# Emergent universe by tunneling in a Jordan-Brans-Dicke theory

## Abstract

In this work we study an alternative scheme for an Emergent Universe scenario in the context of a Jordan-Brans-Dicke theory, where the universe is initially in a truly static state supported by a scalar field located in a false vacuum. The model presents a classically stable past eternal static state which is broken when, by quantum tunneling, the scalar field decays into a state of true vacuum and the universe begins to evolve following the extended open inflationary scheme.

## 1 Introduction

The standard cosmological model (SCM) [1, 2, 3, 4, 5, 6] and the inflationary paradigm [7, 8, 9, 10] are shown as a satisfactory description of our universe [1, 2, 3, 4, 5, 6]. However, despite its great success, there are still important open questions to be answered. One of these questions is whether the 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 inflationary universes, proving that the universe necessarily has a beginning. In other words, according to these theorems, the existence of an initial singularity can not be avoided even if the inflationary period occurs, see Refs. [11, 12, 13, 14, 15]. In theses theorems it is demonstrated that null and time-like geodesics are generally incomplete in inflationary models, regardless of whether energy conditions are maintained, provided that the average expansion condition (\(H> 0\)) is maintained throughout of these geodesics directed towards the past, where *H* is the Hubble parameter.

The search for cosmological models without initial singularities has led to the development of the so-called Emergent Universes models (EU) [16, 17, 18, 19, 20, 21, 22, 23].

In the EU scheme it is assumed that the universe emerged from a past eternal Einstein Static (ES) state to the inflationary phase and then evolves into a hot big bang era. These models do not satisfy the geometrical assumptions of the theorems [11, 12, 13, 14, 15] and they provide specific examples of non-singular 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 the scalar potential (see Fig. 1) with a constant velocity, providing the conditions for a static universe, see for example models [16, 17, 18, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. Other possibility is to consider EU models in which the scale factor only asymptotically tends to a constant in the past [19, 20, 34, 35, 36, 37, 38, 39, 40, 41]. We can note that in these schemes of Emergent Universe are not all truly static during the static regime.

At this respect a new scheme for an EU model was proposed in Ref. [42], where the universe is initially in a truly static state supported by a scalar field located in a false vacuum, also see Refs. [43, 44]. 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 to this new scheme of EU, the model was developed in the context of General Relativity (GR). In particular in Ref. [42] was concluded that this new mechanism for an Emergent Universe is plausible and could be an interesting alternative to the realization of the Emergent Universe scenario.

The ES solution is unstable to homogeneous perturbations, as was early discussed by Eddington in Ref. [45] and more recently studied in Refs. [46, 47, 48, 49]. The instability of the ES solution ensures that any perturbations, no matter how small, rapidly force the universe away from the static state, thereby aborting the EU scenario.

This instability is possible to cure by going away from GR, for example, by consider a Jordan-Brans-Dicke (JBD) theory at the classical level, where it have been found that contrary to general relativity, a static universe could be classically stable, see Refs. [24, 25, 50].

In this work, we are interested in apply the scheme of Emergent Universe by Tunneling of Ref. [42] to EU models which present stable past eternal static regimes. In particular, we study this scheme in the context of a JBD theory, similar to the one studied in Refs. [24, 25], but where the static solution is supported by a scalar field located in a false vacuum. In this case we are going to show that, contrary to what happens in Ref. [42], the ES solution is classically stable.

The Jordan-Brans-Dicke [51, 52] theory is a class of models in which the effective gravitational coupling 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 \(G^{-1}\), the inverse of the Newton’s constant. The origin of Brans-Dicke theory is found in the Mach’s principle according to which the property of inertia of material bodies arises from their interactions with the matter distributed in the universe. In modern context, Brans-Dicke theory appears naturally in supergravity models, Kaluza-Klein theories and in all known effective string actions [53, 54, 55, 56, 57, 58, 59].

