Interacting quintom dark energy with Nonminimal Derivative Coupling

Physics of the Dark Universe 15 (2017) 72–81

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Interacting quintom dark energy with Nonminimal Derivative Coupling Noushin Behrouz, Kourosh Nozari *, Narges Rashidi Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran

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Article history: Received 19 July 2016 Received in revised form 1 November 2016 Accepted 2 November 2016 Keywords: Dynamical systems Dark energy Dark matter Non-minimal derivative coupling Statefinder diagnostic Observational constraints Cosmological perturbations

a b s t r a c t Following our recent work on interacting dark energy models (Nozari and Behrouz, 2016), we study cosmological dynamics of an extended dark energy model in which gravity is non-minimally coupled to the derivatives of a quintessence and a phantom field in a quintom model. There is also a phenomenological interaction between the dark energy and dark matter components. By considering an exponential potential as a self-interaction potential for quintom model, we obtain a scaling solution to alleviate the coincidence problem. The existence and stability of the critical points are discussed in details and it has been shown that in this setup the universe experiences a phantom divide crossing. We compare the model with recent observational data and find some constraints on the model’s parameters. We investigate also perturbations around the homogeneous and isotropic background in our Nonminimal Derivative Coupling (NMDC) quintom model. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Observational data shows that the universe is currently experiencing an accelerating expansion due to the unknown component dubbed ‘‘dark energy’’ [1–10]. Recently the issue of cosmological accelerated expansion has been criticized by some people but as has been shown in [11] all signs point to yes. If we try to address the issue by looking at the General relativity in four dimensions, currently there are two ways to describe this late time cosmic speed up: One way is the modification of the gravitational sector of the Einstein’s field equations (see for instance [12–14]) and another way is the modification of the content of the universe by introducing a dark energy component with negative pressure. The simplest candidate for dark energy is the cosmological constant with the equation of state (EoS) parameter ω = −1 that coincides extraordinarily with the observational data. Unfortunately, cosmological constant suffers from some serious problems such as huge fine-tuning and lack of dynamics [15–18]. In fact, the energy density of the cosmological constant is constant and the vacuum energy density is estimated to be 1074 GeV4 . This is much larger than the observed value of the dark energy, 10−47 GeV4 . If vacuum energy with the energy density of the order of 1074 GeV4 was present in the past, the universe would have entered an eternal stage of cosmic acceleration already, like in the very early Universe. If the cosmological constant is responsible for the present

*

Corresponding author. E-mail addresses: [email protected] (N. Behrouz), [email protected] (K. Nozari), [email protected] (N. Rashidi). http://dx.doi.org/10.1016/j.dark.2016.11.001 2212-6864/© 2016 Elsevier B.V. All rights reserved.

cosmic acceleration, we need to find a mechanism to obtain its tiny value consistent with observations. If cosmological constant had a dynamics, like a field, it could be suppressed [19]. There are other candidates for dark energy including several types of the scalar field. A canonical scalar field, named quintessence field [20–23], a phantom field (a scalar field with negative kinetic term) [24–28], tachyon field emerged from string theory [29–32], a scalar field with a generalized kinetic energy term dubbed k-essence field, [33, 34] and Chaplygin gas component [35,36] are some types of the scalar fields which can be mentioned. Also, a combination of both the quintessence and phantom fields in a unified model is another candidate for dark energy. This combined model is named quintom model [37–45]. As we know, in a quintessence dark energy model, the equation of state parameter remains always larger than −1. Also, in a phantom model, EoS is always smaller than −1. One of the most important properties of the quintom model is that in this model, EoS parameter can cross the phantom divide line. In this respect, a quintom model seems to be an interesting candidate for dark energy. On the other hand, there are some other dark energy models in light of the scalar-tensor theories in which the scalar fields are non-minimally coupled to gravity [46–55]. In a non-minimal coupling scenario, some non-renormalizable operators come out that violate the unitarity bound of the theory during the inflation era [56–59]. However, by considering a non-minimal coupling between the derivatives of the scalar fields and curvature, one can avoid the unitarity bound violation [60–65]. Note that in this framework, Higgs boson would behave like a primordial Inflation. Some authors have considered the coupling between the scalar

N. Behrouz et al. / Physics of the Dark Universe 15 (2017) 72–81

field and the kinetic term as a source of dark energy and have shown that this coupling contributes in the late time cosmic speed up [66,67]. The scenario of the non-minimal coupling between the derivatives of the scalar fields and curvature is regarded as a subset of the most general scalar-tensor theories which appear as the low energy limit of some higher dimensional theories, like superstring theory [68,69]. Also, these theories emerge as part of the Weyl anomaly in N = 4 conformal supergravity [70,71]. From a perturbative viewpoint, this framework has opened a new window on the issue of quantum gravity proposal [72]. Also, some authors have explored the role of the non-minimal derivative coupling during the inflationary stage [73–77]. The possible interaction between gravity and derivatives of the dark energy component are given by [60] L1 = k1 Rϕ,µ ϕ ,µ , L4 = k4 Rµν ϕϕ

;µν

,

L2 = k2 Rµν ϕ ,µ ϕ ,ν ,

L3 = k3 Rϕ □ϕ,

L5 = k5 R;µ ϕϕ ,µ ,

L6 = k6 □Rϕ 2 .

The authors in Refs. [60,61,63,64] have argued that without loss of generality and using the total divergencies one can keep only the two first terms, L1 and L2 . In this regard, in this work, we keep just these two terms. Note that, the coefficients k1 and k2 are the coupling parameters with dimension of the length-squared. Also, to have a ghost free theory, we should set k2 = − 12 k1 [64], which leads to the presence of Einstein tensor in the Lagrangian. To investigate the cosmological coincidence problem, considering the possible interaction between dark energy and dark matter can be useful. Authors have shown that to alleviate the coincidence problem, one can consider a non-minimal interaction between dark energy and dark matter [78–85]. Also, it has been shown that considering this interaction makes interpretation of the observational data better [86–88]. Therefore, it seems that the presence of such an interaction is important (at least theoretically). When there is an interaction between dark energy and dark matter, the energy can flow either from dark energy to dark matter or vice versa. The significant point is that some issues such as the existence of attractor solutions, the growth rate of the perturbations and the stability of cosmological solutions are affected by this direction of the energy flow. On the other hand, a successful dark energy model should be consistent with the observational data. All observational reported in recent years are in the favor of a spatially flat universe. For the sake of comparison with our findings, we note that the value of the matter density parameter, obtained from TT, TE, EE+lowP+lensing+BAO joint data from Planck2015 experiment, is as Ωm = 0.3089±0.0062 and the value of the dark energy density parameter is as Ωde = ±0.6911±0.0062 [86]. It means that a dark energy model to be a successful scenario, should satisfy the constraint Ωm + Ωde = 1 (neglecting the radiation and curvature contributions) with the mentioned values of the matter and dark energy density parameters. With these preliminaries, we consider a quintom cosmological model consisting a quintessence and a phantom field in which there is a non-minimal coupling between the derivatives of these scalar fields and curvature. We also consider an interaction between the dark sectors of the model on the basis of some phenomenological considerations. We are going to find a possible alleviation of the coincidence problem in this setup. For this purpose, in Section 2 we construct the setup of the model and present main equations of cosmological dynamics. In Section 3, we study the cosmological dynamics of this model in phase space via a dynamical system analysis. We find the critical points in the phase space and investigate the stability of each critical points. In Section 4, we investigate the perturbations around the homogeneous and isotropic background solutions. By deriving the general perturbed equations, some specific cases such as the scaling solution and the de Sitter solution are studied. The cosmological viability of

