Non-minimally coupled scalar field cosmology with torsion

# Non-minimally coupled scalar field cosmology with torsion

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July 14, 2019
###### Abstract

In this work we present a generalized Brans-Dicke lagrangian including a non-minimally coupled Gauss-Bonnet term without imposing the vanishing torsion condition. In the resulting field equations, the torsion is closely related to the dynamics of the scalar field, i.e., if non-minimally coupled terms are present in the theory, then the torsion must be present. For the studied lagrangian we analyze the cosmological consequences of an effective torsional fluid and we show that this fluid can be responsible for the current acceleration of the universe. Finally, we perform a detailed dynamical system analysis to describe the qualitative features of the model, we find that accelerated stages are a generic feature of this scenario.

a,b]Antonella Cid c]Fernando Izaurieta d]Genly Leon c]Perla Medina c]Daniela Narbona

Prepared for submission to JCAP

Non-minimally coupled scalar field cosmology with torsion

• Departamento de Física, Grupo Cosmología y Partículas Elementales, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile.

• Observatório Nacional, 20921-400, Rio de Janeiro, RJ, Brasil.

• Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile.

• Departamento de Matemáticas, Universidad Católica del Norte, Avenida Angamos 0610, Casilla 1280, Antofagasta, Chile.

Keywords: Generalized Brans-Dicke theory, Torsion, Non-minimal coupling, Cosmological parameters, Dynamical Systems

## 1 Introduction

Scalar-tensor theories or gravity theories with non-minimally coupled scalar fields constitute an alternative to General Relativity, where the existence of additional fields in the gravitational sector may have important consequences in the description of the gravitational interaction [1]. Nowadays, there are a plethora of scalar-tensor theories including the Brans-Dicke theory, where a single dynamical scalar field is added into the gravitational sector [2], and the the Horndeski theory, the most general scenario in a four-dimensional spacetime yielding second order field equations [3].

In the context of modern cosmology, scalar-tensor theories are very appealing because the accelerated expanding phases that experience our universe can be described by scalar fields. The first work in scalar-tensor cosmology was developed by Peebles and Dicke [4], they consider the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric in the Brans-Dicke theory with the aim of studying the formation of primordial elements in the early universe. Subsequently, La and Steinhardt attempt to solve the graceful exit problem of old inflation with their model of extended inflation [5], however, the value obtained for the Brans-Dicke parameter was in tension with observational limits. In spite of this fact, this work promotes scalar-tensor theories as interesting candidates to construct viable and well-motivated cosmological models. Since then, many scalar-tensor cosmological models have been proposed and studied [6], in particular, scalar-tensor theories have been widely used to model the late-time cosmic acceleration [7, 8].

In General Relativity the spacetime is described by a single rank-2 tensor field, the metric . In the first order formalism, the vierbein (or the metric) and the spin connection (or the affine connection) are independent concepts, so the geometry is not completely determined by the metric and the torsion appears, a geometric quality of spacetime which in the second order formalism is fixed by the constraint [9].

The contribution of torsion into gravity theories and particularly into cosmology has been recently studied. In [10, 11] the author considers a fermionic fluid in the matter content, obtaining a cosmological model where the torsion generates accelerated expansion in the early universe. A different approach is developed in [12], where the authors contemplate the Gauss-Bonnet term coupled to a scalar field and the Einstein-Hilbert term with cosmological constant in the frame of non-vanishing torsion, they find field equations with explicit torsion and present some cosmological scenarios. Furthermore, in [13] it is shown that generic non-minimal couplings of a scalar field and curvature in the Horndeski lagrangian will lead to propagating, non-vanishing torsion. This torsion is “dark” for any Yang–Mills interaction, and would interact with fermions only very weakly. For a chronological revision of non-riemannian cosmological models see ref. [14].

The current work is an exploration of the feasibility of the idea that torsion could play the role of dark energy in a cosmological setting and, whether or not it could lead to some unrealistic consequences. To study the cosmological solutions of the most general minimal couplings in the full-fledged Horndeski lagrangian as source of torsion would be very difficult, instead we made the choice of the lagrangian (2.1), which besides to include the most common non-minimal coupling choice in the literature, the Brans-Dicke scenario, it contains the model studied in [12] as a particular case.

