Accretion, Primordial Black Holes and Standard Cosmology
aa r X i v : . [ g r- q c ] M a r ACCRETION, PRIMORDIAL BLACK HOLESAND STANDARD COSMOLOGY
B. Nayak and L. P. Singh Department of PhysicsUtkal UniversityBhubaneswar-751004India [email protected] lambodar [email protected] Abstract
Primordial Black Holes evaporate due to Hawking radiation. We find thatthe evaporation times of primordial black holes increase when accretion ofradiation is included. Thus, depending on accretion efficiency, more and morenumber of primordial black holes are existing today, which strengthens theconjecture that the primordial black holes are the proper candidate for darkmatter.
PACS numbers : 98.80.-k, 97.60.LfKey words : Primordial Black Hole, accretion, accretion efficiency .1
INTRODUCTION
Black holes which are formed in the early Universe are known as Primordial BlackHoles (PBHs). A comparison of the cosmological density of the Universe at any timeafter the Big Bang with the density associated with a black hole shows that PBHswould have of order the particle horizon mass. PBHs could thus span enormous massrange starting from 10 − gm to more than 10 gm . These black holes are formed asa result of initial inhomogeneities[1,2], inflation[3,4], phase transitions[5], bubblecollisions[6,7] or the decay of cosmic loops[8]. In 1974 Hawking discovered that theblack holes emit thermal radiation due to quantum effects[9]. So the black holes getevaporated depending upon their masses. Smaller the masses of the PBHs, quickerthey evaporate. But the density of a black hole varies as inversely with its mass.So high density is needed for forming lighter black holes. And such high densitiesis available only in the early Universe. Thus Primordial Black Holes are the onlyblack 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 asignificant component of dark matter[11,12,13].Since the cosmological enviornment is very hot and dense in the radiation-dominated era, it is expected that appreciable absorption of the energy-matter fromthe surroundings could take place. Calculation of such PBH accretion in standardcosmology have a long history but are plauged with significant uncertainties. Theearly work by Zel’dovich and Novikov[1] speculated that PBHs might even be ableto grow as fast as the horizon. Subsequent works, especially by Carr and Hawk-ing[2,14], made a convincing case that such growth could not occur and moreoverthat once the PBH became significantly smaller than the horizon, accretion wouldbecome very inefficient. But it has been noticed that such accretion is most effectivein altered gravity scenarios. This accretion is responsible for the prolongation of thelifetime of PBHs in braneworld models[15] as well as in scalar-tensor models[16,17].Using standard cosmology Barrow and Carr[18] have studied the evaporation ofPBHs. They have, however, not included the effect of accretion of radiation whichseems to play an important role in scalar-tensor models. Majumdar, Das Guptaand Saxena[19] have provided a viable solution of the baryon asymmetry problemincluding accretion. In the present work, we include accretion of radiation whilestudying the evaporation of PBHs and have shown that how evaporation times ofPBHs change with accretion efficiency. 2 PBH EVAPORATION IN STANDARD COS-MOLOGY
For a spatially flat(k=0) FRW Universe with scale factor a , the Einstein equationis[20] (cid:16) ˙ aa (cid:17) = 8 πG ρ (1)where ρ is the density of the Universe.The energy conservation equation is˙ ρ + 3 (cid:16) ˙ aa (cid:17) (1 + γ ) ρ = 0 (2)on assuming that the universe is filled with perfect fluid describrd by equation ofstate p = γρ . The parameter γ is for radiation dominated era( t < t ) and is0 for matter dominated era( t > t ), where time t marks the end of the radiationdominated era ≈ sec.Now equation(2) gives ρ ∝ ( a − ( t < t ) a − ( t > t )Using this solution in equation (1), one gets the wellknown temporal behaviour ofthe scale factor a ( t ) as a ( t ) ∝ ( t ( t < t ) t ( t > t ) (3)Due to Hawking evaporation, the rate at which the PBH mass (M) decreases isgiven by ˙ M evap = − πr BH a H T BH (4)where r BH ∼ black hole radius=2 GM with G as Newton’s