A, Patkós

D, Sexty

Sz, Borsányi

English

ELTE Digital Institutional Repository (EDIT)

arXiv:hep-ph/0301117v1  16 Jan 2003January 16, 2019 19:48 WSPC/Trim Size: 9in x 6in for Proceedings proc.1.0NONEQUILIBRIUM GOLDSTONE PHENOMENON INHYBRID INFLATION ∗SZ. BORSÁNYI, A. PATKÓS, D. SEXTYDepartment of Atomic PhysicsEötvös University, BudapestWe study the onset of Goldstone phenomenon in a hybrid inflation scenario. Thephysically motivated range of parameters is analyzed in order to meet the cos-mological constraints. Classical equations of motion are solved and the evolutionthrough the spontaneous symmetry breaking is followed. We emphasize the roleof topological defects that partially maintain the disordered phase well after thewaterfall. We study the emergence of the Goldstone excitations and their role inthe onset of the radiation dominated universe.1. IntroductionThe real time dynamics of phase transitions is one of the most importanttopics of nonequilibrium field theory1 with applications ranging from thedescription of relativistic heavy ion collisions to cosmological scenarios suchas hybrid inflation2. The reheating of the universe in this framework hasbeen addressed by several authors3,4,5. A particularly efficient mechanism,called spinodal (tachyonic) instability sets a sudden end for the inflation bythe so called waterfall mechanism. This framework involves a nonthermalphase transition for the matter field that is coupled to the hypotheticalinflaton field. The formation and decay of topological or domain structuresmay play a crucial role in the further evolution of the fields4,6.In this study we concentrate on the Goldstone phenomenon as it emergesin a hybrid inflation scenario. We consider the coupled system of the(scalar) inflaton field (σ) and a two-component matter field (Φ) in a self-consistently expanding flat and homogeneous FRW metrics with the fol-∗Based on the presentations by Sz. Borsányi and D. Sexty1January 16, 2019 19:48 WSPC/Trim Size: 9in x 6in for Proceedings proc.1.02lowing Lagrangian:L =12(∂νσ(x))2 −12m2σσ2 +12|∂νΦ(x)|2 −12m2Φ|Φ|2 −λ24|Φ|4 −12g2σ2|Φ|2,(1)We solve the classical equations of motion numerically and investigatethe dynamics of the to-be-Goldstone modes. A brief discussion is presentedfirst on the choice of cosmologically viable parameters in Eq. (1) and werecall the most important numerical measurement algorithms. Then theonset of Goldstone phenomenon is addressed with emphasis on the latetime behaviour of the equation of state.2. ParametersLet us concentrate on a GUT-scale (∼ 1015 GeV) scalar field that is re-heated by the spinodal instability induced by the much lighter (∼ 1012GeV) inflaton field. A natural choice for the GUT self coupling is ∼ 1. Thecoupling to the inflaton field is chosen in the range 0.01 < g < 0.1 similarlyto Copeland et al4. This physically motivated range of parameters have tobe confronted to our cosmological expectations.There are two constraints the solution of the model should respect.First, to ensure a successful inflation the total number of e-foldings of thescale factor a(t) should be ∼ 50 . . .100. The second constraint stems fromthe relation of the quantum fluctuations of the inflationary period to thedensity fluctuations measured by the COBE experiment7. This relation isdue to the superhorizon evolution of the fluctuations which left the horizonduring the inflation and reentered again in the recombination era:5× 10−4 =(δρρ)COBE=( δρρ+ p)exit. (2)(p is the pressure of the system). In the terminal phase of the inflationcharacterized by the Hubble constant H the quantum fluctuations of theinflaton (δσ ∼ H/2π) dominate:δρσρσ + pσ∼δσV ′(σ)σ̇2=HV ′(σ)2πσ̇2.During the roll of the inflaton one can directly measure the size of thiscombination. The modes reentering at recombination made their exit nearthe end of inflation, therefore we decided to require the COBE estimate tobe fulfilled at ∆N = 5 e-foldings before the inflation is terminated.The early dynamics of the hybrid inflation is determined by the evolu-tion of the homogeneous mode of the inflaton field σ0(t) and the FRW scaleJanuary 16, 2019 19:48 WSPC/Trim Size: 9in x 6in for Proceedings proc.1.03parameter a(t), since the other modes are unoccupied before the onset ofspinodal instability. At zero time the inflaton homogeneous mode is setto σ0(t = 0) = MPlanck and σ̇0(t = 0) = 0. Then we numerically solvethe Einstein equation and the equation of motion for σ0(t) and a(t) (thescale parameter of the universe) without relying on the slow-roll condition.We scan through the physically motivated range of