Stability of the Einstein Static Universe in Einstein- Cartan Theory

Journal of Cosmology and Astroparticle Physics

Related content

- Corrigendum: Astrophysical and terrestrial Stability of the Einstein static universe in Einstein- probes to test Einstein–Cartan gravity (2001 Class. Quantum Grav. 18 3907) Cartan theory L C Garcia de Andrade - EINSTEIN–CARTAN GRAVITY WITH TORSION FIELD SERVING AS AN To cite this article: K. Atazadeh JCAP06(2014)020 ORIGIN FOR THE COSMOLOGICAL CONSTANT OR DARK ENERGY DENSITY A. N. Ivanov and M. Wellenzohn

- On the stability of the Einstein static View the article online for updates and enhancements. universe John D Barrow, George F R Ellis, Roy Maartens et al.

Recent citations

- Einstein’s 1917 static model of the universe: a centennial review Cormac O’Raifeartaigh et al

- Einstein-Cartan wormhole solutions Mohammad Reza Mehdizadeh and Amir Hadi Ziaie

- Stability of the Einstein static universe in f(R, T) gravity Hamid Shabani and Amir Hadi Ziaie

This content was downloaded from IP address 131.169.4.70 on 28/11/2017 at 11:20 JCAP06(2014)020 hysics P le ic t ar doi:10.1088/1475-7516/2014/06/020 strop A osmology and and osmology C 1401.7639 modiﬁed gravity, cosmic singularity, cosmology of theories beyond the SM, cos-

The existence and stability of the Einstein static solution have been built in the rnal of rnal

ou An IOP and SISSA journal An IOP and 2014 IOP Publishing Ltd and Sissa Medialab srl Department of Physics, AzarbaijanTabriz, Shahid 53714-161 Madani Iran University, E-mail: [email protected] c

