Table Of ContentRelics of cosmological quark-hadron phase transition
Hee Il Kim and Bum-Hoon Lee
Basic Science Institute and Department of Physics, Sogang University, 121-742, Seoul, Korea
Chul H. Lee
Department of Physics, Hanyang University, 133-791, Seoul, Korea
We propose that the amplified density fluctuations by the vanishing sound velocity effect during
the cosmological quark-hadron phase transition lead to quark-gluon plasma lumps decoupled from
the expansion of the universe, which may evolve to quark nuggets (QNs). Assuming power-law
spectrumofdensityfluctuations,weinvestigatetheparameterrangesfortheQNstoplaytheroleof
baryonic dark matter and give inhomogeneities which could affect big-bang nucleosynthesis within
the observational bounds of CMBR anisotropy. The QNs can give the strongest constraint ever
9 found on the spectral index.
9
9 PACS numbers: 98.80.Cq, 12.38.Mh, 95.35.+d
1
As the temperature of the universe cools down to QNs. With a power-law assumption on the density fluc-
n
the critical temperature T 100 200MeV, so called tuation spectrum, we find possible parameter ranges for
a Q ∼ −
quark-hadronphase transitionoccurs in sucha waythat which the QNs form to play the role of dark matter and
J
the high temperature quark gluon plasma (QGP) phase toproducebaryonnumberinhomogeneitiesfortheIBBN
1
2 transformstothelowtemperaturehadronicphase. Ifthe within the observational bounds of CMBR anisotropy.
phase transition is first order, it is described by the nu- The sound velocity vanishes during the phase transi-
1 cleation of hadron bubbles and their growth. As hadron tion as follows. The QGP and the hadronic phase are in
v bubbles occupy more than half of the whole space, the phase equilibrium with constanttemperatureT shortly
Q
6 picture becomes that of QGP bubbles shrinking in the after reheating. In the cosmological situation, chemical
8
seaofthehadronicbubbles. Wittenpointedoutthatthe potential of each fluid is negligible. Thus the pressure
2
shrinking QGP bubbles in the transition may evolve to and the energy density depend only on the temperature
1
0 quarknuggets(QNs)whichcanplaythe roleofbaryonic of the system. Consider the system with a scale smaller
9 dark matter [1]. Baryon concentration in the shrinking than the horizon size at the transition dQ 10km but
H ∼
9 QGP bubbles also give inhomogeneities which could af- largerthanthe meanbubble separationl ,wherethede-
b
h/ fect big-bang nucleosynthesis (BBN) [2]. Even though tails of bubble dynamics are not important. Before and
p recent lattice calculations show that the quark-hadron after the transition, the system is filled with QGP and
- phase transition is weakly first-order [3], the mean bub- hadronicradiationfluid,respectively,sothesoundveloc-
o
r ble separationlb ∼1cm based on the lattice calculations ity vs = 1/3. As the phase transition proceeds, while
t [4] is too small for the transition to achieve the effects the presspure remains constant because the two phases
s
a mentioned above. are in pressure equilibrium, the energy density of the
: Recently,itwasshownthatthe growthofthesubhori- system gradually decreases. Therefore, the sound ve-
v
i zon scale fluctuations in the quark-hadron transition is locity vs = (∂p/∂ρ)1s/2 = 0 during the phase transition
X amplified because the sound velocity during the transi- [5,6]. Since the sound velocity vanishes, the pre-existing
r tion vanishes [5,6]. Amplified subhorizon scale fluctua- density fluctuations grow without any pressure gradient
a tions give gravitational potential wells which will trap and restoring force and the amplitudes of the fluctua-
cold dark matters (CDMs) like axions and primordial tions are amplified. Superhorizon scale fluctuations are
black holes (PBHs) [5]. Resulting CDM lumps have size not affected by the sound velocity transition.
