Bondi accretion onto cosmological black holes

Janusz Karkowski and Edward Malec

Instytut Fizyki Mariana Smoluchowskiego, Instytut Fizyki, Uniwersytet Jagiellonski, Reymonta 4, 30-059 Krakow, Poland(Received 6 November 2012; published 5 February 2013)

In this paper we investigate a steady accretion within the Einstein-Straus vacuole, in the presence of the

cosmological constant. The dark energy damps the mass accretion rate and—above a certain limit—

completely stops the steady accretion onto black holes, which, in particular, is prohibited in the inflation

era and after (roughly) 1012 years from the big bang (assuming the presently known value of the

cosmological constant). Steady accretion would not exist in the late phases of the Penrose’s scenario—

known as the Weyl curvature hypothesis—of the evolution of the Universe.

DOI: 10.1103/PhysRevD.87.044007 PACS numbers: 04.20.�q, 04.25.D�, 04.40.Nr, 98.35.Mp

I. INTRODUCTION

The classical Bondi accretion model [1] describes, inthe test gas and stationarity approximations, the sphericalaccretion/wind of a barotropic fluid onto a Newtoniangravity center. That description has been extended togeneral-relativistic spacetimes. The steady accretion inthe Schwarzschildean geometry has been investigated byMichel [2] and Shapiro and Teukolsky [3], and in a spheri-cal symmetric spacetime with backreaction by Malec [4].A remarkable universal behavior has been discovered forself-gravitating transonic flows, under suitable boundaryconditions: the ratio of respective mass accretion rates_MGR= _MN appears independent of the fractional mass ofthe gas and depends only on the asymptotic temperature. Itis close to 1 in the regime of low asymptotic temperaturesand can grow by 1 order of magnitude at high temperatures[5,6]. Here _MGR and _MN are the general-relativistic and theNewtonian mass accretion rates, respectively. The stabilityof steady accretion has been established by Mach and hiscoworkers [7–9].

In this paper we consider the accretion onto a blackhole that is immersed in a cosmological universe. We adoptthe Einstein-Straus ‘‘Swiss cheese’’ model [10]. A spheri-cally symmetric black hole surrounded with accretinggas sits inside a vacuole that is matched—through theDarmois-Israel junction conditions [11,12] to an externalFriedmann-Lemaıtre-Robertson-Walker (FLRW) space-time. We calculate how the presence of a cosmologicalconstant � would affect the most important systemcharacteristic—the mass accretion rate _M of a fluid ontothe center. We found that the mass accretion rate is dampedby the presence of the cosmological constant. This effect isstronger for negative values than for positive values of thecosmological constant and it depends on the ratio of thedark energy density to the fluid density.

In the next two sections we remind readers of the Kottlersolutions and draw the general picture of the steady accre-tion, together with suitable equations. Sections IV and Vpresent three analytic sets of results. These show that thematter distribution can be affected by the cosmological

system. More importantly, we prove that if the cosmo-logical constant exceeds a certain limit, then the steadyaccretion ceases to exist. Section VI presents results ofnumerical integrations. It appears that the presence of darkenergy, either negative (attractive) or positive (repulsive),damps the mass accretion rate. It appears—in accordancewith analytic predictions—that the steady accretion doesnot exist above certain values of the cosmological constant.The last section summarizes the main results and appliesthem to two cosmological epochs when the dark energy isdominant. It is pointed out that our results do not agreewiththe Penrose’s ‘‘Weyl curvature hypothesis.’’

II. ACCRETION IN ANEINSTEIN-STRAUS VACUOLE

We assume relativistic units with G ¼ c ¼ 1. Let aFLRW spacetime be filled with dust and dark energy. Itsline element reads

ds2 ¼ �d�2 þ a2ð�Þ�

dr2

1� kr2þ r2d�2

�; (1)

where the variables have obvious meaning [13] andk ¼ 0;þ� 1. We do not need to analyze the FLRWequations,for a reason explained below.We cut off, following Einstein and Straus [10], a ball

from this FLRW spacetime and insert instead a sphericalsymmetric inhomogeneity—a black hole surrounded by aspherical cloud of gas—satisfying Einstein’s equation. It isknown that the boundary of the ball must be comovingif the FLRW geometry contains only dust and the darkenergy represented by�, assuming the absence of a bound-ary layer and continuity of pressure [14–16]. Crenon andLake show that the no-boundary layer assumption isnecessary for having a comoving boundary [14].Inside the excision ball itself we can distinguish an inner

accretion region such that the areal radius R does notexceed a limiting radius R1 (that in turn should be smaller,or perhaps much smaller, than the areal radius of theboundary of the ball), and the asymptotic vacuum regionoutside R1. In the asymptotic region the metric is given by