In particular in this work we are going to consider that the universe is initially in a truly static state, which is supported by a scalar field \(\psi \) located in a false vacuum (\(\psi = \psi _F\)), see Fig. 2. The universe begins to evolve when, by quantum tunneling, the scalar field decays into a state of true vacuum. Then, a small bubble of a new phase of field value \(\psi _W\) can be formed, and expands as it converts volume from high to low vacuum energy and feeds the liberated energy into the kinetic energy of the bubble wall. This process was first studied by Coleman and De Luccia in [60, 61] in the context of General Relativity.

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-free inflationary universe. As was discussed in Refs. [11, 12, 13, 14, 15], Eternal Inflation is usually future eternal but it is not past eternal, because in general space-time that allows for inflation to be future eternal, cannot be past null complete. On the other hand Emergent Universe are geodesically complete.

Notice that in the EU by Tunneling scheme, the metastable state which support the initial static universe could exist only a finite amount of time. Then, in this scheme of Emergent Universe, the principal point is not that the universe could have existed an infinite period of time, but that in theses models the universe is non-singular because the background where the bubble materializes is geodesically complete.

This implies that we have to consider the problem of the initial conditions for a static universe. Respect to this point, there are very interesting possibilities discussed for example in the early works on EU [17] and more recently in [42]. One of these options is to explore the possibility of an Emergent Universe scenario within a string cosmology context [67]. Other possibility is that the initial Einstein Static universe is created from “nothing” [68, 69], see Refs. [70, 71, 72] for explicit examples. It is interesting to mention that the study of the Einstein Static 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 state for our universe [47, 48].

In this paper we consider a simplified version of this scheme, with the focus on studying the process of creation and evolution of a bubble of true vacuum in the background of an ES universe in the context of a JBD theory. This is motivated because we are mainly interested in the study of new ways of leaving the static period and begin the inflationary regime for Emergent 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 of the model is a scalar field (inflaton) with a potential similar to Fig. 3 and we study the process of tunneling of the scalar field from the false vacuum \(U_F\) to the true vacuum \(U_T\) and the subsequent creation and evolution of a bubble of true vacuum in the background of an stable ES universe. The simplified model studied here contains the essential elements of the scheme 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. 2 we study a Einstein static universe supported by a scalar field located in a false vacuum and its stability in the context of a JBD theory. In Sect. 3 we study the tunneling process of the scalar field from the false vacuum to the true vacuum and the subsequent creation of a bubble of true vacuum in the background of the Einstein static universe for a JBD theory. In Sect. 4 we study the evolution of the bubble after its materialization. In Sect. 5 we summarize our results.

## 2 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 a classically stable static state. This state is supported by a scalar field located in a false vacuum. The universe begins to evolve when, by quantum tunneling, the scalar field decays to a state of true vacuum. For this reason we will begin by studying the possibility of obtain a static and classically stable ES solution in this theory when the scalar field is in a false vacuum.

*a*(

*t*) is the scale factor and

*t*represents the cosmological time. The content of matter is modeled by a standard perfect fluid with an effective state equation given by \(P_f=\left( \gamma -1\right) \rho _f\), with \(\gamma \) constant, and a scalar field (inflaton) for which

Then, in a purely classical field theory if the universe is static and supported by the scalar field located at the false vacuum \(U_F\), the universe remains static forever. Quantum mechanics makes things different because the scalar field \(\psi \) can tunnel through the barrier and by this process create a small bubble where the field value is \(\psi _T\). Depending on the background where the bubble materializes the bubble could expand or collapse, see Refs. [42, 74, 75].

## 3 Bubble nucleation

In this section we study the tunneling process of the scalar field \(\psi \) from the false vacuum \(U_F\) to the true vacuum \(U_T\), in the potential \(U(\psi )\) shown in Fig. 3 and the consequent creation of a bubble of true vacuum in the background of an Einstein static universe, in the context of a JBD Theory. In particular, we will consider the nucleation of a spherical bubble of true vacuum within the false vacuum. We will assume that the layer which separates the two phases (the wall) has a negligible thickness compared to the size of the bubble (the usual 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. [76] and here we reproduce and adapt these results to the EU scheme. Regarding the study of bubble nucleation in JBD theories in other contexts see Refs. [77, 78].

