Effects of thermal fluctuations on thermal inflation
aa r X i v : . [ h e p - ph ] D ec RESCEU-52/14YITP-14-100
Effects of thermal fluctuations on thermal inflation
Takashi Hiramatsu, ∗ Yuhei Miyamoto,
2, 3, † and Jun’ichi Yokoyama
3, 4, ‡ Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Department of Physics, Graduate School of Science,The University of Tokyo, Tokyo 113-0033, Japan Research Center for the Early Universe (RESCEU),Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU),WPI, TODIAS, The University of Tokyo, Kashiwa, Chiba, 277-8568, Japan
The mechanism of thermal inflation, a relatively short period of accelerated expansion after pri-mordial inflation, is a desirable ingredient for a certain class of particle physics models if they arenot to be in contention with the cosmology of the early Universe. Though thermal inflation is mostsimply described in terms of a thermal effective potential, a thermal environment also gives rise tothermal fluctuations that must be taken into account. We numerically study the effects of thesethermal fluctuations using lattice simulations. We conclude that though they do not ruin the ther-mal inflation scenario, the phase transition at the end of thermal inflation proceeds through phasemixing and is therefore not accompanied by the formations of bubbles nor appreciable amplitude ofgravitational waves.
PACS numbers:
I. INTRODUCTION
The idea of the inflationary Universe [1] is now a key part of the standard model of cosmology. The primordialperiod of accelerated expansion at the beginning of the Universe provides not only a solution to the flatness andhorizon problems, but also the initial density fluctuations that seed the formation of large-scale structure.It has been claimed that a period of accelerated expansion has the potential to reconcile a certain class of particlephysics models with cosmology. The gravitino, a fermionic partner of the graviton with spin 3 /
2, appears in thetheory of supergravity. Its number density per comoving volume is proportional to the reheating temperature afterinflation [2]. Therefore, if the reheating temperature is high, the gravitinos are abundantly produced. The lifetimeof the gravitino is estimated as τ ∼ πM /m / ∼ sec if the gravitino mass takes a value m / = 10 GeV with M Pl = 2 . × GeV being the reduced Planck mass. Namely, they decay after Big-Bang Nucleosythesis (BBN)due to their very weak interactions. Subsequently, the decay products of gravitinos spoil the light elements afterBBN. This is called the gravitino problem. The scalar fields called moduli, with Planck-suppressed couplings, arealso dangerous in a similar way [3]. They start to oscillate when the Hubble parameter becomes as small as theirmass and soon dominate the Universe, since the initial amplitude of such oscillations is expected to be on the orderof M Pl . Driven by the coherent oscillations of the moduli fields the Universe evolves like a matter-dominated one,until the moduli decay to reheat the Universe. The moduli fields are coupled very weakly with other fields, and asa result of their long lifetime the reheating temperature is so low that BBN does not work. Furthermore, in Ref.[4] it is shown that the energy density of moduli is also constrained by X( γ )-ray observations, requiring that thetheoretical prediction does not exceed the observed backgrounds. One can dilute the moduli fields by assuming ashort, low-energy inflationary period after the moduli begin oscillating at H ≈ m moduli [5–7]. This type of temporallyshort inflationary period is called thermal inflation, and is driven by a scalar field with almost flat potential called theflaton [5, 6]. In a similar way, thermal inflation can also evade the gravitino problem [2] by diluting them after theirgeneration. In summary, thermal inflation is needed to dilute the unwanted relics formed after primordial inflation ina similar way that primordial inflation can solve the monopole problem of the big bang model. ∗ Email: hiramatz”at”yukawa.kyoto-u.ac.jp † Email: miyamoto”at”resceu.s.u-tokyo.ac.jp ‡ Email: yokoyama”at”resceu.s.u-tokyo.ac.jp
Thermal inflation has been studied by many authors due to its other interesting properties. First, it is related togravitational waves. A period of accelerated expansion after the generation of tensor perturbations in the primordialinflationary period leads to their dilution [8]. This is a non-negligible effect that must be taken into account whendetermining the value of the primordial tensor-to-scalar ratio and constraining models of inflation using observations.In addition, the collision of bubbles created at the end of thermal inflation can give rise to gravitational waves [9, 10].Second, thermal inflation provides a mechanism for baryogenesis. Though it washes out the baryon number generatedbefore thermal inflation, we can consider mechanisms for generating baryon asymmetry at the end of thermal inflation[11]. Third, effects of thermal inflation on the primordial density fluctuations are studied in Ref. [12].In a similar way as with primordial inflation, the mechanism of thermal inflation is often described in terms of aneffective potential. A key difference with most models of primordial inflation, however, is that there exists a radiationbath during thermal inflation. Interactions with particles in the thermal bath lead to thermal corrections to the flatonpotential, which creates a small dip at the origin of the flaton potential. Thermal inflation is driven by the potentialenergy of the flaton at the origin and we usually assume it ends through a first-order phase transition.Though the existence of a thermal bath is necessary for thermal inflation to occur, it also leads to thermal fluctu-ations that affect the dynamics of the flaton field. Since these effects are not accounted for in the effective potentialapproach, we incorporate the effect of thermal fluctuations separately. In this paper we consider two phases whichare relevant to the thermal inflation scenario. The first phase is before the beginning of thermal inflation. If in somespatial regions the flaton value is kept large even when the Universe cools, thermal inflation never begins. The secondphase is the end of thermal inflation. If thermal inflation ends with a first-order phase transition, bubbles are gener-ated and their collisions induce gravitational waves. Therefore, in order to predict gravitational-wave observables, itis important to study how thermal inflation ends with thermal fluctuations taken into account.This paper is organized as follows. In Section II, we take a brief look at the thermal inflation scenario. Thoughit is often described in terms of an effective potential, we consider the flaton dynamics based on the effective actionin Section III. We study the flaton dynamics further in detail by performing lattice simulations, whose setup issummarized in Section IV, and discuss the results in Section V. In Section VI, we summarize the implications of ourstudy for the thermal inflation scenario.
