Qualitative analysis of Kantowski-Sachs metric in a generic class of f(R) models

# Qualitative analysis of Kantowski-Sachs metric in a generic class of f(R) models

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July 17, 2019
###### Abstract

In this paper we investigate, from the dynamical systems perspective, the evolution of a Kantowski-Sachs metric in a generic class of models. We present conditions (i. e., differentiability conditions, existence of minima, monotony intervals, etc.) for a free input function related to the , that guarantee the asymptotic stability of well-motivated physical solutions, specially, self-accelerated solutions, allowing to describe both inflationary- and late-time acceleration stages of the cosmic evolution. We discuss which theories allows for a cosmic evolution with an acceptable matter era, in correspondence to the modern cosmological paradigm. We find a very rich behavior, and amongst others the universe can result in isotropized solutions with observables in agreement with observations, such as de Sitter, quintessence-like, or phantom solutions. Additionally, we find that a cosmological bounce and turnaround are realized in a part of the parameter-space as a consequence of the metric choice.

a]Genly Leon b]and Armando A. Roque

Qualitative analysis of Kantowski-Sachs metric in a generic class of models

• Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4950, Valparaíso, Chile

• Grupo de Estudios Avanzados, Universidad de Cienfuegos, Carretera a Rodas, Cuatro Caminos, s/n. Cienfuegos, Cuba

Keywords: modified gravity, dark energy, Kantowski-Sachs metric, dynamical analysis

## 1 Introduction

Several astrophysical and cosmological measurements, including the recent WMAP nine year release, and the Planck measurements, suggest that the observable universe is homogeneous and isotropic at the large scale and that it is currently experiencing an accelerated expansion phase [2, 3, 1, 4, 5, 6]. The explanation of the isotropy and homogeneity of the universe and the flatness problem lead to the construction of the inflationary paradigm [7, 8, 9, 10] 111Reference [7] is the pioneer model with a de Sitter (inflationary) stage belonging to the class of modified gravity theories. These models contain as special case the inflationary model which appears to produce the best fit to the recent WMAP9 and Planck data on the value of [5, 6, 11, 12, 13].. Usually in the inflationary scenarios the authors start with a homogeneous and isotropic Friedmann-Robertson-Walker metric (FRW) and then are examined the evolution of the cosmological perturbations. However, the more strong way to proceed is to put from the beginning an arbitrary metric and then examine if the metric tends asymptotically to the flat FRW geometry 222This discussion acquires interest since it may be relevant for the explanation of the anisotropic “anomalies” reported in the recently announced Planck Probe results [13].. This is a very difficult program even using a numerical approach, see for example the references [14, 15, 16]. Thus, several authors investigated the special case of homogeneous but anisotropic Bianchi [19, 17, 18] (see [20] and references therein) and the Kantowski-Sachs metrics [21, 22, 48, 23, 24, 25, 26, 36, 27, 37, 38, 28, 29, 30, 39, 40, 31, 41, 42, 43, 32, 44, 45, 33, 46, 35, 34, 47]. The simplest well-studied but still very interesting Bianchi geometries are the Bianchi I [63, 49, 50, 51, 23, 24, 52, 53, 54, 55, 57, 56, 58, 59, 60, 61, 62, 47] and Bianchi III [64, 23, 65, 66, 47], since other Bianchi models (for instance the Bianchi IX one), although more realistic, they are much more complicated. These geometries have been examined analytically, exploring their rich behavior, for different matter content of the universe and for different cosmological scenarios (e.g., [17, 18, 49, 50, 23, 24, 52, 53, 68, 54, 55, 57, 56, 58, 59, 60, 61, 62, 64, 65, 66, 25, 26, 67, 27, 28, 29, 30, 31, 32, 33, 34, 35, 69, 70, 71, 72, 73, 74, 76, 75, 77, 78, 47]).

On the other hand, to explain the acceleration of the expansion one choice is to introduce the concept of dark energy (see [79, 80, 81] and references therein), which could be the simple cosmological constant, a quintessence scalar field [82, 83, 84, 85], a phantom field [86, 87, 88, 89, 90], or the quintom scenario [91, 92, 93, 96, 97, 94, 95, 98]. The second one is to consider Extended Gravity models, specially the - models (see [99, 100, 101, 102, 105, 103, 106, 104] and references therein), as alternatives to Dark Energy. Other modified (extended) gravitational scenarios that have gained much interests due to their cosmological features are the extended nonlinear massive gravity scenario [107, 108, 109, 110], and the Teleparalell Dark Energy model [111, 112, 113]. However, we follow the mainstream and investigate models.

