Cosmology in Symmetric Teleparallel Gravity and Its Dynamical System

Home , Cosmological constant, Nonmetricity tensor, Teleparallelism

Eur. Phys. J. C (2019) 79:530 https://doi.org/10.1140/epjc/s10052-019-7038-3

Regular Article - Theoretical Physics

Cosmology in symmetric teleparallel gravity and its dynamical system

Jianbo Lua, Xin Zhao, Guoying Cheeb Department of Physics, Liaoning Normal University, Dalian 116029, People’s Republic of China

Received: 28 February 2019 / Accepted: 10 June 2019 / Published online: 20 June 2019 © The Author(s) 2019

Abstract We explore an extension of the symmetric telepa- symmetric teleparallel connection that is not metric compati- rallel gravity denoted the f (Q) theory, by considering a ble. This classification highlights that curvature is a property function of the nonmetricity invariant Q as the gravitational of the connection and not of the metric tensor or the mani- Lagrangian. Some interesting properties could be found in fold. It becomes a property of the metric only through the use the f (Q) theory by comparing with the f (R) and f (T ) the- of the Levi–Civita connection. GR can be equivalently for- ories. The field equations are derived in the f (Q) theory. mulated in terms of either of these connections. All of them The cosmological application is investigated. In this theory can be used to define Lagrangians of which Euler-Lagrange the accelerating expansion is an intrinsic property of the uni- equations coincide with the Einstein equations for a particu- verse geometry without need of either exotic dark energy or lar choice of contributing terms. extra fields. And the state equation of the geometrical dark In recent years teleparallel theories have gained more energy can cross over the phantom divide line in the f (Q) attention as alternative theories of gravity. While one mostly theory. In addition, the dynamical system method are inves- works in the torsion-based setting, there has been interest in tigated. It is shown that there are five critical points in the the direction of symmetric teleparallelism, where instead of STG model for taking f (Q) = Q +αQ2. The critical points curvature or torsion gravity is effectively described by non- P4 and P5 are stable. P4 corresponds to the geometrical dark metricity. It should be mentioned that the mechanism medi- wef f =− energy dominated de Sitter universe ( tot 1), while P5 ating gravity is the affine connection rather than the physi- wef f = corresponds to the matter dominated universe ( tot 0). cal manifold. This is reflected in the fact that in GR curva- Given that P4 represents an attractor, the cosmological con- ture is a property of the connection and not of the manifold stant problems, such as the fine tuning problem, could be itself, and thus can be equally described by other connection solved in the STG model. property such as nonmetricity. Symmetric teleparallel gravity (STG) offers an interesting geometric interpretation of gravitation besides its formulation in terms of a spacetime 1 Introduction metric and Levi–Civita connection or its teleparallel formulation. It describes gravity through a connection which is not The discovery of the cosmic accelerated expansion has moti- metric compatible, however is curvature-free and torsion- vated a vast number of researches on modifications of general free. This connection can be simplified to a partial derivative relativity (GR) (for recent reviews, see [1–3]) to explain this through the so-called coincident gauge [4,5]. This geometri- acceleration. A plethora of theories have been proposed in cally implies that vectors do remain parallel at long distances the literature, essentially based on specific approaches. From on a manifold [6]. By demanding that the curvature vanishes the view point of the connection, theories of gravity can be and that the connection is torsionless, the remaining gravita- classified into three broad classes. The first class uses the tional information is encoded in nonmetricity contributions Levi–Civita connection of the metric and its curvature. The [4,6–9]. second class uses the tetrads of a metric and their curvature In STG, the metricity condition of GR is relaxed and then ¨ free, metric-compatible, Weitzenbock connection with tor- produces teleparallel equivalent of general relativity [10]. sion. And the third class uses a curvature-free and torsion-free One of important properties of STG is their ability to separate gravitational and inertial effects [11] which is not possible in a e-mail: [email protected] GR. It has produced many strains on the theory such as the b e-mail: [email protected] 123 530 Page 2 of 8 Eur. Phys. J. C (2019) 79 :530 issue of defining a gravitational energy-momentum tensor sic property of the universe geometry without need of either [12]. exotic dark energy or extra fields. This theory is identified as More recently, STG was deeply analyzed in [4]. An excep- a metrical formulation of f (T ) gravity. In contrast with f (T ) tional class with a vanishing affine connection was discov- gravity, this new theory respects local Lorentz symmetry and ered. Based on this property, a simpler geometrical formula- harbour no extra degrees of freedom, since the dynamic vari- tion of GR was proposed. It leads to a purely inertial connec- able is the metric instead of the tetrad. At