Specifically, the aim of this work is to investigate if an effective torsional fluid in the framework of the Horndeski theory with torsion, particularly a generalized Brans-Dicke model, can describe the late-time cosmic evolution without considering any additional scalar field. This paper is organized as follows, in section 2 we describe a particular case of the Horndeski theory with torsion in the frame of FLRW geometry, in section 3 we consider a cosmological scenario in order to explain the current acceleration of the universe in terms of an effective torsional fluid. In section 4 we perform a dynamical system analysis to find the qualitative features of the model at hand. Finally, in section 5 we present our final remarks.

## 2 Torsion, non-minimal couplings and FLRW geometry

When the null-torsion constraint is relaxed from the beginning in the Horndeski lagrangian, it is possible to prove that non-minimal couplings between the scalar field and the curvature, along with terms involving second order derivatives (such as ) in the lagrangian, are sources of torsion [13]. In general, the dynamics of the system strongly departs from the classical riemannian case, and it is possible to have non-vanishing torsion even in the absence of fermionic matter. Let us consider an action principle corresponding to a generalized Brans-Dicke theory with a Gauss-Bonnet term as follows,

 S=∫dx4√|g|[NR2κ4−2MX−V+U(R2−4RμνRνμ+RμνρσRρσμν)+LM], (2.1)

where is the lagrangian for matter, and , , and are functions of a scalar field . is the Lorentz curvature for the connection , where corresponds to the Christoffel connection and to the contorsion tensor. The Lorentz curvature is related to the standard Riemann tensor through

 Rρσμν=˚Rρσμν+˚∇μKρσν−˚∇νKρσμ+KρλμKλσν−KρλνKλσμ. (2.2)

The lagrangian (2.1) is a particular case of the Horndeski lagrangian, it corresponds to , and in [13] along with a Gauss-Bonnet term non-minimally coupled through the function . From the lagrangian (2.1) it is possible to recover the Brans-Dicke lagrangian [2] and the scenario studied in [12] by choosing, and , respectively. Moreover, the authors of [15] study the Kaluza Klein dimensional reduction of the Lovelock Cartan theory in a five-dimensional spacetime. A scalar field is naturally introduced in this scenario through the ansatz for Kaluza Klein compactification in . We notice that the dynamics for this model is similar to ours when we consider and the functions fixed to constants.

Since the torsion-less condition is not being imposed from the beginning, metricity and parallelism represent different degrees of freedom, therefore, they must be varied independently, à la Palatini when they are codified by the metric and the connection , or à la Cartan when they are codified by the vierbein and the spin connection . In this context, the equations of motion corresponding to the lagrangian (2.1) are given by

 Eμν = N(Rμν−12gμνR)+κ4(−2MXgμν+2Vgμν−2M∂μϕ∂νϕ−Tμν)=0, (2.3) Eμνλ = N(Tλμν+Tρρμgλν−Tρρνgλμ)+(∂N∂ϕ+4κ4∂U∂ϕR)(gλμ∂νϕ−gλν∂μϕ)−κ4σλμν (2.4) +8κ4∂U∂ϕ(∂ρϕ(Rρμgλν−Rρνgλμ−Rρλμν)+∂μϕRλν−∂νϕRλμ)=0, E = 12κ4∂N∂ϕR−∂M∂ϕX−∂V∂ϕ+M∇μ∂μϕ+M∂νϕTμνμ (2.5) +∂U∂ϕ(R2−4RμνRνμ+RμνρσRρσμν)=0,

where is the energy-momentum tensor associated to , is the spin tensor associated to , and is the torsion.

The equation (2.4) implies that the torsion depends on the derivatives of the scalar field, and it does not vanish even when . The terms generating this feature are precisely the non-minimal couplings. This behavior stands in strong contrast with the standard minimally coupled Einstein–Cartan case, where only the spin tensor associated to fermions can give rise to torsion, in such a way that it can not propagate in vacuum.

