Stability analysis of de Sitter solutions in models with the Gauss-Bonnet term

TL;DR

This paper analyzes the stability of de Sitter solutions in scalar field models with Gauss-Bonnet coupling, deriving conditions for stability and exploring implications for cosmological evolution.

Contribution

It provides a general framework for stability analysis of de Sitter solutions in models with nonminimal scalar-Gauss-Bonnet couplings, including specific conditions and potential cosmological applications.

Findings

Derived effective potential and stability conditions for de Sitter solutions.

Identified specific couplings that allow stable de Sitter configurations.

Discussed implications for early Universe and late-time cosmology.

Abstract

We investigate the scalar field dynamics of models with nonminimally coupled scalar fields in the presence of the Gauss-Bonnet term and derive the structure of the effective potential and conditions for stable de Sitter solutions in general. Specializing to specific couplings, we explore the possibility of realizing the stable de Sitter configurations which may have implications for both the early Universe and late time evolution.

Equations

$$S=\int d^{4}x\sqrt{-g}\left[UR-\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -V-F\mathcal{G}\right],$$

$$\mathcal{G}=R^2-4R_{\mu \nu }{R}^{\mu \nu }+R_{\mu \nu \alpha \beta }{R}^{\mu \nu \alpha \beta }.$$

$$d{s}^2=-d{t}^2+a^2(t)\left(d{x}_1^2+d{x}_2^2+d{x}_3^2\right).$$

$$6H^{2}U+6H{U}^{\prime }\dot{\varphi }=\frac{1}{2}{\dot{\varphi }}^2+V+24H^3{F}^{\prime }\dot{\varphi },$$

$$4\left(U-4H\dot{F}\right)\dot{H}=-{\dot{\varphi }}^2-2\ddot{U}+2H\dot{U}+8H^2\left(\ddot{F}-H\dot{F}\right),$$

$$\ddot{\varphi }+3H\dot{\varphi }-6\left(\dot{H}+2H^2\right)U^{\prime }+V^{\prime }+24H^2{F}^{\prime }\left(\dot{H}+H^2\right)=0,$$

$$\begin{array}{rl}\dot{\varphi }=& \psi ,\\ \dot{\psi }=& \frac{1}{2\left(B-4F^{\prime }H\psi \right)}\{2H\left[3B+4F^{\prime }{V}^{\prime }-6{U^{\prime }}^2-6U\right]\psi \\ -& 2\frac{V^2}{U}X+[12\left[\left(2U^{\prime \prime }+3\right)F^{\prime }+2U^{\prime }{F}^{\prime \prime }\right]H^2\\ -& 96F^{\prime }{F}^{\prime \prime }{H}^4-3\left(2U^{\prime \prime }+1\right)U^{\prime }]{\psi }^2\},\\ \dot{H}=& \frac{1}{4\left(B-4F^{\prime }H\psi \right)}\{8\left(U^{\prime }-4F^{\prime }{H}^2\right)H\psi \\ -& 2\frac{V^2}{U^2}\left(4F^{\prime }{H}^2-U^{\prime }\right)X+\left(8F^{\prime \prime }{H}^2-2U^{\prime \prime }-1\right){\psi }^2\},\end{array}$$

$$B=3{\left(4H^2{F}^{\prime }-U^{\prime }\right)}^2+U,$$

$$X=\frac{U^2}{V^2}\left[24F^{\prime }{H}^4-12U^{\prime }{H}^2+V^{\prime }\right].$$

$$6H_{dS}^2{U}_{dS}=V_{dS},$$

$$12H_{dS}^2{U}_{dS}^{\prime }=V_{dS}^{\prime }+24H_{dS}^4{F}_{dS}^{\prime },$$

$$H_{dS}^2=\frac{V_{dS}}{6U_{dS}}.$$

$$F_{dS}^{\prime }=\frac{3U_{dS}(2U_{dS}^{\prime }{V}_{dS}-V_{dS}^{\prime }{U}_{dS})}{2V_{dS}^2}.$$