the same time it tion that can be completely removed by a coordinate gauge has the advantage over f (R) gravity that its field equations choice [13]. This formulation fundamentally deprives gravity are second-order instead of fourth-order and then is free of from any inertial character. The resulting theory is described pathologies. It may provide an satisfactory alternative to con- by the Hilbert action purged from the boundary term and is ventional dark energy in general relativistic cosmology and a more robustly underpinned by the spin-2 field theory. This new explanation of the acceleration of the cosmic expansion. construction also provides a novel starting point for modified The cosmological constant problem, the fine tuning problem gravity theories, and presents new and simple generaliza- and the problem of phantom divide line crossing are solved tions where analytical self-accelerating cosmological solu- or disappear. tions arise naturally in the early and late time universe. Even though the physical aspects of the dynamical nonmetricity are not easy to interpret, a gauge interpretation 2 Field equations of STG was proposed from a physical point of view [11]. The equivalence principle which allows eliminating locally The symmetric teleparallel connection relies only on non- gravitation makes it to be an integrable gauge theory. In this metricity and does not possess neither curvature nor torsion approach, the metric represents the gravitational field and the which yields some interesting results. One can transform to connection corresponds to the gauge potential [14]. a zero connection gauge and thereby covariantize the partial STG was presented originally in a paper by Nester and Yo derivatives as well as split the Einstein–Hilbert action into [6], where the authors emphasize that the formulation brings the Einstein Lagrangian density and a boundary term [4,6]. a new perspective to bear on GR. However, the formulation In differential geometry, the general affine connection can is geometric and covariant. More recently, some new results always be decomposed into three independent components and important developments have been obtained. An excep- [17], namely, tional class was discovered which is consistent with a vanish- λ λ λ λ μν = μν + K μν + L μν, (1) ing affine connection [7]. Based on this remarkable property, a simpler geometrical formulation of GR was proposed. It where the first term is the Levi–Civita connection of the met- provides a novel starting point for modified gravity theories, ric gμν, given by the standard definition and presents new and simple generalizations where acceler- λ 1 λβ ating cosmological solutions arise naturally in the early and μ ν ≡ g ∂μgβν + ∂ν gβμ − ∂β gμν . (2) 2 late-time universe. λ In a generalization of STG [15], a nonminimal cou- The second term K μν is the contortion: pling of a scalar field to the nonmetricity invariant was λ 1 λ λ K μν ≡ T μν + T(μ ν), (3) introduced. The similarities and differences with analogous 2 scalar-curvature and scalar-torsion theories were discussed. λ λ with the torsion tensor defined as T μν ≡ 2 μν . The third The class of scalar-nonmetricity theories was extended by [ ] term is the disformation considering a five-parameter quadratic nonmetricity scalar λ 1 λρ and including a boundary term [16]. The equivalents for L μν ≡ g −Qμρν − Qνρμ + Qρμν , (4) GR and ordinary (curvature based) scalar-tensor theories 2 were obtained as particular cases. In a recent paper [5], which is defined in terms of the nonmetricity tensor: Qρμν ≡ STG is extended by considering a new class of theories ∇ρ gμν. We define two traces of the nonmetricity tensor: where the nonmetricity is coupled nonminimally to the mat- μ μ Qρ = Qρ μ, Qρ = Q ρμ, (5) ter Lagrangian. The theoretical consistency and motivations on this extension are established. Its cosmological applica- and introduce the superpotential tion provides a gravitational alternative to dark energy. α α α α α α 4P μν=−Q μν+2Q(μ ν) − Q gμν−Q gμν − δ(μ Qν). However, comparing to these complicated general approaches mentioned above, a more special and simple model is (6) more suitable to the cosmological application. In this paper, We have therefore starting from a rather simple Lagrangian a good toy model λ 1 λρ is obtained, where the accelerating expansion is an intrin- L μν ≡ g −∇μgρν −∇ν gμρ +∇ρ gμν , (7) 2 123 Eur. Phys. J. C (2019) 79 :530 Page 3 of 8 530 and can construct an invariant, the quadratic nonmetricity at first. Consider a flat Friedmann–Lemaitre–Robertson– scalar Walker (FLRW) background with the metric μν α β α β 2 2 2 Q =−g L βμL να − L βαL μν , (8) gμν = diag −1, a (t) , a (t) , a (t) , (13) that is special among the general quadratic combination where a(t) is a scale factor. The non-vanishing components because, in addition to being invariant under local general of the Levi–Civita connection are linear transformations, it is also invariant under a transla- · 0 0 0 0 tional symmetry that allows to completely remove the con- 0 0 = 0, 0 i = i 0 = 0, i j = aaδij, nection. In this special gauge with vanishing connection, i = , i = i = δi , λ =− λ 0 0 0 j 0 0 j H j L μν μ ν , and then the nonmetricity scalar Q can be expressed in terms of the Levi–Civita connection as i = 0, i, j, k,...=1, 2, 3. j k μν α β α β Q =−g β μ ν α − β α μ ν . (9) (14)