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the Kottler [Schwarzschild—de Sitter (SdS, �> 0) andSchwarzschild—anti—de Sitter (SAdS,�< 0)] spacetimeline element [17]

ds2 ¼ ��1� 2m

R��

3R2

�dt2 þ dR2

1� 2mR � �

3 R2þ R2d�2:

(2)

The Darmois-Israel gluing conditions [11,12] demand thatalong the comoving boundary the first and second funda-mental forms are continuous. In the spherically symmetriccase the radius R has to be continuous and in addition(i) the excess mass m should be exactly equal to the massof dust in the excised ball and (ii) the boundary shouldbe comoving with the Hubble velocity H. This guaranteesthat the stress-energy tensor T�� does not form the surface

layer at the junction surface. Balbinot et al. have foundan explicit solution that describes the correspondingmatching between the Schwarzschild—de Sitter andFLRW (dustþ�) spacetimes [18]. Therefore we stopdiscussion on FLRW and SdS (SAdS) solutions at thispoint and focus on the description of the accretion zone.

We assume that in the accretion region the Einsteinequations can be approximated by a set of stationaryequations in time intervals that are much smaller than acharacteristic time T that is defined in the next section. LetR1 be the size of the cloud of gas and a1 the asymptoticspeed of sound. Outside R1 the geometry can be connectedsmoothly to the SdS or SAdS spacetime geometry by atransient zone of a negligible mass. Let us remark thatthe assumption of the approximate stationarity can bechecked a posteriori, after finding appropriate solutions.Approximately stationary accretion equations have beenderived in Ref. [4]. A quasistationary solution, of a similaraccretion problem in the asymptotically flat spacetime witha spherical black hole, appears to be stable under axiallysymmetric perturbations [8].

III. EQUATIONS OF STEADYACCRETION

The metric in the accretion zone is spherically symmet-ric and has the form

ds2¼�N2dt2þ adr2þR2ðd�2þsin2ð�Þd�2Þ; (3)

where we use comoving coordinates t, r, 0 � � � �,0 � �< 2�—time, coordinate radius, and two angle var-iables, respectively. R denotes the areal radius and N is thelapse. a is the radial-radial metric component. The radialvelocity of gas is given by U ¼ 1

NdRdt .

The energy-momentum tensor reads TB�� ¼

ð�þ pÞU�U� þ pg�� with the timelike normalized four-

velocity U�, U�U� ¼ �1. A comoving observer would

measure local mass density � ¼ T��U�U�. Let n� be the

unit normal to a coordinate sphere lying in the hypersur-face t ¼ const and let k be the related mean curvature

scalar, k ¼ R2 rin

i ¼ 1ffiffia

p @rR. We assume the perfect gas

equation of state p ¼ ð�� 1Þ�0�, where � is the specificinternal energy and � is a constant. Assuming that theaccretion is isentropic, one can derive the polytropic equa-tion of state p ¼ K��

0 , with a constant K. The internal

energy E ¼ �0� and the rest � and baryonic �0 massdensities are related by � ¼ �0 þ E ¼ �0 þ p=ð�� 1Þ.The matter-related and geometric quantities satisfy

Einstein equations and the baryonic mass conservation.(But let us point out that for isentropic flows the conser-vation of baryonic mass also follows from Einsteinequations [4].) One can find the mean curvature k fromthe Einstein constraint equations G�0 ¼ 8�T�0 [4],

k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2mðRÞ

R��

3R2 þU2

s; (4)

where mðRÞ is the quasilocal mass,

mðRÞ ¼ m� 4�Z R1

Rdrr2�: (5)

In the line element (3) we have the comoving time. Inthe polar gauge foliation one has a new time tSðt; rÞ with@tS ¼ @t � NU@R. The field @tS is tangent to the cylinder

of constant areal radius, @tSR ¼ 0.