II. SCENARIO OF THERMAL INFLATION
We briefly review the scenario of thermal inflation in this section. In considering the dynamics of thermal inflation,we often use the thermal effective potential. Since thermal inflation occurs after primordial inflation and reheating,there is a hot thermal bath and interactions between the flaton and the fields in the bath lead to thermal correctionsto the flaton potential. The flaton is kept at the origin of the potential owing to this correction and the potentialenergy at the origin drives thermal inflation. One example of the flaton potential at zero temperature is V ( φ ) = V TI − m φ φ + λ φ , (1)where the second term represents a tachyonic mass term, whose value is assumed to be set by the soft SUSY breakingscale, m φ ≈ m soft ≈ GeV. The energy scale of thermal inflation is determined by the constant term V TI . Theexactly flat potential is curved due to SUSY breaking, and stabilized by unrenormalizable terms . By requiring thepotential energy at the bottom of the potential to be zero, we obtain λ = m φ V and φ vev = √ V TI /m φ , where φ vev is the vacuum expectation value of the flaton.Let us move on to the thermal corrections. The one-loop effective potential arising from thermal corrections isgiven by V − loop T ( φ ) = T X p g p J p (cid:18) m p ( φ, T ) T (cid:19) , (2)where p labels both the bosonic and fermionic degrees of freedom and the function J p is expressed in terms of anintegral as J ± ( y ) = ± π Z ∞ dx x ln (cid:16) ∓ e − √ x + y (cid:17) , (3) The exact form of the third term and possible higher order terms are unimportant for our study. for bosons and fermions, respectively. Following Ref.[9], the effective mass squared for fields in the bath are m p ( φ, T ) ≈ (cid:26) m + λ φ + ( λ + g ) T boson , λ φ + g T fermion . (4)Here we consider Yukawa couplings between the flaton and scalar boson and fermion, with coupling constants λ b and λ f , respectively. The coupling constants g b and g f are associated with the gauge interactions of the scalar bosonand fermion, respectively. We assume that the masses of other bosons are also determined by m soft ≈ GeV andthat fermions are massless at tree level. Since these corrections lower the potential by O ( T /
10) around | φ | < ∼ T ,there appears a small dip at the origin, which traps the flaton to drive thermal inflation. We show an example flatonpotential in Fig.1. Φ (cid:144)H V TI1 (cid:144) m Φ- L V e ff (cid:144) V T I a typical flaton potential at zero temperature - - - - Φ (cid:144) T V T - l oop (cid:144) T a typical potential correction T = T T = T T = T T = < T < T < T Φ V e ff qualitative evolution of effective potential FIG. 1: The zero-temperature potential of the flaton and its finite-temperature correction.
Thermal inflation begins when the energy density of other components decays to be as small as the potential energyof the flaton. If the Universe is dominated by radiation, the temperature at the beginning of thermal inflation, T begin ,is given by T begin = (cid:16) π g ∗ V TI (cid:17) / .During thermal inflation, the potential energy of the false vacuum phase around the origin is larger than that ofthe true vacuum, meaning that we might expect tunneling from the false to the true vacuum. However, the tunnelingrate is so small [5] that the flaton is assumed to be fixed at the origin until the dip almost disappears. Since theorder of the curvature of the dip is determined by the temperature as V ′′ eff ∼ O ( T ), thermal inflation ends when thetemperature becomes as small as m φ ≈ m s . Therefore, by choosing V TI and m φ , one can tune the duration of thermalinflation. Namely, the number of e -folds of thermal inflation is roughly given by N = log (cid:18) T begin T end (cid:19) ∼ log V TI m φ ! . (5)If we set m φ = 10 GeV and V TI = 10 GeV, we obtain
N ∼ Y / = n / /s to represent the comoving number density of gravitinos, since the entropy density, s , isproportional to a − if there is no entropy production. Before the gravitinos decay, Y / is approximately proportionalto the reheating temperature T R . Hence, if the reheating temperature is high, we have to decrease Y / . Accordingto Ref.[13], T R > ∼ GeV may be problematic. A solution proposed in Ref.[6] is to increase the entropy density viaflaton decay after thermal inflation. The ratio of the entropy densities before and after the flaton decay is s after s before ≈ V TI T R , TI π g ∗ ( T end ) T = 1 . × V TI GeV ! (cid:18) T R , TI (cid:19) − (cid:18) T end (cid:19) − (cid:18) g ∗ ( T end )200 (cid:19) − , (6)where T R , TI is the reheating temperature associated with the flaton decay. Due to this significant entropy productionthe abundance of gravitinos is made harmless.As another possibility, let us consider the case where the Universe transitions to thermal inflation after beingdominated by oscillating moduli. Hereafter we use Φ to represent one of the moduli fields. Since the moduli startoscillating when the Hubble parameter becomes as small as the mass of the moduli ( m Φ ), they start oscillating beforereheating if the reheating temperature is lower than ∼ √ m Φ M Pl . The energy density of the moduli at reheating isestimated as ρ Φ (at reheating) = 12 m Φ × (cid:18) a osc a R (cid:19) = 12 m Φ × (cid:18) H R H osc (cid:19) = 12 Φ H , (7)where Φ is the initial amplitude of the oscillating moduli and the subscript “osc” represents the value at the onsetof oscillation. After reheating, since the temperature scales as T ∝ a − , ρ Φ scales as ∝ T . Therefore T begin isdetermined by 12 Φ H × (cid:18) T begin T R (cid:19) = V TI , (8)then we obtain T begin ≈ . × (cid:16) g ∗ (cid:17) − / (cid:18) T R GeV (cid:19) / V TI GeV ! / (cid:18) Φ M Pl (cid:19) / GeV . (9)On the other hand, if the reheating temperature is high, the oscillations begin in the radiation-dominated Universe,when m = H = π g ∗ T M , (10)is satisfied. If Φ is as large as M Pl , the energy density associated with the coherent oscillation of moduli soon becomesdominant. In this case T begin is determined by12 m Φ × (cid:18) T begin T osc (cid:19) = V TI , (11)combining the above expressions we obtain T begin ≈ . × (cid:16) g ∗ (cid:17) − / V TI GeV ! / (cid:16) m Φ (cid:17) − / (cid:18) Φ M Pl (cid:19) / GeV . (12)Let us move on to the cosmological moduli problem. The moduli abundance Y Φ = n Φ /s should also be small enoughso as not to spoil BBN [14]. Assuming the moduli start oscillating before reheating, during the era when the energydensity associated with the coherent oscillations of the inflaton dominate the Universe, the moduli abundance beforeflaton decay is evaluated as Y Φ = m Φ Φ H T R × M H = 18 T R m Φ (cid:18) Φ M Pl (cid:19) , (13)where we use eq. (7) and assume that there is no entropy production after reheating. After the flaton decays, byusing eq. (6), Y Φ becomes Y Φ , after ≈ π g ∗ ( T end ) (cid:18) Φ M Pl (cid:19) T R T R , TI T m Φ V TI = 8 . × − V TI GeV ! − (cid:18) T R GeV (cid:19) (cid:18) T R , TI (cid:19) (cid:18) T end m Φ (cid:19) (cid:16) m Φ (cid:17) (cid:18) Φ M Pl (cid:19) (cid:18) g ∗ ( T end )200 (cid:19) . (14)Therefore, with appropriate parameters, thermal inflation can make Y Φ small enough for successful BBN. III. FLATON DYNAMICS IN A THERMAL BATH
In this section, we consider the flaton dynamics based on finite-temperature field theory. In order to describe thedynamics of the expectation values of quantum fields in a thermal bath, we use the effective action method, which hasbeen studied in several contexts [15–18] based on the in-in or the closed time-path formalisms. Using this method,we can evaluate the evolution of expectation values by performing path integrals along two time paths, with two fieldvariables φ ± defined on each path. Generally the effective action can be expressed as [15–18]Γ = S + Γ R + Γ I , (15)where S is the tree level action, and Γ R and Γ I , respectively, represent the real and imaginary parts coming frominteractions. The imaginary part has the following structureexp [ i Γ I ] = exp (cid:20) − Z d x d x A a ( x − x ) φ ∆ ( x ) φ ∆ ( x ) + A m ( x − x ) φ ∆ ( x ) φ ∆ ( x ) φ c ( x ) φ c ( x ) (cid:21) , (16)and we can rewrite it as exp [ i Γ I ] = Z D ξ a D ξ m P [ ξ a ] P [ ξ m ] exp [ iS noise ] , (17)where P [ ξ a ] ∝ exp (cid:20) − Z d x d x ξ a ( x ) A − ( x − x ) ξ a ( x ) (cid:21) ,P [ ξ m ] ∝ exp (cid:20) − Z d x d x ξ m ( x ) A − ( x − x ) ξ m ( x ) (cid:21) ,S noise = Z d x [ ξ a ( x ) φ ∆ ( x ) + ξ m ( x ) φ ∆ ( x ) φ c ( x )] ,φ c = φ + + φ − , φ ∆ = φ + − φ − . (18)We can interpret the new variables ξ a and ξ m as stochastic noises whose probability distributions are given by P [ ξ a ]and P [ ξ m ], respectively. Finally the equation of motion for φ c , which is obtained by varying the effective action withrespect to φ ∆ , becomes the Langevin equation (cid:3) φ ( x ) + V ′ eff [ φ ] + Z t −∞ dt ′ Z d x ′ B a ( x − x ′ ) φ ( x ′ ) + φ ( x ) Z t −∞ dt ′ Z d x ′ B m ( x − x ′ ) φ ( x ′ )= ξ a ( x ) + ξ m ( x ) φ ( x ) . (19)We briefly see specific examples studied in Ref. [16]. An interaction term L int = − λ χ φ , where χ is a real scalarfield, leads to both additive and multiplicative noises and the corresponding non-local terms. Functions A and B foradditive noise and the correspondent non-local terms in Fourier space are calculated as A a ( ω, ~k ) = − πiλ Z d q (2 π ) d q (2 π ) d q (2 π ) (2 π ) δ ( ~q + ~q + ~q − ~k ) 18 ω q ω q ω k − q − q × [ { (1 + n q )(1 + n q )(1 + n q ) + n q n q n q } δ ( ω − ω q − ω q − ω q )+ { (1 + n q )(1 + n q ) n q + n q n q (1 + n q ) } δ ( ω − ω q − ω q + ω q )+ { (1 + n q ) n q (1 + n q ) + n q (1 + n q ) n q } δ ( ω − ω q + ω q − ω q )+ { n q (1 + n q )(1 + n q ) + (1 + n q ) n q n q } δ ( ω + ω q − ω q − ω q )+ { (1 + n q ) n q n q + n q (1 + n q )(1 + n q ) } δ ( ω − ω q + ω q + ω q )+ { n q (1 + n q ) n q + (1 + n q ) n q (1 + n q ) } δ ( ω + ω q − ω q + ω q )+ { n q n q (1 + n q ) + (1 + n q )(1 + n q ) n q } δ ( ω + ω q − ω q + ω q )+ { n q n q n q + (1 + n q )(1 + n q )(1 + n q ) } δ ( ω + ω q + ω q + ω q )] , (20) B a ( ω, ~k ) = 8 πλ Z d q (2 π ) d q (2 π ) d q (2 π ) (2 π ) δ ( ~q + ~q + ~q − ~k ) 18 ω q ω q ω q × [ { (1 + n q )(1 + n q )(1 + n q ) − n q n q n q } δ ( ω − ω q − ω q − ω q )+ { (1 + n q )(1 + n q ) n q − n q n q (1 + n q ) } δ ( ω − ω q − ω q + ω q )+ { (1 + n q ) n q (1 + n q ) − n q (1 + n q ) n q } δ ( ω − ω q + ω q − ω q )+ { n q (1 + n q )(1 + n q ) − (1 + n q ) n q n q } δ ( ω + ω q − ω q − ω q )+ { (1 + n q ) n q n q − n q (1 + n q )(1 + n q ) } δ ( ω − ω q + ω q + ω q )+ { n q (1 + n q ) n q − (1 + n q ) n q (1 + n q ) } δ ( ω + ω q − ω q + ω q )+ { n q n q (1 + n q ) − (1 + n q )(1 + n q ) n q } δ ( ω + ω q − ω q + ω q )+ { n q n q n q − (1 + n q )(1 + n q )(1 + n q ) } δ ( ω + ω q + ω q + ω q )] , (21)where ω q i = q | ~q i | + m χ and n q i = e βωqi − . For the multiplicative noise and corresponding non-local term, we find A m ( ω, ~k ) = 2 πλ Z d q (2 π ) ω q ω k − q × [ { (1 + n q )(1 + n k − q ) + n q n k − q } δ ( ω − ω q − ω k − q )+ { (1 + n q ) n k − q + n q (1 + n k − q ) } δ ( ω − ω q + ω k − q )+ { ( n q (1 + n k − q ) + (1 + n q ) n k − q } δ ( ω + ω q − ω k − q )+ { n q n k − q + (1 + n q )(1 + n k − q ) } δ ( ω + ω q + ω k − q )] , (22) B m ( ω, ~k ) = − πiλ Z d q (2 π ) ω q ω k − q × [ { (1 + n q )(1 + n