This gauge choice is called the coincident gauge, and shown and then the nonmetricity scalar is to be consistent in the symmetric teleparallel geometry [4]. =− 2. The difference between the invariant Q and the Ricci Q 6H (15) scalar is a boundary term. The theory described by Q (the · Here H ≡ a/a is the Hubble parameter and a dot represents boundary term in this theory is absent) is a kind of special a derivative with respect to the cosmic time t. Then the ﬁeld STG [4] which is equivalent to an improved version of GR. Eq. (11) take the forms The connection can be fully trivialised and represents a much · 2 simpler geometrical interpretation of gravity, the origins of 1 f κ 3H 2 =− f + 6H 2 f − 18H 2 H + ρ, (16) the tangent space and the spacetime coincide. This theory is 2 f f f called the coincident GR, a symmetric teleparallel equivalent · · 2 1 f κ of GR. −2H − 3H 2= f + 6H 2 f − 18H 2 H + p, 2 f f f We start to reformulate GR using the symmetric teleparal- (17) lel connection and extend it by considering the action deﬁned by a function f (Q): which lead to √ · κ2 · 1 36 2 S = d −gf(Q) + Lm , (10) H =− ρ + p − H Hf , (18) 2κ2 2 f κ2

κ2 = π 0 1 2 3 where 8 G N with the bare gravitational constant G N , where ρ =−T 0, and p = T 1 = T 2 = T 3 are the Lm is the Lagrangian of the matter fields. The variational density and the pressure of the matter fields, respectively. It is principle yields the field equations for the metric gμν: easy to see that, in f (Q) gravity, the gravitational constant κ2 is replaced by an effective (time dependent) κ2 = κ2/f (Q). 2 √ α 1 eff √ ∇α −gf P μν + gμν f On the other hand, it is reasonable to assume that the present −g 2 day value of κ2 is the same as the κ2 so that we get the αβ αβ 2 eff + f Pμαβ Qν − 2Qαβμ P ν =−κ Tμν. (11) simple constraint: where primes () stand for derivatives of the functions with κ2 ( = ) = κ2 → ( ) = , eff z 0 f Q0 1 (19) respect to Q, and where z is the redshift. 2 δLm ( ) Tρσ =−√ , (12) If f Q has the form −g δgρσ f (Q) = Q + (Q) , (20) is the energy-momentum of the matter fields. then the Eqs. (16), (17) and (18) become