73

the model, in confrontation with TT, TE, EE+lowP+lensing+BAO joint data of Planck2015 experiment [86], is studied in Section 5. In Section 6, we study the cosmic history of the model at hand by using the statefinder diagnostic technique. Finally, in Section 7, we summarize the results. 2. The model A quintom model is a hypothetical dark energy model that contains both quintessence and phantom fields simultaneously. In what follows by ‘‘quintom’’ we mean the dark energy that contains both quintessence and phantom field simultaneously. In a cosmological setup in which the derivatives of the quintom as dark energy component are non-minimally coupled to the curvature and there is also an interaction between dark energy and dark matter, the general form of the action is given by [64]

∫ S =

[ √ 1 R d x −g − (gµν − ηGµν )ϕ ,µ ϕ ,ν 4

2

2

]

1

,µ

+ (gµν + ηGµν )σ σ

,ν

2

− V (ϕ, σ ) − F (ϕ, σ )Lm ,

(1)

where R is the curvature scalar, Gµν = Rµν − 21 gµν R is Einstein tensor and V (ϕ, σ ) = V0 e−(mϕ+nσ ) is the quintom potential with positive V0 and constant m and n. ϕ is a quintessence scalar field and σ is phantom scalar field which jointly make the quintom dark energy model. F (ϕ + σ ) = β eα (ϕ+σ ) represents the interaction term between the dark energy and dark matter (where α and β are positive and constant) and Lm is the Lagrangian density of dark matter. We also use the system of units in which 8π G = c = h¯ = 1. Variation of the action (1) with respect to the metric leads to the following gravitational field equations [63,64] (ηϕ )

(ϕ ) (σ ) (ησ ) (m) Gµν = Tµν + Tµν + η(Tµν + Tµν ) + Tµν ,

(2)

with (ϕ ) Tµν = ∇µ ϕ∇ν ϕ −

1

gµν (∇ϕ )2 − gµν V (ϕ, σ ),

2

(σ ) Tµν = −∇µ σ ∇ν σ +

1 2

gµν (∇σ )2 − gµν V (ϕ, σ ),

(3) (4)

1

(η )

−T µνϕ = − ∇µ ϕ∇ν ϕ R + 2∇α ϕ∇(µ ϕ Rαν) + ∇ α ϕ∇ β ϕ Rµανβ 2

1

α

+ ∇µ ∇ ϕ∇ν ∇α ϕ − ∇µ ∇ν ϕ □ϕ − (∇ϕ )2 Gµν 2 [ 1 ] 1 α β 2 + gµν − ∇ ∇ ∇α ∇β ϕ + (□ϕ ) − ∇α ϕ∇β ϕ Rαβ , 2 1

2

(5)

ησ ) −T (µν = − ∇µ σ ∇ν σ R + 2∇α σ ∇(µ σ Rαν) + ∇ α σ ∇ β σ Rµανβ

2

1

α

+ ∇µ ∇ σ ∇ν ∇α σ − ∇µ ∇ν σ □σ − (∇σ )2 Gµν 2 [ 1 ] 1 α β 2 + gµν − ∇ ∇ ∇α ∇β σ + (□σ ) − ∇α σ ∇β σ Rαβ , 2

2

where ∇(µ ϕ Rαν) =

∇ν σ Rαµ ).

(ϕ )

(6)

∇ ϕ Rαν + ∇ν ϕ Rαµ ) and ∇(µ σ Rαν) = 12 (∇µ σ Rαν +

1 ( µ 2 (σ )

(ηϕ )

(η )

Also, Tµν , Tµν , Tµν and Tµνσ are the energy–momentum tensor arisen from the variation of the terms depending on the (m) scalar fields and Tµν is the energy–momentum tensor of the dark matter component. Supposing a spatially-flat Friedmann– Robertson–Walker metric as ds2 = −dt 2 + a2 (t)(dr 2 + r 2 dΩ 2 ),

(7)

the energy density and pressure, derived from Eqs. (3)–(6), take the following forms

ρquin =

1 2

(ϕ˙ 2 − σ˙ 2 ) + V (ϕ, σ ) +

9 2

ηH 2 (ϕ˙ 2 + σ˙ 2 ),

(8)

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N. Behrouz et al. / Physics of the Dark Universe 15 (2017) 72–81

pquin

1

√

(

6 2 x 9 5

˙ ϕ˙ 2 + σ˙ 2 ) = (ϕ˙ − σ˙ ) − V (ϕ, σ ) − η H( 2 ) 3 + H 2 (ϕ˙ 2 + σ˙ 2 ) + 2H(ϕ˙ ϕ¨ + σ˙ σ¨ ) , 2

2

1 2

(ϕ˙ 2 − σ˙ 2 ) + V (ϕ, σ ) +

9 2

˙ 1− H

1 2

1

)

√ 6 2 x 9 5

ηH 2 (ϕ˙ 2 + σ˙ 2 ) + F (ϕ, σ )ρm ,

η(ϕ˙ 2 + σ˙ 2 ) =

2

3

(σ˙ 2 − ϕ˙ 2 ) −

2

H2

1

+ ηH(ϕ˙ ϕ¨ + σ˙ σ¨ ) − γ F (ϕ, σ )ρm

(11) 2 where γ ≡ 1 + wm is the barotropic index. Equations of motion of the scalar fields are obtained by taking variation of the action (1) with respect to each field separately as

˙ + 3H 3 ϕ˙ ) + V,ϕ (ϕ, σ ) (ϕ¨ + 3H ϕ˙ ) + 3η(H 2 ϕ¨ + 2ϕ˙ H H (12)

− (σ¨ + 3H σ˙ ) + 3η(H σ¨ + 2σ˙ H H˙ + 3H σ˙ ) + V,σ (ϕ, σ ) = F,σ (ϕ, σ )ρm , 2

ρ˙ σ + 3H(1 + ωσ )ρσ = Q2 ,

(15)

(F (ϕ, σ )ρm )˙+ 3H γ F (ϕ, σ )ρm = −(Q1 + Q2 )

(16)

respectively. Note that Q1 = α F (ϕ, σ )ϕρ ˙ m and Q2 = α F (ϕ, σ )σ˙ ρm are specific interaction terms between the dark sectors. The sign of Qi (i = 1, 2), demonstrates the direction of the energy transfer. Qi > 0 shows that energy transfers from dark matter to dark energy and Qi < 0 demonstrates the energy transfer from dark energy to dark matter. We choose the direction of the energy transfer from the dark matter to dark energy. 3. The phase space analysis In this section, we present a dynamical system approach for our non-minimal derivative coupling (NMDC) quintom model. The novel feature of this model is that derivatives of the quintom are coupled to the curvature and in the same time the quintom interacts with the dark matter. We argue that the existence of the non-minimal derivative coupling leads to a stable dynamics in late time. We show also that this model can provide a new mechanism for alleviating the coincidence problem. For this purpose, we begin by representing Eq. (10) in terms of the dimensionless variables. In this regard, we introduce the following dimensionless quantities

√ x4 =

,

√

σ˙

x2 = √

6H V (ϕ, σ )

√

3H

,

6H

,

x3 =

F (ϕ, σ )ρm

√

3H

√

, (17)

x5 = 3 η H .

1 = x21 − x22 + x23 + x24 + x25 (x21 + x22 ).

