Thawing models in the presence of a generalized Chaplygin gas

Thawing models in the presence of a generalized Chaplygin gas

Sergio del Campo sdelcamp@ucv.cl Instituto de Física, Pontificia Universidad de Católica de Valparaíso, Casilla 4950, Valparaíso, Chile    Carlos R. Fadragas fadragas@uclv.edu.cu Departamento de Física, Universidad Central de Las Villas, 54830 Santa Clara, Cuba.    Ramón Herrera ramon.herrera@ucv.cl Instituto de Física, Pontificia Universidad de Católica de Valparaíso, Casilla 4950, Valparaíso, Chile    Carlos Leiva cleivas@uta.cl Departamento de Física, Universidad de Tarapacá, Casilla 7-D, Arica, Chile    Genly Leon genly.leon@ucv.cl Instituto de Física, Pontificia Universidad de Católica de Valparaíso, Casilla 4950, Valparaíso, Chile    Joel Saavedra Joel.Saavedra@ucv.cl Instituto de Física, Pontificia Universidad de Católica de Valparaíso, Casilla 4950, Valparaíso, Chile
July 16, 2019
Abstract

In this paper we consider a cosmological model whose main components are a scalar field and a generalized Chaplygin gas. We obtain an exact solution for a flat arbitrary potential. This solution have the right dust limit when the Chaplygin parameter . We use the dynamical systems approach in order to describe the cosmological evolution of the mixture for an exponential self-interacting scalar field potential. We study the scalar field with an arbitrary self-interacting potential using the “Method of -devisers.” Our results are illustrated for the special case of a coshlike potential. We find that usual scalar-field-dominated and scaling solutions cannot be late-time attractors in the presence of the Chaplygin gas (with ). We recover the standard results at the dust limit (). In particular, for the exponential potential, the late-time attractor is a pure generalized Chaplygin solution mimicking an effective cosmological constant. In the case of arbitrary potentials, the late-time attractors are de Sitter solutions in the form of a cosmological constant, a pure generalized Chaplygin solution or a continuum of solutions, when the scalar field and the Chaplygin gas densities are of the same orders of magnitude. The different situations depend on the parameter choices.

pacs:
98.80.Cq

Introduction

It is common knowledge that the expansion of the Universe is currently passing through an accelerated phase R98 (); P99 (). All the observational data from these former references until the current measurements of redshift and luminosity-distance relations of type Ia Supernovae (SNe)S10 () are in agreement with this accelerated era. These observations are indicating the presence of a vacuum energy, the well-known cosmological constant Peebles:2002gy (); S03 (); T04 (). Another possible description is the existence of the scalar field that is evolving in a universe described by a Friedmann-Robertson-Walker (FRW) geometry, the quintessence model Zlatev:1998tr (). The current acceleration of the universe is one of the important problems of modern cosmology. This problem appears in Einstein’s standard general relativity, and one of the proposals to solve it, within this framework, is to consider one exotic component in the matter content of the universe Gabadadze:2007dv (), the so-called dark energy component (for a review about this, see Ref. Copeland:2006wr ()). We would like to focus our attention on models where dark energy is described by a homogeneous scalar field () scalarD-E (). In order to describe the dynamics of dark energy (quintessence), the equation of state parameter (EOS), plays a crucial role. Its current value is close to . Therefore, the cosmic evolution of dark energy through the scalar field is described by the barotropic parameter . In Ref. Caldwell:2005tm (), the authors distinguished between two categories, the freezing () and thawing () models. These quintessence models are characterized by a scalar field potential that asymptotically goes to zero Scherrer:2007pu (). There are several references about this kind of cosmological model (for a summary see Refs. thawing-M ()). On the other hand, Chaplygin gas models have been widely investigated in the literature Bento:2002ps (); Bilic:2001cg (); Bento:2002yx (); Gorini:2002kf (); Debnath:2004cd (); Zhu:2004aq (); Zhang:2004gc (); Chakraborty:2007ui (); Sen:2005sk (); Barreiro:2004bd (); Ali:2011sv (); Fabris:2010yh (); delCampo:2009cz (). The modified Chaplygin gas for the FRW universe is exactly the same as adding a bulk viscosity proportional to a power of the fluid density Barrow:1988yc (); Barrow:1990vx (). The generalized Chaplygin gas (GCG) has been investigated from the dynamical systems viewpoint, for example, in Bhadra:2011ac (); Mazumder:2011ke (); Li:2008uv (); He:2008zzc (); Rudra:2011ku ().

