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Massive Compact Halo Objects from the Relics of the Cosmic Quark-Hadron Transition

The existence of compact gravitational lenses, with masses around 0.5 (M_{\odot}), has been reported 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 nuclear fusion. In this work, we show that they find a natural explanation as being the evolutionary product of the metastable false vacuum domains (the so-called strange quark nuggets) formed in a first order cosmic quark-hadron transition.

One of the abiding mysteries in the so-called standard cosmological model is the nature of the dark matter. It is universally accepted that there is an abundance of matter in the universe which is non-luminous, due to its very weak interaction, if at all, with the other forms of matter, excepting of course the gravitational attraction. The present consensus (for a review, see (Turner 1999;Turner 2000)) based on recent experimental data is that the universe is flat and that a sizable amount of the dark matter is "cold", i.e. nonrelativistic, at the time of decoupling (we are not addressing the issue of dark energy in this work). Speculations as to the nature of dark matter are numerous, often bordering on the exotic, and searches for such exotic matter is a very active field of astroparticle physics. In recent years, there has been experimental evidence (Alcock et al. 1993;Aubourg et al. 1993)  proposed by Paczynski (1986) some years ago. To date, there is no clear picture as to what these objects are made of. In this work, we show that they find a natural explanation as leftover relics from the putative first order cosmic quarkhadron phase transition.
Since the first discovery of MACHO only a few years ago, a lot of effort has been spent in studying them. Based on about 13 -17 Milky Way halo MACHOs detected in the direction of LMC -the Large Magellanic Cloud (we are not considering the events found toward the galactic bulge), the MACHOs are expected to be in the mass range (0.15-0.95) M⊙, with the most probable mass being in the vicinity of 0.5 M⊙ (Sutherland 1999;Alcock et al. 2000), substantially higher than the fusion threshold of 0.08 M⊙. The MACHO collaboration suggests that the lenses are in the galactic halo. Assuming that they are subject to the limit on the total baryon number imposed by the Big Bang Nucleosynthesis (BBN), there have been suggestions that they could be white dwarfs (Fields et al. 1998;Freese et al. 2000). It is difficult to reconcile this with the absence of sufficient active progenitors of appropriate masses in the galactic halo. Moreover, recent studies have shown that these objects are unlikely to be white dwarfs, even if they were as faint as blue dwarfs, since this will violate some of the very well known results of BBN (Freese et al. 2000). There have also been suggestions (Schramm 1998;Jedamzik 1998;Jedamzik & Niemeyer 1999) that they could be primordial black holes c 0000 RAS (PBHs) ( ∼ 1 M⊙ ), arising from horizon scale fluctuations triggered by pre-existing density fluctuations during the cosmic quark-hadron phase transition. The problem with this suggestion is that the density contrast necessary for the formation of PBH is much larger than the pre-existing density contrast obtained from the common inflationary scenarios. The enhancement contributed by the QCD phase transition is not large enough for this purpose. As a result a fine tuning of the initial density contrast becomes essential which may still not be good enough to produce cosmologically relevant amount of PBH (Schmid et al. 1999). Alternately, Evans, Gyuk & Turner (1998) suggested that some of the lenses are stars in the Milky Way disk which lie along the line of sight to the LMC. Gyuk & Gates (1999) examined a thick disk model, which would lower the lens mass estimate. Aubourg et al. (1999) suggested that the events could arise from selflensing of the LMC. Zaritsky & Lin (1997) have argued that the lenses are probably the evidence of a tidal tail arising from the interaction of LMC and the Milky Way or even a LMC-SMC (Small Magellanic Cloud) interaction. These explanations are primarily motivated by the difficulty of reconciling the existence of MACHOs with the known populations of low mass stars in the galactic disks.