-models have been investigated widely in the literature. In particular, Bianchi I models in the context of quadratic and cosmology were first investigated in [63], where the authors showed that anisotropic part of the spacetime metric can be integrated explicitly. In the reference [114] was done a phase-space analysis of a gravitational theory involving all allowed quadratic curvature invariants to appear in the Lagrangian assuming for the geometry the Bianchi type I and type II models, which incorporate both shear and 3-curvature anisotropies. The inclusion of quadratic terms provides a new mechanism for constraining the initial singularity to be isotropic. Additionally, there was given the conditions under which the de Sitter solution is stable, and for certain values of the parameters there is a possible late-time phantom-like behavior. Furthermore, there exist vacuum solutions with positive cosmological constant which do not approach de Sitter at late times, instead, they inflate anisotropically [114]. In the references [115, 116, 117] were investigated the oscillations of the dark energy around the phantom divide line, . The analytical condition for the existence of this effect was derived in [116]. In [117] was investigated the phantom divide crossing for modified gravity both during the matter era and also in the de Sitter epoch. The unification of the inflation and of the cosmic acceleration in the context of modified gravity theories was investigated for example in [118]. However, this model is not viable due the violation of the stability conditions. The first viable cosmological model of this type was first constructed in [119], requiring a more complicated function. In [120] was developed a general scheme for modified gravity reconstruction from any realistic FRW cosmology; another reconstruction method using cosmic parameters instead of the time law for the scale factor, was presented and discussed in [121]. Finally, in the reference [122] is described the cosmological evolution predicted by three distinct theories, with emphasis on the evolution of linear perturbations. Regarding to linear perturbations in viable cosmological models, the more important effect, which is the anomalous growth of density perturbations was, prior to reference [122], considered in [123, 124].

In this paper we investigate from the dynamical systems perspective the viability of cosmological models based on Kantowski-Sachs metrics for a generic class of , allowing for a cosmic evolution with an acceptable matter era, in correspondence to the modern cosmological paradigm. We present sufficient conditions (i.e., differentiability conditions, existence of minima, monotony intervals, an other mathematical properties for a free input function) for the asymptotic stability of well-motivated physical solutions, specially, self-accelerated solutions, allowing to describe both inflationary and late-time acceleration. The procedure used for the examination of arbitrary theories was first introduced in the reference [125] and the purpose of the present investigation is to improve it and extend it to the anisotropic scenario. Particularly, in [125] the authors demonstrated that the cosmological behavior of the flat Friedmann- Robertson-Walker models can be understood, from a geometric perspective, by analyzing the properties of a curve in the plane , where

 m=Rf′′(R)f′(R)=dlnf′(R)dlnR

and

 r=−Rf′(R)f(R)=−dlnf(R)dlnR.

However, as discussed in [126], the approach in [125] is incomplete, in the sense that the authors consider only the condition to define the singular values of and they omit some important solutions satisfying and/or This leads to changes concerning the dynamics, and to some inconsistent results when comparing with [126]. It is worthy to mention that using our approach it is possible to overcome the previously commented difficulties -remarked in Ref. [126]- about the approach of Ref. [125] (see details at the end of section 3.2). However, since the analysis in references [125] and [126], is qualitative, an accurate numerical analysis is required. This numerical elaboration was done in [127] for the case of -gravity, where the authors considered the whole mixture of matter components of the Universe, including radiation, and without any identification with a scalar tensor theory in any frame. There was shown numerically that for the homogeneous and isotropic model, including the case , an adequate matter dominated era followed by a satisfactory accelerated expansion is very unlikely or impossible to happen. Thus, this model seems to be in disagreement with what is required to be the features of the current Universe as noticed in [125]. Regarding the investigations of the it is worthy to note that, in the isotropic vacuum case, this model can be integrated analytically, see [128].

In the reference [129] are investigated, from the dynamical systems viewpoint and by means of the method developed in [125], the so-called theory, where is the trace of the energy-momentum tensor. This theory was first proposed in [130]. In [129] was investigated the flat Friedmann- Robertson- Walker (FRW) background metric. Specially, for the class the only form that respects the conservation equations is where do not depend on T, but possibly depends on We recover the results in [129] for this particular choice in the isotropic regime. However, our results are more general since we consider also anisotropy.