2 2 3H = κ (ρ + ρde) , (21) 3 Cosmological model · 2 2 −2H − 3H = κ (p + pde) , (22) We now investigate the cosmological dynamics for the models based on f (Q) gravity. In order to derive conditions for and ( ) ·· the cosmological viability of f Q models we shall carry a κ2 ( ) =− (ρ + ρ + 3p + 3p ) , (23) out a general analysis without specifying the form of f Q a 6 de de 123 530 Page 4 of 8 Eur. Phys. J. C (2019) 79 :530 where model can be considered as a metric formulation of the f (T ) · 1 6 18 model. ρ =− − H 2 − H 2 H , (24) de 2κ2 κ2 κ2 The Eqs. (28) and (29) become and · 2 =−1 − 2 − 2 + κ2ρ. 1 6 2 · 3H 6H 18H H (33) p = + H 2 + H − H 2 , 2 de 2 2 2 9 (25) · 2κ κ κ 1 −2H − 3H 2 = + 6H 2 are the density and the pressure of the “geometrical dark 2 · ( ) energy” Q , respectively. The state equation of the “geo- +2H − 9H 2 + κ2 p. (34) metrical dark energy” is then · and yield p 2H − 18H 2 w = de =−1 − . (26) de ρ · · 2 de 1 + 2 + 2 κ 2 6H 18H H H =− (ρ + p) . (35) 2 1 + − 18H 2 Let N = ln a. For any function F(a) we have FN =: · dF = . κ2 H F Then the Eq. (26) becomes So, is simply a redeﬁnition of gravitational dN 1+ −18H 2 + constant κ2. w =− + 1 Q N 3Q . de 1 1 (27) Considering a vacuum universe with the density and pres- 3 Q /Q − 2 + Q N 2 sure of universal matter: ρ = 0 and p = 0, Eq. (35)givesa The Eqs. (21) and (22) become universal de Sitter solution 2 · κ 1 1 · H 2 = ρ − + Q + Q H , (28) = , 3 6 3 H 0 (36) κ2 − + − 2 =−2 p Q 2Q . ( ) H (29) for any function Q . Then (26)gives N 3Q + 2 + 2 We see that a constant acts just like a cosmological constant wde =−1, (37) (dark energy), and linear in Q (i.e. = constant) is simply ( ) a redeﬁnition of gravitational constant κ2. Then the Eq. (29) which means that the role of the geometrical quantity Q can be written as can be seen as the “dark energy” or the cosmological constant. 1 3 Q 1 2 1 + − Q Q N =− + − Q + κ p. The Eq. (34)gives 6 2 2 2 (30) · κ2 p + 3H 2 + 1 + 6H 2 Taking a universe with only dust matter so H =− 2 . (38) 2 1 + − 9H 2 p = 0, Substituting (38)into(26) yields

+ 6H 2 + 54H 4 + 2κ2 9H 2 − p wde =− . (39) (1 + ) + 12H 2 − 18H 2 12H 2 + 3H 2 + − 18H 2κ2 p we have According to the Eqs. (16–18) and (26), we can see that the evolutions of cosmological quantities depend on the choice 1 + − 3 Q ( ) − 2 dQ = dN, (31) of f Q , its ﬁrst and second derivatives with respect to Q. 3Q (1 − /Q + 2) As an example, we take a concrete form as ( ) and then we ﬁnd the solution Q a in closed form: (Q) = α (−Q)n = α6n H 2n. (40) Q 3 1 dx 1 + x (x) − xxx (x) a (Q) = exp − 2 . In this case, (38) and (39) become 3 − 2 x 1 − (x) /x + 2 (x) 6H0 x 2 2 1 n 2n (32) · κ p + 3H + − n α6 H H =− 2 , (41) 2 1 + 3n (3n − 5) α6n−2 H 2n−2 The Eqs. (28), (29) and (32) take the same forms as the ( ) ones for f T gravity given by Linder [18]. In a sense our and 123 Eur. Phys. J. C (2019) 79 :530 Page 5 of 8 530

3 (3n − 2)(n − 1) H 2 + n (3n − 1) κ2 p wde =− . −3 (n + 1)(3n − 2) H 2 + 6nn (2n − 1)(3n − 2) αH 2n − 3n (n − 1) κ2 p (42)