$${\dot{\varphi }}_{dS}$$

$${\dot{\psi }}_{dS}$$

$${\dot{H}}_{dS}$$

$$V_{eff}^{\prime }({\varphi }_{dS})=0.$$

$$V_{eff}=-\frac{U^2}{V}+\frac{2}{3}F.$$

$$X({\varphi }_{dS})=\frac{2}{3}{F}_{dS}^{\prime }-2\frac{U_{dS}^{\prime }{U}_{dS}}{V_{dS}}+\frac{V_{dS}^{\prime }{U}_{dS}^2}{V_{dS}^2}=V_{eff}^{\prime }({\varphi }_{dS}),$$

$${\tilde{V}}_{eff}=-\frac{1}{V_{eff}}=\frac{3V}{3U^2-2FV},$$

$$H(t)$$

$$\varphi (t)$$

$$\psi (t)$$

$${\dot{\varphi }}_1$$

$${\dot{\psi }}_1$$

$${\dot{H}}_1$$

$$A=\begin{array}[]{||ccc||}0,&1,&0\\ &&\\ {}-\frac{V^{2}}{UB}X^{\prime}_{,\phi}\,,&H_{dS}\left(1-4\frac{U}{B}\right),&{}-\frac{V^{2}}{UB}X^{\prime}_{,H}\\ &&\\ {}\frac{VX^{\prime}_{,\phi}}{2BU^{2}}\left(V^{\prime}U-U^{\prime}V\right),&\frac{2H_{dS}}{BV}\left(V^{\prime}U-U^{\prime}V\right),&\frac{VX^{\prime}_{,H}}{2BU^{2}}\left(V^{\prime}U-U^{\prime}V\right)\end{array}$$

$$H_1=\frac{V_{dS}^{\prime }{U}_{dS}-U_{dS}^{\prime }{V}_{dS}}{2U_{dS}{V}_{dS}}\left(H_{dS}{\varphi }_1-{\psi }_1\right).$$

$${\dot{\varphi }}_1$$

$${\dot{\psi }}_1$$

$$\tilde{A}=\begin{array}[]{||cc||}0,&1\\ {}-{\frac{{V}^{2}{X^{\prime}_{,\phi}}}{UB}}-{\frac{V{X^{\prime}_{,H}}\,\left({V^{\prime}}\,U-{U^{\prime}}\,V\right)H_{dS}}{2{U}^{2}B}},&H_{dS}\left(1-4\frac{U}{B}\right)+{\frac{V{X^{\prime}_{,H}}\,\left(V^{\prime}\,U-U^{\prime}\,V\right)}{2{U}^{2}B}}\end{array}\,.$$

$${\varphi }_1=c_{11}{\mathrm{e}}^{-{\lambda }_{-}t}+c_{21}{\mathrm{e}}^{-{\lambda }_{+}t},$$

$${\psi }_1=c_{21}{\mathrm{e}}^{-{\lambda }_{-}t}+c_{22}{\mathrm{e}}^{-{\lambda }_{+}t},$$

$$\det (\tilde{A}-\lambda \cdot I)=0,$$

Full text

I Introduction

Among curvature corrections to the Einstein–Hilbert action, the Gauss–Bonnet term is of a special relevance. The Gauss–Bonnet term is a topological invariant quantity in four dimensions but may have dynamical relevance if is coupled to an evolving scalar field. The Gauss–Bonnet term coupled to a scalar field arises naturally in the string theory String and uses in string inspired cosmological models Antoniadis:1993jc ; Torii:1996yi ; Kawai1998 ; Hwang:2005hb ; Tsujikawa:2006ph ; ss , because has interesting cosmological implications both for early Universe vandeBruck:2015gjd ; vandeBruck:2016xvt ; Wu:2017joj ; Nozari:2017rta ; Guo:2010jr ; Oikonomou:2017ppp ; Odintsov:2018zhw ; Fomin:2019yls and for the late time dynamics Nojiri:2005vv ; Sami:2005zc ; Cognola:2006eg ; Cognola:2006sp ; Elizalde:2010jx ; Benetti:2018zhv 111Nonlocal cosmological models with the Gauss–Bonnet term are actively developed as well Capozziello:2008gu ; Cognola:2009jx ; Koshelev:2013lfm ; Buoninfante:2018xiw ; Elizalde:2018qbm ; Tian:2019bla ..