k − q ) − n q n k − q } δ ( ω − ω q − ω k − q )+ { (1 + n q ) n k − q − n q (1 + n k − q ) } δ ( ω − ω q + ω k − q )+ { ( n q (1 + n k − q ) − (1 + n q ) n k − q } δ ( ω + ω q − ω k − q )+ { n q n k − q − (1 + n q )(1 + n k − q ) } δ ( ω + ω q + ω k − q )] . (23)Though the noise terms generally consist of both additive noise, ξ a , and multiplicative noise, ξ m φ , we focus on theadditive noise term since the former is more important to trigger phase transition. This noise term is related to the“friction” term through the fluctuation-dissipation relation [16, 17]noise correlationdissipation coefficient = A a ( ω, ~k ) iB a ( ω, ~k ) / ω = ω e ω/T + 1 e ω/T − → T ( T ≫ ω ) . (24)In Ref. [18] it was shown that the damping scale of the fermionic noise correlation is independent of the mass ofthe fermion, which is different from the bosonic noise whose correlation damps exponentially above the mass scale.Therefore, in the high-temperature regime T ≫ m , the dominant noise component comes from interactions withfermions. More quantitatively, the correlation function for fermionic noise can be expressed as h ξ ( t, ~x ) ξ ( t, ~x ′ ) i ∝ T r e − πrT , for r ≫ πT , ( r = | ~x − ~x ′ | ) . (25)From this expression we take the correlation length of thermal noise as ( πT ) − . This length scale is very important inestimating the typical value of the flaton at finite temperature. Here let us take a quick look at this typical field value,as this will help us to understand the results of numerical simulations later. The form of the effective potential is toocomplicated to be well approximated by a simple polynomial function, so for simplicity let us neglect the potentialhere. Following Ref. [19], the mean square value of the coarse-grained field φ over the spatial scale R is given by h φ i R = 12 π Z ∞ dk k (cid:18)
12 + 1 e kT − (cid:19) W ( k, R ) , (26)where W ( k, R ) is the coarse-graining window function. As an example, if we take the Gaussian function W ( k, R ) = e − k R , (27)we obtain p h φ i ≈ . T for R = ( πT ) − .Since the correlation length of the noise is ∼ ( πT ) − , we can treat the noise as being uncorrelated on larger scales.The same is true for the temporal noise correlation, since it is suppressed exponentially for ∆ t > ( πT ) − . As such,the noise term can be approximated by a white, Gaussian random variable when we consider dynamics on spatial andtemporal scales that are larger than the above correlation length. Hence we use the following simple EoM.¨ φ ( ~x, t ) − ~ ∇ φ ( ~x, t ) + η ˙ φ ( ~x, t ) + V ′ eff [ φ ] = ξ ( ~x, t ) , (28)where the correlation function of the noise term is h ξ ( ~x, t ) ξ ( ~x ′ , t ′ ) i = Dδ ( t − t ′ ) δ ( ~x − ~x ′ ) . (29)The fluctuation-dissipation relation in this simple EoM is Dη = 2 T . (30)Due to the fluctuation-dissipation relation, equilibrium values do not depend on the friction coefficient η . Its value isrelated with the decay rate of φ particle if φ is oscillating [15–17]. On dimensional grounds we can take Γ ∝ T . Sincethe value of η only determines the time scale on which the system approaches equilibrium, here we simply take η = T as strong enough couplings between the flaton and the thermal bath are required for successful thermal inflation.Then the ratio of the equilibration timescale to the cosmic expansion timescale isequilibration timescaleHubble time ∼ η − H − = T − H − ∼ TM Pl (RD era) , V M Pl T (during thermal inflation) . (31)We see that this ratio is much smaller than unity in both the RD era and the period of thermal inflation, from whichwe can conclude that the equilibration time is still much shorter than the Hubble time even if we take other choicesfor the value of η . This huge difference between the two timescales allows us to safely ignore the Hubble expansion insimulations we show later. IV. SETUP OF NUMERICAL SIMULATIONS
In this section we summarize the details of our three-dimensional lattice simulation. We solved the equation ofmotion given by eq. (28) by the second-order explicit Runge-Kutta method with the second-order finite differencesapproximating the spatial derivatives. The basic setup is the same as in Ref.[18]. In numerical calculations we usedimensionless variables like ˜ x = T x , ˜ t = T t , ˜ φ = φ/T , and ˜ ξ = ξ/T since the scale of interest is deeply related to thetemperature.The noise correlation function on the lattice becomes h ξ ( ~x i , t m ) ξ ( ~x j , t n ) i = 2 ηδ ( t m − t n ) δ ( ~x i − ~x j ) → η ∆ t (∆ x ) δ m,n δ i,j , (32)since on the lattice the delta functions are properly replaced as δ ( t m − t n ) → (∆ t ) − δ m,n and δ ( ~x i − ~x j ) → (∆ x ) − δ i,j .The value of noise variable on each lattice is given by ξ ( ~x i , t m ) = (cid:18) η ∆ t (∆ x ) (cid:19) G i,m , (33)where G is a standard Gaussian random variable.We also define approximation function of the potential term, which is shown in Appendix. As can be seen later,the quantitative shape of the effective potential is very sensitive to the temperature, especially at the end of ther-mal inflation. Therefore we use the above approximation function both in the lattice simulation and semi-analyticcalculation.We choose the initial condition for simulations as φ ( ~x, t = 0) = ˙ φ ( ~x, t = 0) = 0 . (34)Although this is an admittedly unrealistic initial condition, we have confirmed that the field quickly reaches thethermal configuration compared to the typical duration of simulation time and the timescale of the temperaturevariation.With the above settings we use the 256 lattice points and m φ (and m b in eq. (4))= 10 and 10 GeV, but thequalitative results do not depend on these mass values.
