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19 November 1998

Ž .Physics Letters B 440 1998 262–268

Cosmic strings from preheating

I. Tkachev a,b, S. Khlebnikov a, L. Kofman c, A. Linde d

a Department of Physics, Purdue UniÕersity, West Lafayette, IN 47907, USAb Institute for Nuclear Research of the Academy of Sciences of Russia, Moscow 117312, Russiac Institute for Astronomy, UniÕersity of Hawaii, 2680 Woodlawn Dr., Honolulu, HI 96822, USA

d Department of Physics, Stanford UniÕersity, Stanford, CA 94305-4060, USA

Received 13 May 1998; revised 20 August 1998Editor: H. Georgi

Dedicated to:the memory of David Abramovich Kirzhnits

Abstract

We investigate nonthermal phase transitions that may occur after post-inflationary preheating in a simple model of aŽ 2 2.2two-component scalar field with the effective potential l f yv r4, where f is identified with the inflaton field. Wei 1

use three-dimensional lattice simulations to investigate the full nonlinear dynamics of the model. Fluctuations of the fieldsgenerated during and after preheating temporarily make the effective potential convex in the f direction. The subsequent1

nonthermal phase transition with symmetry breaking leads to formation of cosmic strings even for v41016 GeV. ThisŽ .mechanism of string formation, in a modulated by the oscillating field f phase transition, is different from the usual1

Kibble mechanism. q 1998 Elsevier Science B.V. All rights reserved.

The theory of cosmological phase transitions pro-posed in 1972 by David Kirzhnits gradually becameone of the most essential parts of modern cosmologyw x1 . Originally it was assumed that such phase transi-tions occur in a state of thermal equilibrium whenthe temperature decreased in the expanding universe.Recently it was found that large fluctuations ofscalar and vector fields produced during preheating

w xafter inflation 2 may lead to specific nonthermalphase transitions which occur far away from the state

w xof thermal equilibrium 3 . To investigate these phasetransitions one should study the self-consistent non-linear dynamics of quantum fluctuations amplifiedby parametric resonance. This is a very complicatedtask. Fortunately, fluctuations of Bose fields gener-ated during preheating have very large occupation

numbers and can be considered as interacting classi-cal waves, which allows one to perform a full studyof all nonlinear effects during and after preheating

w xusing lattice calculations 4 . These calculations, asw xwell as analytical estimates 2–8 have already shown

that fluctuations can grow large enough for cosmo-logically interesting phase transitions to occur. Lat-tice calculations can be used to directly simulatenonthermal phase transitions and formation of topo-logical defects. This made it possible to go beyondthe early attempts to study such phase transitions

w xnumerically 9 , which neglected crucial backreactioneffects beyond the Hartree approximation.

We made a series of lattice simulations of non-thermal phase transitions, focusing primarily on thepossibility of generation of topological defects

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01094-6

( )I. TkacheÕ et al.rPhysics Letters B 440 1998 262–268 263

w x10,11 . The model with one scalar field with theŽ 2 2 .2potential l f yv r4 has a broken discrete sym-

metry f™yf. In this model we have found forma-w xtion of domain structure 10 . If the inflaton field f

strongly couples to another scalar field x , the phasetransition is strongly first order; we observed forma-

w xtion of bubbles of the new phase 11 .In this Letter we will study string formation in the

theory of a two-component scalar field f with theiŽ 2 2 .2 2effective potential l f yv r4, where f s

2 Ž .Ýf . This model has O 2 rotational symmetry andiw x Žallows for string formation 10 for early conference

w x.reports see 12 . Recently Kasuya and Kawasakireported formation of defects in 2d lattice simula-

16 w xtions for vQ3=10 GeV 13 , which confirmedour general conclusion concerning the phase transi-

w xtion in this model 10,12 . However, scattering ofparticles, as well as the nature of topological defectsin two dimensions and in three dimensions are quitedifferent. In particular, there are no strings in thismodel in two dimensions. We have found that in therealistic case of three dimensions the generation offluctuations is much more effective, and string pro-duction is possible even if v is as large as 6=1016