In the reference [131] are investigated FRW metric on the framework of gravity using the same approach as in [126]. Bianchi I universes in cosmologies with torsion have been investigated in [132]. In the reference [133] the authors examine the asymptotic properties of a universe based in a Scalar-tensor theory (and then related through conformal transformation to -theories). They consider an FRW metric and a scalar field coupled to matter, also it is included radiation. The authors prove that critical points associated to the de Sitter solution are asymptotically stable, and also generalize the results in [134]. The analytical results in [133] are illustrated for the important modified gravity models (quadratic gravity) and . For quadratic gravity, it is proved, using the explicit calculation of the center manifold of the critical point associated to the de Sitter solution (with unbounded scalar field) is locally asymptotically unstable (saddle point). In this paper we extent these results to the Kantowski-Sachs metrics. Finally, in the reference [34] were investigated -gravity models for anisotropic Kantowski-Sachs metric. There were presented conditions for obtaining late-time acceleration, additionally, in the range , it is obtained phantom behavior. Besides, isotropization is achieved irrespectively the initial degree of anisotropy. Additionally, it is possible to obtain late-time contracting and cyclic solutions with high probability. In this paper we extent the results in [125, 133, 34] to the Kantowski-Sachs metric. Particularly, we formalize and extent the geometric procedure discussed in [125] in such way that the problems cited in [126] do not arise, and apply the procedure to “generic” models for the case of a Kantowski-Sachs metric. By “generic” we refer to starting with a unspecified , and then deduce mathematical properties (differentiability, existence of minima, monotony intervals, etc) for the free input functions in order to obtain cosmological solutions compatible with the modern cosmological paradigm. We extent the results obtained in the reference [133] related to the stability analysis for the de Sitter solution (with unbounded scalar field) for the homogeneous but anisotropic Kantowski-Sachs metric and we extent to generic models of the results in [34] that were obtained for -cosmologies. Our results are also in agreement with the related ones in [129].

The paper is organized as follows. In section 2 we construct the cosmological scenario of anisotropic -gravity, presenting the kinematic and dynamical variables specifying the equations for the Kantowski-Sachs metric. Having extracted the cosmological equations in section 3 we perform a systematic phase-space and stability analysis of the system. It is presented a general method for the qualitative analysis of -gravity without considering an explicit form for the function . Instead we leave the function as a free function and use a parametrization that allows for the treatment of arbitrary (unspecified) anzatzes. In section 4 we present a formalism for the physical description of the solutions and it is discussed the connection with the cosmological observables. In the section 5 we analyze the physical implications of the obtained results, and we discuss the cosmological behaviors of a generic in a universe with a Kantowski-Sachs geometry. In section 6 are illustrated our analytical results for a number of -theories. Our main purpose is to illustrate the possibility to realize the matter era followed by a late-time acceleration phase. Additionally it is discussed the possibility of a bounce or a turnaround. Finally, our results are summarized in section 7.

## 2 The cosmological model

In this section we consider an -gravity theory given in the metric approach with action [102, 105, 106]

 Smet=∫Vd4x√−g[f(R)−2Λ+Lm], (2.1)

where is the matter Lagrangian. Additionally, we use the metric signature (). Greek indexes run from to , and we impose the standard units in which . Also, in the following, and without loss of generality, we set the usual cosmological constant Furthermore, is a function of the Ricci scalar , that satisfies the following very general conditions [119]:

1. Existence of a stable Newtonian limit for all values of where the Newtonian gravity accurately describes the observed inhomogeneities and compact objects in the Universe, i.e., for , where is the present moment and is the present FRW background value, and up to curvatures in the center of neutron stars:

 |f(R)−R|≪R,|f′(R)−1|≪1,Rf′′(R)≪1, (2.2)

for . The last of the conditions (2.2) implies that its Compton wavelength is much less than the radius of curvature of the background space-time. Additionally, the conditions (2.2) guarantee that non-GR corrections to a space-time metric remain small [119].