˙ H H2

=

3x21

3x22

−

+

1−

1 2 2 x (x 3 5 1

− γ

+

3 2

x22 )

x23

+

−

(x21

4 2 4 x x 9 1 5 1+ 13 x25

+

+

x22 )x25 4 2 4 x x 9 2 5

−1+ 13 x25

√

,

(19)

6x1 x25 −

1 + 13 x25

6 ϕ¨

x′4 = −

2

√

6x2 x25 −

2 3

2 3

√

˙

6x1 x25 HH2

√

,

(20)

˙

6x2 x25 HH2

.

(21)

−

˙ H H2

√

x1 ,

6 σ¨

′

x2 =

6 H2

) ˙ H

(mx1 + nx2 ) +

H2

x4 ,

−

˙ H H2

x′5 =

x2 ,

˙ H H2

(22) x5 ,

(23)

where a prime denotes a derivative with respect to N. Now we should obtain the critical points of the model. To this end, we set the autonomous Eqs. (22)–(23) equal to zero and find the solutions of the resulting equations. To verify the stability around the critical points, we explore the eigenvalues in each fixed point. These eigenvalues are derived from the following Jacobian matrix, evaluated at the critical points

⎛ ∂ x′

1

⎜ ∂ x1 ⎜ ⎜ ⎜ ∂ x′ ⎜ 2 ⎜ ∂x ⎜ 1 M=⎜ ⎜ ′ ⎜ ∂ x4 ⎜ ⎜ ∂ x1 ⎜ ⎜ ⎝ ∂ x′ 5

∂ x1

∂ x′1 ∂ x2

∂ x′1 ∂ x4

∂ x′2 ∂ x2

∂ x′2 ∂ x4

∂ x′4 ∂ x2

∂ x′4 ∂ x4

∂ x′5 ∂ x2

∂ x′5 ∂ x4

⎞ ∂ x′1 ∂ x5 ⎟ ⎟ ⎟ ′ ⎟ ∂ x2 ⎟ ⎟ ∂ x5 ⎟ ⎟ ⎟ ∂ x′4 ⎟ ⎟ ∂ x5 ⎟ ⎟ ⎟ ′ ⎠ ∂x

,

(24)

5

∂ x5

(x1 ,x2 ,x4 ,x5 )=(x1c ,x2c ,x4c ,x5c )

where ‘‘c’’ marks the critical point. If all the real eigenvalues (or the real parts of all the complex eigenvalues) are negative, the fixed points will be attractors. It means that they are asymptotically stable nodes. If all the real eigenvalues or the real parts of all the complex eigenvalues are positive, the critical points will be repellers, meaning that they are asymptotically unstable nodes. However, if at least one of the eigenvalues is negative and the others are positive or vice versa, the fixed points will be saddle points and therefore unstable nodes. The total equation of state parameter ωtot which is defined as ptot

ρtot

=

pquin + (γ − 1)F (ϕ, σ )ρm

ρquin + ρm

,

(25)

in terms of the dimensionless parameters takes the following form

( ωtot =

x21

(18)

Also, by using Eqs. (11), (12) and (13) we get

4 2 4 x x 9 2 5 −1+ 13 x25

3α x23 − 3 6x1 + 3mx24 −

6 √ H2 ( 6

ωtot =

By using these new variables in rewriting the Friedmann equation, the following constraint on the model’s parameters space is obtained

+

−1 + 13 x25

√

(13)

(14)

]

By introducing a new time variable as N = ln a(t) related to the cosmic time through dN = Hdt, and considering x3 as a dependent variable in this setup, we find the autonomous system of equations as follows x1 =

ρ˙ ϕ + 3H(1 + ωϕ )ρϕ = Q1 ,

ϕ˙

H2

4 2 4 x x 9 1 5 1+ 13 x25

√

=

−1+ 31 x25

√

3 6x2 + 3α x23 + 3nx24 −

′

3

where dot shows the time derivative. The continuity equations for the scalar fields and dark matter can be written as

x1 = √

σ¨

√

=

4 2 4 x x 9 2 5

+

3 6x22 +3α x2 x23 +3nx2 x24 − 6x22 x52 −1+ 13 x25

1 − 13 x25 (x21 + x22 ) +

ϕ¨

4 2 4 x x 9 1 5

1+ 13 x25

√

[

+

ηH 2 (ϕ˙ 2 + σ˙ 2 )

= F,ϕ (ϕ, σ )ρm ,

1+ 13 x25

1 − 13 x25 (x21 + x22 ) +

(9)

(10)

(

] √ √ −3 6x21 +3α x1 x23 +3mx1 x24 − 6x21 x52

+

2 respectively. Also, we have the Friedmann equations as 3H 2 =

[

−

x22

−

√ [

+

x24

2 6

ϕ¨

27

H2

+ (γ − x1 +

σ¨ H2

1)x23

−

]) x2

x25 .

[

x21 + x22

][ 2 H˙ 9 H2

+

1

]

3 (26)

The total equation of state parameter has to be restricted as ωtot < − 31 in order to have a positively accelerated expansion in this

N. Behrouz et al. / Physics of the Dark Universe 15 (2017) 72–81

75

Table 1 Properties of the critical points of the NMDC quintom model. Point/Line (x1c , x2c , x4c , x5c )

√

Line A± (x1 , ± x21 − 1, 0, 0) √ 6 3

Point B(

α, −

√

Point D± √

− ±

6 3

,−

6 m 6

Point C± (

√

(√ ( 6 6

α, 0, 0)

Stability

for all α, m, n and x21 ≥ 1

unstable node

for all α, m, n

stable node if α (m − n) > 23 , otherwise saddle point attractor point if n2 < m2 ,

√ 6 n 6

√

,± 1 +

n2 −m2 6

, 0)

2α 2 (m−n)+2α n(n−m)+6α−3m 3α (m−n)−m2 +n2

(

)

2α 2 (m−n)+2α m(n−m)+6α−3n 3α (m−n)−m2 +n2 √ ( )( ) 6 2α (n−m)+3 2α 2 (m−n)−12α+3(m+n) m−n 1 6 −m−n+3α 6 6

Existence

for all α and m − n < 6 2

m2 − n2 + α (n − m) < 3

2

and m2 − n2 < 6, otherwise saddle point

)

,

m ̸ = n, 3α ̸ = m + n, 3α (m − n) ̸ = m2 + n2 ,

,

)

,0

stable point if all eigenvalues

6 2α (n − m) + 3 2α (m − n) − 12α + 3(m + n) > 0

of D± are negative

0 < 2α (m − n) − 3 ×

otherwise saddle point

(

)(

(

)

2

)

( α2 (m−n)−α(m2 −n2 )+12α−3(m+n) ) m−n

(

Point E± − (√

m

n

(√

√ 6mn+

2(m4 −n4 )(m2 +n2 )+6m2 n2

2(m4 −n4 ) ) 2(m4 −n4 )(m2 +n2 )+6m2 n2

√

6mn+

< 3(3α − m − n)2 , )

,

− , 2(m4 −n4 ) √ √ √ (m2 +n2 )(m4 −n4 )+6m2 n2 6mn+6m2 n2 , ± ± 2 2 4 4 √ ) (m +n )(m −n ) 3(m2 −n2 ) ± m2 +n2

m ̸ = n, 3(m2 − n2 ) > m2 + n2 ,

−3m2 n2 < (m2 − n2 )(m2 + n2 )2 , √ √ 2 2 2(m2 −n2 )(m2 +n2 )2 +6m2 n2 (m2 −n2 )

6mn+6m n

see the explanation for

<0

points E± in the text

model. Furthermore, due to the observational constraint Ωde + Ωm ≈ 1, we get the following conditions 0 ≤ Ωm ≤ 1,

0 ≤ Ωde = [x21 − x22 + x24 + (x21 − x22 )x25 ] ≤ 1. (27)

By solving Eqs. (22) and (23), we obtain the critical lines A± and the critical points B, C± , D± , E± in our setup. The results of exploring the stability of the solutions around these critical lines and points are summarized in Tables 1 and 2. Note that in all calculations we have set γ = 1, corresponding to the pressureless dark matter. Here, we investigate the properties of each critical lines and points separately.