Adopting the viewpoint that the lensing MACHOs are indeed in the Milky Way halo, we propose that they have evolved out of the quark nuggets which could have been formed in a first order cosmic quark -hadron phase transition, at a temperature of ∼ 100 MeV during the microsecond era of the early universe. The order of the deconfinement phase transition is an unsettled issue till date. Lattice gauge theory suggests that in a pure (i.e. only gluons) SU (3) gauge theory, the deconfinement phase transition is of first order. In the presence of dynamical quarks on the lattice, there is no unambiguous way to study the deconfinement transition; one investigates the chiral transition. Although commonly treated to be equivalent, there is no reason why, or if at all, these two phase transitions should be simultaneous or of the same order (Alam et al. 1996). The order of the chiral phase transition depends rather crucially on the strange quark mass. If the strange quark is heavy, then the chiral phase transition is probably of first order. Otherwise, it may be of second order. The strange quark mass being of the order of the QCD scale, the situation is still controversial (Blaizot 1999). There are additional ambiguities arising from finite size effects of the lattice which may tend to mask the true order of the transition. Our interest here is in the deconfinement transition. If it is indeed of first order, the finite size effects which could mask it would be negligibly small in the early universe. In such circumstances, Witten (1984) argued, in a seminal paper, that strange quark matter could be the true ground state of Quantum Chromodynamics (QCD) and that a substantial amount of baryon number could be trapped in the quark phase which could evolve into strange quark nuggets (SQNs) through weak interactions. (For a brief review of the formation of SQNs, see .) QCD -motivated studies of baryon evaporation from SQNs have established (Bhattacharjee et al. 1993;Sumiyoshi & Kajino 1991) that primordial SQNs with baryon numbers above ∼ 10 40−42 would be cosmologically stable. We have recently shown that without much fine tuning, these stable SQNs could provide even the entire closure density (Ω ∼ 1) ). Thus, the entire cold dark matter (CDM) ( ΩCDM ∼ 0.3-0.35 ) could easily be explained by stable SQNs.
We can estimate the size of the SQNs formed in the first order cosmic QCD transition in the manner prescribed by Kodama, Sasaki and Sato (Kodama et al. 1982) in the context of the GUT phase transition. For the sake of brevity, let us recapitulate very briefly the salient points here; for details, please see Alam et al (1999) and Bhattacharyya et al (2000). Describing the cosmological scale factor R and the coordinate radius X in the Robertson-Walker metric through the relation one can solve for the evolution of the scale factor R(t) in the mixed phase of the first order transition. In a bubble nucleation description of the QCD transition, hadronic matter starts to appear as individual bubbles in the quark-gluon phase. With progressing time, they expand, more and more bubbles appear, coalesce and finally, when a critical fraction of the total volume is occupied by the hadronic phase, a continuous network of hadronic bubbles form (percolation) in which the quark bubbles get trapped, eventually evolving to SQNs. The time at which the trapping of the false vacuum (quark phase) happens is the percolation time tp, whereas the time when the phase transition starts is denoted by ti. Then, the probability that a spherical † † region of co-coordinate radius X lies entirely within the quark bubbles would obviously depend on the nucleation rate of the bubbles as well as the coordinate radius X(tp, ti) of bubbles which nucleated at ti and grew till tp. For a nucleation rate I(t), this probability P (X, tp) is given by After some algebra (Bhattacharyya et al. 2000), it can be shown that if all the cold dark matter (CDM) is believed to arise from SQNs, then their size distribution peaks, for reasonable nucleation rates, at baryon number ∼ 10 42−44 , evidently in the stable sector. It was also seen that there were almost no SQNs with baryon number exceeding 10 46−47 , comfortably lower than the horizon limit of ∼ 10 50 baryons at that time. Since ΩB is only about 0.04 from BBN, ΩCDM in the form of SQNs would correspond to ∼ 10 51 baryons so that there should be 10 7−9 such nuggets within the horizon limit at the microsecond epoch, just after the QCD phase transition (Alam et al. 1998;Alam et al. 1999). We shall return to this issue later on. It is therefore most relevant to investigate the fate of these SQNs. Since the number distribution of the SQNs is sharply peaked (Bhattacharyya et al. 2000), we shall assume, for our present purpose, that all the SQNs have the same baryon number.
The SQN's formed during the cosmic QCD phase transition at T ∼ 100 MeV have high masses (∼ 10 44 GeV) and sizes (RN ∼ 1m) compared to the other particles (like the usual baryons or leptons) which inhabit this primeaval universe. These other particles cannot form structures until the temperature of the ambient universe falls below a certain critical temperature characteristic of such particles; till then, they remain in thermodynamic equilibrium with the radiation and other species of particles. This characteristic temperature is called the freezeout temperature for the corresponding particle. Obviously the freezeout occurs earlier for massive particles for the same interaction strength. In the context of cosmological expansion of the universe this has important implications ; the 'frozen' objects can form structures. These structures do not participate in the expansion in the sense that the distance between the subparts do not increase with the scale size and only their number increases due to the cosmological scale factor.
