PHYSICAL REVIEW 0 VOLUME 30, NUMBER 2

Cosmic separation of phases

Edwax'd %'ittenInstitute for Aduanced Study, Pnneeton, ¹wJersey 08540

(Received 9 April 1984)

A first-order QCD phase transition that occurred reversibly in the early universe would lead to asurprisingly rich cosmological scenario. Although observable consequences mould not necessarilysurvive, it is at least conceivable that the phase transition mould concentrate most of the quark ex-cess ln dcnsc, lnvlslblc qualk nuggcts, prov1d1ng an cxplanat1on for thc dark matter ln terms ofQCD effects only. This possibility is viable only if quark matter has energy per baryon 1ess than 938MeV. Tvvo Ielated issues are considered in appendices: the possibility that neutron stars generate aquark-matter component of cosmic rays, and the possibility that the QCD phase transition mayhave produced a detectable gravitational signal.

Although the evidence is divided, various theoretical ar-guments Rnd computer s1IQUlat1ons I'R1sc thc posslbi11tythat the early universe may have undeI'gone a first-orderphase transition associated with QCD effects at a tem-perature of order 100 MCV. (A true second-order QCDphase transition in cosmology is implausible because innature there 1s no exact ch1I'Rl symmetry Rnd no exact cn"terion for confinement. There may, however, have been aperiod in which chiral-symmetry breaking turned on con-tinuously but abruptly. If so, the thermal history of thisperiod can be calculated. ) Whether, with the actualvalues of the @CD parameters, the early universe wouldhave undergone a first-order phase transition is somethingthat probably will only be settled by a future computersimulation, In this paper, we will assume that a transitionfrom a state of quasifree light quarks to a state of mesonsand baryons occuI'red at T, —100—200 MCV and we willcxploI'c thc consequences.

Recent investigations of this problem have con-sidered the consequences of a departure from equilibrium„and further thoughts along these hnes will be discussed inan appendix to th1s papcI'. HcI'c, howcvcI, wc w111 assumethat a first-order @CD phase transition occurred smooth-ly, without important departure from equilibrium. Thiswould occur if the rate for nucleating the low-temperaturephase by thermal fluctuations becomes large after relative-ly small supercooling. In this context, "small" supercool-ing means that the transition effectively occurs at a tem-perature at which most of the latent heat between the twophases still remains, so that phase coexistence can be es-tablished after nucleation. Such behavior seems plausiblefor strong interactions. Actually, in nature, first-orderphase tI'Rnsltlons RI'c most typically med1atcd by lmpurl"ties, not thermal fluctuations. Even if the @CD dynamicsis unlucky, equilibrium will be maintained if the impurityabundance 1S adequate. Even wlthoUt R dcta1lcd sccnar1Q~it seems plausible to assume that equilibrium is main-tRIIlcd to lligll RccuIRcy 111 R @CD phase tlaIlsitioll. Aswc w111 scc, th1s RppaI'cntly harmless assumpt1on hassurprising consequences. Some of the points that follow

have been made previously by Suhonen.To fix ideas, consider first how an adiabatic first-order

transition unfolds if we ignore the tiny baryon asymmetry(10 (nz/n& (10 '

) in the Universe. At a temperaturejust below T„bubbles of low-temperature phase appear.Unlike the nonequilibrium situation, however, the bubblesof low-temperature phase do not expand explosively. In afirst-order phase transition there is a difference. betweenthe energy density of the two phases, usually called the la-tent heat. As the bubbles of low-temperature phase ex-pand, they cxpcl heat into thc1I' surroundings, hcat1Ilg thch1gh"tcITlpc1atUrc phase up to T~. At this point thc prcs-sure of the high-temperature phase prevents further ex-pailsioll of tllc low-tclllpcl'Rtlll'c pllasc. Aftcl' Rll, T ls thctcnlpc1atUI'c at which thc two phases have cqUal p1cssuI'csand can coexist (Fig. 1).

As the Universe expands, it loses energy. Normally thisIcsUlts 1Q cooling, but Qot hcIc. Cocx1stcncc bctwccQ thch1gh- Rnd low-tcmpcrRturc phases 1s poss1blc only Rt 1~(ignoring the tiny quark and lepton chemical potentials),and as long as both phases are present the loss of energydocs Qot IcsUlt 1Q GQO11ng, bUt ln an cxpans1QQ of the bub-bles of low-temperature phase at the expense of the re-gions of high-temperature phase. The Universe remains

FICi. 1. Isolated expanding bubbles of low-temperature

phase ln thc h1gh-tcmpcratuI'c phase.

1984 The American Physical Society

COSMIC SEPARATION OF PHASES

at T, until the full latent heat I. of the transition is elim-inated. If p is the energy density of the Universe(p-T, ), then the time scale for this is of order I./ptimes the cosmic expansion time I/Ho-Mpi/T, . Forthe QCD transition, it is plausible that I./p- l.

Thus, the bubbles of low-temperature phase slowly ex-pand. When they occupy roughly 30—50%%uo of the totalvolume, they meet and percolate (Fig. 2). At this pointthe bubbles have a characteristic radius Ro, depending onhow the transition was nucleated. The scenario below issimplest if Ro(10 cm (the baryon diffusion length).ThIs 1s a long dIstancc by hadroMC standards; It corre-sponds to 10 nucleation events per cubic fermi. Wealso will discuss a variant of the scenario which may stilloperate if Ro ~~10 cm.

When bubbles collide, we must take surface tension intoaccount. An isolated bubble is essentially spherical.%hen bubbles Glcct Rnd pcI'colatc, howcvcI', they arrangethemselves into a system of fewer, larger bubbles to mini-mize the surface area. We must estimate the time scalefor this process.

If two bubbles of radius R collide and form a singlebubble of radius 2'~ R, the energy is lowered byhE-crR, where o is the surface tension between the twophases. The force acting is I' -4E/R -oR, since energyb.E is released in a distance R. The fluid mass that mustbe moved is M-pR 3, and it must be moved a distance oforder R. A force Fwill move a mass Ma distance R in atime r -(MR/F)' . If o -T, and p- T, , then

Requiring that this time be less than the expansion timeMpi/T, gives R (Mpi ~ T, ~, which is of order1—10 cm for reasonable T, . Thus, surface tension causesthe bubbles to coalesce until reaching a characteristic scaleR i of a few centimeters.

For a brief period, this bubble coalescence is the fastestimportant process, but once the bubbles are close to thecharacteristic size Ri the bubble coalescence effectivelystops, and we must again take account of the expansion ofthe Universe and the corresponding removal of heat. Thedilute regions of low-temperature phase grow and soonoccupy more than half thc volume. When the high-temperature phase fills only 30—50% of the total volume,the tables are turned; the dense regions of high-

temperature phase detach into isolated, roughly sphericalbubbles (Fig. 3). They have at this point a characteristicsize Ri, since they form from the "holes" between ex-panding regions of low-temperature phase of size Ri.Further expansion results in a loss of heat and a furthershrinking of the dense bubbles of high-temperature phase,until finally they disappear.

%'c must ask whether thermal equilibrium is main-tained in this process. In fact, heat conduction is veryrapid in the epoch under consideration because there areparticles (photons, charged leptons, and especially neutri-nos) with very long mean-free paths. Actually, soundwaves also provide a very fast mechanism for equilibra-tion of temperature. As in any fluid, sound waves rapidlyestablish a pressure equilibrium in the cosmic fluid. Un-like an ordinary liquid or gas, however, the cosmic fluidhas the property that temperature is the only thermo-dynamic variable, since there is no conserved charge (ex-cept the tiny baryon and lepton excesses), so that the pres-sure is a definite function of temperature, and pressureequilibrium means temperature equihbrium. (In an ordi-nary fluid, the pressure depends on the temperature andon the density of atoms. ) Therefore, thermal equilibriumis a IcasonRblc assumption.