In the case of the scalar field having to account for late time cosmic evolution, it is more than desirable to ensure that it does not disturb the thermal history and that the late time dynamics is free from initial conditions of the field. In this case, it is mandatory that dynamics exhibits scaling behavior where the scalar field, in a sense, spends most of its time. In that case, the thermal history remains intact. However, a scaling solution, which is an attractor of dynamics, mimics background matter, and one, therefore, needs to devise an exit mechanism from this regime to acceleration. In the case of the Einstein–Hilbert action, the steep exponential potential with the exponential form of the coupling function gives rise to an exit from the scaling regime to acceleration Tsujikawa:2006ph ; ss . Indeed, the Gauss–Bonnet coupling induces a minimum in the runaway potential where the field can settle such that the de Sitter is a late time attractor of the dynamics Tsujikawa:2006ph ; ss . Let us note that there exist other mechanisms capable of executing the said transition. For instance, coupling with massive neutrino matter to a scalar field proportional to a trace of the energy-momentum tensor of neutrino matter can facilitate an exit from scaling regime. In this case, coupling dynamically builds up only at late stages, resulting in the minimum of the effective potential sss . In general, the addition of the Gauss–Bonnet term modifies the field potential and may induce interesting features in it Tsujikawa:2006ph ; ss .

In order to find out the de Sitter solutions in a model with a minimally coupled scalar field with a potential $V$, it is enough to find zeros of the first derivative of $V$. The sign of the second derivative of the potential $V$ at a de Sitter point determines the stability of the solution. Similar analysis of de Sitter solutions is possible in the case of nonminimal coupling if one introduces an effective potential such that zeroes of its first derivative correspond to de Sitter points whereas the sign of its second derivative determines their stability Skugoreva:2014gka ; Pozdeeva:2016cja .

In this paper, we analyze models with the Gauss–Bonnet term in case of a nonminimally coupled scalar field. We look for the general structure of the corresponding effective potential and investigate the specific cases. Using the framework of the effective potential, our goal is to find out de Sitter solutions in these models and compare them with their counterparts in the corresponding models without the Gauss–Bonnet term.

The paper is organized as follows. In Section II, we remind evolution equations for models with a nonminimally coupled scalar field and the Gauss–Bonnet term. In Section III, we propose the effective potential as a useful tool for the search and stability analysis of de Sitter solutions. In Section IV, we present a few specific examples of models with de Sitter solutions. Section V is devoted to our conclusions.

II Evolution equations for models with the Gauss–Bonnet term

In this paper, we investigate the model with the Gauss–Bonnet term in a general background described by the following action:

[TABLE]

where the functions $U(\phi)$, $V(\phi)$, and $F(\phi)$ are differentiable ones, and $\mathcal{G}$ is the Gauss–Bonnet term,

[TABLE]

In what follows, we specialize to the spatially flat Friedmann–Lemaître–Robertson–Walker universe,

[TABLE]

Variations of action (1) with respect to $g_{\mu\nu}$ and $\phi$ lead to the following evolution equations vandeBruck:2015gjd :

[TABLE]

[TABLE]

[TABLE]

where $H=\dot{a}/a$ is the Hubble parameter, dots and primes denote the derivatives with respect to the cosmic time and $\phi$, respectively.

It would be convenient to cast Eqs. (4) and (5) as a dynamical system,

[TABLE]

where

[TABLE]

[TABLE]

Dynamical system (6) is suitable for the search for de Sitter solutions, which we shall undertake in the following section.

