Connecting The Non-Singular Origin of the Universe, The Vacuum Structure and The Cosmological Constant Problem
aa r X i v : . [ a s t r o - ph . C O ] M a y Connecting The Non-Singular Origin of the Universe, TheVacuum Structure and The Cosmological Constant Problem
Eduardo I. Guendelman ∗ Physics Department, Ben Gurion University of the Negev, Beer Sheva 84105, Israel
Pedro Labra˜na † Departamento de F´ısica, Universidad del B´ıo-B´ıo,Casilla 5-C, Concepci´on, Chile andDepartament d’Estructura i Constituents de la Mat`eria,Institut de Ci`encies del Cosmos, Universitat de Barcelona,Diagonal 647, 08028 Barcelona, Spain. bstract We consider a non-singular origin for the Universe starting from an Einstein static Universe,the so called “emergent universe” scenario, in the framework of a theory which uses two volumeelements √− gd x and Φ d x , where Φ is a metric independent density, used as an additional measureof integration. Also curvature, curvature square terms and for scale invariance a dilaton field φ areconsidered in the action. The first order formalism is applied. The integration of the equations ofmotion associated with the new measure gives rise to the spontaneous symmetry breaking (S.S.B)of scale invariance (S.I.). After S.S.B. of S.I., it is found that a non trivial potential for the dilatonis generated. In the Einstein frame we also add a cosmological term that parametrizes the zeropoint fluctuations. The resulting effective potential for the dilaton contains two flat regions, for φ → ∞ relevant for the non-singular origin of the Universe, followed by an inflationary phase and φ → −∞ , describing our present Universe. The dynamics of the scalar field becomes non linearand these non linearities produce a non trivial vacuum structure for the theory and are responsiblefor the stability of some of the emergent universe solutions, which exists for a parameter range ofvalues of the vacuum energy in φ → −∞ , which must be positive but not very big, avoiding theextreme fine tuning required to keep the vacuum energy density of the present universe small. Thenon trivial vacuum structure is crucial to ensure the smooth transition from the emerging phase,to an inflationary phase and finally to the slowly accelerated universe now. Zero vacuum energydensity for the present universe defines the threshold for the creation of the universe. PACS numbers: 98.80.Cq, 04.20.Cv, 95.36.+x ∗ Electronic address: [email protected] † Electronic address: [email protected], [email protected] . INTRODUCTION One of the most important and intriging issues of modern physics is the so called “Cos-mological Constant Problem” [1–5], (CCP), most easily seen by studying the apparentlyuncontrolled behaviour of the zero point energies, which would lead to a correspondingequally uncontrolled vacuum energy or cosmological constant term. Even staying at theclassical level, the observed very small cosmological term in the present universe is still verypuzzling.Furthermore, the Cosmological Constant Problem has evolved from the “Old Cosmolog-ical Constant Problem”, where physicist were concerned with explaining why the observedvacuum energy density of the universe is exactly zero, to different type of CCP since the ev-idence for the accelerating universe became evident, for reviews see [6, 7]. We have thereforesince the discovery of the accelerated universe a “New Cosmological Constant Problem” [8],the problem is now not to explain zero, but to explain a very small vacuum energy density.This new situation posed by the discovery of a very small vacuum energy density ofthe universe means that getting a zero vacuum energy density for the present universe isdefinitely not the full solution of the problem, although it may be a step towards its solution.One point of view to the CCP that has been popular has been to provide a bound basedon the “anthropic principle” [9]. In this approach, a too large Cosmological Constant willnot provide the necessary conditions required for the existence of life, the anthropic principleprovides then an upper bound on the cosmological constant.One problem with this approach is for example that it relies on our knowledge of lifeas we know it and ignores the possibility that other life forms could be possible, for whichother (unknown) bounds would be relevant, therefore the reasoning appears by its verynature subjective, since of course if the observed cosmological constant will be different, ouruniverse will be different and this could include different kind of life that may be could haveadjusted itself to a higher cosmological constant of the universe. But even accepting thevalidity of anthropic considerations, we still do not understand why the observed vacuumenergy density must be positive instead of possibly a very small negative quantity. Acceptingthe anthropic explanation means may be also giving up on discovering important physicsrelated to the CCP and this may be the biggest objection.Nevertheless, the idea of associating somehow restrictions on the origin of the universe3ith the cosmological constant problem seems interesting. We will take on this point ofview, but leave out the not understood concept of life out from our considerations. Instead,we will require, in a very specific framework, the non-singular origin of the universe. Theadvantage of this point of view is that it is formulated in terms of ideas of physics alone,without reference to biology, which unlike physics, has not reached the level of an exactscience. Another interesting consequence is that we can learn that of a non-singularlycreated universe may not have a too big cosmological constant, an effect that points to acertain type of gravitational suppression of UV divergences in quantum field theory.In this respect, one should point out that even in the context of the inflationary scenario[10–13] which solves many cosmological problems, one still encounters the initial singular-ity problem which remains unsolved, showing that the universe necessarily had a singularbeginning for generic inflationary cosmologies [14–18].Here we will adopt the very attractive “Emergent Universe” scenario, where those con-clusions concerning singularities can be avoided [19–27]. The way to escape the singularityin these models is to violate the geometrical assumptions of these theorems, which assumei) that the universe has open space sections ii) the Hubble expansion is always greater thanzero in the past. In [19, 20] the open space section condition is violated since closed Robert-son Walker universes with k = 1 are considered and the Hubble expansion can become zero,so that both i) and ii) are avoided.In [19, 20] even models based on standard General Relativity, ordinary matter and mini-mally coupled scalar fields were considered and can provide indeed a non-singular (geodesi-cally complete) inflationary universe, with a past eternal Einstein static Universe that even-tually evolves into an inflationary Universe.Those most simple models suffer however from instabilities, associated with the instabilityof the Einstein static universe. The instability is possible to cure by going away from GR,considering non perturbative corrections to the Einstein’s field equations in the context ofthe loop quantum gravity [21], a brane world cosmology with a time like extra dimension[22, 23] considering the Starobinski model for radiative corrections (which cannot be derivedfrom an effective action) [24] or exotic matter [25]. In addition to this, the consideration ofa Jordan Brans Dicke model also can provide a stable initial state for the emerging universescenario [26, 27].In this review we study a different theoretical framework where such emerging universe4cenario is realized in a natural way, where instabilities are avoided and a succesfull infla-tionary phase with a graceful exit can be achieved. The model we will use was studied firstin [28], however, in the context of this model, a few scenarios are possible. For example inthe first paper on this model [28] a special choice of state to describe the present state ofour universe was made. Then in [29] a different candidate for the vacuum that representsour present universe was made. The way in which we best represents the present state ofthe universe is crucial, since as it should be obvious, the discussion of the CCP depends onwhat vacuum we take. In [29] we expressed the stability and existence conditions for thenon-singular universe in terms of the energy of the vacuum of our candidate for the presentUniverse. In [29] a few typos in [28] were corrected and also the discussion of some notionsdiscussed was improved in [29] and more deeper studies will be done in this review.Indeed in this review, all those topics will be further clarified, in particular the vacuumstructure of this model will be extended. A very important new feature that will be presentedin this review is the existence of a “kinetic vacuum”, that produces a vacuum energy statewhich is degenerate with the vacuum choice made in [29], this degeneracy is analyzed andthe dynamical role of this kinetic vacuum in the evolution of the universe and the CCP isanalyzed.We work in the context of a theory built along the lines of the two measures theory (TMT).Basic idea is developed in [30], [31]-[47] [48], [49]-[52], [54]-[58], [59] and more specificallyin the context of the scale invariant realization of such theories [48], [49]-[53], [54]-[58], [59].These theories can provide a new approach to the cosmological constant problem and canbe generalized to obtain also a theory with a dynamical spacetime [61], furthermore, stringand brane theories, as well as brane world scenarios can be constructed using Two MeasureTheories ideas [62]-[67]. We should also point out that the Hodge Dual construction of [33]for supergravity constitutes in fact an example of a TMT. The construction by Comelli [34]where no square root of the determinant of the metric is used and instead a total divergenceappears is also a very much related approach.The two measure theories have many points of similarity with “Lagrange Multiplier Grav-ity (LMG)” [69, 72]. In LMG there is a Lagrange multiplier field which enforces the conditionthat a certain function is zero. For a comparison of one of these lagrange multiplier gravitymodels with observations see [70]. In the two measure theory this is equivalent to the con-straint which requires some lagrangian to be constant. The two measure model presented5ere, as opposed to the LMG models of [69, 72] provide us with an arbitrary constant of inte-gration. The introduction of constraints can cause Dirac fields to contribute to dark energy[71] or scalar fields to behave like dust like in [69] and this dust behaviour can be caused bythe stabilization of a tachyonic field due to the constraint, accompanied by a floating darkenergy component [73, 74]. TMT models naturally avoid the 5th force problem [75].We will consider a slight generalization of the TMT case, where, we consider also thepossible effects of zero point energy densities, thus “softly breaking” the basic structure ofTMT for this purpose. We will show how the stated goals of a stable emergent universecan be achieved in the framework of the model and also how the stability of the emerginguniverse imposes interesting constraints on the energy density of the ground state of thetheory as defined in this paper: it must be positive but not very large, thus the vacuumenergy and therefore the term that softly breaks the TMT structure appears to be naturallycontrolled. An important ingredient of the model considered here is its softly broken con-formal invariance, meaning that we allow conformal breaking terms only though potentialsof the dilaton, which nevertheless preserve global scale invariance. In another models foremergent universe we have studied [76], that rule of softly broken conformal invariance wastaken into account. It is also a perfectly consistent, but different approach.The review will be organized as follows: First we review the principles of the TMT and inparticular the model studied in [48], which has global scale invariance and how this can be thebasis for the emerging universe. Such model gives rise, in the effective Einstein frame, to aneffective potential for a dilaton field (needed to implement an interesting model with globalscale invariance) which has a flat region. Following this, we look at the generalization of thismodel [59] by adding a curvature square or simply “ R term” and show that the resultingmodel contains now two flat regions. The existence of two flat regions for the potentialis shown to be consequence of the s.s.b. of the scale symmetry. We then consider theincorporation in the model of the zero point fluctuations, parametrized by a cosmologicalconstant in the Einstein frame. In this resulting model, there are two possible types ofemerging universe solutions, for one of those, the initial Einstein Universe can be stabilizeddue to the non linearities of the model, provided the vacuum energy density of the groundstate is positive but not very large. This is a very satisfactory results, since it means thatthe stability of the emerging universe prevents the vacuum energy in the present universefrom being very large!. The transition from the emergent universe to the ground state6oes through an intermediate inflationary phase, therefore reproducing the basic standardcosmological model as well. We end with a discussion section and present the point of viewthat the creation of the universe can be considered as a “threshold event” for zero presentvacuum energy density, which naturally gives a positive but small vacuum energy density. II. INTRODUCING A NEW MEASURE
The general structure of general coordinate invariant theories is taken usually as S = Z L √− gd x (1)where g = detg µν . The introduction of √− g is required since d x by itself is not a scalar butthe product √− gd x is a scalar. Inserting √− g , which has the transformation properties ofa density, produces a scalar action S , as defined by Eq.(1), provided L is a scalar.In principle nothing prevents us from considering other densities instead of √− g . Oneconstruction of such alternative “measure of integration”, is obtained as follows: given 4-scalars ϕ a (a = 1,2,3,4), one can construct the densityΦ = ε µναβ ε abcd ∂ µ ϕ a ∂ ν ϕ b ∂ α ϕ c ∂ β ϕ d (2)and consider in addition to the action S , as defined by Eq.(1), S , defined as S = Z L Φ d x (3) L is again some scalar, which may contain the curvature (i.e. the gravitational contribution)and a matter contribution, as it can be the case for S , as defined by Eq.(1). For an approachthat uses four-vectors instead of four-scalars see [68].In the action S defined by Eq.(3) the measure carries degrees of freedom independent ofthat of the metric and that of the matter fields. The most natural and successful formulationof the theory is achieved when the connection is also treated as an independent degree offreedom. This is what is usually referred to as the first order formalism.One can consider both contributions, and allowing therefore both geometrical objects toenter the theory and take as our action S = Z L √− gd x + Z L Φ d x (4)7ere L and L are ϕ a independent.We will study now the dynamics of a scalar field φ interacting with gravity as givenby the following action, where except for the potential terms U and V we have conformalinvariance, the potential terms U and V break down this to global scale invariance. S L = Z L √− gd x + Z L Φ d x (5) L = U ( φ ) (6) L = − κ R (Γ , g ) + 12 g µν ∂ µ φ∂ ν φ − V ( φ ) (7) R (Γ , g ) = g µν R µν (Γ) , R µν (Γ) = R λµνλ (8) R λµνσ (Γ) = Γ λµν,σ − Γ λµσ,ν + Γ λασ Γ αµν − Γ λαν Γ αµσ . (9)The suffix L in S L is to emphasize that here the curvature appears only linearly. Here,except for the potential terms U and V we have conformal invariance, the potential terms U and V break down this to global scale invariance. Since the breaking of local conformalinvariance is only through potential terms, we call this a “soft breaking”.In the variational principle Γ λµν , g µν , the measure fields scalars ϕ a and the “matter” - scalarfield φ are all to be treated as independent variables although the variational principle mayresult in equations that allow us to solve some of these variables in terms of others.For the case the potential terms U = V = 0 we have local conformal invariance g µν → Ω( x ) g µν (10)and ϕ a is transformed according to ϕ a → ϕ ′ a = ϕ ′ a ( ϕ b ) (11)Φ → Φ ′ = J ( x )Φ (12)where J ( x ) is the Jacobian of the transformation of the ϕ a fields.This will be a symmetry in the case U = V = 0 ifΩ = J (13)8otice that J can be a local function of space time, this can be arranged by performing forthe ϕ a fields one of the (infinite) possible diffeomorphims in the internal ϕ a space.We can still retain a global scale invariance in model for very special exponential formfor the U and V potentials. Indeed, if we perform the global scale transformation ( θ =constant) g µν → e θ g µν (14)then (9) is invariant provided V ( φ ) and U ( φ ) are of the form [48] V ( φ ) = f e αφ , U ( φ ) = f e αφ (15)and ϕ a is transformed according to ϕ a → λ ab ϕ b (16)which means Φ → det ( λ ab )Φ ≡ λ Φ (17)such that λ = e θ (18)and φ → φ − θα . (19)We