The QCD phase transition in the inhomogeneous Universe

4 Pages·0.13 MB·English

by  J. Ignatius

HIP-2000-20/TH, TUW-00-14
The QCD phase transition in the inhomogeneous Universe
J. Ignatius1,∗ and Dominik J. Schwarz2,3,†
1 Department of Physics and Academy of Finland, P.O. Box 9, FIN-00014 University of Helsinki, Finland
2 Institut fu¨r Theoretische Physik, Universit¨at Frankfurt, Postfach 111932, D-60054 Frankfurt a.M., Germany
3 Institut fu¨r Theoretische Physik, TU Wien, Wiedner Hauptstraße 8 – 10, A-1040 Wien, Austria
(January 11, 2001)
We investigate a new mechanism for the cosmological QCD phase transition: inhomogeneous nu-
cleation. The primordial temperature fluctuations, measured to be δT/T ∼ 10−5, are larger than
thetinytemperatureinterval,inwhichbubbleswouldforminthestandardpictureofhomogeneous
nucleation. Thus the bubbles nucleate at cold spots. We find the typical distance between bubble
centerstobeafew meters. This exceedstheestimates from homogeneous nucleation bytwoorders
of magnitude. The resulting baryon inhomogeneities may affect primordial nucleosynthesis.
1
98.80.Cq, 12.38.Mh, 64.60.Qb
0
0
Aseparationofcosmicphasesduringafirst-orderQCD andgravitationalwaves[15]hasbeenstudiedpreviously,
2
transition[1]couldgiverisetoinhomogeneousnucleosyn- while we investigate the effect of the density perturba-
n
thesis[2–5]. Duringathermalfirst-orderphasetransition tions on the QCD phase transition here. We conclude
a
J inahomogeneousmediumbubblesnucleateduetostatis- that a first order QCD transition induces an inhomo-
ticalfluctuations(homogeneousnucleation). Theirmean geneity scale of a few meters. In comparison with het-
9
1 separation at nucleation introduces a scale for isother- erogeneous nucleation via ad hoc dirt [16], we do not in-
mal inhomogeneities in the early Universe, which may troduce any new, unknown objects. Our findings might

2 influence the localneutron-to-protonratio, providingin- haveinteresting implications for precisionmeasurements
v
homogeneous initial conditions for nucleosynthesis. The of primordial abundances [4,5].
9
baryon inhomogeneities may survive until the time of First-orderphase transitionsnormally proceedvia nu-
5
2 neutron freeze-out, if the mean bubble nucleation dis- cleation of bubbles of the new phase. When the temper-
4 tance, d , exceeds the diffusion length of the proton. ature is spatially uniform and no significant impurities
nuc
0 Comparing those scales at the time of the QCD tran- are present, the mechanism is homogeneous nucleation.
0
sition, assuming a thermodynamic transition tempera- Theprobabilitytonucleateabubbleofthenewphaseper
0
ture T = 150 MeV, gives d > 2 m [4]. The causal timeandvolumeisapproximatedbyΓ T4exp[ S(T)].
h/ scale isc set by the Hubble dinsutcance at the QCD transi- The nucleation action S is the free en≈ergyc diffe−rence of
p
tion, d c/H 10 km. the system with and without the nucleating bubble, di-
- H ≡ ∼
p The order of the QCD transition and the values of its vided by the temperature.
e parametersarestillunderdebate. Neverthelessthereare Nucleation is a very rapid process, comparedwith the
h
indicationsfromlatticeQCDcalculations[6–10]. Forthe extremely slow cooling of the Universe. The duration of
:
v physicalmassesofthequarkstheorderofthetransitionis the nucleation period, ∆t , is found to be [3,17]
nuc
i
X stillunclear[6,7]. QuenchedQCD(nodynamicalquarks) π1/3
shows a first-order phase transition with a small latent ∆t = . (1)
r nuc −dS/dt
a heat, compared to the bag model, and a small surface tf
(cid:12)
tension, compared to dimensional arguments [8]. We as- The time tf is defined as the mom(cid:12)ent when the fraction
sume that the QCD transition is of first order and that ofspace where nucleationsstill continue equals1/e. The
the values from quenched lattice QCD (scaled appropri- heat flow preceding the deflagration fronts reheats the
ately by the number of degrees of freedom) are typical rest of the Universe. We denote by v the effective
heat
for the physical QCD transition. Based on these values speed by which released latent heat propagates in suf-
and homogeneous bubble nucleation a small supercool- ficient amounts to shut down nucleations. In practice,
ing,∆sc ≡1−Tf/Tc ∼10−4,andatinybubblenucleation vdef < vheat < cs, where vdef is the velocity of the de-
distance, d 1 cm, follow [11]. The actualnucleation flagration front and c is the sound speed [18]. In the
nuc ∼ s
temperature is denoted by T . unlikely case of detonations v should be replaced by
f heat
We argue that the assumption of homogeneous nucle- thevelocityofthephaseboundaryinallexpressionsthat
ation is violated in the early Universe by the inevitable follow.