We have so far assumed that the original scale Ro ofbubble coalescence is small, in which case the effectivebubble size R i is determined by surface tension and is in-dependent of Ro. It is perfectly plausible for Ro to besmall, especially in a system with strong interactions; Romight even be zero if the phase transition occurs by spino-dal decomposition rather than nucleation. However, itmight be that Ro is large; Hogan has shown that in the@CD transition, Ro could plausibly be as big as about 10cm (less than the horizon length Mpi/T by a factor of 4ln Mpi/T, ). In that case, how might the bubbles be ex-pected to grow'7

At zero temperature, bubbles of true vacuum grow atthe speed of light. At nonzero temperature, but far fromequilibrium, bubbles grow explosively, faster than sound,but slower than light. The latter process is similar to adetonation wave in fluid mechanics. ' In strong interac-tions, however, it is quite plausible that the temperature Tat which nucleation occurs is very close to T„at the sametime that the mean distance Ro between nuclei may be

FIG. 2. The expanding bubbles meet.FIG. 3. Isolated shrinking bubbles of the high-temperature

phase.

ED%'ARD %ITTEN

181gc. In th1s case thc bubble growth 1Qvolvcs coQd1t1onsclose to cqu111br1um and BIl cxplos1on oI' dctonat1oIl docsnot seem likely. More likely, one will find a process simi-lar to what Landau and Lifschitz describe as "slow com-bustion, ""

In this context, slow combustion is a process in whichthe velocity of growth of a bubble of low-temperaturephase is of order e=(T, —T)/T„being limited by therate at which heat transport is possibl, As bubbles oflow-temperature phase grow, the liberation of latent heatraises the temperature of the immediate neighborhood upto T, (but no higher; otherwise the bubbles of low-temperature phase would have to contract instead of ex-panding). To avoid a local temperature higher than T„the bubbles of low-temperature phase can expand only ata rate determined by the ability of the liberated heat to becarried Rway. If llcRt ls CRlrlcd Rway mainly Rs R wave,this wave will be propagating a temperature difference oforder c. Since the velocity will be comparable to that oflight, a wave propagating a temperature difference e willcarry away heat at a rate proportional to e, Bnd the veloci-ty of bubble growth will be of order e. (The possible roleof neutrinos, discussed later, does not change the order ofmagnitude. ) Of course, the heat carried off by these wavesheats up the rest of the Universe.

For small e, long before the low-temperature bubblesmeet thc heat gcncl atcd by thc1r gI'owth will heat theUniverse up to T,—after which the bubbles stop expand-ing except insofar as their growth is driven by the expan-sion of the Universe.

In this case, when the bubbles finally meet, surface ten-sion plays no role; one has quasiequilibrium coexistence ofphases on a scale Ro ——R1 &10" cm. As in our previousdiscussion, the loss of heat due to the expansion of theUIl1vcIsc eventually results 1n thc dctach1Ilg of lsolatcdbubbles of the high-temperature phase, which then tend toshrink and disappear.

Thus, we can reasonably expect that for a time compar-able to the Hubble time, there will be coexisting phases ona scale R~, with 1 em&BI &10 cm. %'hat observableconscqucnccs could this lcavc 1n to«Iay s univcrscY

There might in fact be no such consequences. The neteffect of the process might be simply thermal equilibriumin the low-temperature phase below T, . In searching fora possible observable remnant of the epoch of phase coex-istence, we will explore the consequences of the tinybaryon excess in the Universe.

In the high-temperature phase, baryons exist as quasi-free, almost massless quarks. In the low-temperaturephase, baryons are massive particles with a minimummass M=938 MCV. The density of baryons in the low-temperature phase may therefore be less than the densityln thc high-tcmpcratUrc phase by 8 18rgc factof. FoI" 8more accurate estimate, assume equilibrium between thetwo phases with a common chemical potentia1 p. Thebaryon density in either phase is then

(a) =—{ae-~'~')1

y (1)

where V is the volume and ( ) refers to a thermal aver-age. Since the primordial p is tiny, to good approxima-tion

(g)P , (gl) (2)

where ()o is a thermal average at @=0. The ratio of thebaryon densities in the high- and 1ow-temperature phases1S

(3)

If the main contribution is due to the eight spin states ofprotons, neutrons and their antiparticles (a fair approxi-mation if T, (100 MeV so that strange particles aresuppressed), then

' 3/2

{a'),'=s ~ '2m'

with an obvious generalization if T, is larger and otherparticles must be included. As for the high-temperaturephase, we assume that it is an ideal gas of quarks andgluons and that the u, d, and s masses can be neglected.A single quark or antiquark spin state of baryon number+ —, contributes to (8 )0 an amount

1'

de e IP IIT-

(2~)' 1+e- ~P ~"J

T31 g(3),

12~2

where g(3) =1.202. The 36 spin and color states of u, d,and s qUarks and anti«Iuarks g1vc

{a'),=3 (7)

So with these approximations

{& )o 2 ~2m M ll1n;-(gl)0~ 3 g(3) T,

For Tc = 100 MCV, a=0.003. For Tc =200 MCV,@=0.15. (For T, =200 MeV, it is really necessary to in-clude hyperons. Including all particles in the baryon oc-tet, one gets e =0.27.)

Thus, this simple estimate suggests that e is much lessthan 1, and that 6—10 1s possible, but only 1f Tc 1s I'cla-tively low. Hopefully, it will be possible eventually todetermine 6 Icllably by coIIlpUtcl slmulatlons.

The formula (3) for e assumes equilibrium between the

where ( ) ' are averages in the low- and high-temperature phases and ()0' are the averages in thosephases at @=O.

Let us make a simple estimate of e. Treating the low-temperature phase as an ideal gas of mesons and baryons,Bnd BssuIQ1ng ~c Q+~ so 8 QODIclat1v1stlc treatment 1sadequate, a single baryon or antibaryon spin state of massM contributes to (8 )o an amount

3/2~Tc —~/T

82m

30

two phases. Is this reasonable' A sufficient requirementis that Ro is small and that thermal equilibrium existedwhen bubbles originally met at a scale Ro. The subse-quent coalescence of bubbles into larger structures of scaleRI—due to surface tension —involves slow, gross fluidmotions, probably without significant mixing of the twophases, so the Iclatlvc baryon abundance at scale R) w1llbe very close to what it was at scale Ro. The relativeabundance will have its equilibrium value at scale Ro ifthe distance dII across which baryons can diffuse in aHubble time is bigger than E.o. Baryons are strongly in-teracting, so the baryon (or quark) mean-free path at tem-perature T, is of order 1/T, ; this distance is traversed ina time of order 1/T, . Since the cosmic expansion time isOf OldCI' MpI/T~, tlM bal'yoll 11RS time fol MpI/T~ StCpS,each of mean length 1/T„and can diffuse a distance

1/2PI

T~

or about 10 cm if T, —100 MeV. If Ra&10 cm, theequilibrium formula (3) for c is valid.

On the other hand, we have contemplated much largervalues of Ro, as big as 10 cm. In that case, the relativebaryon abundance in the two phases cannot be determinedon grounds of thermal equilibrium only. On the otherhand, expanding bubbles of low-temperature phase willstill tend to expel baryons, so it is conceivable that thebaryon concentration would be higher in the high-tcmpcfaturc phase. %c postpone 8 discUss1on of thispoint, as it is similar to our later discussion of what hap-pens to baryons during the period when bubbles of high-tempeIature phase are beginning to shrink and disappear.

From our discussion so far, when bubbles of high-temperature phase begin to detach themselves and shrink,their radius is of order R I (1—10 cm), they fill roughly50% of the total volume, and they may contain a fractionI/(I+c) or about 80—99% of all the baryons in theUniverse. Although baryons may thus be concentratedmostly 1Il thc high-temperature phRsc, thc Ilct baryon con-centration is very tiny even there. The baryon-to-entropyratio in today's world is observed to be at most 10, andthe baryon-to-entropy ratio in the high-temperature phasein the epoch under discussion is no bigger than that. Inthe low-temperature phase, the baryon-to-entropy ratio isless by a factor of e.

%hat happens next? As the Universe expands, the re-gions of low-temperature phase tend to expand and cool.Equilibrium between the two phases is possible only atT„so to maintain equilibrium, heat must constantly beresupplied from the regions of high-temperature phase toregions of low-temperature phase. What is the principalmechanism for this? The high-temperature phase can loseheat by evaporation of the surface layers, or by emissionof particles of very long mean-free path —neutrinos.

