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