Another interesting feature is that setting into equations (2.3)–(2.5) does not lead to the standard expressions we would get for the standard torsion-less case with the Christoffel connection. Instead, imposing on the equations of motion necessarily freezes the scalar field, leading to .

This behavior reveals that the torsional and torsion-less cases correspond to different dynamical systems when non-minimal couplings are present (see reference [13] for a treatment on the full Horndeski lagrangian111In [13] the results are presented in the first order formalism, in the language of differential forms. The Lagrange multiplier in this article and the 2-form Lagrange multiplier of [13] are related through .). The appropriate procedure to recover the torsion-less case is to include a Lagrange multiplier constraint into the action (2.1),

 ¯S=S+∫dx4√|g|12λλμνTλμν. (2.6)

In our work, the equations of motion for the torsion-less constrained case are recovered:

 ¯¯¯Eμν = Eμν+12∇λ(λμνλ+λνμλ)=0, ¯¯¯Eμνλ = Eμνλ−12(λμνλ−λνμλ)=0, ¯¯¯E = E=0, Tλμν = 0,

where the solution to this system corresponds to the classical riemannian scenario,

 Eμν+∇λ(Eλμν+Eλνμ)∣∣Tαβγ=0 =0, (2.7) E|Tαβγ=0 =0. (2.8)

It is worth to notice that the field equations (2.7) and (2.8), considering the definitions (2.3)–(2.5), are reduced to the field equations of the Brans-Dicke theory in the Jordan frame, presented for instance in reference [16], corresponding to and in our scenario.

The role of torsion in cosmology has been studied in the frame of standard Einstein–Cartan geometry without scalar fields [10, 11]. In this context, torsion can play an important role only inside of a very dense fermionic plasma, as in the Big Bounce model presented in [11]. Inside of standard fermionic matter at usual densities, the effects of torsion are extremely small, see for instance chapter 8 of reference [17]. Torsion generated by non-minimal couplings in the context of cosmology was presented in [12], but only for the Gauss-Bonnet coupling. In [18] it was investigated the cosmology of a higher-order modified teleparallel theory by means of analytical cosmological solutions. In particular, there were determined forms of the unknown potential which drives the scalar field such that the field equations form a Liouville integrable system. The conservations laws were determined using the Cartan symmetries.

On the other hand, in the standard cosmological scenario, the requirement of spatial homogeneity and isotropy is reflected on the condition

 \pounds→ζ gμν=0,

for the vector fields corresponding to spatial translations and rotations, leading to the Friedmann-Lemaitre-Robertson-Walker metric,

 ds2=−c2dt2+a2(t)(dr21−kr2+r2dθ2+r2sin2θ dϕ2),

depending only in the scale factor , where are the coordinates in the co-moving frame. From here on we will consider . A non-vanishing torsion implies new independent degrees of freedom for the geometry, which must also satisfy spatial homogeneity and isotropy in a cosmological setup. These degrees of freedom are codified in the contorsion tensor,

 Kλνμ=Γλμν−˚Γλμν,

and therefore we must require

 \pounds→ζ Kλμν=0,

for vector fields corresponding to spatial translations and rotations. This leads to a “FLRW contorsion” parametrized in four dimensions as

 Kμνλ=(gμλgνρ−gμρgνλ)hρ+√|g|ϵμνλρfρ,

or a FLRW torsion given by

 Tλμν=(gμλgνρ−gμρgνλ)hρ−2√|g|ϵλμνρfρ,

where and are two vectors that in the co-moving frame take the form

 h0=h(t),hi=0,f0=f(t)% andfi=0,

with

In this sense, the cosmological evolution is characterized by the temporal evolution of three functions , and , instead of only as in the classical riemmanian case. For the sake of simplicity, from here on, we will suppose only classical spin-less matter with vanishing spin tensor

 σλμν=0.

It means that we will consider torsion produced only by the scalar field .