For dust matter The Eq. (44) now becomes p = 0, (43) · 3H 2 18αH 2 − 1 H = . (54) 2 6αH 2 + 1 we have Using the formula · 3H 2 + 1 − n α6n H 2n H =− 2 , (44) · 2 1 + 3n (3n − 5) α6n−2 H 2n−2 a 1 · H = =− z, (55) 3 (3n − 2)(n − 1) H 2 a 1 + z wde =− . −3 (n + 1)(3n − 2) H 2 + 6nn (2n − 1)(3n − 2) αH 2n (54) can be rewritten as (45) 2 6αH 2 + 1 dH dz − = , (56) When 3H 18αH 2 − 1 (1 + z) = , n 2 (46) and integrated i.e. for the gravitational Lagrangian 1/3 4/9 2 α 2 − √ H 18 H0 1 2 z = − 1. (57) Lg = −g Q + αQ , (47) 2 α 2 − H0 18 H 1 (39)gives The Eq. (21) indicates that during the evolution of the universe H 2 decreases owing to decreasing of the matter w =− 1 . de (48) ρ w 3 24αH 2 − 1 density . This makes de descend during the evolution of the universe. The evolution of the function w = w (αH 2) α de de Next, we estimate the value of the model parameter . Letting given by (48) is illustrated in Fig. 1. It is shown that the state wde =−1, equation of “geometrical dark energy” can cross over the phantom divide line (wde =−1). According to (48) when and 2 = 1 w =−1 ( ) H 12α , de 3 , Q changes from ”visible” to = / / . × −18 −1, 2 > 1 H0 74km sec Mpc 2 4 10 sec (49) dark as indicated by (23). If H 12α , it decelerates the 2 < 1 expansion, if H 12α , it accelerates the expansion. When we can compute to gain 2 1 H = α , wde crosses the phantom divide line −1. In other − − 18 α = 1. 0145 × 10 5 (km/sec/Mpc) 2 words, the expansion of the universe naturally includes a = 9. 6451 × 1033sec2. (50) decelerating and an accelerating phase. α given by (50) can be seen as a new constant describing the evolution of the Then when universe. 1 w =− , (51) de 3 we obtain 4 Dynamical system approach in the STG theory H = 90. 631km/sec/Mpc. (52) A powerful and elegant way to investigate the universal Observations of type Ia supernovae at moderately large red- dynamics is to recast the cosmological equations into a shifts (z ∼ 0.5 to 1) have led to the conclusion that the dynamical system [28]. Dynamical system approach can be Hubble expansion of the universe is accelerating [19–22]. applied to analyze the stability of system, and has been stud- This is consistent also with microwave background mea- ied in the plenty of cosmological models, such as the canoni- surements [23,24]. According to the result of [25–27], H = cal scalar-field models [29,30], the non-canonical scalar-field 90. 631km/sec/Mpc corresponds to models [31–33], the scalar-tensor theories [34,35], the f (R) gravity theory [36–38], etc [39–43]. For the recent review on z ∼ 0.88, (53) the dynamical system approach, one can see reference [46]. which is consistent with the observations. In this section, we study the cosmological dynamical system 123 530 Page 6 of 8 Eur. Phys. J. C (2019) 79 :530 in the STG theory. Following the above discussion, we consider a specific form, i.e. Eq. (47). Then the cosmological equations (16,18) can be rewritten as 36HH˙ 2α 3H 2 =− 1 − 12H 2α − 2 + 4α + 2( − 2α) − 6H 36H 6H 1 12H 2(1 − 12H 2α) κρ + (58) 1 − 12H 2α κρ + κp H˙ = . (59) 2(−1 + 48H 2α)

We assume that the universal matter include the dust matter Fig. 1 The evolution of wde and the radiation matter. Combining Eqs. (58) and (59), we gain Table 1 The stability and eigenvalues of critical points for STG theory 6H 2α κρ (1 − 24H 2α) ˜ wef f + r Points xym tot Eigenvalues Stability 1 − 12H 2α 3H 2(1 − 12H 2α)(1 − 48H 2α) P − 1 1 5 − 1 (2, 2 ) Unstable κρ ( − 2α) 1 2 4 4 3 5 m 1 30H 1 + = 1, (60) P2 010 (−4, 1) Unstable 2( − 2α)( − 2α) 3 3H 1 12H 1 48H − 1 3 − 2 ( 9 , − 2 ) P3 2 0 2 5 5 5 Unstable where ρm and ρr denote the density of dust matter and radia- − (− 9 , − ) P4 100 1 2 4 Stable tion matter, respectively. The relations between pressure and P 005 0 (−3, −1) Stable = = 1 ρ 5 4 density for two matters: pm 0 and pr 3 r have been used in the above derivation. Deﬁning the following three dimensionless variables eigenvalues λ of M determine the stability of critical points, 6H 2α κρ (1 − 24H 2α) i x = , y = r , i.e. the stability of critical points relate to the symbols of real 1 − 12H 2α 3H 2(1 − 12H 2α)(1 − 48H 2α) parts of λi . The critical point is unstable, if its corresponding κρ ( − 2α) ˜ m 1 30H eigenvalues have one or more positive real part; While the m = , (61) 3H 2(1 − 12H 2α)(1 − 48H 2α) critical point is stable, if its corresponding eigenvalues have |M then Eq. (60) can be rewritten as all the negative real part. Solving the secular equation - λE|=0(E is the unit matrix), we can obtain the eigenvalues ˜ m = 1 − x − y, (62) of the critical points (please see Table 1). From Table 1,we which can be seen as a constraint equation. can see that the critical points: P4 and P5 are stable. We also ˜ The dynamical system in the STG is obtained as follows calculate the values of the dimensionless variable m and the effective state parameter of total matter in our universe x(1 + 2x)[3 + 6x2 + y − 3x(3 + 2y)] wef f =− − 2H˙ = 6x−12x2−6xy+y x =− , (63) tot 1 2 − + 2 for each critical point. N − + 2 3H 3 15x 18x 1 5x 6x After calculation, it is shown that stable critical point P4 cor-