One can show that

_M� � @tSmðRÞ ¼ 4�NUR2ð�þ pÞ: (6)

The mass contained in an annulus (R, R1) changes if thefluxes on the right-hand side, one directed outward and theother inward, do not cancel. The baryonic current densityreads j� � �0U

�. Its continuity equation reads

r�j� ¼ 0: (7)

For stationary flows the local baryonic flux

_M ¼ �4�UR2�0 (8)

is time-independent at a fixed R.We say that the accretion process is stationary (or quasi-

stationary) if all physically relevant observables, which aremeasured at a fixed areal radius R, remain approximatelyconstant during time intervals much smaller than the run-away instability time scale T ¼ M= _M. That means that@tSX � ð@t � NU@RÞX ¼ 0 for X ¼ �0; �; j; U; . . . ; _M.

For quasistationary flows we have @R _M� ¼ 0 and

@R _M ¼ 0 [4]. Therefore both _M� and _M are equal, modulo

a constant factor; they can be identified. The quantity _Mwill be called the mass accretion rate. One obtains from (8)an expression

@RU2 ¼ � 4U2

R� 2U2@R lnð�0Þ: (9)

The speed of sound is defined as a ¼ ffiffiffiffiffiffiffiffiffi@�p

p. It is a useful

and straightforward exercise to express the hydrodynamicquantities in terms of a:

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p ¼ �0

�� 1

�

a2

�� 1� a2;

� ¼ �0

�� 1

�� 1� a2� p;

�0 ¼ �01�a

a1

� 2�1

0@1� a21

��1

1� a2

��1

1A 1

��1

:

(10)

Here (and below) quantities with the suffix 1 do referto their asymptotic values (i.e., at R1). Notice thatp, �0 and a2 show the same monotonicity behavior:@R�0 ¼ C1@Rp ¼ C2@Ra

2, where C1, C2 are strictly posi-tive functions.

There are two conservation equations that originate fromthe contracted Bianchi identities r�T

�� ¼ 0. One of them

can be eliminated, if we choose instead the baryonic massconservation. The case when � ¼ 0 is the relativistic ver-sion of the Euler equation,

Nd

dRpþ ðpþ �Þ d

dRN ¼ 0: (11)

One can solve Eq. (11), using Eq. (10):

N ¼ ~Cð�� 1� a2Þ; (12)

where ~C is a constant. The whole system of algebraicequations (4)–(12) closes with the imposition of theEinstein equation, Grr ¼ 8�Trr, which is the onlyintegro-differential equation. It can be put in thefollowing form:

d

dRlnða2Þ¼���1�a2

a2�U2

k2

� 1

k2R

�mðRÞR

�2U2þ4�R2p��R2

3

�: (13)

Finally the line element, in (t, R) coordinates, is given by

ds2 ¼ �ðN2 �U2Þdt2 � 2N

kdtdRþ dR2

k2þ R2d�2:

(14)

We shall study transonic accretion flows. For them theprincipal object of interest is a sonic point, where both

the denominator a2 � U2

k2and the numerator mðRÞ

R � 2U2 þ4�R2p� �R2

3 of Eq. (13) vanish. The corresponding value

of the areal radius is called the sonic radius, denoted as R�.A closer inspection of the sonic point shows that it is acritical point, with branching pairs of solutions describingaccretion or wind. In the accretion branch, below the sonicpoint the infall velocity jUj=k is bigger than a, whileoutside the sonic sphere the converse is true. This analysisis very much standard, replicating the work done in theNewtonian case by Bondi [1] and in the general-relativisticcase by Malec [4].

IV. QUALITATIVE ANALYSIS OF SOLUTIONS:SDS BLACK HOLES

In this section we shall investigate some properties oftransonic solutions for positive values of �. The speed ofsound a (and thus also �0 and p) decreases as a functionof R—for accreting solutions—when the cosmologicalconstant is absent. We shall show, in Lemma 2, that forlarge values of �R2 the converse is possible.Lemma 1: Assume �< 3m

2R31. Then the speed of sound,

the pressure, and the mass density �0 are decreasing out-side the sonic sphere, d

dR X < 0 for X ¼ a2, p, and �0.