will now work out the equations of motion after introducing V ( φ ) and U ( φ ) and seehow the integration of the equations of motion allows the spontaneous breaking of the scaleinvariance.Let us begin by considering the equations which are obtained from the variation of thefields that appear in the measure, i.e. the ϕ a fields. We obtain then A µa ∂ µ L = 0 (20)where A µa = ε µναβ ε abcd ∂ ν ϕ b ∂ α ϕ c ∂ β ϕ d . Since it is easy to check that A µa ∂ µ ϕ a ′ = δaa ′ Φ, itfollows that det ( A µa ) = − Φ = 0 if Φ = 0. Therefore if Φ = 0 we obtain that ∂ µ L = 0, orthat L = − κ R (Γ , g ) + 12 g µν ∂ µ φ∂ ν φ − V = M (21)where M is constant. Notice that this equation breaks spontaneously the global scale invari-ance of the theory, since the left hand side has a non trivial transformation under the scale9ransformations, while the right hand side is equal to M , a constant that after we integratethe equations is fixed, cannot be changed and therefore for any M = 0 we have obtainedindeed, spontaneous breaking of scale invariance.We will see what is the connection now. As we will see, the connection appears in theoriginal frame as a non Riemannian object. However, we will see that by a simple conformaltranformation of the metric we can recover the Riemannian structure. The interpretationof the equations in the frame gives then an interesting physical picture, as we will see.Let us begin by studying the equations obtained from the variation of the connectionsΓ λµν . We obtain then − Γ λµν − Γ αβµ g βλ g αν + δ λν Γ αµα + δ λµ g αβ Γ γαβ g γν − g αν ∂ µ g αλ + δ λµ g αν ∂ β g αβ − δ λν Φ , µ Φ + δ λµ Φ , ν Φ = 0 (22)If we define Σ λµν as Σ λµν = Γ λµν − { λµν } where { λµν } is the Christoffel symbol, we obtain for Σ λµν the equation − σ, λ g µν + σ, µ g νλ − g να Σ αλµ − g µα Σ ανλ + g µν Σ αλα + g νλ g αµ g βγ Σ αβγ = 0 (23)where σ = lnχ, χ = Φ √− g .The general solution of Eq.(24) isΣ αµν = δ αµ λ, ν + 12 ( σ, µ δ αν − σ, β g µν g αβ ) (24)where λ is an arbitrary function due to the λ - symmetry of the curvature [77] R λµνα (Γ),Γ αµν → Γ ′ αµν = Γ αµν + δ αµ Z, ν (25)Z being any scalar (which means λ → λ + Z ).If we choose the gauge λ = σ , we obtainΣ αµν ( σ ) = 12 ( δ αµ σ, ν + δ αν σ, µ − σ, β g µν g αβ ) . (26)Considering now the variation with respect to g µν , we obtainΦ( − κ R µν (Γ) + 12 φ, µ φ, ν ) − √− gU ( φ ) g µν = 0 (27)solving for R = g µν R µν (Γ) from Eq.(27) and introducing in Eq.21, we obtain M + V ( φ ) − U ( φ ) χ = 0 (28)10 constraint that allows us to solve for χ , χ = 2 U ( φ ) M + V ( φ ) . (29)To get the physical content of the theory, it is best consider variables that have well defineddynamical interpretation. The original metric does not has a non zero canonical momenta.The fundamental variable of the theory in the first order formalism is the connection andits canonical momenta is a function of g µν , given by, g µν = χg µν (30)and χ given by Eq.(29). Interestingly enough, working with g µν is the same as goingto the “Einstein Conformal Frame”. In terms of g µν the non Riemannian contributionΣ αµν dissappears from the equations. This is because the connection can be written as theChristoffel symbol of the metric g µν . In terms of g µν the equations of motion for the metriccan be written then in the Einstein form (we define R µν ( g αβ ) = usual Ricci tensor in termsof the bar metric = R µν and R = g µν R µν ) R µν ( g αβ ) − g µν R ( g αβ ) = κ T effµν ( φ ) (31)where T effµν ( φ ) = φ ,µ φ ,ν − g µν φ ,α φ ,β g αβ + g µν V eff ( φ ) (32)and V eff ( φ ) = 14 U ( φ ) ( V + M ) . (33)In terms of the metric g αβ , the equation of motion of the Scalar field φ takes the standardGeneral - Relativity form 1 √− g ∂ µ ( g µν p − g∂ ν φ ) + V ′ eff ( φ ) = 0 . (34)Notice that if V + M = 0 , V eff = 0 and V ′ eff = 0 also, provided V ′ is finite and U = 0there. This means the zero cosmological constant state is achieved without any sort of finetuning. That is, independently of whether we add to V a constant piece, or whether wechange the value of M , as long as there is still a point where V + M = 0, then still V eff = 0and V ′ eff = 0 ( still provided V ′ is finite and U = 0 there). This is the basic feature that11haracterizes the TMT and allows it to solve the “old” cosmological constant problem, atleast at the classical level.In what follows we will study the effective potential (33) for the special case of globalscale invariance, which as we will see displays additional very special features which makesit attractive in the context of cosmology.Notice that in terms of the variables φ , g µν , the “scale” transformation becomes only ashift in the scalar field φ , since g µν is invariant (since χ → λ − χ and g µν → λg µν ) g µν → g µν , φ → φ − θα . (35)If V ( φ ) = f e αφ and U ( φ ) = f e αφ as required by scale invariance Eqs. (14, 16, 17, 18,19), we obtain from the expression (33) V eff = 14 f ( f + M e − αφ ) (36)Since we can always perform the transformation φ → − φ we can choose by convention α >
0. We then see that as φ → ∞ , V eff → f f = const. providing an infinite flat region asdepicted in Fig. 1. Also a minimum is achieved at zero cosmological constant for the case f M < φ min = − α ln | f M | . (37)In conclusion, the scale invariance of the original theory is responsible for the non appear-ance (in the physics) of a certain scale, that associated to M. However, masses do appear,since the coupling to two different measures of L and L allow us to introduce two indepen-dent couplings f and f , a situation which is unlike the standard formulation of globallyscale invariant theories, where usually no stable vacuum state exists.The constant of integration M plays a very important role indeed: any non vanishingvalue for this constant implements, already at the classical level S.S.B. of scale invariance. III. GENERATION OF TWO FLAT REGIONS AFTER THE INTRODUCTIONOF A R TERM
As we have seen, it is possible to obtain a model that through a spontaneous breakingof scale invariace can give us a flat region. We want to obtain now two flat regions in oureffective potential. A simple generalization of the action S L will fix this. The basic new12eature we add is the presence is higher curvature terms in the action [78]-[86], which havebeen shown to be very relevant in cosmology. In particular he first inflationary model froma model with higher terms in the curvature was proposed in [86].What one needs to do is simply consider the addition of a scale invariant term of theform S R = ǫ Z ( g µν R µν (Γ)) √− gd x (38)The total action being then S = S L + S R . In the first order formalism S R is not onlyglobally scale invariant but also locally scale invariant, that is conformally invariant (recallthat in the first order formalism the connection is an independent degree of freedom and itdoes not transform under a conformal transformation of the metric).Let us see what the equations of motion tell us, now with the addition of S R to theaction. First of all, since the addition has been only to the part of the action that couplesto √− g , the equations of motion derived from the variation of the measure fields remainsunchanged. That is Eq.(21) remains valid.The variation of the action with respect to g µν gives now R µν (Γ)( − Φ κ + 2 ǫR √− g ) + Φ 12 φ, µ φ, ν −
12 ( ǫR + U ( φ )) √− gg µν = 0 (39)It is interesting to notice that if we contract this equation with g µν , the ǫ terms do notcontribute. This means that the same value for the scalar curvature R is obtained as insection 2, if we express our result in terms of φ , its derivatives and g µν . Solving the scalarcurvature from this and inserting in the other ǫ - independent equation L = M we get stillthe same solution for the ratio of the measures which was found in the case where the ǫ terms were absent, i.e. χ = Φ √− g = U ( φ ) M + V ( φ ) .In the presence of the ǫR term in the action, Eq. (22) gets modified so that instead ofΦ, Ω = Φ − ǫR √− g appears. This in turn implies that Eq.