V. RESULTS OF NUMERICAL SIMULATIONSA. phase 1: before thermal inflation
A necessary initial condition for the flaton to drive thermal inflation is that the field value of the flaton shouldbe homogeneously close to zero before thermal inflation begins. However, the form of the 1-loop effective potentialsuggests that there is more than one local minimum, and if the flaton field is trapped in the true vacuum in somespatial regions, the thermal inflation scenario does not work. In order to determine whether or not this problem isencountered, we simulated the time evolution of the flaton from a very high temperature, T , to the temperature atwhich thermal inflation begins.The “high” temperature T is determined by the following consideration. In order to realize a situation where thetypical value of the flaton is φ vev ( ≡ √ V TI /m φ , the vacuum expectation value at T = 0), we first perform a simulationat T = φ vev , expecting p h φ i ≈ T ≈ φ vev . At this temperature the shape of the effective potential becomes like thepotential labelled “ T = T ” in the right panel of Fig.1. We then perform a second simulation, setting the temperatureto half of that in the previous simulation and using the final configuration of the previous simulation to determinethe initial conditions. Since we fix the gridsize of the simulation and the value of the lattice spacing normalized bythe temperature, the physical size of the second simulation box is larger than that of the previous, hotter simulation.We therefore use periodic boundary conditions and define the initial condition for φ and ˙ φ as averaged quantities ofthe previous values of close grids on each new grids. Repeating this procedures N times we can follow the flatondynamics from T = T to T = T × − N ∼ T begin .In the numerical simulations we consider corrections to the potential coming from a single bosonic and singlefermionic degree of freedom. In order to try and establish the importance of the thermal effects we perform simulationswith two choices of the coupling constants appearing in eq. (4). Hereafter we refer to these two choices as the stronglyand weakly coupled cases, and they correspond to taking λ b = g b = λ f = g f = 1 and λ b = g b = λ f = g f = 0 . → TI) and in the second scenario thermal inflation is preceded by radiation domination (RD → TI).The results of one example simulation are shown in Fig.2. For the form of effective potential used in this study, weconfirm that the typical value of the flaton is p h φ i ≈ T , regardless of the temperature before thermal inflation. Inother words, we do not see any spatial regions where the field value remains so large that the flaton potential energybecomes inhomogeneous and ruins the thermal inflation scenario.We close this subsection with comments on the validity of our multistage simulation. The result shown in Fig. 2confirms us that we properly follow the dynamics of the flaton from a high temperature to T begin , with multistagesimulation. Since the equilibration timescale ( ∼ η − ) is much shorter than that of temperature change ( ∼ H − ),the system approaches the equilibrium rapidly enough in each simulation with a fixed temperature. In other words,even though we impose out-of-equilibrium initial condition which is simply connected by the previous simulationwhere the temperature is set twice as hot, we can realize the equilibrium distribution ( p h φ i ∼ T ) by performing asimulation for a longer time than η − (but much shorter than H − ). Therefore repetitive simulations enable us toconsider a system in quasi-equilibrium state for a longer time than Hubble time without including the exact changein temperature. The smooth change of the root mean square (RMS) value obtained in Fig. 2 justifies a factor of 2change of the temperature at each step is small enough to warrant the adiabatic change of the temperature in thesequential simulations. As for the maximum value, we note that for random 256 realization of Gaussian distribution,the probability the maximun exceeds 6.2 σ (5.6T) is 1 % and that it lies lower than 5.2 σ (4.6T) is also 1%. Althoughthe field value at each point is correlated with nearby points, we find one-point distribution function is close to aGaussian distribution. Hence we may conclude the observed maximum values in Fig. 2 are also in accordance withthe entire distribution. Note that the VEV of the zero-temperature potential also depends on V TI as φ vev = √ V TI /m φ . Since the temperature at the beginningof thermal inflation, T begin , is controlled by V TI (see Section II), we choose the value of V TI such that the number of e -folds of thermalinflation becomes about 6. In order to calculate the number of e-folds we also need to know the temperature at the end of thermalinflation, and this can be determined once we have fixed the coupling constants. scenario couplings T begin [GeV] φ vev (= T )[GeV]MD → TI strong 2 . × . × MD → TI weak 1 . × . × RD → TI strong 2 . × . × RD → TI weak 1 . × . × TABLE I: The temperature at the beginning of thermal inflation and VEV of the flaton. Since the ratio of these values is O (10 ) ∼ , we performed about 20 simulations to follow the flaton dynamics from T to T begin . à à à à à à à à à à à ç ç ç ç ç ç ç ç ç ç ç à à à à à à à à à à à ç ç ç ç ç ç ç ç ç ç ç RMS of Φ Maximum value of È Φ È time @ (cid:144) T D Φ (cid:144) T MD ® TI, strong coupling
