Cosmological Dynamics of de Sitter Gravity
aa r X i v : . [ a s t r o - ph . C O ] A ug Cosmological Dynamics of de Sitter Gravity ∗ Xi-chen Ao, Xin-zhou Li, Ping Xi
Shanghai United Center for Astrophysics (SUCA),Shanghai Normal University, 100 Guilin Road, Shanghai 200234
A new cosmological model based on the de Sitter gravity is investigated by dynamical analysisand numerical discussions. Via some transformations, the evolution equations of this model canform an autonomous system with 8 physical critical points. Among these critical points there existone positive attractor and one negative attractor. The positive attractor describes the asymptoticbehavior of late-time universe, which indicates that the universe will enter the exponential expansionphase, finally. Some numerical calculations are also carried out, which convince us of this conclusionderived from the dynamical analysis.PACS: 04.50.Kd, 95.30.Sf, 98.80.Jk
In the last decades, some new cosmological observations such as SNeIa, CMBR and large scale structure all indicatethat our universe is accelerating expanding and there exists a new mystical energy component in our universe,dubbed dark energy.[1] Many heuristic models have been proposed to explain the nature of this new component,which account for almost 74% of the energy density of our universe. Some of them have physical foundations,[2]some are just phenomenological.[3] However, no one is flawless. All these models have their own problems, such ascosmological constant problem and fine tuning problem. Recently, a new kind of model, called torsion cosmology, hasdrawn researchers’ attention, which is basically based on some new gravity theory, the gauge theory of gravity. Amongthese models, Poincare gauge theory is the one that has been investigated widely, which is inspired from the Einsteinspecial relativity and the localization of Poincare symmetry.[4] Nester et al. [5] applied this new gravity theory tocosmology, and obtained a novel model which is likely to be a new explanation of the accelerating expansion. In thatmodel, dynamic connection mimics the contribution of dark energy. Based on the work of Nester and his colleagues,some dynamics analysis and analytical discussion have been conducted in many papers, from which we can know thefate of the universe more clearly.[6]Besides Poincare gauge theory, there is another classical gauge theory of gravity, de Sitter gravity, which can alsobe the alternative gravity theory for Einstein Gravity. This theory is derived from the de Sitter invariant specialrelativity and the localization of de Sitter symmetry.[7] In the de Sitter gravity theory, Lorentz connection andtetrad are combined to form a new connection, i.e. dS connection, which is valued in so (1,4), rather than Lorentzconnection’s so (1,3); and the gravitational action takes the form of Yang-Mills gauge theory. Like Poincare gaugetheory, the spacetime also has a generic Riemann-Cartan structure, U . From the variational principle one can obtainthe gravitational field equation. Analogous to PG theory, de Sitter gravity has also been applied to the cosmologyrecently to explain the accelerating expansion.[8]In this Letter, we transform the cosmological evolution equations of de Sitter gravity model into a set of first-orderdynamics equations, which form an autonomous system. Then, we give some dynamical analysis of this autonomoussystem and obtain all the critical points. We analyze the critical points’ dynamical properties and stabilities, andfind out that there exists a late-time de Sitter attractor. It is concluded that the universe will expand exponentiallyin the end, as the ΛCMD predicted.For a homogeneous, isotropic universe, the space-time is described by the FLRW metric: ds = dt − a ( t )[ dr − kr + r ( dθ + sin θdφ )] , (1)and the isotropic and homogeneous torsion takes the form T = 0 T = T + ϑ ∧ ϑ + T − ϑ ∧ ϑ T = T + ϑ ∧ ϑ − T − ϑ ∧ ϑ (2) T = T + ϑ ∧ ϑ + T − ϑ ∧ ϑ , ∗ Supported by the Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) under Grant No 200931271104,Shanghai Municipal Pujiang Foundation under Grant No 10PJ1408100. ∗∗ Email: [email protected] where T + is the trace part of the torsion, T a a , while T − is the traceless part of the torsion. They are all functionsof time t , with + and − denoting the even and odd parities, respectively. Here ϑ = d t, ϑ = a ( t )d r, ϑ = a ( t ) r d θ and ϑ = a ( t ) sin θ d φ .According to the field equations of de Sitter gravity and Eqs.(1) and (2), one can easily obtain the new evolutionequations of universe,[8] − ¨ a a − (cid:18) ˙ T + + 2 ˙ aa T + − aa (cid:19) ˙ T + + 14 (cid:18) ˙ T − + 2 ˙ aa T − (cid:19) ˙ T − + T − T T − + 116 T − + (cid:18) a a +2 ka − R (cid:19) T − (cid:18)