2. Classical and quantum stability:

 f′(R)>0,f′′(R)>0. (2.3)

The first condition implies that gravity is attractive and the graviton is not ghost. Its violation in an FRW background would imply the formation of a strong space-like anisotropic curvature singularity with power-law behavior of the metric coefficients [135, 63]. This singularity prevents a transition to the region where the effective gravitational constant in negative in a generic, non-degenerate solution. Additionally, note that at the Newtonian regime, the effective scalaron333The scalaron is defined by the scalar field with an effective potential mass squared is . Thus, the second condition in (2.3) implies that the scalaron is not a tachyon, i.e., the scalaron has a finite rest-mass If becomes zero for a finite , then a weak (sudden) curvature singularity forms generically [119].

3. In the absence of matter, exact de Sitter solutions are associated to positive real roots of the functional equation

 Rf′(R)−2f(R)=0. (2.4)

It is well-known that these kind of solutions (and nearby solutions) are very important for the description of early inflationary epoch and the late-time acceleration phase. For the asymptotic future stability of these solutions near de Sitter ones it is required that

 f′(R1)/f′′(R1)>R1,

where satisfies (2.4) [136]. Specific functional forms satisfying all this conditions have been proposed, e.g., in the references [123, 137, 124].

4. Finally, we have additionally considered the condition to get a non-negative scalaron potential.

The fourth-order equations obtained by varying action (2.1) with respect to the metric are:

 f′(R)Rαβ−f(R)2gαβ−∇α∇βf′(R)+gαβ□f′(R)=Tαβ, (2.5)

where the prime denotes differentiation with respect to . In this expression denotes the matter energy-momentum tensor, which is assumed to correspond to a perfect fluid with energy density and pressure , and their ratio gives the matter equation-of-state parameter

 w=pmρm. (2.6)

is the covariant derivative associated with the Levi-Civita connection of the metric and . Taking the trace of equation (2.5) we obtain “trace-equation”

 f′(R)R+3□f′(R)−2f(R)=T, (2.7)

where is the trace of the energy-momentum tensor of ordinary matter.

Our main objective is to investigate anisotropic cosmologies. Let us assume, as usual, an anisotropic metric of form [138]:

 ds2=−N(t)2dt2+[e11(t)]−2dr2+[e22(t)]−2[dθ2+S(θ)2dφ2], (2.8)

where is the lapse function, that we will set , and are the expansion scale factors, which in principle can evolve differently.

Notice that the metric (2.8) can describe three geometric families, that is:

where is the spatial curvature parameter.

From the expansion scale factors are defined the kinematic variables

 H = −13ddtln[e11(e22)2], (2.9a) σ = 13ddtln[e11(e22)−1]. (2.9b)

Furthermore, is the Gauss curvature of the 3-spheres [139] and their evolution equation is given by [140]

 ˙2K = −2(σ+H)(2K). (2.10)

Additionally, the evolution equation for reads (see equation (42) in section 4.1 in [140])

 ˙e11=(−H+2σ)e11. (2.11)

From the trace equation (2.7) for the Kantowski-Sachs geometry () and assuming the matter content described as a perfect fluid we obtain:

 −3d2dt2f′(R)−9Hddtf′(R)+f′(R)R−2f(R)=−ρm+3pm. (2.12)

The Ricci scalar is written as

 R=12H2+6σ2+6˙H+22K. (2.13)

Now, it is straightforward to write the equation (2.5) as [141] 444Alternatively, we can write the field equations (2.14a) as , where , or (if used the trace equation for eliminating second order derivatives of with respect to ), i.e., the recipe I in [142, 143]. The above expression for studied in [142, 143] corresponds exactly to the energy momentum tensor (EMT) of geometric dark energy given by discussed in the references [124, 144, 106, 116].:

 AGαβ=Tαβ+TDEαβ, (2.14a) TDEαβ=(A−f′(R))Rαβ+12gαβ(f(R)−AR)+(∇α∇β−gαβ□)f′(R), (2.14b)

where A is some constant. In order to reproduce the standard matter era () for , we can choose . An alternative possible choice is , where is the present value of . This choice may be suitable if the deviation of from 1 is small (as in scalar-tensor theory with a nearly massless scalar field [86, 145]).

