Emergent Universe by Tunneling in a Jordan-Brans-Dicke Theory
EEmergent Universe by Tunneling in a Jordan-Brans-Dicke Theory
Pedro Labra˜na ∗ and Hobby Cossio † Departamento de F´ısica, Universidad del B´ıo-B´ıo and Grupo deCosmolog´ıa y Part´ıculas Elementales UBB, Casilla 5-C, Concepci´on, Chile. (Dated: April 4, 2019)
Abstract
In this work we study an alternative scheme for an Emergent Universe scenario in the contextof a Jordan-Brans-Dicke theory, where the universe is initially in a truly static state supportedby a scalar field located in a false vacuum. The model presents a classically stable past eternalstatic state which is broken when, by quantum tunneling, the scalar field decays into a state oftrue vacuum and the universe begins to evolve following the extended open inflationary scheme.
PACS numbers: 98.80.Cq ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ g r- q c ] A p r . INTRODUCTION The standard cosmological model (SCM) [1–3] and the inflationary paradigm [4–7] areshown as a satisfactory description of our universe [1–3]. However, despite its great success,there are still important open questions to be answered. One of these questions is whetherthe universe had a definite origin, characterized by an initial singularity or if, on the contrary,it did not have a beginning, that is, it extends infinitely to the past.Theorems about spacetime singularities have been developed in the context of inflationaryuniverses, proving that the universe necessarily has a beginning. In other words, accordingto these theorems, the existence of an initial singularity can not be avoided even if theinflationary period occurs, see Refs. [8–12]. In theses theorems it is demonstrated that nulland time-like geodesics are generally incomplete in inflationary models, regardless of whetherenergy conditions are maintained, provided that the average expansion condition (
H >
0) ismaintained throughout of these geodesics directed towards the past, where H is the Hubbleparameter.The search for cosmological models without initial singularities has led to the developmentof the so-called Emergent Universes models (EU) [13–20].In the EU scheme it is assumed that the universe emerged from a past eternal EinsteinStatic (ES) state to the inflationary phase and then evolves into a hot big bang era. Thesemodels do not satisfy the geometrical assumptions of the theorems [8–12] and they providespecific examples of nonsingular inflationary universes.Usually the EU models are developed by consider a universe dominated by a scalar field,which, during the past-eternal static regime, is rolling on the asymptotically flat part of thescalar potential (see Fig. (1)) with a constant velocity, providing the conditions for a staticuniverse, see for example models [13–15, 19–30]. Other possibility is to consider EU modelsin which the scale factor only asymptotically tends to a constant in the past [16, 17, 31–38].We can note that in these schemes of Emergent Universe are not all truly static during thestatic regime.At this respect a new scheme for an EU model was proposed in Ref. [39], where theuniverse is initially in a truly static state supported by a scalar field located in a falsevacuum, also see Refs. [40, 41]. The universe begins to evolve when, by quantum tunneling,the scalar field decays into a state of true vacuum. For simplicity, in this first approach to2 IG. 1: Schematic representation of a potential for a standard Emergent Universe scenario. this new scheme of EU, the model was developed in the context of General Relativity (GR).In particular in Ref. [39] was concluded that this new mechanism for an Emergent Universe isplausible and could be an interesting alternative to the realization of the Emergent Universescenario.However, as this first model was developed in the context of General Relativity, thepast eternal static period, suffer from classical instabilities associated with the instability ofEinstein’s static universe.The ES solution is unstable to homogeneous perturbations, as was early discussed byEddington in Ref. [42] and more recently studied in Refs. [43–46]. The instability of theES solution ensures that any perturbations, no matter how small, rapidly force the universeaway from the static state, thereby aborting the EU scenario.This instability is possible to cure by going away from GR, for example, by considera Jordan-Brans-Dicke (JBD) theory at the classical level, where it have been found thatcontrary to general relativity, a static universe could be classically stable, see Refs. [21, 22,47].In this work, we are interested in apply the scheme of Emergent Universe by Tunnelingof Ref. [39] to EU models which present stable past eternal static regimes. In particular, westudy this scheme in the context of a JBD theory, similar to the one studied in Refs. [21, 22],but where the static solution is supported by a scalar field located in a false vacuum. Inthis case we are going to show that, contrary to what happens in Ref. [39], the ES solutionis classically stable. 3