*G*is the value of the Newton constant observed in the present. We choose the conformal factor \(\Omega (x)\) to be \(\Omega = \sqrt{8\pi G \phi } \). Then, the action in the rescaled theory can be expressed as follow

In the rescaled formulation of the theory, the gravitational interaction has the standard form, and so we can expect gravitational effects similar to those of the standard theory.

*A*has a net factor of \(b^2\) with respect to a theory in which \(b = 1\), this entails that \(\bar{\Gamma }(t)=b^2 \Gamma _0\), where \( \Gamma _0 \) is the nucleation rate for a normal scalar field theory with potential \( U (\psi ) \). Since

*b*is a function of the time-dependent JBD 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 Eqs. (28) and (29), together with the definition of \(\phi _G \) and \(\Omega \) imply that \( b = \Omega ^2 \). Using the Eq. (34) we find that the nucleation rate of the original theory is, see [76]

## 4 Classical evolution of the bubble on Jordan-Brans-Dicke theories

In this section we study the evolution of true vacuum bubble after its nucleation via quantum 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. [79] where it is assumed that the bubble wall separates spacetime into two parts. The bubble wall is a timelike, spherically symmetric hypersurface, the interior of the bubble is our universe, according to the extended open inflation scheme [62, 63, 64, 65, 66] and the exterior correspond to the static universe discussed in Sect. 2. In particular, we will use the scheme developed in Ref. [80] regarding the Darmois-Israel junction conditions [81, 82] applied to a JBD Theory.

Let us start by summarize the formalism developed by Berezin, Kuzmin and Tkachev in Ref. [79] for the analysis of a thin wall bubble in the context of General Relativity before to study this problem in the context of a Jordan-Brans-Dicke-Theory.

*R*, in the context of General Relativity, as they give us information on how its radius evolves and about 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, see Ref. [80].

We can note from the conditions (51) that if one of the regions of space-time is homogeneous the other region is not. As in our case the outer region (\(V^{+}\)) is a homogeneous ES universe, then the region \(V^{-}\) is generally inhomogeneous.

*p*, \( \rho \) and \(U_{\mu }\) are the pressure, energy density and 4-velocity of the perfect fluid, respectively. Then, by following Ref. [80], we can rewrite Eqs. (59) and (60) as

In the third example, the matter content of the background is dust (\(\gamma _f = 1\)) and we consider \(a_0=10\) and \(\phi _0 =1.3\). The results are shown in Fig. 7.

From these examples we can note that the bubble of the new face, once materialized, grows to fill the background space without collapsing.

## 5 Conclusions

In this paper we study an alternative scheme for an Emergent Universe scenario called Emergent Universe by tunneling. In this scheme the universe is initially in a truly static state supported by a scalar field which is located in a false vacuum. The universe begins to evolve when, by quantum tunneling, the scalar field decays into a state of true vacuum.

The EU by tunneling scheme was originally developed in Ref. [43], in the context of General 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. [43], but where the problem of the classical instability of the static solution is solved by going away from General Relativity and consider a JBD theory.

In particular, in this work we focus our study on the process of tunneling of a scalar field and the consequent creation and evolution of a bubble of true vacuum in the background of a classically stable Einstein Static universe. Our principal motivation is the study of new ways of leaving the static period and begin the inflationary regime in the context of Emergent Universe models.

