On the Stability of Einstein Static Universe in Doubly General Relativity Scenario
aa r X i v : . [ g r- q c ] D ec On the Stability of Einstein Static UniverseinDoubly General Relativity Scenario
M. Khodadi ∗ , Y. Heydarzade † , K. Nozari ‡ and F. Darabi , § Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran Department of Physics, Azarbaijan Shahid Madani University, Tabriz, 53714-161, Iran Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha 55134-441, Iran
August 20, 2018
Abstract
By presenting a relation between average energy of the ensemble of probe photons and energydensity of the Universe, in the context of gravity’s rainbow or doubly general relativity scenario,we introduce a rainbow FRW Universe model. By analyzing the fixed points in flat FRW modelmodified by two well known rainbow functions, we find that the finite time singularity avoidance(i.e. Big-Bang) may still remain as a problem. Then, we follow the “Emergent Universe” scenarioin which there is no beginning of time and consequently there is no Big-Bang singularity. Moreover,we study the impact of a high energy quantum gravity modifications related to the gravity’s rainbowon the stability conditions of an “Einstein static Universe” (ESU). We find that independent of aparticular rainbow function, the positive energy condition dictates a positive spatial curvature for theUniverse. In fact, without raising a nonphysical energy condition in the quantum gravity regimes,we can address an agreement between gravity’s rainbow scenario and basic assumption of modernversion of “Emergent Universe”. We show that in the absence and presence of an energy-dependentcosmological constant Λ( ǫ ), a stable Einstein static solution is available versus the homogeneous andlinear scalar perturbations under the variety of obtained conditions. Also, we explore the stabilityof ESU against the vector and tensor perturbations. Keywords : Doubly General Relativity, Gravity’s Rainbow, Einstein Static Universe, StabilityAnalysis. ∗ email: [email protected] † email: [email protected] ‡ email: [email protected] § email: [email protected]; Corresponding author Introduction
In the framework of general relativity (GR), the gravitational force is explained in terms of the space-time curvature so that the field equations connect the space-time geometry to the matter content. Moretechnically, GR exhibits a Universe modeled by space-time with a mathematical structure formed by fourdimensional differentiable manifold [1]. It is commonly believed that a unique mathematical frameworkto reconcile the quantum mechanics (QM) with GR is highly dependent on our understanding of thespace-time geometry. In other words, there is a comprehensive agreement that the geometry of space-time is fundamentally explained by a quantum theory. It is assumed that the Planck energy ǫ pl addressesa critical energy scale of transition from classical GR to quantum gravity (QG), a theory which issupposed to challenge the most fundamental and long standing issues of modern physics. Despite the lackof a complete theory of QG, it seems a semi-classical or effective phenomenological approach of QG mayguide us to disclose the mysterious nature of QG [2, 3, 4, 5]. It is important to understand how to extractthe testable predictions from this fundamental theory. Interestingly, all semi-classical approaches offeredso far unanimously insist on the existence of a minimal measurement length in the nature, known as thePlanck length l p . Besides, it is believed that the modified mass-shell condition (or dispersion relations)arising from deviation of Lorentz invariance in effective phenomenological approaches [4, 5], may justifysome of the phenomena taking place in astronomical and cosmological scale, like the threshold anomaliesof ultra high energy cosmic rays and TeV photons [6, 7]. It is noticeable that such Lorentz invarianceviolation is already predicted in the context of other approaches to QG such as: non-commutativegeometry [7], spin network in Loop quantum gravity [8], and string field theory [9]. Throughout thepresent work, our attention specifically is focused on a semi-classical formalism knows as “Gravity’sRainbow” which has been designed by Magueijo and Smolin [10]. In fact, it can be said that therainbow gravity is nothing but the doubly special relativity (DSR) [4] in the presence of curved space-time which is knows as the “Doubly General Relativity” (DGR). In this formalism, there is no single fixedspace-time background, namely the space-time background appears as a geometric spectrum in termsof the energy scale of the particle probe. Therefore, in the cosmological setting, we are dealing with therainbow modified metric by introducing a function in the Friedmann-Robertson-Walker (FWR) metricwhich depends on such variable energy scale. Formation of the rainbow metric proposal involve variousmotivations such as the lack of a trivial definition of the dual position space in DSR [10]. However, thisapproach to QG is not problem free and considerable number of works have been focused on its variousaspects [11]. Somehow, the proposal of gravity’s rainbow is similar to the idea of “Running CouplingConstants” in particle physics and field theory, in the sense that at very small distances (high energy) thespace-time geometry is related to the energy scale of the particle probe [12]. It is worth mentioning that,apart from the effective approaches to QG like DSR and DGR, it has been argued that if the physicalspace-time of the standard GR theory is emergent, then it is expected that we face with a radical pictureof the Universe at fundamental quantum scale [12]. The basic idea of “Emergent Universe” dates backto the works of Eddington in 1930 [13], inspired by the proposal of Einstein static Universe (ESU).In simple words, ESU suggests that the content of the Universe as a closed system is made up of atleast cosmological constant and normal matter. While ESU in the context of GR is not stable versusspatially homogeneous perturbations, it predicts that the Universe in past might have been emerged asan static initial states [14]. We know that the data obtained by cosmic microwave background (CMB)observations, generally supports the inflationary scenario. Also, by implementation of some singularitytheorems with the geometrical assumptions that K < K = 0, it has been shown that despite theoccurrence of inflation, the Universe had a violent beginning in the past [15]. Hence, one of the mostimportant motivations for studying QG is to avoid the singularity at the beginning of the Universe.Recovery of ESU in the framework of inflationary Universe by Ellise and Maartens, knows as modernversion of Eddington emergent Universe, was in this direction of research interest [16]. According tothe modern standard cosmology, the inflationary expansion of the Universe has quickly eliminated anyoriginal spatial curvature. On the other hand, the CMB observations implies Ω & . ± .