In an ordinary fluid, evaporation is a slow process be-cause it is diffusion limited. For example, when a waterdrop evaporates, a large concentration of water vaporbuilds up around it, and evaporation proceeds only at therate at which the vapor can diffuse away. There is noanalogous effect in the cosmic fiuid, since (as the baryon

cxccss Is tlIIy), tllcrc Is no I'clcvRIIt conserved quantityanalogous to the number of water molecules. Evaporationof a water drop is also hmited by heat conduction, sincethe evaporation cools the surface layers and the evapora-tion rate is low at low temperatures. There is no analo-gous effect in the cosmic fiuid, since (as we have noted),the latent heat of the transition can be efficiently carriedaway by soUnd waves.

As for neutrino emission, one might naively think thatit would be very slow, proportional to G~ . However,neutrino emission can occur anywhere within 8 distanceA,„of the surface, A, being the mean-free path. In fact, A,

is of order I/(Gp T, ), or about 10 cm for T, = 100 MeV.Since A,„ is proportional to 1/GF, the fact that neutrinoemission can occur a distance k„ from the surface justcancels the fact that the neutrino emission rate per unitvolume is proportional to A,~, so there is no real suppres-sion associated with the weakness of weak interactions.

One may therefore reasonably expect that energy lossesdue to surface evaporation and due to neutrinos will becomparable. Their implications are completely different,however. Evaporation involves a loss of the surfacelayers, including baryons and everything else. Neutrinoemission from a distance A,„ofthe surface involves loss ofheat without loss of baryons. If it dominates, the bubblesof h1gh-temperature phRsc CRII shrlIlk, developing R stcadI-ly increasing baryon excess; the baryons cannot diffuseout since A, is much bigger than the baryon diffusionlength dII. In such a scenario, the high-temperature bub-bles would undergo a steady process of shrinking, with thebRryon cxccss trapped 1ns1dc.

In such a scenario, the region of high-temperaturephase behaves like a leaky balloon, unable to suport itselfagainst outside pressure because of steady neutrino losses.The ba11oon shrinks, losing neutrinos without losingbaryons; the shrinking balloon does not build up a pres-sure excess because excess qq pairs annihilate into neutri-nos, which cscapc. Thc bRr'yon cxccss 1S continually coIl-centrated in a smaller volume. There is a rich element ofwishful thinking here, since this picture assumes neutrinolosses are the main way for the high-temperature phase tolose energy, while in fact neutrino losses and surfaceevaporation appear comparable. However, this scenario isworth exploring because —as we will see—it leads to fas-c1natlng conscgUcnccs.

If such a scenario develops, then as the dense regionsshrink, eventually the excess baryons in those regions willexert a pressure that will help stab111ze those regionsagainst further shrinkage. This is sketched in Fig. 4; thetemperature at which the dense regions can coexist withthe low-temperature phase outside depends on the chemi-cal potential p in ihe dense regions. Figure 4 is not 8thermodynamic equilibrium curve (which might or mightnot be qualitatively similar, as we will discuss). Rather,assuming that the chemical potential in the low-temperature phase is negligible, Fig. 4 indicates the tem-perature T, as a function of the chemical potential p inthe dense phase, at which the two phases exert the samepressure. This corresponds to an idealized situation inwhich the two phases are separated by a membrane thattransmits heat but no baryons, and all the baryon excess is

ED%'ARD WITTEN

RKS

survive only if there can exist a stable state of so-called"quark matter, " a dense state of matter consisting of de-generate quaI'ks at a density somewhat above ordinary nu-clear matter density; such a state would differ fromoldlnary QUclcar InattcI' 1Il that thc 1ndlvldUR1 qURrk wRvcfUnct1ons would bc delocalizcd, spreading thIoughoUt R

macroscopic volume. Qf course, there is at present no ex-perimental evidence on the existence of quark matter, andconsiderable theoretical uncertainty, so let us sketch thepossibiHties, ranging from absolutely stable to absolutelyunstable quark matter.

A. Stable macroscopic quark matter

FIG. 4. A sketch of the coexistence temperature for quarkmatter of chemical potential p coexisting with the meson-baryonphase of p=0. %hat is shown is the temperature, as a functionof p, at which the two phases exert equal pressure.

in the dense region; the problem of interest approximatesthat situation if diffusion of baryons is negligible and ifheat 1S carried almost entirely by neutrinos.

As indicated in Fig. 4, once p is not negligible, theUniverse will begin to cool since the coexistence of phasesnow requires this. The baryon-to-entropy ratio in thedense regions is of order 1 when p-T; at this point T iscomparable to but significantly less than T„p reh pas

The dense regions now occupy a tiny fraction of the to-tal volume but contain 80—99% of the baryon excess.How large are they' Let f be the primordial baryon to en-tropy ratio (so 10 (f(10 ', experimentally). Thebaryon-to-photon ratio of the dense regions when theyhave radius R, is f, so this ratio becomes 1 when the ra-dius shrinks to r=R,f ' . If, for instance, f=10RIld 1 CIn Q R1 Q 10 cm~ thcIl r 1s bctwccIl 10 Rnd 10cm. We have formed, in effect, lumps of hot quarkmatter with a density of 10' g/cm and a mass from109 1018 g

%'hat will happen next~ The most extreme possibility isthat these lumps might manage to survive to the presentand still exist in today's universe. Because of their ex-tremely large 1nert1a they would not have been incorporat-ed in stars or planets and would be floating in interstellarspace, affected significantly only. by gravity. The baryonsthey conta1Q would not part1c1patc 1Q QUclcosyQthcs1s, soone could assume a closure density of baryons withoutdisturbing the successes of big bang nucleosynthesis. Ingalaxy formation they would behave like the planetarymass black holes that have recently been discussed. " Ifthc rclat1vc baryon abUIldancc 1Il thc two phRscs wasdetermined by thermal equilibrium, they could contain areasonable fraction of the total mass in the Universe, andmight account for the "missing mass" in the Universe'z ifg —10

These lumps could survive to the present only in theform of some exotic, presently unknown state of matter,since ordinary nuclear rnatter is unstable at zero pressure,and ordinary atomic matter would not survive the condi-tions 1Q thc carly UI11vcI'sc. Thc lUIIlps coUld plaUs1bly

The most extreme possibility is that macroscopic quarkmatter 1s boUnd Rnd stable at zcIo tcInpcI'RtuI'c Rnd pIcs-sure. At first sight, one might think that this is excludedby the fact that ordinary nuclei do not spontaneously turninto the hypothetical dense quark state. This is Qot quiteso, however. Observations of nuclear physics only showthat in the absence of strange quarks, nuclear matter ismore stable than quark matter. Addition of strangenessdocs not help stab111zc QUclcR1 IIlattcI ~ bccRUsc stl angebaryons are heavier than nonstrange baryons. For quarkmatter, the story is different. (This point has been notedbefore in Refs. 19 and 26.) The likely Fermi momentum1n quark IIlattcI 1s 300—350 McV, more thRQ thc strangc-quark inass, so it is energetically favored for some of thenonstrange quarks to become strange quarks, lowering theFcrml momentum Rnd thc cIlc1'gy.

This effect can easily be estimated in the simplest formof the bag model. ' A single quark flavor of Fermimomentum p~ exerts a pressure pz /4m . Let p be theFermi momentum of up quarks in quark matter of zerostrangeness. For electrical neutrality, the down-quarkchemical potential must then be p2'~, and the pressure is(1+2 ~ )p /4ir . If we ignore the strange-quark mass,then three flavors of quarks can exert the same pressure ata common Fermi momentum P=[—,(1+2 )]' p of &,

d, and 5 quarks.Thc Rvcragc qURI'k klnctlc cIlcrgy 1s proportional 'to p,

so (with a common pressure in the two cases) it is smallerin the three-flavor case by a factor

P/( —,'p+ —', 2'i p)=[3/(1+2 i )] =0.89 .

IIl equilibrium, thc cIlcrgy pcI' qURI'k cqURls thc chemicalpotential, so the energy per quark in strange-quark matteris less than the energy per quark in zero strangeness quarkiiiattei' by tliis factoi' of 0.89; iii this idealizatioil, stiailge-quark matter is more tightly bound than nonstrange-quark matter by about 100 MCV per baryon. Thestrange-quark mass will reduce this effect, but it is stiHplausible that strange quarks lower the energy per baryonof quark matter by 50—70 MeV per baryon. This issoInewhat paradoxical, because in nuclear matter strangequarks have the opposite effect.