In the first part of the paper, Sect. 2, we study an Einstein static universe supported by 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 against isotropic perturbations if some general conditions are satisfied, see Eqs. (22)–(24). This modification of the stability behavior has important consequences for the emergent universe by tunneling scenario, since it ameliorates the fine-tuning that arises from the fact that the ES model is an unstable saddle in GR and it improves the preliminary model studied in Ref. [42]. In this study, for simplicity, we have not considered inhomogeneous or anisotropic perturbations. At this respect, the stability of the ES solution under anisotropic, tensor and inhomogeneous scalar perturbations have been studied in the context of JBD theories in Refs. [25, 50]. It was found for theses JBD models that different from General Relativity [49] and others modified theories of gravity as *f*(*R*) [88] or modified Gauss-Bonnet gravity [89], that a static universe which is stable against homogeneous perturbations, could be also stable against anisotropic and inhomogeneous perturbations. We expect a similar behavior for our JBD model, were the static universe is supported by a scalar field located in a false vacuum. Then, we expect that in our case the inhomogeneous and anisotropic perturbations do not lead to additional instabilities. Nevertheless, we intend to return to these points in the near future by working an approach similar to that followed in Refs. [25, 49, 50, 84, 85, 86, 87].

In Sect. 3 we study the tunneling process of the scalar field from the false vacuum to the true vacuum and the consequent creation of a bubble of true vacuum in the background of Einstein static universe for a JBD theory. In particular we determinate the nucleation rate of the true vacuum bubble using the approaches developed in Ref. [76] and previous results obtained in Ref. [42].

The classical evolution of the bubble after its nucleation is studied in Sect. 4 where we found that once the bubble has materialized in the background of an ES universe, it grows filling the background space. This demonstrates the viability of our EU model, since there is the possibility of having an open inflationary universe inside the bubble. During this study we consider the gravitational back-reaction of the bubble by using the formalism developed in Ref. [80] applied to a JBD theory. At this respect we found a system of coupled differential equations, which we solved numerically. Three specific examples of these solutions were shown in Sect. 4 concerning to different background material contents.

It is worth to note that once the bubble has materialized, from conditions (51), it follows that if one of the regions of spacetime separated by the wall is homogeneous, then the other region is, in general, inhomogeneous [80]. Given that in our case the exterior of the 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 in the energy density of the interior and the exterior of the bubble, it is possible in our case to decrease this inhomogeneity by adjusting the parameters of the static solution as was discussed in Ref. [80]. Then in our model, it is possible to study the feasibility of having an open inflationary universe inside the bubble. Nevertheless, given the similarities, we expect that the behavior inside the bubble of the EU by tunneling, will be similar to the models of single-field open and extended open inflation, as the ones studied in Refs. [62, 63, 64, 65, 66]. We expect to return to this point in the near future.

## Notes

### Acknowledgements

P. L. is supported by Dirección de Investigación de la Universidad del Bío-Bío through Grants no. 182707 4/R, and GI 172309/C. H. C. was supported by Dirección de Postgrado de la Universidad del Bío-Bío and by Research Assistant Grant of Escuela de Graduados Universidad del Bío-Bío.