parameters and selectthose models that meet the cosmological constraints. In Fig. 2 we variedthe inflaton-matter coupling g keeping (λ/6)3/2/g = 1 fixed (note that thiscombination appears in the slow-roll approximation to Eq. (2).Figure 1. Ranges of parameters that eventuate density fluctuations expected upon theCOBE results. The bands refer to different number of e-foldings (N). In the presentstudy we concentrate on the set of parameters marked by the dark cross.In this presentation we discuss numerical results corresponding to theset of parameters marked by the dark cross in Fig. 2. (g = 0.1, λ = 1.29,MH = 2.2 · 1015 GeV, mσ = 3.6 · 1011 GeV.)3. Measurement techniques for the physical degrees offreedomIn the O(2) hybrid inflation there are three degrees of freedom at each latticesite: the inflaton, and the two components of the O(2) symmetric scalarfield. In the broken phase, our intuition is that the independent degrees offreedom are the radial (ρ(x) =√Φ21(x) + Φ22(x)), and the angular (ρφ(x) =ρ arctan(Φ2(x)/Φ1(x)) fields. (One may think of the radial component asthe GUT scale Higgs boson.) In order to test the onset of this idea, weJanuary 16, 2019 19:48 WSPC/Trim Size: 9in x 6in for Proceedings proc.1.04measured the velocity correlation matrix (Wij) which is defined byWij =W̃ij√W̃iiW̃jj,W̃ij = 〈vivj〉 − 〈vi〉〈vj〉, (3)where vi = (σ̇, ρ̇, ρφ̇) stands for velocity-like field components and thebrackets mean spatial averaging. The inflaton-Higgs cross correlation canbe seen in Fig. 2. It shows a maximum after the tachyonic instability, butit is negligible even then. Its height strongly depends on the parametersbut not on the system size. The other two correlators (angular-Higgs andangular-inflaton) are smaller and vanish with increasing volume.The change of sign of the effective Higgs massM2eff = m2Φ + g2〈σ2〉+ λ〈ρ2〉/2 (4)coincides with the exponential growth of the 〈ρ〉 field (Fig. 2). Later thiseffective mass relaxes close to the measured gap value of the radial compo-nent. 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 1  10  100  1000time [1/MHiggs (bare)]g=0.1N=128Higgs zero mode / vHiggs-Inflaton correlationFigure 2. Two stage symmetry breaking: in the first stage, in tachyonic instability,the O(2) field gets excited, only later in the second stage is the symmetry breakingcompleted. (The vertical lines limit the time interval when the effective mass is negative(see text).) The inflaton-higgs correlation squared (W 212) is also displayed.The masses of the angular and radial field components were extractedfrom their dispersion relation8,9. The method requires uncorrelated fieldJanuary 16, 2019 19:48 WSPC/Trim Size: 9in x 6in for Proceedings proc.1.05components in Fourier space Xk. We measure the quantity |Ẋk|2/|Xk|2. Ifonly a single frequency (ωk) dominates the motion of a mode, than this ratiois equal to ω2k. Extrapolating this quantity to k = 0 we get the squared massgap of the component X . In practice we average over different directionsin k space, over microscopical time scales and over different runs withrandom initial conditions. We measured the radial (Higgs) mass choosingX = ρ =√Φ21+Φ22, and the angular one with X = 〈ρ〉 exp(iϕ). Wefitted the dispersion relation with a polinomial in k2. The radial mass wasactually extrapolated from the k > 0.75 region, because of the observedlow-k region bending-down (See Fig. 5).4. Mechanism of Symmetry BreakingIn our simulations we observe a delay in the completion of the symmetrybreaking (SB) relative to the tachyonic instability. At the instant of thetachyonic preheating the Higgs expectation value (vev) reaches a finite butvery small value. The symmetry is broken in the strict sense but the Higgsvev has not reached the value one would expect from the effective potentialyet. At later times the Higgs vev approaches its final value. (In fact, theinitial conditions break the O(2) symmetry so that we can observe the SBat the length scale of the lattice size. (L ≈ 90×M−1H )).The sudden growth of the radial component induces large number ofdomains with different directions of SB. The domain walls are characterizedby small values of the radial component. One may also observe extendedvolumes of small ρ, called hot spots4. The volume ratio of these hot spotsis reflected by the actual value of the ρ0(t) in the period of the incompleteSB.In momentum space one can easily understand the oscillatory behaviorof the OP before the completion of the SB. The inflaton field (still oscil-lating) has a direct influence on the mass squared of the matter field (seeEq. (4)). The time scale