For the SQN's, however, the story is especially interesting. Even if they continue to be in kinetic equilibrium due to the radiation pressure (photons and neutrinos) acting on them, their velocity would be extremely non relativistic. Also their mutual separation would be considerably larger than their radii; for example, at T ∼ 100MeV, the mutual separation between the SQN's (of size ∼ 10 44 baryons) is estimated to be around ∼ 300m. It is then obvious that the SQN's do not lend themselves to be treated in a hydrodynamical framework; they behave rather like discrete bodies in the background of the radiation fluid. They thus experience the radiation pressure, quite substantial because of their large surface area as well as the gravitational potential due to the other SQN's.
In such a situation, one might be tempted to assume that since the SQN's are distributed sparsely in space and interact only feebly with the other SQN's through gravitational interaction, they might as well remain forever in that state. This, in fact, is quite wrong, as we demonstrate below.
The fact that the nuggets remain almost static is hardly an issue which requires justification. The two kinds of motion that they can have are random thermal motion and the motion in the gravitational well provided by the other SQN's. This other kind of motion is typically estimated using the virial theorem, treating the SQN's as a system of particles moving under mutual gravitational interaction (Bhatia 2001;Peebles 1980). The kinetic energy ( K ) and potential energy V of the nuggets at temperature T = 100 MeV can be estimated as, where k b is the Boltzmann constant, Mi, Mj are the masses of the ith and jth nugget, Ri,j is the distance between them and Rav is the average inter-nugget distance. Substituting the number of nuggets N = 10 7 , the baryon number of each nugget to be 10 44 and Rav = 300m, one gets K = 2.4 × 10 −4 and V = 3.09 × 10 35 (in MKS units) so that the ratio of K and V 2 becomes ∼ 10 −39 . 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 shows that gravitational collapse might be a possible fate.
Such, of course, would not be the case for any other massive particles like baryons; their masses being much smaller than SQN, the kinetic energy would continue to be very large till very low temperatures. More seriously, the Virial theorem can be applied only to systems whose motion is sustained. For SQNs, a notable property is that they become more and more bound if they grow in size. Thus SQNs would absorb baryons impinging on them and grow in size. Also, if two SQNs collide, they would naturally tend to merge. In all such cases, they would lose kinetic energy, making the Virial theorem inapplicable.
One can argue that the mutual interaction between uniformly dispersed particles would prevent these particles from forming a collapsed structure, but that argument holds only in a static and infinite universe, which we know our universe is not. Also a perfectly uniform distribution of discrete bodies is an unrealistic idealization and there must exist some net gravitational attraction on each SQN. The only agent that can prevent a collapse under this gravitational pull is the radiation pressure, and indeed its effect remains quite substantial until the drop in the temperature of the ambient universe weakens the radiation pressure below a certain critical value. In what follows, we try to obtain an estimate for the point of time at which this can happen.
It should be mentioned at this juncture that for the system of discrete SQN's suspended in the radiation fluid, a detailed numerical simulation would be essential before any definite conclusion about their temporal evolution can be arrived at. This is a quite involved problem, especially since the number of SQN's within the event horizon, as also their mutual separation, keeps increasing with time. Our purpose in the present work is to examine whether such an effort would indeed be justified.