K. Atazadeh Received March 14, 2014 Accepted May 21, 2014 Published June 10, 2014 Abstract. Einstein-Cartan gravity. We show thatdensity this satisfying solution the in Weyssenhoff the restriction presencepoint. is of cyclically Thus, perfect stable study fluid around with ofcosmological a spin this models center solution equilibrium in is whichEinstein interesting static the because solution early it and universe supports is oscillates non-singular thus indeterminately emergent past about eternal. Keywords: an initial Stability of the Einsteinin static Einstein-Cartan universe theory mological perturbation theory ArXiv ePrint: J JCAP06(2014)020 ) and the connection. The µν g – 1 – = 0, between a metric ( µν g ν ∇ ´ Elie Cartan wrote a series of papers on geometrical aspects of the theory of relativity. Einstein-Cartan theory is highly symmetric and non-singular, but its significance is In the context of alternative theories of gravity, cosmological solutions in the EC theory In the framework of Einstein’s general relativity (GR), a new scenario, called an emer- probably limited because even abalance small amount the of anisotropy repulsive and gravitationaltheory shear is interaction is sufficient due to viable, counter- to butgravitational aligned its effects spins predictions of [10, differ spinspin11 ]. very become squared little is comparable Any from of to way the those those this same of of order the mass asof Einstein the only gravity theory, density have when the been of the considered energy. the density in geometrical which of structure the of spin space-timespinning properties are matter studied. of in matter In cosmological and refs. contextof their [12–18], such the influence the as on effects universe inflationary of scenarios, andstudied torsion late removing and the time the acceleration flatness singularities andRecently, are in horizon investigated. [23] problems Also, the withoutgravity can in authors exponential lead [19–22] have inflation to have shown in signature been that changing the the solutions. EC spin-spin theory. gent contact universe interaction was in introduced the EC in [24, 25 ] to remove initial singularity. Observation from generalization due to Cartanand was a more new significant,displacement law reflect he the of role introduced played parallel connections bylinear the transport with group group to of torsion (the affine translations. transformations, CartanCartan Attempting an was displacement). extension to of led treat the torsion to Torsionmodification, and a and known curvature slight on the today modification the as Cartan to same of the footing, the Einstein’s Einstein-Cartan relativistic density theory theorybe of (EC), of zero, intrinsic consists gravitation. as angular in donemore The relating momentum in clear, torsion (spin) the if of one Einsteinrelativity matter, tries theory [3]. to [2]. instead In incorporate EC of The thenot theory, assuming importance the spinor a spin field it of dynamical sources into quantity the to canconsider [4–6]. the Cartan be a torsion-free Weyssenhoff theory noted fluid general exotic in becomes with theory perfect terms intrinsic of fluid of spin is torsion, density thus one [7–9]. torsion of is the usual ways to 1 Introduction In 1923 The papers contained importantof new differential mathematical geometry and, ideas in which particular,was influenced led remarkably, to aware the the of development general the theoryphysics. of importance connections H. of [1]. Weyl the Cartan wasthe notion the condition of first of a to compatibility, connection introduce and its non-Riemannian relevance linear for connections by relaxing Contents 1 Introduction1 2 Friedmann equation in spin-dominated3 Einstein-Cartan2 gravity The Einstein static solution4 and5 stability Conclusion and remarks6 JCAP06(2014)020 , the (2.1) (2.2) (2.3) (2.4) ρσ µ ˜ Γ − and Λ is the , , ρσ µ µν ρσ ) gravity [44, 45] ˜ µνρ Γ µ R ˜ Γ = ( f πGT πGτ ρσ µ = 8 Q ) = 8 x, 4 µνρ d . ρµν T T M µ + L σ ρ + Q , νρµ T ρσ − µ 2Λ − K µ − ρ σ + µνρ ˜ R Q T ρσ ( µ − – 2 – × πG via ρσ 1 µ = Γ σ µ 16 ] Q ρσ ρσ − ρσ [ µ Q 2 1 ˜ Γ ˜ Γ g = is the Lagrangian density of matter fields. Utilizing of the = − + 2 µ ρ √ ρσ M µ ∇ L = 0 [4–6] and the definition