1. Introduction
The analysis of the cosmological observations suggests that our Universe, on large scales, is isotropic and homogeneous, as described by the four-dimensional Friedmann–Lemaître–Robertson–Walker (FLRW) geometry. The primary theoretical mechanism proposed to explain the observations is the so-called cosmic inflation [
1,
2], which solves the flatness and homogeneity problems [
3,
4].
In the context of Einstein’s General Relativity, inflation is described by a scalar field, known as “inflaton”. Specifically, the inflationary mechanism introduces a scalar field in the cosmic fluid, and the cosmic expansion appears when the scalar field potential dominates to drive the dynamics [
5,
6,
7,
8,
9]. The additional degrees of freedom the scalar field provides can describe higher-order geometric invariants introduced in the Einstein–Hilbert Action. Indeed, in the Starobinsky model for inflation [
10] inspired by field theory, a quadratic term of the Ricci scalar has been introduced to modify the Einstein–Hilbert Action. The higher-order derivatives are attributed by a scalar field which can provide an inflationary epoch, see also the recent studies [
11,
12].
Furthermore, at present, the Universe is under a second acceleration phase [
13], attributed to an exotic matter source with negative pressure known as dark energy. The nature of the dark energy is unknown. The two acceleration phases of the Universe challenge the theory of General Relativity, and cosmologists have proposed various modified and alternative theories of gravity in the last decades [
14,
15,
16,
17,
18], including stringy inspired theories [
19,
20,
21,
22,
23].
General Relativity’s main characteristic is a second-order theory of gravity. Moreover, according to Lovelock’s theorem, General Relativity is the unique second-order gravitational theory in the four dimensions where the field equations are generated from an Action Integral [
24]. However, General Relativity is only a case of Lovelock gravity in higher dimensions. The latter is a second-order theory of gravity in higher dimensions where higher-order invariants are introduced in the gravitational Action Integral [
25,
26]. The Gauss–Bonnet invariant is the only invariant derived by the Riemann tensor quadratic products that does not introduce any terms with higher-order derivatives into the field equations [
25]. Conversely, in the case of four dimensions, the Gauss–Bonnet invariant is a topological invariant, a total derivative that, when introduced in the gravitational Lagrangian, does not affect the field equations. The Einstein–Gauss–Bonnet theory is the most straightforward extension of Einstein’s General Relativity and belongs to Lovelock’s theories.
The Einstein Gauss-Bonnet terms have been widely studied in higher-order theories of gravity (see, for instance [
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44,
45,
46,
47,
48,
49] and references therein). In particular, quintessence in five-dimensional Einstein–Gauss–Bonnet black holes was examined in [
28], anisotropic stars in Einstein–Gauss–Bonnet theory were examined in [
29,
30,
31,
32,
33,
34], wormholes in 4D Einstein–Gauss–Bonnet in [
35], black holes in 4D Einstein–Gauss–Bonnet gravity, and the thermodynamics were considered in [
36,
37,
38]. Quasinormal modes of the Dirac field in the consistent 4D Einstein–Gauss–Bonnet gravity were studied in [
39]. Furthermore, the Gauss–Bonnet term can describe the quantum corrections to gravity, mainly related to the heterotic string [
50]. An essential property of the Einstein–Gauss–Bonnet theory is that it is a ghost-free theory of gravity [
51].
In the case of four dimensions, because the Gauss–Bonnet is a topological invariant, it can be introduced in gravitational Action Integral only with modifications. Indeed, there is a family of theories known as
theories of gravity, where nonlinear functions of the Gauss–Bonnet invariant are introduced in the Gravitational Integral [
52,
53,
54]. Another attempt is to introduce a scalar field coupled to the Gauss–Bonnet invariant. In that case, an a coupling function exists between the Gauss–Bonnet term and the scalar field. The cosmological scenario we deal with in this work is the Einstein–Gauss–Bonnet scalar field theory [
55]. The properties of astrophysical objects in this theory were the subject of various studies [
16,
56,
57,
58,
59,
60].