ofabout1 10AUtoday. Alsothevanishingsoundveloc- Thesubhorizonfluctuationswhichenteredthehorizon
−
ity effect enables, though it needs fine-tuned initial con- longbeforethe quark-hadrontransitionweredampedby
ditions[7],theformationofPBHsof1M⊙ 1033gwhich the neutrino diffusion. Only the fluctuations with the
∼
can be related to MACHOs [6]. The amplified fluctua- comoving wavenumber k > 104k can retain their hori-
H
tions may produce nonlinear lumps which are decoupled zoncrossingamplitude δ ∼[9],where k is the comoving
H H
from the expansion of the background universe. If the wavenumber of the fluctuation which enters the horizon
decoupled lumps were hadronic, however, they would be
atthetransition. Duringthewholetimeofthetransition,
washed out by neutrino damping [8] before the big-bang the sound velocity transition amplify the amplitudes of
nucleosynthesisera[5,9],socouldnotgiveanysignificant
the subhorizon fluctuations with δ and k upto
H
effect on dark matter problem as well as the inhomoge-
neous big-bang nucleosynthesis (IBBN) model. In this ζ
k
letter, we propose that the fluctuations amplified by the δHamp =(cid:18)k (cid:19) δH for kH <k<104kH (1)
vanishing sound velocity effect may produce nonlinear H ∼ ∼
QGP lumps which will survive till today in the form of
whereζ =1forthebagmodelandζ =3/4forlatticecal-
1
culation [5]. The exponent ζ is obtained for k k but sition. For k 104k , the minimum scale relevant to
H H
we simply regard it valid for k > k . The flu≫ctuations the sound velo≈city transition, N 1037. The QGP
H B
with δamp > 1, are nonlinearize∼d by the sound velocity lump can evolve to a QN if the q≈uarks in the lump
H
transition a∼nd are decoupled from the expansion of the become degenerate. It occurs if the baryon density to
backgrounduniverse. Soitneedsδ >δc =(k /k)ζ for entropy density ratio, n /s 1. For comparison, the
the nonlinearization. Since this coHnd∼itioHn is faHr weaker background universe hasB(n ∼/s) 3.8 10−9(Ω h2)
B bg B
≃ ×
than δc 1 in ordinary case without the sound veloc- where Ω is the density fraction of the baryonic mat-
H ∼ B
ity transition,the nonlinearobjectswith massbelow the ter at present and h is the Hubble parameter in units
horizon mass at the transition MHQ ≈1033g (even PBHs of 100km sec−1 Mpc−1. The resulting QN mass MQN
with 1033g [6]) are formed much more copiously during is determined by how and when the lump get enormous
the quark hadron phase transition. increase in nB/s and transform to a QN. Here, we do
Whatwillhappenasthedensityfluctuationsgrowand notattempttoanswerthesequestionsbutwillgivesome
become nonlinear? We propose that some nonlinearized qualitative descriptions. For this we introduce a param-
fluctuations lead to the formation of QNs. The non- eter κ, the QN to the QGP lump volume ratio. If one
linearization of fluctuations with scale λ in the linear takes the quark matter energy density the QGP en-
≃
analysis can be interpreted in another sense as the over- ergy density at the transition, then MQN =κMQ. From
denseregionsofsizeλevolvingintogravitationallybound a chromoelectric flux-tube model (CFTM), it was found
lumpsdecoupledfromtheexpansionoftheuniverse. The that the baryon penetration at the surface of the QGP
temperatureofthedecoupledlumpswillnotdecreaseany lump is very low and the baryon number density in the
more by the expansion of the universe. Instead, the in- lump increases as the surface hadronizes [10]. With this
creased pressure and energy density of the lumps by the view, QNs may form by increased nB during the sur-
nonlinearization (or the gravitational binding) will in- face hadronization [10]. If there were no efficient ways
duce the transition to the QGP phase in the hadronic to get rid of entropy and the QN formed solely by the
phase regions. If the overdense regions were composed surface hadronization (only by the increase of nB), then
of a single phase, the temperature of the overdense re- κ (nB/s)bg and nB should be increased to about the
gion, in general, would increase as the energy density ba∼ryon number density of a nucleon 0.3fm−3. How-
increases. For a region with mixed phases of QGP and ever, for the QGP lump with N <∼1034(n /s)−1
hadronic phases, however,the temperature of the region 1042(Ω h2)−1, the lump becomesBsm∼aller thanBthebgneu∼-
B
does not increase until the hadronic phase is converted trino mean free path l 10(T/100MeV)−5cm before
ν
entirely to the QGP phase. The phase of the nonlin- reaching κ (n /s) . T∼hen the entropy in the lump
B bg
earizedlump is determined by comparing two important willberedu∼cedrapidlybyneutrinos. Ifthisoccursfaster
but uncertain time scales. One is tnonl, the time needed than the surface hadronization, then κ can be approx-
for the overdense regionto become nonlinear. The other imated as κ (N /1034)−1. In fact, the entropy loss
B
is tback, the time for the mixed phase to return to QGP by neutrinos∼also occur at distances < lν from the sur-
phase. Whatisneededforthedecoupledlumpstoreturn face and it needs more detailed calcu∼lations to find the
to the QGP phase is tnonl tback. If tback > tnonl, the conditionforQNformation. However,wesimplyassume
∼
decoupled lumps are made of almost hadronic phase so κ (N /1034)−1 as the upper limit for the QN forma-
B
that they will be washed out by the neutrino damping. tio≃n. ResultingQNshavemasses<1018g. Inthemodels
Also, if tback < tnonl, the lump is still in linear regime generatinglargeentropybefore th∼e quark-hadrontransi-
whenitisentirelyconvertedbacktotheQGPphaseand tion[11],the QNs canbe formedeasilywith κ 1. This
it experiences ordinary phase transition to the hadronic situation is, however, not relevant to our work∼because
phase due to the cooling by the expansion of the uni- there can be no sound velocity transition.