02) for thetotal density parameter. This means that the geometry of Universe is not exactly flat, rather it couldhave non-zero spatial curvature at the beginning, resulting in negligible late-time effects. In this sense,2he emergent Universe scenario allowed the existence of a positive curvature Universe that emanatesasymptotically as a static initial state known as ESU which afterwards experienced the inflation. Thestable ESU addresses the existence of a fixed point around which our Universe in the early times waseternally fluctuating. In the case of the Universe filled with a massless scalar field φ with a suitableinflaton potential, these fluctuations had been disappeared and the stable state of Universe had beenturned to inflationary phase. Within the context of standard cosmology, the prerequisites of a oneor further flat wings (of course with a little positive gradient) in the inflaton potential, can make anemergent Universe. This transmission from a stable ESU to inflationary phase takes place around thedifferent values of e-folds. For instance, in [14] by analyzing the spectrum of inflationary perturbationsof some stable cosmological models in the context of GR, it has been demonstrated that the transmissionoccurs in over 60 e-folds. Also, it should be noted that while these inflationary emergence GR-basedmodels respect to CMB constraints, they suffer from a fine-tuning problem. This problem can beresolved in the context of modified emergence models of GR. However, recently in [17] some argumentsare presented which tries to show the emergent cosmological scenario can not really be past-eternal. Asa prominent feature of emergent Universe scenario, one can point to the removal of initial singularityand also the horizon problem. This is a strong motivation to study the ESU models along with theirstability conditions in the presence of high energy corrections of GR. For instance, one can point tothe works done in the context of massive gravity [18], Hoˇrava-Lifshitz model of gravity [19], braneworldscenarios [20], induced matter theory [21], loop quantum cosmology [22], f ( R ), f ( T ) and f ( G ) gravity[23]. As discussed at the beginning of this section, gravity’s rainbow or DGR is also considered asan alternative of GR with high energy corrections. Besides, it has been shown that in the study ofFRW cosmology and isotropic quantum cosmological perfect fluid model in rainbow gravity setup, someconditions are derived which prevent the initial singularity [25]. Inspired by this introduction, our maingoal in this paper is to study the stability of ESU against the homogeneous scalar, vector and tensorperturbations in the presence of rainbow’s gravity setup as a QG modification of GR. But before doingthis, by following the phase space method we will analyze the status of the Big-Bang singularity in theframework of a flat rainbow FRW Universe model.The present paper is arranged as follows: In section 2, we have derive rainbow modified Friedmanequations in details for a general non-flat universe. In section 3, by introducing two different forms ofrainbow functions depending on energy density, firstly in absence of cosmological constant, we obtainthe stability conditions of ESU against the homogeneous linear scalar perturbations. In the follow ofthis section, we pursue our goal in the presence of a cosmological constant related to energy density viaintroducing a new rainbow function. In section 4, we analysis the stability against the vector and tensorperturbations. Finally in section 5, we give the summery and conclusions. In this section, firstly we have a quick review on the gravity’s rainbow theory, known also as DGR.Then, with note to the importance of spatial curvature in Emergent Universe scenario, we derive themodified Friedman equations for general non-flat universes. In DSR, the modified dispersion relationsof a massive particle with mass m reads as ǫ f ( ǫ/ǫ p , η ) − p g ( ǫ/ǫ p , η ) = m , (1)where f ( ǫ/ǫ p , η ) and g ( ǫ/ǫ p , η ) are the well known energy dependent rainbow functions and η is adimensionless parameter. In the low energy limit, the modified dispersion relation (1) reduces to therelativistic dispersion relations. So, the rainbow functions f ( ǫ/ǫ p , η ) and g ( ǫ/ǫ p , η ) satisfy the followingconditions lim ǫ/ǫ p → f ( ǫ/ǫ p , η ) = 1 , lim ǫ/ǫ p → g ( ǫ/ǫ p , η ) = 1 . (2)Indeed, the condition (2) points to the correspondence principle. Based on the discussion carried outin [26], for established position space in DSR, theories including free fields must also lead to the plane3ave solutions in flat space-time, despite satisfying the modified dispersion relations (1). For this reason,contraction between infinitesimal displacement dx a and momentum p a , must be linearly invariant, i.e dx a p a = dtǫ + dx i p i . (3)In fact, the linear contraction (3) guarantees the existence of the plane wave solutions. The rainbowmetric generally has the form of ds = − f ( ǫ/ǫ p , η ) dt + 1 g ( ǫ/ǫ p , η ) dx . (4)Using a one parameter family of energy momentum tensors, the gravity’s rainbow modified Einsteinequations will be written as G αβ ( ǫ/ǫ p ) = 8 πG ( ǫ/ǫ p ) T αβ ( ǫ/ǫ p ) + g αβ ( ǫ/ǫ p )Λ( ǫ/ǫ p ) , (5)so that G ( ǫ/ǫ p ) = h ( ǫ/ǫ p ) G and Λ( ǫ/ǫ p ) = h ( ǫ/ǫ p )Λ where h ( ǫ/ǫ p ) and h ( ǫ/ǫ p ) are energy dependentrainbow functions. This means that in DGR setup, the Newtonian gravitational constant and thecosmological constants are energy dependent such that in low energy limit, we recover G ( ǫ/ǫ p ) = G andΛ( ǫ/ǫ p ) = Λ. In order to achieve our main goal, in this section we will set Λ = 0 for simplicity, andbegin with the following modified FRW metric of a homogeneous and isotropic Universe ds = − N ( t ) f ( ǫ/ǫ p , η ) dt + 1 g ( ǫ/ǫ p , η ) a ( t ) h ij dx i dx j , (6)where h ij represents the spatial part of the metric. Commonly, it is assumed that the energy ǫ of particleprobe for each measurement is constant and independent of space-time coordinates. Nevertheless, suchassumption for the measurements at early Universe seems to be far from