In cosmological studies, the four-dimensional Einstein–Gauss–Bonnet scalar field theory has been applied to describe various epochs of cosmological evolution. It has been found that the Gauss–Bonnet invariant and the coupling function introduce non-trivial effects on the early inflationary stage of the universe [
61], and that a small transition exists to Einstein’s General Relativity at the end of the inflationary epoch. Some exact solutions describing cosmic inflation were derived in [
62]. On the other hand, inflationary models with a Gauss–Bonnet term were constrained in the view of the GW170817 event in a series of studies [
63,
64,
65,
66], and the GW 190814 event [
66,
67,
68,
69]. In the presence of a nonzero spatial curvature for the background space, exact solutions in Einstein–Gauss–Bonnet scalar field theory were derived before in [
70]. It was found that the quadratic coupling function of the scalar field to the Gauss–Bonnet term is essential because the singularity-free theory provides inflationary solutions.
In [
71], the dynamics of the cosmological field equations were investigated for the four-dimensional Einstein–Gauss–Bonnet scalar field theory, where the authors have assumed that the Hubble function is that of a scaling solution; however, in [
72], the most general case was studied, and the equilibrium points of the field equations were investigated. The analysis in [
72] shows that the only equilibrium points where the Gauss–Bonnet term contributes to the cosmological fluids are that of the de Sitter universe. However, as we shall show in this research, additional equilibrium points exist that describe scaling solutions to which the Gauss–Bonnet term contributes. These points have the equation of state
, interpreted as the equation of state of cosmic strings (
), where
,
N is the number of strings in our cosmic horizon,
is the linear density of the strings, and
is the length of each string). Cosmic strings have the effect that they do not contribute to the “non-inertial” expansion of the Universe. Similarly, for other topological defects such as domain walls,
, where the surface tension of the wall is
, with superficial area
, we have
, which leads to an accelerated expansion of the Universe [
73,
74]. In particular, we perform a detailed analysis of the phase for the cosmological field equations in the Einstein–Gauss–Bonnet scalar field theory to understand the evolution of the cosmological parameters. Such analysis provides essential information about the significant cosmological eras provided by the theory. Simultaneously, important conclusions about the viability of the theory can be made. It is desirable to have complete cosmological dynamics [
75]; namely, it should describe an early radiation-dominated era, later entering into an epoch of mater domination, and finally reproducing the present speed-up of the Universe. In the dynamical systems language, complete cosmological dynamics can be understood as an orbit connecting a past attractor, also called a source, with a late-time attractor, also called a sink, that passes through some saddle points, such that radiation precedes matter domination. These are often the extreme points of the orbits; therefore, they describe asymptotic behavior. Some solutions interpolating between critical points can provide information on the intermediate stages of the evolution, with interest in orbits corresponding to a specific cosmological history [
76,
77,
78,
79].
The paper is organized as follows. In
Section 2, we present the gravitational Action integral for the Einstein–Gauss–Bonnet scalar field theory in a four-dimensional, spatially flat FLRW geometry. We present the field equations where we observe that they depend on two functions, the scalar field potential
, selected as the exponential function
, and the coupling function
of the scalar field with the Gauss–Bonnet scalar. Moreover, the scalar field can be a quintessence or a phantom field. We perform a global analysis of the field equations’ phase space to reconstruct the cosmological parameters’ evolution. In
Section 3, we study the equilibrium points for linear function
, while in
Section 4, we perform the same analysis for the exponential function
. In
Section 3.1 and
Section 3.2 for the linear case, and in
Section 4.1 and
Section 4.2 for the exponential one, we obtain additional equilibrium points in the finite region as compared with the analysis in [
72]. Those new points describe scaling solutions to which the Gauss–Bonnet term contributes, which differ from de Sitter points.
Section 3.3 and
Section 4.3 are devoted to the analysis at infinity for the linear and exponential functions, respectively, where the equilibrium points dominated by Gauss–Bonnet terms are also present. Finally,
Section 5 discusses our results and presents our conclusions.
2. Einstein–Gauss–Bonnet Scalar Field 4D Cosmology
The gravitational Action Integral for the Einstein–Gauss–Bonnet scalar field theory of gravity in a four-dimensional Riemannian manifold with the metric tensor
is defined as follows
where
R is the Ricci scalar of the metric tensor,
is the scalar field,
the scalar field potential, and
G is the Gauss–Bonnet term
Function
is the coupling function between the scalar field and the Gauss–Bonnet term, and
indicates if the scalar field
is quintessence
or phantom
. In the case where
is a constant function, the gravitational Action Integral (
1) reduces to that of General Relativity with a minimally coupled scalar field.