In this paper we would like to extend the analysis in Scherrer:2007pu () by considering a more general matter component, that is a GCG and a scalar field characterized by its self-interacting scalar potential. We obtain an exact solution for a flat arbitrary potential, that have the right dust limit when the Chaplygin parameter Scherrer:2007pu (). In order to motivate the analysis for a general (arbitrary) potential, we first consider the simple case of an exponential potential Copeland:1997et (); Halliwell:1986ja (); Barreiro:1999zs (); Burd:1988ss (); Sami:2002fs (); Rubano:2001su (); Heard:2002dr (); Coley:1997nk (); Liddle:1988tb (); Rubano:2003et (); Guo:2003eu (); Barrow:1994nt (); Aguirregabiria:1993pm (); Aguirregabiria:1993pk (); Ibanez:1995zs (); Pavluchenko:2003ge (); Goheer:2002ac (); Piedipalumbo:2011bj (); Fang:2006zq (); Aguirregabiria:1996uh (); Copeland:2009be (). Then, we study an arbitrary self-interacting scalar field potential. In this case we use the “Method of -devisers” presented in Escobar:2013js (). This method allows to perform the phase space analysis without specifying the potentials ad initium, and then one just substitute the desired forms, instead of repeating the whole procedure for every distinct potential. The method is a refinement of a method that has been applied to isotropic (FRW) scenarios Copeland:2009be (); Fang:2008fw (); Matos:2009hf (); Leyva:2009zz (); UrenaLopez:2011ur (); Dutta:2009yb (), and that has been generalized to several cosmological contexts Escobar:2012cq (); Escobar:2011cz (); Farajollahi:2011ym (); Xiao:2011nh ().

This article is organized as follow, in section I we present the cosmological model under consideration. Section II is devoted to the study of the exponential potential an the corresponding dynamical system. In section III we study the phase space of the cosmological model for general potential . Finally, we conclude in the section IV.

I The cosmological model

In this section we would like to describe a cosmological setting compose by a minimally-coupled scalar field that describes the dark energy and a Chaplygin gas that behaves as dark matter in the appropriate limit.

The cosmological equations are given by

(1)
(2)

where we work in units in which , is the Hubble constant, and are the Chaplygin gas and scalar field densities, respectively. For simplicity, the radiation component is neglected. and represent the Chaplygin gas and scalar field pressures, respectively. For the scalar field we have:

(4)
(5)

On the other hand, the EoS for the GCG is

(6)

where is a positive constant and is a constant with an upper bound, . In particular, when corresponds to the original Chaplygin gas.In the framework of FRW cosmology, this EoS leads, after inserted into the relativistic energy conservation equation, to an evolution of the energy density as

(7)

Here, is the scale factor and B is a positive integration constant. In this way, the GCG is characterized by two parameters, and . Here, is the current value of , considering that at the present. These parameter has been confronted with the observational data, see Refs.const (); const1 (). In particular, the values of and were obtained in Ref.const1 (). Also, in Ref.Bento:2002ps () the values and were found from the observational data arising from different colaborations, such that Archeops (by using the first peak localization) and BOOMERANG (by using the third peak localization). Recently, the values and were obtained from Markov Chain Monte Carlo method const2 (). For the phase space simulations implemented in the present paper we select the value first in agreement with Archeops and BOOMERANG collaborations, and second, following the reference DelPopolo:2013bpa (). This value seems to be large in comparison with the observational values in const2 (), however, if we consider large values for , and include the effect of shear and rotation, then when studying the evolution of the perturbations in GCG universes, it is found that that the joint effect of shear and rotation is that of slowing down the collapse with respect to the simple spherical collapse model. The described effect allows to solve the instability problems of the so-called unified dark matter models at the linear perturbation level DelPopolo:2013bpa ().

The evolution of the energy density shows the behaviors of GCG at different times. At early times, the energy density behaves as matter while at late times it behaves like a cosmological constant. Then, this GCG in principle describes both dark matter and dark energy in a single matter component.

Now we can define the barotropic index :

(8)

At late times, the universe is dominated by the scalar field and the GCG, neglecting the radiation component.

