shibaji banerjee et al- massive compact halo objects from the relics of the cosmic quark-hadron...

8/3/2019 Shibaji Banerjee et al- Massive Compact Halo Objects from the Relics of the Cosmic Quark-Hadron Transition

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Mon. Not. R. Astron. Soc. 000, 000000 (2002)

Massive Compact Halo Objects from the Relics of the

Cosmic Quark-Hadron Transition

Shibaji Banerjee

a,b

, Abhijit Bhattacharyya

c,d

, Sanjay K. Ghosh

a,e

, Sibaji Raha

a,a Department of Physics, Bose Institute, 93/1, A.P.C.Road, Kolkata 700 009, INDIAb Physics Department, St. Xaviers College, 30, Park Street, Kolkata 700 016, INDIAc Theory Division, Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata 700 064, INDIAd Physics Department, Scottish Church College, 3 Cornwallis Sq. Kolkata 700 006, INDIAe Variable Energy Cyclotron Centre, 1/AF, Bidhannagar, Kolkata 700 064, INDIAf Research Center for Nuclear Physics, Osaka University, Mihogaoka 10-1, Ibaraki City, Osaka 567-0047, JAPAN

27 November 2002

ABSTRACT

The existence of compact gravitational lenses, with masses around 0.5 M, has been re-ported in the halo of the Milky Way. The nature of these dark lenses is as yet obscure,

particularly because these objects have masses well above the threshold for nuclearfusion. In this work, we show that they find a natural explanation as being the evolu-tionary product of the metastable false vacuum domains (the so-called strange quarknuggets) formed in a first order cosmic quark-hadron transition.

Key words: early universe dark matter gravitational lensing cosmology : mis-cellaneous

One of the abiding mysteries in the so-called standardcosmological model is the nature of the dark matter. It isuniversally accepted that there is an abundance of matterin the universe which is non-luminous, due to its very weak

interaction, if at all, with the other forms of matter, ex-cepting of course the gravitational attraction. The presentconsensus (for a review, see (Turner 1999; Turner 2000))based on recent experimental data is that the universe isflat and that a sizable amount of the dark matter is cold,i.e. nonrelativistic, at the time of decoupling (we are notaddressing the issue of dark energy in this work). Specula-tions as to the nature of dark matter are numerous, oftenbordering on the exotic, and searches for such exotic matteris a very active field of astroparticle physics. In recent years,there has been experimental evidence (Alcock et al. 1993;Aubourg et al. 1993) for at least one form of dark matter- the Massive Astrophysical Compact Halo Objects (MA-CHO) - detected through gravitational microlensing effects

proposed by Paczynski (1986) some years ago. To date, thereis no clear picture as to what these objects are made of. In

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this work, we show that they find a natural explanation asleftover relics from the putative first order cosmic quark -hadron phase transition.

Since the first discovery of MACHO only a few yearsago, a lot of effort has been spent in studying them. Basedon about 13 - 17 Milky Way halo MACHOs detected in thedirection of LMC - the Large Magellanic Cloud (we are notconsidering the events found toward the galactic bulge), theMACHOs are expected to be in the mass range (0.15-0.95)M, with the most probable mass being in the vicinity of0.5 M (Sutherland 1999; Alcock et al. 2000), substantiallyhigher than the fusion threshold of 0.08 M. The MACHOcollaboration suggests that the lenses are in the galactichalo. Assuming that they are subject to the limit on thetotal baryon number imposed by the Big Bang Nucleosyn-thesis (BBN), there have been suggestions that they couldbe white dwarfs (Fields et al. 1998; Freese et al. 2000). It isdifficult to reconcile this with the absence of sufficient ac-tive progenitors of appropriate masses in the galactic halo.Moreover, recent studies have shown that these objects areunlikely to be white dwarfs, even if they were as faint asblue dwarfs, since this will violate some of the very wellknown results of BBN (Freese et al. 2000). There have alsobeen suggestions (Schramm 1998; Jedamzik 1998; Jedamzik& Niemeyer 1999) that they could be primordial black holes(PBHs) ( 1 M ), arising from horizon scale fluctuationstriggered by pre-existing density fluctuations during the cos-mic quark-hadron phase transition. The problem with thissuggestion is that the density contrast necessary for the for-