IG. 2: Scalar field potential U ( ψ ). Here U F = U ( ψ F ) and U T = U ( ψ T ). The Jordan-Brans-Dicke [48] theory is a class of models in which the effective gravitationalcoupling evolves with time. The strength of this coupling is determined by a scalar field,the so-called Brans-Dicke field, which tends to the value G − , the inverse of the Newton’sconstant. The origin of Brans-Dicke theory is found in the Mach’s principle according towhich the property of inertia of material bodies arises from their interactions with the matterdistributed in the universe. In modern context, Brans-Dicke theory appears naturally insupergravity models, Kaluza-Klein theories and in all known effective string actions [49–55].In particular in this work we are going to consider that the universe is initially in a trulystatic state, which is supported by a scalar field ψ located in a false vacuum ( ψ = ψ F ), seeFig. (2). The universe begins to evolve when, by quantum tunneling, the scalar field decaysinto a state of true vacuum. Then, a small bubble of a new phase of field value ψ W can beformed, and expands as it converts volume from high to low vacuum energy and feeds theliberated energy into the kinetic energy of the bubble wall. This process was first studiedby Coleman & De Luccia in [56, 57] in the context of General Relativity.If the potential has a suitable form, inflation and reheating may occur inside the bubbleas the field rolls from ψ W to the true minimum at ψ T , in a similar way to what happensin models of Open Inflationary Universes and Extended Open Inflationary Universes, seefor example [58–62], where the interior of the bubble is modeled by an open Friedmann-Robertson-Walker universe.The advantage of the EU by Tunneling scheme (and of the Emergent Universe in general),over the Eternal Inflation scheme is that it correspond to a realization of a singularity-free4 U ( ψ ) U F U T FIG. 3: Potential with a false and true vacuum. inflationary universe. As was discussed in Refs. [8–12], Eternal Inflation is usually futureeternal but it is not past eternal, because in general space-time that allows for inflation tobe future eternal, cannot be past null complete. On the other hand Emergent Universe aregeodesically complete.Notice that in the EU by Tunneling scheme, the metastable state which support the initialstatic universe could exist only a finite amount of time. Then, in this scheme of EmergentUniverse, the principal point is not that the universe could have existed an infinite period oftime, but that in theses models the universe is non-singular because the background wherethe bubble materializes is geodesically complete.This implies that we have to consider the problem of the initial conditions for a staticuniverse. Respect to this point, there are very interesting possibilities discussed for examplein the early works on EU [14] and more recently in [39]. One of these options is to explore thepossibility of an Emergent Universe scenario within a string cosmology context [63]. Otherpossibility is that the initial Einstein Static universe is created from ”nothing” [64, 65], seeRefs. [66] for explicit examples. It is interesting to mention that the study of the EinsteinStatic solution as a preferred initial state for our universe have been considered in the past,where it has been proposed that entropy considerations favor the ES state as the initial statefor our universe [44, 45].In this paper we consider a simplified version of this scheme, with the focus on studyingthe process of creation and evolution of a bubble of true vacuum in the background of an ESuniverse in the context of a JBD theory. This is motivated because we are mainly interested5n the study of new ways of leaving the static period and begin the inflationary regime forEmergent Universe models which present a classically stable static state period.In particular, in this paper we consider a JBD theory where one of the matter content ofthe model is a scalar field (inflaton) with a potential similar to Fig. (3) and we study theprocess of tunneling of the scalar field from the false vacuum U F to the true vacuum U T andthe subsequent creation and evolution of a bubble of true vacuum in the background of anstable ES universe. The simplified model studied