Equations (2) and (I) can be rewritten in terms of the auxiliary variables , and , defined by

(9)

where the prime denotes derivatives with respect to . Considering that the contribution for the kinetic and potential energy are given by and respectively, the density parameter of scalar field is given by

(10)

therefore the equation of state is

(11)

Thus, inserting the auxiliary (9) into the equations of motions (1),(2) and (I) we arrive to the following system

(12)
(13)
(14)

where

(15)

Besides, the Friedmann constraint equation can be written as , and this implies , for a non-negative density. Therefore the dynamical evolution of (12)-(14) leave the coordinates within the upper-half unit disc.

Furthermore, the system (12)-(14) is non-autonomous since

(16)

and it is in general not closed since does not depends a priori on the state variables

Before to perform a detailed analysis of the stability of this dynamical system we rewrite it in terms of the observable quantities and , and the new equations are given by

(17)
(18)
(19)

Equation (17) together with eq. (19) encode the exact description of the dynamic evolution of the scalar field. In any case to find an exact solution it is a difficult task, and in order to proceed we consider two assumptions. First, we consider that the barotropic parameter of the scalar fluid is near to , therefore . Second, we consider a near flat potential that is is approximately constant, say Scherrer:2007pu ().

In the limit we obtain from (18) an approximated equation with solution

(20)

satisfying where

and Thus, in the limit is recovered the solution described by the expression (25) in Ref. Scherrer:2007pu (): corresponding to standard quintessence. From the solution (20) and from equation (8) we obtain the key formula

(21)
    
(a) (b)
Figure 1: (a) The value of vs. assuming a nearly flat potential and The dotted (red) line corresponds to the approximated solution discovered in Scherrer:2007pu () (equation (23) in Scherrer:2007pu ()) and the dash-dotted (dark) line corresponds to the approximated solution, (28), for presented here. (b) EoS parameter of the scalar field for the exponential potential with slope We have considered the initial conditions The scale factor is normalized to 1 at present. The continuous (green) line corresponds to the exact value of for model with solely a scalar field; the dashed (blue) line corresponds to the exact value of for model that includes the Chaplygin gas; the dotted (red) line corresponds to the approximated solution found in Scherrer:2007pu () (equation (23) in Scherrer:2007pu ()) and the dash-dotted (dark) line corresponds to the approximated solution, (28), presented here (assuming ).

On the other hand, let us introduce the auxiliary function Assuming that is a monotonic function of the scale factor (in order to avoid that at any value ), we obtain from (17) and (18)

(22)

where we have used the approximation . Using the hypothesis one is able to use the Taylor-expand the above differential equation around up to second order to obtain the approximated equation

(23)

Substituting into the equation (23) the expression given by (21) and integrating the resulting equation with the initial condition at (which is true for the models we are considering here) we obtain the exact solution for given by

(24)

where

(25)

and are the elliptic integral of the first and second kind respectively, with

(26)

and

(27)

Finally we obtain the expression

(28)

Expression (28) allows to obtain an exact solution for the dynamical evolutions of the barotropic index and using (20) we obtain . We would like to note that our solutions have the right dust limit when Scherrer:2007pu (), this behavior is shown in Figures 1 (a) and (b).

In fact, in the limit (i.e., ), the deviation between our solution (28) and the solution (23) in Scherrer:2007pu () is given by the term

(29)

where

(30)

But

(31)

Thus,

(32)

This means that That is, in the limit our solution (28) and the solution (23) in Scherrer:2007pu () coincides.

Particularly, in the Figure 1 (a) depict the value of vs. assuming a nearly flat potential and The dotted (red) line corresponds to the approximated solution discovered in Scherrer:2007pu () (equation (23) in Scherrer:2007pu ()) and the dash-dotted (dark) line corresponds to the approximated solution, (28), for presented here. In Figure 1 (b) are displayed the EoS parameter of the scalar field for the exponential potential with constant slope. The scale factor is normalized to 1 at present. The continuous (green) line corresponds to the exact value of for model with solely a scalar field; the dashed (blue) line corresponds to the exact value of for model with the addition of the Chaplygin gas; the dotted (red) line corresponds to the approximated solution discovered in Scherrer:2007pu () (equation (23) in Scherrer:2007pu ()) and the dash-dotted (dark) line corresponds to the approximated solution, (28), presented here (assuming ).