• Critical Lines A± : Kinetic energy dominated solutions The critical lines A± show the solutions with a quintom scalar field’s kinetic energy term domination. These solutions belong to the stiff-fluid (ωtot = 1), corresponding to a decelerating phase of the universe expansion. These lines are unstable and cannot denote the final stage of the universe evolution.

• Critical Point B: Matter dominated solution The critical point B represents an attractor point in the phase space if α (m − n) > 32 , otherwise it is a saddle point. This point belongs to the matter domination era which is expected to be relevant at early times. Nevertheless, in both cases (attractor and saddle) there is no possibility to have an accelerating phase of expansion.

• Critical Points C± : Scalar field dominated solution The critical points C± denote the solution with either a scalar field’s kinetic energy term dominated or potential energy term dominated (quintom dark energy dominated) solutions. Our study shows that this contribution depends on the values of the free parameters m and n. Although these points are attractor if n2 < m2 , m2 − n2 + α (n − m) < 3 and m2 − n2 < 6, there is no scaling solution because the contribution of the matter is zero. These points can show an attractor solution that belongs to the future. The universe experiences an accelerated phase if m2 −n2 < 2 and also this

Fig. 1. The 3-dimensional phase space trajectories of the interacting NMDC quintom model with α = 1.5, m = −1.47, n = 0.5. Critical points D± (−0.28, −0.38, ±0.86) are stable spirals so that the scaling dominated solutions are the late time attractor which can alleviate the coincidence problem. The critical points B(1.22, −1.22, 0) and C± (−0.6, −0.2, 0.82) are saddle points. All the phase space trajectories diverge from the unstable points and converge towards the attractors.

condition leads to the attractor solutions. We note that if we set m = n, the critical points C± are attractor points in which the cosmological constant with ωtot = −1 is dominated.

• Critical Points D± : Scaling solutions Physically, the most acceptable points in our setup, generated by considering m opposite to n in the power of the potential, are critical points D± . By choosing the proper values of m, n and α , leading to all four eigenvalues being negative (see Table 2), we obtain some attractor solutions which are scaling solutions with accelerating expansion. In this situation, the coincidence problem can be alleviated. Note that, in

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N. Behrouz et al. / Physics of the Dark Universe 15 (2017) 72–81

Fig. 2. The left panel shows that the equation of state parameter of quintom dark energy crosses the phantom divide in N = −0.23 (z = 0.26). The right panel illustrates that the deceleration parameter of the model becomes negative in the past at N = −0.61 (z = 0.84). Note that the panels have been plotted for critical points D± with initial values x1 = −0.28, x2 = −0.38, x4 = ±0.86, x5 = 0 and α = 1.5, m = −1.47, n = 0.5. Table 2 The values of the Ωde , ωtot , existence of the expanding phase (a¨ c > 0) and the eigenvalues (σi ’s) of the critical points.

Ωde

ωtot

a¨ c > 0

σ1c , σ2c , σ3c , σ4c

1

1

No

0, √ −3, 3 −

√

A±

± B

0

C±

m2 −n2 3

D±

0 m 2 − n2 3

−1

No

−1

2m3 α 2 −2m2 nα 2

α −2mn α + 3−(n2m −m)(m−3 α+n)2 2 3

3 2

2 3

3m2 α−21mα 2

(2α (m−n)−3)α 3(−m−n+3α )

(2α (m−n)−3)α (−m−n+3α )

√ 6 (m 4

+ 2α )+

x21 −1+(48α 2 +(−24m−24n)α+6m2 +6n2 )x21 −6(n−2α )2 4

3 2

+ α (n − m)

n2 −m2 2

,

m 2 − n2 2

m2 −n2 2

− 3,

− 3, α (n − m) + m2 − n2 − 3

(m−n)α 2 −6α+ 32 (m+n) 3α−m−n

(n−m)α 2 +3α− 32 (m+n)

−3α+m+n

,

⎛ ( ( ) 21 ∓⎝2(m−n) − 87 (m−n)2 +(m−n) α 2 −6α+(m−n)(m2 −n2 − 49 α )− 16 (m2 −n2 )

3n2 α+21nα 2 −9m 3(n−m)(m−3 α+n)2

+36α+4mnα + 3−(n9n −m)(m−3 α+n)2 3

1

√

< −1

+ 3(n−m)(m−3 α+n)2 −

E±

12(n−2α )(m−2α )x1

− 23 , − 32 , − 32 ,

2 2

α −2n α + 3(n2n −m)(m−3 α+n)2

(n + 2α ) −

4

Yes if m2 − n2 < 2 Yes if

3(n−m)(m−3 α+n)2

6(x21 −1)

+ 23 (m + n) + −1

Yes

the absence of the non-minimal derivative coupling, there is no scaling solution (see [84] and references therein). By considering the existence condition as mentioned in Table 1, we choose the values m = −1.47, n = 0.5, and α = 1.5, so that the current value of the dark matter density parameter Ωm , is in agreement with Ωm = 0.3089±0.0062 from TT, TE, EE+lowP+lensing+BAO data of Planck2015 experiment [7] (in Section 5 we perform a numerical analysis on the model parameter space and show that these values of the constant parameters, that lead to the attractor solutions, are in agreement with the observational data). Therefore, this result Ωm verifies that Ω < 1 and ωtot < − 13 . The 3-dimensional and de 2-dimensional phase portrait for this case are illustrated in the left and right panels of Fig. 1 respectively. These figures show that all trajectories converge to the attractor points D± . A parameter which gives us a suitable background to understand the dynamics of the universe is the equation of state parameter. The left panel of Fig. 2 demonstrates that equation of state parameter of the quintom dark energy has

9 2

))

1 /2

(m−n)(−3α+m+n)

+

−2(m2 +n2 −2mn)α 2 +12(m−n)α−3(m2 −n2 ) (m−n)(−3α+m+n)

σ1E± , σ2E± , σ3E± , σ4E± cannot be calculated analytically

crossed the phantom divide line (PDL) in the near past at N = −0.23 or equivalently z = 0.26 and its value in the present time is in agreement with the recent observational 0.075 data, ωde = −1.019+ −0.080 [7]. By using the following definition for deceleration parameter

˙ H , (28) H2 we have plotted the deceleration parameter versus N. The result is illustrated in the right panel of Fig. 2. We can see from this figure that in the past, at N = −0.61 or z = 0.84, the deceleration parameter has become negative, meaning that the universe has entered to an accelerating phase at this value of the redshift. q = −1 +

• Critical Points E± : Scalar field dominated solution The critical points E± denote the solution with either the scalar field’s kinetic energy term domination or potential energy term domination. For these points there is no contribution from the matter. In these cases, we have ωtot = −1

N. Behrouz et al. / Physics of the Dark Universe 15 (2017) 72–81

77

and the behavior is similar to the one with cosmological constant domination. These points can be an attractor solutions that belong to the future. Also, the universe experiences the accelerated phase of expansion. However, these solutions cannot solve the coincidence problem in the present universe. Unfortunately, we cannot obtain the analytical form of the eigenvalues of these critical points, but we can calculate them numerically. For some values such as m = −1.47, n = 0.5, α = 1.5, there is a future attractor point and for some values there are the saddle points. We note that these points have two important differences with critical points D± . The first difference is that x5 is zero for critical points D± , and 1.54 for E± . The second difference is that critical points E± have future attractor but critical points D± have a scaling solution to address the coincidence problem.