No observational argument seems to exclude the possi-bility that quaI'k matter with up and down quarks only isunbound, but by less than 50—70 MCV per baryon, andthat quark matter with strangeness is bound and stable.Ordinary nuclei will not convert to the dense, strange state

ISMIC SEPARATION OF PHASES

because of the difficulty in creating strange quarks byweak interactions.

If quark matter with strangeness is boumi, then energet-ically it can grow indefinitely by absorbing nucleons. Onemight at first think that quark matter would be dangerousin contact with ordinary matter, but this is not the case.Ordinary nuclei (except iron) are also energetically capableof undergoing nuclear reactions, with a big release of ener-

gy; this is prevented at low temperature by Coulomb bar-riers, because the nuclei are positively charged. A nuggetof quark matter will have a net positive charge on the sur-face (not in the bulk) for the same reason that nuclei arepositively charged, For instance, in the quark mattermodel of Freedman and McLerran' (their model I, ex-trapolated to zero pressure) the ratio of up-to-down-to-strange-quark abundance is roughly 2:3:l. This meansthat the up and down quark chemical potentials p„andpd are in the ratio p„/pd =(—,

' )'~, so that p~ —JM„=50McV.

To balance the chemical potential difference, there willbe an almost equal electrostatic potential of + 50 MeV atthe surface of (and throughout) a nugget of quarkmatter; this potential will be more than adequate to re-pel positively charged nuclcl at ordinary temperatures orat reasonably low velocities. Quark matter is in thisrespect no more dangerous than oxygen.

Electrostatic potentials, however, would not preventquark matter from absorbing neutrons. (This is somewhatretarded, however, by the need to equilibrate strangeness,if nonstrange quark matter is unbound. ) If quark matter isstable, it is probably necessary to assume that ordinaryneutron stars are really quaik stars. However, there areconventional models of neutron stars in which these starshave a large quark core (see, for instance, Ref. 24), and ithas been claimed that the macroscopic properties ofquark stars are hard to distinguish from those of neutronstars. This point is discussed further in the Appendix.

In the models of McLerran and Freedman, quarkmatter with strangeness is unbound by about. 25 MCV perbaryon (I am here extrapolating their data to zero pres-sure); without strangeness, it would be unbound by about100 McV per baryon. As they note, however, the energyper baryon of quark matter is quite sensitive to the as-sumed QCD parameters, with an uncertainty of order 100MCV per baryon.

In short, there does not seem to be any conclusivetheoretical or experimental argument ruling out the possi-bility that there exists a stable state of quark rnatter withlarge strangeness. Qne experiment that might shed somelight on this question would be the search for the II a-hypothetical AA bound state whose binding energymight be 100—200 MCV.

B. Stsblc drops

If quark matter, even with strangeness, is unstable inbulk, it is possible that it would have been stable in the ab-sence of electromagnetism, and is unstable only becausethe constraint of dectrical neutrality forces an incon-vclllcllt 1atlo of qllark abundance. (This ls tIllc of ordi-nary nuclear matter, w'hich with equal numbers of neu-

trons and protons would be bound as far as QCD effectsare concerned, and is unbound only because electromag-netic effects prevent a nucleus from having large electriccharge. ) If electromagnetism is the only reason that bulkquark matter with strangeness is unbound, then suchmatter will probably be stable in droplets small enoughthat the electrostatic energy is not too large. (In the con-ventional case the corresponding droplets are ordinary nu-clei. ) The possibility of metastable (but not truly stable)droplets of quark matter has been considered previouslyas a possible explanation for Centauro events. 27

C. Quark matter unstable because of weak interactions

It is possible that quark matter is unstable (both in bulkand in drops) but only because of weak interactions. Forinstance, in model I of Freedman and McLerran, quarkmatter (at zero pressure) has an energy per baryon ofabout 965 MeV and the quantum numbers of a roughlyequal mix of neutrinos and A' s. Since 965 MeV is abovethe neutron mass, quark matter is unstable in this model;but since 965 MeV is well below the average of the neu-tron and A masses, quark matter can decay in this modelonly with the hdp of weak interactions. Bulk quarkmatter would be relatively long-lived under these conch-tions, with evaporation of neutrons at the surface creatinga strangeness excess that must be equilibrated by weak de-cays or else by diffusion of strangeness. The stabilityagainst strong interactions of quark drops of varyingstrangeness has been discussed previously.

D. Quark matter unstable against strong interactions

Finally, there is the possibility that even with equilibri-um strangeness content, quark matter is unstable againstdecay by strong interactions; if so it would probably decayrather quickly.

It is unfortunate that at present there is no reliable wayto choose between these possibilities. There is, in addi-tion, another major uncertainty. Even if lumps of quarkmatter form in the manner that has been conjectured attemperatures of order T„and even if there is a stableground state at low temperatures that the lumps couldlogically settle into, they may evaporate at intermediatetemperatures before settling into their stable groundstates.

At, temperatures near T„ the evaporation of baryonsfrom the dense lumps is suppressed because of the rela-tively low diffusion length dII of the outgoing baryonsand because of the relatively low equilibrium vapor pres-sure of baryons ln thc low-temperature phase, proportion-—M/Tal to e '. (Even if the full equilibrium pressure ofbaryons would leak out of the dense regions to a distancedII, this would be a minor loss of baryons. ) At very lowtemperatures, evaporation of baryons from quark matterwill be suppressed because there simply is not availableenough energy to power the evaporation. (At low tem-peratures the thermal excitation energy of a degenerateFermi gas is of order T /pz, and less if a superconductinggap forms. This means that by the time neutrinos decou-ple at TD —1—2 MeV, the thermal excitation energy of

ED%'ARD %'ITTEN

quark matter is perhaps a tenth of an MeV per baryon, fartoo llttlc to cIIablc quark Inattcl to cvRpol'Rtc If It ls boundby, say, 10 McV per baryon. Under such conditionsquark matter is thermodynamically stable. )

However, at temperatures below T, and well above TD,qURI'k mattci has 8 large qual k chemical potential Rndhcncc 8 I'clRtivcly large vapor prcssUre Rnd 1s not, stRblcthermodynamically, in contact with the dilute low-temperature phase. It will tend to evaporate, but theevaporation will be limited by the time scales for equili-bration of heat, baryon number, and strangeness. (For in-stance, the evaporatIon of a lump of quark matter wIll In-volve mainly the emission of neutrons at the surface,creating a strangeness excess at the surface; the time scalefor diffusion or weak decay of strange quarks will be oneof tllc coIISIdcl'Rtlolls llnlltIIlg tllc cvapol'ation IR'tc. ) Aserious study of the evaporation process between T, andTD will Qot be attempted here.

HowcvcI', 8 naive cstiIQRtc, similar to oUr' plcv1ous esti-mate of the baryon diffusion length at T„ is that atT=10 MeV, the diffusion scale is about 10 cm, so thatquaI'k lumps Rt thc lower cQd of OUI' estimated range(10 —10 cm) are in danger of evaporating before theybecome thermodynamically stable at temperatures some-what below T~.

The scenario proposed here is subject to many uncer-tainties. Various QCD uncertainties have been noted.Even more serious is the lack of a clear reason for neutri-no losses to be the main way that bubbles of low-temperature phase lose energy. If the QCD situation isfavorable and the scenario can somehow be made to work,1t, would have oIlc ma)or advRQtagc. In many partlclcphysics explanations of the dark matter, one must postu-late a neutral particle with a lucky combination of massRnd abundance, Rnd 1t is 8 coincidence tilat this particlehas a mass abundance coInparable to that of baryons. Inthe scenario suggested here, the luminous and dark matterhave the same origin —baryons —and the ratio of lumi-nous to dark matter is computable, in principle, as a QCDeffect—if the scenario somehow works.