reality. Therefore, it is expectedthat the background metric of space-time throughout its evolution is affected by the energy ǫ of particleprobe. Then, it is reasonable to consider the evolution of energy ǫ with the cosmological time, denotedas ǫ ( t ). In this regard, we will derive the modified FRW equations. In doing so, the following ansatz isusually suggested [27] η = 1 , g ( ǫ, η ) = 1 , (7)where f has the form of f ( ǫ/ǫ p ). Therefore, the modified Einstein field equation can be written as R αβ = − πG ( ǫ ) S αβ ( ǫ ) , (8)where R αβ is Ricci tensor defined as follows R αβ = ∂ Γ λλα ∂x β − ∂ Γ λαβ ∂x λ + Γ λαµ Γ µβλ − Γ λαβ Γ µλµ , α, β = 0 , , , , (9)and S αβ ( ǫ ) is written in terms of the energy momentum tensor T αβ ( ǫ ) as S αβ ( ǫ ) = T αβ ( ǫ ) − g αβ T µµ ( ǫ ) . (10)To continue our calculations, we need the non-zero components of the affine connection as [27]Γ = − ˙ ff , Γ ij = f ˙ aaδ ij , Γ i j = δ ij ˙ aa i, j = 1 , , . (11)It is seen that unlike the usual FRW metric, for modified FRW metric (6), the components of the affineconnection with two time indices remain non-zero. By putting (11) into (9) and after a straightforwardcalculation, we obtain R = 3 ¨ aa + 3 ˙ aa ˙ ff , (12)4nd R ij = ˜ R ij − (cid:16) ( a ¨ a + 2 ˙ a ) f − f ˙ f ˙ aa (cid:17) δ ij , (13)where ˜ R ij denotes the purely spatial Ricci tensor defined as˜ R ij = ∂ Γ nni ∂x j − ∂ Γ nij ∂x n + Γ min Γ mjm − Γ mij Γ nnl . (14)The spatial components Γ ijk of the four dimensional affine connection are identical with those of affineconnection computed in three dimensions from the spatial three-metric h ij , i.e.Γ ijk = 12 h in (cid:18) ∂h jn ∂x k + ∂h kn ∂x j − ∂h jk ∂x n (cid:19) ≡ ˜Γ ijk , (15)where ˜Γ ijk denotes purely spatial affine connection and h ij is the inverse of the 3 × h ij . Therefore,we have Γ nij = kx n h ij which results in ˜ R ij = − kh ij . (16)Here, k has a geometrical interpretation. Indeed, it measures the spatial curvature with zero, negativeand positive k values corresponding to flat, open and closed Universes respectively [28]. The Ricci tensorexpression (13) can be rewritten as R ij = − (cid:16) ( a ¨ a + 2 ˙ a ) f − f ˙ f ˙ aa + 2 k (cid:17) h ij . (17)Now, for obtaining the components of S αβ , we consider the perfect fluid with the energy-momentumtensor as T αβ = ( ρ + p ) u α u β + pg αβ , (18)so that ρ and p represent the energy density and the pressure, respectively. Also, u α is the velocity fourvector defined by u α = ( f − , , , g αβ u α u β = −
1. We should note that the diagonalcomponents of the modified FRW metric tensor g αβ is (cid:0) − f − , h ii (cid:1) with the signature ( − , + , + , +) sothat the components of spatial tensor h ii are independent of the rainbow function f . Using the equation(10), one can decompose the time and spatial components of S αβ tensor as follows S tt = T tt + 12 g tt T, S ij = T ij − h ij a T , (19)where T tt = ρf − , T ij = h ij a p , (20)and T = g αβ T αβ = ( − ρ + 3 p ) . (21)By substituting expressions (20) and (21) into (19), we have S tt = 12 f ( ρ + 3 p ) , S ij = 12 ( ρ − p ) a h ij . (22)Ultimately, the first and second rainbow modified Friedman equations for the metric parameterized bythe varying energy probe (6), takes the form of( ˙ aa ) + ka f = 8 πGρ f , (23)¨ aa = − πGρ (1 + 3 ω )3 1 f − ˙ aa ˙ ff . (24)5ere, also for simplicity, it is assumed that gravitational constant G is independent of energy ǫ . Itshould be noted that the existence of ˙ f term in the second rainbow modified Friedman equations, doesnot mean the explicit dependence of rainbow function to the time i.e. f ( t ), at all. As explained above, f is an explicit function of the energy of the test particles which are probing the geometry of space-timein early Universe. Given the fact that ǫ can vary with respect to the evolution of Universe, so f can bean implicit function of cosmic time and not explicit. We note that, the explicit dependence of f to thetime, may be leads to this wrong result from equations (23) and (24) that they are nothing but the usualFriedman equations of GR so that the rainbow function f in this case is merely the gauge parameterdetermining the choice of time. Hence, based on the gauge freedom, one may choose the gauge of f = 1.In contrast to this misconception, the gravity’s rainbow theory is a high energy modified theory of GRin which according to correspondence principle, for the case of low energy limit i.e. ǫǫ pl → f → t may play the role of proper timebased on the gauge choice N = f ( ε ) f ω ( ε ) a ω [25]. As expected, in the limit of GR, t represents the propertime indicated by the choice of N = a ω . Also, by combining the modified Friedman equations (23) and(24), we get the following energy conservation equation˙ ρ + 3 ˙ aa ρ (1 + ω ) = 0 . (25)It can be mentioned that in framework of DGR, the form of equation of state (EOS) p = ωρ , remainsunchanged for massless prob particles such as photons while for the massive ones, this equation of statewill be modified, see [29, 10] for more discussion. Λ In this section, we plan to apply the linear homogeneous scalar perturbations in the vicinity of theEinstein static Universe and explore its stability against these perturbations. To achieve our goal, weneed to fix the rainbow function f . To this end, using dispersion relation offered in [30], we introducerainbow functions as follows f ( ǫ ) = (1 − ǫǫ P l ) − , (26)where via suggesting the average energy ¯ ǫ = c ρ ( c is some constant) [31], this rainbow function willbe rewritten as f ( ρ ) = (1 − ξρ ) − , (27)where ξ = cǫ Pl . The authors of [31], to reach the above relation considered a large ensemble of probphotons which are in thermal equilibrium. This assumption is reasonable since based on the standardmodel of cosmology, early universe passed a radiation dominated era with p = ρ . One may ask thequestion that why the identification ǫ ∼ ¯ ǫ is used to obtain the rainbow function (27)?. The answer isthat, we are deal with the energy of prob particle in the rainbow metric as the statistical mean value ofall prob photons in radiation