Substituting the Kantowski-Sachs metric into the equations (2.14a) with the definition for given by (2.14b), and after some algebraic manipulations, we obtain

 A(3H2−3σ2+2K) = ρm+ρDE, (2.15a) A(−3(σ+H)2−2˙σ−2˙H−2K) = pm+pDE−2πDE, (2.15b) A(−3σ2+3σH−3H2+˙σ−2˙H) = pm+pDE+πDE. (2.15c)

where

 ρDE = −3Hddtf′(R)+12(Rf′(R)−f(R))+(3H2−3σ2+2K)(A−f′(R)), (2.16a) pDE = d2dt2f′(R)+2Hddtf′(R)+12(f(R)−Rf′(R))+ (2.16b) πDE = −ddtf′(R)f′(R)Aσ, (2.16c)

denote, respectively, the isotropic energy density and pressure and the anisotropic pressure of the effective energy-momentum tensor for Dark Energy in modified gravity. Note that for the choice of an FRW background, where , we recover the equations (4.94) and (4.95) in the review [141].

The advantage of using the expression (2.14b) for the definition of the effective energy-momentum tensor for Dark Energy in modified gravity, instead of using the alternative “curvature-fluid” energy-momentum tensor [146, 147]

 Teffμν=1f′(R)[12gμν(f(R)−Rf′(R))+∇μ∇νf′(R)−gμν□f′(R)],

is that by construction (2.14b) is always conserved, i.e., , leading to the conservation equation for the effective Dark Energy, which in anisotropic modified gravity is given by [140]:

 ˙ρDE+3H(ρDE+pDE)+6σπDE=0. (2.17)

On the other hand, is not conserved in presence of mater (it is conserved for vacuum solutions only).

Now, combining the equations (2.15a) and (2.16a), we obtain

 f′(R)ρtot=−3AHddtf′(R)+12A[Rf′(R)−f(R)]+Aρm, (2.18)

where we have defined the total (effective) energy density Eliminating between (2.15b) and (2.15c), and using (2.13), we acquire

 ptot =−A3[9(H2+σ2)+6˙H+2K] =A3[3(H2−σ2)−R+2K], (2.19)

where we have defined the total (effective) isotropic pressure Thus, we can define the effective equation of state parameter

 wtot≡ptotρtot=2f′(R)(−2K−3H2+R+3σ2)3[6H˙Rf′′(R)−Rf′(R)+f(R)−2ρm] (2.20)

Additionally, we define the observational density parameters:

• the spatial curvature:

 Ωk≡−2K3H2, (2.21)
• the matter energy density:

 ~Ωm≡ρm3AH2, (2.22)
• the “effective Dark Energy” density:

 ~ΩDE≡ρDE3AH2, (2.23)
• the shear density:

 Ωσ≡(σH)2, (2.24)

satisfying .

Now, for the homogeneous but anisotropic Kantowski-Sachs metric, the Einstein’s equations (2.5) along with (2.12) can be reduced with respect to the time derivatives of ,   and , leading to: the equation for the shear evolution:

 ˙σ=−σ2−3Hσ+H2−ρm3f′(R)−16[R−f(R)f′(R)]+(H−σ)f′(R)ddtf′(R), (2.25)

the Gauss constraint:

 2K=3σ2−3H2−ρmf′(R)+12[R−f(R)f′(R)]−3Hf′(R)ddtf′(R), (2.26)

and the Raychaudhuri equation:

 (2.27)

Using the trace equation (2.12) it is possible to eliminate the derivative in (2.27), obtaining a simpler form of the Raychaudhuri equation:

 ˙H=−H2−2σ2−ρm3f′(R)+f(R)6f′(R)+1f′(R)Hddtf′(R). (2.28)

Furthermore, the Gauss constraint (2.26) can alternatively be expressed as

 (2.29)

Finally, the evolution of matter conservation equation is:

 ˙ρm=−3γHρm, (2.30)

where the perfect fluid, with equation of state , satisfies the standard energy conditions which implies .

In summary, the cosmological equations of -gravity in the Kantowski-Sachs background are the “Raychaudhuri equation” (2.28), the shear evolution (2.25), the trace equation (2.12), the Gauss constraint (2.29), the evolution equation for the 2-curvature (2.10) and the evolution equation for (2.11). Finally, these equations should be completed by considering the evolution equation for matter source (2.30). These equations contains, as a particular case, the model investigated in [34].

## 3 The dynamical system

In the previous section we have formulated the -gravity for the homogeneous and anisotropic Kantowski-Sachs geometry. In this section we investigate, from the dynamical systems perspective, the cosmological model without considering an explicit form for the function . Instead we leave the function as a free function and use a parametrization that allows for the treatment of arbitrary (unspecified) anzatzes.