If it turns out that quark lumps of size estimated hereare responsible for the dark matter, what is their flux atthe earth? The density of dark matter at our locationseems to be' about 10 g/cm . Lumps of quark matterof radius 10 —10 cm weigh about 2X10 —2X10' g, sothe number abundance is about 5 X 10 —5 X 10 " /cm .They would have typical galactic velocities of order2X10 cm/sec, corresponding to a flux 10 —10 /cm /sec. With the cross sectional area of the earthSX10' cm, the rate of collisions is 10 —10 '7/sec orbetween almost one pcr year (if the radius is 10 cm) andalmost one per 10 yr (if the radius is 10 cm).

These collisions would apparcn. tly not be as spectacularas one might think. Quark lumps would not initiate nu-clear reactions (for reasons discussed earlier). Lumps ofthe radius we have estimated would plunge all the waythrough the earth, displacing everything they met in atube of 0.01—10 cm radius, since their mass per unit area(10' —10' g/cm ) is much greater than that of the earth(5 X 10 g/cm ). Such events might be detectable if theyoccur Rs often Rs QIlcc a ycal' 0I' Ollcc a decade, but tllcy

apparently would not be conspicuous.If quark matter is not produced cosmologically, there

might still be a quark matter flux in cosmic rays. This isdiscussed 1Il Rn appcnd1x.

We have tried in this paper to explain dark matter pure-ly on the basis of known physics. Departing from thisrigorous rule, one intriguing variant of the scenario wehave considcicd 1s to sUpposc that thc primordial baryon-to-photon ratio nII/nr was much bigger than today' svalue of 10 —10

Suppose, for instance, that primordially nII/nr —1. InthIs case, fhc prtmordlal cosIIllc flui would behave I'Rtllcl'

like a normal fluid, with baryon number conservationplaying the role that conservation of the number of mole-cules plays in an ordinary fluid. Pressure equilibriumwould be maintained in the cosmic fluid by sound waves,but this would Qot preserve temperature equilibrium since(Rs III watcl or RIr) tcIIlpcratuic gladlcllts coUld bc c0111-pensated for by gradients in the baryon concentration,while keeping a constant prcssure. In such a situation,heat transport is likely to be by diffusion (as in any nor-mal fluid near equilibrium), and since neutrino mean-freepaths are enormous, heat transport would be via neutri-nos.

If so, shrinking bubbles of high-temperature phasewouki lose heat w1thoUt los1ng baryons, Rnd qURrk InattcI'would form. In fact, if nII/nr —1 primordially, onewould form quark matter lumps 10 times heavier thanwe previously estimated (so m —10' —10 g), since thehigh-temperature bubbles would not need to shrink veIymuch before the quarks would become degenerate. Insuch a picture, one would of course need a mechanism forgenerating a large entropy prior to nucleosynthesis. Onepossible picture is the idea that the decay of a 10 GeVgravitino generates a large entropy and reignites nu-cleosynthesis well after the QCD epoch.

ACKNOVPI. EDGMENTS

I would like to thank T. Hanks, C. G. Callan, F. Dyson,E. Farhi, G. Farrar, D. Gilden, S. Glashow, H. Hamber,J. Harvey, R. Jaffe, V. Kaplunovsky, C. Nappi, J. Taylor,J. Weisberg, and V. Weisskopf for useful discussions. Iespecially wish to thank E. Farhi and R. Jaffe for discus-sions of their new calculations about quark matter. Fi-nally, I wish to express my gratitude to the Institute forAdvanced Study for hospitality while this work was inprogress. The author's research was supported in part bythe U.S. National SC1cncc FoundatioQ UIldcr Grant No.PHYSO-19754 and by the U.S. Department of Energyunder Grant No. DE-AC02-76ER02220.

This paper has really considered two independent ideas:the possibility that three-flavor quark matter is stable, andthe possibility that it was manufacturIx1 in the big bang.It is possible that quark matter is stable, but was not pro-duced cosmologically. If so, is there any hope of observ-ing quark matter today~ Th1s question natuI'ally leads onc

30 COSMIC SEPARATION OF PHASES

to ask whether there exists a mechanism for producingquark matter in today's universe.

There is another good reason for asking this question.It has been claimed that the Centauro cosmic-ray eventsmay be evidence for a quark-matter component in cosmicrays. (The Centauros are events ' in which the primaryseems to fragment into hundreds of baryons and almostnothing else. This is a plausible fate for a small drop ofquark matter that strikes the atmosphere at, high energy.The Centauro primaries seem to penetrate much furtherin the atmosphere than a conventional heavy nudeusmould. )

If the Centauro events are real and are to be explainedin terms of quark matter, then we must explain a flux ofquark-matter droplets with roughly 10 baryons per dropand roughly 10 —10 GeV per baryon. The actual flux issmall, of order 10 /m /yr. One cannot account for thisflux by synthesizing quark matter in the big bang, sincedrops of quark matter produced in the big bang, regard-less of their velocities at that time, would have long sincered-shifted to low velocities.

To discuss contemporary production of quark matter,we must bear in mind the following: regardless of wheth-er quark matter is stable at zero pressure and temperature,it is certainly the ground state of a large assembly ofbaryons if the external pressure is high enough. At highpressure (or density}, the strong interactions are asymptot-ically free, and the ground state of a dense collection ofquarks can be found perturbatively.

We are a.ssuming in this paper that at zero externalpressure (and temperature), quark matter is stable if, andonly. if, the strangeness is large enough.

If strange quarks are not available in reasonablenumbers —which is true except in the big bang —the onlyway to create quark matter is to compress protons andneutrons to a pressure at which two-Oavor quark matterbecomes stable. Such high pressures are not attained inordinary stars or planets. In addition, a small nucleus ofquark matter that might be present in an ordinary star orplanet would not grow spontaneously, since Coulomb bar-riers would prevent it form absorbing charged nuclei.

Bui there is one conspicuous astrophysical environmentin which quark matter can readily form and grow. This isthe neutron star. It may be that when neutron stars formthey already contain a small nucleus of quark matter ofcosmological or cosmic-ray origin. Even if no nucleus ofquark matter is present to begin with, it is very likely thatthe pressure at the center of a neutron star is so large thatin that region quark matter will form spontaneously evenwithout strangeness. (If three-flavor quark matter isstable at zero pressure, then two-flavor quark matter isrelatively close to being stable even at zero pressure and isprobably stable under the high pressure in the center of aneutron star. ) One way or another, neutron stars almostsurely contain a quark-matter component.

Once a quark-matter component is present in a neutronstar, two things will rapidly happen. It will quicklydevelop the equilibriuin strangeness content via weak in-teractions, such as ud ~us. The energy will be lowered asstrange quarks are created one by one until equilibrium isreached. Even more important, the quark-matter com-

ponent can rapidly grow in a neutron star by absorbingfree neutrons, since there is no Coulomb barrier. The en-tire star will turn into quark matter, except perhaps for athin outer crust in which there are no free neutrons.

This outer crust, consisting of nuclei and degenerateelectrons, is rather thin and dilute. It is usually estimatedto have a thickness of a few tenths of a kilometer (com-pared to the 10-km radius of the neutron star) and a den-sity of at most 4 X 10"g/cm (10 of the average densityof the star). Actually, if quark matter is stable at zeropressure, this conventional estimate is only an upperbound on the thickness of the "normal" outer crust. Onecan envisage a star that consists entirely of quark matter,with no crust, the surface density of the star being simplythe density of quark matter at zero pressure, or about3—4&& 10' g/cm .

It is estimated thai about 10 of the mass in a typicalgalaxy such as ours is in the form of neutron stars (corre-sponding to 10 —10' neutron stars in the galaxy). Thus,if quark matter is stable, it is a fairly abundant com-ponent of our galaxy. What are the prospects that someof this neutron star material is ejected from its place oforigin and is incident on earth~ This wouM depend onviolent events in which part of the contents of a neutronstar are expelled into the galaxy.

For instance, Gilden and Shapiro recently made a nu-merical study of a head-on collision of two neutron starsat moderately relativistic velocities. They found thatabout 13% of the mass was ejected from the star system.If quark matter is stable, this 13% would probably mostlyescape as lumps of quark matter of various sizes—thespectrum of sizes being difficult to estimate. Since thismass loss occurs in violent conditions, some part of theoutgoing matter might be accelerated to relativistic veloci-ties, perhaps furnishing a candidate for Centauros.