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2 Banerjee et al.

mation of PBH is much larger than the pre-existing densitycontrast obtained from the common inflationary scenarios.The enhancement contributed by the QCD phase transitionis not large enough for this purpose. As a result a fine tuningof the initial density contrast becomes essential which maystill not be good enough to produce cosmologically relevantamount of PBH (Schmid et al. 1999). Alternately, Evans,Gyuk & Turner (1998) suggested that some of the lenses arestars in the Milky Way disk which lie along the line of sight

to the LMC. Gyuk & Gates (1999) examined a thick diskmodel, which would lower the lens mass estimate. Aubourget al. (1999) suggested that the events could arise from self-lensing of the LMC. Zaritsky & Lin (1997) have argued thatthe lenses are probably the evidence of a tidal tail arisingfrom the interaction of LMC and the Milky Way or even aLMC-SMC (Small Magellanic Cloud) interaction. These ex-planations are primarily motivated by the difficulty of recon-ciling the existence of MACHOs with the known populationsof low mass stars in the galactic disks.

Adopting the viewpoint that the lensing MACHOs areindeed in the Milky Way halo, we propose that they haveevolved out of the quark nuggets which could have beenformed in a first order cosmic quark - hadron phase transi-tion, at a temperature of 100 MeV during the microsec-ond era of the early universe. The order of the deconfine-ment phase transition is an unsettled issue till date. Latticegauge theory suggests that in a pure (i.e. only gluons) SU(3)gauge theory, the deconfinement phase transition is of firstorder. In the presence of dynamical quarks on the lattice,there is no unambiguous way to study the deconfinementtransition; one investigates the chiral transition. Althoughcommonly treated to be equivalent, there is no reason why,or if at all, these two phase transitions should be simulta-neous or of the same order (Alam et al. 1996). The order ofthe chiral phase transition depends rather crucially on thestrange quark mass. If the strange quark is heavy, then thechiral phase transition is probably of first order. Otherwise,it may be of second order. The strange quark mass being ofthe order of the QCD scale, the situation is still controver-sial (Blaizot 1999). There are additional ambiguities arisingfrom finite size effects of the lattice which may tend to maskthe true order of the transition. Our interest here is in thedeconfinement transition. If it is indeed of first order, thefinite size effects which could mask it would be negligiblysmall in the early universe. In such circumstances, Witten(1984) argued, in a seminal paper, that strange quark mattercould be the true ground state of Quantum Chromodynam-ics (QCD) and that a substantial amount of baryon numbercould be trapped in the quark phase which could evolve intostrange quark nuggets (SQNs) through weak interactions.(For a brief review of the formation of SQNs, see Alam,Raha & Sinha (1999).) QCD - motivated studies of baryonevaporation from SQNs have established (Bhattacharjee etal. 1993; Sumiyoshi & Kajino 1991) that primordial SQNswith baryon numbers above 104042 would be cosmologi-cally stable. We have recently shown that without much finetuning, these stable SQNs could provide even the entire clo-sure density ( 1) (Alam et al. 1999). Thus, the entirecold dark matter (CDM) ( CDM 0.3-0.35 ) could easilybe explained by stable SQNs.

We can estimate the size of the SQNs formed in the firstorder cosmic QCD transition in the manner prescribed by

Kodama, Sasaki and Sato (Kodama et al. 1982) in the con-text of the GUT phase transition. For the sake of brevity,let us recapitulate very briefly the salient points here; fordetails, please see Alam et al (1999) and Bhattacharyya etal (2000). Describing the cosmological scale factor R andthe coordinate radius X in the Robertson-Walker metricthrough the relation

ds2 = dt2 + R2dx2

= dt2 + R2{dX2 + X2(sin2d2 + d2)}, (1)one can solve for the evolution of the scale factor R(t) in themixed phase of the first order transition. In a bubble nucle-ation description of the QCD transition, hadronic matterstarts to appear as individual bubbles in the quark-gluonphase. With progressing time, they expand, more and morebubbles appear, coalesce and finally, when a critical fractionof the total volume is occupied by the hadronic phase, acontinuous network of hadronic bubbles form (percolation)in which the quark bubbles get trapped, eventually evolv-ing to SQNs. The time at which the trapping of the falsevacuum (quark phase) happens is the percolation time tp,whereas the time when the phase transition starts is de-

noted by ti. Then, the probability that a spherical regionof co-coordinate radius X lies entirely within the quark bub-bles would obviously depend on the nucleation rate of thebubbles as well as the coordinate radius X(tp, ti) of bubbleswhich nucleated at ti and grew till tp. For a nucleation rateI(t), this probability P(X, tp) is given by