here contains the essential elements of thescheme we want to present, so we postpone the detailed study of the inflationary period,which occurs after the tunneling, for future work.The paper is organized as follow. In Sect. II we study a Einstein static universe supportedby a scalar field located in a false vacuum and its stability in the context of a JBD theory.In Sect. III we study the tunneling process of the scalar field from the false vacuum to thetrue vacuum and the subsequent creation of a bubble of true vacuum in the background ofthe Einstein static universe for a JBD theory. In Sect. IV we study the evolution of thebubble after its materialization. In Sect. V we summarize our results. II. FALSE VACUUM AND THE ES STATE IN JBD THEORIES
In this paper we consider a scheme for EU scenario where the universe is initially in aclassically stable static state. This state is supported by a scalar field located in a falsevacuum. The universe begins to evolve when, by quantum tunneling, the scalar field decaysto a state of true vacuum. For this reason we will begin by studying the possibility of obtaina static and classically stable ES solution in this theory when the scalar field is in a falsevacuum.We consider the following JBD action for a self-interacting potential and matter, givenby [67] S = (cid:90) d x √− g (cid:20) φ R − ωφ ( ∇ φ ) + V ( φ ) + ( ∇ ψ ) − U ( ψ ) + L m (cid:21) , (1)where R is the Ricci scalar curvature, φ is the JBD field, ω is the JBD parameter, V ( φ ) isthe potential associated to the JBD field, ψ is the scalar field (inflaton), U ( ψ ) is the scalarpotential and L m denotes the Lagrangian density of a barotropic perfect fluid. In this theory1 /φ plays the role of the gravitational constant, which changes with time. This action alsomatches the low energy string action for ω = −
1, see [55].6ollowing the EU scheme we consider a closed Friedmann-Robertson-Walker metric: ds = dt − a ( t ) (cid:20) dt − r + r (cid:0) dθ + sin θdϕ (cid:1)(cid:21) , (2)where a ( t ) is the scale factor and t represents the cosmological time. The content of matter ismodeled by a standard perfect fluid with an effective state equation given by P f = ( γ − ρ f ,with γ constant, and a scalar field (inflaton) for which P ψ = 12 ˙ ψ − U ( ψ ) , ρ ψ = 12 ˙ ψ + U ( ψ ) . (3)The scalar field potential U ( ψ ) es depicted in Fig.(2). The global minimum of U ( ψ ) is U T ,but there is also a local minimum U F (the false vacuum).We have considered that the early universe is dominated by two fluids because in ourscheme of EU scenario, during the static regime, the scalar field ψ remains static at the falsevacuum, in contrast to standard EU models where the scalar field rolls on the asymptoticallyflat part of the scalar potential [22]. Then, in order to obtain a static universe under theseconditions, we have to included a standard perfect fluid. For simplicity we will considerthat there are no interactions between the standard perfect fluid and the scalar field. TheFriedmann-Raychaudhuri equations become H + 1 a + H ˙ φφ = ρ φ + ω (cid:32) ˙ φφ (cid:33) + V φ , (4)2 ¨ aa + H + 1 a + ¨ φφ + 2 H ˙ φφ + ω (cid:32) ˙ φφ (cid:33) − Vφ = − Pφ . (5)The field equation for the JBD field is¨ φ + 3 H ˙ φ = ρ − P ω + 3 + 22 ω + 3 [2 V − φV (cid:48) ] , (6)where V (cid:48) = dV ( φ ) /dφ , ρ = ρ f + ρ ψ , P = P f + P ψ and dots represent derivatives with respectto cosmological time.The conservation equations for the scalar field and perfect standard fluid are¨ ψ + 3 H ˙ ψ = − ∂U ( ψ ) ∂ψ , (7)7nd ˙ ρ f + 3 H ( ρ f + P f ) = 0 , (8)respectively.The static universe is characterized by the conditions a = a = Const . , ˙ a = ¨ a = 0 and φ = φ = Const . , ˙ φ = ¨ φ = 0, ψ = ψ F . From Equations (4), (5) and (6) the static solutionfor a universe dominated by a scalar field placed in a false vacuum and a standard perfectfluid is obtained if the following conditions are satisfied: a = 3 V (cid:48) , (9) ρ f = − U F − V + V (cid:48) φ , (10)and γ = 2 φ a ρ f , (11)where V = V ( φ ) and V (cid:48) = ( dV ( φ ) /dφ ) φ = φ and ρ f is the energy density of the perfectfluid present in the static universe. These equations connect the equilibrium values ofthe scale factor and the JBD field with the energy density and the JBD potential at theequilibrium point. We now study the stability of this solution against small homogeneousand isotropic perturbations. In order to do this, we consider small perturbations around thestatic solutions for the scale factor, JBD field and inflaton field.We set a ( t ) = a [1 + α ( t )] , (12) φ ( t ) = φ [1 + β ( t )] , (13) ψ ( t ) = ψ F [1 + λ ( t )] . (14)Then, we have ρ f ( t ) = ρ f (cid:18) a a ( t ) (cid:19) γ = ρ f (cid:18) a a (1 + α ( t )) (cid:19) γ ≈ ρ f (cid:16) − γ α ( t ) (cid:17) = ρ f + δρ f , (15)where α ( t ) (cid:28) β ( t ) (cid:28) λ ( t ) (cid:28) λ δρ ψ = 2 ˙ ψ F δ ˙ ψ + U (cid:48) F δψ , (16) δP ψ = 2 ˙ ψ F δ ˙ ψ − U (cid:48) F δψ , (17)8here δψ = ψ F λ ( t ) and δ ˙ ψ = ψ ˙ λ ( t ). As in our case (inflaton field in a local minimum ofits potential) ˙ ψ F = 0 and U (cid:48) F = 0 then we obtain that δρ ψ = δP ψ = 0, at linear order. Then,we have that the evolution of λ is determined only by the following equation obtained fromEq. (7) and Eq. (14) ¨ λ + U (cid:48)(cid:48) F λ = 0 , (18)where U (cid:48)(cid:48) F = ( d U ( ψ ) /dψ ) ψ = ψ F . We can note that the evolution of λ is completely decoupledfrom the evolution of α and β and correspond to an oscillatory behavior given that U (cid:48)(cid:48) F > α , β and λ , we obtain the following set of coupled equations¨ α − (cid:20) a + 3 ( γ − a (cid:21) α + ¨ β − βa = 0 , (19)and (3 + 2 ω ) ¨ β − (cid:18) a − φ V (cid:48)(cid:48) (cid:19) β + (4 − γ ) 6 a α = 0 , (20)where V (cid:48)(cid:48) = ( d V ( φ ) /dφ ) φ = φ .From the system of equations (19) and (20) we can obtain the frequencies for small oscilla-tions w ± = 1 a (3 + 2 ω ) (cid:2) a φ V (cid:48)(cid:48) − ω (2 − γ ) ± (cid:113) [ − a φ V (cid:48)(cid:48) + 2 ω − ωγ ] + 2 (3 + 2 ω ) ( − a φ V (cid:48)(cid:48) [3 γ − (cid:21) . (21)Note that the static solution is stable if the inequality w ± > ω satisfies the constraint, (3 + 2 ω ) >
0, we find that thefollowing inequalities must be fulfilled in order to have a stable static solution23 < γ < , γ >
43 (22) − < ω < −
18 ( γ − − γ ) , (23)and 2 (6 + ω ) − ω ) γ + √ | − γ | √ ω < a φ V (cid:48)(cid:48) < γ − . (24)From these inequalities we can conclude that for a universe dominated by a scalar field in afalse vacuum and a perfect standard fluid, it is possible to find a solution where the universeis static and stable. 9 t (cid:45) a (cid:45) a o a o t (cid:45) (cid:45) (cid:45) (cid:70) (cid:45) (cid:70) o (cid:70) o t (cid:45) (cid:45) (cid:45) Ρ (cid:45) Ρ o Ρ o t (cid:45) (cid:45) (cid:45) Ψ (cid:45) Ψ o Ψ o FIG. 4: Behavior of the scale factor a , the JBD field φ , the perfect fluid density ρ and the inflatonfield ψ as a function of cosmological time. In particular, we study numerical solutions to the equations (5), (6), (7) and (8), forinitial conditions close to the static solution. We address the case in which the parameterstake the following values that satisfy the stability conditions (22), (23) and (24) γ = 1; φ = 0 . a = 1; U F = 1; ψ F = 1 , (25)where we have used units in which 8 πG = 1. From the conditions of the static solution (9),(10), (11) and the stability condition (24) we obtain the values for V , V (cid:48) and V (cid:48)(cid:48) .In particular, for the numerical solution, we consider the following inflaton potential U ( ψ ) = U F + ( ψ − ψ F ) , (26)where we have used that around the local equilibrium point ψ ∼ ψ F the inflaton potentialpotential can be approximated to a quadratic function.Also, we consider the following JBD potential which satisfies the static and stabilityconditions discussed above V ( φ ) = V + V (cid:48) ( φ − φ ) + V (cid:48)(cid:48) φ − φ ) . (27)10n Fig. (4) it is shown one numerical solution corresponding to a universe starting from aninitial state not in the static solution but close to it. We can note that while the scalar fieldmakes small oscillations around the false vacuum, the scale factor, JBD field and perfectfluid density present small oscillations around their equilibrium values. This tells us thatthe static solution is classically stable.Then, in a purely classical field theory if the universe is static and supported by the scalarfield located at the false vacuum U F , the universe remains static forever. Quantum mechanicsmakes things different because the scalar field ψ can tunnel through the barrier and by thisprocess create a small bubble where the field value is ψ T . Depending on the backgroundwhere the bubble materializes the bubble could expand or collapse, see Refs. [39, 68, 69]. III. BUBBLE NUCLEATION
