Cosmic Strings and Domain Walls in Models with Goldstone and Pseudo-Goldstone Bosons

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    Cosmic Strings and Domain Walls in Models with Goldstone and Pseudo-Goldstone Bosons

    Alexander Vilenkin and Allen E. EverettPhysics DePartment, Tufts University, Medford, Massachusetts 02155

    (Received 11 January 1982)

    Topological vacuum structures are investigated in models with spontaneous breaking ofexact and approximate global symmetries. The cosmological. evolution of the structureis discussed. A spontaneous breaking of an exact global U|,'1) symmetry gives rise tovacuum strings which can produce cosmological density fluctuations leading to gal. axyformation. In a simplified axion model, the vacuum structures are strings connected bydomain walls. They decay before they can dominate the universe.

    PAGS numbers: 98.80.Bp, 12.10.-g, 11.30.c

    Gauge theories of elementary particles withspontaneous symmetry breaking predict that theearly universe passes through a sequence ofphase transitions as it cools after the big bang.These phase transitions can give rise to vacuumstructures vacuum domain walls, strings, ormonopoles, depending on the topology of the gaugetheory. ' Domain walls are produced if M, themanifold of degenerate vacuums after symmetrybreaking, consists of two or more disconnectedcomponents. Strings are formed if iM containsincontractible loops (that is, if M is not simplyconnected), and monopoles are formed if M con-tains incontractible two-dimensional surf aces.The cosmological evolution of these structureshas been discussed previously. ' '

    Besides gauge symmetries, some particlemodels have exact or approximate global sym-metries which can also be spontaneously broken.Well-known examples are the U(1) 8-L symmetryof SU(5), ' and the approximate U(1) chiral sym-metry in models with axions. " In the presentpaper, we shall discuss the vacuum structuresarising in such models and their cosmologicalevolution. We shall concentrate on the mostinteresting case of a global U(1) symmetry.

    Let us first consider the case of an exact U(1)symmetry. The character of vacuum structuresis insensitive to the details of the model, and allrelevant physics is present in a simple Goldstonemodel described by the Lagrangian

    I, = 8 C+8t C -'hs(~C ~s &s)2

    where 4 is a complex scalar field. The manifoldM of degenerate vacuum states in this model isa circle. It is not simply connected, and thus wemust have strings. The strings can be classifiedaccording to their winding number, n= b, 6/2v,where 68 is the change of phase of the vacuumexpectation value (VEV) (4) as we go around thestring. The lowest energy strings have n = 1.

    The magnitude of the VEV ~ (4) ~ is substantiallydiffexent from g only inside the string core ofwidth A, -(hti) ' and is equal to zero on the stringaxis. The energy per unit length of the core is

    -h'q z'- q'. All this applies equally to thecases of local and global U(l) symmetries. Thebasic differences are to be found outside thecore. Consider an infinite straight string par-allel to the z axis. In cylindrical coordinates x,y, z, and for ~A. ,

    where we have assumed that n =1. In the case ofa local U(1) symmetry, the variation of (4) isbalanced by that of the gauge field, so that thefield outside the core is pure gauge. All the en-ergy of the string is in its core, and the energyper unit length is t rp. For a global symmetry,there is no gauge field, and the energy per unitlength of the string diverges logarithmically,

    tt-ti'+ f~ ~r '(8/8p)(4))'2~rdr-~. (3)For thi. s reason, such strings are usually thoughtto be illegitimate. "

    We note, however, that the divergence of theintegral (3) does not introduce any difficulties ina cosmological context. If the scale of the systemof strings is - $ (that is, the typical curvatureradius of the strings and the typical distance be-tween adjacent strings are - $), then $ providesa natural cutoff in the integral (3) and we obtain

    p. - tl' ln(]/x) .[Note that the energy of a closed loop is alwaysfinite, E-rPRln(A/A. ), where R is the size of theloop and we have assumed that the loop is momen-tarily at rest. We note also that global-symme-try strings are well known to condensed matterphysicists in the form of quantized vortex linesin liquid helium. ] Two approximately parallelstrings with opposite signs of b6 attract one

    1982 The American Physical Society 1867


    p, /p- Gp - (q/m p)' In(t/Z) . (6)It has been shown in Refs. 3 and 4 that stringscan produce cosmological density fluctuationssufficient to explain galaxy formation" if" 6p.-10 '. Then Eq. (6) gives g-10"-10"GeV. Weconclude that a model in which a global U(l) sym-metry is spontaneously broken at the grand unifi-cation mass scale can provide a solution to theproblem of cosmological density fluctuations. ItshouM be noted that grand unified models withstable strings resulting from a gauge-symmetrybreaking are difficult to construct"; no realisticmodel of this kind has yet been suggested.

