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VOLUME 78, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 24 MARCH 1997
02155
t
2288Cosmic Strings and the String Dilaton
Thibault DamourInstitut des Hautes Etudes Scientifiques, F91440 Bures sur Yvette, France
and DARC, CNRSObservatoire de Paris, F92195 Meudon, France
Alexander VilenkinInstitute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts
(Received 3 October 1996)
The existence of a dilaton (or moduli) with gravitationalstrength coupling to matter imposes stringenconstraints on the allowed energy scale of cosmic strings,h. In particular, superheavy gauge stringswith h , 1016 GeV are ruled out unless the dilaton massmf * 100 TeV , while the currentlypopular valuemf , 1 TeV imposes the boundh & 3 3 1011 GeV . Some nonstandard cosmologicalscenarios which can avoid these constraints are pointed out. [S00319007(97)027798]
PACS numbers: 98.80.Cq, 11.25.w, 11.27.+du
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ry;alSuperstring theory predicts the existence of liggaugeneutral scalar fields (the dilaton and the modwith gravitationalstrength couplings to ordinary matteOf particular interest among those fields is the modindependent dilaton, whose treelevel couplings are wunderstood. Because of their weak couplings, the litimes of the moduli can be very long. In a cosmologiccontext, if moduli are created in the early Universe, thslow decay rate is the source of serious potential conflwith observations [14]. To simplify the discussion, wshall refer to moduli as the dilaton, but most of thfollowing treatment is applicable,mutatis mutandis,togeneric moduli.
Several mechanisms of cosmological dilaton producthave been discussed in the literature. First, the valuethe dilaton field in the early universe can be set away frthe minimum of its potential [14]. [This is usually thcase because the minima of the dilaton effective potenat late and early times generically differ byOsmPlanckd.]Coherent oscillations of the field about the minimum athen equivalent to a condensate of nonrelativistic particAnother mechanism is the production of dilatons in binaparticle collisions in a hot plasma [5]. A third productiomechanism is the amplification of quantum fluctuationsthe dilaton field in early cosmology [6,7]. Requiring consistency between the cosmological production of dilatoand observations leads to very stringent, anda priori unnatural, constraints on the dilaton mass and couplingsSeveral mechanisms have been proposed to solve thismological moduli problem: e.g., a late stage of secondinflation [5,8], or the presence of a symmetry of moduspace ensuring the coincidence of the minima of thefective potential at early and late times [9,7].
In this paper we shall discuss another mechanismdilaton production. Oscillating loops of cosmic strinwhich could be formed at a symmetrybreaking phatransition in the early Universe, will copiously emit dilatons, as long as the characteristic frequency of oscillatis greater than the dilaton mass. Cosmic strings are00319007y97y78(12)y2288(4)$10.00htli)r.l
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dicted in a wide class of elementary particle models [10Their mass per unit lengthm, which is equal to the stringtension, is determined by the symmetrybreaking enerscaleh, m , h2 [11]. Of particular interest are grandunification strings withh , 1016 GeV which could beresponsible for the formation of galaxies and largescstructure. We shall calculate the dilaton density producby the strings and explore the constraints it imposesthe dilaton and string parameters.
We assume that the string thickness is small compato the loop size and to the Compton wavelength of tdilaton, so that the string can be regarded as an infinitthin line. We work in the Einstein conformal framewhere tensor gravity decouples from the dilaton anddescribed by the standard EinsteinHilbert action. Ththe interaction of the dilaton fieldf with the string isdescribed by the action
S 21
4pG
Zd4x
"12
s=fd2 1 V sfd
#2
Zmsfd dS .
(1)Here, msfd is the fdependent string tension,dS isthe surface element on the string world sheet,G isNewtons constant, and we have used the Minkowsmetric assuming the spacetime to be approximately flat
We choose the origin off so that the minimum of thedilaton potentialV sfd is at f 0. Then, forf in thevicinity of the minimum, the dilaton field equation takethe form
s=2 2 m2fdfsxd 24pGaT sxd , (2)where mf fV 00s0dg1y2 is the dilaton mass, andTsxdis the trace of the string energymomentum tensThe dimensionless parametera ; ln
pmsfd yf
m0s0dy2ms0d measures the strength of the coupling offto cosmic strings. One generically expectsa , 1.