of this oscillation is much slower than the GUTscale so the elementary oscillators of the matter field are tuned adiabati-cally. The amplitude squared times the frequency is an adiabatic invariantof these oscillators. For σ0(t) ≈ 0 we have a smaller effective mass by Eq. (4)and hence, larger amplitudes and larger value for the OP. Opposingly, forlarge inflaton values the OP will come closer to zero, this is reflected bya considerably higher volume ratio of hot spots (see Fig. 3). Note that inthis regime the value of the effective mass squared is always positive. Thegrowth of the OP during the completion of SB is not of tachyonic nature!January 16, 2019 19:48 WSPC/Trim Size: 9in x 6in for Proceedings proc.1.06-2-1 0 1 2 3 4 5 0  50  100  150  200  250  300Inflaton zero mode / crit. 0.0001 0.001 0.01 0.1 1A BOrder Parameter / vABFigure 3. Influence of inflaton zero mode oscillations (dotted, left-hand-side scale) onthe OP (solid, right-hand-side scale). At different inflaton values lattice snapshots aredisplayed. The dark zones refer to large ρ(x) values. We may interpret the light zonesas hot spots.The OP cannot reach its broken phase value as long as the domainstructure is present. An easy way to characterize the density of topologicaldefects is to monitor the average of the angular twist Σxδϕ(x) calculatedalong straight lines. This number is large right after the tachyonic excita-tions of the matter field, since domains are formed with random orientation.In the completely broken phase it should be zero. The periodic reappear-ance and dissolution of hot spots erases the domain walls, and eventuallythe OP departs from the vicinity of zero. As a support of this interpretationwe show the coincidence of the vanishing of the twist number and the finalgrowth of the OP in Fig. 4.After the SB is complete, the virial equilibrium of the angular andinflaton modes is soon accomplished (Ekinetic = Egradient +Epotential).Before the onset of virial equlibrium, there is an energy excess in the angulargradient energy, which is due to topological excitations6.In the phase of completely broken symmetry the angular matter fieldcomponents are the Goldstone excitations. In the period of the incompleteSB, however, one should be more careful with the identification. In thefollowing we study how the angular modes start obeying the Goldstonetheorem.We use now the methods reviewed in Sec. 3 to monitor the evolution ofthe dispersion relation and the onset of the Goldstone theorem.January 16, 2019 19:48 WSPC/Trim Size: 9in x 6in for Proceedings proc.1.07-4 4-4  4Φ1Φ2t=200-4 4-4  4Φ1Φ2t=300 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0  100  200  300  400  500  600 0.001 0.01 0.1 1 10TwistOrder Parameter / vFigure 4. The twist number (related to the topological defect density, see text) indicatesthat the decay of the domain structure coincides with the complete breaking of symmetry.The side plots show the matter field values for the ensemble of lattice sites right beforeand after the completion of SB. 0 1 2 3 4 5 6 0  0.5  1  1.5  2  2.5  3ω2  [M2 0,Higgs]k2 [M20,Higgs]t= 90’Higgs’ modesfit for ’Higgs’ modesAngular modes 0 1 2 3 4 5 6 0  0.5  1  1.5  2  2.5  3ω2  [M2 0,Higgs]k2 [M20,Higgs]t=300’Higgs’ modesfit for ’Higgs’ modesGoldstone modesFigure 5. Dispersion relations for the different matter field components before andafter the complete breaking of symmetry. On the early time plot the angular componentdefinitely show a nonvanishing gap, which is absent at late times. This fact encouragedus to refer to the angular modes as “Goldstone” in the second plot.In Fig. 5 we display the dispersion relation for the radial and angu-lar field components in the partially and completely broken phase. Thesecurves would be straight lines for the physical quasiparticle excitations ofthe equilibrium theory. The curvature of the dispersion relation mostly in-dicate the far-from-equilibrium field configuration. The method obviouslyfails with the radial (Higgs) mode for k2 < 0.5Fig. 5 (left) clearly demonstrates a nonvanishing gap at early times.January 16, 2019 19:48 WSPC/Trim Size: 9in x 6in for Proceedings proc.1.08This gap, however, is absent in the dispersion relation measured in thebroken phase. The gap for both field components may be displayed as afunction of time if the measurements are carried out frequently through theevolution. The resulting plot is shown in Fig. 6.