P(X, tp) = exp

4

3

tpti

dtI(t)R3(t)[X + X(tp, ti)]3

. (2)

After some algebra (Bhattacharyya et al. 2000), it can beshown that if all the cold dark matter (CDM) is believed toarise from SQNs, then their size distribution peaks, for rea-sonable nucleation rates, at baryon number 104244, evi-dently in the stable sector. It was also seen that there werealmost no SQNs with baryon number exceeding 104647,comfortably lower than the horizon limit of 1050 baryonsat that time. Since B is only about 0.04 from BBN, CDMin the form of SQNs would correspond to 1051 baryons sothat there should be 1079 such nuggets within the horizonlimit at the microsecond epoch, just after the QCD phasetransition (Alam et al. 1998; Alam et al. 1999). We shallreturn to this issue later on.

It is therefore most relevant to investigate the fate ofthese SQNs. Since the number distribution of the SQNs issharply peaked (Bhattacharyya et al. 2000), we shall assume,for our present purpose, that all the SQNs have the samebaryon number.

The SQNs formed during the cosmic QCD phase tran-sition at T 100 MeV have high masses ( 1044GeV) andsizes (RN 1m) compared to the other particles (like theusual baryons or leptons) which inhabit this primeaval uni-verse. These other particles cannot form structures until thetemperature of the ambient universe falls below a certaincritical temperature characteristic of such particles; till then,

For the QCD bubbles, it is believed that there is a sizable

surface tension which would facilitate spherical bubbles.


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MACHOS from QH Phase Transition 3

they remain in thermodynamic equilibrium with the radia-tion and other species of particles. This characteristic tem-perature is called the freezeout temperature for the corre-sponding particle. Obviously the freezeout occurs earlier formassive particles for the same interaction strength. In thecontext of cosmological expansion of the universe this hasimportant implications ; the frozen objects can form struc-tures. These structures do not participate in the expansionin the sense that the distance between the subparts do not

increase with the scale size and only their number increasesdue to the cosmological scale factor.

For the SQNs, however, the story is especially inter-esting. Even if they continue to be in kinetic equilibriumdue to the radiation pressure (photons and neutrinos) actingon them, their velocity would be extremely non relativistic.Also their mutual separation would be considerably largerthan their radii; for example, at T 100MeV, the mutualseparation between the SQNs (of size 1044 baryons) isestimated to b e around 300m. It is then obvious that theSQNs do not lend themselves to be treated in a hydrody-namical framework; they behave rather like discrete bodiesin the background of the radiation fluid. They thus expe-rience the radiation pressure, quite substantial because oftheir large surface area as well as the gravitational potentialdue to the other SQNs.

In such a situation, one might be tempted to assumethat since the SQNs are distributed sparsely in space andinteract only feebly with the other SQNs through gravita-tional interaction, they might as well remain forever in thatstate. This, in fact, is quite wrong, as we demonstrate below.

The fact that the nuggets remain almost static is hardlyan issue which requires justification. The two kinds of mo-tion that they can have are random thermal motion andthe motion in the gravitational well provided by the otherSQNs. This other kind of motion is typically estimated us-ing the virial theorem, treating the SQNs as a system of par-ticles moving under mutual gravitational interaction (Bhatia2001; Peebles 1980). The kinetic energy ( K ) and potentialenergy V of the nuggets at temperature T = 100 MeV canbe estimated as,

K=3

2NkbT

V =i,j

GMiMjRi,j

=GM2N2

2Rav(3)

where kb is the Boltzmann constant, Mi,Mj are the massesof the ith and jth nugget, Ri,j is the distance between themand Rav is the average inter-nugget distance. Substitutingthe number of nuggets N = 107, the baryon number of eachnugget to be 1044 and Rav = 300m, one gets K= 2.4

104

and V = 3.09 1035 (in MKS units) so that the ratio of Kand V

2becomes 1039. Thus it is impossible for these

objects to form stable systems, orbiting round each other.On the other hand the smallness of the kinetic energy showsthat gravitational collapse might be a possible fate.