Let us now consider the possibility of two nuggets coalescing together under gravity, overcoming the radiation pressure. The mean separation of these nuggets and hence their gravitational interaction are determined by the temperature of the universe. If the entire CDM comes from SQNs, the total baryon number contained in them within the horizon at the QCD transition temperature ( ∼ 100 MeV) would be ∼ 10 51 (see above). For SQNs of baryon number bN each, the number of SQNs within the horizon at that time would be just 10 51 /bN . Now, in the radiation dominated era the temperature dependence of density nN ∼ T 3 , horizon volume VH varies with time as t 3 , i.e. VH ∼ T −6 and hence the variation of the total number inside the horizon volume will be NN ∼ T −3 . So at any later time, the number of SQNs within the horizon ( NN ) and their density ( nN ) as a function of temperature would be given by : where the time t and the temperature T are related in the radiation dominated era by the relation : with g * being ∼ 17.25 after the QCD transition ). From the above, it is obvious that the density of SQNs decreases as t −3/2 so that their mutual separation increases as t 1/2 . Therefore, the force of their mutual gravitational pull will decrease as t −1 . On the other hand, the force due to the radiation pressure (photons and neutrinos) resisting motion under gravity would be proportional to the radiation energy density, which decreases as T 4 or t −2 . It is thus reasonable to expect that at some time, not too distant, the gravitational pull would win over the radiation pressure, causing the SQNs to coalesce under their mutual gravitational pull. The expression for the gravitational force as a function of temperature T can written as : where bN is the baryon number of each SQN and mn is the baryon mass.rnn(T ) is the mean separation between two nuggets and is given by the cube root of the ratio κ of total volume available and the total number of nuggets The force due to the radiation pressure on the nuggets may be roughly estimated as follows. We consider two objects (of the size of a typical SQN) approaching each other due to gravitational interaction, overcoming the resistance due to the radiation pressure. The usual isotropic radiation pressure is 1 3 ρc 2 , where ρ is the total energy density, including all relativistic species. The nuggets will have to overcome an additional pressure resisting their mutual motion, which is given by 1 3 ρc 2 (γ − 1); the additional pressure arises from a compression of the radiation fluid due to the motion of the SQN. The moving SQN would become a prolate ellipsoid (with its minor axis in the direction of motion due to Lorentz contraction), whose surface area is given by 2πR 2 N (1+ sin −1 ǫ γǫ ). The eccentricity ǫ is related to the Lorentz factor γ as ǫ = √ γ 2 −1 γ . For small values of ǫ (small γ), sin −1 ǫ ∼ ǫ, so that the surface area becomes 2πR 2 N γ+1 γ . Thus the total radiation force resisting the motion of SQNs is where ρ rad is the total energy density at temperature T , v fall or βc is the velocity of SQNs determined by mutual gravitational field and γ is 1/ 1 − β 2 . The quantities F rad , β and γ all depend on the temperature of the epoch under consideration. (It is worth mentioning at this point that the t dependence of F rad is actually t −5/2 , sharper than the t −2 estimated above, because of the v fall , which goes as t −1/4 .) The ratio of these two forces is plotted against temperature in figure 1 for two SQNs with initial baryon number 10 42 each. It is obvious from the figure that ratio Fgrav/F rad is very small initially. As a result, the nuggets will remain separated due to the radiation pressure. For temperatures lower than a critical value T cl , the gravitational force starts dominating, facilitating the coalescence of the SQNs under mutual gravity.
Let us now estimate the mass of the clumped SQNs, assuming that all of them within the horizon at the critical temperature will coalesce together. This is in fact a conservative estimate, since the SQNs, although starting to move toward one another at T cl , will take a finite time to actually coalesce, during which interval more SQNs will arrive within the horizon.
In table 1, we show the values of T cl for SQNs of dif-  ferent initial baryon numbers along with the final masses of the clumped SQNs under the conservative assumption mentioned above. It is obvious that there can be no further clumping of these already clumped SQNs; the density of such objects would be too small within the horizon for further clumping. Thus these objects would survive till today and perhaps manifest themselves as MACHOs. It is to be reiterated that the masses of the clumped SQN's given in table 1 are the lower limits and the final masses of these MACHO candidates will be larger. (The case for bN = 10 46 is not of much interest, especially since such high values of bN are unlikely for the reasonable nucleation rates (Bhattacharyya et al. 1999;Bhattacharyya et al. 2000); we therefore restrict ourselves to the other cases in table 1 in what follows.) A more detailed estimate of the masses will require a detailed simulation, but very preliminary estimates indicate that they could be 2-3 times bigger than the values quoted in table 1.
The total number of such clumped SQNs ( N macho ) within the horizon today is evaluated in the following way. With the temperature ∼ 3 0 K and time ∼ 4 × 10 17 seconds, the total amount of visible baryons within the horizon volume can be evaluated using photon to baryon ratio η ∼ 10 −10 . The amount of baryons in the CDM will be Ω CDM Ω B times the total number of visible baryons. This comes out to be ∼ 1.6 × 10 79 , ΩCDM and ΩB being 0.3 and 0.01 respectively. The total number of baryons in a MACHO is bN ×NN i.e. 2.44 × 10 56 and 1.13 × 10 55 for initial nugget sizes 10 42 and 10 44 respectively. The quantities bN and NN are taken from the Table 1. So dividing the total number of baryons in CDM by that in a MACHO, the N macho comes out to be in the range ∼ 10 23−24 .