FIG. 2: The results of one example multistage lattice simulation that was performed assuming moduli-domination beforethermal inflation and strong coupling to the fields in the thermal bath. In other cases the results are qualitatively thesame. The root mean square of φ and the maximum value of | φ | in the first and the last simulation at each referencetime are shown. The red lines with square vertices are the results of the first (hot) simulation and the dashed blue lineswith circular vertices are those of the last simulation, where T ∼ T begin . Since we impose the initial conditions φ = ˙ φ = 0in the first simulation and the initial conditions for the following simulations are determined sequentially by the finalconfiguration of the preceding, higher temperature simulation, the flaton distribution at each first reference time is notthe equilibrium configuration. B. phase 2: at the end of thermal inflation
It is believed that thermal inflation ends with a first-order phase transition accompanied by the formation of bubbles,and that the collision of these bubbles then leads to gravitational wave production. Here we briefly review the theoryof tunneling at a finite temperature and define the percolation temperature at which the bubbles collide and startgenerating gravitational waves.The tunneling rate per unit volume at temperature T is estimated as [20]Γ( T ) ∼ T e − S T , (35)where S is the Euclidean action after performing the time integral, S = Z d x (cid:18)
12 ( ∇ φ ) + V ( φ ) (cid:19) . (36)The dominant contribution to the tunneling rate comes from the solution of the equation of motion, d φdr + 2 r dφdr − dVdφ = 0 , ( r = | ~x | ) (37)0under the boundary conditions φ ( r = ∞ ) = 0 and dφdr (cid:12)(cid:12)(cid:12) r =0 = 0.The fraction of spatial regions occupied by bubbles can be written as [21] F ( t ) = 1 − e − P ( t ) , (38)where the function P ( t ) is given by P ( t ) = Z t dt ′ Γ( t ′ ) 4 π (cid:18)Z tt ′ dt ′′ a ( t ) a ( t ′′ ) (cid:19) = 4 π Z t dt ′ Γ( t ′ ) 1 H (cid:16) e H ( t − t ′ ) − (cid:17) . (39)Making use of eq.(35) we can rewrite this in terms of temperature as P ( T ) = 4 π Z ∞ T dT ′ T ′ H (cid:18) T ′ T − (cid:19) e − S T ) T . (40)In this paper we define the percolation temperature as F ( T = T p ) = 0 . . Note that since the exponential factorexp[ − S ( T ) /T ] is very sensitive to the temperature and quickly becomes small when we take a large value of T , itis sufficient to take the upper limit of the integral to be some finite value. For example, it is enough to take it as2 T curv , where T curv is the temperature at which the curvature of the potential becomes zero. After evaluating theabove quantities numerically, we find that the difference between the percolation temperature T p and T curv is tiny, sothat the Universe becomes filled with critical bubbles almost immediately after bubble formation effectively begins.From the above consideration based on the shape of the flaton effective potential, we may expect that thermalinflation ends with a first-order phase transition characterized by critical bubble formation. However, this descriptionis based on the assumption that the flaton is well within the false vacuum phase before bubble nucleation occurs.We see from Fig.3 that around the percolation temperature the potential barrier is located at φ ≪ T and theheight of the barrier is much smaller than T . Taking thermal fluctuations into account, since the width of the fielddistribution is p h φ i ≈ T , we conclude that the small potential barrier cannot trap the flaton in the false vacuumphase until the temperature becomes as small as the temperature at which critical bubble nucleation occurs. Thismeans that the two phases coexist well before the percolation epoch in the bubble nucleation picture, and the phasetransition proceeds with phase-mixing. As such, the standard description of the end of thermal inflation in terms ofa strong first-order phase transition which is accompanied with bubble formation is inappropriate.Now let us investigate more quantitatively the failure of critical bubble formation as a description of the end ofthermal inflation. The width of the wall trapping the flaton is broad at high temperatures and gradually becomesthin as the temperature drops. We define the width in field space, φ wid , at temperature T , as V eff [ φ = φ wid , T ] = V eff [ φ = 0 , T ] . (41)Since the shape of the effective potential depends on temperature, we obtain φ wid ( T ) by solving the above equation.As a typical temperature at which phase-mixing occurs, we define the temperature T sub as φ wid ( T = T sub ) = T sub , (42)i.e. T sub is the temperature at which the width of the potential wall becomes as small as the temperature. As we seefrom the simulations in the previous subsections and the analytical estimation (eq. (26)), the typical value of φ is aslarge as T . Therefore, at T = T sub , and if the height of the potential barrier is small enough, spatial regions in whichthe flaton lies outside of the potential dip are ubiquitous in the Universe. We call such regions subcritical bubbles[22], which are continuously created and destroyed by thermal fluctuations and hence differ from the critical bubbleswhich only grow after being nucleated by tunneling. For the effective potential we study in this paper, the relations T sub > T p and F ( T sub ) ≪ T = T sub the flaton is no longer trapped at the local minimum at theorigin, meaning that there are practically no critical bubbles. Specific values are shown in Table II. We would like tomake a comment on the temperature at the end of thermal inflation, T end quantitatively. In Section II we estimated T end ∼ m φ . Table II, however, shows that while T curv , T p , and T sub coincide with each other within 5% they deviatefrom m φ by a factor of 5 - 40. Hence we should use T end ∼ T sub to estimate the proper duration of thermal inflation. The qualitative conclusion ( T curv ≈ T p < T sub ) remains unchanged if we employ other definitions such as F ( T p ) = 0 .