Such, of course, would not be the case for any other mas-sive particles like baryons; their masses being much smallerthan SQN, the kinetic energy would continue to be verylarge till very low temperatures. More seriously, the Virialtheorem can be applied only to systems whose motion is sus-tained. For SQNs, a notable property is that they become

more and more bound if they grow in size. Thus SQNs wouldabsorb baryons impinging on them and grow in size. Also,if two SQNs collide, they would naturally tend to merge. Inall such cases, they would lose kinetic energy, making theVirial theorem inapplicable.

One can argue that the mutual interaction between uni-formly dispersed particles would prevent these particles fromforming a collapsed structure, but that argument holds onlyin a static and infinite universe, which we know our universe

is not. Also a perfectly uniform distribution of discrete bod-ies is an unrealistic idealization and there must exist somenet gravitational attraction on each SQN. The only agentthat can prevent a collapse under this gravitational pull isthe radiation pressure, and indeed its effect remains quitesubstantial until the drop in the temperature of the ambi-ent universe weakens the radiation pressure below a certaincritical value. In what follows, we try to obtain an estimatefor the point of time at which this can happen.

It should be mentioned at this juncture that for thesystem of discrete SQNs suspended in the radiation fluid, adetailed numerical simulation would be essential before anydefinite conclusion about their temporal evolution can bearrived at. This is a quite involved problem, especially sincethe number of SQNs within the event horizon, as also theirmutual separation, keeps increasing with time. Our purposein the present work is to examine whether such an effortwould indeed be justified.

Let us now consider the possibility of two nuggets co-alescing together under gravity, overcoming the radiationpressure. The mean separation of these nuggets and hencetheir gravitational interaction are determined by the tem-perature of the universe. If the entire CDM comes fromSQNs, the total baryon number contained in them withinthe horizon at the QCD transition temperature ( 100MeV) would be 1051 (see above). For SQNs of baryonnumber bN each, the number of SQNs within the horizonat that time would be just 1051/bN. Now, in the radia-tion dominated era the temperature dependence of densitynN T3, horizon volume VH varies with time as t3, i.e.VH T6 and hence the variation of the total number in-side the horizon volume will be NN T3. So at any latertime, the number of SQNs within the horizon ( NN ) andtheir density ( nN ) as a function of temperature would begiven by :

NN(T) = 1051

bN

100MeV

T

3(4)

nN(T) =NNVH

=3NN

4(2t)3(5)

where the time t and the temperature T are related in the

radiation dominated era by the relation :

t = 0.3g1/2mplT2

(6)

with g being 17.25 after the QCD transition (Alam et al.1999).

From the above, it is obvious that the density of SQNsdecreases as t3/2 so that their mutual separation increasesas t1/2. Therefore, the force of their mutual gravitational pullwill decrease as t1. On the other hand, the force due to theradiation pressure (photons and neutrinos) resisting motionunder gravity would be proportional to the radiation energy


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4 Banerjee et al.

density, which decreases as T4 or t2. It is thus reasonableto expect that at some time, not too distant, the gravita-tional pull would win over the radiation pressure, causingthe SQNs to coalesce under their mutual gravitational pull.The expression for the gravitational force as a function oftemperature T can written as :

Fgrav =GbN

2mn2

rnn(T)2

(7)

where bN is the baryon number of each SQN and mn is thebaryon mass. rnn(T) is the mean separation between twonuggets and is given by the cube root of the ratio of totalvolume available and the total number of nuggets

=1.114 1012c3

T3(8)

The force due to the radiation pressure on the nuggets maybe roughly estimated as follows. We consider two objects(of the size of a typical SQN) approaching each other due togravitational interaction, overcoming the resistance due tothe radiation pressure. The usual isotropic radiation pres-sure is 1

3c2, where is the total energy density, includ-

ing all relativistic species. The nuggets will have to over-come an additional pressure resisting their mutual motion,which is given by 1

