Stability of the Einstein static universe in presence of vacuum energy
aa r X i v : . [ a s t r o - ph . C O ] A ug Stability of the Einstein static universe in presence of vacuumenergy
Saulo Carneiro , ∗ and Reza Tavakol † Astronomy Unit, School of Mathematical Sciences,Queen Mary University of London,Mile End Road, London E1 4NS, UK Instituto de F´ısica, Universidade Federal da Bahia, Salvador, BA, 40210-340, Brazil
Abstract
The Einstein static universe has played a central role in a number of emergent scenarios recentlyput forward to deal with the singular origin of the standard cosmological model. Here we study theexistence and stability of the Einstein static solution in presence of vacuum energy correspondingto conformally-invariant fields. We show that the presence of vacuum energy stabilizes this solutionby changing it to a centre equilibrium point, which is cyclically stable. This allows non-singularemergent cosmological models to be constructed in which initially the universe oscillates indefinitelyabout an initial Einstein static solution and is thus past eternal. ∗ ICTP Associate Member. E-mail address: [email protected]. † E-mail address: [email protected]. . INTRODUCTION Recent accumulation of high resolution observations is compatible with the so calledstandard model of cosmology which has a number of intriguing features. In addition tothe early and late accelerating phases, which are difficult to account for within the classicalrelativistic framework with non-exotic matter sources, this model also possesses an initialsingular state at which the laws of physics break down. To deal with this latter shortcoming,a number of attempts have recently been made to construct models which are non-singularand/or past eternal . These fall into a number of groups, including emergent scenarios[3, 4, 5] (see also [6, 7]) and cyclic/ekpyriotic models [8]. There are also a number ofother cosmological models which are not necessarily recurrent, but nevertheless are non-singular in the past. These include the pioneering non-singular model by Bojowald based onmodifications due to Loop Quantum Gravity [9], the model based on polymer matter [10]and others [11].The so called emergent scenarios, which are non-singular and past eternal, come in twovarieties: those that employ classical general relativity [3, 4], and those that incorporatequantum effects which are expected to be present at early phases of the universe [5]. Animportant ingredient in the construction of these models is the Einstein static solution,which in the classical general relativistic setting has long been known to be unstable . Thisinstability makes the construction of emergent models within the framework of classicalrelativity difficult. Interestingly though, quantum effects which are known to be operativeat early phases of the universe have recently been shown to be able to stabilise the Einsteinstatic solution by changing it from a hyperbolic equilibrium point (which is unstable) toa centre equilibrium point, which is cyclically stable. This was first shown to be the casein presence of quantum modifications due to Loop Quantum Gravity effects in [5] (seealso [13, 14] and other related works in this connection [15, 16]). In most cases, wherethe singularity is removed, the overall effect is to change the effective evolution equationsin such a way which allow the singularity theorems to be circumvented. For example, in The history of attempts at constructing non-singular/oscillatory universes which are past eternal goesmuch further back to at least the work of Tolman [1] (see also [2] and references therein for a review ofnon-singular models). This solution has, however, been shown to be stable with respect to inhomogeneous perturbations [12]). < − a − , where a is the scale factor.This result has been confirmed more recently by other studies which show that the Casimirenergy in such backgrounds is proportional to a − [18]. We study the effects of presence ofthis vacuum energy on the dynamics of the universe and show that it has the consequenceof stabilising the Einstein static solution by changing it into a centre equilibrium