In this section we study the tunneling process of the scalar field ψ from the false vacuum U F to the true vacuum U T , in the potential U ( ψ ) shown in Fig. (3) and the consequentcreation of a bubble of true vacuum in the background of an Einstein static universe, in thecontext of a JBD Theory. In particular, we will consider the nucleation of a spherical bubbleof true vacuum within the false vacuum. We will assume that the layer which separates thetwo phases (the wall) has a negligible thickness compared to the size of the bubble (theusual thin-wall approximation). The energy budget of the bubble consists of latent heat(the difference between the energy densities of the two phases) and surface tension.In order to study the tunneling process in this model, we have to deal with the non-standard gravitational interaction of the JBD theory. This was done in Ref. [70] and herewe reproduce and adapt these results to the EU scheme. Regarding the study of bubblenucleation in JBD theories in other contexts see Refs. [71, 72].We begin with the JBD action (1) described in Sect. II and perform a weyl rescaling inorder to remove the non-standard coupling of φ to R . g µν = Ω − ( x )˜ g µν , (28)˜ φ = (cid:114) ω + 38 πG ln ( φ/φ G ) , (29)where φ − G = G , here G is the value of the Newton constant observed in the present. We11hoose the conformal factor Ω( x ) to be Ω = √ πGφ . Then, the action in the rescaled theorycan be expressed as follow S [˜ g, ˜ φ, ψ ] = (cid:90) d x (cid:112) − ˜ g (cid:20) πG ˜ R −
12 ˜ g µν ˜ ∇ µ ˜ φ ˜ ∇ ν ˜ φ + W ( ˜ φ ) + b g µν ∇ µ ψ ∇ ν ψ − b U ( ψ ) + b L m (cid:21) , (30)where we have defined b ≡ exp (cid:32) − (cid:114) πG ω + 3 ˜ φ (cid:33) , (31) W ( ˜ φ ) = V ( φ ( ˜ φ ))(8 π ) G φ = V ( φ ( ˜ φ ))(8 π ) exp (cid:32) − (cid:114) πG ω + 3 ˜ φ (cid:33) . (32)In the rescaled theory the probability of nucleation per unit of physical volume per timeis given by ¯Γ(¯ t ) ≡ dP (¯ t ) d ¯ V (¯ t ) d ¯ t , (33)where P (¯ t ) is the bubble probability of nucleation and ¯ V (¯ t ) is the physical volume measuredin the rescaled system at time ¯ t . Therefore, the equivalent rate in the original theory isgiven by Γ( t ) ≡ dP ( t ) dV ( t ) dt = dP (¯ t ) (cid:0) Ω d ¯ V (cid:1) (Ω d ¯ t ) = Ω − ( t )¯Γ(¯ t ) , (34)where we have used that P ( t ) = P [¯ t ( t )], see Ref. [70].In the rescaled formulation of the theory, the gravitational interaction has the standardform, and so we can expect gravitational effects similar to those of the standard theory.Nevertheless as it is usually, in order to eliminate the problem of predicting the reactionof the geometry to an essentially a-causal quantum jump (the materialization of the bubble),we neglect during this computation the gravitational back-reaction of the bubble onto thespace-time geometry. The gravitational back-reaction of the bubble will be consider in thenext Section when we study the evolution of the bubble after its materialization. Thus, inthis approach only the matter field ψ remains as a dynamical quantity in the action (30).Then, by following Ref.[70] we determinate the rate of nucleation of a bubble of true vacuumin the remaining dynamical theory for the field ψ . The nucleation rate ¯Γ is given by¯Γ( t ) = A exp ( − S E ) (35)where S E is the Euclidean action. 12t is shown in Ref. [70] that the coefficient A has a net factor of b with respect to atheory in which b = 1, this entails that ¯Γ( t ) = b Γ , where Γ is the nucleation rate for anormal scalar field theory with potential U ( ψ ). Since b is a function of the time-dependentJBD field, then we have a time-dependent nucleation rate in the rescaled theory. However,this time dependence disappears if we return to the original theory. The equations (28) and(29), together with the definition of φ G and Ω imply that b = Ω . Using the equation (34)we find that the nucleation rate of the original theory is, see [70]Γ( t ) = Ω − ( t )¯Γ(¯ t ) = (cid:0) b − (cid:1) (cid:0) b Γ (cid:1) = Γ . (36)The computation of Γ in the context of the EU scheme, that is, the tunneling processof the scalar field ψ in a background of Einstein static universe in the context of GeneralRelativity, was developed in Refs. [39–41], where it was found thatΓ ≈ exp (cid:20) − σ π ε (cid:18) − σ ε a (cid:19)(cid:21) , (37)where ε is the difference of energy density between the two phases (latent heat) and σ is theenergy density of the wall. IV. CLASSICAL EVOLUTION OF THE BUBBLE ON JORDAN-BRANS-DICKETHEORIES
In this section we study the evolution of true vacuum bubble after its nucleation viaquantum tunnel effect. During this study we are going to consider the gravitational back-reaction of the bubble.We follow the approach and notation used in Ref. [73] where it is assumed that thebubble wall separates spacetime into two parts. The bubble wall is a timelike, sphericallysymmetric hypersurface, the interior of the bubble is our universe, according to the extendedopen inflation scheme [58–62] and the exterior correspond to the static universe discussed inSec. II. In particular, we will use the scheme developed in Ref. [74] regarding the Darmois-Israel junction conditions [75, 76] applied to a JBD Theory.Let us start