The string world history can be represented asxmsz ad,a 0, 1, where z 0 and z 1 are the world sheet coordinates. The choice of these coordinates is largely arbitrait is convenient for most purposes to use a conform 1997 The American Physical Society
VOLUME 78, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 24 MARCH 1997
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gauge, specified by the conditionsx ? x0 0, x2 1 x02 0 , (3)
where dots and primes stand for differentiation wirespect toz 0 andz 1, respectively. The residual freedomof coordinate transformations can be used to setz 0 x0 ; t, which allows us to describe the string trajectorusing the threevectorxsz , td, wherez ; z 1. The stringenergymomentum tensor is then given by
Tmnsx, td mZ
dz s xm xn 2 x0mx0ndds3dfx 2 xsz , tdg ,
(4)and its trace is
T sx, td 22mZ
dz x02sz , tdds3dfx 2 xsz , tdg , (5)
wherem ; msf 0d. Disregarding dilatonic and gravitational back reaction, the string equation of motion in thgauge (3) has a simple form,
x 2 x00 0 . (6)It can be shown from Eqs. (3) and (6) that the motion ofclosed loop of string is periodic with a periodLy2, whereL ; Mym and M is the loops mass. The quantityL isoften called the length of the loop, although the actulength varies with time.
The rates of dilaton energy loss and of dilaton numbproduction by a periodic source of angular frequencyvcan be found from the following general equations
Ef Xn
Pn, Nf Xn
Pnyvn , (7)
Pn Ga2
2pvnkn
ZdV jT sk, vndj2, (8)
T sk, vnd 1
Tn
Z Tn0
dtZ
d3x eivnt2ik?xT sx, td , (9)
wherevn nv, Tn 2pyvn, n 1, 2, . . .; kn ; jkj sv2n 2 m
2fd1y2, dV is the solid angle element, and th
angular integration is over the directions ofk. The dilatonmomentumkn has to be real; hence, only terms witvn . mf are included in the sums (7).
For a loop of lengthL, vn 4pnyL, and the sums aretaken overn . LyLc, where
Lc 4pymf . (10)For L Lc, vn mf for all values of n, and wecan approximately setmf 0. Then, dilaton radiationfrom specific loop trajectories [described by solutionsEqs. (3) and (6)] can be analyzed using the techniqudeveloped for the gravitational case in Ref. [12]. Detaof such an analysis will be given in a separate pap[13]; here we shall only summarize the results. We fouthat the energy spectrum and angular distribution of tdilaton radiation are very similar to the gravitational ca(and very different from the electromagnetic radiation bsuperconducting strings [14]). The energy and particradiation rates can be represented as
Ef Gfa2Gm2, Nf Gfa
2Gm2yv , (11)where the numerical coefficientsGf and Gf depend onh
y
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a
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feslserdeeyle
the loop trajectory (but not on its size). Typically,Gf ,30 andGf , 13. The total radiation power from the loopis E GGm2 with G Gg 1 a2Gf, where Gg , 65.The highfrequency asymptotic of the spectrum for a lowith cusps isPn ~ n24y3, and for a cuspless loop withkinks is Pn ~ n22. This can be used to estimate thradiation rates from loops withL Lc,
Ef , Gfa2Gm2sLyLcd21y3,Nf , Gfa2Gm2m21f sLyLcd
21y3. (12)
Here, we used a cuspy loop spectrum,Pn ~ n24y3, andintroduced a lower cutoff atn , LyLc.
To estimate the cosmological density of dilatons prduced by oscillating string loops, we shall adopt a simpmodel in which the loops radiate allf quanta in the fundamental mode,v1 ; v 4pyL. This approximationhas been proven to give accurate results (within a facof ,3) for the gravitational wave power spectrum. Moreover, the largen contribution to Nf in Eq. (7) convergesfaster than that to the powerEf, and thus we expect ourestimate for the particle density to be no less accurate tthat for the power spectrum.
Loops of initial lengthL are chopped off the stringnetwork at a cosmic timeti , Lyb [15] and decay attime tf , sGGmd21L. They have number densityni ,z b2yL3 at the time of birth and
nf ,
titf
!3y2, ni ,
k1y2z
GGmt3f(13)
at the time of decay. Here,z is a parameter characterizingthe density of long strings (its definition can be foundRef. [10]), k ; byGGm, and we have used the radiatioera expansion law,astd ~ t1y2. Numerical simulations ofstring evolution indicate thatz , 14 andb & 1023. Theexact value ofb is not known, but it is bounded frombelow byb * GGm, so thatk * 1. From Eq. (11), thetotal number of dilatons emitted by a loop decayingt , tf is
N , Ntf , s4pd21GGfa2G2m3t2f . (14)The quantity of interest to us will beYf nfstdysstd.