point.The outline of the paper is as follows. In Section II we perform a general analysis of theequilibrium of Einstein solution in the presence of matter and the vacuum energy. In SectionIII we consider a numerical example, in which the energy content is assumed to consist ofrelativistic matter plus a vacuum term with negative pressure. In Section IV we study theexistence of Einstein static solution in presence of a single scalar field, with the vacuumcontribution represented by a self-interaction potential. We then briefly discuss possiblealternatives for the exit from the initial Einstein phase to an inflationary one. We concludewith a discussion in Section V. II. STUDY OF THE EINSTEIN UNIVERSE IN THE GENERAL SETTING
In this section we study the effects of vacuum energy due to conformal fields on thedynamics of the universe, and in particular the way this affects the existence and the stabilityof the Einstein static solution. As was mentioned above the vacuum energy of a conformalscalar field in a spatially closed static universe of radius a has been shown to have a densitygiven by ρ Λ = C/a , where C is a positive constant [17]. Initially it was also proposed thatthe corresponding pressure should have the form p Λ = ρ Λ /
3, to ensure the vacuum energy-momentum tensor is traceless in order to respect the conformal symmetry. However, with3he discovery of trace anomaly such a requirement is not necessary. In the de Sitter space-time the equation-of-state parameter of the vacuum is ω Λ = −
1, due to the symmetry of thebackground. This result, however, cannot be assumed to hold in other more general space-times. Here, therefore, we shall proceed by first considering the general case, p Λ = ω Λ ρ Λ ,where ω Λ is allowed to take arbitrary values. We shall then find the conditions for thestability of the Einstein static universe in terms of ω Λ .Starting with a closed isotropic and homogeneous Friedmann-Lemaˆıtre-Robertson-Walkermodel sourced by a general fluid with total density ρ and total pressure p , the evolutionequations are given by 3 H = ρ − a , (1)˙ ρ + 3 H ( ρ + p ) = 0 , (2)where H is the Hubble parameter. We shall assume the fluid to consist of a combinationof the above vacuum energy plus matter with p m = ω m ρ m , where ρ m , p m and ω m are thecorresponding density, pressure and equation-of-state parameter. Substituting these in theabove evolution equations and letting ρ Λ = C/a , the Raychadhuri equation can be writtenas ¨ a = − ˙ a + 12 a (1 + 3 ω m ) + C a ( ω m − ω Λ ) . (3)The Einstein static solution is given by ¨ a = 0 = ˙ a . To begin with we obtain the conditionsfor the existence of this solution. The scale factor in this case is given by a ES = C ( ω m − ω Λ )3 ω m + 1 . (4)The existence condition reduces to the reality condition for a ES , which for a positive C takesthe forms ω m > − / ω Λ < ω m , (5)or ω m < − / ω Λ > ω m . (6)Therefore, in the case of ordinary matter ( ω m ≥
0) plus a positive vacuum energy withnegative pressure (i.e. with ω Λ < . It is interesting to note that conditions (5)-(6) exclude the case ω Λ = 1 /
3, originally proposed in [17].
4o study the stability of this solution, it is helpful to cast the equation (3) as a 2-dimensional dynamical system by introducing the phase-space variables x = a and x = ˙ a ,˙ x = x , (7)˙ x = − x + 12 x (1 + 3 ω m ) + C x ( ω m − ω Λ ) . (8)In these variables the Einstein static solution corresponds to the fixed point ( x = a ES , x =0). The stability of this equilibrium point is readily found by looking at the eigenvalues, λ ,of the Jacobian matrix J ij = ∂ ˙ x i /∂x j evaluated at this point, which are found to be λ = − C ( ω m − ω Λ ) a ES . (9)The stability depends on the sign of λ . For λ >
0, the Einstein static solution isa hyperbolic fixed point and hence unstable, in the sense that trajectories starting in theneighbourhood of such a point exponentially diverge from it (this is the same as the classicalrelativistic case). For λ <