4. Perturbations Now we investigate perturbations around the homogeneous and isotropic solutions of our NMDC quintom model. First we derive the general perturbed equations and then the specific cases such as the scaling solution and de Sitter solution will be studied. To this end, we suppose Hubble parameter H = H0 (t) as a general solution for the background of Friedmann–Robertson–Walker metric which satisfies the background equation (10). We consider small variations of the energy density, background Hubble parameter, quintessence and phantom fields according to the following equations [89,90]

(

)

(

) ρm = ρm0 1 + δm (t) , H = H0 1 + δ (t) , ( ) ( ) ϕ = ϕ0 1 + δϕ (t) , σ = σ0 1 + δσ (t)

(29)

where δm (t), δ (t), δϕ (t), and δσ (t) show the perturbation of the matter, background, quintessence and phantom fields respectively. In order to get the linear regime of the perturbations, we use the first two terms of the Taylor expansion of the interaction term and potential as follows

(

) F (ϕ, σ ) = β 1 + α (ϕ + σ ) , ( ) V (ϕ, σ ) = V0 1 − (mϕ + nσ ) .

(30)

We begin our investigation for the scaling cosmological solutions related to the critical points D±. Taking derivative of Eq. (35) and replacing the result in the Eqs. (32) and (33), we can omit the perturbations of the background dynamics and obtain the following equations a11 δ˙ m (t) − a2 δm (t) − a3 δ˙ ϕ (t) − a4 δϕ (t)

− a5 δ˙σ (t) − a6 δσ (t) = 0,

(36)

b11 δ¨m (t) − b22 δ˙ m (t) − b3 δm (t) − b4 δ¨ϕ (t) − b5 δ˙ϕ (t) − b6 δϕ (t) = 0,

(37)

c11 δ¨m (t) − c22 δm (t) − c3 δm (t) − c4 δ¨ϕ (t)

− c5 δ˙ϕ (t) − c6 δϕ (t) = 0.

a1 δ (t) − a2 δm (t) − a3 δ˙ ϕ (t) − a4 δϕ (t) − a5 δ˙ σ (t) − a6 δσ (t) = 0, (32) b1 δ˙ (t) − b2 δ (t) − b3 δm (t) − b4 δ¨ϕ (t) − b5 δ˙ ϕ (t) − b6 δϕ (t) = 0, (33) (34)

where the explicit forms of the coefficients a1 , . . . , a6 , b1 , . . . , b6 and c1 , . . . , c6 are presented in the Appendix. These coefficients depend on the form of the interaction term, the scalar fields and their derivatives evaluated in the background solution. Furthermore, by perturbing the continuity equation for matter as ρ˙ m + 3H(1 + ωm )ρm = 0 and using Eq. (29) we get

δ˙m (t) = −3H0 δ (t).

4.1. Scaling solution

(38)

(31)

Using Eqs. (29)–(31), the perturbed Friedmann and Klein–Gordon equations take the following forms up to the linear approximation

c1 δ˙ (t) − c2 δ (t) − c3 δm (t) − c4 δ¨ϕ (t) − c5 δ˙ ϕ (t) − c6 δϕ (t) = 0,

Fig. 3. The growth of matter perturbation versus the cosmic time for scaling solutions D± corresponding to α = 1.5, β = 1, η = 0, m = −1.47, n = 0.5, µ = n 0.1, ν = V0 = σ0 = H0 = 1, ϕ0 = − m σ0 . At late time the growth is stable but in future it turns out to be unstable.

(35)

The issue of stability can be investigated by solving Eqs. (32)–(35). In the following subsections we shall study the stability of the cosmological solutions.

Coefficients a11 , b11 and c11 are presented in Appendix. To solve the above equations, we define the scalar fields as ϕ = ϕ0 e−µt , 2t

σ = σ0 e−ν t and the scale factor as a0 e 3(1+ωtot ) . While analytical solutions of the above equations in general is very complicated, by fixing the values of the free parameters, such as the coupling constants and the initial values, Eqs. (36)–(38) have the following solutions

δm (t) = C1 e(n1 +iµ1 )t + C2 e(n2 +iµ2 )t + C3 e(n3 +iµ3 )t + C4 e(n4 +iµ4 )t + C5 e(n5 +iµ5 )t ,

(39)

where C1,...,5 are integration constants. The values of n1,...,5 and µ1,...,5 depend on the free parameters. For instance, for scaling solution D± corresponding to α = 1.5, β = 1, η = 0, m = −1.47, n σ0 we get n = 0.5, µ = 0.1, ν = V0 = σ0 = H0 = 1, ϕ0 = − m

δm (t) = C1 e0.65t + C2 e(−11+1.4i)t + C3 e−0.34t + C4 e−4.6t + C5 e(−11−1.6i)t .

(40)

The first term in this equation shows a growing amplitude. The second and fifth terms each contain a rapidly damping factor times an oscillating part resulting together rapidly damping perturbations. The third and fourth terms indicate fast decaying amplitudes.

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N. Behrouz et al. / Physics of the Dark Universe 15 (2017) 72–81

Table 3 The ranges of n+ and n− in which the values of the scaled H0 parameter is viable. H0

0.2

0.8

1

2

n+ n−

0

0.19 −2

−0.26

−5.4

0.25

0.6

−0.58

Since the first term grows exponentially and is unstable, it seems that the scaling solution is unstable. However, as Fig. 3 shows, by choosing appropriate integration constants, the growth rate of the perturbations is stable at late time. If we change the free parameters and initial values, (39) takes some other values for n1,...,5 and µ1,...,5 . However, at least three terms of the Eq. (39) have damped oscillation with decreasing amplitudes and by choosing appropriate integration constants we can reach the previous result regarding existence of scaling solutions. 4.2. de Sitter Solution Now we consider the stability of the de Sitter solution in this setup. In this case we have H(t) = H0 , a(t) = a0 eH0 t .

(41)

For the de Sitter solution the matter part is absent in Eqs. (32)–(34) leading to a1 δ (t) − a3 δ˙ ϕ (t) − a4 δϕ (t) − a5 δ˙ σ (t) − a6 δσ (t) = 0,

(42)

b1 δ˙ (t) − b2 δ (t) − b4 δ¨ϕ (t) − b5 δ˙ ϕ (t) − b6 δϕ (t) = 0,

(43)

c1 δ˙ (t) − c2 δ (t) − c4 δ¨ϕ (t) − c5 δ˙ ϕ (t) − c6 δϕ (t) = 0.