of torsion Z ρσ K Q = − µν g S ρ µν ˜ g ∇ are the Christoffel symbol and the contorsion tensor, respectively, which Λ ρσ − µ can be written as K µν ρσ G µ ˜ Γ and is the Ricci scalar associated to the asymmetric connection ) gravity [49], from Horava-Lifshitz gravity [46, 48] to brane gravity32] [ and massive ρσ µ ˜ R T ( In this paper, we consider the stability of the Einstein static universe in the Friedmann- By variation of the action with respect to the metric and contorsion, one can find the f where cosmological constant, also metric compatibility where Γ are related to the torsion eaır-oeto-akrspace-time inLemaˆıtre-Robertson-Walker the framework of EChof gravity with perfect exotic fluid. Weyssen- Inin section3 thewe presence present ofthe an matter analysis energy of and contain the thepressure. relativistic equilibrium spin. The matter of paper and Einstein Next, ends solution a we with a spin consider brief a fluid conclusions numerical with2 in example, negative section4. in energy which and Friedmann equation negative inEinstein-Cartan spin-dominated gravity Einstein-Cartan can be gravity started by writing the following action WMAP7 [26] supportsuniverse the is positivity favored of at spaceEinstein the static curvature 68% state in confidence and level, then whichgent evolves and it theory to the a is of universe subsequent gravitation, found stays, inflationarystate the phase. that rather past-eternally, universe than in a According a might an to closed big have emer- stein’s bang been general singularity. relativity originated Nevertheless, is the from unstable, Einsteinto which an static means maintain Einstein universe that its in it static stability the is insuch Ein- almost a as impossible long for the time the quantum universe dose because not fluctuations. of resolve the the existence big Thus,epoch, bang of it singularity the varieties problem seems universe of properly is perturbations, to asstate apparently expected. us under may be However, fanatical that in affected physical the the by conditions,or early novel original the a physical study emergent effects, modified of such model gravity the asproblem or those initial of even resulting the other from cosmological new quantum constantsymmetric gravity, physics. in Einstein the static In context universe. [27], ofstudied Finally, the EC the in author theory stability various for has of cases a the consideredand [28–48], static Einstein the static and from sign state spherically loop has quantum beengravity gravity [50]. 29–31] [ to connection equations of motion [4–6] JCAP06(2014)020 , ν u (2.7) (2.8) (2.9) (2.6) (2.5) (2.11) (2.10) (2.12) (2.13) µν ω , µ ) σ S ρ ( µνλ u τ + µ ) µνλ σ τ S ρ + ( ] µ µ ω , λν − M τ µν ν L ) [ µ µν , , ρ δg δ λ σ is associated with the quantum τ ρσ Q g , 4 g σ µσ − ) ( 2 2 µν µ Q S √ µν ρ , ν S ρσ , δ τ ν g i ρσ = , πGσ u S 1 2 − + ρ µν T − u µν σ S + + 2 µν + T νµ µν νσ σ σ T µν is the antisymmetric spin density [51]. Then, S S ( u S ρσ ) Q µν 1 2 ρ h F σ ρ µ τ µν – 3 – u δ T 1 2 u πG 2 S = ρ ( µνρ + = = M µν ρνµ τ ρ 2 L µνρ Q ρσ πGσ δ µν σ + τ theory. In the case of the spin fluid the canonical spin δK + T Q , as (Γ) = 8 4 g µ µν ; = 4 U ) − 1 ρσ = µν σ i S µ τ ) ρ √ G T σ ρσ µν τ S = ρµν h T ρ τ ( 2 u µνρ − τ +( ] ν µ ; µ σν u τ ν ν [ u are the usual Einstein tensor and covariant derivative for the full nonsymmet- µ ρµ ) ˜ Γ, respectively. Also, the canonical spin-density and the energy-momentum ρ τ σ 4 ∇ S ρ − ( (Γ) is the known symmetric Einstein tensor and is the 4-velocity of the fluid and u represents a fluid of spinning particles in the early Universe minimally coupled to and semicolon denote the angular velocity associated with the intrinsic spin and and µν ρ = = ω G u M µν = 0 and if we consider ρσ and an intrinsic-spin part L ρσ i G τ S µν ρσ T F S mechanical spin of microscopich particles [12–18], thus for unpolarized spinning field we have where covariant derivative with respectusual to explanation Levi-Civita of connection, EC respectively. gravity According we to can the assume that tensor is given by [7–9] the energy-momentum tensor canT be separated into the two parts: the usual perfect fluid the metric and the torsion of the where is a kind of