Head-on neutron star collisions may be rare events.However, neutron star collisions of some kind are almostcertainly not rare. Of less than 10 known pulsars, thereseems to be one known binary neutron star system, thebinary pulsar PSR 1913+ 16. The gravitational radiationfrom this system gives it a lifetime of 10 yr; in thattime —short compared to the age of the Galaxy —the twoneutron stars will spiral inward and collide. Naively ex-trapolating from one known example out of 10, it is like-ly that 10 —10 of the 10 —10' neutron stars in ourgalaxy have undergone this fate.

The decay of a binary system seems less promising formass ejection than a head-on collision. However, Clarkand Eardley have speculated that also in this case, dur-ing mass transfer between the two stars, some mass will beejected to infinity by hydrodynamic effects or neutrinopressure; in fact, they suggest that this could be as muchas 10 ' solar masses. VA'th 10 binary systems ejecting10 'M~ each, there could be 10 Mg of quark matter freein our galaxy, or 10 of all the luminous galaxy. Thisfigure is probably an extreme upper bound.

Actually, to explain Centauro events 'the flux requiredis rather small. The Centauro flux, if real, is about10 /m /yr. If the Centauro primaries are relativisticparticles of baryon number 10, the baryon density inCentauro primaries is about 10 ' baryons/cm. The

ED%ARD %'ITTEN

average universal abundance of baryons in luminous formseems to be about 10 /cm, so on this interpretationCentauro primaries are 10 ' of all baryons if they areuniformly distributed in the Universe; if Centauro pri-maries are trapped in galaxies by magnetic fields they are10 of all baryons.

To explain Centauros, a typical galaxy with 10"Mo ofluminous matter must convert 10 MD or 10 Mo ofbaryons into Centauro primaries. Of course, the Centauroprimaries are highly relativistic (10 —10 GCV/baryon),and there may be a much bigger, yet unobserved flux oflower-energy particles of the same composition as Centau-ro primaries. Even so, it would appear that the mass re-quirement for explaining Centauros is quite moderate; allobserved Centauros might even be the product of a singleenergetic neutron star collision in this history of theGalaxy. To develop a detailed theory of Centauros alongthese lines, one would have to understand the mechanismsof mass ejection from neutron stars, the fraction of ejectedmatter which escapes in drops of order 10 baryons, andthe fraction of those drops which are accelerated to rela-tivistic velocities.

Finally, one may worry that it violates basic observa-tions about neutron stars to suppose that neutron stars aremade of quark matter except for a thin outer crust. Thatthis is not so can be seen from work by Fechner andJoss, who considered an equation of state for whichquark matter (at zero pressure) was slightly unbound andshowed that the bulk properties of quark stars were verysimilar to those of neutron stars.

This point can be made in the following terms. Simpli-fylllg to lgllol'c qllal'k IIlasscs (which RI'c lIllportallt 111 VRC-

uum, but not in neutron stars), the relation between pres-sure p and density p of quark matter is roughly

p = —,(p —48), where 8 is a phenomenological constant—essentially the MIT bag constant. The density of zero-pressure quark matter is just 48. With this form of thepressure-density relation, a simple numerical integrationof the standard equations of stellar structure shows thatthe maximum mass of a stable quark star isI=O.OZ5S/(6'"8'"), (Al)

where 6 is Newton's constant. The corresponding radiusand critical density are

8 =O.O95/G'"8'",

p(0) =19.28 .(A2)

A typical value of 8 is 8O ——56 MCV/cm =(145 MeV) .Ill coIlvclltlollal ulllts, (Al) Rnd (A2) glvc

1/280

M =2.0OMO 8

(A3)

p(0) = 1.91X 10'cm +o

for the stable quark star of maximum mass. The surface

TABLE I. The table shows the mass and radius of a quarkstar of various central densities„ for 8=So——(145 MeV}4. Thecentral density is given in units of p*=48=4)&10' g/cm,which is the density of quark matter at zero pressure. The max-imum central density of a stable star is 4.8 p . The radius andmass are again given in kilometers and solar masses, respective-ly. If 8&So, then R and Mmust be multiplied by (Bo/8}'».

R(km}M/Mo

10.51.11

12.01.58

12.31.89

4.1

11.21.98

11.12.00

density is 48=(4.0X10' g/cm )(8/8O).If there is an outer crust of normal matter, it is only the

radius that changes appreciably (by a few percent). Thevalues of (11) are within the range of conventional neu-tron star models. For further details of the radius andcentral density of quark stars for various mass, seeTable I.

APPENDIX 8: GRAVITATIONAL SIGNALFROM THE @CD EPOCH

In this paper, we have assumed that the QCD phasetransition occurred in near-equilibrium conditions. It ispossible, instead, that this transition occurred explosively,with the nucleation of the low-temperature phase trigger-ing detonation waves —in which much of the latent heatwould go into acceleration of bubble walls —as opposed tothe process of slow combustion that we have discussed.The collision of detonation waves would leave a rathercomplicated universe on scales less than the horizon scaleat the QCD epoch. What observable consequences mightremain in today's world~

Various possibilities have been discussed in the litera-ture. Here we will consider the possibility that gravita-tional waves produced in the violent bubble collisionswould be detectable today.

As we will see, these waves would have very long wave-lengths in today's universe. Long-wavelength gravitation-al waves can be detected by Doppler tracking of space-craft or by pulsar timing measurements. The twomethods are similar in principle, but the latter method issensitive at the longest wavelength and is therefore (forreasons we will see) most suited to exploring the condi-tions that prevailed during the QCD epoch.

To understand how pulsar timing can be used to detectgravitational waves, consider a pulsar which (prior to thepassage of a gravitational wave) is separated from theearth by a distance I. in the z direction. If a weak gravi-tational wave with A~=II cos(k.x —cot) passes by, thelight travel time from earth to pulsar is not precisely I.,but is

L(t)=L+ —,'

JI J dz cos[k,z co(t z)] . (Bl)— —

If k,L (1, then the time-dependent part of L is of orderhL. If k,L &&1, then the time-dependent part of L is oforder h k, where A, =2~/k.

The time dependence of I. causes a correction to the ar-

30

rival time of the pulsar pulses. A random gravitationalwave background would show up as a residual "noise" inthe pulse arrival times which could not be removed by fit-ting for various physical effects that are relevant (the pul-sar period and phase, its spin-down rate, its velocity, andnumerous solar system effects).

Limits on pulsar timing noise have already been used toset cosmologically interesting bounds on gravitationalwaves of period 1—10 yr. If it turns out someday that agravitational wave background is really the cause of partof the pulsar timing noise, then —as originally noted byDetweiler —general relativity predicts definite correlationsin the timing noise of different pulsars; observations ofsuch correlations would. be convincing evidence that grav-itational waves are really being detected. %@hat makes thissubject exciting is the fact that the newly discoveredmillisecond pulsar 7 is far more stable than previouslyknown pulsars, and offers the hope of a dramatic im-provernent in the bounds on a cosmological background ofgravitational waves.

An experiment conducted for a period r is sensitiveonly to gravitational waves of period co&2m. /r. (Thetechnical reason for this perhaps obvious fact is that theinfluence on timing measurements of the first few timederivatives of L can be absorbed in a fit for the unknownpulsar phase, period, and spin-down rate. ) In practice, fornormal experiments r is at most a few years or decades,while pulsars are thousands of light years away. Sok,L &&I, and the time-dependent part of L is of orderhA, . This modifies the pulse arrival time by 5t=hA, /c.VA'th X&cv the timing fluctuations due to gravitationalwaves in an experiment of duration ~ are of order

5t-h~. (82)For 6t to be detectable, it must be greater than the timingnoise At due to other causes; the smallest detectable h ish -b, r/~.

The remarkable fact is then that (for r less than thelight travel time to the pulsar) one can measure smallerand smaller h by studying the pulsar for a longer andlonger ~. But at long periods one may reasonably expect alarger amplitude h. The reason for this is that in generalrelativity the energy density of a gravitational plane waveof amplitude h and period P is

(83)8m 6 I'

This means that gravitational waves of period P and ener-

gy density

(84)

are detectable by means of pulsar timing observations.For the millisecond pulsar, a timing stability At-10

sec has been demonstrated, and it is plausible thatht-10 sec may be attainable. If so, at a period P=1yr =3 &( 10 sec, a detectable gravitational wave amplitudeis h & 3&10 . This corresponds to an energy density ofabout 10 g/cm or 10 times closure density. Moredetailed and precise theoretical treatments can be found inRef. 35.

