19 November 1998
.Physics Letters B 440 1998 262268
Cosmic strings from preheatingI. Tkachev a,b, S. Khlebnikov a, L. Kofman c, A. Linde d
a Department of Physics, Purdue Uniersity, West Lafayette, IN 47907, USAb Institute for Nuclear Research of the Academy of Sciences of Russia, Moscow 117312, Russiac Institute for Astronomy, Uniersity of Hawaii, 2680 Woodlawn Dr., Honolulu, HI 96822, USA
d Department of Physics, Stanford Uniersity, Stanford, CA 94305-4060, USAReceived 13 May 1998; revised 20 August 1998
Editor: H. Georgi
Dedicated to:the memory of David Abramovich Kirzhnits
We investigate nonthermal phase transitions that may occur after post-inflationary preheating in a simple model of a 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 subsequent1nonthermal phase transition with symmetry breaking leads to formation of cosmic strings even for v41016 GeV. This
.mechanism of string formation, in a modulated by the oscillating field f phase transition, is different from the usual1Kibble 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 28 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. .PII: S0370-2693 98 01094-6
( )I. Tkache et al.rPhysics Letters B 440 1998 262268 263
w x10,11 . The model with one scalar field with the 2 2 .2potential l f yv r4 has a broken discrete sym-
metry fyf. 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 thei 2 2 .2 2effective potential l f yv r4, where f s
2 .f . This model has O 2 rotational symmetry andiw x allows 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=1016GeV.
In the model with the two-component field f weican always rotate fields in such a way that initiallyf s0, and the field f plays the role of the2 1classical oscillating inflaton field. We denote by .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
. . .conformal time t , where l f 0 dtsa t dt , a 0s1, and perform conformal transformation of the
. . .fields, wsf t a t rf 0 . We also will use the .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
.time is zero. This happens when f 0 f0.35MPl . w xand a t f0.51tq1 5 .
In the new variables, the equations of motionbecome
w y=2w q w 2 y2a2 w s0 , 1 . . i i i . .where svrf 0 . Eq. 1 contains only one param-
eter, . The coupling constant l is hidden in the
initial conditions for fluctuations; these were chosenw xas described in Ref. 4 .
.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 i .:w x,t for each of the components is1,2 of thei
.2:field w, and variances dw , which measure thei .average magnitude of fluctuations dw sw x,t yi i
w . Since the system is homogeneous on large0 iscales, averages in these relations can be understoodas volume averages. This is equivalent to taking an
average 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=1016GeV 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 i : .2:f , and of the variances, f yf , for eachi i 0 iof 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
Fig. 1. Variances of the field components f and f upper and2 1.lower solid curves respectively as well as the zero modes of these
:fields upper dotted curve is f and lower dotted curve is1 :.f are shown as functions of conformal time.2
( )I. Tkache et al.rPhysics Letters B 440 1998 262268264
2 2 . 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 , 2 . .i k i 01 i k .where the background inflaton oscillations w t01
are given by an elliptic function. The resonanceparameter is different for different components: q s13 and q s1. For both components there is a single2instability band, but the locations of the bands and
w xthe strengths of the resonance are different 14 . Theresonance in the inflaton direction w is weak,1the maximal value of the characteristic exponent ofthe fluctuations w Ae m1t is m f0.036; the reso-1k 1nance in the perpendicular direction w is much2stronger and broader, m f0.147.2
However, the actual growth rate of the fluctua-tions w is not smaller but larger than that of the1kw fluctuations. Indeed, the cross-interaction term2 kw 2w 2 leads to the production of dw fluctuations in1 2 1
w 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
than 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-
.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 is1shown in Fig. 2. We immediately see some peculiar-
2 2 .2ity. In the potential of the form f yv the fieldcannot oscillate near fs0 with amplitude smaller
than f 2 v. This critical value of the amplitudecis 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-ctions 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.
.We can reconstruct the effective potential V weff . :using the already calculated function w t s w0 i i
and the definitionw qdV rdw s0 . 3 . 0 i eff 0 iThis definition of the effective potential is perfectlylegitimate in the case when corrections to terms withthe time derivative are small, which appears to be the
case. Expansion of the universe will be properlyincluded when we work in the conformal coordi-
.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.effThe reconstruction along w direction is shown for1several 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 we1are 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, and1symmetry is restored, at least in the f direction.1
( )I. Tkache et al.rPhysics Letters B 440 1998 262268 265
Fig. 3. Reconstruction of the effective potential in f direction at1several 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 f1implies that symmetry between states with f )01and f -0 is not exact. Thus one wonder whether1one can say that symmetry is restored if the effectivepotential has the minimum at f s0, or one should1not 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 no1spontaneous symmetry breaking, so in this sense thesymmetry f yf is restored. From a more prag-1 1matic 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
.down to the O 2 symmetric valley of minima of theeffective potential in all possible directions in the
:field space, which, after averaging, gives f
( )I. Tkache et al.rPhysics Letters B 440 1998 262268266
surfaces where f s0. Intersection of these surfaces1with the domain walls f s0 form strings, on which2f sf s0. One can easily find out that when one1 2moves around such a string, the phase a s
( )I. Tkache et al.rPhysics Letters B 440 1998 262268 267
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 when :f passes through zero. If that expectation is1correct, 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-
.sition which happens at different times for vs3=1016 GeV, 5=1016 GeV, 6=1016 GeV, and 1017GeV, 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.
.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
.larger box Ls32p , see Fig. 7. At the time shownin this figure, the physical size of the box is L ;phys
2 y1 y1 .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
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 this
w xmodel in Ref. 11 for the case of a one-componentfield 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 xrather 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
( )I. Tkache et al.rPhysics Letters B 440 1998 262268268
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.
This work was supported in part by DOE grant . .DE-FG02-91ER40681 Task B S.K. and I.T. , NSF . grants PHY-9219345 A.L. , PHY-9501458 S.K.
. .and I.T. , AST95-29-225 L.K. and A.L. , and by the .Alfred P. Sloan Foundation S.K. .
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