(23) keeps its form but where σ is replaced by ω = ln ( Ω √− g ) = ln ( χ − κǫR ), where once again, χ = Φ √− g = U ( φ ) M + V ( φ ) .Following then the same steps as in the model without the curvature square terms, wecan then verify that the connection is the Christoffel symbol of the metric g µν given by g µν = ( Ω √− g ) g µν = ( χ − κǫR ) g µν (40)13 µν defines now the “Einstein frame”. Equations (39) can now be expressed in the“Einstein form” R µν − g µ ν R = κ T effµν (41)where T effµν = χχ − κǫR ( φ ,µ φ ,ν − g µν φ ,α φ ,β g αβ ) + g µν V eff (42)where V eff = ǫR + U ( χ − κǫR ) (43)Here it is satisfied that − κ R (Γ , g ) + g µν ∂ µ φ∂ ν φ − V = M , equation that expressed interms of g αβ becomes − κ R (Γ , g ) + ( χ − κǫR ) g µν ∂ µ φ∂ ν φ − V = M . This allows us to solve for R and we get, R = − κ ( V + M ) + κ g µν ∂ µ φ∂ ν φχ κ ǫg µν ∂ µ φ∂ ν φ (44)Notice that if we express R in terms of φ , its derivatives and g µν , the result is the same asin the model without the curvature squared term, this is not true anymore once we express R in terms of φ , its derivatives and g µν .In any case, once we insert (44) into (43), we see that the effective potential (43) willdepend on the derivatives of the scalar field now. It acts as a normal scalar field potentialunder the conditions of slow rolling or low gradients and in the case the scalar field is nearthe region M + V ( φ ) = 0.Notice that since χ = U ( φ ) M + V ( φ ) , then if M + V ( φ ) = 0, then, as in the simpler modelwithout the curvature squared terms, we obtain that V eff = V ′ eff = 0 at that point withoutfine tuning (here by V ′ eff we mean the derivative of V eff with respect to the scalar field φ ,as usual).In the case of the scale invariant case, where V and U are given by equation (15), itis interesting to study the shape of V eff as a function of φ in the case of a constant φ , inwhich case V eff can be regarded as a real scalar field potential. Then from (44) we get14 = − κ ( V + M ), which inserted in (43) gives, V eff = ( f e αφ + M ) ǫκ ( f e αφ + M ) + f e αφ ) (45)The limiting values of V eff are:First, for asymptotically large positive values, ie. as αφ → ∞ , we have V eff → f ǫκ f + f ) .Second, for asymptotically large but negative values of the scalar field, that is as αφ →−∞ , we have: V eff → ǫκ .In these two asymptotic regions ( αφ → ∞ and αφ → −∞ ) an examination of the scalarfield equation reveals that a constant scalar field configuration is a solution of the equations,as is of course expected from the flatness of the effective potential in these regions.Notice that in all the above discussion it is fundamental that M = 0. If M = 0 thepotential becomes just a flat one, V eff = f ǫκ f + f ) everywhere (not only at high values of αφ ). All the non trivial features necessary for a graceful exit, the other flat region associatedto the Planck scale and the minimum at zero if M < M = 0 implies the we are considering a situationwith S.S.B. of scale invariance.These kind of models with potentials giving rise to two flat potentials have been appliedto produce models for bags and confinement in a very natural way [60]. IV. A NOTE ON THE THE “EINSTEIN” METRIC AS A CANONICAL VARI-ABLE OF THE THEORY
One could question the use of the Einstein frame metric g µν in contrast to the originalmetric g µν . In this respect, it is interesting to see the role of both the original metric and thatof the Einstein frame metric in a canonical approach to the first order formalism. Here wesee that the original metric does not have a canonically conjugated momentum (this turnsout to be zero), in contrast, the canonically conjugated momentum to the connection turnsout to be a function exclusively of g µν , this Einstein metric is therefore a genuine dynamicalcanonical variable, as opposed to the original metric. There is also a lagrangian formulationof the theory which uses g µν , as we will see in the next section, what we can call the actionin the Einstein frame. In this frame we can quantize the theory for example and considercontributions without reference to the original frame, thus possibly considering breaking15he TMT structure of the theory through quantum effects, but such breaking will be done“softly” through the introduction of a cosmological term only. Surprisingly, the remainingstructure of the theory, reminiscent from the original TMT structure will be enough tocontrol the strength of this additional cosmological term once we demand that the universeoriginated from a non-singular and stable emergent state. V. GENERALIZING THE MODEL TO INCLUDE EFFECTS OF ZERO POINTFLUCTUATIONS
The effective energy-momentum tensor can be represented in a form like that of a perfectfluid T effµν = ( ρ + p ) u µ u ν − p ˜ g µν , where u µ = φ ,µ (2 X ) / (46)here X ≡ ˜ g αβ φ ,α φ ,β . This defines a pressure functional and an energy density functional.The system of equations obtained after solving for χ , working in the Einstein frame withthe metric ˜ g µν can be obtained from a “k-essence” type effective action, as it is standard intreatments of theories with non linear kinetic terms or k-essence models[87]-[90]. The actionfrom which the classical equations follow is, S eff = Z p − gd x (cid:20) − κ R ( g ) + p ( φ, R ) (cid:21) (47) p = χχ − κǫR X − V eff (48) V eff = ǫR + U ( χ − κǫR ) (49)where it is understood that, χ = 2 U ( φ ) M + V ( φ ) . (50)We have two possible formulations concerning R : Notice first that R and R are differentobjects, the R is the Riemannian curvature scalar in the Einstein frame, while R is a differentobject. This R will be treated in two different ways:1. First order formalism for R . Here R is a lagrangian variable, determined as follows, R that appear in the expression above for p can be obtained from the variation of the pressure16 - Φ- - V eff - - Φ- - V eff FIG. 1: The form of effective potential V eff ( φ ) versus the scalar field φ . We consider unit where κ = 1, α = 1, Λ = 0 .
35 and ǫ = −
1. Left panel: M = − f = 1 / f = 1. Right panel: M = 1, f = 1 / f = 1. functional action above with respect to R , this gives exactly the expression for R that hasbeen solved already in terms of X, φ , etc, see Eq. (44).2. Second order formalism for R . R that appear in the action above is exactly theexpression for R that has been solved already in terms of X, φ , etc. The second orderformalism can be obtained from the first order formalism by solving algebraically R fromthe Eq. (44) obtained by variation of R , and inserting back into the action.One may also use the method outlined in [91] to find the effective action in the Einsteinframe, in [91] the problem of a curvature squared theory with standard measure was studied.The methods outlined there can be also applied in the modified measure case [92], thusproviding another derivation of the effective action explained above.The problem that we have to solve to find the effective lagrangian is basically finding thatlagrangian tat will produce the effective energy momentum tensor in the Einstein frame bythe variation of the g µν metric T eff µν = g µν L eff ( h ) − ∂L eff ∂g µν (51)In contrast to the simplified models studied in literature[87–90], it is impossible hereto represent p ( φ, X ; M ) in a factorizable form like ˜ K ( φ )˜ p ( X ). The scalar field effectiveLagrangian can be taken as a starting point for many considerations.In particular, the quantization of the model can proceed from (47) and additional termscould be generated by radiative corrections. We will focus only on a possible cosmologicalterm in the Einstein frame added (due to zero point fluctuations) to (47), which leads then17 - Φ V eff - - Φ V eff FIG. 2: The effective potential V eff ( φ ) versus the scalar field φ . We consider unit where κ = 1, α = 1, Λ = 0 .
35 and ǫ = 1. Left panel: M = 1, f = 1 / f = 1. Right panel: M = − f = 1, f = 1. to the new action S eff, Λ = Z p − gd x (cid:20) − κ R ( g ) + p ( φ, R ) − Λ (cid:21) (52)This addition to the effective action leaves the equations of motion of the scalar fieldunaffected, but the gravitational equations aquire a cosmological constant. Adding the Λterm can be regarded as a redefinition of V eff ( φ, X ; M ) V eff ( φ, R ) → V eff ( φ, R ) + Λ (53)As we will see the stability of the emerging Universe imposes interesting constraints on ΛAfter introducing the Λ term, we get from the variation of R the same value of R , unaf-fected by the new Λ term, but as one can easily see then R does not have the interpretationof a curvature scalar in the original frame since it is unaffected by the new source of energydensity (the Λ term), this is why the Λ term theory does not have a formulation in theoriginal frame, but is a perfectly legitimate generalization of the theory, probably obtainedby considering zero point fluctuations, notice that quantum theory is possible only in theEinstein frame. Notice that even in the original frame the bar metric (not the original met-ric) appears automatically in the canonically conjugate momenta to the connection, so wecan expect from this that the bar metric and not the original metric be the relevant one forthe quantum theory.In Figure 1 and 2 we have plotted the effective potential as a function of the scalar field,for ǫ = − ǫ = 1 respectively. We consider unit where κ = 1, α = 1, Λ = 0 .