(44)

In our model, depending on the values of m and n, the de Sitter solution is corresponding to points C± and E± in Table 1. If m = −n, points C± and otherwise points E± , are corresponding to the de Sitter solution. The solution of the polynomial system of Eqs. (42)– (44) for points C± has the following form

δ (t) = C1 en+ t + C2 en− t ,

(45)

which is independent of the constant coefficients (except for µ and ν ) and the initial values. However, it depends on H0 . The result for some values of H0 with µ = ν = 1 are presented in Table 3. As we see from this table, the parameters n+ and n− have different values for different H0 and one of them always grows exponentially leading to unstable de Sitter solution. If we solve Eqs. (42)–(44) for points E± , both terms in (45) will be unstable and grow exponentially. 5. Observational constraints In this section, we compare the results of our model with TT, TE, EE+lowP+lensing+BAO joint data of Planck2015 experiment. In this regard, we obtain some observational constraints on the model’s parameters space. To this end, we consider the relation between the matter density parameter Ωm and the dark energy density parameter Ωde and we plot Ωde versus Ωm based on the TT, TE, EE+lowP+lensing+BAO joint data of Planck2015 experiment. In a dynamical system analysis, we have found some critical points and studied their stability in details previously in this paper. Our study have shown that in some ranges of the parameters, the critical points D± are attractor points with scaling solutions and capability to alleviate the coincidence problem. Now we verify that the adopted values of the constant parameters are consistent with recent observational data. We use the constraint Ωm + Ωde = 1 and Eq. (27) and replace the values of the parameters xi (i = 1, 2, 4, 5) from Table 1. These values represent the attractor points D± and we

Fig. 4. The dark energy density parameter versus matter density parameter for a model in which derivatives of the quintom are non-minimally coupled to gravity and there is an interaction between the dark energy and dark matter, in comparison with the Planck2015 TT, TE, EE+lowP+lensing+BAO joint data. Figure is plotted for β = 0.985, 0.995, 1, 1.005, 1.02. The dashed line is corresponding to β = 1.

want to check the cosmological viability of this case. We adopt m = −1.47 and n = 0.5, as Section 3. By these values, we explore the values of the parameters α and β which lead to the cosmologically viable values of Ωm and Ωde . Fig. 4 shows the behavior of Ωde versus Ωm based on Planck2015 observational data. We have plotted the figure for β = 0.985, 0.995, 1, 1.005, 1.02 and varying values of α . The ranges of the observationally viable values of α are given in Fig. 5. As we see, the values of the parameters, which have been used to obtain an attractor point with scaling solution, are consistent with TT, TE, EE+lowP+lensing+BAO joint data of the Planck2015 experiment. 6. Statefinder diagnostic In this section, following [91], we investigate the model in more details by statefinder diagnostic technique. In this regard, we introduce a pair of parameters {r , s}, named statefinder parameters, by using the second and third derivatives of the scale factor. Statefinder diagnostic is a ‘‘geometrical’’ diagnostic as the statefinder parameters depend on the scale factor. By considering the deceleration parameter as defined in (28), these statefinder diagnostic pair of parameters, {r , s}, are given by r = s=

˙˙a˙

=

¨ H

− 3q − 2,

aH 3 H3 r −1 3(q − 12 )

(46)

.

(47)

Now we explore the expansion history of the universe in our setup by using the third derivative of the scale factor (˙˙a˙) and describe the trajectories in the {r , q}, {s, q}, {r , s} phase planes. We want to distinguish our model from the ΛCDM scenario phase diagrams with {r , q}ΛCDM = {1, −1}, {s, q}ΛCDM = {0, −1}, {r , s}ΛCDM = {1, 0} [92]. In this regard, we write Eq. (46) as r =

[ ˙ ]′ H H2

[ ˙ ]2 H

+2

H2

[ ˙ ] H

+3

H2

+ 1.

(48)

N. Behrouz et al. / Physics of the Dark Universe 15 (2017) 72–81

79

Fig. 5. The contour plots for the model’s parameters α and β . In the left panel the region between the dashed lines shows the values of the parameters leading to the observationally viable values of Ωde . Similarly, in the right panel the region between the dashed lines shows the values of the parameters leading to the observationally viable values of Ωm .

where a prime denotes the derivative with respect to N = ln a. Also, x′1 , x′2 , x′4 , x′5 are given by Eqs. (22) and (23). The explicit form of the Eq. (47) in terms of the critical points can be derived similar to Eq. (46) by replacing Eqs. (49), (28) and (19) into Eq. (47) (we do not show the lengthy result here). Fig. 6 shows the trajectories of {q, r }, {q, s} and {r , s} in phase planes for the late time stable attractor point D± . Note that the initial values used here are as x1 = −0.28, x2 = −0.8, x4 = 0.86 and x25 = 0 (for α = 1.5, m = −1.47 and n = 0.5). With these values we get Ωm = 0.31 and Ωde = 0.69 in agreement with Planck2015 experimental data. It is obvious that when the dark sectors are dominated, our trajectories in phase space will be different from the ΛCDM’s trajectories. In a special case, if we plot point C for m = n it will be coincide with ΛCDM model, that is, ΛCDM is in the history of the model in this case.

which by using Eq. (19) can be rewritten as follows

[

1

r = 1−

[[ ×

−

]

2 3

1 2 2 x (x 3 5 1

+

x22 )

+

x5 x′5 (x21 + x22 ) +

4 2 4 x x 9 1 5

2 3

4 2 4 x x 9 2 5

−1+ 13 x25

x25 (x1 x′1 + x2 x′2 )

(8x1 x′1 x45 + 16x21 x35 x′5 )(1 + 9(1 +

−

+

1+ 13 x25

x25 3

) − ( 83 x5 x′5 )(x21 x45 )

x25 2 3

)

(8x2 x′2 x45 + 16x22 x35 x′5 )(−1 + 9(−1 +

x25 3

x25 3

) − ( 83 x5 x′5 )(x21 x45 )

]

)2

[ ˙ H × 2 + −3γ x3 x′3 − 6(x1 x′1 − x2 x′2 ) H ] − 2(x1 x′1 + x2 x′2 )x25 − 2x5 x′5 (x21 + x22 ) √ √ √ [ [ 3α x23 + 3mx24 − 3 6x1 − 6x1 x25 ] 2 6 + x5 x′5 x1 9 1 + 13 x25 √ ( 3α x2 + 3nx2 + 3 6x − √6x x2 )] 2 2 5 4 3 + x2 −1 + 13 x25 √ √ √ [ [ 6 2 ′ 3α x23 + 3mx24 − 3 6x1 − 6x1 x25 ] + x x 9 5 1 1 + 13 x25 [ 6α x + 6mx x′ − 3√6x′ − √6(x′ x2 + 2x x x′ ) 3 4 4 1 5 5 1 1 5 + x1 1 + 13 x25 √ √ ( 23 x5 x′5 )(3α x23 + 3mx24 − 3 6x1 − 6x1 x25 ) ] − (1 + 13 x25 )2 √ √ [ 2 2 6x2 x25 ] ′ 3α x3 + 3nx4 + 3 6x2 − + x2 −1 + 13 x25 [ 6α x + 6nx x′ + 3√6x′ − √6(x′ x2 + 2x x x′ ) 3 4 4 2 5 5 2 2 5 + x2 −1 + 31 x25 √ √ ]] ( 23 x5 x′5 )(3α x23 + 3mx24 + 3 6x2 − 6x2 x25 ) ] − (−1 + 13 x25 )2 ˙ H + 3 2 + 1, H

7. Summary

(49)