gravitational constant. a H ∼ black body constantand T BH ∼ Hawking Temperature= πGM .Now equation (4) becomes ˙ M evap = − a H π G M (5)3ntegrating the above equation, we get M = h M i + 3 α ( t i − t ) i (6)where α = a H π G and M i is the black hole mass at its formation time t i . Itis worthwhile to remark that we assume M i to be same as the horizon mass asconjectured in [21]. We will, however, demonstrate in the following that two masseswill have different temporal growth. When a PBH passes through radiation dominated era, the accretion of radiationleads to increase of its mass with the rate given by˙ M acc = 4 πf r BH ρ r (7)where ρ r is the radiation energy density of the sorrounding of the black hole= πG (cid:16) ˙ aa (cid:17) and f is the accretion efficiency. The value of the accretion efficiency f depends uponcomplex physical processes such as the mean free paths of the particles comprisingthe radiation sorrounding the PBHs. Any peculiar velocity of the PBH with respectto the cosmic frame could increase the value of f [19,22]. Since the precise value of f is unknown, it is customary[23] to take the accretion rate to be proportional tothe product of the surface area of the PBH and the energy density of radiation with f ∼ O (1).After substituting the expressions for r BH and ρ R equation(7) becomes˙ M acc = 6 f G (cid:16) ˙ aa (cid:17) M (8)Using equation(3), we get ˙ M acc = 32 f G M t (9)On integration, the above eqution gives M ( t ) = h M − i + 32 f G (cid:16) t − t i (cid:17)i − (10)Using horizon mass which varies with time as M H ( t ) = G − t , as initial mass ofPBH, we get M ( t ) = M i h f (cid:16) t i t − (cid:17)i − (11)4e draw two important conclusions from equation (11).First we obtain the variation of accreting mass with time for different f as shownin Figure-1. The figure clearly indicates that the mass of the PBH increases withaccretion efficiency.Figure 1: Variation of accreting mass for f = 0 . , . , . , . t , M BH of equation (11) asymptotes to its maximum value as M max = M i − f (12)which leads to an upperbound, f <
23 (13)The second conclusion is with regard to variation of PBH mass visavis that ofhorizon with time.Since the horizon mass grows as M H ( t ) ∼ G − t , from equation (11) one findsthat M H grows faster than the black hole mass M BH . This is graphically shown inFigure-2. Thus enough radiation density is available within the cosmological horizonfor a PBH to accrete causally, making accretion effective in this scenario.5igure 2: Variation of M P BH having f = 0 . M H with t Primordial Black Holes, as discussed before, are only formed in radiation dominatedera. So depending on their evaporation, we can divide PBHs into 2 categories.(i) PBHs evaporating in radiation dominated era ( t < t )(ii) PBHs evaporating in matter dominated era ( t > t ). CASE-I ( t < t )Black hole evaporation equation (6) implies M = M i h αM i ( t i − t ) i (14)If we consider both evaporation and accretion simultaneously, then the rate at whichprimordial black hole mass changes is given by˙ M P BH = 32 f G M t − α M (15)This equation can not be solved analytically. So we have solved it by using numer-ical methods. 6or PBHs with formation mass M i > a H fG , the magnitude of the first term(accretion) exceeds that of the second term (evaporation). In the radiation dom-inated era for a PBH whose formation mass satisfies the above relation, accretionis dominant upto a value of t , say t c , at which accretion rate equals evaporationrate (the PBH mass rises to a maximum value M max at this stage), and after thatevaporation dominates over accretion. For our calculation purpose, we have used α ≈ G − = 10 ( gm sec ) and G = 10 − ( secgm ).For a given M i , the solution as given by equation(14) and the solution of the equa-tion(15) are shown in Figure-3. The figure clearly shows that the evaporation timeof PBH increases with accretion efficiency.Figure 3: Variation of PBH mass for f = 0 , . , . , . CASE-II ( t > t )Since there is no accretion in matter dominated era, so the first term in the combinedequation (15) for variation of M P BH with time needs to be integrated only upto t .Based on numerical solution