density perturbations from inflation or from other seeds The mean distance between nucleation centers, mea-
for structure formation. Those fluctuations in density sured immediately after the transition completed, is
and temperature have been measured by COBE [12]
d =2v ∆t . (2)
to have an amplitude of δT/T 10−5. The effect nuc,hom heat nuc
∼
of the QCD transition on density perturbations [13,14] This nucleationdistance sets the spatialscale for baryon
number inhomogeneities.
1
Lattice simulations [9,10] imply that in real-world We find [19] for the COBE normalized [12] rms temper-
QCD the energy density must change very rapidly in ature fluctuation of the radiation fluid (not of cold dark
a narrow temperature interval. This can be seen from matter) ∆rms = 1.0 10−4 for a primordial Harrison-
T ×
the microscopic sound speed in the quark phase, c Zel’dovichspectrum. Thechangeoftheequationofstate
s
≡
(∂p/∂ε)1/2. Lattice QCD indicates that 3c2(T ) = prior to the QCD transition modifies the temperature-
S s c
(0.1)[10]. Thus,thecosmologicaltime-temperaturere- energy density relation, ∆ = c2δε/(ε+p). We may ne-
s
O
lationisstronglymodifiedalreadybeforethenucleations, glect the pressure p near the critical temperature since
due to p εq at Tc. On the other hand the drop of the sound
≪
speedenhancesthe amplitude ofthe densityfluctuations
dT T
= 3c2 , (3) proportional to c−1/2 [14]. Putting all those effects to-
dt − st s
H gether and allowingfor a tilt in the powerspectrum, the
where t 1/H = (3M2/8πε )1/2 with ε being the COBE normalized rms temperature fluctuation reads
H ≡ pl q q
energy density in the quark phase. This behavior of the
(n−1)/2
k
soundspeed increases the nucleationdistance because of ∆rms 10−4(3c2)3/4 , (7)
the proportionality ∆tnuc ∝1/[3c2s(Tf)] [11]. T ≈ s (cid:18)k0(cid:19)
In the thin-wall approximation the nucleation action
where k = (aH) . For a Harrison-Zel’dovich spectrum
has the following explicit expression: 0 0
(n=1) and 3c2 =0.1, we find ∆rms 2 10−5.
s T ≈ ×
C2 π σ3/2 A small scale cut-off in the spectrum of primordial
S(T)= , C 4 , (4)
(1 T/T )2 ≡ r3l√T temperature fluctuations comes from collisional damp-
− c c ing by neutrinos [20,14]. The interaction rate of neu-
for small supercooling. Assuming further that cs does trinos is ∼ G2FT5. This has to be compared with the
notchangeverymuchduring supercooling,the following angular frequency c k of the acoustic oscillations. At
s ph
relationholds for the supercoolingandnucleationscales: the QCD transition neutrinos travel freely on scales
l 4 10−6d . Fluctuations below the diffusion scale
∆tsc = ∆sc = 2 S¯. (5) oνf ≈neut×rinos areHwashed out,
∆t ∆ π1/3
nuc nuc
1
Here we denote by ∆ a relative (dimensionless) temper- tc 2
l = l (t¯)dt¯ 7 10−4d . (8)
ature interval and by ∆t a dimensionful time interval. diff (cid:20)Z ν (cid:21) ≈ × H
0
S¯ S(T ) is the critical nucleation action, S¯= (100).