On very large scales, the universe is considered isotropic and homogeneous. The FLRW metric tensor describes the physical space with line element
The three-dimensional surface is a maximally symmetric space and admits six isometries. Moreover, we assume that the scalar field inherits the symmetries of the background space, which means that .
For the line element (
3), the Ricci scalar is derived
where a dot means derivative with respect to
t,
and
is the Hubble function. Moreover, the Gauss–Bonnet term is calculated as
By replacing the latter in the Action Integral (
1) and by integrating by parts, we end with the point-like Lagrangian function
while the field equations are
where the comma means derivative with respect to the argument of the function.
The effective density and pressure of the scalar field are given by
And we also define the effective equation of state (EoS)
In the following, we shall perform a detailed analysis of the phase-space for the exponential scalar field potential and for two coupling functions , the linear and the exponential , where and are constants.
3. Phase-Space Analysis for Linear :
The field equations for the linear coupling function
read
In order to study the phase space, we introduce the following normalized variables
With these definitions, the first modified Friedmann equation is written in the algebraic form
Moreover, the rest of the field equations are described by the following system of first-order ordinary differential equations
We define the function , and introduce the time derivative .
Since
, we can solve Equation (
16) for
y and reduce the dimension of the system; the expression for
y is
The dynamics of the model with linear
f and
is given by
The effective equation of state parameter (
10) can be expressed in terms of
x and
as
whereas the deceleration parameter,
, can be expressed as
3.1. Dynamical System Analysis of 2D System for
In this section, we perform the stability analysis for the equilibrium points of system (
21) and (
22) taking
. The stability results and physical observables are summarized in
Table 1.
The equilibrium points in the coordinates are the following:
, with eigenvalues . The asymptotic solution is that of the Minkowski spacetime.
, with eigenvalues The asymptotic solution describes a universe dominated by the Gauss–Bonnet term with deceleration parameter . This equilibrium point is a source.
, with eigenvalues This equilibrium point is a sink. The asymptotic solution is similar to that of point .
, with eigenvalues This equilibrium point is a sink. We derive that . The asymptotic solution is similar to that of point .
, with eigenvalues This equilibrium point is a source. Moreover, for the deceleration parameter, it follows . The asymptotic solution is similar to that of point .
, with eigenvalues . The deceleration parameter is calculated ; hence, the asymptotic solution describes the de Sitter universe. This equilibrium point is a saddle that exists for or
, with eigenvalues . Point describes a de Sitter universe, i.e., . This equilibrium point is a saddle that exists for or
Phase-space diagrams for the dynamical system (
21) and (
22) where the scalar field is a quintessence are presented in
Figure 1.
Figure 2 depicts
,
, and
evaluated at the solution of system (
21) and (
22) for
and initial conditions
(i.e., near the saddle point
).
The solution is past asymptotic phantom regime , and then remains near the de Sitter point approaching the phantom solution (whence ), then crosses from below of (zero acceleration), decelerating and tending asymptotically to from above. This evolution, in which the equation of state parameter of the scalar field interpolates between (saddle point, de Sitter solution) and (attractor dominated by the Gauss–Bonnet term), corresponds to an inflationary solution, which does not eliminate the topological defect of the cosmic string. This behavior is due to the linear coupling between the scalar field and the Gauss–Bonnet term.
3.2. Dynamical System Analysis of 2D System for
In this section, we perform the stability analysis for the equilibrium points of system (
21) and (
22), taking
.
The stability results and physical observable results for system (
21) and (
22) are summarized in
Table 2.
The equilibrium points in the coordinates are the following.
, with eigenvalues . The asymptotic solution corresponds to the Minkowski spacetime.
, with eigenvalues The deceleration parameter is . That means the asymptotic solution describes a universe dominated by the Gauss–Bonnet term. This equilibrium point is a source.
, with eigenvalues with . This equilibrium point is a sink. The asymptotic solution is similar to that of point .
, with eigenvalues This equilibrium point is a sink. Moreover, means that the asymptotic behavior is similar to that of
, with eigenvalues , while the deceleration parameter is calculated . This equilibrium point is a source. As before, the asymptotic solution is similar to point .
, with eigenvalues . This equilibrium point corresponds to a de Sitter solution, i.e., . This equilibrium point exists for , or and is a saddle.
with eigenvalues , is a de Sitter point that is . This equilibrium point exists for or , and is a saddle.