    One possible candidate for the role of U(1) 8&,b, &is the B-L symmetry of SU(5). Models withspontaneous breaking of this symmetry can bereadily constructed. ' It can also be shown' thatthe resulting Goldstone boson is virtually unob-servable and does not give rise to a 1/~' forcecompeting with gravity.

    We now turn to the case of an approximate glob-al U(1) symmetry. Again we sha, ll consider asimple model with one complex scalar field,

    L = 84 '&"C - -,'h'( p' rP)'+2m'(cos& 1), (7)where

    As a prototype for this model we used modelswith "invisible" axions' designed to solve thestrong CP problem. In such models g-10" GeVand m -0.1 GeV. (We shall use these values forestimations below. ) The approximate U(1) sym-metry of (7) corresponds to the anomalous ehiralPeecei-Quinn symmetry" which is explicitlybroken by instantons at the strong interactionmass scale.

    The field 4 acquires a nonzero VEV when theuniverse cools to a temperature T- g. At suchtemperatures the last term in (7) is negligibleand the behavior of the model is the same as that

    another, the force per unit length being E;,p/The force of tension in curved strings is of

    the same order of magnitude. It appears that thepresence of interaction between the strings doesnot substantially alter the scenarios of Refs. 1,3, and 4; thus the scale of the system of stringsat cosmic time t is -t. The energy density due tothe strings is

    p, - pt/t'- p, t '.The energy density of matter is p 1/-Gt', andthus

    QH+m, 'sino=0, (10)where m, =m'/q-10 ' eV is the axion mass. 'Equation (10) is the so-called sine-Gordon equa-tion which is known to have domain-wall solutions(solitons) ":

    e(x) =4tan 'exp(m, x),where the x axis is perpendicular to the wall.The thickness of the wall is 6-m, '-10' cm andthe energy per unit area is


    (the exact expression is" 0 =16m'g).The domain walls can be considered as "formed"

    when their thickness becomes smaller than thehorizon, i.e., at" f&m, '-IO ' s, T&I GeV. Atlater times the system consists of strings con-nected by domain walls. The linear mass densityof the strings is

    p, -rfln(m, x) '. (13)Strings form the boundaries of the walls and ofthe holes in the walls. One wall surface can bestretched between a number of strings and canhave a complicated topological structure. In addi-tion, there can be closed or infinite wall surfacewithout strings.

    The frictional force acting on the walls as a re-sult of their interaction with particles is negligi-ble, since the width of the walls is much greaterthan the thermal wavelength of the particles. "The friction of strings is also negligible. " The

    of the Goldstone model (1). The phase transitiongives rise to strings which evolve as discussedabove. At sufficiently low temperatures the 9-dependent term in (7) becomes important. Theminimum of energy corresponds to 6 = 0 (mod2E'). However, the phase 8 cannot become equalto zero everywhere, since it changes by 40 = 2~around the strings. It is natural to assume that6) settles down to zero at all points around astring, except within a relatively thin wall, sothat 6 changes by 2~ across the wall. " The pos-sibility that strings can get connected by domainwalls has been first suggested, in a different con-text, by Kibble, Lazarides, and Shafi. "

    To see that the model (7) indeed has domain-wall solutions, we note that, away from thestring cores, the VEV of C is {4)= ge', andthe effective Lagrangian for 6 is

    L = q'&& &"0+2m'(cos& -1) .The corresponding wave equation is



    force of tension in a string of curvature radiusR, E- p/R, is greater than the wall tension, v,for R & p/v. Therefore, at f & p/v-10 ' s theevolution of the strings will not be qualitativelydifferent from that in model (1). At t& g/v thetypical curvature radius of the strings becomesgreater than p/v, and the dynamics of the systemis dominated by the wall tension. Domain wallswill tend to shrink, pulling the strings towardone another, and the holes in the walls will in-crease in size. As the strip of mall connectingtwo strings shrinks, its energy is transferredto the kinetic and potential (rest) energy of thestrings; energetic strings intersect and passthrough one another, and a wall surface startsstretching between them again. The whole sys-tem thus violently oscillates. The strings fre-quently intersect and occasionally intercommute;as a result, the strip of domain wall connectingthe strings is cut into pieces. If the probabilityof intercommuting for intersecting strings is p- 1, then the whole system breaks into pieces ofsize -p/v at f - p/v. (For p 1 the size of thepieces is larger. )