## References

- 1.S. Weinberg,
*Gravitation and Cosmology: Principle and Application of the General Relativity*(Wiley, New York, 1972)Google Scholar - 2.ChW Misner, K.S. Turner, J.A. Wheeler,
*Gravitation*(W. H.: Freeman and Company, San Francisco, 1973)Google Scholar - 3.P.J.E. Peebles,
*Principles of Physical Cosmology*(Princeton University Press, Princeton, 1993)Google Scholar - 4.J.A. Peacock,
*Cosmological Physics*(Cambridge University Press, Cambridge, 1998)CrossRefGoogle Scholar - 5.S. Weinberg,
*Cosmology*(Oxford University Press, Oxford, 2008)zbMATHGoogle Scholar - 6.E. Kolb, M. Turner,
*The Early Universe*(Addison-Wesley Publishing, Boston, 1989)Google Scholar - 7.A. Guth, Inflationary universe: a possible solution to the horizon and flatness problems. Phys. Rev. D
**23**, 347 (1981)ADSCrossRefGoogle Scholar - 8.A. Albrecht, P.J. Steinhardt, Cosmology for grand unified theories with radiatively induced symmetry breaking. Phys. Rev. Lett.
**48**, 1220 (1982)ADSCrossRefGoogle Scholar - 9.A.D. Linde, A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems. Phys. Lett.
**108B**, 389 (1982)ADSCrossRefGoogle Scholar - 10.A.D. Linde, Chaotic inflation. Phys. Lett.
**129B**, 177 (1983)ADSCrossRefGoogle Scholar - 11.A. Borde, A. Vilenkin, Eternal inflation and the initial singularity. Phys. Rev. Lett.
**72**, 3305 (1994)ADSCrossRefGoogle Scholar - 12.A. Borde, A. Vilenkin, Violation of the weak energy condition in inflating spacetimes. Phys. Rev. D
**56**, 717 (1997)ADSMathSciNetCrossRefGoogle Scholar - 13.A. Guth, Eternal inflation. Annals N. Y. Acad. Sci.
**950**, 66 (2001). ArXiv:astro-ph/0101507 ADSCrossRefGoogle Scholar - 14.A. Borde, A.H. Guth, A. Vilenkin, Inflationary space-times are incompletein past directions. Phys. Rev. Lett.
**90**, 151301 (2003)ADSCrossRefGoogle Scholar - 15.Vilenkin A., Quantum cosmology and eternal inflation. arXiv:gr-qc/0204061 (2002)
- 16.G.F.R. Ellis, R. Maartens, The emergent universe: inflationary cosmology with no singularity. Class. Quant. Grav.
**21**, 223 (2004)ADSMathSciNetCrossRefGoogle Scholar - 17.G.F.R. Ellis, J. Murugan, C.G. Tsagas, The emergent universe: an explicit construction. Class. Quant. Grav.
**21**, 233 (2004)ADSMathSciNetCrossRefGoogle Scholar - 18.D.J. Mulryne, R. Tavakol, J.E. Lidsey, G.F.R. Ellis, An emergent universe from a loop. Phys. Rev. D
**71**, 123512 (2005)ADSCrossRefGoogle Scholar - 19.S. Mukherjee, B.C. Paul, S.D. Maharaj, A. Beesham, Emergent universe in Starobinsky model, arXiv:gr-qc/0505103 (2005)
- 20.S. Mukherjee, B.C. Paul, N.K. Dadhich, S.D. Maharaj, A. Beesham, Emergent universe with exotic matter. Class. Quant. Grav.