Proof of Lemma 1: Let us define CðRÞ � mðRÞR �

2U2ðRÞ þ 4�R2p� �R2

3 and let C1 � CðR1Þ. One easily

obtains from this definition, using Eq. (9) in order toeliminate @RU

2, that

d

dRCðRÞ ¼ �4C

Rþ 4�R

��þ 6p� �

2�

�þ 3m

R2

þ 4U2ddR �0

�0

þ 4�R2 d

dRp: (15)

The condition �< 3mR31

guarantees that C1 > 0 and—using

the argument of continuity—the positivity of the functionCðRÞ in an open interval (Rs, R1). We show that Rs ¼ R�.Notice that while at the sonic point CðR�Þ ¼ 0, the

function @RC must be strictly positive. Indeed, assumethe opposite; from continuity, C would have to be negativejust above R�. Then from Eq. (15) it would follow that bothterms with derivatives must be negative, that is—seethe remark below (10)—@Ra

2 < 0. But Eq. (13) gives@Ra

2 > 0, if C< 0 and R> R�. This contradictionproves our statement and implies the positivity of C in aninterval (R�, Rb).Now, assume that C changes sign at Rb, that is, CðRbÞ ¼

0 and ddR CðRÞR¼Rb

< 0. But if CðRbÞ ¼ 0 then—from

Eq. (13)—the derivative @Ra2 would have to vanish. From

(10) it would then follow @R�0 ¼ @Rp ¼ 0. Thus Eq. (15)yields at Rb

d

dRC ¼ 4�Rb

��ðRbÞ þ 6pðRbÞ � �

2�

�þ 3m

R2b

; (16)

but this implies ddR C > 0 at Rb, due to the condition

�< 3mR3b

. This contradiction proves that CðRÞ is strictly

positive outside the sonic sphere.The positivity of C allows one to conclude—from

Eq. (13)—that @Ra2 is strictly decreasing outside the sonic

sphere. This in turn implies the decrease of the pressurep and the baryonic mass density �0 [see the remark underEq. (10)].

Lemma 2: Assume �R21 > 18a21��1�a21

and define R0 ¼ð6m� Þ13. The necessary condition for the existence of solu-

tions is � 1. If > 1=3 then R0 � R1 and the speed of

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sound, the pressure, and the mass densities are increasingin the interval (R0, R1): d

dR X > 0 for X ¼ a2, p, and �0.

Proof of Lemma 2: The estimate R0 � R1 followsimmediately from the boundary condition a21 � m=R1and the assumed value of �. In the asymptotic end(R0, R1) the function �CðRÞ can be bounded from belowas follows,

�CðRÞ ¼ �mðRÞR

þ 2U2ðRÞ � 4�R2pþ�R2

3

� �2m

Rþ�R2

3: (17)

The second line follows from U2 � 0 and from the obser-

vation that p � �—thus 4�R2p � 4�RR1R drr�—and

4�RR1R drr� � m

R .

Taking into account the last estimate, we can estimatefrom below the right-hand side of Eq. (13) by

DðRÞ � �� 1� a21a2

�� 2m

R2þ�R

3

�: (18)

We obtain from Eqs. (13) and (18) the inequality

d

dRa2 � ð�� 1� a21Þ

�� 2m

R2þ�R

3

�: (19)

It is clear that (i) R0 is the null point of DðRÞ and (ii) thefunction DðRÞ is strictly positive for R 2 ðR0; R1Þ.Therefore a2 (and consequently the remaining gas charac-teristics p and �0) does increase in this interval, providedthat a solution does exist.

The necessary condition for the existence of a solutionis that aðR0Þ should be strictly positive. Integration ofEq. (19), between R0 and R1, yields

a21 � a2ðR0Þ � ð�� 1� a21Þ

��� 2m

R0

��R20

6þ 2m

R1þ�R21

6

�: (20)

We can replace 2mR0

by�R2

0

3 , using the definition of R0, and

get the following:

a2ðR0Þ � ð�� 1� a21Þ�R2118

�6R2

0

R21� 1

�

þ a21 � ð�� 1� a21Þ 2mR1

� a21 þ ð�� 1� a21Þ�R2118

6� 36

13ð mR1

Þ23ð�R21Þ23

� 1

!:

(21)