by summarize the formalism developed by Berezin, Kuzmin and Tkachev inRef. [73] for the analysis of a thin wall bubble in the context of General Relativity before tostudy this problem in the context of a Jordan-Brans-Dicke-Theory.13e begin by defining a time-like spherically symmetric hypersurface Σ, representing theworld surface of the wall dividing space-time into two regions, V + (outside) and V − (inside).Also, we define a space-like vector N µ , which is orthogonal to Σ and points from V − to V + .It is convenient to introduce a normal Gaussian coordinate system ( n, x i ) such that thehypersurface n = 0 corresponds to Σ. If we assume that the wall is infinitely thin, itssurface energy-momentum tensor is S ij ≡ lim δ → (cid:90) δ − δ T ij dn. (38)Using the extrinsic curvature tensor defined by K ij ≡ N i ; j and the Einstein’s equationswe can express the joint conditions on the wall as follows, see [75][ K ij ] ± = − κ (cid:18) S ij − h ij T r S (cid:19) , (39) − S ji | j = [ T ni ] ± , (40) K i + j + K i − j S ji = [ T nn ] ± , (41)where κ ≡ πG , h ij is the 3-metric on Σ, and | denotes the three-dimensional covariantderivative. We use the brackets to represent the difference between the outer and innervalues of any field variable, for example: [ F ] ± ≡ F + − F − . By using Eq.(39) and Eq.(41)we eliminate K i − j obtaining: K i + j S ji + κ (cid:26) S ij S ji −
12 (
T r S ) (cid:27) = [ T nn ] ± . (42)Since Eq.(40) and Eq.(42) do not contain K i − j , they may be useful when the geometry ofinterior of the bubble is unknown.We are going to assume that the bubble wall Σ is spherically symmetric, then the intrinsicmetric on the shell Σ is given by ds (cid:12)(cid:12) Σ = dτ − R ( τ ) (cid:0) dθ + sin θ dϕ (cid:1) , (43)where R ( τ ) is the radius of circumference of the wall and τ the proper time of the wall. Thesurface energy-momentum tensor S ij can be written as a perfect fluid: S ij = ( σ − ˜ ω ) u i u j + ˜ ωh ij , (44)14here σ , ˜ ω and u i = (1 , ,
0) are the surface energy density, surface pressure and a unittime-like vector tangent to Σ, respectively. Then we can rewrite Eq.(40) and Eq.(42) asfollow dσdτ + 2 dRdτ ( σ + ˜ ω ) R = [ T nτ ] ± , (45) − σK τ + τ + 2˜ ωK θ + θ + κ σ (cid:16) σ ω (cid:17) = [ T nn ] ± . (46)As was discussed in Ref.[73], Eq.(45) and Eq.(46) determine the evolution of σ and R , inthe context of General Relativity, as they give us information on how its radius evolves andabout the energy density that accumulates on the outer side of the wall.In our case, in order to derive the equations of motion of the bubble wall in the Jordan-Brans-Dicke theory we are going to follow a scheme similar to that discussed above, seeRef.[74].From the JBD action Eq. (1) we obtained the following field equations R µν − g µν R = 1 φ ˜ T µν , (47) (cid:3) φ = 12 ω + 3 [ T r T − φV (cid:48) ( φ ) + 4 V ( φ )] , (48)where ˜ T µν ≡ ωφ ∇ µ φ ∇ ν φ − ω φ g µν ( ∇ φ ) + ∇ µ ∇ ν φ − g µν (cid:3) φ + V ( φ ) g µν + T µν , (49)and primes denote derivatives with respect to φ . In this context it is useful to introduce thefollowing surface energy-momentum tensor defined over Σ˜ S ij ≡ lim (cid:15) → (cid:90) (cid:15) − (cid:15) ˜ T ij dn = S ij − (cid:18) ω + 3 (cid:19) h ij T r S , (50)where the second equality is obtained from Eq.(48) and Eq. (49). With these definitionswe can use the formalism described above (for General Relativity) in order to study theevolution of the bubble in the Jordan-Brans-Dicke context, we only have to replace T µν , S ij and κ by ˜ T µν , ˜ S ij y 1 /φ , respectively. The junction condition for the Jordan-Brans-Dickefield φ is obtained from Eq(48), see Ref.[77], in particular we obtain: φ + = φ − , [ φ ,n ] ± = 12 ω + 3 T r S . (51)We note from this condition that in general V + and V − can not be both homogeneous.15ow we rewrite Eq. (39), Eq. (40) and Eq. (42) in the context of a Jordan-Brans-Dicketheory [ K ij ] ± = − φ (cid:18) ˜ S ij − h ij T r ˜ S (cid:19) , (52) − (cid:40) ˜ S ji φ (cid:41) | j = (cid:104) ˜ T ni (cid:105) ± φ , (53) K i + j ˜ S ji + 12 φ (cid:26) ˜ S ij ˜ S ji − (cid:16) T r ˜ S (cid:17) (cid:27) = (cid:104) ˜ T nn (cid:105) ± . (54)By using the equations (49) and (50), we obtain[ K ij ] ± = − φ (cid:18) S ij − h ij (cid:26) − ω + 3 (cid:27) T r S (cid:19) , (55) − S ji | j = [ T ni ] ± , (56) K i + j S ji + 12 φ (cid:18) S ij S ji −
12 (