**23**, 6927 (2006)ADSMathSciNetCrossRefGoogle Scholar - 21.A. Banerjee, T. Bandyopadhyay, S. Chakraborty, Emergent universe in brane world scenario. Grav. Cosmol.
**13**, 290 (2007)ADSzbMATHGoogle Scholar - 22.N.J. Nunes, Inflation: a graceful entrance from loop quantum cosmology. Phys. Rev. D
**72**, 103510 (2005)ADSCrossRefGoogle Scholar - 23.J.E. Lidsey, D.J. Mulryne, A graceful entrance to braneworld inflation. Phys. Rev. D
**73**, 083508 (2006)ADSMathSciNetCrossRefGoogle Scholar - 24.S. del Campo, R. Herrera, P. Labrana, Emergent universe in a Jordan-Brans-Dicke theory. JCAP
**0711**, 030 (2007). arXiv:0711.1559 [gr-qc]CrossRefGoogle Scholar - 25.S. del Campo, R. Herrera, P. Labrana, On the stability of Jordan-Brans-Dicke static universe. JCAP
**0907**, 006 (2009). arXiv:0905.0614 [gr-qc]CrossRefGoogle Scholar - 26.S. del Campo, E. Guendelman, R. Herrera, P. Labrana, Emerging universe from scale invariance. JCAP
**1006**, 026 (2010). arXiv:1006.5734 [astro-ph.CO]CrossRefGoogle Scholar - 27.S. del Campo, E.I. Guendelman, A.B. Kaganovich, R. Herrera, P. Labrana, Emergent universe from scale invariant two measures theory. Phys. Lett. B
**699**, 211–216 (2011). arXiv:1105.0651 [astro-ph.CO]ADSCrossRefGoogle Scholar - 28.E.I. Guendelman, Non singular origin of the universe and its present vacuum energy density. Int. J. Mod. Phys. A
**26**:2951–2972. arXiv:1103.1427 [gr-qc] (2011) - 29.E.I. Guendelman, Non singular origin of the universe and the cosmological constant problem (CCP). Int. J. Mod. Phys. D
**20**2767. arXiv:1105.3312 [gr-qc] (2011) - 30.E.I. Guendelman, P. Labrana, Int. J. Mod. Phys. D
**22**, 1330018 (2013). arXiv:1303.7267 [astro-ph.CO]ADSCrossRefGoogle Scholar - 31.E. Guendelman, R. Herrera, P. Labrana, E. Nissimov, S. Pacheva, Gen. Rel. Grav.
**47**(2), 10 (2015). arXiv:1408.5344 [gr-qc]ADSCrossRefGoogle Scholar - 32.E. Guendelman, R. Herrera, P. Labrana, E. Nissimov, S. Pacheva, Astron. Nachr.
**336**(8/9), 810 (2015). arXiv:1507.08878 [hep-th]ADSCrossRefGoogle Scholar - 33.S. del Campo, E.I. Guendelman, R. Herrera, P. Labrana, JCAP
**1608**, 049 (2016). arXiv:1508.03330 [gr-qc]CrossRefGoogle Scholar - 34.A. Banerjee, T. Bandyopadhyay, S. Chakraborty, Gen. Rel. Grav.
**40**, 1603–1607 (2008). arXiv:0711.4188 [gr-qc]ADSCrossRefGoogle Scholar - 35.
- 36.B.C. Paul, S. Ghose, Gen. Rel. Grav.
**42**, 795–812 (2010). arXiv:0809.4131 [hep-th]ADSCrossRefGoogle Scholar - 37.A. Beesham, S.V. Chervon, S.D. Maharaj, Class. Quant. Grav.
**26**, 075017 (2009). arXiv:0904.0773 [gr-qc]ADSCrossRefGoogle Scholar - 38.U. Debnath, S. Chakraborty, Int. J. Theor. Phys.
**50**, 2892–2898 (2011). arXiv:1104.1673 [gr-qc]CrossRefGoogle Scholar - 39.S. Mukerji, N. Mazumder, R. Biswas, S. Chakraborty, Int. J. Theor. Phys.
**50**, 2708–2719 (2011). arXiv:1106.1743 [gr-qc]CrossRefGoogle Scholar - 40.P. Labrana, Phys. Rev. D
**91**(8), 083534 (2015). arXiv:1312.6877 [astro-ph.CO]ADSCrossRefGoogle Scholar - 41.Q. Huang, P. Wu, H. Yu, Phys. Rev. D
**91**(10), 103502 (2015)ADSMathSciNetCrossRefGoogle Scholar - 42.P. Labrana, Emergent universe by tunneling. Phys. Rev. D
**86**, 083524 (2012). arXiv:1111.5360 [gr-qc]ADSCrossRefGoogle Scholar - 43.P. Labrana, Tunneling and the emergent universe scheme. Astrophys. Space Sci. Proc.