Ii Exponential potential

In order to do the analysis from the dynamical systems viewpoint of the mixture of a scalar field with exponential potential and a Chaplygin gas, we need to consider the variables defined in the previous section plus the new variable

Then, from the equations of motions (1),(2) and (I) we obtain the following autonomous system

(33)

We note, that in the dust limit () the variable becomes automatically zero, the last equation in the system (33) is satisfied identically ( defines an invariant set) and the remaining equations (33) corresponds to the usual exponential quintessence scenario Copeland:1997et ().

Now, it is convenient to express the observable magnitudes in terms of the phase space variables. These observable magnitudes are the dimensionless Dark Energy density, given by (10); the equation of state (EoS) parameter of the dark energy given by

(34)

the EoS of the Chaplygin gas

(35)

the total (effective) EoS given by

(36)

and the deceleration parameter

(37)

These expressions are valid not only at the fixed points but also they are valid in the whole phase space. Thus, we evaluate them at the fixed points in order to determine the type of solution that they represent.

Cr. P. Existence
0 0 0 always 0 arbitrary 0 0
1 0 0 always 1 1 arbitrary 1 2
-1 0 0 always 1 1 arbitrary 1 2
0 1 arbitrary
0 0 0 0
0 1 0 1 arbitrary
0
0 0 1 always 0 arbitrary
Table 1: The real and physically meaningful critical points of the autonomous system (33) associated to the exponential potential. Stability conditions and the values of the dark-energy density parameter , of the dark-energy EoS parameter , of the total EoS parameter and of the deceleration parameter .
Cr. P. Eigenvalues Stability
saddle
unstable
unstable
saddle
saddle
stable (see Appendix A.1)
NH
stable
Table 2: The real and physically meaningful critical points of the autonomous system (33) associated to the exponential potential. Stability conditions, NH stands for non-hyperbolic. Observe that the critical points and belong to the singular surface In this case both denominator and numerator of (33) vanish simultaneously. In this case the additional eigenvalue due to the extra -coordinate could be finite positive or infinite with undefined sign depending of how the point is approached. Thus, linear approximation fails and we need to resort to numerical investigation.

In table (1) we show the existence conditions for the real and physically meaningful (curves of) critical points of the autonomous system (33) associated to the exponential potential and also the values of the dark-energy density parameter , of the dark-energy EoS parameter , of the total EoS parameter and of the deceleration parameter evaluated at them.

Now, let us discuss in more details the stability conditions for the corresponding critical points. The critical point associated to a matter dominated universe is a saddle point. Observe that the singular points and belong to the singular surface In this case, both, the denominator and the numerator of Eq. (33) vanish simultaneously. In this case the additional eigenvalue due to the extra -coordinate could be finite positive or infinite with undefined sign depending of how the point is approached. Thus, linear approximation fails and we need to resort to numerical investigation. The critical points and corresponding to stiff solutions, are always unstable. ( resp.) is a local source for (, resp.), otherwise they are saddles. This argument is supported by numerical studies as shown in Figures 2 (a)-(d), for the values of the parameters in the typical intervals (that are determined by the bifurcation values). Critical points and are the usual quintessence solutions widely investigated in the literature (see for instance Copeland:1997et ()). Then, the main difference with respect to the results found in Ref. Copeland:1997et () is that, for none of them can be a late time attractor. For this argument is based on numerical analysis, since belongs to the singular surface and in this case, both the numerator and the denominator of the the system (33) vanish, and then, the linear approximation is not valid. In the case of the point we support this result in the fact that there exist one eigenvalues with positive real part. This means that if we include a GCG in the background, we cannot get a stable solution dominated by the scalar field () or an scaling solution (). If we restrict our attention to the invariant set , then we have that is a stable one for and thus it can be the late time state of the universe. In this case the equation for is vanished identically, and we do not require to include this variable in the analysis. corresponds to a dark-energy dominated universe, with a dark-energy EoS in the quintessence regime, which can be accelerating or not according to the -value. Additionally, this solution is free of instabilities. This point is quite important, since it is stable and possesses and compatible with observations Copeland:1997et (). Point is stable in the invariant set It can attract the universe at late times (in case of a GCG behaving as dust), and it is free of instabilities. It has the advantage that the dark-energy density parameter lies in the interval , that is it can alleviate the coincidence problem, but it has the disadvantage that it is not accelerating and possesses , which are not favored by observations Copeland:1997et (). However, let us remark that they are saddles for the full vector field. The solution exists for It represents a de Sitter solution which is stable but not asymptotically stable (see the Appendix A.1). The curve of critical points (which exists only for ) is stable but not asymptotically stable, whereas, is asymptotically stable (see details of the center manifold calculations for both and in the Appendix B). To finish this section let us proceed to the discussion of some numerical elaborations:

(a) (b)
(c) (d)
Figure 2: The phase space of the system (33). Without lack of generality we use : (a) and are local sources; is a saddle. coincides with (contained in the curve ) and it is stable but not asymptotically stable for otherwise it is a saddle (see Appendix A.2). does not exist. Any arbitrary point in the curve (that exist only for ) is stable, attracting an open set of orbits. The center manifold of is stable, also is. (b) and are local sources; and are saddles and is the attractor. is a local attractor in the invariant set The solutions at the - plane correspond to those at Figure 2 in Copeland:1997et (). (c) and are local sources; and are saddles and is the attractor. is a local attractor in the invariant set The solutions at the - plane correspond to those at Figure 3 in Copeland:1997et (). (d) (the kinetic-dominated solution with ) is a saddle point; (the kinetic-dominated solution with ) is the local source (unstable node); does not exists; and are saddles and is the attractor. is a local (spiral) attractor in the invariant set The solutions at the - plane corresponds to those at Figure 4 in Copeland:1997et ().
  • Fig. 2 (a) shows several orbits for the values of the parameters and are local sources; is a saddle. coincides with (contained in the curve ) and it is stable but not asymptotically stable for otherwise it is a saddle (see Appendix A.2). does not exist. Any arbitrary point in the curve (that exist only for ) is stable, attracting an open set of orbits. The center manifold of is stable.

  • In Fig. 2 (b) we presented several orbits in the phase space for The kinetic-dominated solution and are local sources; the matter dominated solution and the scalar-field dominated solution are saddles and the Chaplygin gas dominated solutions (which also mimics a de Sitter solution) is the attractor. is a local attractor in the invariant set

  • Fig. 2 (c) shows several orbits in the phase space for and are local sources; and are saddles and is the attractor. is a local attractor in the invariant set

  • Finally, in Fig. 2 (d) we display several orbits in the phase space for (the kinetic-dominated solution with ) is a saddle point; (the kinetic-dominated solution with ) is the local source (unstable node); does not exists; and are saddles and is the attractor. is a local (spiral) attractor in the invariant set

Now, as commented before, it is a fact that the dynamical evolution leave the coordinates within the upper-half unit disc. However, the only restriction on is that This means that in priciple the -coordinate could be unbounded and, then, there migh exist critical points at infinity (which would correspond to ). In order to determine the fixed points at infinity and study their stability, we need to compactify the phase space using the Poincaré method. Transforming to polar coordinates Lefschetz (); Carloni:2004kp (); Abdelwahab:2007jp ():

(38)

where and substitutig the regime corresponds to . The points are mapped onto

(39)

thus, the points at infinity are mapped on the unitary sphere

Using this coordinate transformation, introducing the new time variable which preserves the time orientation, the leading terms of the system (33) as are

(40)
(41)
(42)

where now the comma denotes derivative with respect In this case the radial equation does not contain the radial coordinate, thus, the fixed points can be obtained using just the angular equations. Setting and we obtain that the fixed point with physical sense () must satisfy i.e., In this case the eigenvalues of the Jacobian matrix associated to the angular coordinates are and at the equilibrium point. Then, we cannot obtain information on their stability using the linearization. The complete analysis is outside the scope of the present investigation.