modificationthe to spin the zero space-time in equation curvaturethat that (2.7) we stems will from have the the spin standard [7–9]. Einstein field If equations. we set We suppose where thus, the explicit form of intrinsic-spin part is given by we get where respectively. Thus, by meansfield of equations equations as (2.4) and (2.5) one can write generalized Einstein tensors are given by ric connection and JCAP06(2014)020 (2.18) (2.20) (2.21) (2.22) (2.14) (2.19) (2.15) (2.16) (2.17) , . ρσ 2 g ) . s ) p s πGσ ρ 2 − ρσ , − , p − ) pg i ( s s p 2 . , p ρ . Thus, from the conservation equa- , − σ − n w h = w 2 u , − σ − σ ρσ 2 ) ρ 1+ u u p ~ ρ w Θ u ρ ρ tot ρ ( 1 8 2 u w u + 2 ) ) πG ρ 1+ p = 3(1+ πG s ( p − i − w πGσ + 2 i a H – 4 – 8 B , p 0 = 8 − ρ 3 S 2 2 ρσ − ρ h s 8 2 − ~ τ 3 2 1 ρ a h (Γ) = 8 = = ( = = − ρσ = ρ πGσ + i i + ) = 2 2 p i s 2 G 2 ρσ ρσ σ ρ σ F − + ρσ S H T − 3 T ρ T ρ h h h ρ = ( = = ( d dt tot ρσ ρ . Finally, we have Θ wρ = . In comparison with the usual GR, we can conclude that equations (2.15) p 2 is the Hubble parameter. The conservation equation gives πGσ ˙ a a explains the effective macroscopic limit of matter field is a dimensional constant dependent on = = 2 is the particle number density, and averaging process gives [52] w ρσ s B H n ρ where where where tion (2.19) we can write and This leads to the simplification of EC generalization of standard gravity as follow Equation (2.19) is athe generalized spin. form To of continue,equation the we of covariant take energy state the conservation matter law field by as including a unpolarized fermionic perfect fluid with where Θ It is considerable tosemi-classical recall models that of spin the fluid correctionspin which are terms in negative EC signs [7–9, theory in12–18]. playsIn (2.17) In the such are other role a words compatible model of the with a the effectInserting of perfect Einstein the closed static fluid isotropic universe with and occurs negativement homogeneous line and energy into Friedmann-Lemaˆıtre-Robertson-Walker it ele- density the is and (2.15) stable pressure. and around (2.16) equilibrium point. gives the field equations and (2.16) show the equality betweena EC fluid field equations with and a the particular Einstein equationconcept equations of the coupled state to contribution as the of matter the source. torsion Actually, can in a be hydrodynamical done by a spin fluid such that where JCAP06(2014)020 = 1, (3.5) (3.6) (3.8) (3.3) (3.4) (3.7) (3.1) (3.2) takes (2.23) (2.24) s w D 1 where s ρ s w = s p . ) . , which for a positive ) w w Es − , . = 0 describes the Einstein static a s s s − 2 , w s ( w w 2 5 1 , x . ( to obtain the eigenvalue we have ) can be written as . 1+ x 0 5 D a. w < w w > w ) Es 2 2 ρ w s a D a λ w ρ 2 w = ˙ . To begin with we obtain the conditions . 2 − w a 6 2 = 1+ − ) + s − 1 − s w ) + w w x ( w B w Da 1 + 3 ( ~ D – 5 – D 6 Es 2 = 0 = ˙ = a a, x (1 + 3 s − πG 32 a = 8 ρ (1 + 3 3 and 3 and = 1 = / / 4 Es + 1 = 1 x 2 1 1 a a 2 2 2 x + 1 λ 2 − − , x D 2 2 ˙ a x − − w > w < = = ) stemming from linearizing the system explained in details by = 1 2 i j ˙ ˙ ˙ x ¨ a x x ∂ ∂x = 1. = ij πG J is present value of energy density. For simplicity, we define , denotes for stiff matter or brane effects with dust on a brane with negative tension. 0 6 ρ − a Here, we are going to study the stability of the critical point. For convenience, we We have set units 8 ∝ 1 s It is then easy to obtain the following equations above two equations near the critical point. Using introduce two variables for the existence of this solution. The scale factor in this case is given by the forms or According to these variables, the fixed point, The Einstein static solution is given by ¨ The existence condition reduces to the reality condition for solution properly. Thecoefficient stability matrix of ( the critical point is determined by the eigenvalue of the where therefore, from equations (2.17), (2.21) and (2.22) Note that effectsadditional of non-interacting spin fluid for are which dynamically the equivalent equation to of state introducing is into the model some 3 The Einstein staticBy solution using equations and (2.18) stability and (2.19) the Raychadhuri equation can be written as ρ JCAP06(2014)020