This equilibrium point exists for
, has eigenvalues
and is a sink for
, a saddle for
or nonhyperbolic for
Moreover,
is from where we infer that the asymptotic solution is that of the de Sitter universe. The numerical analysis of the real part of the eigenvalues for
is presented in
Figure 3.
describes a de Sitter solution because
. This equilibrium point exists for
, has eigenvalues
, and is a source for
, a saddle for
or nonhyperbolic for
As before, the numerical analysis of the real part of the eigenvalues for
is presented in
Figure 3.
This equilibrium point exists for
, it describes a de Sitter solution because
, has eigenvalues
and is a source for
, a saddle for
or nonhyperbolic for
The numerical analysis of the real part of
and
for
is presented in
Figure 3.
Finally, the de Sitter point
This equilibrium point exists for
, has eigenvalues
and is a sink for
, a saddle for
or nonhyperbolic for
As before, the numerical analysis of the real part of
and
for
is presented in
Figure 3.
Figure 3.
Real part of the eigenvalues where for points .
Figure 3.
Real part of the eigenvalues where for points .
Phase-space diagrams for the dynamical system (
21) and (
22) where the scalar field is a phantom field, that is,
are presented in
Figure 4 for various values of the free parameters.
Figure 5 depicts
,
, and
evaluated at the solution of system (
21) and (
22) for
and initial conditions
(i.e., near the saddle point
).
The solution is past asymptotic to (zero acceleration), then remains near the de Sitter point approaching a quintessence solution , and then tending asymptotically to (de Sitter point ) from above. The past attractor is dominated by the Gauss–Bonnet term, and then the saddle point corresponds to an inflationary solution, eliminating the topological defect of the cosmic string. The late-time attractor is a de Sitter solution. Therefore, this solution connects inflation with late-time acceleration. This behavior is due to the linear coupling between the scalar field and the Gauss–Bonnet term.
3.3. Analysis of System (21) and (22) at Infinity
The numerical results in
Figure 1 and
Figure 4 suggest non-trivial dynamics when
. For that reason, we introduce the compactified variable
and the new time variable
we obtain the compactified dynamical system
where
The limit
corresponds to
.
The equilibrium points of system (
27) and (
28) at the finite region are the same as (
21) and (
22) by the rescaling
.
Table 3 summarizes the equilibrium points of system (
27) and (
28) for
with their stability conditions.
The equilibrium points at infinity are those satisfying , say
, with eigenvalues This equilibrium point is a saddle or nonhyperbolic for The value of the deceleration parameter is . That means the asymptotic solution describes a universe dominated by the Gauss–Bonnet term.
, with eigenvalues This equilibrium point is a saddle or nonhyperbolic for The value of the deceleration parameter is The asymptotic behavior is the same as
, with eigenvalues This equilibrium point is a saddle or nonhyperbolic for The value of the deceleration parameter is The asymptotic behavior is the same as
, with eigenvalues This equilibrium point is a saddle or nonhyperbolic for The value of the deceleration parameter is The asymptotic behavior is the same as
, with eigenvalues . This equilibrium point is a source for , a sink for or nonhyperbolic for Note that for
- (a)
, the equilibrium point has This equilibrium point exists for or or
- (b)
, the equilibrium point has This equilibrium point exists for or or
The value of the deceleration parameter is The asymptotic solution is a de Sitter universe.
, with eigenvalues . This equilibrium point is a sink for , a source for or nonhyperbolic for The existence conditions for are the same as The value of the deceleration parameter is The asymptotic behavior is the same as
, with eigenvalues . This equilibrium point is a source for , a sink for ; or nonhyperbolic for The existence conditions for are the same as The value of the deceleration parameter is The asymptotic behavior is the same as
, with eigenvalues . This equilibrium point is a sink for , a source for or nonhyperbolic for The existence conditions for are the same as The value of the deceleration parameter is The asymptotic behavior is the same as
The phase-space of the field equations at the new compactified variables is presented in
Figure 6 and
Figure 7 for different values of the free parameters. As far as the physical properties of the asymptotic solutions are concerned, we find that
,
,
, and
are Gauss–Bonnet points with deceleration parameter
, while points
,
,
, and
are de Sitter points with
.
Figure 8 depicts
,
, and
evaluated at the solution of system (
27) and (
28) for initial conditions
(i.e., near the saddle point
).