f
≡SurfacetensionandlatentheatareprovidedbOylattice In Ref. [14] the damping scale from collisional damping
simulations with quenched QCD only, giving the values by neutrinos has been calculated to be kνph = 104H at
σ = 0.015T3, l = 1.4T4 [8]. Scaling the latent heat for T = 150 MeV. The estimate (8) is consistent with this
c c
the physical QCD leads us to take l =3T4. damping scale. We assume lsmooth =10−4dH. The com-
c
With these values for the latent heat and surface ten- pression timescale for a homogeneous volume ∼ ls3mooth
sFiroonm, tEhqe.(a5m)oitufnotlloowf ssuthpaetrc∆oolin=g i1s.5∆s1c0=−6.2.S3u×bs1ti0t−ut4-. tisemδtpe=ratπulrsemoflouthc/tucsat∼ion1s0a−r3etHfr.ozSenincweitδhtr≫espe∆cttnutoc tthhee
nuc
ifnorg 3thc2se=du0r.a1tiionntooEfqt.h(e3)n,uwceleafitnidon∆ptne×urcio=d.1.5T×he1n0−uc5lteH- ttihmeeFesrcmaliesocfalenuhcolemaotigoennse.ouAssbulobnbgleansuclslemaotoitohneaxpcpeelidess
ation distance depends on the unknown velocity v in within these small homogeneous volumes. This is a cru-
heat
Eq. (2). With the value 0.1 for v , the nucleation dis- cialdifferencetothescenarioofheterogeneousnucleation
heat
tance d would have the value 2.9 10−6d . One [16], where bubbles nucleate at ad hoc impurities.
nuc,hom H
should take these values with caution, d×ue to large un- LetusnowinvestigatebubblenucleationinaUniverse
certaintiesinlandσ. Asourreferencesetofparameters, with spatially inhomogeneous temperature distribution.
we take: ∆ =10−4, ∆ =10−6, ∆t =10−5t . Bubble nucleation effectively takes place while the tem-
sc nuc nuc H
In the real Universe the local temperature of the ra- perature drops by the tiny amount ∆nuc. To determine
diation fluid fluctuates. We decompose the local tem- the mechanism of nucleation, we compare ∆nuc with the
peratureT(t,x)intothemeantemperatureT¯(t)andthe rms temperature fluctuation ∆rTms:
perturbation δT(t,x). The temperature contrast is de- 1. If∆rTms <∆nuc,theprobabilitytonucleateabubble
noted by ∆ δT/T¯. On subhorizon scales in the radi- ata giventime is homogeneous inspace. Thisis the case
ation domina≡ted epoch, each Fourier coefficient ∆(t,k) of homogeneous nucleation.
oscillates with constant amplitude, which we denote by 2. If∆rTms >∆nuc,theprobabilitytonucleateabubble
∆ (k). Inflation predicts a Gaussian distribution, at a given time is inhomogeneous in space. We call this
T
inhomogeneous nucleation.
1 1 ∆2 The quenchedlattice QCD data anda COBE normal-
p(∆)d∆= exp d∆ . (6)
√2π∆rTms (cid:18)−2(∆rTms)2(cid:19) ized flat spectrum lead to the values ∆nuc ∼ 10−6 and
2
as deflagrations. They merge within ∆t if ∆ <
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) cool nuc
(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)H(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)H(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) Q(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) fi(vlldeedf/fvohreaotu)∆rrTrmefse.reTncheissectonodfiptiaornamsheoteurlsd. bTehuclseathrleycfoulld-
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)Q(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) H
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) spots have fully been transformedinto the hadron phase
(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)Ht1 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)H(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) t2 wTdihhreielceltaitothenenst,rewhsethaioctfhrtephlreeoaUvseinddievsienrthsaeecslotelindllgsitsphoinstcapthlreoepqaugaartkespihnasaell.
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) l 2v ∆t . (11)
H (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) heat ≡ heat cool
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) Q(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) If the typical distance from the boundary of a cold spot
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) to the boundary of a neighboring cold spot is less than
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) lheat, then no hadronic bubbles can nucleate in the in-
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) t3 tervening space. In this case the nucleation process is
totallydominatedbythecoldspots,andtheaveragedis-
FIG.1. Sketch of a first-order QCD transition in the in- tance between their centers gives the spatial scale for
homogeneous Universe. At t1 the first hadronic bubbles (H) the resulting inhomogeneities. In the following analysis
nucleate at the coldest spots (light gray), while most of the for a more realistic scenario we concentrate in this case,
Universe remains in the quark phase (Q). At t2 the bubbles l >l .
heat smooth
insidethecoldspotshavemergedandhavegrowntobubbles The real Universe consists of smooth patches of typ-
as large as thetemperaturefluctuation scale. Att3 thetran- ical linear size l , their temperatures given by the
smooth
sitionisalmostfinished. Thelastquarkdropletsarefoundin
distribution(6). As discussedabove,the mergingoftiny
thehottest spots (dark gray).