0, on the other hand, the Einstein static solution becomes a centreequilibrium point, which is circularly stable, in the sense that small departures from the fixedpoint results in oscillations about that point rather than exponential deviation from it. Inthis case the universe stays (oscillates) in the neighbourhood of the Einstein static solutionindefinitely. Thus the condition for stability is given by λ <
0. For
C >
0, this impliesthat ω Λ < ω m . Comparing this inequality with the conditions for existence of the Einsteinstatic solution, (5)-(6), we conclude that, given ω Λ < ω m , the Einstein universe is stablefor ω m > − /
3. In particular, it is stable in presence of ordinary matter ( ω m ≥
0) plus apositive vacuum energy with negative pressure .To close this section, we note that perturbations about the fixed point imply perturbationsin a which in turn imply perturbations in ρ Λ . In the case of vacuum energy with ω Λ = − ω Λ = −
1, the stability is In the case of a negative vacuum energy, that is
C <
0, the conditions for existence and stability of theEinstein solution are given by ω Λ > ω m > − /
3, which are satisfied by any ordinary matter, provided thenegative vacuum term has negative pressure. - - FIG. 1: The scale factor as a function of time (left) and a typical trajectory in the phase space( a, ˙ a ) (right). achieved only for ω m < − / III. NUMERICAL STUDY OF THE COSMOLOGICAL DYNAMICS
In this section we make a brief quantitative study of the effects of the vacuum energyon the dynamics of the universe. As an example, we consider the case where the energycontent consists of vacuum energy, which we assume to have ω Λ = − ρ Λ = C/a , plusa relativistic matter with an equation-of-state parameter ω rm = 1 /
3. Using these equation-of-state parameters in Eq. (3) we obtain3 a ¨ a + 3 a ˙ a + 3 a − C = 0 . (10)For the Einstein static solution the corresponding scale factor is given by a = 2 C/ a (0) = 1 and ˙ a (0) = 0,together with 2 C = 3 . ρ Λ and a constant ρ rm a for relativistic matter. In our case, however, the effect of the coupling is to make thesequantitities oscillate with time, as can be seen from Fig. 2.6 - - - FIG. 2: The time dependence of ρ rm a (left) and ρ Λ (right). IV. THE SCALAR FIELD ANALOGY
As discussed above, a perturbed Einstein universe with ρ Λ = C/a and ω Λ = − φ with aself-interaction potential V . This can be done by interpreting the usual expressions for theenergy density and the pressure of the scalar field, ρ φ = ˙ φ / V, (11) p φ = ˙ φ / − V, (12)as a sum of a vacuum energy component with density V and pressure − V , plus a stiff fluidcomponent with density and pressure equal to ˙ φ /
2. Now, when the scale factor is perturbed,there is an energy exchange between the potential energy V (here representing the vacuumand taken to be V = C/a ) and the kinetic energy ˙ φ / L = √− g (cid:20) R −
12 ( ∂φ ) − V ( φ ) (cid:21) . (13)For a FLRW universe with positive curvature, the evolution equations take their usual forms3 H = V + ˙ φ − a , (14)˙ φ − ˙ H + 1 a . (15)Using (11)-(12), it is not difficult to cast the above system in the form of the evolutionequations (1)-(2). Taking for the potential the expression V = C/a , we can derive the7aychaudhuri equation in the form a ¨ a + 2 a ˙ a + 2 a − C = 0 . (16)The behaviour of this system can be readily studied using the general analysis given inSection II, by recalling that in this case we have a mixture of a vacuum term (with ω Λ = − ω m = 1). The Einstein static solution again corresponds to ¨ a = 0 = ˙ a ,which in this case gives the corresponding scale factor to be a = C/
2. A similar analysisto that used above shows that this fixed point is again a centre.It is also instructive to briefly re-visit the original classical emergent model proposed in[3] in terms of this alternative formulation of the dynamics in terms of a scalar field. Inthat scenario also, the initial Einstein static phase has a scalar field as its energy content.We can see from (14)-(15) that, for the Einstein static solution, ˙ φ and V are both non-zeroconstants. Therefore, while φ is changing with time, V is not. In other words, the scalarfield is rolling along a potential plateau. As discussed in [3], this plateau may be consideredas the past-asymptotic limit of a smoothly decreasing potential, which eventually leads toan exit from the Einstein static regime into an inflationary phase. Specific forms of such apotential have been considered in [3, 4, 5].Finally, another possibility that may be considered is that of a complex scalar field. Forexample, with a harmonic field φ = φ e iωt we have ˙ φ = ω φ , and equations (14)-(15) aresimultaneously satisfied, with V = ω φ . Therefore, V remains constant while φ rotates inthe complex plane. The stability of the solution indicates that this is a local minimum of V , and the exit to the inflationary phase may involve, for example, a tunnelling to a globalminimum. V. CONCLUSION