verse before it becomes nonlinear. Therefore, it needs OncetheQNsform,theyareunstabletosurfaceevap-
tnonl tback for nonlinear QGP lump formation. This oration [12] and boiling [13]. Only the QNs with the
can be∼achievedfor the overdense regions with δH ∼δHc . baryon number larger than the critical baryon number
For such regions tnonl ∆tQ that is, roughly the time Nc can survive till today. With small baryon penetra-
∼ B
needed for the phase transition between two phases. So bility in the CFTM, Nc is lowered and two results of
wecanconceivethattheoverdenseregionswithδH ∼δHc NBc =1039,1044 are knoBwn [14,15]. However, as claimed
evolve to nonlinear QGP lumps. Though the overdense by the authors, their works overestimated the baryon
regionswith δH ≫δHc whichhavetnonl ≪∆tQ may also evaporation. Especially,they assumedflavorequilibrium
satisfy the condition tnonl tback, the number of such in their calculations. The flavor non-equilibrium will re-
∼
regionsisnegligibleascanbeseeninthe probabilitydis- duce Nc. So, we take Nc as a free parameterbearing in
B B
tribution of overdense regions. mind the above values. The baryon number of the QNs
The mass and size of the QGP lump are specified by needed to affect the IBBN is smaller than Nc, but the
the baryon number in the lump NB k−3 as MQ = difference will not be large considering the raBpid evapo-
∝
(NNB−BH)MHQ whereNB−H =1049(TQ/100MeV)−3 isthe ratNioonwr,awtee bhaelvoewfoNuBncd[1t4h,e15c]o.nditions for the QN forma-
totalbaryonnumbercontainedinthehorizonatthetran-
2
tion, δ > δc and N > Nc. To estimate the number whichisusedforcalculatingthenumberfractionofPBHs
H H B B
of the QN∼s, we just cou∼nt how many overdense regions formedbythedensityfluctuationswithblue-shiftedspec-
satisfying the conditions exist. We would like to empha- trum.
size that the number density of the QNs depends much Since the QNs can be regarded as pressureless dust,
on the details of density fluctuation. Though the details the density fraction at present is
of the phase transition may rule out the QN formation
ρ Mc a(t ) −3
fromthe shrinking QGPbubbles, whether the QNs from Ω (t )=Ω (t )= i QN eq β (5)
QN 0 QN eq i
the sound velocity transition can be produced enough ρeq(cid:18) Mc (cid:19)(cid:18) a(ti) (cid:19)
is another problem. One thing to note further is that
where the subscript ‘eq’ represents the values at matter-
theQNformationproposedhereoccursevenwithoutthe
radiation equal time. M is the mass contained in the
c
soundvelocitytransitionbutwithmuchseverecondition
overdense region with k at t . The density fraction can
δ >1. Thesoundvelocitytransitionrelaxthecondition c i
H be arrangedinto
and∼enables more copious QN formation.