-0.5 0 0.5 1 1.5 2 2.5 50  100  200  400  800mass2 [1/M2 0,Higgs]time [1/M0,Higgs]Higgs Mass2Angular Gap2Goldstone Mass2Figure 6. Evolution of the mass gap for the radial (Higgs) and angular field components.The decay of the domain structure coincides with the onset of the Goldstone theorem.As soon as the OP departs from zero (see Fig. 3) the mass of the angularmode vanishes and stays further in the vicinity of zero. In the presence ofdomains the mass scale is set by the domain size9. To have a gaplessexcitation this size should grow beyond the lattice size.5. Equation of stateEquilibrium fields of definite mass obey well defined equations of state(EOS): ultrarelativistic particles obey ρ = 3p, while for the nonrelativisticmatter p = 0. (Here ρ stands for the energy density, p for the pressure.)The appeareance of well-defined EOS gives a hint how the system arrivesto the state of the Hot Universe from the inflaton.In a nonequilibrium evolution the EOS varies strongly with time. InFig. 7 (left) we displayed the effective EOS for the full system, which iswildly oscillating. These oscillations are completely averaged out by thesolution of the Einstein equations.The early evolution of the energy density is dominated by the inflatonzero mode. If one is interested in the bulk thermodynamics of the micro-scopic degrees of freedom, this oscillating contribution has to be removed.January 16, 2019 19:48 WSPC/Trim Size: 9in x 6in for Proceedings proc.1.09-0.5 0 0.5 1 1.5 2 2.5 3 3.5 1  2  3  4  5  6  7pressureenergy densityfull systemParticle Gasp=ρ/3-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8  2pressureenergy densityGoldstoneInflatonHiggsp=ρ/3Figure 7. Effective equations of state for the full system and the different field com-pontents (see text).The contribution of the inflaton zero mode has to be substracted with greatcare since the potential energy for the matter field is mostly determined bythe inflaton zero mode. By making use of the separation of time scales theoscillating part of the inflaton energy is considered to be a contribution ofthe zero mode, the rest is regarded as the energy contribution of the micro-scopic degrees of freedom. The equation of state for the disordered motionof the system is also shown in Fig. 7 (left). In Fig. 7 (right) we show theEOS for the different microscopic field components. The curves give thep − ρ paths the field components move along, they are displayed startingshortly before the completion of the SB.A massive field is expected to fulfil ρ > 3p, i.e. the corresponding curveshould be below the marked straight line in Fig. 7 (right). The curve for theangular component lies slightly above this line. Note that extrapolating thedispersion relation from the high-k modes one would get a negative squaredeffective mass (see Fig. 5). It is not the gap itself but the entire dispersionrelation that determines the EOS the system follows.Remarkably, the effective EOS for the light fields rapidly converge tothe expected straight line behaviour even though the spectral equilibrium isfar from being reached. To that time the virial equilibrium is estabilished,although the mode temperatures are different. In fact, in a weakly coupledmassless theory each of the elementary oscillators provide the same contri-bution to ρ/p even if they would assume different mode temperature6. Inconclusion we can say that the Goldstone excitations begin to play theircosmological role right after the complete SB by yielding a dominant contri-bution to the pressure and therefore leading the universe into the radiationdominated epoch.January 16, 2019 19:48 WSPC/Trim Size: 9in x 6in for Proceedings proc.1.010References1. D. Bodeker, Nucl. Phys. Proc. Suppl. 94 (2001) 612. A. D. Linde, “Lectures on inflationary cosmology,” arXiv:hep-th/9410082;A. D. Linde, Phys. Rev. D 49 (1994) 7483. J. Garcia-Bellido and A. D. Linde, Phys. Rev. D 57 (1998) 60754. E. J. Copeland, S. Pascoli and A. Rajantie, Phys. Rev. D 65 (2002) 1035175. J. Skullerud, J. Smit and A. Tranberg, W particle distribution in elec-troweak tachyonic pre-heating, (contribution to this Workshop), arXiv:hep-ph/0210349.6. M. Yamaguchi, Phys. Rev. D 60 (1999) 1035117. E. W. Kolb and M. S. Turner, “The Early Universe,” Addison-Wesley Pub-lishing Company, 19908. Sz. Borsányi, A. Patkós, D. Sexty Phys. Rev. D 66 (2002) 0250149. Sz. Borsányi, A. Patkós, D. Sexty and Zs. Szép, Phys. Rev. D 64 (2001)12501110. G. N. Felder, J. Garcia-Bellido, P. B. Greene, L. Kofman, A. D. Linde andI. Tkachev, Phys. Rev. Lett. 87 (2001) 011601

Non-Equilibrium Goldstone Phenomenon in Hybrid Inflation

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Non-Equilibrium Goldstone Phenomenon in Hybrid Inflation

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