a (3.9) 0.03 . Comparing 0.02 s 0.01 w < w a 1.03 1.02 1.01 0.01 - . 0.02 - = 0 0, this means that D 3. Especially, it is stable in the presence 0.03 / − - 1 4 D > − a t + 3 – 6 – w > 2 ˙ a 0. For 4 a 3 < 2 λ + 3 23 has been plotted in figure1. . ¨ a 2 5 a = 3 3 , in addition to a relativistic matter with an equation-of-state 6 D 1 D/a = 0 the Einstein static solution has a center equilibrium point, so it 3. ) plus a spin fluid with negative energy density and negative pressure. a / s w ρ < 3. Obviously, phase space trajectories which is beginning precisely on 1.03 1.02 1.01 3. Using these equation of state parameters in equation (3.1) we obtain = 1 2 / D/ 1 λ w - (0) = 0, with a = 1 = w 4 = 1 and 2 a s - . The evolutionary curve of the scale factor with time (left) and the phase diagram in space w From the above equation the corresponding scale factor of Einstein static solution is To continue, we study the effects of spin field in EC on the dynamics of the universe. ) (right) for 3 a - , ˙ (0) = 1 and ˙ a 4 Conclusion and remarks We have discussed the existenceof and spin stability of fieldsfluid. the coupled Einstein We have to static shown universe gravity thatinverse in the through sixth the spin power presence the energy of densitythe EC in the equation Einstein gravity scale universe of with is factor. stateis proportional exotic parameters to Also, stable, related Weyssenhof the we while to perfect havestudy it the determined of spin is the such energy dynamically a allowed such belongingemergent solution intervals that oscillatory to for is the models the a Einstein resultproblem center which universe of in equilibrium are its the point. past essential standard eternal, role cosmological The scenario. in and motivation the hence construction can of resolve non-singular the singularity has circular stability, whichoscillations means about that that smalluniverse point perturbation oscillates rather in from the than thestability neighborhood exponential fixed condition of is deviation point the determined Einstein from results by static in it. solution indefinitely. In Thus, the this case, the given by the Einstein static fixedtrajectories point which are remain creating therethis in indeterminately. solution. the From vicinitya An another of example point this point of of would view, such oscillate a indefinitely universe near trajectory using initial conditions given by In the case of Figure 1 ( of ordinary matter ( As an example, wewe consider put the caseparameter where the energy content consists of spin fluid, which this inequality with the conditions forwe existence find of that the the Einstein Einstein static solution, universe (3.3) is and stable (3.4), JCAP06(2014)020 9 ]. ]. 56 (2010) B 112 Phys. Acta SPIRE 27 , , IN SPIRE (1973) 7 IN [ (1985) 451 ]. 242 ]. 15 (2009) 1652 Phys. Rev. Lett. (1973) 96[ SPIRE Gen. Relativ. Gravit. , Nuovo Cimento IN , , Nature SPIRE A 24 , 244 IN ][ gr-qc/0409029 Class. Quant. Grav. , , General Relativity with Spin and Found. Phys. Nature , , (1976) 393[ Metric affine gauge theory of gravity: 48 (1923) 325. Int. J. Mod. Phys. 40 , arXiv:0706.2367 Weyssenhoff fluid dynamics in a 1+3 covariant ]. On the cosmological effects of the Weyssenhoff – 7 – SPIRE IN Rev. Mod. Phys. ]. , ][ (2007) 6329[ General relativity with torsion Torsion formulation of gravity 24 Relativistic dynamics of spin-fluids and spin-particles SPIRE Can spin avert singularities? ]. IN Ann. Ec. Norm. Sup. ][ , SPIRE ]. IN gr-qc/9402012 ]. ][ Friedmann-like cosmological models without singularity SPIRE Spin Dominated Inflation in the Einstein-Cartan Theory Spin and torsion may avert gravitational singularities SPIRE (1947) 7. IN The cosmological model with scalar, spin and torsion field IN On the Kinematics of the Torsion of Space-time Sur affine, les et variétésàconnexion la théoriede la relativitégénéralisée 9 (1995) 1[ Class. Quant. Grav. ]. ]. , arXiv:0907.5583 258 SPIRE SPIRE IN arXiv:0907.1701 IN We can go through the classical spin in general relativity in two approaches. The first (1978) 511. peieepartie)(première (Suite). 