We assume a simple power-law spectrum of density TQ
Ω (t )=κ β . (6)
fluctuationwith δ 2 kn,whereδ istheFouriertrans- QN 0 (cid:18)T (cid:19) i
k k eq
form of δ(x) (|ρ(x|)∝ ρ )/ρ and the spectral index n
≡ − b b Fig. 1 shows the relation between κ and Nc to satisfy
is a constant. The initial power spectrum of the fluc- B
Ω = 1 for possible range of the spectral index, 0.9
tuation amplitude is defined by the rms amplitude for QN ≤
n 1.5. Itseems impossiblefor QNsto be formedsolely
a given logarithmic interval in k, δ¯2(k) k3 δ 2/2π2. ≤
From the linear analyses, the rms amplitu≡de δ¯|i(kk|) at ti, bluymtphsebsuecrfoamceehsamdarlolenriztahtaionnl(κw≃he(nnBκ/s=)bg()N. cT/h1e03Q4)G−P1
the time when the fluctuations develop, is related to the ν B
(the bold line in Fig. 1). So, in the left side of the bold
rms horizon crossing amplitude δ¯ (k) as follows
H line, the QGP lumps can evolve to QNs (n /s 1) by
B
∼
−2 (n−1)/2 further losing their entropy by neutrinos. The QN for-
k k
δ¯ (k)= δ¯(k)= δ¯ (k ) (2) mation is relevant for only Nc < 1042. For n 1, it
H (cid:18)kHi(cid:19) i (cid:18)k0(cid:19) H 0 needs Nc < 1038 for ζ = 1, wBh∼ich is smaller th≈an the
B
where k is the wave number of the horizon scale at t CFTM resu∼lts [14,15]. With ζ =0.75,the QN formation
Hi i
andthesubscript‘0’denotesthevaluesatpresent. From is possible if n > 1.1. Assuming κ = (Nc/1034)−1, the
B
0th.3e[C16O].BEThmeenausumrbemerendte,nδs¯iHty(ko0f)o≈ve1r0d−en5saenrdegnio=ns1w.2it±h IufpNpecr<lim1i0t4s0o,nth∼neaQreNfocuanndgifvroemstrtohnegceosntdciotniosntrΩaiQnNts≤eve1r.
B
[δ1H7]>∼whδiHcchciasnwibdeelfyouunseddbtyo ethsteimParteesst-hSechneucmhtbeerrmdeenthsiotdy fPoBunHds,∼c0a.n9g<∼ivenautppberest<∼nu1p.2per(t>he1.b2o3ldevleinnewseiankeFriga.s t2h)e.
of structures like galaxies. The difference here is in the reheating temperature increas∼es [18,20]. Note that the
sTchaeleindietipaelnmdeansscespoefcttrhuemcorifttichaelQaGmPplliutumdpeiδnHcth∝e rakn−gζe. uinpgpteermlipmeritastuorne inn dinoflantoitondaerpyemndodoenls.tiDoermtahnedirnegheleasts-
(M ,M +dM )whereM M2/3isthehorizonmass ΩQN,theparameterrangesbecomebroaderandthecon-
H H H H ∝ Q straints on n becomes stronger.
of scale with k
SinceQNshavepositiveelectricsurfacepotentialofor-
n(MH)dMH MH(α−10)/4exp δHc 2 dMH der∼MeV,theyabsorbonlyneutrons. Thisreducesthe
∝ (cid:20)−(cid:18)√2σ (cid:19) (cid:21) neutron to proton ratio, so lowering the helium produc-
H
M α/2 tion. Not to violate the observations, the QNs should
∝MH(α−10)/4exp(cid:20)−(cid:18)MHH∗ (cid:19) (cid:21)dMH (3) hbaavryeonsizpeen>∼etr1a0b−il6itcym(eavsesnumsminaglleΩrQwNith=sm1alalenrdpepneertfercat-
twhheefireltαer=edna+m2pζli−tu1deanodfδ¯σH(k=)σanHd0(σMH/=M9H0)1(01−−n5)[/148i]s. bcoilnitsyid)e[r2e1d].heTrheeeacsoirlyressaptoisnfdyinthgeNcoBcnd≃it1io0n1.3,MsoortehseigQniNfis-
The cutoff scale M∗ MHc. Due tHo0the s×cale depen- canteffects onthe IBBNcanbe inducedby the evapora-
dence of δc , the mHass≪specHtrum has exponential cutoff tionoftheQNs. TheminimalrequisitionfortheIBBNis
H that the mean separation between QNs should be larger
evenwithn=1Harrison-Zel’dovichspectrum. Forcom-
parison, n(M ) M (n−11)/4exp( M (n−1)/2) with- than proton diffusion length when the BBN starts. It
H ∝ H − H requires the mean separation of QNs at the transition,
out the sound velocity transition [19], so if n = 1, the
l > 3m [22] ruling out the QNs from the shrinking
mass spectrum could have very broad mass ranges with QN