3c2( 1); the additional pressure arises

from a compression of the radiation fluid due to the mo-tion of the SQN. The moving SQN would become a pro-late ellipsoid (with its minor axis in the direction of motiondue to Lorentz contraction), whose surface area is given by

2R2N(1+sin1

). The eccentricity is related to the Lorentz

factor as =

21

. For small values of (small ),

sin1 , so that the surface area becomes 2R2N+1 .Thus the total radiation force resisting the motion of SQNsis

Frad =1

3radcvfall(R

2

N) (9)

where rad is the total energy density at temperature T, vfallor c is the velocity of SQNs determined by mutual grav-itational field and is 1/

1 2. The quantities Frad,

and all depend on the temperature of the epoch underconsideration. (It is worth mentioning at this point that thet dependence ofFrad is actually t

5/2, sharper than the t2

estimated above, because of the vfall, which goes as t1/4.)

The ratio of these two forces is plotted against tempera-ture in figure 1 for two SQNs with initial baryon number1042 each. It is obvious from the figure that ratio Fgrav/Fradis very small initially. As a result, the nuggets will remainseparated due to the radiation pressure. For temperatureslower than a critical value Tcl, the gravitational force startsdominating, facilitating the coalescence of the SQNs undermutual gravity.

Let us now estimate the mass of the clumped SQNs,assuming that all of them within the horizon at the criticaltemperature will coalesce together. This is in fact a conser-vative estimate, since the SQNs, although starting to movetoward one another at Tcl, will take a finite time to actuallycoalesce, during which interval more SQNs will arrive withinthe horizon.

In table 1, we show the values of Tcl for SQNs of dif-ferent initial baryon numbers along with the final masses of

10-2

10-1

100

101

102

10-6

10-4

10-2

100

102

104

106

108

Fgrav

/Frad

T (Temperature in MeV)

Figure 1. Variation of the ratio Fgrav/Frad with temperature.

The dot represents the point where the ratio assumes the value1.

Table 1. Critical temperatures ( Tcl ) of SQNs of different initial

sizes bN, the total number NN of SQNs that coalesce togetherand their total final mass in solar mass units.

bN Tcl (MeV) NN M/M

1042 1.6 2.44 1014 0.24

1044 4.45 1.13 1011 0.01

1046 20.6 1.1 107 0.0001

the clumped SQNs under the conservative assumption men-tioned above.

It is obvious that there can be no further clumping ofthese already clumped SQNs; the density of such objectswould be too small within the horizon for further clump-ing. Thus these objects would survive till today and perhapsmanifest themselves as MACHOs. It is to be reiterated thatthe masses of the clumped SQNs given in table 1 are thelower limits and the final masses of these MACHO candi-dates will be larger. (The case for bN = 10

46 is not of much


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MACHOS from QH Phase Transition 5

interest, especially since such high values of bN are unlikelyfor the reasonable nucleation rates (Bhattacharyya et al.1999; Bhattacharyya et al. 2000); we therefore restrict our-selves to the other cases in table 1 in what follows.) A moredetailed estimate of the masses will require a detailed sim-ulation, but very preliminary estimates indicate that theycould be 2-3 times bigger than the values quoted in table 1.

The total number of such clumped SQNs ( Nmacho )within the horizon today is evaluated in the following way.

With the temperature 30K and time 4 1017 sec-onds, the total amount of visible baryons within the hori-zon volume can be evaluated using photon to baryon ratio 1010. The amount of baryons in the CDM will be CDM

Btimes the total number of visible baryons. This comes outto be 1.61079, CDM and B being 0.3 and 0.01 respec-tively. The total number of baryons in a MACHO is bNNNi.e. 2.44 1056 and 1.13 1055 for initial nugget sizes 1042and 1044 respectively. The quantities bN and NN are takenfrom the Table 1. So dividing the total number of baryonsin CDM by that in a MACHO, the Nmacho comes out to bein the range 102324.