T rS ) (cid:26) − ω + 3 (cid:27)(cid:19) = [ T nn ] ± . (57)As was mentioned, the exterior of the bubble ( V + ) is described by the metric of a closedFriedmann-Robertson-Walker universe. We write this metric as follows: ds = g + µν dx µ + dx ν + = dt − a ( t + ) (cid:8) dχ + r ( χ + ) (cid:0) dθ + sin θ dϕ (cid:1)(cid:9) , (58)where r ( χ + ) = sin( χ + ).We can note from the conditions (51) that if one of the regions of space-time is homoge-neous the other region is not. As in our case the outer region ( V + ) is a homogeneous ESuniverse, then the region V − is generally inhomogeneous.Substituting Eq. (44) into Eq. (56) and Eq. (57), we get − σK τ + τ + 2˜ ωK θ + θ + 14 φ σ ( σ + 4˜ ω ) + 12 ω + 3 ( − σ + 2˜ ω ) = (cid:104) ˜ T nn (cid:105) ± , (59) dσdτ + dRdτ ( σ + ˜ ω ) R = [ T nτ ] ± . (60)The conditions of continuity for the metric on Σ are obtained from (43) and (58) as R ( τ ) = a ( t + ( τ )) r ( χ + ( τ )) | Σ , dτ = dt − a dχ | Σ . (61)As usual, we consider that the content of matter in the outer region V + and in the innerregion V − is a perfect fluid, with the following energy-momentum tensor T ± µν = (cid:0) p ± + ρ ± (cid:1) U ± µ U ± ν + p ± g ± µν , (62)16here p , ρ and U µ are the pressure, energy density and 4-velocity of the perfect fluid,respectively. Then, by following Ref. [74], we can rewrite Eq.(59) and Eq.(60) as d ( β + v + ) dt + = − β + (cid:26)(cid:18) − ωσ (cid:19) v + H − R drdχ + ˜ ωσ (cid:27) + 14 φ (cid:40) σ + 4˜ ω + 12 ω + 3 ( − σ + 2˜ ω ) σ (cid:41) − [ β ( v ρ + p )] ± σ , (63) dσdt + = − dRdt + ( σ + ˜ ω ) R + [ βv ( ρ + p )] ± , (64) dRdt + = drdχ + v + + HR, (65)where v + ≡ a dχ + dt , β + ≡ dt + dτ ≡ (cid:112) − v , H ≡ da/dt + a . (66)So far we have summarized the calculations of Ref. [74] related to the evolution of a bubble ina Jordan-Brans-Dicke theory in a general background. Now we will concentrate on our par-ticular case, where the bubble materializes and then evolves in a background correspondingto a closed and static universe, as the one described in Sec. II.We assume that the outer region of the bubble is static, that is a ( t ) = a , φ ( t ) = φ , (67)with a and φ constants, defined in Sec.II.In the outer region, as content of matter we consider a scalar field ψ localized in a falsevacuum, and a perfect fluid, see discussion in Sec. II. Then we have ρ + = ρ f + U F . (68) p + = ( γ f − ρ f − U F . (69)In the inner region, the matter content is described by the scalar field in the true vacuum( γ − = 0). Then, in the inner region we have p − = − ρ − = − U T . (70)Now, we develop the derivative of the left hand side of the equation (63) d (cid:18) v + √ − v (cid:19) dt + = v dv + dt + (1 − v ) / + dv + dt + (1 − v ) / = dv + dt + (1 − v ) / . (71)17hen the equation (63) takes the form dv + dt + (1 − v ) / = 1 (cid:112) − v (cid:26) R (˜ γ − drdχ + (cid:27) + 14 φ (cid:26) γσ − σ + σ ω + 3 ( − γ ) (cid:27) + εσ + γ + ρ + (1 − v ) σ , (72)where we have defined ε = ( ρ + − ρ − ) /σ .By using r ( χ + ) = sin( χ + ) and R = a r , we obtain d r ( χ + ) dχ + = cos( χ + ) = (cid:113) − sin ( χ + ) = (cid:113) − ( R/a ) . (73)If we introduce this into the Eq. (65), we have dRdt + = (cid:113) − ( R/a ) v + . (74)We are going to consider that the matter content of the bubble wall is the same as thatof the outer region ( V + ), then we have ˜ γ = γ f .Therefore, we write Eq. (72) as follows dv + dt + (1 − v ) / = 1 (cid:112) − v (cid:26) R ( γ f − (cid:18)(cid:113) − ( R/a ) (cid:19)(cid:27) + 14 φ (cid:26) γ f σ − σ + σ (cid:18) ω + 3 (cid:19) ( − γ f ) (cid:27) + εσ + γ f ρ + (1 − v ) σ . (75)Finally, Eq. (64) can be written as follows dσdt + = − (cid:16) γ f R (cid:17) dRdt + + γ f ρ + v + (cid:112) − v . (76)The evolution of the bubble wall is completely determined, in the outside coordinates,by the equations (74)-(76). We solved these equations numerically by consider differentkind of matter content for the background, satisfying the stability conditions discussed inSec. II. From these numerical solutions we found that once the bubble has materializedin the background of an ES universe in the context of a JBD theory, it grows filling thebackground space.In particular in Figs. (5)-(7) we show three examples. In the first example we consider γ f = 0 . a = 10 , φ = 0 . . (77)18 t R (a) × × × t σ (b) FIG. 5: Evolution of the true vacuum bubble for γ f = 0 .
7. (a) Radius of the bubble as a functionof time t + . (b) Surface energy density as a function of time t + . t R (a) × × × t σ (b) FIG. 6: Evolution of the true vacuum bubble for γ f = 1 .
3. (a) Radius of the bubble as a functionof time t + . (b) Surface energy density as a function of time t + . The results for this case are shown in the Figure (5).In the second example, we consider γ f = 1 . a = 10 , φ = 1 . . (78)The results for this case are shown in Figure (6).In the third example, the matter content of the background is dust ( γ f = 1) and weconsider a = 10 and φ = 1 .