35 anddifferent values for
M, f , f . 18 I. ANALYSIS OF THE EMERGENT UNIVERSE SOLUTIONS
We now want to consider the detailed analysis of The Emerging Universe solutions andin the next section their stability in the TMT scale invariant theory. We start consideringthe cosmological solutions of the form ds = dt − a ( t ) ( dr − r + r ( dθ + sin θdφ )) , φ = φ ( t ) (54)in this case, we obtain for the energy density and the pressure, the following expressions.We will consider a scenario where the scalar field φ is moving in the extreme right region φ → ∞ , in this case the expressions for the energy density ρ and pressure p are given by, ρ = A φ + 3 B ˙ φ + C (55)and p = A φ + B ˙ φ − C (56)It is interesting to notice that all terms proportional to ˙ φ behave like “radiation”, since p ˙ φ = ρ ˙ φ is satisfied. here the constants A, B and C are given by, A = f f + κ ǫf , (57) B = ǫκ κ ǫf /f ) = ǫκ A , (58) C = f f (1 + κ ǫf /f ) + Λ = f f A + Λ . (59)It will be convenient to “decompose” the constant Λ into two pieces,Λ = − κ ǫ + ∆ λ (60)since as φ → −∞ , V eff → ∆ λ . Therefore ∆ λ has the interesting interpretation of thevacuum energy density in the φ → −∞ vacuum. As we will see, it is remarkable that thestability and existence of non-singular emergent universe implies that ∆ λ >
0, and it isbounded from above as well.The equation that determines such static universe a ( t ) = a = constant , ˙ a = 0, ¨ a = 0gives rise to a restriction for ˙ φ that have to satisfy the following equation in order to19uarantee that the universe be static, because ¨ a = 0 is proportional to ρ + 3 p , we mustrequire that ρ + 3 p = 0, which leads to3 B ˙ φ + A ˙ φ − C = 0 , (61)This equation leads to two roots, the first being˙ φ = √ A + 12 BC − A B . (62)The second root is: ˙ φ = −√ A + 12 BC − A B . (63)It is also interesting to see that if the discriminant is positive, the first solution hasautomatically positive energy density, if we only consider cases where
C >
0, which isrequired if we want the emerging solution to be able to turn into an inflationary solutioneventually. One can see that the condition ρ > w > (1 − √ − w ) /
2, where w = − BC/A >
0, since we must have
A >
B < w <
1, it is always true that this inequality is satisfied.Before going into the subject of the small perturbations and stability of these solutions,we would like to notice the “entropy like” conservation laws that may be useful in a nonperturbative analysis of the theory.In fact in the φ → ∞ region, we have the exact symmetry φ → φ + constant . andconsidering that the effective matter action here is a p , we have the conserved quantity π φ = a ( A ˙ φ + 4 B ˙ φ ) (64)It is very interesting to notice that π φ = S = a s (65)where s assumes the “entropy density” form s = ( ρ + p ) /T (66)provided we identify the “Temperature” T with ˙ φ .20 II. STABILITY OF THE STATIC SOLUTION
We will now consider the perturbation equations. Considering small deviations of ˙ φ thefrom the static emerging solution value ˙ φ and also considering the perturbations of thescale factor a , we obtain, from Eq. (55) δρ = A ˙ φ δ ˙ φ + 12 B ˙ φ δ ˙ φ (67)at the same time δρ can be obtained from the perturbation of the Friedmann equation3( 1 a + H ) = κρ (68)and since we are perturbing a solution which is static, i.e., has H = 0, we obtain then − a δa = κδρ (69)we also have the second order Friedmann equation1 + ˙ a + 2 a ¨ aa = − κp (70)For the static emerging solution, we have p = − ρ / a = a , so2 a = − κp = 23 κρ = Ω κρ (71)where we have chosen to express our result in terms of Ω , defined by p = (Ω − ρ , whichfor the emerging solution has the value Ω = . Using this in 69, we obtain δρ = − ρ a δa (72)and equating the values of δρ as given by 67 and 72 we obtain a linear relation between δ ˙ φ and δa , which is, δ ˙ φ = D δa (73)where D = − ρ a ˙ φ ( A + 12 B ˙ φ ) (74)21e now consider the perturbation of the eq. (70). In the right hand side of this equationwe consider that p = (Ω − ρ , with Ω = 2 (cid:16) − U eff ρ (cid:17) , (75)where, U eff = C + B ˙ φ (76)and therefore, the perturbation of the Eq. (70) leads to, − δaa + 2 δ ¨ aa = − κδp = − κδ ((Ω − ρ ) (77)to evaluate this, we use 75, 76 and the expressions that relate the variations in a and ˙ φ (73). Defining the “small” variable β as a ( t ) = a (1 + β ) (78)we obtain, 2 ¨ β ( t ) + W β ( t ) = 0 , (79)where, W = Ω ρ " B ˙ φ A + 12 ˙ φ B − C + B ˙ φ ) ρ − κ Ω + 2 κ , (80)notice that the sum of the last two terms in the expression for W , that is − κ Ω + 2 κ vanish since Ω = , for the same reason, we have that 6 ( C + B ˙ φ ) ρ = 4, which brings us tothe simplified expression W = Ω ρ " B ˙ φ A + 12 ˙ φ B − , (81)For the stability of the static solution, we need that W >
0, where ˙ φ is defined eitherby E. (62) ( ˙ φ = φ ) or by E. (63) ( ˙ φ = φ ). If we take E. (63) ( ˙ φ = φ ) and use this inthe above expression for W , we obtain, W = Ω ρ " √ A + 12 BC − √ A + 12 BC − A , (82)to avoid negative kinetic terms during the slow roll phase that takes place following theemergent phase, we must consider A >
0, so, we see that the second solution is unstableand will not be considered further. 22ow in the case of the first solution, E. (62) ( ˙ φ = φ ), then W becomes W = Ω ρ " − √ A + 12 BC √ A + 12 BC − A , (83)so the condition of stability becomes 2 √ A + 12 BC − A <
0, or 2 √ A + 12 BC < A , squaringboth sides and since
A >
0, we get 12
BC/A < − /
4, which means
B <
0, and therefore ǫ <
0, multiplying by −
1, we obtain, 12( − B ) C/A > /
4, replacing the values of
A, B, C ,given by 57 we obtain the condition ∆ λ > , (84)Now there is the condition that the discriminant be positive A + 12 BC > λ < − ǫ ) κ (cid:20) f f + κ ǫf (cid:21) , (85)since A = h f f + κ ǫf i > B <
0, meaning that ǫ <
0, we see that we obtain a positiveupper bound for the energy density of the vacuum as φ → −∞ , which must be positive, butnot very big. VIII. INFLATION AND ITS GRACEFUL EXIT