with above provision, we construct the Table-1 for thePBHs which are evaporating at present time.7 evap = t = 4 . × sf t i M i . × − s 2 . × g0 . . × − s 2 . × g0 . . × − s 1 . × g0 . . × − s 1 . × g0 . . × − s 0 . × g0 . . × − s 0 . × g0 . . × − s 0 . × gTable 1: The formation times and initial masses of the PBHs which are evaporatingnow are displayed for several accretion efficiencies.It is clear from the table that accretion makes it possible for the PBHs evapo-rating now to be formed at earlier times with smaller initial masses. The fraction of the Universes’ mass going into PBHs at time t is[2] β ( t ) = h Ω P BH ( t )Ω R i (1 + z ) − (16)where Ω P BH ( t ) is the density parameter associated with PBHs formed at time t , z isthe redshift associated with time t . Ω R is the microwave background density havingvalue 10 − .For t < t , redshift defination implies, (1 + z ) − = (cid:16) tt (cid:17) (cid:16) t t (cid:17) .So β ( t ) = (cid:16) tt (cid:17) (cid:16) t t (cid:17) Ω P BH ( t ) × (17)Using M = G − t , we can transcribe the equation (17) to write the fraction of theUniverse going into PBHs’ as a function of mass M is β ( M ) = (cid:16) MM (cid:17) (cid:16) t t (cid:17) Ω P BH ( M ) × (18)Observations of the cosmolgical deceleration parameter imply Ω P BH ( M ) < M ∗ ) generate a γ -ray background whose most of the energy is appear-ing at around 100 Mev[24]. If the fraction of the emitted energy which goes into8 evap = t f M ∗ β ( M ∗ ) < . × g 5 . × − . . × g 4 . × − . . × g 3 . × − . . × g 1 . × − .
666 2 . × g 1 . × − . . × g 5 . × − Table 2: Upper bounds on the initial mass fraction of PBHs that are evaporatingtoday for various accretion efficiencies f .photons is ǫ γ , then the density of the radiation at this energy is expected to beΩ γ = ǫ γ Ω P BH ( M ∗ ). Since ǫ γ ∼ . γ -ray background densityaround 100 Mev is Ω γ ∼ − , one gets Ω P BH < − .Now equation (18),therefore, becomes β ( M ∗ ) < (cid:16) M ∗ M (cid:17) × (cid:16) t t (cid:17) × − (19)The variation of β ( M ∗ ) with f drawn from variation of M ∗ with f is shown in theTable-2. The bound on β ( M ∗ ) is strengthened as f approaches its maximum value2 / Consideration of evaporation alone makes the Primordial Black Holes which are cre-ated on or before 2 . × − sec completely evaporate by the present time. How-ever, we found that if we include accretion, then the Primotdial Black Holes whichare created at the same instant of time will live longer depending on their accretionefficiency. Our analysis also leads to an upperbound on the accretion efficiency as f < . Further, the constraint on the fraction of the Universes’ mass going intoPBHs’ obtained by us is consistent with previous results[25,26] that β ( M ∗ ) < − .Thus accretion increases the number of existing PBHs depending on accretionefficiency, which lends support to the proposal of considering PBHs as the viablecandidate for dark matter. We, thus, provide within standard cosmology a possiblerealisation of the speculation advanced earlier[11,12,13].In the present context, one may consider back reaction of primordial black holeevaporation which can lead to non-trivial consequences[27]. Back reaction modifies9he radius and temperature of PBH [28] which ultimately affects the accretion andevaporation rates. Thus it might be interesting to see in what way resulting modi-fication could in turn impact the evolution of black holes. Such effects, it is argued[29], may make the Hawking process terminate while the PBH still has macroscopicmass. There are also competing speculations that blackholes completely evaporateleaving no remnants [30] or that blackholes cease to evaporate as they approachPlanck mass [31]. Whatever may be the cause of the stability of final remnant ofradiating PBHs, the finite mass relics would provide a possible cold dark mattercandidate [32]. ACKNOWLEDGEMENT
We are thankful to Institute of Physics, Bhubaneswar, India, for providing the li-brary and computational facility. B.Nayak would like to thank the Council of Sci-entific and Industrial Research, Government of India, for the award of SRF, F.No.09/173(0125)/2007-EMR-I .
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