065014[ [ Field equations, Noether identities,Rept. world spinors and breaking of dilation invariance Phys. Pol. spinning fluid in the Einstein-Cartan framework (1986) 2873[ (1997) 363 [ [ approach [ Torsion: Foundations and Prospects [1] E. Cartan, [2] M.A. Lledóand L. Sommovigo, [6] F.W. Hehl, J.D. McCrea, E.W. Mielke and Y. Ne’eman, [7] J. Weyssenhoff and A. Raabe, [8] G. de Berredo-Peixoto and E.A. De Freitas, [3] T. Watanabe and M.J. Hayashi, [4] F.W. Hehl, P. von der Heyde, G.D. Kerlick[5] and J.M.F.W. Nester, Hehl, [9] S.D. Brechet, M.P. Hobson and A.N. Lasenby, [13] M. Gasperini, [14] S.-W. Kim, [12] B. Kuchowicz, [10] A. Trautman, [11] J.M. Stewart and P. Hajicek, one is that one cantheories define without spin as modifying ametric the dynamical [53–55]. quantity geometry, This in i.e., both kindspin special of without and and introduced changing general spin also relativity leads theto to to introduce space-time the a the Dirac structure similarity intrinsic to angular(EC theory the momentum of theory). is quantum the mechanics generalization ofmatter electron. To the fields consider structure with The the of intrinsic second, angular space-time is influences momentum. to more set of In interesting a order torsion approach fluidFinally, to which the in gain has results this an EC of intrinsic purpose, this gravity spinassumption. paper the density are one usual called essentially way the may a Weyssenhoff consequence equipped exotic of perfect the the fluid. Weyssenhoff perfect fluid Acknowledgments The author is grateful to F. Darabi for usefulReferences discussion and recommendation. JCAP06(2014)020 , , D D B , Phys. D 83 , D 27 B 694 ]. D 73 ] ]. (2011) 291 ]. Phys. Rev. SPIRE Phys. Rev. Phys. Lett. , 523 , (2011) 18 SPIRE IN , Phys. Rev. IN ][ Phys. Rev. , Phys. Rev. ]. Phys. Lett. SPIRE ][ ]. , , , 192 ]. ]. IN ][ SPIRE ]. arXiv:1201.5498 SPIRE SPIRE IN SPIRE IN IN ][ IN ][ Annalen Phys. ][ ][ , -cosmologies containing Einstein SPIRE ]. ) ]. IN R gr-qc/0310058 ( arXiv:0706.4431 ][ f An Emergent Universe from a loop SPIRE (2012) 3186[ SPIRE ]. The Einstein static universe in Loop IN arXiv:0809.5247 Astrophys. J. Suppl. IN 51 , ][ ][ ]. SPIRE Stability of the Einstein static universe in ]. gr-qc/0211082 arXiv:0905.3116 ]. arXiv:0706.1663 (2004) 1119[ IN arXiv:1007.0587 (2007) 6243[ ][ Dynamics of SPIRE The Emergent universe: An Explicit construction ]. ]. 21 Emergent universe in a Jordan-Brans-Dicke theory 24 Seven-Year Wilkinson Microwave Anisotropy Probe IN – 8 – ]. SPIRE arXiv:0907.4795 SPIRE ][ IN IN (2009) 105003[ Torsion Cosmology and the Accelerating Universe ][ SPIRE SPIRE ][ gr-qc/0307112 (2004) 223[ SPIRE (2009) 007[ IN IN 26 IN Inverse volume corrections to emergent tachyonic inflation in (2011) 672] [ ][ ][ astro-ph/0502589 21 05 ][ Cosmological model with macroscopic spin fluid Int. J. Theor. Phys. (2007) 084005[ A Graceful entrance to braneworld inflation The emergent universe: Inflationary cosmology with no , ]. ]. Stability of the Einstein static universe in presence of vacuum Signature transition in Einstein-Cartan cosmology Spinning Fluids in the Einstein-Cartan Theory arXiv:0805.3834 A modified theory of gravity with torsion and its applications to B 701 (2009) 043528[ JCAP D 76 (2004) 233[ , Class. Quant. Grav. SPIRE SPIRE arXiv:0711.1559 Class. Quant. Grav. ]. , 21 , IN IN D 80 arXiv:1111.4595 astro-ph/0305364 ][ ][ (2005) 123512[ Cosmological constant from quarks and torsion Matter-antimatter asymmetry and dark matter from torsion Cosmology with torsion: an alternative to cosmic inflation Nonsingular, big-bounce cosmology from spinor-torsion coupling The Einstein static universe with torsion and the sign problem of the arXiv:1308.2877 hep-th/0601203 arXiv:1101.4012 SPIRE Class. Quant. Grav. Phys. Rev. IN , , Erratum ibid. Class. Quant. Grav. (2008)[ 023522 D 71 , (2007) 030[ collaboration, E. Komatsu et al., ]. Phys. Rev. 11 , gravity D 78 (2013) 28[ ) (2012) 107502[ (2004) 043510[ SPIRE R ( arXiv:1001.4538 IN arXiv:1005.0893 [ JCAP energy cosmological constant Class. Quant. Grav. WMAP (WMAP) Observations: Cosmological Interpretation Phys. Rev. Quantum Cosmology (2006) 083508[ f