We can also mention here that if the MACHOs are in-deed made up of quark matter, then they cannot grow toarbitrarily large sizes. Within the (phenomenological) Bagmodel picture (Chodos et al. 1974) of QCD confinement,where a constant vacuum energy density (called the Bagconstant) in a cavity containing the quarks serves to keepthem confined within the cavity, we have earlier investigated(Banerjee et al. 2000) the upper limit on the mass of astro-physical compact quark matter objects. It was found thatfor a canonical Bag constant B of (145 MeV) 4, this limitcomes out to be 1.4 M. The collapsed SQNs are safely be-low this limit. (It should be remarked here that althoughthe value ofB in the original MIT bag model is taken tobe B 1/4 = 145 MeV from the low mass hadronic spectrum,there exist other variants of the Bag model (Hasenfratz &Kuti 1978) , where higher values of B are required. Evenfor B 1/4 = 245 MeV, this limit comes down to 0.54 M(Banerjee et al. 2000), which would still admit such SQN.

As a consistency check, we can perform a theoreticalestimate of the abundance of such MACHOs in the galactichalo which is conventionally given by the optical depth. Theoptical depth is the probability that at any instant of time agiven star is within an angle E of a lens, the lens being themassive body (in our case MACHO) which causes the de-flection of light. In other words, optical depth is the integralover the number density of lenses times the area enclosedby the Einstein ring of each lens. The expression for opticaldepth can be written as (Narayan & Bartelmann 1999):

=

4G

c2 D2

s

(x)x(1 x)dx (10)where Ds is the distance between the observer and thesource, G is the gravitational constant and x = DdDs

1,Dd being the distance between the observer and the lens. Inparticular is the mass-density of the MACHOs, which is ofthe form = 0

1r2

in the naive spherical halo model, whichwe have adopted in our calculations. In the present case 0is given by

0 =Mmacho Nmacho

4R(11)

where R =

D2e + D2s + 2DeDs cos, and De being theinclination of the LMC and the distance of observer (earth)from the Galactic centre respectively. Mmacho and Nmachoare the mass of a MACHO and the total number of MACHOsin the Milky Way halo.

The total visible mass of the Milky Way ( 1.6 1011M ) corresponds to 2 1068 baryons. This corre-sponds to a factor of 2 109 of all the visible baryonswithin the present horizon. Scaling the number of clumped

SQNs within the horizon by the same factor yields a to-tal number of MACHOs, Nmacho 101314 in the MilkyWay halo for the range of baryon number of initial nuggetsbN = 10

4244. The value of De and Ds are taken to be 10and 50 kpc, respectively. The value of the inclination angleused here is 40 degrees. Using these values for a naive in-verse square spherical model comprising such objects uptothe LMC, we obtain an optical depth of 106107. Theuncertainty in this value is mainly governed by the valueof , CDM and B, and to a lesser extent by the specifichalo model. This value compares reasonably well with theobserved value and may be taken as a measure of reliabilityin the proposed model.

As an interesting corollary, let us mention that the sce-nario presented here could have other important astrophys-ical significance. The origin of cosmic rays of ultra-high en-ergy 1020 eV continues to be a puzzle. One of the pro-posed mechanisms (Bhattacharjee & Sigl 2001) envisages atop-down scenario which does not require an accelerationmechanism and could indeed originate within our galactichalo. For our picture, such situations could easily arise fromthe merger of two or more such MACHOs, which would shedthe extra matter so as to remain within the upper mass limitmentioned above. This is currently under active investiga-tion.

We thus conclude that gravitational clumping of the pri-mordial SQNs formed in a first order cosmic quark - hadronphase transition appears to be a plausible and natural expla-nation for the observed halo MACHOs. It is quite remark-able that we obtain quantitative agreement with the exper-imental values without having to introduce any adjustableparameters or any fine-tuning whatsoever. We may finishby quoting a famous teaching of John Archibald Wheeler,One should never do a calculation unless one knows an an-swer. Our attempt in this work has been to find an answerso that a calculation (in this case, a detailed simulation ofthe collapsing SQNs) can be embarked upon.

The work of SB was supported in part by the Council ofScientific and Industrial Research, Govt. of India. SR wouldlike to thank the Nuclear Theory group at Brookhaven Na-tional Laboratory, USA, where this work was initiated, fortheir warm hospitality.

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Alam J., Raha S., Sinha B., 1999, ApJ, 513, 572Alcock C. et al., 1993, Nat, 365, 621Alcock C. et al., 2000, ApJ, 542, 281

Aubourg E. et al., 1993, Nat, 365, 623Aubourg E. et al., 1999, A&A, 347, 850

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