**38**, 95 (2014)ADSCrossRefGoogle Scholar - 44.P. Labrana, The emergent universe scheme and tunneling. AIP Conf. Proc.
**1606**, 38 (2014). arXiv:1406.0922 [astro-ph.CO]ADSCrossRefGoogle Scholar - 45.A.S. Eddington, Mon. Not. R. Astron. Soc.
**90**, 668 (1930)ADSCrossRefGoogle Scholar - 46.E.R. Harrison, Rev. Mod. Phys.
**39**, 862 (1967)ADSCrossRefGoogle Scholar - 47.G.W. Gibbons, The entropy and stability of the universe. Nucl. Phys. B
**292**, 784 (1987)ADSMathSciNetCrossRefGoogle Scholar - 48.G.W. Gibbons, Sobolev’s inequality, Jensen’s theorem and the mass and entropy of the universe. Nucl. Phys. B
**310**, 636 (1988)ADSMathSciNetCrossRefGoogle Scholar - 49.J.D. Barrow, G.F.R. Ellis, R. Maartens, C.G. Tsagas, On the stability of the Einstein static universe. Class. Quant. Grav.
**20**, L155 (2003). arXiv:gr-qc/0302094 ADSMathSciNetCrossRefGoogle Scholar - 50.H. Huang, P. Wu, H. Yu, Phys. Rev. D
**89**(10), 103521 (2014)ADSCrossRefGoogle Scholar - 51.P. Jordan, The present state of Dirac’s cosmological hypothesis. Z. Phys.
**157**, 112 (1959)ADSCrossRefGoogle Scholar - 52.C. Brans, R.H. Dicke, Mach’s principle and a relativistic theory of gravitation. Phys. Rev.
**124**, 925 (1961)ADSMathSciNetCrossRefGoogle Scholar - 53.P.G.O. Freund, Kaluza-Klein cosmologies. Nucl. Phys. B
**209**, 146 (1982)ADSMathSciNetCrossRefGoogle Scholar - 54.T. Appelquist, A. Chodos, P .G .O. Freund,
*Modern Kaluza-Klein theories*(Addison-Wesley, Redwood City, 1987)zbMATHGoogle Scholar - 55.E.S. Fradkin, A.A. Tseytlin, Effective field theory from quantized strings. Phys. Lett. B
**158**, 316 (1985)ADSMathSciNetCrossRefGoogle Scholar - 56.E.S. Fradkin, A.A. Tseytlin, Quantum string theory effective action. Nucl. Phys. B
**261**, 1 (1985)ADSMathSciNetCrossRefGoogle Scholar - 57.C.G. Callan, E.J. Martinec, M.J. Perry, D. Friedan, Strings in background fields. Nucl. Phys. B
**262**, 593 (1985)ADSMathSciNetCrossRefGoogle Scholar - 58.G. CallanC, I.R. Klebanov, M.J. Perry, String theory effective actions. Nucl. Phys. B
**278**, 78 (1986)ADSMathSciNetCrossRefGoogle Scholar - 59.M.B. Green, J.H. Schwarz, E. Witten,
*Superstring Theory*(Cambridge, UK: Univ. Pr., Cambridge Monographs On Mathematical Physics, 1987)Google Scholar - 60.S.R. Coleman, The Fate of the False Vacuum 1. Semiclass. Theory. Phys. Rev. D
**15**, 2929 (1977). Erratum: [Phys. Rev. D**16**, 1248 (1977)]ADSGoogle Scholar - 61.S.R. Coleman, F. De Luccia, Gravitational effects on and of vacuum decay. Phys. Rev. D
**21**, 3305 (1980)ADSMathSciNetCrossRefGoogle Scholar - 62.A. Linde, Phys. Rev. D
**59**, 023503 (1998)ADSCrossRefGoogle Scholar - 63.A. Linde, M. Sasaki, T. Tanaka, Phys. Rev. D
**59**, 123522 (1999)ADSCrossRefGoogle Scholar - 64.S. del Campo, R. Herrera, Phys. Rev. D
**67**, 063507 (2003)ADSMathSciNetCrossRefGoogle Scholar - 65.S. del Campo, R. Herrera, J. Saavedra, Phys. Rev. D
**70**, 023507 (2004)ADSCrossRefGoogle Scholar - 66.L. Balart, S. del Campo, R. Herrera, P. Labrana, J. Saavedra, Tachyonic open inflationary universes. Phys. Lett. B
**647**, 313–319 (2007)ADSCrossRefGoogle Scholar - 67.I. Antoniadis, C. Bachas, J. Ellis, D.V. Nanopolous, Phys. Lett. B
**211**, 4 (1988)CrossRefGoogle Scholar - 68.E.P. Tryon, Nature (Lond.)