III The effective potential and de Sitter solutions

III.1 The search for de Sitter solutions

Our first goal is to find de Sitter solutions in the model described by action (1) and to compare them with de Sitter solutions in the corresponding model without the Gauss–Bonnet term. In principle, solutions with a constant Hubble parameter $H$ and the field $\phi$ depending on time can exist, but these solutions correspond to a varying effective gravitational constant, which is proportional to $1/U(\phi)$, and have properties different from the standard de Sitter solutions. For this reason, we restrict ourselves to de Sitter solutions with a constant $\phi$. Substituting $\phi=\phi_{dS}$ and $H=H_{dS}$ into Eqs. (3) and (5), we have

[TABLE]

[TABLE]

where $V_{dS}=V(\phi_{dS})$, $U_{dS}=U(\phi_{dS})$, and $F_{dS}=F(\phi_{dS})$. We see that the value of the Hubble parameter at the de Sitter point is the same as in the corresponding model without the Gauss–Bonnet term,

[TABLE]

The value of $F^{\prime}(\phi_{dS})$ is fixed by Eq. (10):

[TABLE]

Therefore, we come to the conclusion that for arbitrary functions $U$ and $V$ with $VU>0$, we can choose $F(\phi)$ such that the corresponding point becomes a de Sitter solution, with the Hubble parameter defined by Eq. (11).

Let us note that at a de Sitter point, the system of equations (6) has the following form:

[TABLE]

It would be convenient if all the necessary information on the existence and stability of de Sitter solutions is obtained from a single combination of functions $U$, $V$, and $F$ dubbed effective potential $V_{eff}$. The de Sitter solutions would correspond to zeros of the first derivative of $V_{eff}$, and the stability of the solutions would correspond to its second derivative being positive.

Let us emphasize that we require only these two conditions for the definition of the effective potential: this is a function $V_{eff}(\phi)$ such that $V_{eff}^{\prime}=0$ at de Sitter point $\phi_{dS}$, and this de Sitter solution is stable if the second derivative of the effective potential with respect to $\phi$ is positive for $\phi=\phi_{dS}$. Such a definition is a direct generalization of a scalar field potential $V(\phi)$ in the theory with a minimally coupled scalar field, since, in this theory, the de Sitter solution requires $V^{\prime}=0$ and is stable in the point where the scalar field potential reaches its local minimum. We do not require that any other properties of the physical system considered should be expressed through properties of the effective potential.

In the present paper, we show that while the general situation is more complicated, we achieve the above-mentioned goal in the case of $U>0$.

De Sitter solutions in the model with nonminimal coupling and without the Gauss–Bonnet term have been considered in Skugoreva:2014gka , where the effective potential has been introduced. In the model with the Gauss–Bonnet term, we can also introduce such $V_{eff}(\phi)$ that

[TABLE]

Indeed, let

[TABLE]

Using Eq. (11), we get

[TABLE]

therefore, Eq. (16) is satisfied, and de Sitter solutions correspond to extremum points of the effective potential $V_{eff}$.

It should be noted that the first term in Eq. (17) can be expressed through the effective potential without the Gauss–Bonnet term $\tilde{V}_{eff}$ introduced in Skugoreva:2014gka as $-\tilde{V}_{eff}^{-1}$. A disadvantage of the form (17) for the effective potential is that it diverges at $V=0$. However, in such a case we have a Minkowski solution instead of de Sitter one, so this feature does not spoil the analysis of de Sitter solutions.

Note that the effective potential $V_{eff}$ (17) is not a unique function suitable to define the values of $\phi_{dS}$ and the stability of de Sitter solutions. For example, one can use

[TABLE]

as an effective potential. It is easy to see that $\tilde{V}_{eff}^{\prime}(\phi_{dS})=0$, and the sign of $\tilde{V}_{eff}^{\prime\prime}(\phi_{dS})$ coincides with the sign of $V_{eff}^{\prime\prime}(\phi_{dS})$. In the case of the absence of the Gauss–Bonnet term, $F=0$, $\tilde{V}_{eff}$ is the effective potential, introduced in Ref. Skugoreva:2014gka . Moreover, the function $\tilde{V}_{eff}$ is proportional to $V(\phi)$ in the case of $F=0$ and a constant $U$.