QGP∼bubbles (l 1cm). We find
n(MH) MH−5/2. With the exponentialcutoff,the QN b ≈
∝
wsthuiemthienoiδtn-ilafyulnnNcutBmiobn≃ertNyfprBcaecftmoiroamnssosfsigptnheicefitcrQuaNnmtslyaw.nitdShoaNpwpceroacxsainmaatse- lQN ≈nQ−1N/3 ≈108(cid:18)NNB−BcH(cid:19)1/3βi−1/3m . (7)
B
The upper values of n for the IBBN (l 3m) are
β (Mc)=σ (Mc)exp δHc 2 (4) shown in Fig. 2. It can be seen thatQNthe≈the mini-
i H H H (cid:20)−(cid:18)√2σ (Mc)(cid:19) (cid:21) mal condition is easily satisfied. The closure condition
H H
3
ΩQN 1 (the bold lines in Fig. 2) corresponds to de- J.Madsen,H.Heiselberg,andK.Riisager, Phys.Rev.D
mandi≤ng l >50m. It goes to l >100m demanding 34, 2947 (1986).
QN QN
Ω 0.1 no∼t shown in Fig. 2. Inte∼restingly, demand- [13] C. Alcock and A. Olinto, Phys. Rev.D 39, 1933 (1989);
QN
ing 0.≤1<Ω <1, l lies in the ranges for the IBBN J. Madsen and M. L. Olesen, Phys. Rev. D 43, 1069
QN QN
to be eff∼ective [∼22,23]. So, the QNs can contribute the (1991); Erratum ibid 44, 566 (1991); M. L. Olesen and
densityoftheuniverseintheformofbaryonicdarkmat- J. Madsen, Phys. Rev.D 47, 2313 (1993).
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formation can be the signature for the QNs [23]. Suppl.) 24, 80 (1991).
[15] P. Bhattacharjee, et al, Phys.Rev.D 48, 4630 (1993).
In summary, we propose that QNs can be formed by
[16] K. Gorski et al., Astrophys.J. Lett. 464, L11 (1996).
the sound velocity transition during the quark-hadron
[17] W. H. Press and P. Schechter, Astrophys. J. 187, 425
phase transition. The difficulties of the QNs formed
(1974); T. Padmanabhan, Structure Formation in the
by the shrinking QGP bubbles can be avoided by the
Universe(CambridgeUniversityPress,Cambridge,Eng-
QNsfromthesoundvelocitytransitionbecausetheirfor-
land, 1993).
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[18] A. M. Green and A. R. Liddle, Phys. Rev. D 56, 6166
tions. So, they can dominate the density of the universe
(1997).
ΩQN ≈ 1 and affect the standard BBN. Also, the QN [19] H. I.Kim and C. H. Lee, Phys.Rev.D 54, 6001 (1996).
formationcanstronglyconstrainthespectralindex. Our
[20] H. I. Kim, C. H. Lee, and J. H. MacGibbon, to be pub-
analyses are so far rather qualitative and include unde-
lished in Phys. Rev.D (1999).
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This work was supported in part by the Korean Min- phys.J. 422, 423 (1994).
istry of Education and the KOSEF. HIK was also sup-
ported by the KOSEF.
0
n=1.1
−1
−2
−3 n=1.0
−4
κog() −5 n=1.1
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−8 n=1.4
(P1h9y8s5.)R;Je.v.HD.A3p5p,l1e1g5a1te(,1C9.87J).;HGo.gaMn.,FaunldleRr,.GJ..SJc.hMerartehr-, −9 n=1.5 ζζ==10..075
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(1993). FIG. 1. The parameter ranges for κ and NBc within the
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T.Blum et al.,Phys.Rev.D51,5153 (1995); Y.Iwasaki
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53, 5446 (1996). 1.4 20 m 40 m 100 m
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n 1.3
Lett.78, 791 (1997). ex
d
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(1994). 0.9
37 38 39 40 41 42
[9] C. Schmid, D. J. Schwarz, and P. Widerin, astro- critical baryon number (Nc)
B
ph/9807257. FIG. 2. The relation between the spectral index and the
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B
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4