Here, nfstd is the dilaton density,sstd is the entropydensity, which during the radiation era is given bysstd 0.0725fN stdg1y4smpytd3y2, N std is the effective numberof spin degrees of freedom at timet, andmp is the Planckmass. Apart from dilaton production and decay and oofequilibrium phase transitions (such as thermalizatiafter inflation), Yf is conserved in the course of thcosmological evolution. The contribution toYf fromloops decaying att , tf can be estimated as
Yfstfd , nfNysstfd
, k1y2z Gfa2sGmd2smptfd1y2N 21y4f , (15)whereNf ; N stfd. Equations (14) and (15) are validas long astf & tc ; 4pyGGmmf, so that the loop sizesare smaller than the critical size (10). From Eq. (15) wsee that larger values oftf give a greater contribution,2289
VOLUME 78, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 24 MARCH 1997
l
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ict

r
r
h

g
tsay
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ly
,
ly
of
ss
he
f
e
tentsn
nand thus the dominant contribution toYf is given bytf , tc [16]. With z , 14, N stcd , 100, Gf , 13,andG , 100, we have
Yf , Yfstcd , 20k1y2a2sGmd3y2smpymfd1y2. (16)Equation (16) is the main result of the present paper.
Strings of energy scaleh are typically formed at timets , tpyGm, wheretp m21p is the Planck time. Longstrings are initially overdamped and begin to move retivistically at time tp , tpysGmd2. Small loops becomerelativistic at an earlier time, but damping due to interation with the surrounding plasma remains a significant eergy loss mechanism untiltp. In the derivation of Eqs. (11)and (16) we assumed damping to be negligible, and tthe condition of validity of (16) istc . tp, which gives
mfymp , 4pGmyG . (17)The analysis of the cosmological implications of th
dilaton density (16) is similar to that for any weakly interacting relic particles [2,17]. The resulting constrainare sensitive to the lifetime of the dilatont, which is determined by its mass and couplings. The dilaton coup(in the Einstein frame) to spin0 and spin12 particles onlythrough the mass terms, so that decays into such partare suppressed by powers of their mass [18]. The inaction Lagrangian responsible for decays into light gaubosons isLint 12 aFfF2mn , and the corresponding lifetime is
t 4m2pyNFa2Fm
3f
3.3 3 1013s12yNFda22F m23G s . (18)
Here, mG mfy1 GeV , NF is the number of gaugebosons with massesmf, and the value ofa2F is averaged over all such bosons. Formf * 1 TeV , allstandardmodel gauge bosons should be included (NF 12). The coupling constantaF is normalized so thataF 1 for a treelevel superstring dilaton. It is geneically expected thataF , 1 for all moduli. For numerical estimates below we setaF a k 1. (Note thatsinceYf ~ k1y2 andk * 1, settingk 1 will result inconservative bounds onm andmf.)
A multitude of astrophysical constraints on unstabrelic particles have been discussed in the literatuShortrange Cavendish experiments [19] exclude ultlight dilatons of mass smaller than1.6 3 1023 eV [20].For quasistable dilatons, with lifetimes larger than tpresent age of the Universe,t . t0 . 4 3 1017 s (corresponding tomf & 40 MeV ), one has the usual uppebound on the cosmological dilaton mass densityVfh2
, 1 [21], where Vf nfmfyrcritical and h ; H0y100 km s21 Mpc21. This yieldsYf , 3.6 3 1029m
21G .
For t * tdec , 1013 s, very stringent constraints follow from the limits on the diffusegray backgroundthat would result from dilaton decays [21]:Yf , 2.9 310216m21G for tdec & t & t0 andYf , 1.3 3 10220m
24G
for t * t0. For 1021 s & t & tdec, the bounds areobtained by requiring that the decay products do nsignificantly change the abundances of4He , 3He , and2290a
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D. The relevant processes are the interaction of ambient nucleons with the hadronic showers resultinfrom hadronic decays for0.1 & t & 107 s [22,23], andphotodissociation and photoproduction of light elemenby electromagnetic cascades initiated by the decproducts for 104 s & t & tdec [17,24] (both processesbeing important for 104 & t & 107 s [22]). The tdependence of the resulting bound on the dilaton densis rather complicated, but roughlyYf & 1.4 3 10212m
21G
for 107 s & t & tdec and Yf & 10213 10214 for 1 &t & 107 s. For t , 0.1 s, dilatons decay well beforethe onset of nucleosynthesis, and the bound is rapidweakened as we move towards smaller values oft.