static models loop quantum cosmology cosmology and particle physics Rev. [ (2010) 181[ 85 [ (2011) 084033[ 726 singularity (1983) 1383[ 70 [28] S. Carneiro and R. Tavakol, [29] D.J. Mulryne, R. Tavakol, J.E. Lidsey and G.F.R. Ellis, [27] C.G. Boehmer, [26] [30] L. Parisi, M. Bruni, R. Maartens and K. Vandersloot, [32] J.E. Lidsey and D.J. Mulryne, [33] C.G. Boehmer, L. Hollenstein and F.S.N. Lobo, [35] S. del Campo, R. Herrera and P. Labrana, [18] L. Fabbri and S. Vignolo, [19] N.J. Poplawski, [21] N.J. Poplawski, [24] G.F.R. Ellis and R. Maartens, [25] G.F.R. Ellis, J. Murugan and C.G. Tsagas, [31] P. Wu, S.N. Zhang and H.W. Yu, [34] N. Goheer, R. Goswami and P.K.S. Dunsby, [22] N.J. Poplawski, [23] B. Vakili and S. Jalalzadeh, [17] K.-F. Shie, J.M. Nester and H.-J. Yo, [20] N.J. Poplawski, [16] J.R. Ray and L.L. Smalley, [15] M.lowski Szyd and A. Krawiec, JCAP06(2014)020 ]. , , , (2008) (1939) (1939) SPIRE Phys. models D 81 IN , ]. Class. ) 25 ]. , R 114 112 Gravity ( f ) T SPIRE ]. ( SPIRE Class. Quant. f IN IN , ][ ]. Phys. Rev. ][ ]. Z. Phys. (1937) 163 [ Z. Phys. , ]. , SPIRE , 6 IN ]. (2008) 044011 ][ SPIRE SPIRE IN SPIRE IN IN ][ Class. Quant. Grav. ][ SPIRE D 78 ][ , ]. IN ]. ]. Stability of the Einstein static ][ ]. On the stability of the Einstein static arXiv:1305.0025 arXiv:0902.2982 SPIRE SPIRE SPIRE IN ]. IN IN Acta Phys. Polon. ]. SPIRE Phys. Rev. ][ , ][ ][ arXiv:1207.3922 IN , gr-qc/0302094 Einstein static Universe in hybrid SPIRE arXiv:1001.1266 SPIRE arXiv:0809.4131 IN The Existence of Einstein Static Universes and IN ]. ][ Proceedings of the Twelfth Marcel Grossmann – 9 – ][ (2013) 104019[ arXiv:0909.3986 (2009)[ 067504 , in (1983) 378[ On the stability of the Einstein Static Universe in ]. SPIRE The earliest evolutionary stages of the universe and Einstein Static Universe in Exponential ]. D 88 IN (2012) 024035[ The Weyssenhoff fluid in Einstein-Cartan theory D 79 (2003) L155[ (2010) 795[ arXiv:1302.2688 ][ Stability of the Einstein static universe in modified Stability of the Einstein static universe in IR modified Hoˇrava arXiv:0909.4198 arXiv:0901.0892 Einstein static universes are unstable in generic On the Stability of Static Ghost Cosmologies B 130 SPIRE 20 42 SPIRE , T. Damour, R.T. Jantzen and R. Ruffini eds., World, Scientific ¨ ¨ Uber die innere Bewegung des Elektrons. II Uber die innere Bewegung des Elektrons. I. IN D 86 ]. The Existence of Godel, Einstein and de Sitter universes (2010) 1111[ IN gr-qc/0511076 ][ Emergent Universe Scenario in the Einstein-Gauss-Bonnet Gravity arXiv:0904.1340 Phys. Rev. Phys. Rev. , SPIRE C 70 Emergent universe from the Hoˇrava-Lifshitzgravity , IN (2013) 2315[ Phys. Lett. ][ , (2009)[ 064009 (2009) 103515[ Phys. Rev. Holes in the static Einstein universe and the model of the cosmological voids , (1987) 1633[ arXiv:0909.2821 Neue mechanik materieller systemes C 73 4 Gen. Rel. Grav. Emergent Universe and Phantom Tachyon Model , (2005) 123003[ D 79 D 80 (2009) 195003[ Class. Quant. Grav. , arXiv:0808.2379 Eur. Phys. J. , 26 D 72 arXiv:0804.3528 Grav. 205019[ Phys. Rev. Gauss-Bonnet gravity Rev. universe in modified theories of gravity metric-Palatini gravity Phys. Rev. gravity 512. with Dilaton universe Meeting on General Relativity Singapore (2012) [ISBN: 978–981–4374–51–4] [ [ (2010) 103522[ Eur. Phys. J. Massive Gravity Quant. Grav. space-time torsion 478. their Stability in Fourth order Theories of Gravity [40] C.G. Boehmer and F.S.N. Lobo, [44] C.G. Boehmer, L. Hollenstein, F.S.N. Lobo and S.S. Seahra, [45] C.G. Boehmer, F.S.N. Lobo and N. Tamanini, [46] A. Odrzywolek, [47] C.G. Boehmer and F.S.N. Lobo, [48] P. Wu and H.W. Yu, [38] B.C. Paul and S. Ghose, [43] J.D. Barrow and C.G. Tsagas, [36] R. Goswami, N. Goheer and P.K.S. Dunsby, [39] S.S. Seahra and C.G. Boehmer, [41] J.D. Barrow, G.F.R. Ellis, R. Maartens and C.G.[42] Tsagas, T. Clifton and J.D. Barrow, [50] L. Parisi, N. Radicella and G. Vilasi, [51] Y.N. Obukhov and V.A. Korotkii, [52] I.S. Nurgalev and W.N. Ponomarev, [53] M. Mathisson, [54] H. Honl and A. Papapetrou, [55] H. Honl and A. Papapetrou, [37] U. Debnath, [49] J.-T. Li, C.-C. Lee and C.-Q. Geng,