    A piece of wall of size R & p/v oscillates at atypical frequency ~ -R ' and loses its energy bygravitational radiation at a rate

    dM/dt- GM R &o - GvM.

    strings to that of matter never exceedsZ/2

    w8 G~ (G~)1/2 10 3p max k/v

    and thus the vacuum structures never dominatethe universe. This is in striking contrast withthe conclusions of Refs. 1 and 2, where it isshown that "regular" domain walls (which cannotbe bounded by strings and can only be infinite orclosed) lead to disastrous cosmological conse-quences. A more detailed discussion of the evolu-tion of strings connected by domain walls willbe given elsewhere.

    To summarize, we have shown that models inwhich an exact global U(1) symmetry is sponta-neously broken give rise to vacuum strings. Ifthe symmetry breaking occurs at the grand unifi-cation mass scale, such strings can producedensity fluctuations sufficiently large to explainthe galaxy formation. If the spontaneously bro-ken global U(1) symmetry is only approximate,then the corresponding vacuum structures arestrings connected by domain walls. In our modelthese structures rapidly decay and never dom-inate the universe. "

    This work was stimulated by a discussion of oneof us (A.V.) with Alan Guth, who suggested thatglobal, as well as local, bxoken symmetries maygive rise to strings.

    The lifetime of the piece is independent of itssize,

    The lifetime ean be smaller if the pieces rapidlydecay as a result of multiple self-intersections.When the size of the piece becomes smaller thanp/v, its mass is determined mostly by the string,and the decay time is' T &(Gp) 'R~ (Gv) '. (Itis assumed that sufficiently small pieces decayinto elementary particles. )

    We still have to discuss what happens to closedand infinite walls without strings. Like intersect-ing strings, intersecting domain walls can alsointercommute. One can easily convince oneselfthat intereommuting of a large wall with a small-er wall bounded by a string results in for mationof a hole in the bigger wall. By this process,closed and infinite walls become holey and arealso cut in pieces by intereommuting strings. Ifthe probabilities of intercommuting processesare -1, the whole system decays at T-(Gv) '-100 s. The ratio of the density of walls and

    T. W. B. Kibble, J. Phys. A9, 1387 (1976), and Phys.Rep. 67, 183 (1980).

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    S. Coleman, .in Neu' Phenomena in Subnuclear Phys



    its, edited by A. Zichichi (Plenum, New York, 1977).'In the case of a global U(1) symmetry, additional

    fluctuations will be produced by large-scale variationsof the Goldstone field [see A. Vilenkin, Phys. Rev.Lett. 48, 59 (1982)]. However, the fluctuations pro-duced by the strings are greater by a factor of in(t/Aj-100 and therefore dominate. The Goldstone fieldmechanism in its pure form (without strings) operateswith spontaneously broken, simply connected globalsymmetry groups, such as SU(N) with N) 2.

    ' The required magnitude of &p, depends on the behav-ior of closed loops (Ref. 4). In different scenarios thevalue of Gp, may vary from 10 to 10 5.

    A. Vilenkin, Nucl. Phys. 8196, 240 (1982); G. Laz-arides, Q. Shafi, and T. Walsh, Nucl. Phys. 8195, 157(1982) .

    Another way to see the character of vacuum struc-tures of the model (7) is to note that the ground stateconsists of an infinite number of disconnected points,0 = 2~n, and we must have domain walls. However, ~is defined only modulo 2', and thus different domains

    do not have to be completely separated by the walls.Domain walls can have boundaries on which ~ is unde-fined and, therefore, p=0. These boundaries are vacu-um strings.

    Q. Shafi, private communication; T. %. B. Kibble,G. Lazarides, and Q. Shafi, to be published.' We d.isregard the possible dependence of m on tem-

    perature.A. E. Everett, Phys. Rev. D 10, 361 (1974).

    ' After this paper was submitted, we noticed a paperby P. Sikivie I Phys. Hev. Lett. 48, 1156 (1982)] dis-cussing the vacuum structures in axion models. Sikivieappears not to have noticed the presence of strings;however, he shows that in axion models with N quarkflavors there are N types of domain walls, so that theHiggs phase changes by 2n/N across each wall. (Ourmodel of Eq. (7) corresponds to N=1.] In this caseeach string is connected to N different domain walls;the cosmological evolution of the system is more com-plicated and may lead to conflict with standard cosmol-ogy