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

Regular Article - Theoretical

Cosmology in symmetric teleparallel 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

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 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 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 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 . 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 mod- els 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 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 con- stant. 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 2κ2 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) − xxx (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 con- sider 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 : 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 ), 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 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.]