### 3.1 Parametrization of arbitrary f(R) functions

For the treatment of arbitrary models, we introduce, following the idea in [125], the functions

 m=Rf′′(R)f′(R)=dlnf′(R)dlnR, (3.1a) r=−Rf′(R)f(R)=−dlnf(R)dln(R). (3.1b)

Now assuming that is a singled-valued function of say , and leaving the function still arbitrary, it is possible to obtain a closed dynamical system for and for a set of normalized variables. On the other hand, given or

 M(r)=r(1+r+m(r))m(r), (3.2)

as input, it is possible to reconstructing the original function as follows. First, deriving in both sides of (3.1b) with respect to , and using the definitions (3.1a) and (3.1b), is deduced that

 drdR=r(1+m(r)+r)R. (3.3)

Separating variables and integrating the resulting equation, we obtain the quadrature

 R(r)=R0exp[∫drr(1+m(r)+r)]. (3.4)

Second, using the definition of we obtain:

 r=−dlnf(R)dlnR.

Reordering the terms at convenience are deduced the expressions

 −rdlnR = dlnf(R), −∫rdlnR = ln[f(R)f0], f(R) = f0exp(−∫rdlnR). (3.5)

Substituting (3.4) in (3.1) we obtain

 f(r)=f0exp(−∫11+m(r)+rdr). (3.6)

Finally, from the equations (3.1) and (3.4) we obtain by eliminating the parameter . In the table 1 are shown the functions and for some -models.

Therefore, following the above procedure, we can transform our cosmological system into a closed dynamical system for a set of normalized, auxiliary, variables and . Such a procedure is possible for arbitrary models, and for the usual ansatzes of the cosmological literature it results to very simple forms of , as can be seen in Table 1. In summary, with the introduction of the variables and , one adds an extra direction in the phase-space, whose neighboring points correspond to “neighboring” -functions. Therefore, after the general analysis has been completed, the substitution of the specific for the desired function gives immediately the specific results. Is this crucial aspect of the method the one that make it very powerful, enforcing its applicability.

To end this subsection let us comment about anisotropic curvature (strong)- and weak-singularities.

It is well-known that the violation of the stability condition during the evolution of a FRW background results in the immediate lost of homogeneity and isotropy, and thus, an anisotropic curvature singularity is generically formed [135, 63]. On other hand, at the instance where becomes zero for a finite also an undesirable weak singularity forms [119]. In this section we want to discuss more about this kind of singularities.

• At the regime where reaches zero for finite , an anisotropic curvature (strong) singularity with power-law behavior of the metric coefficients is generically formed [135, 63]. This singularity prevents a transition to the region where the effective gravitational constant in negative in a generic, non-degenerate solution. Now let us characterize this singularity in terms of our dynamical variables. Observe that

 f′(R)=−rf(R)R. (3.7)

Hence, for and , the singularity corresponds to the value

• At the regime where becomes zero, for a non-zero , a weak singularity develops generically [119]. Now let us characterize this singularity in terms of our dynamical variables. Observe that

 f′′(R)=−rm(r)f(R)R2. (3.8)

Hence, for the singularity corresponds to the values such that 555 must be different to zero since it is required at the singularity.

A more complete analysis of all possible singularities requires further investigation and is left for future research.

### 3.2 Normalization and Phase-space

In order to do a systematic analysis of the phase-space, as well as doing the stability analysis of cosmological models, it is convenient to transform the cosmological equations into their autonomous form [148, 133, 131]. This will be achieved by introducing the normalized variables:

 Q=HD,Σ=σD,x=12Df′(R)df′(R)dt,y=16D2[R−f(R)f′(R)], (3.9a) z=ρm3D2f′(R),K=2K3D2, (3.9b)

where we have defined the normalization factor:

 D =  ⎷(2K)3+(H+12f′(R)df′(R)dt)2≡ ⎷(2K)3+(H+f′′(R)2f′(R)dRdt)2. (3.10)

From the definitions (3.9) we obtain the bounds and . However, can in principle take values over the whole real line.

Now, from the Gauss constraint (2.29) and the equation (3.10) follows the algebraic relations

 x2+y+z+Σ2=1, (3.11a) K+(Q+x)2=1, (3.11b)

that allow to express to and in terms of the other variables. Thus, our relevant phase space variables will be and the variables (3.9a).