The only general theoretical limit on a cosmologicalbackground of gravitational waves comes from considera-tions of nucleosynthesis. This argument was first givenby Carr and Zeldovich and Novikov. Let Q& and A&be the fraction of closure density in gravitational wavesand photons, respectively. The ratio Q~/Qr is approxi-mately conserved in time, since gravitons and photonsred-shift in the same way. At the time of nucleosynthesisone must have 0+ & 0&, otherwise the extra cosmic expan-sion due to gravitational waves would spoil conventionalnucleosynthesis calculations, the effect being similar tothe effect of including extra neutrino types. Since todayQ&-10, one has a bound Q (10

If a cosmological background of gravitational radiationis discovered, it may have been produced in the conven-tional matter or radiation-dominated eras, or it may havebeen produced even earlier, perhaps in the "inflationaryuniverse. Star otHQsky has calculated the gfav1tatlonalradiation produced in a transition from a de Sitter infla-tionary phase to a radiation-dominated era. Observationof the spectrum computed by Starobinsky, with its I/cofrequency dependence, would be an extraordinary confir-mation of the inflationary universe. We will have nothingto add here to Starobinsky's beautiful calculation, some ofwhose implications have been discussed by Rubakov, Sa-zhin, and Veraskin. Here we will consider only the caseof gravitational radiation produced in the radiation-dominated era.

In a model-independent sense, what can one learn byobservations sensitive to waves of period P and amplitudeh =(10 sec)/P7 If gravitational waves are generatedduring the radiation-dominated era at a time when thetemperature of the Universe is T, their wavelength whenthey are produced would be no longer than the horizonscale at that epoch. In order of magnitude this scale isMp&/T2; a more precise value is

1/2

(85)4maN( T)

where a=+ /45 is the Stefan-Boltzmann constant andX(T) is the effective number of particle spin states attemperature T. Let T* be the current cosmic tempera-ture„T*=2.7'K. Because of the cosmic expansion, thewavelength of a wave produced at temperature T is biggertoday by a factor of T/T* than it was when the wave wasgenerated. The maximum possible wavelength today ofwaves generated when the temperature was T is thus

' 1/2Mp)

A,(T)= 34maN( T)

Mp)=(0.088 cm) T (86)

For T=100 MeV (a plausible QCD temperature) and%=70 {areasonable value just before the @CD transition)we get

A, =1.28& 10' em=1. 37 lj.ght years . {87)

We see, then, that gravitational waves produced in the@CD epoch would have periods as long as about one year.

ED%'ARD %'ITTEN

[However, the fact that X of Eq. (87) is miniscule bygalactic standards means that the QCD transition couldinfluence galaxy formation only indirectly, if at all. ]

If a gravitational wave is created at temperature T withamplitude ho, its amplitude today is h =ho(1 /T). (Thisfollows from standard formulas for propagation of gravi-tational waves in the expanding universe. See, for in-stance, pp. 584 and 585 of Ref. 43.) With T*=2.7'K andT=100 MeV, h =2.3X10 ' ho. From our previous dis-cussion, we know that for waves of period 1.37 yr, thedetectable range is crudely (and perhaps optimistically)h &10 sec/(1. 37 yr)=2. 3&&10 ' . So a gravitationalwave created in the QCD epoch with primordial wave-length equal to the horizon scale and primordial ampli-tude ho & 10 may be detectable. This potential sensi-tivity is truly remarkable.

In general, the wavelength of waves created at the QCDepoch may not be the maximum A, allowed by the horizonscale. If the actual wavelength is A, , the sensitivity is 10sec/A, instead of 10 sec/X, and a detectable primordialamplitude could be

ho&10-3

Let us now estimate —under reasonable but optlm1stlcassumptions —the amplitude of gravitational waves thatmight be generated in the QCD phase transition.

A body of mass M and radius R has a gravitationalfield of order GM/R. If two such bodies collide atmoderately relativistic velocities, and are brought to a haltor scattered through a large angle, the gravitational wavesgenerated have amplitude of order GM/R.

In our case the colliding objects are expanding bubblesof low-temperature phase. We suppose these bubbles areexpanding explosively, as a detonation wave, with the la-tent heat going into the bubble walls, whose velocities aremoderately relativistic. The latent heat is a large fractionof the cosmic energy density p. When a bubble has radiusR its expanding walls have energy (4~/3) R p soGM /R = (4n /3) GpR .

The horizon radius is R~=(~81T/3)Gp, so 1f we setR =R~, then GM/R —l. In such a ease, abundant gravi-tational waves would be produced; however, if the ex-panding bubble walls have GM/R —1, then their col-lisions would be likely to convert much of the mass of theUniverse into black holes. A gross contradiction withmeasurements of the total mass density of today' suniverse would arise if more than about 10 of the totalmass is converted into black holes during the QCD epoch.

Let us suppose that the bubbles when they meet have atypical radius R that is less than the horizon scale RH, lete=R/RH. Then GM/R -e . Since the expanding bub-bles fill all space when they collide, they produce an all-pervasive gravitational wave signal of mean primordialamplitude ho —e and wavelength R. Since A, /X=R/RH ——e, we see from Eq. (88) that this signal may bedetectable if ho&10 e '. In other words, we requiree) 1/10; @=1/10 corresponds to waves of one monthperiod 1n today s un1vclsc.

What value of Els plausible? He're we must, make as-sumptions about how the phase transition was nucleated.

In particle physics it is often assumed that phase transi-tions are nucleated by therma1 fluctuations. In practice,experience shows that except in very pure, homogeneoussamples, phase transitions are often nucleated by variousforms of impurities and inhomogeneities of nonthermalorigin.

Hogan has recently estimated the plausible range of efor the case in which the QCD transition is nucleated bythermal fluctuations. He finds e(10 . The origin ofthis estimate is as follows: If S(T) is the free energy for abubble that forms at temperature T, then the nucleationrate is of order T e . There is at least one bubblecreated per space-time horizon volume if1'e """-&(T'/M»)', or if S(T)/T &41nM» /1=180. If S(T)/T=180, then (without very lucky finetuning) dS/dT) 10 . This means that if the first bubbleforms in a typical horizon volume at temperature T&,then many will have formed by temperature

1Tp —— 1 —— T) .

10

From T~ to T2 a bubble can expand only 10 horizonlengths, and this is then a reasonable upper bound on e.More detail can be found in Hogan's paper.

Although this upper hmit on e is temptingly close tothe a=10 that can give a detectable gravitational wavesignal, we must note that the ratio of signal to sensitivityscales like e . If the QCD transition was nucleatedthermally, it would not be likely that the gravitationalwave signal is detectable with 1984 techno1ogy.

What if the transition was nucleated by impurities'7 Inthis case the mean spacing between bubbles has nothing todo with free energies of nucleation and is simply the spac-ing r between the relevant impurities. Impurities ofr )R~ are too few to play a role, while if r &&R~/10, theresulting gravitational wave signal is again too small,since we need e=r/R~ & 1/10. The interesting case is10 RH &r&R~.

An example of a type of impurity that could nucleatethe QCD transition is a magnetic monopole. The core ofa monopole is a region of strongly broken chiral symme-try, and this region could easily serve as a nucleus forthe low-temperature QCD phase, rather as in weakly cou-pled gauge theories. However, it would be quite a Aukefor the mean monopole spacing to be between 10 'R~and R~ during the epoch considered.

Fa1 morc p1omis1ng 1s thc poss1b111ty that cosmicstrings could serve as nucleation sites. Unlike monopoles(or any other poinilike particles), strings have the propertythat the number of strings per horizon volume is roughlyconstant as the Universe expands. (Just this propertymakes strings plausible as agents of galaxy formation. ) Itis quite natural for the mean separation between strings tobe comparable to, but a bit less than, the horizon scale.%'hether strings arising in a given context would actuallyserve as nuclei for the QCD phase transition is somewhatmodel dependent, but if so, there is a good chance of adetectable gravitational signal.