Iii Phase-space analysis without potential specification

In order to transform the system (12)-(14) to an autonomous one, first, it is necessary to determine a specific potential form of the scalar field . However, using the above example as a motivation, one could alternatively handle the potential differentiations using the auxiliary variable given by

(43)

while keeping the potential still arbitrary 111The variable is just a constant () for the exponential potential .. The next step is to introduce the function

(44)

to be an arbitrary function of In fact, if can be expressed as an explicit one-valued function of , that is , then, it is possible to write a closed dynamical system for and a set of normalized-variables. On the other hand, by giving non identically equal to zero, we obtain the expressions

(45)
(46)

where the integration constants satisfy , 222We would like to note that the requirement that must be different to zero exclude of this analysis the case of the exponential potential and for this reasons we studied the exponential potential in the previous section separately.. Thus, it is possible to reconstruct the potential by the elimination of between (45) and (46). For the usual cosmological cases the potential can be written explicitly, that is . The details of the method, coined “Method of -devisers”, were presented in Escobar:2013js (). This method has the significant advantage, that one can first perform the analysis for arbitrary potentials and then just substitute the desired forms, instead of repeating the whole procedure for every distinct potential (see Escobar:2013js () and references therein).

Then, from the equations of motions (1), (2) and (I) we result in the following autonomous system

(47)
Cr. P. Existence
0 0 0 always 0 arbitrary 0 0
1 0 0 always 1 1 arbitrary 1 2
-1 0 0 always 1 1 arbitrary 1 2
0 1 arbitrary
0 0 0 0
0 1 0 0 always 1 arbitrary
0 0
0 0 1 always 0 arbitrary
Table 3: The real and physically meaningful (curves of) critical points of the autonomous system (47). Existence conditions and the values of the dark-energy density parameter , of the dark-energy EoS parameter , of the total EoS parameter and of the deceleration parameter . We use the notation for the values of such that and for denoting arbitrary values of at equilibrium.
Cr. P. Eigenvalues Stability
saddle
unstable
unstable
saddle
saddle
stable (see Appendix A.2)
NH, stable for , saddle otherwise
NH (unstable)
Table 4: The real and physically meaningful critical points of the autonomous system (47). Stability conditions, NH stands for non-hyperbolic. We introduce the notations and Observe that the critical points and belong to the singular surface In this case both denominator and numerator of (47) are vanished simultaneously. In this case the additional eigenvalue due to the extra -coordinate could be finite positive or infinite with undefined sign depending of how the point is approached. For there are two eigenvalues whose nature depends on the way that is approached. For that reason they are undefined. Thus, linear approximation fails and we need to resort to numerical works.

In table 3 we present the existence conditions for the real and physically meaningful (curves of) critical points of the autonomous system (47). We use the notation for the values of such that and for denoting arbitrary values of at equilibrium. We display also the corresponding values of the dark-energy density parameter , of the dark-energy EoS parameter , of the EoS of Chaplygin gas , of the total EoS and of the deceleration parameter . In table 4 are presented the stability conditions for the the critical points.

Now, let us comment briefly on the stability and physical interpretation of the critical points of (47).

The curve of critical point is always a saddle. It represents cosmological solutions dominated by the Chaplygin gas mimicking dust, this solution correlates with the transient matter dominated epoch of the universe. Observe that the critical points with such that and belong to the singular surface In this case both denominator and numerator of (47) are vanished simultaneously. In this case the additional eigenvalue due to the extra -coordinate could be finite positive or infinite with undefined sign depending of how the point is approached. For there are two eigenvalues whose nature depends on the way that is approached, for that reason they are undefined.

For , the solutions and are past attractors or saddle points under the same conditions of the standard quintessence scenario Copeland:1997et () with the identification (see table 4). They represent solutions dominated by the kinetic energy of the scalar field mimicking a stiff fluid. The solutions and represents the scalar field dominated solution, the matter-scalar scaling solution and de Sitter solutions dominated by the potential energy of the scalar field, respectively. The main difference here with respect the standard quintessence scenario Copeland:1997et () is that are saddle points (we are considering ). So, the standard quintessence solutions are not late time solutions in this scenario. For analyzing the important critical point , the linear approximation fails and we need to resort to numerical studies or include higher order terms in the analysis. In fact, following our approach in the Appendix A.2, we find that actually is the late-time attractor for and a saddle for

Combining expressions (16) and (35) we find that as Thus at late times we can approximate the system (47) by

(48)

and the decoupled equation

(49)

If as then from equation (49) follows that increases without bound in contradiction with the boundedness of Thus, as time goes forward, Hence where By calculating the critical points of the system (III) and analyzing their linear stability we find that the only candidates to be the late-time attractors are:

  • the curve which have the following system of eigenvalues and eigenvectors:

    where Since the center subspace is tangent to the -axis, follows that the curve is normally hyperbolic 333Recall that a set of non-isolated critical points is said to be normally hyperbolic if the only eigenvalues with zero real parts are those whose corresponding eigenvectors are tangent to the set. In this case the stability of the set can be deduced by examining the signs of the remaining non-null eigenvalues (i.e., for a curve, in the remaining directions) Aulbach1984a ().. Then follows the stability of on the space This argument is not complete, since we have forget about what happens in the -direction. In fact, in the general case (when the -direction is included in the analysis), this curve is actually non-hyperbolic and it is not normally hyperbolic anymore, thus we cannot obtain information about its stability looking at the linearization. This one is the main difference that appears when considering the extra direction .

  • The other candidate is the curve which have the following system of eigenvalues and eigenvectors

    which is also normally hyperbolic (the center subspace is the plane - is tangent to the line ).

  • Finally, both numerical simulations and analytical methods suggest that (contained in the curve ) is an attractor for and for it is a saddle (see Appendix A.2).

The above heuristic reasoning suggest that the future attractor of the system (47) is located at the curve , which contains the especial point , or it is located at the curve .

With the exception of the point which is dominated by a constant potential, represents a class of solutions where neither the potential energy of the scalar field nor the Chaplygin gas dominates. On the other hand the curve corresponds to purely Chaplygin gas dominated solutions (which also mimics a de Sitter solution).

Indeed, using the Center Manifold Theory it can be proved that if the condition is satisfied, the curve of fixed points is stable but not asymptotically stable. Applying the same procedure to the curve we find that also the curve is stable but not asymptotically stable. The details of the calculation are presented in the Appendix C.

To investigate the dynamics at infinity one introduces the Poincaré variables Lefschetz (); Carloni:2004kp (); Abdelwahab:2007jp ():

(50)

where and and the new time variable which preserves the time orientation. The region at infinity corresponds to the region in the space. Then, we take te limit in where now the comma denotes derivative with respect to and preserve the leading terms. In the case that the radial equation does not contain the radial coordinate, the fixed points can be obtained using just the angular equations. Setting , and , are obtained the fixed points. The stability of these points is studied by analyzing first the stability of the angular coordinates and then deducing, from the sign of , the stability on the radial direction Lefschetz (); Carloni:2004kp (); Abdelwahab:2007jp (). That is, it is required at equilibrium. This means that the radial coordinate increases in value to reaching the boundary from below. To do the analysis it is required to provide the functional form of however, the complete analysis is outside the scope of the present study.

iii.1 An example: Cosh-like potential

The cosh-like potential has been widely studied in the literature (see for example Sahni:1999qe (); Sahni:1999gb (); Lidsey:2001nj (); Pavluchenko:2003ge (); Copeland:2009be (); Ratra:1987rm (); Wetterich:1987fm (); Matos:2000ng (); Sahni:1999qe (); Wetterich:1987fm (); Ratra:1987rm (); Copeland:2009be (); Matos:2009hf (); Leyva:2009zz (); Escobar:2013js ()). For this potential,

(51)

Observe that This is the sufficient condition for the stability of the class of de Sitter solutions represented by the curve of critical points For this choice Also thus For this choice the system (47) admits twelve (curves of) critical points denoted by , and

Figure 3: Projections of some orbits in the phase space of the system (47) for . Without lack of generality we use Observe that and are local sources and the point located at the curve is a local attractor. The curve K is stable, but not asymptotically stable. The scalar field-dominated solution are of saddle type, as well as the rest of the (curves of) fixed points. The scaling solutions do not exist.
Figure 4: Projections of some orbits in the phase space of the system (47) for . Without lack of generality we use Note that and are local sources and the point located at the curve is a local attractor. The curve K is stable, but not asymptotically stable. The scalar field-dominated solution and the scaling solution are of saddle type, as well as the rest of the (curves of) fixed points.
Figure 5: Projections of some orbits in the phase space of the system (47) for . Without lack of generality we use and are local sources and the point located at the curve is a local attractor. The curve K is stable, but not asymptotically stable. The scalar field-dominated solution and the scaling solution are of saddle type, as well as the rest of the (curves of) fixed points.

To finish this section let us discuss some numerical simulations:

  • In the Fig.