01 or 0 . T = T sub we are able to verify that the height of the potential barrier issmall enough for the flaton to escape from φ = 0. In some cases we found that the flaton rolls down to the bottomof the potential – meaning that thermal inflation ends at T > T sub – and in other cases we found that the flatonremained around the origin, but with a distribution width that was broader than the potential well. We thus seethat all cases deviate from the standard scenario in which thermal inflation ends as the result of a strong first-orderphase transition. We summarize the dependence of the potential shape on temperature in Fig.4 schematically. scenario couplings T curv [GeV] T p [GeV] T sub [GeV] F ( T sub ) simulated p h φ i at T sub MD → TI strong 5230 5239 (2 × − ) 5502(5 × − ) 10 − φ vev MD → TI weak 41216.96 41216.97(4 × − ) 41378(4 × − ) less than 10 − . T RD → TI strong 5230 5252 (4 × − ) 5502(5 × − ) 10 − φ vev RD → TI weak 41216.96 41216.97(4 × − ) 41378(4 × − ) less than 10 − . T TABLE II: Specific temperature values for four different scenarios. In all four scenarios we take m φ = 1 TeV. Since thetemperatures themselves are almost the same, we also show the relative differences ( T p − T curv ) /T curv and ( T sub − T curv ) /T curv in brackets. In evaluating T p and F ( T ), we fix the value of V TI so that thermal inflation begins at T = T curv × e . The RMSvalues of φ at T = T sub , obtained by simulations with duration t = 2000 /T , are also shown. In the two strongly-coupled casesthe flaton leaves the origin and settles in its VEV. In the two weakly-coupled cases the flaton stays at the origin, but the widthof its distribution function is as broad as the barrier. Though the potential barrier is negligible, the potential force arising fromthe tachyonic mass term is also so weak that it may take a long time to displace the flaton from the origin. T = T curv T = T p - ´ - - ´ - ´ - ´ - Φ (cid:144) T H V e ff @ Φ D - V e ff @ D L (cid:144) T strong coupling case H MD ® TI L T = T curv T = T sub - - Φ (cid:144) T H V e ff @ Φ D - V e ff @ D L (cid:144) T strong coupling case H MD ® TI L T = T curv T = T p ´ - ´ - - ´ - - ´ - ´ - ´ - Φ (cid:144) T H V e ff @ Φ D - V e ff @ D L (cid:144) T weak coupling case H MD ® TI L T = T curv T = T sub - ´ - - ´ - ´ - ´ - ´ - ´ - Φ (cid:144) T H V e ff @ Φ D - V e ff @ D L (cid:144) T weak coupling case H MD ® TI L FIG. 3: Some examples of the effective potential at T = T curv , T p , and T sub are shown. Since at T = T p the local maximum islocated at φ < T and its height is much smaller than T , the flaton is able to escape the local minimum and critical bubbleformation theory is not applicable. This may be explained as an effect of surface tension, which is stronger than the potential force pulling the flaton away from the origin. T = T p T = T sub T = T curv PDF of Φ at T = T sub ææ Φ (cid:144) T V e ff @ Φ D - V e ff @ D schematic summary of potential shapes around the origin H not to scale L FIG. 4: A schematic relation of the potential shapes at T = T sub , T p , and T curv . We also show the probability distributionfunction of the flaton at T = T sub , which indicates that subcritical bubbles are abundant in the Universe at T = T sub . VI. CONCLUSION
In this paper we studied the effect of thermal fluctuations on the thermal inflation scenario. Thermal inflation is ashort period of accelerated expansion after reheating and provides a way to dilute dangerous moduli and gravitinosin order to make theories based on supersymmetry compatible with cosmological observations. Thermal inflation isdriven by the flaton potential energy at the origin with the help of thermal corrections. Since the thermal environmentgives rise to thermal fluctuations as well, we used lattice simulations to study the dynamics of the flaton taking intoaccount the 1-loop effective potential, thermal fluctuations and the dissipation term. First we studied the effects ofthermal fluctuations before thermal inflation. Though the effective potential contains multiple local minima duringthe course of the evolution of the Universe, the flaton settles at the origin before thermal inflation even when thermalfluctuations are taken into account. Therefore the scenario of thermal inflation may be feasible. Second, we find thatthermal inflation ends with a cross over phase transition. The tunneling rate of the flaton from the origin of thepotential is so small that the tunneling does not occur until the position of the potential barrier becomes very closeto the origin. However, since the height of the barrier is much smaller than T , the flaton can escape over the barrierbefore tunneling occurs. Though the form of the effective potential suggests that thermal inflation ends with a first-order phase transition accompanied by bubble formation, thermal fluctuations make the transition to proceed throughphase mixing, which is characterized by subcritical bubbles. As such, we cannot expect critical bubble formation andthe production of gravitational waves. Acknowledgments
We would like to thank Jonathan White for helpful comments. Y.M. also thanks Kohei Kamada and DaisukeYamauchi for informative discussions. This work was supported in part by MEXT SPIRE and JICFuS (T.H.), JSPSResearch Fellowships for Young Scientists (Y.M.), and JSPS Grant-in-Aid for Scientific Research No.23340058 (J.Y.)3
Appendix : constructing approximation functions of the potential term