In the second inequality we dropped the positive termwith m=R1 and used the definition of R0. We alreadyassumed that m=R1 � a21 which ensures that the firstterm within the second bracket is small and the right-hand side of the last inequality in (21) is approximated

by EðRÞ � a21 � �R2118 ð�� 1� a21Þ. Therefore if the cos-

mological constant is given by �R21 ¼ 18 a21��1�a21

, with

larger than 1, then EðRÞ must be negative and the squareof the speed of sound at R0 must be nonpositive, whichmeans that solutions are absent. On the other hand solu-tions can exist for 0:01<< 1; in this case CðRÞ< 0between (R0, R1), which implies that d

dR X < 0 for X ¼ a2,

p, �0, and �.The two lemmas proven in this section have a transpar-

ent physical interpretation. Lemma 1 assumes that the darkenergy density �=4� is smaller than one half of theaveraged matter density 3m=4�R31. The presence of � isexpected to have some quantitative impact onto accretionmass rate _M, but qualitative features of the flow are notinfluenced—in particular, the gas characteristics a, jUj, �0,and � are all decreasing functions.Lemma 2 assumes the opposite—that the dark energy

density �=4� is much larger than the average matterdensity 3m=4�R31. Above a particular value—thatdepends on the boundary characteristics of the flow, itsvolume, and the black hole mass—steadily accreting solu-tions do not exist. This is intuitively understandable—large� implies a large Hubble expansion velocity which canobstruct or even prohibit accretion, through the junctioncondition on the boundary of the vacuole. Therefore onecan expect a significantly smaller mass accretion rate _Mand even its absence.

V. QUALITATIVE ANALYSIS OF SOLUTIONS:SADS BLACK HOLES

Lemma 3: Assume �< 0. Then,(i) For R> R� the speed of sound, the pressure, and the

mass densities �0 and � are decreasing, ddR X < 0 for

X ¼ a2, p, and �0.

(ii) Solutions are absent if �R21 � �6 ��1�a21��1�a2�

.

Proof of Lemma 3:Part (i). The key observation is that the function C is

strictly positive, if �< 0. Indeed, let Rb > R� be a small-est radius at which CðRbÞ vanishes, and assume C< 0 inthe open interval (R�, Rb) [notice that CðR�Þ vanishes]. ButEq. (15) would imply in such a case that both terms withderivatives become negative in a subinterval (R�, R1) of(R�, Rb), that is—see the remark below (10)—@Ra

2 < 0.But this in turn would contradict Eq. (13)—if C< 0 then@Ra

2 > 0 outside the sonic sphere. Thus we arrive at thecontradiction in this subinterval (R�, R1) and the argumentof continuity and the definition of Rb imply the positivityof C in the open interval (R�, Rb).In the next step we show thatC cannot vanish outside the

sonic sphere. Indeed, (13) and (15) imply that dCdR is strictly

positive whenever C ¼ 0 for R> R�; that in turn yields(by the argument of continuity) the strict positivity of Coutside the sonic sphere. Equation (13) allows oneto conclude that the speed of sound is monotonically

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decreasing, and that implies also decrease of the massdensity �0 and the pressure.

Part (ii). In order to prove the necessary existencecondition, notice that Eq. (13) gives, after integration,

ja21 � a2ðRÞj �Z R1

Rdr

��������ð�� 1� a2Þ 1r

��2U2 ��r2

3

���������:(22)

Notice that a2 < �� 1; thus ja21 � a2ðRÞj � �� 1� a21.The conservation of the mass accretion rate _M andddR �0 � 0 lead to the estimate d

dR ðU2R4Þ � 0, and this in

turn implies inequality

U2 � U21R41R4

: (23)

Collecting this information, one finally obtains

��1�a21�ð��1�a2ðRÞÞ��U21

2��R21

6

�1� R2

R21

��:

(24)

Let R ¼ R� and �R216 <� ��1�a21

��1�a2�; then—taking into

account that R2

R21is negligibly small, and the same is true

for the velocity term U2—we conclude that the right-handside of Eq. (22) exceeds its left-hand side. This contra-diction proves the necessary condition of Lemma 3.

VI. DESCRIPTION OF NUMERICS

Our aim is to find transonic accretion flows. We assumethe total massm ¼ 1, � ¼ 4=3, andR1 ¼ 106. The bound-

ary values at R1 are k ¼ N ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2m

R1�U21 � �

3 R21

qand

a21 ¼ 2� 10�4. There are three remaining parameters thatare free: the cosmological constant �, the mass accretionrate _M, and the asymptotic density �01—after specifyingthem, equations are integrated inward, starting from R1.These quantities should be such as to ensure the asymptotic

conditions: jU1j �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m=R1

p � a1.A numerical run, with the above boundary data, has been

regarded as successful, if(i) the integration started from R1 and reached the

outermost apparent horizon, that the point Ri such

that 2mðRiÞRi

þ �R2i

3 ¼ 1, along the accretion branch

(with the increasing—in the direction to thecenter—infall velocity jUj);