GeV.In the model with the two-component field f wei

can always rotate fields in such a way that initiallyf s0, and the field f plays the role of the2 1

classical oscillating inflaton field. We denote byŽ .f 0 the value of the field f at the moment when1

inflation ends and the inflaton field begins to oscil-late. It is convenient to work with the rescaled

' Ž . Ž . Ž .conformal time t , where l f 0 dtsa t dt , a 0s1, and perform conformal transformation of the

Ž . Ž . Ž .fields, wsf t a t rf 0 . We also will use the' Ž .rescaled spatial coordinates x™ l f 0 x. The ini-

tial conditions at the beginning of preheating aredetermined by the preceding stage of inflation. Wedefine the beginning of preheating as the momentts0 when the velocity of the field f in conformal

Ž .time is zero. This happens when f 0 f0.35MPlŽ . w xand a t f0.51tq1 5 .

In the new variables, the equations of motionbecome

w y=2w q w 2 yÕ2a2 w s0 , 1Ž .Ž .¨ i i i

Ž . Ž .where Õsvrf 0 . Eq. 1 contains only one param-eter, Õ. The coupling constant l is hidden in the

initial conditions for fluctuations; these were chosenw xas described in Ref. 4 .

Ž .The full nonlinear equations of motion 1 weresolved numerically directly in the configurationspace. The computations were done on 1283 latticeswith the box size Ls16p , with the expansion ofthe universe assumed to be radiation dominated.There are several important quantities that can bemeasured in the simulations during the evolution ofthe system. One can define the zero mode w s0 i² Ž .:w x,t for each of the components is1,2 of thei

²Ž .2:field w, and variances dw , which measure theiŽ .average magnitude of fluctuations dw sw x,t yi i

w . Since the system is homogeneous on large0 i

scales, averages in these relations can be understoodas volume averages. This is equivalent to taking an

Žaverage over realizations of the initial data the.ensemble average . The averaged quantities depend

only upon time and do not depend upon spatialcoordinates x.

We performed calculations for various v and l. Inthe beginning we present results for vs3=1016

GeV and ls10y12. In the last part of the paper wealso present result for different values of v and forls10y13. The behavior of the zero modes f s0 i² : ²Ž .2:f , and of the variances, f yf , for eachi i 0 i

of the field components is shown in Fig. 1.Let us explain the evolution of the fluctuations.

For such a small value of v, preheating is completedat tf100, well before the phase transition withsymmetry breaking, which occurs at tf240. Duringthe stage of parametric resonance tF100, the term

ŽFig. 1. Variances of the field components f and f upper and2 1.lower solid curves respectively as well as the zero modes of these

Ž ² :fields upper dotted curve is f and lower dotted curve is1² :.f are shown as functions of conformal time.2

( )I. TkacheÕ et al.rPhysics Letters B 440 1998 262–268264

2 2 Ž .Õ a in 1 can be neglected. The equations for themode functions of fluctuations in the directions off and f are1 2

w q k 2 qq w 2 w s0 , 2Ž .¨ Ž .i k i 01 i k

Ž .where the background inflaton oscillations w t01

are given by an elliptic function. The resonanceparameter is different for different components: q s1

3 and q s1. For both components there is a single2

instability band, but the locations of the bands andw xthe strengths of the resonance are different 14 . The

resonance in the ‘‘inflaton’’ direction w is weak,1

the maximal value of the characteristic exponent ofthe fluctuations w Ae m1t is m f0.036; the reso-1k 1

nance in the perpendicular direction w is much2

stronger and broader, m f0.147.2

However, the actual growth rate of the fluctua-tions w is not smaller but larger than that of the1k

w fluctuations. Indeed, the cross-interaction term2 k

w 2w 2 leads to the production of dw fluctuations in1 2 1w xthe process of rescattering 6 . As a result, the ampli-

tude w grows as w Ae2 m 2t, where 2m s0.2941, k 1k 2w x12 , as clearly seen in Fig. 1. This growth is much

w xfaster than in the 2d lattice simulations of Ref. 13 .We see that during the time interval between

completion of preheating and the phase transition,100QtQ240, the zero mode of f decreases faster1

Žthan the variances both are decreasing due to theexpansion of the universe, but in addition the zero

.mode continues to decay into field fluctuations , andby the time tf240 the field variance is as big as itszero mode.