III.2 The stability analysis

To investigate the Lyapunov stability of a de Sitter solution we use the following expansions:

[TABLE]

where $\delta$ is a small parameter. From (6), in the first order in $\delta$ we get the following linear system

[TABLE]

where

[TABLE]

and all functions are taken at $\phi=\phi_{dS}$.

Since $\det(A)=0$, functions $H_{1}(t)$, $\phi_{1}(t)$ and $\psi_{1}(t)$ are not independent. From Eq. (3), we obtain

[TABLE]

Substituting (23) into (20) and (21), we get:

[TABLE]

where

[TABLE]

Solutions of Eqs. (24) and (25) have the following form:

[TABLE]

[TABLE]

where $c_{ij}$ are some constants whose values are not important for the stability analysis. The numbers $\lambda_{\pm}$ are roots of the determinant of the characteristic matrix. Solving the equation

[TABLE]

we get:

[TABLE]

where

[TABLE]

and

[TABLE]

To get this result, we assume that $H_{dS}>0$, because only such solutions correspond to the observable Universe, and put $H_{dS}=\sqrt{\frac{V}{6U}}$.

A de Sitter solution is stable if real parts of both $\lambda_{-}$ and $\lambda_{+}$ are negative. In the case of a positive $U$ and, therefore, a positive $V$, we see that $Z<0$ and $B>0$. This means that the condition $Z/B<0$ is satisfied at any de Sitter point, and, therefore, $\Re e(\lambda_{-})<0$.

At $U>0$, the condition $\Re e(\lambda_{+})<0$ is equivalent to $Y<0$ and, hence, $V^{\prime\prime}_{eff}(\phi_{dS})>0$. Thus, we finally reach a conclusion that for any $U(\phi_{dS})>0$, a de Sitter solution is stable if $V^{\prime\prime}_{eff}(\phi_{dS})>0$ and unstable if $V^{\prime\prime}_{eff}(\phi_{dS})<0$.

In the next section, we consider several examples of models and explore the existence and stability of de Sitter solutions.

IV Examples

IV.1 Models with exponential potential

The string theory inspired cosmological model with

[TABLE]

where $U_{0}$, $\alpha$, $c$, $\tilde{\lambda}$, and $\mu$ are positive constants, has been considered in Tsujikawa:2006ph . In this model, the effective potential is

[TABLE]

The condition $V_{eff}^{\prime}(\phi_{dS})=0$ gives

[TABLE]

There exists a de Sitter solution for all $\mu\neq\tilde{\lambda}$. It is easy to see that $V_{eff}^{\prime\prime}=0$ at

[TABLE]

and $\phi_{dS}>\phi_{2}$ for any $\tilde{\lambda}\neq\mu$.

If $\mu>\tilde{\lambda}$, then $V_{eff}^{\prime\prime}$ is positive at large $\phi$, so the second derivative is positive at the de Sitter point and this point is stable. In the opposite case, $\mu<\tilde{\lambda}$, $V_{eff}^{\prime\prime}<0$ at large $\phi$ and the de Sitter solution is unstable. This result coincides with the result obtained in Tsujikawa:2006ph by another method.