Finally, by considering the energy-momentum tensor for a perfect fluid, (where is the energy density, is the pressure and the four-velocity of an observer co-moving with the fluid), the generalized equations for the FLRW geometry are given by

 3N((H+h)2+ka2−f2)−⎛⎝M˙ϕ22+V⎞⎠ = ρ, (2.9) N(2(˙H+˙h)+(3H+h)(H+h)+ka2−f2)+M˙ϕ22−V = −p, (2.10) Nh−˙ϕ(12∂N∂ϕ+4∂U∂ϕ((H+h)2+ka2−f2)) = 0, (2.11) f(N−8∂U∂ϕ˙ϕ(H+h)) = 0, (2.12)

where we have considered the Hubble expansion rate and dots denote derivatives with respect to the cosmic time . In the same way, the equation of motion for the scalar field is given by

 3∂N∂ϕ(˙H+˙h+H(H+h)+(H+h)2+ka2−f2)−12∂M∂ϕ˙ϕ2−∂V∂ϕ−M(¨ϕ+3˙ϕH) +24∂U∂ϕ((˙H+˙h+H(H+h))((H+h)2+ka2−f2)−2f(H+h)(˙f+Hf))=0, (2.13)

and the conservation equation for the energy-momentum tensor become

 ˙ρ+3H(ρ+p)=0, (2.14)

where the equation of state will be provided by a barotropic fluid where , for a constant state parameter . Notice that equation (2.14) is obtained by combining equations (2.9)–(2), and in this sense it is not an independent equation.

## 3 Cosmological Scenario

In cosmology it is common to consider effective fluids in the study of scalar-tensor theories of gravity [19], given that in our scenario the torsion is closely related to the scalar field, we define an effective fluid associated to the torsion by

 ρT = M˙ϕ22+V+3(H2+ka2)(1−N)−3N(2Hh+h2−f2), (3.1) pT = M˙ϕ22−V−(3H2+ka2+2˙H)(1−N)+N(2˙h+4Hh+h2−f2), (3.2)

in such a way that the field equations (2.9), (2.10), (2) and (2.14) can be cast in the standard form for a universe filled with two non-interacting fluids, and , as

 3(H2+ka2) = ρ+ρT, (3.3) 2˙H+3H2+ka2 = −p−pT, (3.4) ˙ρT+3H(ρT+pT) = 0, (3.5) ˙ρ+3H(ρ+p) = 0. (3.6)

Furthermore, equations (2.11) and (2.12) have to be satisfied.

Notice that we can recover the expressions for the effective density and pressure defined in [12] by assuming , and into equations (3.1) and (3.2), where we have generalized the cosmological constant to the potential of the scalar field , including it as part of the total energy-momentum tensor instead of the gravitational sector as in [12].

On the other hand, following reference [12], we find conditions to analytically solve the system of equations (2.9)–(2), where an effective torsional fluid defined through (3.1) and (3.2), drives the acceleration in a cosmological setup. In order to do this we rewrite the set of equations (2.9)–(2) in the following way:

 Z2+W−V3N−13N(ρ+12M˙ϕ2) = 0, (3.7) 2˙Z+2HZ+Z2+W−VN+1N(p+12M˙ϕ2) = 0, (3.8) −Nh+4U′˙ϕ(Z2+W)+12N′˙ϕ = 0, (3.9) f(N−8U′˙ϕZ) = 0, (3.10) (˙Z+HZ+Z2+W)N′−13(12M′˙ϕ2+V′+M(¨ϕ+3H˙ϕ)) +8U′((Z2+W)(˙Z+HZ)−2fZ(˙f+Hf)) = 0, (3.11)

where prime denotes derivative with respect to and we have defined:

 Z = H+h, (3.12) W = ka2−f2, (3.13)

Notice that the cosmological scenario presented in [12] is recovered by considering , , , into equations (3.7)–(3). In the following subsections we present analytical solutions to the set of equations (3.7)–(3).

### 3.1 Solution for f≠0 with pressureless matter

In the late-time universe the energy-momentum tensor must consider baryonic and dark matter, both of which behave as pressureless matter () at cosmological scales.