y[−1 + x − 144x5 + y − 20xy − 48x3(−2 + 3y) + 48x4(1 + 3y) + 8x2(−5 + 13y)] yN = , (64) (1 − 2x)2(1 − 9x + 18x2) where subscript “N” denotes the derivative with respect to N = ln a. Solving the equations xN = 0 and yN = 0, we responds to the geometrical dark energy dominated de Sitter wef f =− can get ﬁve critical points of the dynamical system for the universe ( tot 1), while P5 corresponds to the matter STG theory, which are listed in Table 1. Next, we investigate dominated universe. the stability of critical points. Taking x j = xcj +δx j , we can We plot the phase space portraits on the x − y plane for δ = Mδ linearize the dynamical equations to obtain x j x j , STG, which are illustrated in Fig. 2.FromFig.2, we can read where x j denote the dimensionless variable (x, y) for j = 1 that P4 and P5 represent attractor solutions of the dynamical and j = 2 respectively, xcj denote critical points, δx j denote system, while there are not attractor solutions corresponds the linear perturbations of the dynamical variables, and M to P1, P2 and P3 critical points. Given that P4 represents an denotes the coefﬁcients matrix. It is well known that the attractors for the universe at late times (at where surrounding 123 Eur. Phys. J. C (2019) 79 :530 Page 7 of 8 530

as indicated by (24)an(25). This is a geometrical dynamic 1.5 model of dark energy and thus is different from the CDM model essentially. As shown in Eqs. (26)or(48), the state P2 1.0 equation of the geometrical dark energy can cross over the phantom divide line (wde =−1) in the STG model. Furthermore, the dynamical system method are investi- 0.5 gated in the STG theory. It is shown that there are ﬁve criti- 2 P1 cal points in the STG model for taking f (Q) = Q + αQ . y The critical points P and P are stable. P corresponds to P3 P5 P4 4 5 4 0.0 the geometrical dark energy dominated de Sitter universe wef f ( tot =-1), while P5 corresponds to the matter dominated wef f universe ( tot =0). Given that P4 is an attractor, the cosmo- –0.5 logical constant problems, such as the ﬁne tuning problem, could be solved in the STG model.

–1.0 Acknowledgements We thank the anonymous referee for his/her very instructive comments, which improve our paper greatly. The research –1.0 –0.5 0.0 0.5 1.0 1.5 2.0 work is supported by the National Natural Science Foundation of China x (11645003,11705079).

Fig. 2 Phase-space trajectories on the x − y plane for STG. The red Data Availability Statement This manuscript has no associated data or dot corresponds to the critical point the data will not be deposited. [Authors’ comment: The data supporting the ﬁndings of this study are available within the article.]

Open Access This article is distributed under the terms of the Creative points are attracted into it irrespective of the initial condi- Commons Attribution 4.0 International License (http://creativecomm tions), it is shown that the cosmological constant problems, ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, such as the ﬁne tuning problem, could be solved in this STG and reproduction in any medium, provided you give appropriate credit model. to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3.

5 Discussion and conclusions

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