(ii) there appears a sonic point at a radius R� � Ri.More specifically, in order to find a single transonic flow,

we fix � and the mass accretion rate _M, and treat theasymptotic value of the baryonic mass density as a freeparameter. For a random choice of �01 there are threepossibilities, one exceptional and two generic:

(a) an almost unlikely event, that the numerical run issuccessful in the sense defined above and we find thesought accretion flow;

(b) the solution numerically fails—a singularity appearsduring integration;

(c) there exists a numerical solution without a sonicpoint—a piece of the solution describes acceleratingaccreting flow, while the other corresponds to adecelerating accreting gas.

In the case (a) the job is done. If one of the remaininggeneric alternatives occurs—say (b), with the asymptoticdensity �bÞ01—we change �01 until we get the situation

described in (c) with the asymptotic density �cÞ01. Now we

use the bisection method in order to find an intermediatevalue �i01 between the two values �bÞ01 and �cÞ01, forwhich the numerical run is successful and the transonicflow exists.The numerics itself is standard—we use commonly

known algorithms of the 8th order.The calculations have been performed for a few dozen

values—negative and positive—of the cosmological con-stant�. In the sector of negative cosmological constant wetake � 2 ð�5� 10�13;�10�19Þ; the absolute value overthe bound is larger by 1 order of magnitude than the valuefound in Lemma 3 as a necessary existence condition.In the positive part of the spectrum we choose � 2ð10�19; 7:2� 10�15Þ; this is quite close to the value estab-lished as the necessary condition in Lemma 2. For eachfixed value of�we find about 100 solutions correspondingto different values of _M.The obtained information is summarized in Figs. 1 and 2.

The abscissa shows the ratio x ¼ Mg=m of the massMg ¼4�

RR1Ri

drr2� of gas to the total massm. The total mass has

been normalized to 1. We obtain, although in a less trans-parent form, the same result as in Ref. [5]—the maximum

-40

-39

-38

-37

-36

-35

-34

-33

-32

-31

-30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1567

7.15

FIG. 1. The ordinate shows the mass accretion rate _M and theabscissa shows 1� x, where x is the relative mass of gas in thesystem. The various lines correspond to �R21=10�3 ¼ 1, 5, 6, 7,7.15 in the order from the top.

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of the mass accretion rate corresponds to x 1=3. For afixed value of _M there exist two flows, one with heaviercenter and lighter gas, and the other with opposite charac-teristics. Just around the maximum of _M there are blankpoints—we find that in this region it is difficult to find anumerical solution. We think that this is a purely numericalartifact. There is a slight shift of this maximum towardssmaller values of xwith the increase of the absolute value ofthe cosmological constant.

The only new physically relevant information is thatconcerning the mass accretion rate _M. It appears that thepresence of the cosmological constant manifests only whenthe dark energy density �=ð4�Þ exceeds the averagedmatter density 3m=ð4�R31Þ 4� 10�19 by at least 3orders of magnitude. The mass accretion rate _M is decreas-ing with the increase of the absolute value of the cosmo-logical constant. It is clear that the rate of the falloffdepends on the sign of �—it is faster for positivevalues—but the falloff itself occurs for both positive andnegative cosmological constant.

VII. COSMOLOGICAL IMPLICATIONS

Results that are reported in the preceding section dem-onstrate that while the mass accretion rate depends on thecosmological constant, this effect becomes significant onlywhen the dark energy fraction �� is much larger than thematerial fraction�m. On the other hand, this impact can bedramatic, a sevenfold increase of �—starting from itsvalue corresponding to ��=�m 103—leads to thediminishing of _M by 7 orders in magnitude, as seen in