By assuming , we can replace from equation (3.10) into equation (3.9) to get

 W=Z2−2HZ−N′NZ˙ϕ, (3.14)

then, by considering , replacing (3.14) into equation (3.8) and rewriting in terms of through equation (3.10) we obtain

 2(˙Z+Z2)=N′8U′+VN−MN128U′2Z2, (3.15)

where we notice that by setting the following conditions:

 M(ϕ)=0,V(ϕ)=βN(ϕ),U′(ϕ)=αN′(ϕ), (3.16)

we are able to find an analytical solution for and constants. This scenario corresponds to a generalized Brans-Dicke theory without the kinetic term and including a non-minimally coupled Gauss-Bonnet term. By replacing (3.16) into equation (3.15) we get:

 Z(τ)=Ze tanh(τ), (3.17)

where , is a rescaled dimensionless time, is an integration constant and we have imposed the condition . Since , is positive for , and negative for . Besides, by integrating equation (3.10) we have

 N(τ)=N∗ sinh(τ)2−3x, (3.18)

where is an integration constant, and for convenience we define the dimensionless parameter . Also, by integrating (3.6) for we get , where the subscript 0 indicates current values. By using this last result into equation (3.7) and considering equation (3.14) we obtain

 (3.19)

Finally, by inserting equations (3.17) and (3.18) into equation (3.19) we are able to integrate, obtaining the following dimensionless scale factor

 ¯a(τ)=aa0=¯a0 cosh(τ) sinh(τ)−1+x(1−y tanh(τ))1/3, (3.20)

where we define the dimensionless parameter and is a positive defined integration constant. Notice that the behavior of the scale factor is independent of the exact functional form of when and are defined by (3.16).

In the same way, we obtain and from equations (3.12) and (3.13), respectively:

 h(τ) = Ze((1−x) coth(τ)+y sech2(τ)3(1−y tanh(τ))), (3.21) f(τ) = ± ⎷k tanh2(τ)(1−y tanh(τ))−2/3¯a20 sinh(τ)2x−Z% 2e(x−Y tanh(τ))1−y tanh(τ), (3.22)

where .

Without loss of generality, we can assume in order to show the behavior of the scale factor depending on the parameters and as we see in table 1.

From the cosmological point of view, an interesting behavior is associated with , where the scale factor starts with a null value at and subsequently expands. In figure 1 we show different possible behaviors for , depending on the values for the parameters. Among these models, the most interesting scenario is the one with a transition from decelerated expansion to accelerated expansion, the case in table 1.

For the scenario , we obtain the Hubble expansion rate and the deceleration parameter by deriving (3.20),

 H = Ze(tanh(τ)+(x−1)coth(τ)−y sech2(τ)3−3y tanh(τ)), (3.23) q = −1−Z2e3H2⎛⎝2 sech2(τ)−3(x−1) csch2(τ)+(1−y2)% sech2(τ)(1−y tanh(τ))2⎞⎠. (3.24)

From (3.24) we notice that in the limit we have , a positive value for , meanwhile in the limit we get , then we can see that for necessarily exist a sign-transition going from decelerated expansion to accelerated expansion, scenario favored by current data.

On the other hand, if we want to impose the conditions and , as in the standard cosmological scenario [20], we find that the following constraints

 (1−x)(ycoth(τ)−coth2(τ))−ycsch% (τ)sech(τ)/3y−coth(τ) < tanh(τ), (3.25) 3x+2sech2(τ)+1−2ysinh(τ)(cosh% (τ)−ysinh(τ))(cosh(τ)−ysinh(τ))2 > 6, (3.26)

must be satisfied, respectively. Nevertheless, in figure 2 we show some numerical examples indicating that these conditions are inconsistent with and . In figure 2 the curves represent the lower limit of the parameter satisfying inequality (3.25) (left panel) and inequality (3.26) (right panel) for , the shaded regions represent . We notice that during the evolution the condition becomes inconsistent for , then we conclude that if a transition from decelerated to accelerated expansion exists, necessarily occurs a period with . This behavior of the Hubble expansion rate has been considered in the literature before, in order to describe the transition from a static state to an inflationary phase in Emergent Universe scenarios [21], where the regime is transitory. By contrast, in our scheme, this behavior corresponds to the final stage of evolution.