The emerging phase owes its existence to a strictly constant vacuum energy (which hereis represented by the value of A ) at very large values of the field φ . In fact, while for M = 0the effective potential of the scalar field is perfectly flat, for any M = 0 the effective potentialacquires a non trivial shape. This causes the transition from the emergent phase to a slowroll inflationary phase which will be the subject of this section.Following [28], we consider now then the relevant equations for the model in the slow rollregime, i.e. for ˙ φ small and when the scalar field φ is large, but finite and we consider thefirst corrections to the flatness to the effective potential. Dropping higher powers of ˙ φ in thecontributions for the kinetic energy and in the scalar curvature R , we obtain ρ = 12 γ ˙ φ + V eff , (86) γ = χχ − κǫR , (87) R = − κ ( V + M ) . (88)23ere, as usual χ = U ( φ ) M + V ( φ ) . In the slow roll approximation, we can drop the second derivativeterm of φ and the second power of ˙ φ in the equation for H and we get3 Hγ ˙ φ = − V ′ eff , (89)3 H = κV eff , (90)where V ′ eff = dV eff dφ . The relevant expression for V eff will be that given by (45), i.e., where allhigher derivatives are ignored in the potential, consistent with the slow roll approximation.We now display the relevant expressions for the region of very large, but not infinite φ ,these are: V eff = C + C exp ( − αφ ) , (91) χ = 2 f f exp ( αφ ) − M f f , (92)and γ = γ + γ exp ( − αφ ) . (93)The relevant constants that will affect our results are, C , as given by (59) and C and γ given by C = − ǫκ f M (4 f + 4 κ ǫf ) + 2 f M f + 4 κ ǫf , (94)and γ = f f + κ ǫf , (95)respectively.Using Eq. (91) we can calculates the key landmarks of the inflationary history: first, thevalue of the scalar field where inflation ends, φ end and a value for the scalar field φ ∗ biggerthan this ( φ ∗ > φ end ) and which happens earlier, which represents the “horizon crossingpoint”. We must demand then that a typical number of e-foldings, like N = 60, takes placebetween φ ∗ , until the end of inflation at φ = φ end .To determine the end of inflation, we consider the quantity δ = − ˙ HH and consider thepoint in the evolution of the Universe where δ = 1, only when δ <
1, we have an acceleratingUniverse, so the point δ = 1 represents indeed the end of inflation. Calculating the derivative24ith respect to cosmic time of the Hubble expansion using (90) and (89), we obtain thatthe condition δ = 1 gives δ = 12 γ ( V ′ eff /V eff ) = 1 , (96)working to leading order, setting γ = γ , V eff = C and V ′ eff = − αC exp ( − αφ end ), this givesas a solution, exp ( αφ end ) = − αC C √ κγ , (97)notice that if M and f have different signs and if ǫ < C < − C represents the absolute value of C . Wenow consider φ ∗ and the requirement that this precedes φ end by N e-foldings, N = Z Hdt = Z H ˙ φ dφ = − Z H γV ′ eff dφ, (98)where in the last step we have used the slow roll equation of motion for the scalar field (89)to solve for ˙ φ . Solving H in terms of V eff using (90), working to leading order, setting γ = γ and integrating, we obtain the relation between φ ∗ and φ end , exp ( αφ ∗ ) = exp ( αφ end ) − N α C Cκγ , (99)as we mentioned before C < φ ∗ > φ end as it should be for everything to make sense. Introducing Eq.(97) into Eq. (99), we obtain, exp ( αφ ∗ ) = − C C √ κ ( α √ γ + N α √ κγ ) . (100)We finally calculate the power of the primordial scalar perturbations. If the scalar field φ had a canonically normalized kinetic term, the spectrum of the primordial perturbationswill be given by the equation δρρ ∝ H ˙ φ , (101)however, as we can see from (86), the kinetic term is not canonically normalized because ofthe factor γ in that equation.In this point we will study the scalar and tensor perturbations for our model where thekinetic term is not canonically normalized. The general expression for the perturbed metricabout the Friedmann-Robertson-Walker is ds = − (1 + 2 F ) dt + 2 a ( t ) D , i dx i dt + a ( t )[(1 − ψ ) δ ij + 2 E ,i,j + 2 h ij ] dx i dx j , F , D , ψ and E are the scalar type metric perturbations and h ij characterizes thetransverse-traceless tensor perturbation. The power spectrum of the curvature perturbationin the slow-roll approximation for a not-canonically kinetic term becomes Ref.[93](see alsoRefs.[94]) P S = k (cid:18) δρρ (cid:19) = k H c s δ , (102)where it was defined “speed of sound”, c s , as c s = P , X P , X + 2 XP , XX , with P ( X, φ ) an function of the scalar field and of the kinetic term X = − (1 / g µν ∂ µ φ∂ ν φ .Here P , X denote the derivative with respect X . In our case P ( X, φ ) = γ ( φ ) X − V eff , with X = ˙ φ /
2. Thus, from Eq.(102) we get P S = k H γ ( φ ) ˙ φ . (103)The scalar spectral index n s , is defined by n s − d ln P S d ln k = − δ − η − s, (104)where η = ˙ δδ H and s = ˙ c s c s H , respectively.On the other hand, the generation of tensor perturbations during inflation would producegravitational wave. The amplitude of tensor perturbations was evaluated in Ref.[93], where P T = 23 π (cid:18) XP , X − PM P lanck (cid:19) , and the tensor spectral index n T , becomes n T = d ln P T d ln k = − δ, and they satisfy a generalized consistency relation r = P T P S = − c s n T . (105)Therefore, the scalar field (to leading order) that should appear in Eq. (101) should be √ γ φ and instead of Eq. (103) , we must use δρρ = H √ γ ˙ φ . (106)26 IG. 3: The plot shows r versus n s for three values of α . For α = 1 solid line, α = 0 . α = 0 .
01 dots line, respectively. Here, we have fixed the values M = − ǫ = − f = 1 / f = 1, λ = 1 /
10 and κ = 1, respectively. The seven-year WMAP data places stronger limits onthe tensor-scalar ratio (shown in red) than five-year data (blue) [95]. The power spectrum of the perturbations goes, up to a factor of order one, which we willdenote k as ( δρ/ρ ) , so we have, P S = k (cid:18) δρρ (cid:19) = k H γ ˙ φ , (107)this quantity should be evaluated at φ = φ ∗ given by (100). Solving for ˙ φ from the slow rollequation (89), evaluating the derivative of the effective potential using (91) and solving for H from (90), we obtain, to leading order, P S = k κ γ C α C exp (2 αφ ∗ ) , (108)27sing then (100) for exp( αφ ∗ ), we obtain our final result, P S = k κ C √ N α √ γ κ ) , (109)it is very interesting first of all that C dependence has dropped out and with it all depen-dence on M . In fact this can be regarded as a non trivial consistency check of our estimates,since apart from its sign, the value M should not affect the results. This is due to the factthat from a different value of M (although with the same sign), we can recover the originalpotential by performing a shift of the scalar field φ .In Fig.3 we show the dependence of the tensor-scalar ratio r on the spectral index n s .From left to right α = 1 (solid line), α = 0 . α = 0 .
01 (dots line), respectively.From Ref.[95], two-dimensional marginalized constraints (68% and 95% confidence levels)on inflationary parameters r and n s , the spectral index of fluctuations, defined at k =0.002 Mpc − . The seven-year WMAP data places stronger limits on r (shown in red) thanfive-year data (blue)[96], [97]. In order to write down values that relate n s and r , we usedEqs.(104) and (105). Also we have used the values M = − ǫ = − f = 1 / f = 1, λ = 1 /
10 and κ = 1, respectively.From Eqs.(98), (104) and (105), we observed numerically that for α = 1, the curve r = r ( n s ) (see Fig.3) for WMAP 7-years enters the 95% confidence region where the ratio r ≃ . N ≃
32. For α = 0 . r ≃ . N ≃
227 and for α = 0 . r ≃ .
136 corresponds to N ≃ α = 1, r ≃ . N ≃
34. For α = 0 . r ≃ . N ≃
240 and for α = 0 . r ≃ .
109 corresponds to N ≃ α , which lies in the range 1 > α >
0, the model is well supported by thedata as could be seen from Fig.3.