We generalize this result assuming that the constants can be negative. For

[TABLE]

the de Sitter point $\phi_{dS}={\frac{1}{N_{1}-N_{2}}\ln\left({}-{\frac{c_{1}\,{N_{1}}}{c_{2}\,{N_{2}}}}\right)}$ exists only if ${\frac{c_{1}\,{N_{1}}}{c_{2}\,{N_{2}}}}<0$ and $N_{2}\neq N_{1}$. If we assume functions $U$ and $V$ in the form given by (30), then $c_{1}<0$. At the same time, one and the same the effective potential corresponds to a different choice of functions $F$, $V$, and $U$. If two of these functions are given, then we can get the third function using the given form of the effective potential. It is a way of constructing models with de Sitter solutions. For example, the model with a nonminimally coupled scalar field defined by functions

[TABLE]

has the effective potential given by (34) if

[TABLE]

In this model, $c_{i}$ and $N_{i}$ are arbitrary constants. The analysis of the second derivative of $V_{eff}$ gives the following stability conditions:

• •

if $c_{1}>0$ and $c_{2}>0$, then the de Sitter solution is stable;

• •

if $c_{1}<0$ and $c_{2}<0$, then the de Sitter solution is unstable;

• •

if $c_{1}>0$ and $c_{2}<0$, then the de Sitter solution is stable at $|N_{1}|>|N_{2}|$ and unstable at $|N_{1}|<|N_{2}|$;

• •

if $c_{1}<0$ and $c_{2}>0$, then the de Sitter solution is stable at $|N_{1}|<|N_{2}|$ and unstable at $|N_{1}|>|N_{2}|$.

The effective potential can be used not only to simplify the analysis of the stability of de Sitter solutions in a given model, but also to construct a new model with de Sitter solutions.

IV.2 Models with $V=CU^{2}$

Let us consider the case $V=CU^{2}$, where $C$ is a positive constant. In this case, a model without the Gauss–Bonnet term transforms to a model with a constant potential in the Einstein frame. If the Gauss–Bonnet term is presented, then the function $F(\phi)$ plays a role of the effective potential, fully determining the position and stability of the de Sitter solutions, because

[TABLE]

So, values of $\phi_{dS}$ satisfy the condition $F^{\prime}(\phi_{dS})=0$. From Eq. (29), it follows:

[TABLE]

For $U(\phi_{dS})>0$, a de Sitter solution is unstable at $F^{\prime\prime}<0$ and stable at $F^{\prime\prime}>0$.

Note that the only difference between minimal and nonminimal coupling cases is that values of the Hubble parameter at de Sitter points

[TABLE]

can be different if $U$ is not a constant.

In what follows, we shall demonstrate the working of a de Sitter search algorithm through concrete examples.

For $F={A_{4}}{\phi}^{4}+{A_{2}}{\phi}^{2}$, de Sitter points defined by the condition $F^{\prime}=0$ are

[TABLE]

It is evident that ${\phi_{dS}}_{\pm}$ are real only if constants $A_{2}$ and $A_{4}$ have different signs. The values of the second derivative of $F$ at the de Sitter points are

[TABLE]

Thus, the de Sitter solution in points ${\phi_{dS}}_{\pm}$ is unstable for any $A_{2}>0$ and $A_{4}<0$ and is stable for any $A_{2}<0$ and $A_{4}>0$. At the point ${{\phi_{dS}}_{0}}$, the de Sitter solution is stable for any $A_{2}>0$ and unstable at $A_{2}<0$. At $A_{2}=0$, the only de Sitter point is $\phi_{dS}=0$ and we get ${\lambda}_{+}=0$ and ${\lambda}_{-}={}-\sqrt{6CU}/2$. Figure 1 illustrates these different possibilities.

A more complicated example is

[TABLE]

where $C$ and $\omega$ are constants. Pictures in Fig. 2 demonstrate that independence of values of the parameters $C$ and $\omega$ the number of de Sitter solutions changes. In the left picture, the blue and black lines correspond to models with one stable de Sitter solution, whereas the green solid line corresponds to a model with 3 stable and 2 unstable de Sitter solutions. In the right picture, the red dash-dot curve corresponds to a model with 2 stable and 1 unstable de Sitter solutions, the black dash line corresponds to models with one stable de Sitter solution, and the green solid line corresponds to a model with 4 stable and 3 unstable de Sitter solutions. Therefore, using a graphical representation of the function $\tilde{F}$, one can get the structure and stability properties of de Sitter solutions.