By convenience, we choose to measure the parameter in terms of the Hubble parameter as , where is a dimensionless parameter. On the other hand, by imposing the value of the expansion rate today to be , we find a constraint among the parameters , and , that is, there exist a maximum value for the parameter over which the consistency condition can not be reached at any time. For instance, in figure 3 we see that for and , the maximum possible value of is around 0.6.

In figures 46 we have used the dimensionless scale factor to give account of the evolution in our cosmological scenario, where is related to the physical redshift by . Besides, given that we are interested in the periods of matter and dark energy domination, we focus on the range or , when the radiation contribution in the standard cosmological scenario is of order . Note that we are not including a radiation component in this work.

Taking into account the model dependency on three parameters, namely , and , it is complicated to qualitatively describe the evolution of the relevant functions, therefore, in order to show the possible behavior of the density parameter associated to the torsional fluid we plot in figure 4 some possible evolution of for given values of , and . In the left panel the curves represent the evolution of for different values of the parameter for fixed and . We observe that becomes negative in the past for , which is not forbidden given that this density parameter corresponds to an effective fluid. For the torsional fluid contribution can remain important in the past and the matter domination period would experience important differences compared to the standard cosmological scenario. For , the torsional fluid becomes the dominant contribution during a period that should correspond to matter domination. We notice that in general, the behaviors observed in this figure are common for different values of and in the allowed ranges. The right panel of figure 4 is a closer view of and for different values of , we observe that it is possible to have a small enough contribution of in the past for some values of , for instance, .

As a particular example we have chosen a cosmological scenario given by the parameters , and . The evolution of the relevant cosmological functions of this scenario is shown in figures 5 and 6. In figure 5 we see the sign transition of the deceleration parameter at . The asymptotic limits of are in the future and in the past, consistent with the standard cosmological scenario. On the other hand, we note that the Hubble expansion rate assumes the value twice because becomes positive during the evolution with the sign-transition of occurring in the future ().

Note that equation (3.20) imposes for in order to have today. Besides, this scenario corresponds to an approximated evolution of 13.8 Gyr since , consistent with the standard cosmological evolution for [km/s/Mpc] [22].

In the left panel of figure 6 we show the behavior of the dimensionless density parameters, and , where we have chosen the current value of the density parameter for pressureless matter to be , according with the last measurements of the Planck satellite [22]. We observe that around (or ) the matter contribution corresponds to approximately of the overall, consistent with the standard cosmological scenario. The right panel of figure 6 shows the behavior of the state parameter for the torsional effective fluid , this parameter is negative during the entire evolution and asymptotically tends to a cosmological constant in the future. It is interesting to note that among the many possible behaviors shown in figure 4, we are able to find a combination of parameters where the standard cosmological scenario at late time seems to be reproduced at background level and the role of dark energy is played by an effective torsional fluid.

It is worth to pointing out that a scalar-tensor cosmological scenario as a single parameter extension of the standard cosmological model (CDM) was presented in [23]. The authors found similar behaviors to those shown in the left panel of figure 4, but representing the effective fluid for a non-torsional dark energy component. In this sense, it would be interesting to clarify if there is any distinctive imprint of the torsion on the scenarios here presented, which allows us to differentiate from the standard CDM scenario and any other scalar-tensor cosmological scenario.

### 3.2 Solution for f=0 with pressureless matter

In this section we impose conditions (3.16) along with into equations (3.7)–(3). Specifically, by considering (3.12), and assuming the matter content is a pressureless fluid, equations (3.7)–(3.9) become

 Z2 = −ka2+β3+ρ3N, (3.27) ˙Z = 12(β−ka2−Z(2H+Z)), (3.28) ˙N = 2a2N(Z−H)a2(8αZ2+1)+8αk. (3.29)

Then, by replacing (3.28) and (3.29) into (3), for , we obtain

 (H−Z)(24α[Z2+ka2]2−3[Z2+ka2](8αβ+1)−β)=0, (3.30)

which leads to the following two possibilities.