Using the variables (3.9a), and the new time variable defined as

 dτ=Ddt,

we obtain the autonomous system

 r′=2xM(r), (3.12a) Q′=−12(1+r){2(1+r)−2(1+r)x2−2ry+2Q3(1+r)Σ+2(1+r)Σ2+ +Q2(1+r)(2+3x2(−2+γ)+4xΣ−6Σ2+3γ(−1+y+Σ2))+ +Q(1+r)(−2Σ+x(−2+3x2(−2+γ)+2xΣ−6Σ2+ +3γ(−1+y+Σ2)))}, (3.12b) Σ′=12{−2+2(Q+x)2−3(Q+x)(2+x2(−2+γ)+(−1+y)γ)Σ− −2(−1+Q+x)(1+Q+x)Σ2−3(Q+x)(−2+γ)Σ3}, (3.12c) x′=−12(1+r){−4−4r+6Qx+6Qrx+10x2+10rx2−6Qx3− −6Qrx3−6x4−6rx4+2ry+3γ+3rγ−3Qxγ− −3Qrxγ−6x2γ−6rx2γ+3Qx3γ+3Qrx3γ+ +3x4γ+3rx4γ−3yγ−3ryγ+3Qxyγ+3Qrxyγ+ +3x2yγ+3rx2yγ+2(1+r)x(−1+Q+x)(1+Q+x)Σ+ +(1+r)(4+3x(Q+x)(−2+γ)−3γ)Σ2}, (3.12d) y′=−11+ry{3(1+r)x3(−2+γ)+(1+r)x2(3Q(−2+γ)+2Σ)+ +x(4+2r−3γ−3rγ+3yγ+3ryγ+4Q(1+r)Σ+ +3(1+r)(−2+γ)Σ2)+(1+r)(−2Σ+Q(2(Q−3Σ)Σ+ +3γ(−1+y+Σ2)))} (3.12e)

where ( ) denotes derivative with respect to the new time variable , defined in the phase space:

 Ψ = {(r,Q,Σ,x,y)|r∈R,Q∈[−2,2],Σ∈[−1,1],|Q+x|≤1,x∈[−1,1], (3.13) y∈[0,1],x2+y+Σ2≤1}.

Observe that the phase space (3.13) is in general non-compact since . Additionally, in the cases where has poles, i.e., given there are r-values, such that then cannot take values on the whole real line, but in the disconnected region where runs over the set of poles of In these cases, due the fact that defines a singular surface in the phase space, the dynamics of the system is heavily constrained. In particular, it implies that do not exist global attractor, so it is not possible to obtain general conclusions on the behavior of the orbits without first providing information about the initial conditions and the functional form of [126]. The study is more difficult to address if and vanish simultaneously, since this would imply the existence of infinite eigenvalues. For example, as discussed in [126], for the model , some of the eigenvalues diverge for . This is a consequence of the fact that for these two values of the parameter the cosmological equations assume a special form and is not a pathology of the system.

It is worth to mention that our variables are related with those introduce in the reference [125] by the relation:

 Qdτ=dN,r=x3x2,DQ=H,x=−12Qx1,y=Q2(x2+x3). (3.14)

So, in principle, we can recover their results in the absence of radiation (), by taking the limit

Finally, comparing with the results in [125] we see that our equation (3.12a) reduces to the equation

 drdlna=r(1+r+m)˙RHR (3.15)

presented in [125] for the choice (up to a time scaling, which of course preserves the qualitative properties of the flow). From (3.12a) it follows that the asymptotic solutions (corresponding to fixed points in the phase space) satisfy or or both conditions. Due to the equivalence of (3.12a) and (3.15) when , it follows that the solutions having and/or contain as particular cases those solutions satisfying and/or that where omitted in [125] according to [126]. In fact, is one of the values that annihilates the function and due the identity it implies that, for finite and nonzero, the points having also satisfy . For the above reason the criticism of [126] is not applicable to the present paper.

### 3.3 Stability analysis of the (curves of) critical points

In order to obtain the critical points we need the set the right hand side of (3.12) equal to the vector zero. From the first of equations (3.12) are distinguished two cases: the first are the critical points satisfying , and the second one are those corresponding to an -coordinate such that We denote de -values where is zero by , i.e., . In the tables 2 and 3, are presented the critical points for the case () and respectively.