Time interval 100QtQ240 is dominated byw xrescattering of all modes 4 . During this time inter-

Ž .val, the symmetry at least in one direction is re-stored by large fluctuations. To see this, we study thetime dependence of the zero mode of f , which is1

shown in Fig. 2. We immediately see some peculiar-Ž 2 2 .2ity. In the potential of the form f yv the field

cannot oscillate near fs0 with amplitude smaller'than f ' 2 v. This critical value of the amplitudec

is shown in Fig. 2 by the dashed line, and we seethat the field does oscillate with an amplitude smallerthan f . At some point, the amplitude of the oscilla-c

tions even becomes smaller than v, i.e. the field isoscillating on the top of the tree potential withoutrolling down to its minimum. This means that the

Fig. 2. Time dependence of the zero mode of f .1

tree potential is significantly altered by the interac-tion with the background of created fluctuations, andthe symmetry is restored.

Ž .We can reconstruct the effective potential V weffŽ . ² :using the already calculated function w t s w0 i i

and the definition

w qdV rdw s0 . 3Ž .¨ 0 i eff 0 i

This definition of the effective potential is perfectlylegitimate in the case when corrections to terms withthe time derivative are small, which appears to be the

Žcase. Expansion of the universe will be properlyincluded when we work in the conformal coordi-

Ž .nates, as in Eq. 3 , and then ‘‘rotate’’ potential back.to the synchronous frame and to the original fields.

This method allows one to find V up to a constant.eff

The reconstruction along w direction is shown for1

several moments of time in Fig. 3. At tf90 theeffective potential, shown by dots, coincides to avery good accuracy with the tree potential, shown bythe solid line. This moment is not far away from theend of the exponential growth of fluctuations, whichoccurs at tf100, but fluctuations at tf90 are stillrather small. At tf110 fluctuations are large, andthe effective potential is completely different: it hasonly one minimum. Note that the potential is slightlyasymmetric with respect to the change of sign of thefield f . This happens because in our simulations we1

are sampling the effective potential in the time-de-pendent background, when the points with positiveand negative f correspond to different moments oftime. The ‘‘instantaneous’’ potential would be ex-actly symmetric, with a minimum at f s0, and1

symmetry is restored, at least in the f direction.1


Fig. 3. Reconstruction of the effective potential in f direction at1

several moments of time. Dots correspond to t f90, diagonalcrosses to t f110, larger stars to t f180 and smaller stars tot f215. Solid line is the tree potential.

Still, the existence of an oscillating zero mode f1

implies that symmetry between states with f )01

and f -0 is not exact. Thus one wonder whether1

one can say that symmetry is restored if the effectivepotential has the minimum at f s0, or one should1

not say so until the amplitude of oscillations of thefield f completely vanishes.1

In our opinion, if the effective potential has aminimum at f s0, this implies that there is no1

spontaneous symmetry breaking, so in this sense thesymmetry f ™yf is restored. From a more prag-1 1

matic point of view, the issue of symmetry restora-tion and subsequent symmetry breaking is importantmainly because it is related to production of topolog-ical defects. Therefore instead of debating whichdefinition of symmetry restoration is better, we will

try to find whether the topological defects are pro-duced. Indeed we will see that cosmic strings areproduced as a consequence of preheating.