• , which implies and constant from (3.12) and (3.29), respectively. No torsional fluid is present in this case and given that the non-minimal coupling to the gravitational sector disappears in , we recover the standard CDM scenario, where the field equations, considering , become

 ˙Z=12(−ka2+β−3Z2),˙a=aZ,˙ρ=−3Zρ,Z2+ka2=β3+ρ3. (3.31)
• For (or ), equation (3.30) implies

 Z2+ka2+β3−8α(Z2+ka2)(Z2+ka2−β)=0, (3.32)

corresponding to a polynomial equation for , where is constant. By using this last result into (3.8) we get the following expansion rate,

 H=±(β−Z202Z20)√Z20−ka2, (3.33)

valid for and . By integrating equation (3.33) for we obtain the following scale factor,

 a(~τ)=e~τ+ke−~τ4Z20,where ~τ=β−Z202Z0(t−t0), (3.34)

where is an integration constant. Furthermore, from equations (3.6) and (3.27) we respectively obtain

 ρ(~τ)=ρ0(e~τ+ke−~τ4Z20)−3,N(~τ)=(ρ03Z20−β)(e~τ+ke−~τ4Z20)−3, (3.35)

where is an integration constant.

We notice that the scale factor (3.34) corresponds to eternal accelerated expansion, that is, this scenario does not allow a transition from decelerated to accelerated expansion even when pressureless matter is considered.

## 4 Dynamical system analysis

In the section 3 we find an exact solution to equations (3.7)–(3), for some specific model parameters and initial conditions. We noticed that the solution can correspond to a period of matter domination followed by accelerated expansion. Now, we are interested in studying the cosmological behavior in a general sense, independently of the initial conditions and the specific universe evolution. For this purpose we apply the dynamical systems approach, which allows to extract global features of the scenario at hand. In this procedure, the first step is to transform the cosmological equations into an autonomous system, then find the fixed points and finally study their stability. For this end, linear perturbations around the critical points are performed, and the system is rewritten in terms of a perturbation matrix from which the type and stability of each critical point can be obtained. In the case of non-hyperbolic critical points, we should use the center manifold theorem or rely on numerical inspection [24, 25].

### 4.1 Dynamical system analysis for f≠0 with presureless matter

We perform a dynamical systems analysis of the scenario defined by equations (3.7)–(3) considering and pressureless matter. By replacing from equation (3.7), using the chain rule (for any function of ) and imposing conditions (3.16), from equations (3.8), (3.6), (3.10) and (3.9) we obtain respectively,

 ˙Z = 116(1α+8β−16Z2), (4.1) ˙ρ = ρ(N(8αβ−48αZ2+3)+8αρ)16αZN, (4.2) ˙N = N8αZ, (4.3) 0 = 38α+β+6Z(H−Z)+ρN, (4.4)

where the last equation constitutes a constraint.

From equation (3.17) we can easily note that, by defining a new variable , it takes values over the interval and it tends to as and to as . Note that here we are considering .

By defining the following dimensionless variables,

 U1=−ka2Z2e(1+ζ2),U2=ρ3NZ2e(1+ζ2),U3=ζ√1+ζ2, (4.5)

and a new time variable , in such a way that for any generic function we have

 G′=dGd¯τ=ζ2(ζ2+1)3/2Ze˙G, (4.6)

we obtain the following autonomous system

 U′1 = U1U3(−2x+U2+4U43+2xU23−6U23+2), (4.7) U′2 = 12U2U3(3U2+8U43−12U23+2), (4.8) U′3 = U23(1−U23)(1−2U23), (4.9)

which defines a flow on the phase space defined by

 {(U1,U2,U3)∈R3:U1+U2≤(x+1)U23−x,−√22≤U3≤√22}, (4.10)

where the first inequality comes from the reality condition imposed into equation (3.7) and the second from the definition of . Note that, we use the dimensionless parameter (see table 1 for a description of its role in the dynamics).

#### 4.1.1 Fixed points at the finite region

In this section we find the fixed points and study their stability by examining the eigenvalues of the perturbation matrix, the main results are summarized in table 2. The fixed points corresponding to are not included in table 2 because they lie outside of the physical portion of the phase space (4.10).