domination. Indeed, we deal with the average effect of photon particles inradiation dominated era and no with a special elected photon from the radiation [27]. It is worthwhileto remind that the form of ¯ ǫ in terms of ρ is independent of the modified dispersion relation picked for aspecific model [24]. We also mention that the above rainbow function is valuable from the theoreticalviewpoint so that in [30] it is shown that in the absence of the varying speed of light (VSL) proposal, itcan removes the horizon problem in early Universe. Moreover, it is seen that when ρ or ¯ ǫ get their largestvalues, then f ( ρ ) or f (¯ ǫ ) becomes infinite which causes the time-like component of the rainbow metricvanishing. This means that the FRW metric which is modified with rainbow function (27) becomesdegenerate at that time and has no inverse. Indeed, a degenerate metric addresses the existence of6ther distinct possibility for lightlike dimension (other than timelike and spacelike dimensions). Also,remember that in the Palatini formulation of standard GR, degenerate metric also appears. One ofthe consequences the degenerate metric is that the curvature remains bounded and the topology ofspace-time can change. Overall, it is believed that singularities arisen from degenerate metric have amilder manner than other type of singularities and seem appropriate for a QG proposal [32]. Here, it isnecessary to review the finite time singularity issue, including the Big-Bang singularity, in the presenceof rainbow function (27). Indeed, we want to investigate whether the presence of rainbow function (27)in this system will lead to resolve the singularity problem. Inspired by the idea proposed in [33, 31],we want to resolve this problem by finding an upper bound on the density of energy ρ . In other words,according to [33, 31] the finite time singularity issue will be eliminated via the presentation of a fixedpoint as ρ f for energy density ρ which is reached at an infinite time. More exactly, the author of [31], byfollowing the terminology used for stability analyzing the dynamical systems in [33], demonstrated thatfor any first order system as ˙ ρ = O ( ρ ), the finite time singularity will be solved through the fulfillmentof either of the following conditions: 1) the function O ( ρ ) be a continuous and differentiable on a rangeenclosed by zeroes of function O ( ρ ). 2) asymptotically, function O ( ρ ) growths like a linear function as K ( ρ ) or slower than it, i.e K ( ρ ) ≥ O ( ρ ). Therefore, for cancelation of the finite time singularity, it issufficient that one of these conditions is satisfied. We begin our analysis in this way by substitutingthe rainbow function (27) into energy conservation equation (25) and modified first Friedmann equation(23) (for case k = 0) to obtain the ordinary differential equation (ODE) as ˙ ρ = O ( ρ ) where O ( ρ ) = − ρ (1 − ξρ ) (cid:18) πG ρ (cid:19) / . (28)The advantage of the dynamical system analysis method followed in [33, 31] is that by having fixedpoints and regarding the asymptotic behavior of O ( ρ ), one is able to predict the demeanor of the systemwithout need to have a detailed form of the solutions. Hence, by setting ω = (since our attentionis on radiation dominated state in order to study the initial singularity), equation (28) results in thefollowing two fixed points ρ f = 0 , ρ f = ( ǫ P l c ) . (29)However, to understand the qualitative manner of a solution, we should know that how long it takes toget a fixed point. By a straightforward calculation, one can show that the time required to reach thesefixed points, can be obtained as t = − Z ρ f ρ ∗ dρ (cid:0) πG ρ (cid:1) / (cid:16) ρ − ξ ρ / (cid:17) , (30)where ρ ∗ represents an arbitrary initial finite value for density which lies in the intervals between thefixed points. The negative sign in the back of integral relation, denotes a backward in direction of time.We find that for the case of ρ f = 0, the integral (30) does not converge i.e. | t | → ∞ , while for the fixedpoint ρ f = ( ǫ Pl c ) the integral (30) is not solvable and its numerical values fail to converge. At thispoint, let us to draw the plot O ( ρ ) − ρ (or ˙ ρ − ρ ) in the figure 1, in order to get a qualitative analysisof the situation. As it is seen from the figure 1, O ( ρ ) has two fixed points ρ f = 0 and ρ f = 1. Infact, any of these fixed points are equivalent to a de Sitter space since in these points we are dealingwith constant solution ( ˙ ρ = 0). The first ρ f is a future fixed point because for the case ρ ∗ > O ( ρ ) <
0, while the second ρ f is an early (or past) fixed point since for case of ρ ∗ < ρ ∗ > O ( ρ ) < O ( ρ ) >
0, respectively. It is noteworthy that for distinction between the type of thissingularities, we follow Ref. [31]. These fixed points classify the possible solutions into two classes: 1) asolution belongs to interval ρ ∈ [0 , ρ f < ρ ∗ < ρ f and 2) a solution belongs to interval ρ ∈ [1 , ∞ ),if ρ ∗ > ρ f . Then, to get the fixed point ρ f by beginning from some initial value as ρ ∗ and also from ρ f to ρ ∗ , the time required is infinite. In this interval, O ( ρ ) is continuous and differentiable. Then,according to approach in [31] by pursuing the terminology of [33], one can say that the first solution7 .0 0.2 0.4 0.6 0.8 1.0 1.2 - - Ρ O H Ρ L Figure 1:
The behavior of O ( ρ ) (28) in terms of ρ for flat rainbow FRW Universe model. To simplify, we set ǫ pl = c and 8 πG = 3. Fixed points are located in ρ f = 0 and ρ f = 1. is free of physical singularity so that it interpolates steadily from ρ f to ρ f . It is clear that the firstcondition already is not usable for the second solution, which represents a clear violation of the secondcondition. On the other hand, for the case ρ ∗ > ρ f , the function (28) is growing quicker than a linearfunction. So, we conclude that the second solution is not free of a finite time singularity. Because weare interested in studying initial singularity problem, so let us mention to stability status of early fixedpoint ρ f . A fixed point is stable if by putting a neighboring initial value, the trajectory of the solutionremains always near to the fixed point. Equivalently, a fixed point is unstable if for any point at vicinityof the fixed point, one can find some solution that starts near the fixed point but go away from it in afinite time. The fixed point ρ f is an unstable point because dO ( ρ ) dρ | ρ = ρ f > . Then, although it takesan infinite time to get a fixed point, the issue of finite time singularities avoidance remains unsolved andeventually this fixed point will collapse.Now, we investigate the DGR theory form the emergent universe point of view in which there isno beginning of time and consequently, there is no big bang singularity in the early Universe. In thisscenario, the Universe did not born from a Big-Bang singularity in a past finite time and rather itpossesses an eternal Einstein static state. In this context, the key point is the required conditions forthe stability of the existing ESU, in which we will explore in the following of the present paper. TheESU in the DGR scenario with rainbow functions varying with cosmological time can be obtained bythe conditions ¨ a = ˙ a = 0 through the equations (23) and (24) as ka = 8 πGρ , (31)and − πGρ (1 + 3 ω )3 f ( ρ ) = 0 , (32)where a , ρ and ω denote the scale factor, the energy density of the Einstein static Universe andbarotropic equation of state parameter ( p = ωρ ), respectively. Considering the positive energy condition ρ > k >
0. Also, Eq.(32), shows that for the Einstein static Universe in the At a fixed point where O ( ρ f ) = 0, if dO ( ρ ) dρ | ρ = ρ f >
0, we have increasing O at ρ f , or equivalently f ( ρ f − δ ) <
0, we conclude that the fixed point ρ f is unstable.Similarly, one can show that if dO ( ρ ) dρ | ρ = ρ f <
0, the fixed point will be stable. ω = − or f ( ρ ) → ∞ corresponding to ρ = (cid:0) ξ (cid:1) − .The perturbations in the cosmic scale factor a ( t ) and the energy density ρ ( t ) can be written as a ( t ) → a (1 + δa ( t )) ,ρ ( t ) → ρ (1 + δρ ( t )) . (33)Substituting (33) into the equation (23) after linearizing the perturbation terms, we obtain the followingequation − ka δa = 8 πG ρ δρ . (34)This indicates, with respect to the positiveness of k , that the sign of variation of the scale factor mustbe opposite to the sign of variation of the matter density. We must also apply the same method onthe equation (24), however before doing this let us introduce the following replacement due to theperturbation in the rainbow function f ( ρ ) f − ( ρ ) → (cid:18) f − ( ρ ) − ξρ δρ (cid:19) , (35)such that f − ( ρ ) = 1 − ξρ . (36)Also, by combining the energy conservation equation (25) with the first rainbow modified FRW equation(23), we obtain − ˙ aa ˙ ff = − ˙ aa dfdρ . ˙ ρf = 3(1 + 3 ω )4 ξf − ρ (cid:18) πG ρ − ka (cid:19) . (37)Then, after applying perturbation (33), we have − ˙ aa ˙ ff = 3(1 + 3 ω )4 ξρ (1 + 14 δρ ) (cid:18) f − ( ρ ) − ξ ρ δρ (cid:19) (cid:18) πG ρ (1 + δρ ) − ka (1 − δa ) (cid:19) . (38)By substituting (31) and (34) into the above expression, one finds that − ˙ aa ˙ ff = 0. It seems that thisresult is independent of any specific form of rainbow function f . Therefore, it can be seen that the lastterm of the second rainbow modified FRW equation (24) does not contribute in derivation of stabilityconditions of Einstein static Universe in the framework of DGR scenario. Now, putting perturbationequations (33) into (24) and using (35), we get δ ¨ a = − πG ω ) (cid:18) f − ( ρ ) − ξρ (cid:19) ρ δρ , (39)where inserting the equation (34) into (39) and neglecting the non-linear perturbation terms, results inthe following differential equation δ ¨ a − ka (1 + 3 ω ) (cid:18) − ξρ (cid:19) δa = 0 . (40)It is obvious that, for the cases ξ →
0, i.e. ǫ P l → ∞ , Eq.(40) will take the following form δ ¨ a − ka (1 + 3 ω ) δa = 0 , (41)which refers to the oscillatory modes of ESU in the framework of standard GR for ω < − /
3. In orderto have the general oscillating perturbation modes in the framework of the DGR scenario, the followingcondition should be satisfied − ka (1 + 3 ω ) (cid:18) − ξρ (cid:19) > , (42)9hich results in the following solution for the equation (40) δa = α e iγ t + α e − iγ t , (43)where α and α are integration constants and γ refers to the frequency of oscillation around the stableESU as γ = (cid:18) − ka (1 + 3 ω ) (cid:18) − ξρ (cid:19)(cid:19) (44)Now, we can analysis the stability condition (42). Eqs.(31) and (32) will play the role of two fundamentalconstraints to achieve our goal. There are two possibility in order to satisfy the equation (32); the firstone is that ω = − and the second one is that the matter density of the ESU be ρ = (cid:0) ξ (cid:1) − . If weset ω = − , then the condition (42) is automatically violated and we obtain δ ¨ a = 0 implying that thereis no stable ESU against the linear scalar perturbations in the form of relations (33). As it is clearfrom Eq.(41), in the framework of the standard GR for ω = − , there is no stable ESU. For the secondpossibility as ρ = (cid:0) ξ (cid:1) − , with respect to the positivity of the spatial curvature, due to the positiveenergy condition through Equation (31), we need that the barotropic EOS parameter satisfies ω > − / ω > − / ρ < ǫ pl . There are no such similar results in the framework of the standard GRtheory. Unlike GR, here there is a stable solution against small scalar perturbations for a closed universefilled by the matter fields respecting the energy conditions. Also, within standard GR theory and evenin many of the modified gravity theories, there is no such upper bounds on energy density of ESU ρ .In the following, we want to investigate the stability conditions of Einstein static Universe against thelinear homogeneous scalar perturbations in the context of DGR, in terms of another rainbow functionproposed by Ling and et al in [35], which is evolving with cosmic time as f = q − l p ¯ ǫ . (45)Considering previous arguments after placing the average energy ¯ ǫ = cρ [31], it can be rewritten interms of energy density ρ as f ( ρ ) = r − χρ , (46)where χ = l p c is a constant. Let us