In this paper we have considered a cosmological quintom model in which the derivatives of both the quintessence and phantom fields are non-minimally coupled to the Einstein tensor. We have also considered possible interaction between the dark sectors phenomenologically. Cosmological dynamics in this setup has been studied in details and the background equations are derived in order to see trajectories and critical points in phase space via a dynamical system approach. The critical lines A± and critical points B, C± , D± and E± are obtained and the stability of cosmological solutions around these critical states are studied. Our study has demonstrated that if n2 < m2 , the critical points C± , denoting the solution with either a scalar field’s kinetic energy term domination or potential energy term domination, are attractors and in this case the universe experiences an accelerating phase of expansion. However, since in this case there is no contribution from the matter, there is no scaling solution. The critical points E± are attractor points and show an accelerating universe. However, in this case there is no scaling solution and the model cannot solve or even alleviate the coincidence problem. The best (that is, cosmologically viable) critical points in our model are the critical points D± . These points, by choosing the proper values of m, n, and α , are attractors that are scaling solutions with accelerating expansion. In this case, the model alleviates the coincidence problem. Note that, in the absence of the non-minimal derivative coupling, there is no scaling solution. Then perturbations around the homogeneous and isotropic background solutions of this NMDC quintom model

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N. Behrouz et al. / Physics of the Dark Universe 15 (2017) 72–81

Fig. 6. The trajectories in {q, r } phase plane (left panel), in {q, s} phase plane (middle panel) and in {r , s} phase plane (right panel). Points T , F and ΛCDM denote respectively the late time values, future values and the values of ΛCDM model for {q, r }, {q, s} and {r , s} diagrams. Note that the panels have been plotted for stable spirals D± , with the initial values x1 = −0.28, x2 = −0.8, x4 = 0.86 and x25 = 0 and with α = 1.5, m = −1.47 and n = 0.5.

are studied in details. We have derived the general perturbed equations and studied the specific cases such as the scaling and de Sitter solutions. First we have studied the scaling solution and derived the perturbations of the background dynamics. Looking at Eq. (39) we see that the growth rate of the perturbations depends on the overall behavior of the solution that contains several terms. Nevertheless, we have shown that by choosing suitable integration constants, the growth of the perturbations is stable at late time. Then we have studied the de Sitter solution in this setup. Depending on the values of m and n, this solution corresponds to points C± or E± . If we consider the case corresponding to the points C± , one term of the solution always grows exponentially leading to the unstable de Sitter solution. If we consider the case corresponding to E± , both terms of the solutions grow exponentially and are unstable. The cosmological viability of the model in confrontation with the Planck2015 TT, TE, EE+lowP+lensing+BAO joint data has been studied carefully. It has been shown that the constant values of the parameters used in describing the attractor points D± are compatible with observational data. In this respect, we have found some constraints on the parameters space of the model. We have checked the model at hand by statefinder diagnostic technique. In this regard and in order to study the expansion history of the universe in this setup, we have introduced the state finder parameters {r , s} and described the trajectories in {r , q}, {s, q} and {r , s} phase plane. To be more clarified, we have plotted the trajectories of the phase plane for the late time stable attractor points D± . Our numerical analysis shows that when the dark sectors are dominated, the trajectories in phase space will be different from the ΛCDM’s trajectories, which have no attractor solution for the late time. Note that, in a special case, if we plot point C for m = n, it coincides with ΛCDM model. In summary, in this paper we have shown that an interacting quintom dark energy model with nonminimal derivative coupling between quintom and Einstein tensor has the potential to alleviate the cosmological coincidence problem via existence of the scaling solutions. Appendix. Coefficients of perturbations equations The explicit forms of the coefficients of perturbations Eqs. (32)– ˙ , ϕ, ϕ˙ , ϕ, (34) as functions of H , H ¨ σ , σ˙ , σ¨ are given as follows:

(

)

a1 = 3H02 2 − 3η(ϕ˙ 02 + σ˙ 02 ) ,

(

)

a2 = β 1 + α (ϕ0 + σ0 )ρm0 , a3 = ϕ˙0 ϕ0 (1 + 9ηH02 ), a4 = (1 + 9ηH02 )ϕ˙0 2 + (αβρm0 − V0 m)ϕ0 ,

a5 = σ˙0 σ0 (9ηH02 − 1), a6 = (9ηH02 − 1)σ˙0 2 + (αβρm0 − V0 n)σ0 . b1 = 6ηH02 ϕ˙ 0 ,

(

)

˙ 0 + 9ηH02 )ϕ˙ 0 , b2 = 3H0 2ηH0 ϕ¨ 0 + (1 + 4ηH b3 = αβρm0 , b4 = (1 + 3ηH02 )ϕ0 ,

˙ 0 + 3ηH02 )3H0 ϕ0 , b5 = (2 + 6ηH02 )ϕ˙ 0 + (1 + 2ηH ˙ 0 + 3ηH02 )3H0 ϕ˙ 0 . b6 = (1 + 3ηH02 )ϕ¨ 0 + (1 + 2ηH c1 = 6ηH02 σ˙ 0 ,

(

)

˙ 0 + 9ηH02 − 1)σ˙ 0 , c2 = 3H0 2ηH0 ϕ¨ 0 + (4ηH c3 = αβρm0 , c4 = (3ηH02 − 1)σ0 ,

˙ 0 + 3ηH02 − 1)3H0 σ0 , c5 = (6ηH02 − 2)σ˙ 0 + (2ηH ˙ 0 + 3ηH02 − 1)3H0 σ˙ 0 . c6 = (3ηH02 − 1)σ¨ 0 + (2ηH The coefficients of Eqs. (36)–(38) are given as follows

(

)

a11 = 3η(ϕ˙0 2 + σ˙0 2 ) − 2 H0 . b11 = −2ηH0 ϕ˙0 ,

˙ 0 + 9ηH02 − 2ηH˙ 0 )ϕ˙ 0 . b22 = 2ηH0 ϕ¨ 0 + (1 + 4ηH c11 = −2ηH0 σ˙0 ,

˙ 0 + 9ηH02 − 2ηH˙ 0 − 1)σ˙ 0 . c22 = 2ηH0 σ¨ 0 + (4ηH References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

A.G. Riess, et al., Astron. J. 116 (1998) 1009. S.J. Perlmutter, et al., Astrophys. J. 517 (1999) 565. A.G. Riess, et al., Astrophys. J. 607 (2004) 665. P. Astier, et al., Astron. Astrophys 447 (2006) 31. M. Kowalski, et al., Astrophys. J. 686 (2008) 749. P.A.R. Ade, et al., Planck Collaboration, A and A 571 (2014) A16. P.A.R. Ade, et al., (2015), arXiv:astro-ph/1502.01589. W.J. Percival, et al., arXiv:0907.1660[astro-ph.CO]. D.N. Spergel, et al., WMAP Collaboration, Astrophys. J. Suppl. 170 (2007) 377. E. Komatsu, et al., WMAP Collaboration, Astrophys. J. Suppl. 180 (2009) 330. D. Rubin, B. Hayden, Is the expansion of the universe accelerating? All signs point to yes, arXiv:1610.08972. T.P. Sotiriou, V. Faraoni, Rev. Modern Phys. 82 (2010) 451. S. Nojiri, S.D. Odintsov, Phys. Rep. 505 (2011) 59. S. Capozziello, M. De Laurentis, Phys. Rep. 509 (2011) 167. V. Sahni, A. Starobinsky, Internat. J. Modern Phys. D 9 (2000) 373. S.M. Carroll, Living Rev. Rel 4 (2001) 1. T. Padmanabhan, Phys. Rep. 380 (2003) 235. P.J.E. Peebles, B. Ratra, Rev. Modern Phys. 75 (2003) 559.

