VOLUME 76, NUMBER 20 P H Y S I C A L R E V I E W L E T T E R S 13 MAY 1996

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Comment on “Brown Dwarfs, Quark Stars, andQuark-Hadron Phase Transition”

In a recent Letter, Cottingham, Kalafatis, and Vinh M[1] studied, within the Lee-Wick model, the formatioof quark stars in the cosmological quark-hadron phtransition. They conclude that the formation of quastars with the solar baryon numberNØ , 1057 requires ahigh degree of supercooling, which can be achieved wa reasonable choice of the parameters of the Lee-Wmodel.

We show below that the cosmological quark-hadphase transition in the Lee-Wick model with such a hdegree of supercooling cannot be completed and thuquark stars can be formed in this scenario. Conclusof Ref. [1] are based on the assumption that the expanof the Universe is dominated by radiation during tphase transition, with the scale factorRstd ,

pt. This

assumption is, however, unjustified.For cosmological application, the potential energy of

Lee-Wick model has to be chosen properly. The vsmall value of the cosmological constant requires tthe energy of the true vacuum,s ­ svac, is essentiallyzero,Ussvacd ­ 0. Then the energy of the false vacuus ­ 0, isUs0d ­ B . 0. With this choice ofU, when thedegree of supercooling is as high as required in Ref.the phase transition is slow and the expansion ofUniverse soon becomes dominated by the vacuum enB. As a result, bubbles of a new phase do not percola

To show that the phase transition is slow we compthe bubble nucleation rateGstd and the expansion ratHstd by calculating the dimensionless quantityestd ­GstdyHstd4 [2]. The expansion rate,Hstd ­ ÙRstdyRstd,satisfies the equation

H2 ­8p

3G

√rc

R41 B

!,

whererc is the energy density of massless particles atcritical temperatureTc, rc ­ geffaT 4

c y2, with geff ø 32in the Lee-Wick model. Normalizing the scale factortc, Rstcd ­ 1, where timetc corresponds to the criticatemperatureTc, we find the solution

Rstd ­ s1yp

2lde2atytc

pe4atytc 2 1 ,

wherel2 ­ Byrc ø 1y5 and a ­ lnsp

l2 1 1 1 ldy2 ø0.2.

The bubble nucleation rateGstd for adiabatic expansionTR ­ Tc is calculated using Eqs. 6 and 7 of Ref. [

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with TstdyTc ­ 1yRstd. As shown in [1],Gstd is alwaysstrongly suppressed except in a narrow time interaround timetn when almost all the bubbles are nucleateCorrespondingly, the values ofestd are always lower thanthe maximum valueemaxstnd ­ 7.5 3 10210 occurring attn ø 2tc when the scale factor isRstnd ø

p3. The values

of estd are always below the percolation threshold [2The nucleation of bubbles through the quantum tunneis negligible here.

The density of nucleated bubbles attn is of the orderof 10230 cm23, and a typical distance between bubblesof the order of 1010cm. It exceeds the Hubble distanc1yHstnd by a factor of the order of 1000. The separatiof neighboring bubbles is so big that there is no chafor them to coalesce during the intermediate evolutiwhenRstd approaches the exponential behavior. Afterexponential expansion sets in, the bubble density vanisexponentially. Similarly, as in the old inflationary moddue to Guth [3], the phase transition is never comple[2].

The Lee-Wick model with a high degree of supercoing leads to the unphysical scenario for the cosmolical quark-hadron phase transition. However, moderchanges of the Lee-Wick model parameters can drasticlower the degree of supercooling. The phase transiwith the low degree of supercooling is quickly completand no unphysical behavior, found for a strongly supcooled system, in encountered. In this case, however,smaller aggregates of quark matter, with the baryon nuberNB ø 1025NØ, could be formed.

This research is partially supported by the Polish StCommittee for Scientific Research (KBN), Grants NoP03B 083 08 and No. 2 P03D 001 09.

S. Kubis and M. KutscheraH. Niewodniczan´ski Institute of Nuclear Physicsul. Radzikowskiego 152, 31-342 Kraków, Poland

Received 7 November 1995 [S0031-9007(96)00065PACS numbers: 97.20.Vs, 12.38.Mh, 95.35.+d

[1] W. N. Cottingham, D. Kalafatis, and R. Vinh Mau, PhyRev. Lett.73, 1328 (1994).

[2] A. H. Guth and E. J. Weinberg, Nucl. Phys.B212, 321(1983).

[3] A. H. Guth, Phys. Rev. D23, 347 (1981).

© 1996 The American Physical Society