52 ˙ a a + ka − R (cid:19) T − + 2 ˙ aa (cid:18) ¨ aa − a a − ka + 3 R (cid:19) T + − ˙ aa (4 T − T − ) T + + ˙ a a (cid:18) ˙ a a + 2 ka − R (cid:19) + k a − R ka + 2 R = − πGρ R , (3)¨ a a + (cid:18) ˙ T + + 2 ˙ aa T + − aa + 6 R (cid:19) ˙ T + − (cid:18) ˙ T − + 2 ˙ aa T − (cid:19) ˙ T − − T + 32 T T − − T − + ˙ aa (4 T − T − ) T + − (cid:18) a a + 2 ka + 3 R (cid:19) T + 12 (cid:18)
52 ˙ a a + ka + 3 R (cid:19) T − − aa (cid:18) ¨ aa − a a − ka − R (cid:19) T + − R ¨ aa − ˙ a a (cid:18) ˙ a a + 2 ka (cid:19) + 2 R − k a − R ka + 6 R = − πGpR , (4)¨ T − + 3 ˙ aa ˙ T − + (cid:18) T − − T + 12 ˙ aa T + + ¨ aa − a a − ka + 6 R (cid:19) T − = 0 , (5)¨ T + + 3 ˙ aa ˙ T + − (cid:18) T − T − − aa T + − ¨ aa + 5 ˙ a a + 2 ka − R (cid:19) T + −
32 ˙ aa T − − ... aa − ˙ a ¨ aa + 2 ˙ a a + 2 ˙ aa ka = 0 . (6)Equations (3) and (4) are the 0-0 and 1-1 component of Einstein-like equations, respectively; and Eqs.(5)-(6) are twoindependent Yang-like equations, which are derived from the ( r, θ, φ ) and ( t, r, r ) components. Here we also assumethat the spin density is zero.Also, the energy momentum tenor is conserved by the virtue of Bianchi identities, leading to the continuity equation˙ ρ = − aa ( ρ + p ) . (7)Equation (7) can be derived from Eqs.(3)–(6), which means 4 of Eqs.(3)–(7) are independent. These four in-dependent equations with the EOS of matter content comprise a complete system of equations for five variables a ( t ) , T − ( t ) , T + ( t ) , ρ ( t ) and p ( t ). By some algebra and differential calculations, we could simplify these five equationsto ˙ H = − H − ka + 2 R + 4 πG ρ + 3 p ) + 32 (cid:18) ˙ T + + 3 HT + − T + T − (cid:19) + (1 + 3 w ) ρ, (8)¨ T + = − (cid:18) H + 32 T + (cid:19) ˙ T + − T − ˙ T − − πG ρ + 3 p ) . − HT − + (cid:20)
132 ( T + − H ) T + + 6 H + 3 ka + 9 T − − R − πG ρ + 3 p ) (cid:21) T + , (9)¨ T − = − H ˙ T − − (cid:20) − T + 33 HT + − H − ka + 8 R + 54 T − + 32 ˙ T + + 4 πG ρ + 3 p ) (cid:21) T − , (10)˙ ρ = − H ( ρ + p ) , (11) w = pρ , (12)where H = ˙ a/a denotes the Hubble parameter. In order to make these equations dimensionless, we can rescale thevariables and parameters as t → t/l ; H → l H ; k → l k ; R → R/l ; T ± → l T ± ; ρ → πGl ρ ; p → πGl p, (13)where l = c/H is the Hubble radius.By some further calculations, we find that if the equation of state is constant, then these equations would turn outto form a six-dimensional autonomous system, which takes the forms˙ H = − H + 2 R − k (cid:18) ρρ (cid:19) w ) + 32 (cid:18) P + 3 HT + − T + T − (cid:19) + (1 + 3 w ) ρ, (14)˙ P = − (cid:18) H + 32 T + (cid:19) P − T − Q − HT − + "
132 ( T + − H ) T + + 6 H + 3 k (cid:18) ρρ (cid:19) w ) − R − ρ T + +6 H (1 + w ) ρ, (15)˙ T + = P, (16)˙ Q = − HQ − " − T + 33 HT + − H − k (cid:18) ρρ (cid:19) w ) + 8 R + 54 T − + 32 P + (1 + w ) ρ T − , (17)˙ T − = Q, (18)˙ ρ = − H (1 + w ) ρ, (19)where ρ is a dimensionless parameter denoting the current energy density. For such an autonomous system, i.e.Eqs.(14)–(19), we can use the qualitative method of ordinary equations with respect to the new set of variables,( H, P, Q, T + , T − , ρ ). In order to find the critical points of this system, we should set the left-hand side ofEqs.(14)–(19) to zero and solve these algebra equations. Next, we just consider the matter dominant case withspatial flatness. We find that there are nine critical points ( H c , P c , Q c , T + c , T − c , ρ c ) of this system, as shown in Table 1.Furthermore, we analyze the stability of these critical points by means of first-order perturbation. By the Taylorexpansion, we could obtain the perturbation equation around the critical points, i.e. δ ˙ x = A x , A = ∂ f ∂ x | x = x c , (20)where x means the six variables of this autonomous system and f denotes the six vector functions on the right-handside of Eqs. (14)–(19). According to the dynamical analysis theory, we could classify these critical points by thecoefficient matrix A ’s eigenvalue. The classification of these critical points are shown in Table 2. It is easy to findthat there are only one positive attractor, i.e. point 1, whose eigenvalues all are negative, and only one negativeattractor, i.e. point 2, whose eigenvalues all are positive. Positive attractors attract the solutions nearby it, whilenegative attractors repel them. The phase lines connecting a positive attractor and a negative attractor are calledthe heteroclinic lines as shown in Fig. 1.Critical points are actually some exact solutions of a dynamic system, especially the