At tf240 the field fluctuations become dilutedby the expansion of the universe to the extent suffi-cient for symmetry breaking. This event can be seenboth in Fig. 1 and in Fig. 2. At that time, the zeromode nearly vanishes. This happens not because ofsymmetry restoration, but because the field rolls

Ž .down to the O 2 symmetric valley of minima of theeffective potential in all possible directions in the

² :field space, which, after averaging, gives f <v.1

A useful quantity to consider is the probabilityŽ .distribution function P f ,f ,t shown in Fig. 4.1 2

The first of these two figures shows that at t;200the maximum of the probability distribution oscil-lates near f s0 in the f direction. This distribu-1 1

tion has only one maximum in the f direction.1

When this maximum approaches f s0, the distri-1Ž .bution P f ,f ,t is approximately symmetric with1 2

respect to the change f ™yf . On the other hand,1 1

there are two maxima of the probability distributionwith respect to the field f . They are concentrated2

near f f"v, which means that the symmetry f2 2

™yf is broken.2

This implies that the universe at that time be-comes divided into domains filled with the fieldf f"v. These domains are separated by two-di-2

mensional domain walls, the surfaces where f s0.2

When the distribution of the scalar field f oscil-1

lates, the center of this distribution moves, but if it iswide enough, there always will be two-dimensional

Fig. 4. The joint probability distribution function of f and f before and after the phase transition. Fields f and f are shown in units1 2 1 2

of v.


surfaces where f s0. Intersection of these surfaces1

with the domain walls f s0 form strings, on which2

f sf s0. One can easily find out that when one1 2

moves around such a string, the phase a sŽ < <.arccos f r f changes by 2p . Thus these strings1

are topologically stable. Most of them form stringŽ .loops which move expand, shrink, and expand again

when the distribution of the field f oscillates. This1

mechanism of string formation is different from theKibble mechanism.

Gradually the amplitude of fluctuations of thefields f decreases, and symmetry f ™yf alsoi 1 1

breaks down. The choice of direction of the symme-try breaking will depend on the shape of the effec-tive potential, but also on the amplitude and phase of

² :the oscillations of the zero mode f , and on the1Ž .width of the probability distribution P f ,f ,t . If1 2

Ž .the width of the probability distribution P f ,f ,t1 2

in the f direction is sufficiently large, then this1

phase transition modulated by the oscillations of thefield f may lead to formation of long strings,1

which may have interesting cosmological conse-quences.

After the modulated phase transition, which oc-Ž .curs at ts240, we see an O 2 symmetric ring inŽ .the probability distribution P f ,f ,t , superim-1 2

Žposed with a peak in some random direction which.does not coincide either with f or with f , see the1 2

second figure in Fig. 4. The existence of the ringshows that the absolute value of the field afterspontaneous symmetry breaking is close to v, and

Ž .that in different points of space the vector f ,f1 2

looks in all possible directions, which indicates thepresence of strings. The peak along the randomdirection represents additional spontaneous symme-try breaking, which appears because it is energeti-

Ž .cally preferable for all vectors f ,f to look in the1 2

same direction. With time, the ring will disappear,and the width of the peak will decrease.

Strings can be detected directly. For that purpose,we had plotted 3D coordinates of the grid points

< <where f is close to zero. A series of such imagesdescribing several different stages of this process isshown in Fig. 5. Strings are clearly seen, and we canobserve the formation of one large loop.

At times 100-t-240, while the zero mode wasstill oscillating, we had observed abundant formationand subsequent annihilation of string loops. Many

Fig. 5. The process of string formation for v s3=1016 GeV andls10y12.

strings appear when the oscillating zero mode² Ž .:f t passes through f s0. Then they disappear1 1

² :when f grows and appear when it becomes small1

again. Fig. 5, on the other hand, shows the behaviorof strings after the phase transition, when the fieldf is not capable of rolling over the top of the1

effective potential. At this stage the new strings arenot produced, and the old ones move slowly, chang-ing their position because of the string tension.