Combining these bounds onYf with the expression(16) for the dilaton density produced by cosmic stringswe obtain constraints onmf andm which are representedin Fig. 1. We see that the excluded domain cuts deepinto the region of physically interesting values of theparameters. In particular, the most popular valuesGm , 1026 and mf , 1 TeV are incompatible withone another. If the dilaton mass is indeed,1 TeV ,then the string tension is bounded byGm & 6 3 10216,which corresponds to symmetrybreaking scalesh & 3 31011 GeV . On the other hand, if grand unified theory(GUT)scale strings are discovered, then the dilaton mamust satisfymf * 100 TeV .
These conclusions are rather robust with respect to tvariation of the numerical coefficient in Eq. (16) (whichwe expect to be accurate only within a factor of,3).If, for example, the coefficient is changed by 1 order omagnitude, then the bound onGm at a fixedmf is modifiedby a factor of,5, and the bound onmf with Gm in thegrandunification range remains essentially unchanged.
In the derivation of Eq. (16) forYf we assumed thatgravitational and dilaton radiation were the dominant energy loss mechanisms of strings. This is justified for gaug
FIG. 1. Constraints on log10sGmd versus log10smfy1 Gev d.The region above the solid curve is forbidden. Labels indicathe source of the various constraints: Cavendish experime(C), Vfh2 , 1sOd, gammaray background (G), photodissociation (P), combined hadroproduction and photodissociatio(HP), hadroproduction (H). The dashed line indicates the codition of validity (17). (The constraints apply only above thedashed line.)
VOLUME 78, NUMBER 12 P H Y S I C A L R E V I E W L E T T E R S 24 MARCH 1997
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strings, formed as a result of a gauge symmetry breing. In the case of global strings, oscillating loops losmost of their energy by Goldstone boson radiation.will be shown in a subsequent publication [13] that stronconstraints on the symmetrybreaking scale can alsoobtained in this case, as well as in the case of globmonopoles and textures (which have all been suggesas possible seeds of structure formation [25,26]).
We finally mention some ways of avoiding the abovconstraints. The main contribution to the dilaton desity in Eq. (16) comes from the timet , tc, which corresponds to the temperatureTc , 109sGmd1y2m
1y2G GeV .
Our analysis, therefore, is not directly applicable to moels in which the Universe has never been heated upsuch temperatures. For example, in inflationary scenios the thermalization temperature after inflation canbelow Tc. Alternatively, string formation can be delayed until afterTc: in some supersymmetric models GUTscale strings can be formed at temperatures as low aselectroweak scale [27]. Another possibility is to invokmodels where topological defects are produced duringflation [28]. Then the defects begin emitting dilatons onafter their characteristic scale comes within the horizowhich can happen att . tc. In all three cases the resulting dilaton density is very model dependent.
Once dilatons are produced, they can be diluteda brief period of inflation. Models of this kind havebeen suggested [5,8] to resolve the usual Polonyimodproblem: overproduction of dilatons and other modudue to a mismatch of the minima of their potentialearly and late times. The same models can be usedrelax the constraints on topological defects discussed hWe note, however, that another proposed solution tomoduli problem will not work in our case. Dine, Randaland Thomas [9] suggested that moduli production duriinflation can be suppressed if the potential in moduspace has some symmetry which enforces that the potenminima before and after inflation coincide. Clearly, thdoes not resolve the conflict between moduli and defecall defects formed after inflation will produce dilatons, anthus the defect parameters are subject to all constrawe discussed earlier in this paper. The only exceptiis the model suggested in Ref. [20] (whose cosmologicconsequences were further studied in Ref. [7]) in which tminimum of the potential is a point of enhanced symmetfor all dilaton couplings. Then, near the minimum, thdilaton is essentially decoupled from all other fields (particular,a 1), and dilaton production by topologicadefects is suppressed.
A. V. is grateful to IHES (France), where most othis work was done, for hospitality, and to the NationScience Foundation for partial support.
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