We have studied the existence and stability of the Einstein static universe in presenceof vacuum energy corresponding to conformally-invariant fields. Using the result that thevacuum energy density in Einstein universe is proportional to the inverse fourth power ofthe scale factor, we have found the range of equation of state parameters for the vacuumenergy such that the Einstein universe is stable, in the sense of dynamically correspondingto a centre equilibrium point. The importance of such a solution is due to the central role8t plays in the construction of non-singular emergent oscillatory models which are pasteternal, and hence can resolve the singularity problem in the standard cosmological scenario.Given that the oscillatory universe discussed above is close to but not exactly an Einsteinuniverse, the form of the vacuum energy density in the initial oscillatory phase of the universemay depart from the above inverse fourth power form. To partially answer what happensif vacuum density takes other forms, we considered, as a first step in this direction, a moregeneral functional form of the type ρ Λ = C/a n . Proceeding in a similar manner to that used above we have been able to show that, in thecase where the content of the universe consists of radiation (with ω m = 1 /
3) plus vacuumenergy (with ω Λ = − n > Acknowledgements
Saulo Carneiro was partially supported by CAPES (Brazil). [1] R.C. Tolman. Relativity, Thermodynamics and Cosmology (Clarendon Press, Oxford, 1934).[2] M. Novello and S.E. Perez Bergliaffa, Phys. Rept. , 127, 2008.[3] G.F.R. Ellis and R. Maartens, Class. Quant. Grav. , 223, 2004.[4] G.F.R. Ellis, J. Murugan and C.G. Tsagas, Class. Quant. Grav. , 233, 2004.[5] D.J. Mulryne, R. Tavakol, J.E. Lidsey and G.F.R. Ellis, Phys. Rev. D 71 , 123512, 2005.[6] S. Carneiro, Phys. Rev.
D 61 , 083506, 2000.[7] S. Carneiro and R. Tavakol, arXiv:0905.3131, to appear in Gen. Rel. Grav.[8] J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, Phys. Rev.
D 64 , 123522 (2001); P.J.Steinhardt and N. Turok, Science , 1436 (2002); P.J. Steinhardt and N. Turok, Phys.Rev.
D 65 , 126003 (2002); J. Khoury, P.J. Steinhardt and N. Turok, Phys. Rev. Lett. ,031302 (2004); Y. Shtanov and V. Sahni, Phys. Lett. B 557, 1 (2003); J. D. Barrow, D. imberly and J. Magueijo, Class. Quant. Grav. , 4289 (2004); J. D. Barrow and C. G.Tsagas, arXiv:0904.1340.[9] M. Bojowald, Phys. Rev. Lett. , 5227 (2001).[10] G.M. Hossain, V. Husain and S.S. Seahra, arXiv:0906.2798 [astro-ph].[11] V. Sahni and L. A. Kofman, Phys. Lett. A 117 , 275 (1986); R. Ferraro and F. Fiorini, Phys.Rev.
D 78 , 124019 (2008).[12] J.D. Barrow, G.F.R. Ellis, R. Maartens and C. Tsagas, Class. Quant. Grav. , L155, 2003.[13] D.J. Mulryne, N.J. Nunes R. Tavakol and J.E. Lidsey, Int. J. Mod. Phys. A 20 , 2347, 2005.[14] G.V. Vereshchaign, JCAP, , 013 (2004).[15] C. G. Boehmer, L. Hollenstein and F. S. N. Lobo, Phys. Rev.
D 76 , 084005 (2007); R.Goswami, N. Goheer and P.K.S. Dunsby, Phys. Rev.
D 78 , 044011, 2008; U. Debnath, Class.Quant. Grav. , 205019, 2008; S.S. Seahra and C.G. Boehmer, Phys. Rev. D 79 , 064009,2009; C.G. Boehmer and F.S.N. Lobo, Phys. Rev.
D 79 , 067504, 2009.[16] L. Parisi, M. Bruni, R. Maartens, K. Vandersloot, Class. Quant. Grav. , 6243, 2007.[17] L.H. Ford, Phys. Rev. D 11 , 3370, 1975; Phys. Rev.
D 14 , 3304, 1976.[18] E. Elizalde and A.C. Tort, Mod. Phys. Lett.
A 19 , 111, 2004; I.H. Brevik, K.A. Milton andS.D. Odintsov, hep-th/0210286., 111, 2004; I.H. Brevik, K.A. Milton andS.D. Odintsov, hep-th/0210286.