = 0, 0.1, 0.15, 0.2 Figure-3 clearly shows that the evaporation time of PBH increases with accretion efficiency. 6
CASE-II (t > t1 ) Since there is no accretion in matter dominated era, so the first term in the combined equation (15) for variation of MP BH with time needs to be integrated only upto t1 . Based on numerical solution with above provision, we construct the following table-1 for the PBHs which are evaporated today i.e. tevap = t0 . Table-1 tevap = t0 = 4.42 × 1017 s f ti Mi −23 0 2.3669 × 10 s 2.3669 × 1015 g 0.1 2.0119 × 10−23 s 2.0119 × 1015 g 0.2 1.6568 × 10−23 s 1.6568 × 1015 g 0.3 1.3018 × 10−23 s 1.3018 × 1015 g 0.4 0.9467 × 10−23 s 0.9467 × 1015 g 0.5 0.5916 × 10−23 s 0.5916 × 1015 g 0.6 0.23669 × 10−23 s 0.23669 × 1015 g
5
CONSTRAINTS ON PBH
The fraction of the Universe’s mass going into PBHs at time t is [2] β(t) =
hΩ
P BH (t)
ΩR
i
(1 + z)−1
(16)
where ΩP BH (t) is the density parameter associated with PBHs formed at time t, z is the redshift associated with time t and ΩR is the microwave background density having value 10−4 . 1 2 t t1
Again for t < t1 , (1 + z)−1 = So β(t) =
t1 t0
2
3
.
t 12 t1 23 ΩP BH (t) × 104 t1 t0
(17)
Again using M = G−1 t, we can write the fraction of the Universe going into PBHs’ of mass M is β(M) =
M 1 t 2 2
M1
1
3
t0
ΩP BH (M) × 104
(18)
Observations of the cosmolgical deceleration parameter imply ΩBH (M) < 1 over all mass ranges for which PBHs have not evaporated. But presently evaporating PBHs(M∗ ) generate a γ-ray background whose most of the energy is appearing at around 100 Mev[19]. If the fraction of the emitted energy which goes into 7
photons is ǫγ , then the density of the radiation at this energy is expected to be Ωγ = ǫγ ΩP BH (M∗ ). Since ǫγ ∼ 0.1 and the observed γ-ray background density around 100 Mev is Ωγ ∼ 10−9 , we infer ΩP BH < 10−8 . Now equation (18) becomes β(M∗ )