bubbles within a cold spot can here be treated as an in-
stantaneousprocess. Thefractionofspacethatisnotre-
∆rms 10−5. We conclude that the cosmological QCD heated by the released latent heat (and not transformed
T ∼
transitionmayproceedviainhomogeneousnucleation. A to hadron phase), is given at time t by
sketch of inhomogeneous nucleation is shown in Fig. 1.
t
The basic idea is that temperature inhomogeneities de- f(t) 1 Γ (t′)V(t,t′)dt′, (12)
ihn
termine the location of bubble nucleation. Bubbles nu- ≈ −Z
0
cleate first in the cold regions.
where we neglect overlapand merging of heat fronts. At
The temperature change at a given point is governed
time t heat, coming from a cold spot which was trans-
bytheHubbleexpansionandbythetemperaturefluctua-
formedintohadronphaseattimet′,occupiesthevolume
tions. Forthefastestchangingfluctuations,withangular
V(t,t′) = (4π/3)[l /2+v (t t′)]3. The other
frequency cs/lsmooth, we find smooth heat −
factor in Eq. (12), Γ , is the volume fraction converted
ihn
dT(t,x) T¯ t into the new phase, per physical time and volume as a
= 3c2+ ∆ H . (9)
dt tH (cid:20)− s O(cid:18) T δt(cid:19)(cid:21) function of the mean temperature T =T¯(t). Γihn is pro-
portional to the fraction of space for which temperature
The Hubble expansion is the dominant contribution, as
is in the interval [T ,T (1+d∆)]. This fraction of space
typical values are 3c2 = 0.1 from quenched lattice QCD f f
s is givenby Eq. (6)with ∆=T /T 1. Rewriting d∆ by
and ∆rmst /δt 0.01 from the discussion above. This f −
T H ≈ means of Eq. (3) leads to the expression
means that the local temperature does never increase,
exceptbythe releasedlatentheatduringbubble growth. T 1 T
Γ =3c2 f p(∆= f 1), (13)
To gain some insight in the physics of inhomogeneous ihn s T t T −
H smooth
nucleation, let us first inspect a simplified case. We V
have some randomly distributed cold spheres of diam- where the relevant physical volume is smooth =
V
eter lsmooth with equal and uniform temperature, which (4π/3)(lsmooth/2)3.
is by the amount∆rTmsTc smallerthan the againuniform The end of the nucleation period, tihn, is defined
temperature in the rest of the Universe. When the tem- through the condition f(tihn) = 0. We introduce the
perature in the cold spots has dropped to Tf, homoge- variables N ≡(1−Tf/T)/∆rTms and N ≡N(tihn). Since
neousnucleationtakesplaceinthem. DuetotheHubble cs may be assumed to be constant during the tiny tem-
expansion the rest of the Universe would need the time perature interval where nucleations actually take place,
we find from Eq. (3): 1 t/t 2/(3c2)∆rms(N ).
∆rms − ihn ≈ s T −N
∆t = T t (10) Putting everything together we determine from
cool 3c2 H N
s
to cool down to Tf. Inside each cold spot there is a lh3eat ∞dNe−21N2 lsmooth +N 3 =1. (14)
large number of tiny hadron bubbles, assumed to grow ls3mooth ZN √2π (cid:18) lheat −N(cid:19)
3
The COBE normalized spectrum gives l /l = most probable range of cosmological and QCD parame-
heat smooth
2v (3c2)−1/4(k/k )(n−1)/2. For l /l = ters.
heat s 0 heat smooth
1,2,5,10 we find 0.8,1.4,2.1,2.6,respectively. We acknowledge Willy the Cowboy for valuable en-
N ≈
Theeffectivenucleationdistanceininhomogeneousnu- couragement. We thank H. Kurki-Suonio and K. Rum-
cleationis defined fromthe numberdensity ofthose cold mukainen for references to the literature, and J. Mad-
spots that acted as nucleation centers, d n−1/3. sen for correspondence. D.J.S. would like to thank the
nuc,ihn
≡
We find Alexander von Humboldt foundation and the Austrian
Academy of Sciences for financial support.
tihn −1/3
d Γ (t)dt (15)
nuc,ihn ihn
≈(cid:20)Z (cid:21)
0
3
= [1 erf( /√2)] −1/3l . (16)
smooth
{π − N }
With the above values l /l = 1,2,5,10 we get
heat smooth
d =1.4,1.8,3.0,4.8 l , where l 1 m. ∗ Email address: janne.ignatius@iki.fi
nuc,ihn smooth smooth
× ≈ † Email address: [email protected]
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4