From Table 1 we can see that the critical point A1 is dominated by the kinetic energy of the scalar field (), with corresponding to ”stiff” matter, and
is unstable critical point that could describe early time dominance of the scalar field.
The critical point A2 which is dominated by the kinetic coupling of the scalar field is unstable and gives an effective EoS that mimics dust-like matter.
The fixed point A3 is dominated by the scalar field and is a de Sitter solution with . The negative sign of indicates phantom behavior and the eigenvalues indicate that at least the point is saddle. The three zero eigenvalues make difficult to analyze the stability, but since the rest of the eigenvalues are negative, the point is saddle.
This solution could correspond to an unstable inflationary phase which evolves towards a matter or dark energy dominated phase.
The point A4 is controlled by the non-minimal coupling and gives a solution that leads to an equation of state corresponding to radiation . At this critical point the potential and the kinetic coupling are absent and is a saddle point, depending on the values of the parameters and . Thus for instance, if , , and all the eigenvalues except one are negative. For background radiation () or dust matter () three of the eigenvalues might take negative values. In the case of background matter given by radiation, this critical point presents a scaling behavior. At this point, despite the presence of the background matter in form of radiation or dust, the universe becomes radiation dominated, but due to the saddle character, this point could describe a transient phase of radiation dominated universe.
The critical point A5 is dominated by the potential and the non-minimal coupling with
and . The effective EoS describes different regimes depending on the parameters associated with the non-minimal coupling, the potential and the kinetic coupling. Note that for the scalar field dominated universe the effective EoS and the dark energy EoS take the same value. From (3.9) follows that in the case we obtain the de Sitter solution with , with eigenvalues given by
This solution is a stable fixed point for any type of matter with , whenever and or and . The quadratic potential and the standard non-minimal coupling, corresponding respectively to () and (), lead to de Sitter solution, but in this case the eigenvalues are and the solution is marginally stable since four eigenvalues are negative (whenever ). The Higgs-type potential, , corresponding to with non-minimal coupling (), leads to de Sitter stable solution whenever . The cubic non-minimal coupling, , and cubic potential , also give stable de Sitter solution with eigenvalues , for any . The de Sitter solution can also be obtained for with the eigenvalues , which contain three zeros, making difficult the stability analysis by the centre manifold method.
We can also consider values for the effective EoS in the region of quintessence (), or in the phantom region () , which are consistent with observations for in the interval . The conditions for the existence of stable quintessence fixed point are , and or , and . Thus, , give a stable critical point with eigenvalues and . The conditions for the existence of stable phantom solutions are , and or , and . The parameters give a stable phantom solution with and eigenvalues .
The coordinates of this fixed point give the behavior of the physical quantities related to the model. From defined in (2.8) and the solution (3.6) it is found
The last solutions leads to the known Big Rip singularity characteristic of the phantom power-law expansion. To find the scalar field we use the dynamical variables and defined in (2.8) and (2.18) taking into account their values at A5
which gives after integration gives
with these solutions, the asymptotic value at () is obtained by using (), and from (3.13) ()
The inequalities and or and lead to
Thus, the conditions for stable quintessence solution satisfy this limit. And the conditions, and or and , lead to the limits
These conditions are satisfied by stable phantom solutions.
From the expression for (assuming )
for stable quintessence solutions takes place in the two cases: and or and . In the first case , and in the second case . For stable phantom solutions, the limit
takes place in two cases: and , where according to Eq. (3.13), , or and , where .
Concerning the effective Newtonian coupling, as follows from the definition
we see that the restrictions on quintessence solutions lead to vanishing effective Newtonian coupling at , which also takes place for the phantom solutions, where at , , indicating that the gravity reaches an asymptotic freedom regime (see ) as the universe evolves towards the Big Rip singularity.
According to the EoS (3.9), the de Sitter solution takes place for , where the Hubble parameter becomes constant and the universe expands exponentially
The scalar field can be found from the relation at the fixed point
where we have replaced . Integrating this equation gives
Taking into account that the de Sitter solution is stable in the cases () and (), then the scalar field takes the asymptotic values
To find the constant Hubble parameter in this case, we consider the critical value of the -coordinate given by and the definition of the variable given by the Eq. (2.8)
thus, according to this equation, the critical value of the -coordinate (i.e. ) can be reached at , independently of the parameter since the power cancels with the denominator in the expression for the scalar field (3.23) . Thus, we find the Hubble parameter as
Since we assume that the potential is positive (i.e. ), then this solution exists whenever .
These results give us the behavior of from
using (3.21) and (3.23) for and we can see that
hence we find that for , when , we have , and for the case
, when , then , which are consistent with the solution (3.25). The coordinate from (3.17) satisfies the limit , for any as follows form the expression for the scalar field (3.23). From the expression (3.20) for we can conclude that when the fixed point becomes a de Sitter solution, the gravitational interaction reaches the asymptotic freedom regime, i.e. at .
As seen from Table 1, the critical point A6 is dominated by the non-minimal and kinetic couplings, and from the expression for the effective EoS follows that the de Sitter solution takes place for . The expressions for the eigenvalues are too large to be displayed, and therefore we limit ourselves to the specific case of de Sitter solution, where we presented the real part of the eigenvalues. As follows from the eigenvalues for the point A6, the first eigenvalue prevents the stability of this point.
From Table 1 for the point A6 it can be seen that can not provide values in the interval , and takes only values in the interval , which are interesting for early time cosmology where the behavior includes scaling solutions. These values take place for and . So, the critical point A6 can not describe solutions with accelerated expansion.
Analyzing the stability in the relevant case , and taking into account the above conditions for , it is found that the scaling solution with is stable in the case , and , and the scaling solution with is unstable.
The critical point A7 is also dominated by the non-minimal and kinetic couplings, and presents the same characteristics and eigenvalues as the point A6, leading to the same cosmological solutions.
To the fixed point A8 the matter and the non-minimal coupling contribute giving with and . The positivity of the density parameters and impose the restriction , which excludes the pressureless dust matter. If the background matter consists of radiation (), then the universe becomes radiation-dominated with and , and the scaling solution mimics the radiation. At this saddle point with eigenvalues the system reaches the conformal invariance (given ) and can be considered as a transient phase of radiation dominated universe. In Fig. 1 we show the behavior of some trajectories around the critical point A5, corresponding to de Sitter solution, for , .
Fig. 1. The projection of the phase portrait of the model on the -plane for and , assuming . The attractor character of the de Sitter solution for the point A5 on the -plane is shown. The graphic shows trajectories evolving from the saddle points A4 and A8 to the attractor A5.
The de Sitter solution shown in Fig. 1 corresponds to the standard non-minimal coupling and the quadratic potential (, ), assuming . The trajectories that converge to the de Sitter point A5, evolve from the points A4 (saddle point, which attracts from the -direction, corresponding to radiation dominated universe with ) and A8 (which is not physical since in this case ).
There are two more critical points, namely
which are not of cosmological interest, since the density parameters fall out of the physical range.
2. Exponential function for couplings and potential
Here we impose the restrictions on the on the couplings and potential by redefining the constant parameters , and as
with the new dynamical variable defined as
Integrating the equations (3.28) with respect to the scalar field, one finds
where , and are real numbers. The only equation of the autonomous system (2.13), (2.16) and (2.19)-(2.22) that changes is the one related with the variable which reduces to
The critical points of the system are displayed in Table 3, with the respective eigenvalues given in Table 4.