The solution is past asymptotic to a phantom regime , then remains near the de Sitter point approaching the phantom solution (whence ), then it crosses from below (zero acceleration), decelerating and tending asymptotically to from above. As before, this evolution corresponds to an inflationary solution, which does not eliminate the topological defect of the cosmic string. This behavior is due to the linear coupling between the scalar field and the Gauss–Bonnet term.
Figure 9 depicts
,
, and
evaluated at the solution of system (
27) and (
28) for
and initial conditions
(i.e., near the saddle point
).
The solution is past asymptotic to (zero acceleration), then remains near the de Sitter point approaching a quintessence solution , and then tending asymptotically to (de Sitter point ) from above. The past attractor is dominated by the Gauss–Bonnet term, and then the saddle point corresponds to an inflationary solution, eliminating the topological defect of the cosmic string. The late-time attractor is a de Sitter solution. Therefore, this solution connects inflation with late-time acceleration. This behavior is due to the linear coupling between the scalar field and the Gauss–Bonnet term.
5. Conclusions
In this work, we considered a four-dimensional FLRW geometry and a second-order modified gravitational theory with a scalar field coupled to the Gauss–Bonnet scalar. The Gauss–Bonnet term does not contribute to the gravitational Action Integral in the limit where the scalar field is constant. The theory reduces to General Relativity with a cosmological constant term. However, for a dynamical scalar field, the physical properties of the present cosmological model are distinct from that of the minimally coupled scalar field theory.
In order to study the dynamical properties of the phase-space and physical variables, we introduced dimensionless variables different from that of the H-normalization. The latter is because, from the field equations, we observed that it is possible in Einstein–Gauss–Bonnet scalar field theory that the Hubble function can change its sign during its evolution, which means it can vanish. Hence, the H-normalization, widely applied before, must be validated for global analysis and the complete reconstruction of the cosmological history and epochs. Additionally, we observed that the dynamical variables are not bounded in a finite regime, which means that to perform a complete study of the phase-space, we assumed compactified variables to investigate the asymptotic behavior of the model at infinity.
The phase-space analysis of the gravitational field equations is a novel mathematical approach to the model’s asymptotic description and evolution of the physical variables. In cosmological studies, such analysis provides essential information about the significant cosmological eras the theory provides. Simultaneously, important conclusions about the viability of the theory can be made. According to cosmological observations, some particular forms of matter at each stage seem to dominate evolution. The required dominance should be translated into different critical points, around which cosmological solutions remain a lapse of time before approaching a stable late-time configuration. In the dynamical systems language, complete cosmological dynamics [
75] can be understood as an orbit connecting a past attractor, also called a source, with a late-time attractor, called a sink, that passes through some saddle points such that radiation precedes matter domination. These are often the extreme points of the orbits and therefore describe the asymptotic behavior. However, some solutions interpolate between critical points and then provide information on the intermediate stages of the evolution, with interest in orbits corresponding to a specific cosmological history [
76,
77,
78,
79]. However, according to our setup, we have partial cosmological dynamics. Therefore, the analysis incorporating additional matter fields to complete cosmological dynamics is left to a forthcoming investigation.
The gravitational Action Integral depends on two functions, which are the coupling function of the scalar field with the Gauss–Bonnet scalar and the scalar field potential. For the coupling functions, we consider two functional forms, the exponential function, a power-law function, while the potential we assume to be the exponential functional form. Moreover, a parameter has been introduced in the kinetic part of the scalar field, such that the scalar field is a quintessence field, , or a phantom field, . The two functional forms for the coupling function of the scalar field with the Gauss–Bonnet scalar provide different cosmological evolution. Last, but not least, the stability properties of the asymptotic solutions were investigated.
For the linear coupling function with a quintessence scalar field, we have found the following equilibrium points for the system (
21) and (
22) for
, as it was summarized in
Table 1. Say,
, which does not appears in the reference [
72], because it corresponds to
. It is nonhyperbolic, and the effective EoS and deceleration parameters are indeterminate.
The sources points are and , which have , and which are related to cosmic strings.
The sinks are and , which have , and , therefore, they are related to cosmic strings.
The de Sitter solutions ( and ), are saddle.