We can also mention here that if the MACHOs are indeed made up of quark matter, then they cannot grow to arbitrarily large sizes. Within the (phenomenological) Bag model picture (Chodos et al. 1974) of QCD confinement, where a constant vacuum energy density (called the Bag constant) in a cavity containing the quarks serves to keep them confined within the cavity, we have earlier investigated (Banerjee et al. 2000) the upper limit on the mass of astrophysical compact quark matter objects. It was found that for a canonical Bag constant B of (145 MeV) 4 , this limit comes out to be 1.4 M⊙. The collapsed SQNs are safely below this limit. (It should be remarked here that although the value of B in the original MIT bag model is taken to be 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. Even for 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 theoretical estimate of the abundance of such MACHO's in the galactic halo which is conventionally given by the optical depth. The optical depth is the probability that at any instant of time a given star is within an angle θE of a lens, the lens being the massive body (in our case MACHO) which causes the deflection of light. In other words, optical depth is the integral over the number density of lenses times the area enclosed by the Einstein ring of each lens. The expression for optical depth can be written as (Narayan & Bartelmann 1999): where Ds is the distance between the observer and the source, G is the gravitational constant and x = D d Ds −1 , D d being the distance between the observer and the lens. In particular ρ is the mass-density of the MACHOs, which is of the form ρ = ρ0 1 r 2 in the naive spherical halo model, which we have adopted in our calculations. In the present case ρ0 is given by where R = D 2 e + D 2 s + 2DeDs cos φ, φ and De being the inclination of the LMC and the distance of observer (earth) from the Galactic centre respectively. M macho and N macho are the mass of a MACHO and the total number of MACHOs in the Milky Way halo.
The total visible mass of the Milky Way ( ∼ 1.6 × 10 11 M⊙ ) corresponds to ∼ 2 × 10 68 baryons. This corresponds to a factor of ∼ 2 × 10 −9 of all the visible baryons within the present horizon. Scaling the number of clumped SQNs within the horizon by the same factor yields a total number of MACHOs, N macho ∼ 10 13−14 in the Milky Way halo for the range of baryon number of initial nuggets bN = 10 42−44 . The value of De and Ds are taken to be 10 and 50 kpc, respectively. The value of the inclination angle used here is 40 degrees. Using these values for a naive inverse square spherical model comprising such objects upto the LMC, we obtain an optical depth of ∼ 10 −6 − 10 −7 . The uncertainty in this value is mainly governed by the value of η, ΩCDM and ΩB, and to a lesser extent by the specific halo model. This value compares reasonably well with the observed value and may be taken as a measure of reliability in the proposed model.
As an interesting corollary, let us mention that the scenario presented here could have other important astrophysical significance. The origin of cosmic rays of ultra-high energy ≥ 10 20 eV continues to be a puzzle. One of the proposed mechanisms (Bhattacharjee & Sigl 2001) envisages a top-down scenario which does not require an acceleration mechanism and could indeed originate within our galactic halo. For our picture, such situations could easily arise from the merger of two or more such MACHOs, which would shed the extra matter so as to remain within the upper mass limit mentioned above. This is currently under active investigation.
We thus conclude that gravitational clumping of the primordial SQNs formed in a first order cosmic quark -hadron phase transition appears to be a plausible and natural explanation for the observed halo MACHOs. It is quite remarkable that we obtain quantitative agreement with the experimental values without having to introduce any adjustable parameters or any fine-tuning whatsoever. We may finish by quoting a famous teaching of John Archibald Wheeler, "One should never do a calculation unless one knows an answer". Our attempt in this work has been to find an answer so that a calculation (in this case, a detailed simulation of the collapsing SQNs) can be embarked upon.
The work of SB was supported in part by the Council of Scientific and Industrial Research, Govt. of India. SR would like to thank the Nuclear Theory group at Brookhaven National Laboratory, USA, where this work was initiated, for their warm hospitality.