IX. THE VACUUM STRUCTURE OF THE THEORY, INCLUDING THE “KI-NETIC VACUUM STATE”
For the discussion of the vacuum structure of the theory, we start studying V eff for thecase of a constant field φ , given by, V eff = ( f e αφ + M ) ǫκ ( f e αφ + M ) + f e αφ ) + Λ (110)28his is necessary, but not enough, since as we will see, the consideration of constant fields φ alone can lead to misleading conclusions, in some cases, the dependence of V eff on thekinetic term can be crucial to see if and how we can achieve the crossing of an apparentbarrier.For a constant field φ the limiting values of V eff are (now that we added the constant Λ):First, for asymptotically large positive values, ie. as αφ → ∞ , we have V eff → f ǫκ f + f ) + Λ.Second, for asymptotically large but negative values of the scalar field, that is as αφ →−∞ , we have: V eff → ǫκ + Λ = ∆ λ .In these two asymptotic regions ( αφ → ∞ and αφ → −∞ ) an examination of the scalarfield equation reveals that a constant scalar field configuration is a solution of the equations,as is of course expected from the flatness of the effective potential in these regions.Notice that in all the above discussion it is fundamental that M = 0. If M = 0 thepotential becomes just a flat one, V eff = f ǫκ f + f ) + Λ everywhere (not only at high valuesof αφ ).Finally, there is a minimum at V eff = Λ if M < f > A > V eff ( αφ → −∞ ) = ∆ λ < V eff ( min, M <
0) = Λ < V eff ( αφ → ∞ ) = C (111)where C = f f (1+ κ ǫf /f ) + Λ = f f A + Λ. notice that we assume above that f > M <
0, but f < M > f /M <
0. We could have a scenario where we start thenon-singular emergent universe at φ → ∞ where V eff ( αφ → ∞ ) = f ǫκ f + f ) + Λ, whichthen slow rolls, then inflates [28] and finally gets trapped in the local minimum with energydensity V eff ( min, M <
0) = Λ, that was the picture favored in [28], while here we want toargue that the most attractive and relevant description for the final state of our Universe isrealized after inflation in the flat region φ → −∞ , since in this region the vacuum energydensity is positive and bounded from above, so its a good candidate for our present state ofthe Universe. It remains to be seen however whether a smooth transition all the way from φ → ∞ to φ → −∞ is possible.Before we discuss the transition to the φ → −∞ , it is necessary to discuss another29acuum state, which we may call the “kinetic vacuum state” which is in fact degeneratewith this one. The “kinetic vacuum state” that, with time dependence and say for no spacedependence and ˙ φ given by ˙ φ = − ǫκ (112)which can be solved for ˙ φ in the real domain for ǫ <
0. For this case R (which is nota Riemannian curvature), as given by 44 diverges, the Riemannian scalar derived from theEinstein frame metric is perfectly regular. In this case then V eff = ǫR + U ( χ − κǫR ) + Λ → ǫκ + Λ = ∆ λ (113)that is, for this value of ˙ φ , regardless of the value of the scalar field, the value of V eff becomes degenerate with its value for constant and arbitrarily negative φ , which is ourcandidate vacuum for the present state of the Universe.Notice that this value for ˙ φ is also the one obtained by extremizing the pressure func-tional in the region of very large scalar field values, so in this limit it is obvious that suchconfiguration satisfy the Euler Lagrange equations, but indeed it is a general feature, theequations of motion for the kinetic vacuum are satisfied, regardless of what value we takefor the scalar field. X. EVOLUTION OF THE UNIVERSE TO ITS PRESENT SLOWLY ACCEL-ERATING STATE AT, CROSSING “BARRIERS” AND DYNAMICAL SYSTEMANALYSIS.
In order to discuss the possibility of transition to φ → −∞ . In our case, since we areinterested in a local minimum between φ → ∞ or φ → −∞ , we can take M of either sign.Taking for definitness f > f > A > ǫ <
0, we see that there will be a point,given by 110, defined by ǫκ ( f e αφ + M ) + f e αφ = 0 where V eff as will spike to ∞ , gothen down to −∞ and then asymptotically its positive asymptotic value at φ → −∞ . Thishas the appearance of a potential barrier. However, this is deceptive, such barrier existsfor constant φ , but can be avoided by considering a transition from any φ , but with theappropriate value of ˙ φ that defines the kinetic vacuum. A detailed dynamical analysis will30e presented now concerning these issues, The field equations become in the cosmologicalcase: ˙ H = − k ρ + p ) − H + k ρ , (114)˙ ρ = − H ( ρ + p ) . (115)By using the definitions of V ( φ ) and U ( φ ) we can express ρ and p as follow: ρ = (1 + κ ǫ ˙ φ ) U [ U + κ ǫ ( V + M ) ] ˙ φ V eff , (116) p = (1 + κ ǫ ˙ φ ) U [ U + κ ǫ ( V + M ) ] ˙ φ − V eff , (117) V eff = (1 + κ ǫ ˙ φ ) ( V + M ) U + κ ǫ ( V + M ) ] " U + ǫ − κ ( V + M ) + κ ˙ φ χ (1 + κ ǫ ˙ φ ) ! + Λ (118)We can note that independent of ˙ φ there are a singularity in the potential (also in ρ and p ) when φ = φ ∗ , where U ( φ ∗ ) + κ ǫ ( V ( φ ∗ ) + M ) = 0.The only way to avoid this situation is consider ˙ φ = − κ ǫ before φ arrive to φ ∗ . Wecan note that, in this case, the effective potential becomes flat (i.e. independent of φ ) andeverything is finite.It is interesting to note that for the case where this model admit an static and stableuniverse solution in the region φ → ∞ the kinetic vacuum state solution is an attractor inthe region φ >
0, see discussion below.This situation was already found in the limit φ → ∞ in [28] where the stability of thestatic solution was studied.In particular in the limit φ → ∞ the set of equations (114, 115) could be written as anautonomous system of two dimensions respect to H and y = ˙ φ as follow, see [28]:˙ H = κ h C + B y − A (1 + κ ǫ y ) y i − H , (119)˙ y = − A (1 + κ ǫ y ) y A + A κ ǫ y + 2 B y H , (120)31s was mentioned in [28] this system has five critical points where one of these pointscorrespond to the ES universe discussed previously, but there are also the critical point H = r κ ǫ + Λ ≡ H (121) y = − κ ǫ ≡ y . (122)This critical point is, precisely, the kinetic vacuum state which avoid the singularityproblem of V eff discussed above. After we linearize the equations (119, 120) near this criticalpoint we obtain that the eigenvalues of the linearized equations are negative λ = − H and λ = − H , then, this critical point is an attractor. The Fig. (4) top left panel show partof the Direction Field of the system and four numerical solutions where we can note thatthe kinetic vacuum state is an attractor solution.It is interesting to note that this solution is in fact an attractor not only in the limit φ → ∞ , but also in others regions.In order to study this point in more details let us write the set of equations (114, 115) asan autonomous system of three dimensions as follow:˙ H = − H + Λ3 + κ
12 ( M + V ) − U y (4 + 3 κ ǫ y ) U + κ ǫ ( M + V ) , (123)˙ y = 1 + κ ǫ y κ ǫ y − H y + α M ( M + V )( − κ ǫ y ) √ yU + κ ǫ ( M + V ) ! , (124)˙ φ = −√ y (125)Where we have defined y = ˙ φ . We are consider ˙ φ < −∞ to positive values, following the Emergent Universescheme. We can can note that, in general, the solutions H = H , y = y is stable. Thissolution correspond to a flat effective potential and φ rolling with constant ˙ φ . In this case, wecan past over the point φ = φ ∗ , see numerical solutions Fig.(5). Also, we can observed thatsolutions near the kinetic vacuum solution can pass over this point , because this solutionis an attractor. The general behaviour could be see in the Fig. (4) where it is plot the Direction Field for the effective two dimensional autonomous system in variables H and y φ . Thefirst plot correspond to the limit φ → ∞ the second is for φ & φ ∗ , and the third correspondto φ ≪ φ ∗ .In order to study in a more systematic way the nature of the kinetic vacuum state welinearize the Eqs.(123, 124,125) near the critical point H , y leaving φ arbitrary. We obtainthe following two dimensional effective autonomous system with variables δH and δy : δ ˙ H = − H δH + χ δy ǫ h κ ( V + M ) + χ κ ǫ i , (126) δ ˙ y = − H + 2 α M ǫ κ ( M + V ) √ y U + κ ǫ ( M + V ) ! δy . (127)The eigenvalues of equations (126), (127) are: λ = − H , (128) λ = − H + 2 α M ǫ κ ( M + V ) √ y U + κ ǫ ( M + V ) . (129)The equilibrium point is stable (attractor) if the eigenvalues are negative. Then, depend-ing on the values of the parameters of the models, this is the case for a large set of values of φ , not only for the case φ → + ∞ discussed previously in [28]. In particular for the numericalvalues used in [28], which are consistent with the stability of the ES universe, the criticalpoint is stable for φ > φ ∗ , see Fig. (4).It is interesting to note that when φ → −∞ , then λ → − H + 2 α √ y . (130)For example, if we consider the numerical values used in [28] this is a positive number.This means that at some value of φ < φ ∗ , when the scalar field moves to −∞ the stable(attractor) equilibrium point becomes unstable (Focus), see Fig. (4).In order to study numerical solutions we chose the following values for the free parametersof the model, in units where κ = 1; f = 1, f = 1 / ǫ = − α = 1, Λ = 0 .