Fig. 1. This is so for the accretion onto the SdS black hole,but the phenomenon is strong also in the case of SAdS(Fig. 2). When applying these results to our Universe, wehave to take into account that �� � �m in two epochs—during inflation and after 1011 years from the big bang.Inflation strictly excludes steady accretion onto primor-

dial black holes, due to Lemmas 2 and 3, because during theinflation era�� exceeds�m by something like 50 orders ofmagnitude. This fact is concordant with the well knownfreezing of structures that are bigger than a particle horizon[19], but the difference is that now it applies to smallaccretion systems that are located well within the horizon.In the present Universe the dark energy and material

densities are roughly the same, which suggests that theeffect of dark energy is negligible. The mass density oflargest bound structures—galactic superclusters—exceedsthe cosmological mass density by 1 order of magnitude.Notice, however, that the tenfold ageing of the Universewould increase the ratio ��=�m by a factor of 100. Thatmeans that in the Universe older than 1011 years the steadyaccretion would become less efficient and at the timet ! 1012 years its efficiency goes to zero.Penrose conjectured the so-called Weyl curvature

hypothesis [20]. It asserts, in its informal version, that aFriedmann-Lemaıtre-Robertson-Walker spacetime, wherethe Ricci curvature is nonzero but the Weyl curvaturevanishes, evolves towards a vacuum spacetime filled withblack holes and gravitational radiation—with nonvanish-ing Weyl curvature and negligible Ricci tensor. (A moreformal scenario has been worked out by Tod [21].) Thispicture assumes that initially small inhomogeneities of theFLRW universe accrete matter and transform themselvesinto black holes, which gradually merge, leaving at the enda net of huge black holes and a lot of gravitational radia-tion. A cautious interpretation of our findings would be tosay that the role of steady accretion in the realization of thisscenario, in the presence of dark energy, is insignificant. Itmight well happen, however, that this property of darkenergy of diminishing the steady spherical accretion sig-nals a more general feature—that dark energy damps anyaccreting process. In such a case there arises anotherfundamental question: how does dark energy impact thisscenario outlined in the Weyl curvature hypothesis?

ACKNOWLEDGMENTS

This research was carried out with the supercomputerDeszno purchased thanks to the financial support of theEuropean Regional Development Fund in the frameworkof the Polish Innovation Economy Operational Program(Contract No. POIG. 02.01.00-12-023/08).

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FIG. 2. The ordinate shows the mass accretion rate _M and theabscissa shows 1� x, where x is the relative mass of gas in thesystem. The various lines correspond to �R21=10�3 ¼ �1,�25,�50, �100, �250, �500 in the decreasing order from the top.

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[1] H. Bondi, Mon. Not. R. Astron. Soc. 112, 195 (1952).[2] F. C. Michel, Astrophys. Space Sci. 15, 153 (1972).[3] S. Shapiro and S. Teukolsky, Black Holes, White Dwarfs

and Neutron Stars (Wiley, New York, 1983).[4] E. Malec, Phys. Rev. D 60, 104043 (1999).[5] J. Karkowski, B. Kinasiewicz, P. Mach, E. Malec, and

Z. Swierczynski, Phys. Rev. D 73, 021503(R) (2006).[6] E. Malec and T. Rembiasz, Phys. Rev. D 82, 124005

(2010).[7] P. Mach, Rep. Math. Phys. 64, 257 (2009).[8] P. Mach and E. Malec, Phys. Rev. D 78, 124016 (2008).[9] P. Mach, Acta Phys. Pol. B 38, 3935 (2007).[10] A. Einstein and A.G. Straus, Rev. Mod. Phys. 17, 120

(1945).[11] G. Darmois, Mm. Sci. Math. XXV, 1 (1927).[12] W. Israel, Nuovo Cimento B 44, 1 (1966); 48, 463

(1967).

[13] S. Weinberg, Cosmology (Oxford University Press,Oxford, UK, 2008).

[14] C. Grenon and K. Lake, Phys. Rev. D 84, 083506 (2011).[15] N. P. Humphreys, R. Maartens, and D. R. Matravers, Gen.

Relativ. Gravit. 44, 3197 (2012).[16] D. Giulini and M. Carrera, Rev. Mod. Phys. 82, 169

(2010).[17] F. Kottler, Ann. Phys. (Berlin) 361, 401 (1918); H. Weyl,

Phys. Z. 20, 31 (1919); see also a concise description inV. Perlick, Living Rev. Relativity 7, 9 (2004).

[18] R. Balbinot, R. Bergamini, and A. Comastri, Phys. Rev. D38, 2415 (1988).

[19] V. Mukhanov, Physical Foundations of Cosmology(Cambridge University Press, Cambridge, England, 2005).

[20] R. Penrose, The Road to Reality: A Complete Guide to theLaws of the Universe (Jonathan Cape, London, 2005).

[21] P. Tod, Classical Quantum Gravity 20, 521 (2003).

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