positive attractors which areoften the extreme points of the orbits in phase space. Therefore, they would describe the asymptotic behaviors ofsolutions. In this model, the positive attractor point 1 would show us the late time evolution of our universe. It isindicated that all quantities tend to zero, except the Hubble constant, which will remain at a fixed value, and thereforethe whole universe will approach the exponential expansion, just like the ΛCDM model. H ∞ = ˙ aa = 1 R ⇒ a ( t ) ∝ exp (cid:18) tR (cid:19) (21)As we know, the critical points can only describe the local properties. If we want to know the global properties,we have to resort to the numerical calculations. We solve Eqs.(14)–(19) numerically, and show some generic solutionsin Fig. 2. From these numerical results, we could see easily that the late-time positive attractor covers a wide rangeof initial values, and therefore alleviate the fine-tuning problem. Also for its insensitivity to the initial conditions,this autonomous system has no chaotic features. From this perspective, the cosmology based on de Sitter gravity isquite different from the one based on the PG theory, which suggests that the expansion will asymptotically come toa halt.[6] However, this discrepancy is only due to the existence of the de Sitter radius R . If we set R → ∞ , the deSitter gravity would degenerate to the PG theory, and have the same conclusion.In summary, we have studied the torsion cosmology based on de Sitter gravity. According to Ref.[8], we have rewrit-ten the evolution equations as a set of first-order dimensionless equations which form a six-dimensional autonomous Critical points Eigenvalues(1) ( R , , , , , − R , − R , − R , − R , − R , − R (2) ( − R , , , , , R , R , R , R , R , R (3) ( − R , , , − R , , − R , R , R , − R , R , R (4) ( R , , , R , , − R , R , R , − R , − R , − R (5) ( − R , , , R , , R , R , − R , R , R , R (6) ( R , , , − R , , − R , R , − R , − R , − R , − R (7) (0 , , , − √ / R , , R ) − . R , . R , − . − . R , − . . R , . R , , , , √ / R , , R ) . R , − . R , . − . R , . . R , − . R , , , , , , − R ) Not physicalTABLE I: The critical points and their corresponding eigenvalues. The point 9 is not physical, for its negative energy density.Critical points Property Stability(1) Positive-attractor Stable(2) Negative-attractor Unstable(3) Saddle Unstable(4) Saddle Unstable(5) Saddle Unstable(6) Saddle Unstable(7) Spiral-saddle Unstable(8) Spiral-saddle UnstableTABLE II: The stability properties of critical points. system. We find out that among all the eight physical critical points, there are one positive attractor and one negativeattractor. The positive attractor implies that the universe will expand exponentially in the end and all other physicalquantities will turn out to vanish. Also we present some numerical analysis of this model, and find out that thelate-time evolution is not sensitive to the initial values and parameter and, for a large range of parameter choice, thedynamical system would approach to the positive attractor. Therefore, in this sense, the de Sitter gravity model looksmore like the ΛCDM model,[2] rather than the PG theory.[6]If we want to know deeper on whether this model can explain the accelerating expansion, we have to settle the initialvalues and parameter choice. It requires us to do some further analytical and numerical calculations and examinethem with the current observations, such as SNeIa and CMB etc. These issues will be considered in the upcomingpapers. [1] Peebles P J E 2003 Rev. Mod. Phys. Phys. Rep. , 235 Li X Z, Hao J G and Liu D J 2002
Chin. Phys. Lett. Phys. Rev. D , 107303[3] Copeland E J, Sami M and Tsujikawa S 2006 Int. J. Mod. Phys. D Phys. Rev. D Phys. Lett. B J. Math. Phys.
212 Utiyama R 1956
Phys. Rev.
FIG. 1: The (
H, T + , ρ ) section of the phase diagram with R = 4 /
3. The heteroclinic orbits connect the critical points 1 and 2. t H H a L H - - LH - - LH L t H H b L H LH - LH - L FIG. 2: Evolution of Hubble constant H with respect to some initial values and parameter choice ( R, H , P , Q , T +0 ,T − , ρ ). According to the transformations (13), the unit of time here is the Hubble Time. (a) R is fixed and T ± is changed.(b) R is changed.Blagojevi`c M 2002 Gravitation and Gauge Symmetries (Bristol: IoP Publishing)[5] Shie K F, Nester J M and Yo H J 2008
Phys. Rev. D Phys. Rev. D Phys. Rev. D J. Cosmol. Astropart. Phys.
015 Ao X C, Xi P and Li X Z 2010
Phys. Lett. B Mod. Phys. Lett.
A 1701 Guo H Y, Huang C G, Xu Z and Zhou B 2004
Phys. Lett. A Chin. Phys. Lett. Acta Phys. Sin. Europhys. Lett. J. Cosmol. Astropart. Phys.0810