point out that rainbow function (46) has application in theframework of black hole physics leading to valuable phenomenological outcomes, e.g. see [35, 36].Moreover, it also as previous rainbow function has a theoretical trait. By reaching the energy density ρ or ¯ ǫ to its largest value, f ( ρ ) in (46) becomes zero which leads to vanishing spatial-like components of therainbow metric. This means that FRW metric modified by (46) or (45), is a smooth and differentiablemetric but is not invertible. Moreover, by applying the rainbow functions (45) or (46), we will be facedwith a degenerate metric at energy levels close to the Planck energy scale. Also, based on the factthat there is no observational evidence for lightlike dimension in nature, there might be a suppressionmechanism for appearance of lightlike dimension at the quantum level of Universe [32].As before, we first examine the Big-Bang singularity problem for such a choice of rainbow functionwithin a flat rainbow FRW model. In the same way, for the rainbow function (46), we get˙ ρ = − s πGρ − χρ , (47)which has only one fixed point ρ f = 0 and is not bounded, see Figure 2. As it is seen from the figure,the function O ( ρ ) begins from the fixed point ρ f and ends with a singularity. So we must separately10alculate the time needed from an infinite to a finite value ρ ∗ as well as the time needed from ρ ∗ to thefixed point ρ f = 0, i.e, t = − Z ρ ∗ ∞ dρ s − χρ πGρ , (48)and t = − Z ρ ∗ dρ s − χρ πGρ . (49)Here, we are dealing with a different situation compared to the previous one. We found that the integral(48) converges which provide ρ ∗ ≥ χ while the integral (49) does not converge anyway on the giveninterval. This means that from ρ ∗ to fixed point ρ f = 0 an infinite time is needed, i.e | t | → ∞ , whileto get an infinite to a finite value ρ ∗ it takes a finite time. Because the rainbow FRW metric includes anatural cutoff as the energy Planck, the constraint obtained for integral (48) should lie in the interval χ ≤ ρ ∗ ≤ χ . Then, the solution derived from the rainbow function (46) for the flat rainbowUniverse model can not be free of finite time singularity. We conclude that even assuming that integral - - - - - - - Ρ O H Ρ L Figure 2:
The behavior of O ( ρ ) (47) in terms of ρ for the flat rainbow FRW Universe model. To simplify, weset 8 πG = 3 and χ = 1 / (48) results in an infinite time, the Big-Bang singularity issue is not canceled, yet. Because accordingto the discussion given in the previous case, one can realize that ρ f is a future fixed point and is not apast one.In the following, we study the DGR theory from the emergent universe point of view and investigatethe required conditions for the stability of an ESU with respect to scalar perturbations, in the presenceof rainbow function (46). By applying the scalar perturbation terms on the rainbow function (46), ittakes the form f − ( ρ ) → f − ( ρ ) + 89 χρ δρ , (50)where f − ( ρ ) = 1 + 169 χρ . (51)Now, inserting the equations (50), (51) and the perturbation equations (33) into the second rainbowmodified Friedmann equation (24), we have δ ¨ a = − πG ω ) (cid:18) ( ρ + 1) f − ( ρ ) + 89 χρ (cid:19) ρ δρ , (52)where similar to the previous analysis, we neglected non-linear terms. Finally, by applying the equations(32) and (34), the above differential equation takes the following form δ ¨ a − ka (1 + 3 ω ) (cid:18) χρ (cid:19) δa = 0 . (53)11s can be seen, for the limit χ →
0, the above equation reduces to the oscillatory modes (41). Then,in order to have a stable ESU in the framework of DGR scenario, the following condition should besatisfied − ka (1 + 3 ω ) (cid:18) χρ (cid:19) > , (54)where the frequency of oscillatory modes γ for the rainbow function (46) reads as γ = − ka (1 + 3 ω ) (cid:18) χρ (cid:19) . (55)It is seen that for ω = − , the condition (54) is automatically violated and we get δ ¨ a = 0 throughthe equation (53), which indicates that there is no stable ESU against the linear scalar perturbationsfor this rainbow function. For the second possibility as ρ = (cid:0) χ (cid:1) − , with respect to the positivityof the spatial curvature due to the positive energy condition through equation (31), we need that thebarotropic EOS parameter ω satisfies the condition ω < − /
3. Therefore, here we have a stable ESU forthe baratropic EOS parameter ω < − /
3, denoting the phantom matter fields, with the bounded energydensity of ESU ρ as ρ < l − pl . As mentioned before, the existence of such constraint on energy densityof ESU may be absent in other modified gravity theories. Here, such a bound on initial density ρ isarising from the energy dependent metric in gravity’s rainbow proposal which is one of the common cutoffs of quantum gravity theories. Finally, it should be mentioned that choosing the rainbow function(46) leads to the same result with GR so that a closed universe filled by usual non-relativistic matterfields, is not stable against small linear scalar perturbations. Λ In this subsection, we want to consider the possible modification by the cosmological constant Λ( ǫ ) andinvestigate its effects on stability condition of an ESU. In presence of cosmological constant Λ( ǫ ), thesecond modified FRW equation (24) remain without change. But, the first modified FRW equation (23)and the conservation law (25), take the form of( ˙ aa ) + ka f = 8 πGρ f + Λ( ǫ )3 f , (56)and ˙ ρ + 3 ρ ˙ aa (1 + ω ) = − ˙Λ( ǫ )8 πG . (57)One may consider another rainbow function as h ( ǫ ) so that Λ( ǫ ) = h ( ǫ ) Λ where Λ is the usual cosmo-logical constant. By this consideration, the modified FRW equation (56) will be( ˙ aa ) + ka f = 8 πGρ f + h ( ǫ ) Λ3 f . (58)By setting this new rainbow function as h ( ǫ ) = (1 + λρ ), see second Ref. of [27], for the ESU describedby ˙ a = ¨ a = 0, we get the following equation from equation (56) as ka = 8 πGρ λρ ) , (59)where λ is a dimensional parameter. By keeping positive energy condition, the above equation impliesthat positive spatial curvature k >
0, for two cases Λ > <