In this section, we consider the approximation of Eq. (3), which determines the functional shape of the thermalcorrection to the flaton potential. Expanding the integrand of Eq. (3), we can perform the integration term by term, J ± ( y ) = ∓ π ∞ X n =1 ( ± n n Z ∞ dx x e − n √ x + y , = ∓ y π ∞ X n =1 ( ± n n K ( ny ) , (43)where K ( x ) is the modified Bessel function of the second kind. The derivative of J ( y ) with respect to y , whichappears in the field equation, (28), is calculated as dJdy = ± y π ∞ X n =1 ( ± n n K ( ny ) . (44)For convenience, we define the shape function, S ± ( y ) ≡ ∞ X n =1 ( ± n n K ( ny ) . (45)The modified Bessel function K ( z ) for small z can be approximated as K ( z ) ≈ z . (46)Therefore, the shape function for small y becomes S ± ( y ) ≈ y ∞ X n =1 ( ± n n = 1 y × ζ (2) , for + − ζ (2)2 . for − (47)Away from y = 0 this approximation breaks down almost immediately. Moreover, it is difficult to achieve betteraccuracy by simply retaining more terms in the expansion in Eq. (46), since there are logarithmic terms like ln z ,meaning that we cannot take the infinite summation analytically. Instead, we use the following ansatz, e S (0)+ ( y ) = e − y y (cid:0) ζ (2) + a y + a y + a y (cid:1) , (48) e S (0) − ( y ) = e − y y (cid:18) − ζ (2)2 + b y + b y + b y + b y (cid:19) , (49)where a i and b i are determined by requiring a good fit with the shape function in the limited region 0 ≤ y ≤
2; weobtain a i = (0 . , . , − . b i = ( − . , . , − . , . y we can truncate the infinite summation in Eq. (45) at relatively small n thanks tothe asymptotically exponential decay of K ( ny ). Here we take the summation up to n = 2. We also use the asymptoticexpansion of the modified Bessel functions. To guarantee accuracy, we expand K ( y ) up to y − and K (2 y ) up to y − . Eventually we obtain e S ( ∞ ) ± ( y ) = ± r π y e − y (cid:18) y − y + 1051024 y (cid:19) + r π y e − y (cid:18) y (cid:19) . (50)Finally, we approximate the shape function given in Eq. (45) as S ± ( y ) ≈ e S (0) ± ( y ) , for y < , e S ( ∞ ) ± ( y ) , for y ≥ . (51)The partitioned fitting curve for the shape function constructed here has an accuracy E = 1 . × − for S − and E = 2 . × − for S + , where E ≡ || − e S ± ( y ) /S ± ( y ) || ∞ . Note that, as a result of the naive matching of the4two functions, dV − loop T /dφ is discontinuous at y = 2 by construction. However, this is not problematic, since theamplitude of the discontinuity in dV T /dφ at y = 2 is on the order of 0 . [1] A. A. Starobinsky, Phys. Lett. B , 99 (1980); K. Sato, Mon. Not. Roy. Astron. Soc. , 467 (1981); A. H. Guth, Phys.Rev. D , 347 (1981).[2] M. Y. Khlopov and A. D. Linde, Phys. Lett. B , 265 (1984); J. R. Ellis, J. E. Kim and D. V. Nanopoulos, Phys. Lett.B , 181 (1984).[3] G. D. Coughlan, W. Fischler, E. W. Kolb, S. Raby and G. G. Ross, Phys. Lett. B , 59 (1983); T. Banks, D. B. Kaplanand A. E. Nelson, Phys. Rev. D , 779 (1994) [hep-ph/9308292]; B. de Carlos, J. A. Casas, F. Quevedo and E. Roulet,Phys. Lett. B , 447 (1993) [hep-ph/9308325].[4] M. Kawasaki and T. Yanagida, Phys. Lett. B , 45 (1997) [hep-ph/9701346].[5] K. Yamamoto, Phys. Lett. B , 341 (1986).[6] D. H. Lyth and E. D. Stewart, Phys. Rev. Lett. , 201 (1995) [hep-ph/9502417]; D. H. Lyth and E. D. Stewart, Phys.Rev. D , 1784 (1996) [hep-ph/9510204].[7] T. Asaka, J. Hashiba, M. Kawasaki and T. Yanagida, Phys. Rev. D , 083509 (1998) [hep-ph/9711501]; T. Asaka andM. Kawasaki, Phys. Rev. D (1999) 123509 [hep-ph/9905467]; K. Choi, W. I. Park and C. S. Shin, JCAP , 011(2013) [arXiv:1211.3755 [hep-ph]].[8] L. E. Mendes and A. R. Liddle, Phys. Rev. D , 063508 (1999).[9] R. Easther, J. T. Giblin, Jr., E. A. Lim, W. I. Park and E. D. Stewart, JCAP , 013 (2008) [arXiv:0801.4197 [astro-ph]].[10] A. Kosowsky, M. S. Turner and R. Watkins, Phys. Rev. Lett. , 2026 (1992); M. Kamionkowski, A. Kosowsky andM. S. Turner, Phys. Rev. D , 2837 (1994) [astro-ph/9310044].[11] E. D. Stewart, M. Kawasaki and T. Yanagida, Phys. Rev. D , 6032 (1996) [hep-ph/9603324]; D. -h. Jeong, K. Kadota,W. -I. Park and E. D. Stewart, JHEP , 046 (2004) [hep-ph/0406136]; M. Kawasaki and K. Nakayama, Phys. Rev.D , 123508 (2006) [hep-ph/0608335]. S. Kim, W. I. Park and E. D. Stewart, JHEP , 015 (2009) [arXiv:0807.3607[hep-ph]]; K. Choi, K. S. Jeong, W. I. Park and C. S. Shin, JCAP , 018 (2009) [arXiv:0908.2154 [hep-ph]];[12] M. Kawasaki, T. Takahashi and S. Yokoyama, JCAP , 012 (2009) [arXiv:0910.3053 [hep-th]].[13] M. Kawasaki, K. Kohri and T. Moroi, Phys. Rev. D (2005) 083502 [astro-ph/0408426].[14] J. R. Ellis, G. B. Gelmini, J. L. Lopez, D. V. Nanopoulos and S. Sarkar, Nucl. Phys. B (1992) 399; E. Holtmann,M. Kawasaki, K. Kohri and T. Moroi, Phys. Rev. D , 023506 (1999) [hep-ph/9805405][15] M. Morikawa, Phys. Rev. D , 3607 (1986); M. Gleiser and R. O. Ramos, Phys. Rev. D , 2441 (1994) [hep-ph/9311278].[16] J. Yokoyama, Phys. Rev. D , 103511 (2004) [hep-ph/0406072].[17] C. Greiner and B. Muller, Phys. Rev. D , 1026 (1997) [hep-th/9605048].[18] M. Yamaguchi and J. Yokoyama, Phys. Rev. D , 4544 (1997) [hep-ph/9707502].[19] M. Yamaguchi and J. Yokoyama, Nucl. Phys. B , 363 (1998) [hep-ph/9805333].[20] A. D. Linde, Phys. Lett. B , 37 (1981); A. D. Linde, Contemp. Concepts Phys. , 1 (1990) [hep-th/0503203].[21] A. H. Guth and E. J. Weinberg, Phys. Rev. D , 876 (1981).[22] M. Gleiser, E. W. Kolb and R. Watkins, Nucl. Phys. B , 411 (1991); M. Gleiser and E. W. Kolb, Phys. Rev. Lett. ,1304 (1992); M. Gleiser and E. W. Kolb, Phys. Rev. D , 1560 (1993) [hep-ph/9208231]; T. Shiromizu, M. Morikawa andJ. Yokoyama, Prog. Theor. Phys.94