3. The results are shown in Figure (7).In all these examples we have assumed the following initial conditions σ init = 10 − , R init = 10 − , v init = 10 − , (79)and units where 8 πG = 1. 19 t R (a) × × × × t σ (b) FIG. 7: Evolution of the true vacuum bubble for γ f = 1 .
0. (a) Radius of the bubble as a functionof time t + . (b) Surface energy density as a function of time t + . From these examples we can note that the bubble of the new face, once materialized,grows to fill the background space without collapsing.
V. CONCLUSIONS
In this paper we study an alternative scheme for an Emergent Universe scenario calledEmergent Universe by tunneling. In this scheme the universe is initially in a truly staticstate supported by a scalar field which is located in a false vacuum. The universe begins toevolve when, by quantum tunneling, the scalar field decays into a state of true vacuum.The EU by tunneling scheme was originally developed in Ref. [40], in the context ofGeneral Relativity, where it was concluded that this mechanism is feasible as an EU scheme.Nevertheless, this first model present the problem that the ES solution is classically instable.The instability of the ES solution ensures that any perturbation, no matter how small,rapidly force the universe away from the static state, thereby aborting the EU scenario.The present work is the natural extension of the idea presented in Ref. [40], but wherethe problem of the classical instability of the static solution is solved by going away fromGeneral Relativity and consider a JBD theory.In particular, in this work we focus our study on the process of tunneling of a scalar fieldand the consequent creation and evolution of a bubble of true vacuum in the backgroundof a classically stable Einstein Static universe. Our principal motivation is the study ofnew ways of leaving the static period and begin the inflationary regime in the context ofEmergent Universe models. 20n the first part of the paper, Sect. II, we study an Einstein static universe supportedby a scalar field located in a false vacuum and its stability in the context of a JBD theory.Contrary to General Relativity, we found that this static solution could be stable againstisotropic perturbations if some general conditions are satisfied, see Eqs. (22)-(24). Thismodification of the stability behavior has important consequences for the emergent universeby tunneling scenario, since it ameliorates the fine-tuning that arises from the fact that theES model is an unstable saddle in GR and it improves the preliminary model studied inRef. [39]. In this study, for simplicity, we have not considered inhomogeneous or anisotropicperturbations. At this respect, the stability of the ES solution under anisotropic, tensorand inhomogeneous scalar perturbations have been studied in the context of JBD theoriesin Refs. [22, 47]. It was found for theses JBD models that different from General Relativity[46] and others modified theories of gravity as f ( R ) [79] or modified Gauss-Bonnet gravity[80], that a static universe which is stable against homogeneous perturbations, could be alsostable against anisotropic and inhomogeneous perturbations. We expect a similar behaviorfor our JBD model, were the static universe is supported by a scalar field located in a falsevacuum. Then, we expect that in our case the inhomogeneous and anisotropic perturbationsdo not lead to additional instabilities. Nevertheless, we intend to return to these points inthe near future by working an approach similar to that followed in Refs. [22, 46, 47, 78].In Sect. III we study the tunneling process of the scalar field from the false vacuum tothe true vacuum and the consequent creation of a bubble of true vacuum in the backgroundof Einstein static universe for a JBD theory. In particular we determinate the nucleationrate of the true vacuum bubble using the approaches developed in Ref. [70] and previousresults obtained in Ref. [39].The classical evolution of the bubble after its nucleation is studied in Sect. IV where wefound that once the bubble has materialized in the background of an ES universe, it growsfilling the background space. This demonstrates the viability of our EU model, since there isthe possibility of having an open inflationary universe inside the bubble. During this studywe consider the gravitational back-reaction of the bubble by using the formalism developed inRef. [74] applied to a JBD theory. At this respect we found a system of coupled differentialequations, which we solved numerically. Three specific examples of these solutions wereshown in Sect. IV concerning to different background material contents.It is worth to note that once the bubble has materialized, from conditions (51), it follows21hat if one of the regions of spacetime separated by the wall is homogeneous, then theother region is, in general, inhomogeneous [74]. Given that in our case the exterior ofthe bubble is a homogeneous universe, then the interior of the bubble will be, in general,inhomogeneous. However, since the degree of inhomogeneity depends on the difference inthe energy density of the interior and the exterior of the bubble, it is possible in our caseto decrease this inhomogeneity by adjusting the parameters of the static solution as wasdiscussed in Ref. [74]. Then in our model, it is possible to study the feasibility of having anopen inflationary universe inside the bubble. Nevertheless, given the similarities, we expectthat the behavior inside the bubble of the EU by tunneling, will be similar to the modelsof single-field open and extended open inflation, as the ones studied in Refs. [58–62]. Weexpect to return to this point in the near future. VI. ACKNOWLEDGEMENTS
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