To present one possible evolution of the physical model,
Figure 2 depicts
,
and
evaluated at the solution of system (
21) and (
22) for
for initial conditions near the saddle point
. The solution is past asymptotic to a phantom regime
, then remains near the de Sitter point
approaching the phantom solution
(whence
), and then crosses from below
(zero acceleration), decelerating
and tending asymptotically to
from above. This evolution, in which the equation of state parameter of the scalar field interpolates between
and
, corresponds to an inflationary solution, which does not eliminate the topological defect of the cosmic string. This behavior is due to the linear coupling between the scalar field and the Gauss–Bonnet term.
For the linear coupling function with a phantom scalar field, we have found the following equilibrium points for the system (
21) and (
22) for
, as it was summarized in
Table 2.
As for the quintessence field, the stability conditions and the physical interpretation of (with and q indeterminate), and (which have , and ) which are related to cosmic strings, and the Sitter solutions ( and ) are the same as for quintessence.
Because of the phantom’s negative kinetic energy, we obtain new points compared with the quintessence case. For example is sink for , a saddle for , or nonhyperbolic for .
Moreover, is a source for , a saddle for , or nonhyperbolic for .
Furthermore, the equilibrium point is source for , a saddle for , or nonhyperbolic for .
Finally, is a sink for , a saddle for , or nonhyperbolic for . The four solutions are de Sitter solutions with and which can be late-time, early-time solutions, or intermediate stages in the evolution.
To present one possible evolution of the physical model,
Figure 5 depicts
,
, and
evaluated at the solution of system (
21) and (
22) for
and initial conditions near the saddle point
. The solution is past asymptotic to
, then remains near the de Sitter point
approaching a quintessence solution
, and then tending asymptotically to
(de Sitter point
) from above. The past attractor is dominated by the Gauss–Bonnet term, and then the saddle point
corresponds to an inflationary solution, eliminating the topological defect of the cosmic string. The late-time attractor is a de Sitter solution. Therefore, this solution connects inflation with late-time acceleration. This behavior is due to the linear coupling between the scalar field and the Gauss–Bonnet term.
Because there is non trivial dynamics as
, we have introduced a compactified variable
. The equilibrium points of system (
27) and (
28) at the finite region are the same as of (
21) and (
22) by the re-scaling accordingly; whereas, the equilibrium points at infinity, which are those satisfying
, are summarized in
Table 3. They are the following.
and are saddle for , nonhyperbolic for . They corresponds to cosmic strings with , .
The de Sitter solutions with infinity x are , which are sink for , source for , or nonhyperbolic for , and , which are source for , sink for , nonhyperbolic for .
As per the evolution of the observables regards, we have produced
Figure 8 and
Figure 9, which retain the information of
Figure 2 and
Figure 5, respectively, as
.
For the exponential coupling function with a quintessence scalar field, we have found the following equilibrium points in the coordinates
of system (
35)–(
37) for
. Say, we obtain the following equilibrium points at the finite region. The stability analysis of the equilibrium point of the system is summarized in
Table 4.
The line of equilibrium points that is nonhyperbolic with and q indeterminate.
The equilibrium point is a source with and .
The equilibrium point is sink and .
The equilibrium point is source for , a saddle for or , or nonhyperbolic for or . The cosmological observables are , . It is a stiff-matter solution.
The equilibrium point is sink for , a saddle for or , or nonhyperbolic for or . The cosmological observables are , . It is a stiff-matter solution.
The equilibrium point . It is a source for , a saddle for or , or nonhyperbolic for or . The cosmological observables are , . It is a stiff-matter solution.
The equilibrium point . It is a sink for , a saddle for or , or nonhyperbolic for or .
The equilibrium point is nonhyperbolic for , or a saddle for , or , or , or .
The equilibrium point is nonhyperbolic for , or a saddle for , or , or , or . For , the cosmological observables are and , from where we infer that acceleration occurs for .
The equilibrium point
, with
x given by Equation (
41)
and
is source or saddle, according to
Figure 10, because
and
are decelerating solutions.
The equilibrium point
, with
x given by Equation (
42)
and
is saddle, as shown in
Figure 12, because
and
are decelerating solutions.
Finally, the equilibrium point
, with
x given by Equation (
43)
and
is a sink or a saddle, as shown in
Figure 14, because
and
correspond to a solution with superluminal behavior.