One last possibility is that the cosmic fluid prior to theQCD epoch might not be uniform, but could have a lowamplitude of turbulent motion. Such turbulent motion

30 COSMIC SEPARATION OF PHASES 283

might be, for instance, a remnant of bubble collisions inthe earlier SU(2) )& U(1) transitions. (Such turbulencemight be large initially, but could be largely damped byneutrinos prior to the QCD epoch. ) In a fluid in a tur-bulent state, the temperature and pressure are not uni-form. Preexisting turbulence could play a crucial role innucleating the QCD transition since the regions of mostextreme temperature depression would act as nucleationsites for the QCD transition. Under such conditions, topredict the mean sparing between bubbles one would needto know the detailed spectrum of the fluid motion, so it isvery hard to estimate the possible gravitational signal.

Let us conclude this discussion of the gravitational sig-nal from the QCD transition with a comment of a modelindependent sort. It has been suggested that the QCDtransition produces black holes which constitute the "darkmatter" in today's universe.

For this to be sa, the QCD transition must convertabout 10 of the mass into black holes. Black holes canpossibly be formed if bubbles of low-temperature phaseare so far apart when they nucleate that by the time theycollide, the mass in colliding bubble walls is adequate toform black holes.

If the bubbles expand to radius R before colliding, thecolliding walls carry a mass M-T 8, since the energydensity is of order T . For coHiding objects of mass Mand radius R to form a black hole, one needs GM/R & 1,or in this case, 8 & 1/G'~ T, which is the horizon scale.

Nucleated bubbles cannot be farther apart when theyform than the horizon scale, or a cosmological disasterwould ensue. If we hope to form black holes, we mustsuppose R-1/G'~ T . In this case, the mass of theblack holes is M-T R —1/G ~ T, which is of order asolar mass for T-100 MeV.

We do not want a typica/ horizon volume to react into ablack hole, because this would lead to by far an excessivemass density in today's universe. Rather, it must be thata given horizon volume has a 10 probability to turninto a black hole. Where a black hole forms, the metricperturbation is of order 1. If a black hole was producedin 10 of all harizon volumes —, then even if nothing hap-pened elsewhere, the root mean square metric perturbationin primordial gravitational waves is of order(ho )'~ -(10 )'~ —10 . This is not quite detectable,because we saw earlier that for a detectable signal today,the mean primordial gravitational wave signal at the QCDhorizon scale should be ho&10 . However, if blackholes farmed in 10 of all horizon volumes, a metric dis-turbance that is large but not large enough to make ablack hole must have been the typical occurrence. Ifblack holes of QCD genesis are the explanation of thedark matter in the Universe, the gravitational radiationproduced in their formation is almost certainly detectable,but it is impossible to be more quantitative without betterunderstanding of how the QCD transition occurs.

As for the possibility that the nucleation scale R in theQCD transition was comparable to the horizon scale, thisdepends on how the QCD transition was nucleated. If itwas nucleated thermally, then as we discussed earlier, thenucleation scale was almost certainly at least about 10times less than the horizon scale, and black hole forma-

tion seems quite unlikely. Of the scenarios we consideredearlier for nonthermal nucleation of the transition, theonly one that is promising for making black holes is thepossibility that the transition was nucleated by cosmicstrings (or walls bounded by strings); it is natural for themean string separation to be comparable to but slightlyless than the horizon scale.

This completes our discussion of gravitational wavegeneration in the QCD transition. Of course, there aremany other possible sources of a cosmological gravitation-al wave background. We have already notedStarobinsky's calculation in the context of the inflationaryuniverse model. There are many galactic processes thatmight possibly generate long period gravitational waves.Without claiming to be comprehensive, we will here con-sider several scenarios which have been considered in theliterature and which are of interest as possible sources ofgravitational waves.

Stecker and Shafi" suggested that axion walls boundedby strings detach themselves at T-100 MeV. At thistime their radius is of order the horizon scale, and theybegin oscillating at the speed of light with a period of or-der 10 —10 sec.

The oscillating walls decay by emitting gravitational ra-diation; this process takes 10 —10 sec, and is completedwhen the temperature of the Universe is of order 10 keV.Most of the radiation is emitted near that temperature.Although these gravitational waves were generated with aperiod of order 10 —10 sec, the period today would belarger by a factor of (10 keV)/(2. 7'K) and would be of or-

der 10 —10 sec.To estimate the amplitudes of these waves, let us note

that in the model of Steeker and Shafi, the walls were onthe verge of dominating the mass in the Universe whenthey disappeared into gravitational waves. This meansthat, when produced, the waves have energy comparableto the energy in photons (which are a large component ofthe energy content of the Universe in that epoch). Sincegravitons and photons red-shift in the same way, the ener-

gy content in gravitational waves would still be compar-able today to the energy content of the blackbody radia-tion, which is about 4&10 of closure density. Themodel of Stecker and Shafi therefore corresponds to10 —10 of closure density in gravitational waves ofperiod 10 —10 sec. The period of these waves is tooshort to be favorable for detection by pulsar timing mea-surements, and detection in that way appears farfetched.However, this signal might ultimately become detectableas a result of future developments in the related area ofDoppler tracking of spacecraft.

Another scenario that leads to an interesting gravita-tional wave flux is the idea that topologically stablestrings served as seeds for galaxy formation. In this case,the strings are still oscillating and generating gravitationalwaves in today's universe.

In such a context, Turok recently estimated the gravi-tational wave flux in today's universe would have a typi-cal period P —5000 yr and a typical amplitudeh-2&&10 ". Such a flux can definitely be detected bypulsar timing measurements if one is patient enough.Over a 5000-yr period, the gravitational wave would cause

ED%'ARD %'ITTEN 30

a discrepancy in the pulse arrival time of about(2X10 ")X(5000 yr)=3 sec, while an effect of 10 secmight be detectable with 1984 technology for pulsar ob-servation.

If one is not so patient, how long will it take to observethis signals If the measurements are conducted not for5000 yr, but for a smaller time t, the amplitude of the de-Iay ln pulse arrival tIIDc duc to thc gravltatlonaI wave 1snot 3 sec, but (3 sec)cos(t/T+q&), where p is an unknownphase and T=5000 yr. Expanding this in powers of t/Tfor t « T, the terms of order (t/T), (t/T)', and (t/T)can be absorbed in the fact that one does not knowa prion the phase, pertod, and spm-down rate of the pul-sar. The first term from which a gravitational wave sig-nal can be extracted is the term of order (t/T) . Thegravitational wave signal is hence of order (3sec)(1/3!)( t/T), and requiring this to be at least 10 " sec,one finds it would take about 25 yr to detect the effects ofcosmic strings in this way, in the absence of technological1ITlprovemcnts.

If the spin-down rate of the binary pulsar were knownQ pI"EOI'E~ onc would obtain 8 gravltatlonal wave signal oforder (3 sec)(1/2!)( t /T ), which would be detectablewithin a few years. Bertotti, Carr, and Rees pointed out

that thc orbital motion of tbc binary pulsar 1s 8 clockwhose behavior can be predicted (if one believes generalrelativity) and used to search for gravitational waves ofperiod up to 10 yr, but on the basis of their numbers, theknown binary pulsar PSR I913 + 16 could not be used todetect a signal of the magnitude estimated by Turok.

Evidently, 8 great many lnechanisrns might generate 8cosmic background of long period gravitational waves. Ifsuch a background is found, its frequency dependence willplay Rn important role ln dctcrlmmng whether thc signalwas generated in the inflationary universe, in the QCDphase tI'Rnsltlon, by osci08ting stnngs, oI' in some othcl%'Ry.

Note added in proof. N. Turok and A. Vilenkin haveindependently pointed out (private communications) thatthc tllTlc IcqulI'cd to dctcct gl Rvltatlonal waves fromcosIIllc strings 1s IIluch less than thc 25 years estimatedhI'c. The estimate herc corresponds to gravitational wavesthat are curI"ently being generated, but red-shifted wavesemitted at large z are more favorable. The spectrum ofgravitational waves emitted bp strings Rt lalgc z %'Rs con™puted by Vilenkin. The question of detection of thesewaves via the millcsecond pulsar has been discussed in-dcpcndc11tly by Hogan Rnd Rccs.

On lcavc from DcpartIIlcnt of Physics„Princeton Un1vcrsity,Princeton, N.J.

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