It is interesting that at time ts280 there is onlyŽone string loop in the integration box upon account

. Žof the periodic boundary conditions , see Fig. 5. Asingle spatial configuration, like this one, of coursecorresponds to a particular realization of random

.initial conditions. This loop stays intact, up to smallvibrations, for a long time, being in quasi-equi-librium. However, later on, different segments of theloop quickly approach each other and reconnect atts328, forming another loop configuration at ts330. This final loop collapses almost to a point,bounces once, collapses again and disappears. 1 Thisprocess of reconnection and final collapse confirms

1 The movie which shows this process can be seen at˜http:rrwww.physics.purdue.edurtkachevrmovies.html or at

˜http:rrphysics.stanford.edurlinde.


that loops are dynamical strings, not accidental linesin space.

To find the range of v for which strings areproduced we had run simulations with different val-ues of v in the interval from 4=1015 to 1017 GeV.In all these cases we observed formation of stringloops. From cosmological perspective, however, themost interesting possibility would be formation of aninfinite string. One can expect that, at least when v islarge, the probability of formation of long strings ishigher in the case when the moment of the symmetrybreaking nearly coincides with the moment when² :f passes through zero. If that expectation is1

correct, the number of long strings should be anon-monotonic function of v.

To verify this conjecture we plotted the stringdistribution at the time Dts10 after the phase tran-

Ž .sition which happens at different times for vs3=

1016 GeV, 5=1016 GeV, 6=1016 GeV, and 1017

GeV, for ls10y13, see Fig. 6. As we see, forvs5=1016 GeV and 1017 GeV all loops are short,whereas for vs3=1016 GeV and 6=1016 GeVthere are many large loops. This indicates a possibil-ity of formation of a network of infinite strings forvs3=1016 GeV and 6=1016 GeV.

Fig. 6. The string distribution at the time Dt s10 after themoment of the phase transition for v s1017 GeV, 6=1016 GeV,5=1016 GeV, and 3=1016 GeV respectively. The plot for v s3=1016 GeV corresponds to a different realization of the initialconditions than Fig. 5.

Fig. 7. The string distribution with two ‘‘infinite’’ strings forv s3=1016 GeV in a box of a larger size.

Ž .Additional evidence for or against formation ofinfinite strings after preheating may be obtained ifone uses lattices of a greater size. In particular, weperformed simulations for vs3=1016 GeV in a

Ž .larger box Ls32p , see Fig. 7. At the time shownin this figure, the physical size of the box is L ;phys

2 y1 y1 'Ž .10 m , where m ;1r l v is the typicalthickness of the string. The ratio of the horizon sizeto the physical size of the box at large t is trL, sofor Ls32p and t;300 these sizes are comparable.In Fig. 7, we see two ‘‘infinite’’ strings and a largecollapsing string loop. Note that on a lattice withperiodic boundary conditions ‘‘infinite’’ strings canonly be created in pairs, because the winding numberfor our initial conditions is zero.

Also, one may consider models where one mayhave independent reasons to expect production ofinfinite strings. For example, one may add to our

Ž 2 . 2 2model the term g r2 f x describing interactionof the inflaton field f with the scalar field x withthe coupling constant g 2

4l. We have studied thisw xmodel in Ref. 11 for the case of a one-component

field f and found a first-order phase transition. Weexpect that this result remains valid for the two-com-ponent field as well, because the main reason for the

w xfirst order phase transition found in Ref. 11 was thestrong interaction of the field f with the field x

rather than the self-interaction of the field f. If thisis indeed the case, then nonthermal fluctuations ofthe field x will lead to symmetry restoration withrespect to both of the fields f and f . These two1 2


fields will be captured in the local minimum of theeffective potential at fs0 until the moment of thefirst order phase transition. In this case, there will bestrings formed by the usual Kibble mechanism, in-cluding infinite ones. We hope to return to thediscussion of this model in a separate publication.

Acknowledgements

This work was supported in part by DOE grantŽ . Ž .DE-FG02-91ER40681 Task B S.K. and I.T. , NSFŽ . Žgrants PHY-9219345 A.L. , PHY-9501458 S.K.

. Ž .and I.T. , AST95-29-225 L.K. and A.L. , and by theŽ .Alfred P. Sloan Foundation S.K. .

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