To present one possible evolution of the physical model,
Figure 19 depicts
evaluated at the solution of system (
35)–(
37) for
and initial conditions
. The solution is past asymptotic to
(zero acceleration, cosmic string fluid), then crosses the line
(superluminal evolution) twice, remaining near the de Sitter point
(de Sitter point, inflation) from above, following an era where
(domain wall), before an accelerated de Sitter solution
(late-time acceleration). The past attractor is dominated by the Gauss–Bonnet term. Then, the saddle point corresponds to an inflationary solution with
, eliminating the topological defect of the cosmic string. At the latter stage, the solution has
corresponding to a domain wall. The late-time attractor is a de Sitter solution that allows the latter cosmological defect to exit. Therefore, this solution connects inflation with late-time acceleration. This behavior is due to the exponential coupling between the scalar field and the Gauss–Bonnet term.
For the exponential coupling function with a phantom scalar field, we have found the following equilibrium points of system (
35)–(
37) for
summarized in
Table 5. They are the following.
The line of equilibrium points is nonhyperbolic with EoS and deceleration parameters indeterminate.
The equilibrium point is a source with and , and is a sink with and . They have the same behavior as the analogous point in the quintessence case.
Because of the phantom’s negative kinetic energy, we obtain new points compared with the quintessence case. For example is nonhyperbolic for or , or a sink for , or , or a saddle for , or .
The equilibrium point is nonhyperbolic for or , a source for , or . It is a saddle for , or . The cosmological observables for are and ; they are always phantom accelerated solutions.
The equilibrium point
with
x given by Equation (
43),
and
is a source or saddle, according to
Figure 20. The cosmological observables are
and
, see
Figure 21.
The equilibrium point
with
x given by Equation (
44),
and
is a sink or a saddle, see
Figure 22. The cosmological observables are
and
, see
Figure 23.
The equilibrium point
with
x given by Equation (
45),
and
is a saddle, see
Figure 24. The cosmological observables are
and
, see
Figure 25.
The equilibrium point
with
x given by Equation (
46),
and
is a source, see
Figure 26. The cosmological observables are
and
, see
Figure 27.
The equilibrium point
with
x given by Equation (
47),
and
is a sink or a saddle, see
Figure 28. The cosmological observables are
and
, see
Figure 29.
The equilibrium point
with
x given by Equation (
48),
and
is a saddle, see
Figure 30. The cosmological observables are
and
, see
Figure 31.
To present one possible evolution of the physical model,
Figure 34 depicts
evaluated at the solution of system (
35)–(
37) for
and initial conditions
. The solution is past asymptotic to
and future asymptotic to a phantom solution with
(late-time acceleration). The late-time attractor is a phantom solution that allows the latter cosmological defect to exit. This behavior is due to the exponential coupling between the scalar field and the Gauss–Bonnet term.
Because there are non-trivial dynamics at infinity, we have introduced the Poincaré compactification variables along with the definition of
as given by (
49) which leads to the system (
50)–(
52). The equilibrium points are the following.
. It is nonhyperbolic. The cosmological observables are , and .
The equilibrium point
is a saddle. The EoS parameter and the deceleration parameter are represented in
Figure 35.
The equilibrium point
is a saddle. The EoS parameter and the deceleration parameter are represented in
Figure 36.
The equilibrium point
is a saddle. The EoS parameter and the deceleration parameter are represented in
Figure 37.
The equilibrium point
is a saddle. The EoS parameter and the deceleration parameter are represented in
Figure 38.
The equilibrium point is a saddle.
The equilibrium point is a saddle.
The equilibrium point is a source.
The equilibrium point is a sink.
The equilibrium point is a saddle.
The equilibrium point is a saddle.
The equilibrium point is nonhyperbolic. The cosmological observables of to are and . They correspond to cosmic string solutions.
As in the quintessence case, the equilibrium points from to and to are the same. Their stability conditions and cosmological interpretations are identical to the analogous quintessence points.
Because of the phantom’s negative kinetic energy, we obtain new points compared with the quintessence case. For example , where , , , , , . By analyzing numerically and q, we conclude that to cannot correspond to the current accelerated universe since and as .
This study extends and completes previous results in the literature in Einstein–Gauss–Bonnet scalar field cosmology [
71,
72]. The analysis indicates that the theory can explain the main eras of cosmological history. We plan to extend the further analysis in future work by introducing matter source components and new functional forms for the scalar field potential and the coupling function.