0, is guaranteed only under thefollowing constraints, respectively λ > − ρ Λ − ρ , (60)12nd λ < | ρ Λ | − ρ , (61)so that ρ Λ ≡ Λ8 πG and | ρ Λ | ≡ | Λ | πG . With note to the our previous result on upper bounds on energydensity of ESU as ρ < l − pl , the above constraints reads as follows λ > − ρ Λ − l pl , (62)and λ < | ρ Λ | − l pl . (63)In DGR formalism of gravity, it is expected that parameter λ to have the order of magnitude | λ | ∼ l pl .As it seems, this value of λ be satisfied both the above constraints. Also, in the presence of cosmologicalconstant, we recover Eq. (32) from the second modified FRW equation (24). Similar to the previousanalysis, by inserting equation (33) into the equation (56) and neglecting the non-linear perturbationterms, we get − ka δa = (cid:18) πG λ (cid:19) ρ δρ (64)It is seen that for the case of positive spatial curvature k >
0, the sing of variation of the scale factor δa is in contrast to the variation of matter density δρ , provided that λ > − ρ Λ , (65)and λ < | ρ Λ | , (66)where are related to the cases Λ > <
0, respectively. For the case of positive cosmologicalconstant, Λ >
0, by comparing the constraints (60) and (65), one realizes that constraint (60) can besatisfied via constraint (65) while its inverse is not true and so relation (65) will be a tighter constraint.With the same reason, one can say that for the case of negative cosmological constant, Λ <
0, constraint(61) is more tighter than (66). Finally, by using equation (64) and considering equation (33) into thesecond modified FRW equation (24), after neglecting the non-linear perturbation terms, we arrive tofollowing differential equation δ ¨ a − ka (cid:18)
11 + λρ Λ (cid:19) (cid:18) − ξρ (cid:19) (1 + 3 ω ) δa = 0 . (67)By putting ρ = (cid:0) ξ (cid:1) − , the above differential equation takes the form of δ ¨ a + 3 k a (cid:18)
11 + λρ Λ (cid:19) (1 + 3 ω ) δa = 0 , (68)In order to have a stable ESU against homogeneous and linear scalar perturbation described by theoscillatory modes, we have two possibilities as • For the case of Λ > ω > − , λ > − ρ Λ , or ω < − , λ < − ρ Λ . (69) • For the case of Λ < > − , λ < | ρ Λ | , or ω < − , λ > | ρ Λ | . (70)Now, in the same way as above by using the rainbow function (27), by choosing the rainbow function(46), we obtain the following differential equation δ ¨ a − ka (cid:18)
11 + λρ Λ (cid:19) (cid:18) χρ (cid:19) (1 + 3 ω ) δa = 0 , (71)where by inserting ρ = (cid:0) χ (cid:1) − , coming from constraint (32) for the rainbow function (46), it can berewritten as δ ¨ a − k a (cid:18)
11 + λρ Λ (cid:19) (1 + 3 ω ) δa = 0 . (72)This differential equation possesses the stable oscillatory modes under the following possibilities • The case of Λ > ω > − , λ < − ρ Λ , or ω < − , λ > − ρ Λ . (73) • The case of Λ < ω > − , λ > | ρ Λ | , or ω < − , λ < | ρ Λ | . (74)We note that since the cosmological constant does not appear in constraint equation (32), the energydensity of ESU is bounded as ρ < ǫ pl or ρ < l − pl , for the both rainbow functions (27) and (46). It isfound that in order to have stable solutions for an ESU in the presence of a cosmological constant, besidethe cutoff on energy density ρ , the conditions (69), (70) and (73), (74) for each of rainbow functions (27)and (46), should also be satisfied respectively. For the case of positive cosmological constant Λ >
0, bycomparing conditions mentioned in (69) and (73) with the constraint (65), one realize that with choiceof rainbow functions (27) and (46), the solutions with ω > − , λ > − ρ Λ and ω < − , λ > − ρ Λ are allowed, respectively. For the case of negative cosmological constant Λ <
0, constraint (61), is moretighter than (66). In fact, we recall that the constraint (61) is important in the sense that with respectto physical energy condition in quantum gravity regimes, ensures that k >
0. So, it is clear that allthe constraints on the parameter λ in (70) and (74) violate the constraint (61). Overall, this meansthat in the presence of a quantum gravity modifications such as what is done by gravity’s rainbow,with respect to positive energy condition and positive spatial curvature k > , which is one of the basicassumptions in modern version of emergent universe scenario, for the cosmological models possessing anegative cosmological constant, there is no stability for an ESU. In the cosmological context, the vector perturbations of a perfect fluid having energy density ρ andbarotropic pressure p = ωρ are governed by the co-moving dimensionless vorticity defined as ̟ a = a̟ .The vorticity modes satisfy the following propagation equation [37]˙ ̟ κ + (1 − c s ) H̟ κ = 0 , (75)where c s = dp/dρ and H are the sound speed and the Hubble parameter, respectively. This equation isvalid in our treatment of Einstein static Universe in the framework of the rainbow gravity through themodified Friedmann equations (23) and (24). For the Einstein static Universe with H = 0, the equation(75) reduces to ˙ ̟ κ = 0 , (76)14here indicates that the initial vector perturbations remain frozen and consequently we have neutralstability against the vector perturbations. Tensor perturbations, namely gravitational-wave perturba-tions, of a perfect fluid is described by the co-moving dimensionless transverse-traceless shear tensorΣ ab = aσ ab , whose modes satisfy the following equation¨Σ κ + 3 H ˙Σ κ + (cid:20) K a + 2 ka − (1 + 3 ω ) ρ − ǫ )3 (cid:21) Σ κ = 0 , (77)where K is the co-moving index ( D → −K /a in which D is the covariant spatial Laplacian)[37]. Forthe Einstein static Universe ( H = 0), this equation reduces to¨Σ κ + (cid:20) K a + 2 ka − (1 + 3 ω ) ρ − λρ )3 (cid:21) Σ κ = 0 . (78)Note that in the general tensor perturbation equation (77), the Hubble parameter H contains therainbow function f through equation (23). But, because for the Einstein static universe we have H = 0,then the rainbow function f did not appear in the equation (78). Then, in order to have stable modesagainst the tensor perturbations, the following inequality should be satisfied K a + 2 ka > (1 + 3 ω ) ρ − λρ )3 . (79)Where, for a closed universe with k = 1, considering the eigenvalue spectra K = n ( n + 2) with n = 1 , , , ... [38], will takes the form of n + 2 n + 2 a > (1 + 3 ω ) ρ − λρ )3 . (80)This inequality gives a restriction on the scale factor of an ESU in terms of its matter density andbackground cosmological constant. Even though the existence of open and flat solutions are possiblein some modified gravity models, in the present modified gravity model it is forbidden by regardingthe weak energy condition. Therefore, the stability analysis in the present model is restricted to thephysically viable closed cosmological model. For the the eigenvalue spectra K for open and flat models,one is referred to [38]. According to a modern version of emergent cosmological scenario proposed by Ellis and Maartens[16], the early Universe before passing to the inflationary phase, has experienced an eternal Einsteinstatic state rather than a Big-Bang singularity. More exactly, it refers to a static closed space inthe asymptotic past (before entering the Universe to the period of inflation) known as “Einstein staticUniverse” (ESU). The initial conditions as the quantum gravity effects in the early high energy Universe,will influence the stability of this static state . For this reason, in the present work, we have examinedthe effects of an effective approach to quantum gravity proposed by Magueijo and Smolin [10], knownas ”gravity’s rainbow” or ”doubly general relativity (DGR)”, on the stability of the Einstein staticstate against the linear homogeneous scalar, vector and tensor perturbations. In order to following ouraim in the framework of DGR, we have needed to introduce appropriate rainbow functions. Then, wehave considered two appropriate well known rainbow functions (27) and (46). First, by following thephase space mechanism represented in [31], we have investigated the fixed points belonging to a flatFRW model which is modified by the presence of the rainbow function (27). Indeed, these fixed pointsare de Sitter space solutions of typical flat rainbow FRW model. Our results indicate that althoughthe needed time to get a fixed point is infinite, it may not lead to the elimination of the initial finitetime singularities (Big-Bang). This is because of the fact that the fixed points corresponding to past15ingularities (Big-Bang) are unstable and may be collapse. Then, we studied the DGR theory in thecontext of the emergent universe scenario and tried to find its stable Einstein static universe with therequired conditions. In order to find a stable ESU, as the first result, we realized that similar to standardGR scenario and independent of the choice of rainbow function, the positive spatial curvature k >
0, isthe only option respecting to positive energy condition ρ > ρ = (cid:0) ξ (cid:1) − and ω > − , respectively. Similarly, for therainbow function (46), we also verified that Big-Bang singularity, can still exists. Unlike the previouscase, the time needed to get the fixed point is not infinite in this case. Furthermore, the only fixedpoint which is revealed in the presence of rainbow function (46), is a future fixed point and is not apast fixed point. Then, by looking at the DGR theory from the emergent universe point of view, westudied the stability of the Einstein static universe and its required conditions. For this case, a stablesolution for ESU is guaranteed under these conditions ρ = (cid:0) χ (cid:1) − and ω < − describing the exoticmatter fields. As it is seen, stability conditions ρ = (cid:0) ξ (cid:1) − and ρ = (cid:0) χ (cid:1) − are direct result from theidea of an energy dependent metric in the framework of gravity’s rainbow scenario. These results areequivalent to an explicit cutoff on the energy density of an ESU as ρ < ǫ pl and ρ < l pl , respectively.Given that, ESU point out a initial static state (or static closed space) of universe before getting intoinflationary phase, so the existence of such explicit cutoff (or upper bounds) on ρ could be interpretedas a result of initial dominate quantum gravity effects such as “gravity’s rainbow”. In the following ofour analysis, we take an energy dependent cosmological constant via the introduction of new rainbowfunction as h ( ǫ ) = (1 + λρ ). It is seen that the positive spatial curvature, k > , dictates an oppositesing for δa relative to energy density δρ through the equation (64). We find that in order to havestable solutions for an ESU,beside the cutoff on energy density ρ , the conditions (69), (70) and (73),(74) for each of rainbow functions (27) and (46), should also be satisfied respectively. In particular, forthe case of positive cosmological constant Λ >
0, by comparing conditions mentioned in (69) and (73)with the constraint (65), one realize that with choice of rainbow functions (27) and (46), the solutionswith ω > − , λ > − ρ Λ and ω < − , λ > − ρ Λ are allowed, respectively. For the case of negativecosmological constant Λ <
0, constraint (61), is more tighter than (66). In fact, we recall that theconstraint (61) is important in the sense that with respect to physical energy condition in quantumgravity regimes, ensures that k >
0. So, it is clear that all the constraints on the parameter λ in (70)and (74) violate the constraint (61). Overall, this means that in the presence of a quantum gravitymodifications such as what is done by gravity’s rainbow, with respect to positive energy condition andpositive spatial curvature k > , which is one of the basic assumptions in modern version of emergentuniverse scenario, for cosmological models with a negative cosmological constant, there is no stabilityfor an ESU. Finally, we investigated the the stability of an ESU in the framework of DGR versus vectorand tensor perturbations. It is found that there is a neutral stability against the vector perturbations.In order to have the stability against the tensor perturbations, the scale factor of an ESU is restrictedby its matter density and background cosmological constant. Acknowledgment
This work has been supported financially by Research Institute for Astronomy and Astrophysics ofMaragha (RIAAM) under research project NO.1/3720-3.