35 and M = − φ >>
0, seeRef.[28]. Under this assumptions we obtain that:33 ∗ = − .
40 (131) H = 0 .
18 (132) y = 1 (133)Numerical solution to the Eqs.(123, 124, 125) are show in Fig.(5), where we can notethat the point φ = φ ∗ is passed on during the evolution of the scalar field. This situation isachieved by the kinetic vacuum solution, but also by others solutions which decays to thekinetic vacuum solution before arrive to the point φ = φ ∗ , see Fig. (5).The figure (5) left panel shown a projection of the axis H and φ and the evolution ofsix numerical solutions. The right panel shown a projection of the axis y and φ and theevolution of six numerical solutions. XI. DISCUSSION, THE CREATION OF THE UNIVERSE AS A “THRESHOLDEVENT” FOR ZERO PRESENT VACUUM ENERGY DENSITY, WHEN DOESTHE BOUND RESTRICT US TO A SMALL VACUUM ENERGY DENSITY FORTHE LATE UNIVERSE?
We have considered a non-singular origin for the Universe starting from an Einstein staticUniverse, the so called “emergent universe” scenario, in the framework of a theory whichuses two volume elements √− gd x and Φ d x , where Φ is a metric independent density, usedas an additional measure of integration. Also curvature, curvature square terms and forscale invariance a dilaton field φ are considered in the action. The first order formalismwas applied. The integration of the equations of motion associated with the new measuregives rise to the spontaneous symmetry breaking (S.S.B) of scale invariance (S.I.). AfterS.S.B. of S.I., using the the Einstein frame metric, it is found that a non trivial potentialfor the dilaton is generated. One could question the use of the Einstein frame metric g µν incontrast to the original metric g µν . In this respect, it is interesting to see the role of boththe original metric and that of the Einstein frame metric in a canonical approach to the firstorder formalism. Here we see that the original metric does not have a canonically conjugatedmomentum (this turns out to be zero), in contrast, the canonically conjugated momentum tothe connection turns out to be a function exclusively of g µν , this Einstein metric is therefore34 .2 0.4 0.6 0.8 1.0 y - - - H y - - - H y - - - H FIG. 4: Plots showing part of the
Direction Field of the system for different values of φ and somenumerical solutions. a genuine dynamical canonical variable, as opposed to the original metric.There is also a lagrangian formulation of the theory which uses g µν , what we can callthe action in the Einstein frame. In this frame we can quantize the theory for exampleand consider contributions without reference to the original frame, thus possibly consideringbreaking the TMT structure of the theory, but such breaking will be done “softly” throughthe introduction of a cosmological term only. Surprisingly, the remaining structure of thetheory, reminiscent from the original TMT structure will be enough to control the strengthof this additional cosmological term once we demand that the universe originated from anon-singular and stable emergent state. 35 - -
50 0 50 100 150 Φ H -
100 100 200 300 Φ y FIG. 5: Plots showing some numerical solution of the Eqs.(123, 124, 125).
In the Einstein frame we argue that the cosmological term parametrizes the zero pointfluctuations.The resulting effective potential for the dilaton contains two flat regions, for φ → ∞ relevant for the non-singular origin of the Universe, followed by an inflationary phase andthen transition to φ → −∞ , which in this paper we take as describing our present Universe.An intermediate local minimum is obtained if f /M <
0, the region as φ → ∞ has a higherenergy density than this local minimum and of course of the region φ → −∞ , if A > f > A > φ → ∞ (after the emergent phase). The dynamics of the scalar field becomes non linear and thesenon linearities are instrumental in the stability of some of the emergent universe solutions,which exists for a parameter range of values of the vacuum energy in φ → −∞ , which mustbe positive but not very big, avoiding the extreme fine tuning required to keep the vacuumenergy density of the present universe small. A sort of solution of the Cosmological ConstantProblem, where an a priori arbitrary cosmological term is restricted by the consideration ofthe non-singular and stable emergent origin for the universe.Notice then that the creation of the universe can be considered as a “threshold event”for zero present vacuum energy density, that is a threshold event for ∆ λ = 0 and we canlearn what we can expect in this case by comparing with well known threshold events. Forexample in particle physics, the process e + + e − → µ + + µ − , has a cross section of the form36ignoring the mass of the electron and considering the center of mass frame, E being thecenter of mass energy of each of the colliding e + or e − ), σ e + + e − → µ + + µ − = πα E (cid:20) m µ E (cid:21) r E − m µ E (134)for E > m µ and exactly zero for E < m µ . We see that exactly at threshold this crosssection is zero, but at this exact point it has a cusp, the derivative is infinite and thefunction jumps as we slightly increase E . By analogy, assuming that the vacuum energycan be tuned somehow (like the center of mass energy E of each of the colliding particles inthe case of the annihilation process above), we can expect zero probability for exactly zerovacuum energy density ∆ λ = 0, but that soon after we build up any positive ∆ λ we willthen able to create the universe, naturally then, there will be a creation process resultingin a universe with a small but positive ∆ λ which represents the total energy density forthe region describing the present universe, φ → −∞ or by the kinetic vacuum (which isdegenerate with that state).One may ask the question: how is it possible to discuss the “creation of the universe” inthe context of the “emergent universe”?. After all, the Emergent Universe basic philosophyis that the universe had a past of infinite duration. However, that most simple notionof an emergent universe with a past of infinite duration has been recently challenged byMithani and Vilenkin [98], [99] at least in the context of a special model. They have shownthat an emergent universe, although completly stable classically, could be unstable undera tunnelling process to collapse. On the other hand, an emergent universe can indeed becreated from nothing by a tunnelling process as well.An emerging universe could last for a long time provided it is classically stable, thatis where the constraints on the cosmological constant for the late universe discussed herecome in. If it is not stable, the emergent universe will not provide us with an appropriate“intermediate state” connecting the creation of the universe with the present universe. Theexistence of this stable intermediate state provides in our picture the reason for the universeto prefer a very small vacuum energy density at late times, since universes that are created,but do not make use of the intermediate classically stable emergent universe will almostimmediately recollapse, so they will not be “selected”. Finally, it could be that we arriveto the emergent solution not by quantum creation from nothing, by the evolution fromsomething else, for example by the production of a bubble in a pre-existing state [100], from37ere we go on to inflation.Notice that the bound gives a small vacuum energy density, without reference to thethreshold mechanism mentioned before. For this notice that upper the bound on the presentvacuum energy density of the universe contains a 1 /ǫ suppression. If we think of the R term as generated through radiative corrections, ǫ is indeed formally infinite, in dimensionalregularization goes as ǫ = K/ ( D − D − ǫ means a very strict bound on thepresent vacuum energy density of the universe. XII. ACKNOWLEDGEMENTS
We would like to thank Sergio del Campo and Ramon Herrera, our coauthors in ref. [28],which is central for our review, for our crucial collaboration with them and for discussionson all the subjects in this review, in particular we have benefited from our discussionsconcerning the aspects related to inflation, slow roll, etc. in the context of the model studiedhere. We also want to thank Zvi Bern, Alexander Kaganovich and Alexander Vilenkinfor very important additional discussions. UCLA, Tufts University and the University ofBarcelona are thanked for their wonderful hospitality. P. L. has been partially supported byFONDECYT grant N
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