**246**, 396 (1973)ADSCrossRefGoogle Scholar - 69.A. Vilenkin, Phys. Rev. D
**32**, 10 (1985)MathSciNetCrossRefGoogle Scholar - 70.A.T. Mithani, A. Vilenkin, JCAP
**1201**, 028 (2012)ADSCrossRefGoogle Scholar - 71.A.T. Mithani, A. Vilenkin, JCAP
**1405**, 006 (2014)ADSCrossRefGoogle Scholar - 72.A.T. Mithani, A. Vilenkin, JCAP
**1507**(07), 010 (2015)ADSCrossRefGoogle Scholar - 73.M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory Vol. 1: 25th Anniversary Edition. (2018). https://doi.org/10.1017/CBO9781139248563
- 74.D. Simon, J. Adamek, A. Rakic, J.C. Niemeyer, JCAP
**0911**, 008 (2009). arXiv:0908.2757 [gr-qc]ADSCrossRefGoogle Scholar - 75.W. Fischler, S. Paban, M. Zanic, C. Krishnan, JHEP
**0805**, 041 (2008). arXiv:0711.3417 [hep-th]ADSCrossRefGoogle Scholar - 76.R. Holman, E.W. Kolb, S.L. Vadas, Y. Wang, E.J. Weinberg, False vacuum decay in Jordan-brans-dicke cosmologies. Phys. Lett. B
**237**, 37 (1990)ADSCrossRefGoogle Scholar - 77.B.H. Lee, W. Lee, Class. Quant. Grav.
**26**, 225002 (2009). https://doi.org/10.1088/0264-9381/26/22/225002. arXiv:0809.4907 [hep-th]ADSCrossRefGoogle Scholar - 78.H. Kim, B.H. Lee, W. Lee, Y.J. Lee, D.H. Yeom, Phys. Rev. D
**84**, 023519 (2011). https://doi.org/10.1103/PhysRevD.84.023519. arXiv:1011.5981 [hep-th]ADSCrossRefGoogle Scholar - 79.V.A. Berezin, V.A. Kuzmin, I.I. Tkachev, Phys. Rev. D
**36**, 2919 (1987)ADSMathSciNetCrossRefGoogle Scholar - 80.N. Sakai, K i Maeda, Bubble dynamics in generalized Einstein theories. Prog. Theor. Phys.
**90**, 1001 (1993)ADSCrossRefGoogle Scholar - 81.W. Israel, Singular hypersurfaces and thin shells in general relativity. Nuovo Cim. B
**44S10**, 1 (1966) [Nuovo Cim. B**44**, 1 (1966)] Erratum: [Nuovo Cim. B**48**, 463 (1967)]Google Scholar - 82.G. Darmois, Memorial des Sciences Mathematiques. Fasc. 25, Gauthier-Villars (1927)Google Scholar
- 83.K.G. Suffern, J. Phys.
**A15**, 1599 (1982)ADSMathSciNetGoogle Scholar - 84.M. Bruni, P.K.S. Dunsby, G.F.R. Ellis, Astrophys. J.
**395**, 34 (1992)ADSCrossRefGoogle Scholar - 85.P.K.S. Dunsby, M. Bruni, G.F.R. Ellis, Astrophys. J.
**395**, 54 (1992)ADSCrossRefGoogle Scholar - 86.M. Bruni, G.F.R. Ellis, P.K.S. Dunsby, Class. Quant. Grav.
**9**, 921 (1992)ADSCrossRefGoogle Scholar - 87.P.K.S. Dunsby, B.A.C.C. Bassett, G.F.R. Ellis, Class. Quant. Grav.
**14**, 1215 (1997)ADSCrossRefGoogle Scholar - 88.S.S. Seahra, C.G. Boehmer, Phys. Rev. D
**79**, 064009 (2009)ADSMathSciNetCrossRefGoogle Scholar - 89.H. Huang, P. Wu, H. Yu, Phys. Rev. D
**91**(2), 023507 (2015)ADSMathSciNetCrossRefGoogle Scholar

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