N. Behrouz et al. / Physics of the Dark Universe 15 (2017) 72–81 [19] L. Amendola, S. Tsujikawa, Dark Energy: Theory and Observations, Cambridge University Press, 2010. [20] B. Ratra, P.J.E. Peebles, Phys. Rev. D 37 (1988) 3406. [21] C. Wetterich, Nuclear Phys. B 302 (1988) 668. [22] R.R. Caldwell, R. Dave, P.J. Steinhardt, Phys. Rev. Lett. 80 (1998) 1582. [23] I. Zlatev, L. Wang, P.J. Steinhardt, Phys. Rev. Lett. 82 (1999) 896. [24] R.R. Caldwell, Phys. Lett. B 545 (2002) 23. [25] R.R. Caldwell, M. Kamionkowski, N.N. Weinberg, Phys. Rev. Lett. 91 (2003) 071301. [26] S.i. Nojiri, S.D. Odintsov, Phys. Lett. B 562 (2003) 147. [27] V.K. Onemli, R.P. Woodard, Phys. Rev. D. 70 (2004) 107301. [28] E.J. Copeland, M. Sami, S. Tsujikawa, Internat. J. Modern Phys. D 15 (2006) 1753. [29] T. Padmanabhan, T.R. Choudhury, Phys. Rev. D. 66 (2002) 081301. [30] A. Sen, JHEP 0207 (2002) 065. [31] A. Sen, Mod. Phys. Lett. A 17 (2002) 1797. [32] K. Nozari, N. Rashidi, Phys. Rev. D. 90 (2014) 043522. [33] C. Armendariz-Picon, V. Mukhanov, P.J. Steinhardt, Phys. Rev. Lett. 85 (2000) 4438. [34] T. Chiba, T. Okabe, M. Yamaguchi, Phys. Rev. D. 62 (2000) 023511. [35] N. Bilic, G.B. Tupper, R.D. Viollier, Phys. Lett. B 535 (2002) 17. [36] K. Karami, Mubasher Jamil, S. Ghaffari, K. Fahimi, Can. J. Phys. 10 (2013) 91. [37] Z.K. Guo, Y.S. Piao, X.M. Zhang, Y.Z. Zhang, Phys. Lett. B 608 (2005) 177. [38] K. Nozari, M.R. Setare, T. Azizi, N. Behrouz, Phys. Scr. 80 (2009) 025901. [39] K. Nozari, N. Rashidi, Adv. High Energy Phys. 2013 (2013) 803735. [40] B. Feng, X.L. Wang, X.M. Zhang, Phys. Lett. B 607 (2005) 35. [41] B. Feng, M. Li, Y.S. Piao, X. Zhang, Phys. Lett. B 634 (2006) 101. [42] X.F. Zhang, H. Li, Y.S. Piao, X.M. Zhang, Modern Phys. Lett. A 21 (2006) 231. [43] Y.F. Cai, J. Wang, Classical Quantum Gravity 25 (2008) 165014. [44] H. Wei, R.G. Cai, D.F. Zeng, Classical Quantum Gravity 22 (2005) 3189. [45] H. Wei, R.G. Cai, Phys. Lett. B 634 (2006) 9. [46] J.P. Uzan, Phys. Rev. D. 59 (1999) 123510. [47] N. Bartolo, M. Pietroni, Phys. Rev. D. 61 (2000) 023518. [48] O. Bertolami, P.J. Martins, Phys. Rev. D. 61 (2000) 064007. [49] B. Boisseau, G. Esposito-Farese, D. Polarski, A.A. Starobinsky, Phys. Rev. Lett. 85 (2000) 2236. [50] V. Faraoni, Phys. Rev. D. 62 (2000) 023504. [51] R. Gannouji, D. Polarski, A. Ranquet, A.A. Starobinsky, JCAP 0609 (2006) 016. [52] G. Gupta, E.N. Saridakis, A.A. Sen, Phys. Rev. D. 79 (2009) 123013. [53] K. Nozari, N. Behrouz, T. Azizi, B. Fazlpour, Progr. Theoret. Phys. 122 (2009) 3. [54] A.R. Gomes, L. Amendola, J. Cosmol. Astropart. Phys. 1403 (2014) 041. [55] S. Tsujikawa, Lecture Notes in Phys. 892 (2015) 97. [56] C.P. Burgess, H.M. Lee, M. Trott, JHEP 0909 (2009) 103.

[57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92]

81

J.L.F. Barbon, J.R. Espinosa, Phys. Rev. D. 79 (2009) 081302. T. Han, S. Willenbrock, Phys. Lett. B 616 (2005) 215. R.N. Lerner, J. Mc Donald, JCAP 1004 (2010) 015. L. Amendola, Phys. Lett. B 301 (1993) 175. S. Capozziello, G. Lambiase, Gen. Relativity Gravitation 31 (1999) 1005. S. Capozziello, G. Lambiase, H.-J. Schmidt, Ann. Phys. 9 (2000) 39. V. Sushkov, Phys. Rev. D. 80 (2009) 103505. J.B. Dent, S. Dutta, E.N. Saridakis, J.Q. Xia, J. Cosmol. Astropart. Phys. 1311 (2013) 058. C. Germani, A. Kehagias, Phys. Rev. Lett. 105 (2010) 011302. L.N. Granda, JCAP 07 (2010) 021. L.N. Granda, Classical Quantum Gravity 28 (2011) 025006. M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory, Cambridge University Press, 1987. B. Gumjudpai, P. Rangdee, Gen. Relativity Gravitation 47 (2015) 140. H. Liu, A.A. Tseytlin, Nuclear Phys. B 533 (1998) 88. S. Nojiri, S.D. Odintsov, Phys. Lett. B 444 (1998) 92. J.F. Donoghue, Phys. Rev. D. 50 (1994) 3874. E.N. Saridakis, S.V. Sushkov, Phys. Rev. D. 81 (2010) 083510. S. Tsujikawa, Phys. Rev. D. 85 (2012) 083518. H.M. Sadjadi, P. Goodarzi, JCAP 02 (2013) article 038. K. Nozari, N. Rashidi, arxiv:1509.06240. K. Nozari, M. Shoukrani, N. Rashidi, Adv. High Energy Phys. 2014 (2014) 343819. L. Amendola, Phys. Rev. D. 62 (2000) 043511. W. Zimdahl, D. Pavon, L.P. Chimento, Phys. Lett. B 521 (2001) 133. L. Amendola, C. Quercellini, Phys. Rev. D. 68 (2003) 023514. T. Gonzalez, G. Leon, I. Quiros, Classical Quantum Gravity 23 (2006) 3165. C.G. Boehmer, G. Caldera-Cabral, R. Lazkoz, R. Maartens, Phys. Rev. D. 78 (2008) 023505. K. Nozari, N. Behrouz, N. Rashidi, Adv. High Energy Phys. 2014 (2014) 569702. K. Nozari, N. Behrouz, Phys. Dark Univ. 13 (2016) 92. Y.L. Bolotin, et al., Internat. J. Modern Phys. D 24 (03) (2015) 1530007. P.A.R. Ade, et al. (2015), arXiv:astro-ph/1502.01590. J. Valiviita, E. Palmgren, (2015), arXiv:astro-ph/1504.02464. M.G. Richarte, L. Xu, (2015) arXiv:astro-ph/1506.02518. A. de la Cruz-Dombriz, D. Sáez-Gómez, Classical Quantum Gravity 29 (2012) 245014. F. Kiani, K. Nozari, Phys. Lett. B 728 (2014) 554–561. V. Sahni, T.D. Saini, A.A. Starobinsky, U. Alam, JETP Lett. 77 (2003) 201. U. Alam, V. Sahni, T.D. Saini, A.A. Starobinsky, Mon. Not. R. Astron. Soc. 344 (2003) 1057.