Thus, for an arbitrary positive $U(\phi)$, we obtain de Sitter solutions given by an arbitrary function $V_{eff}$, if we choose $V=CU^{2}$ and $F=V_{eff}$.

IV.3 Models with a massive scalar field

In the previous subsection, the de Sitter solutions appear due to properties of the coupling function $F$ solely. It is more interesting to get de Sitter solutions via an interplay between the scalar field potential $V$ and the coupling function $F$ so these two functions, being simple monomials, give rise to a de Sitter solution. Using the conception of the effective potential, it is easy to create such models. For example, let the potential be of the simplest massive form

[TABLE]

with the coupling function

[TABLE]

for the coupling function with curvature. In this situation, the effective potential is

[TABLE]

Without the Gauss–Bonnet contribution the effective potential is a monotonic function, so there are no de Sitter solutions222In the point $\phi=0$ the function $U=0$, so this extremum of $V_{eff}$ does not correspond to a de Sitter solution.. However, it is clear that the addition of any monomial $F=F_{0}\phi^{n}$ with $n>2$ and $F_{0}>0$ gives us a stable de Sitter solution. Straightforward calculation shows that

[TABLE]

and, consequently, the de Sitter solution exists if $n\neq 2$. Using the second derivative of the effective potential, we easily obtain

[TABLE]

which implies that the de Sitter solution is unstable for $n<2$. Looking at the plot of the effective potential in Fig. 3, one can clearly distinguish between cases corresponding to $n>2$ $\&$ $n<2$.

IV.4 Models with the Higgs potential

Let us consider model with

[TABLE]

where $U_{0}$, $\xi$ and $V_{0}$ are positive constants. The corresponding model without the Gauss–Bonnet term is the physically motivated inflationary model dubbed Higgs-driven inflation Higgs perfectly consistent with cosmological observations. The inflationary scenario proposed in Ref. vandeBruck:2015gjd uses the functional form of $F$ given by $F=F_{0}/\phi^{4}$. In this case, the effective potential has the following form:

[TABLE]

The de Sitter solutions correspond to real values of $\phi$ only if $F_{0}>3U_{0}^{2}/(2V_{0})$:

[TABLE]

At the de Sitter points, the second derivative of the effective potential is

[TABLE]

Thus, evidently, all de Sitter solutions of the considered model are stable. In Fig. 4, the red solid curve corresponds to a model without a de Sitter solution, whereas the blue dash curve corresponds to a model with de Sitter solutions.

In another interesting case of the function

[TABLE]

extrema of the effective potential are $\phi_{1,2}=\pm\sqrt{-\xi_{2}/(2\xi_{3})}$ and $\phi_{0}=0$. The point $\phi_{0}=0$ is a singular point of the function $F$ and we do not consider this point. Evidently, the de Sitter points $\phi_{1,2}$ are real only if the parameters $\xi_{3}$ and $\xi_{2}$ have different signs.

In the points $\phi_{1,2}$ the second derivative is $V^{\prime\prime}_{eff}={}-8\xi_{2}/3$. So, the points $\phi_{1,2}$ are unstable at $\xi_{2}>0$ and stable at $\xi_{2}<0$.

IV.5 Six degree potential

In this subsection, we consider more general functions $U$ and $V$, than in the previous subsections, and power-law functions $F$:

[TABLE]

where $U_{0}$, $V_{0}$ and $\xi$ are positive constants333A similar model has been considered in Granda:2017oku . The effective potential of the model has the following form:

[TABLE]

In the case $n>2N$, at $F_{0}=0$ the function $V_{eff}$ is a monotonically increasing one, so there is no de Sitter solution for $U>0$ that has been mentioned in Sami:2012uh .

Let us consider the case of $N=2$ and $n=6$. If we add the function $F=F_{0}/\phi^{2}$, then an unstable de Sitter can be obtained for some values of parameter (see the blue dash and cyan dash-dot curves in Fig. 5). The analysis of $V_{eff}^{\prime}$ shows that no more than two values of $\phi_{dS}$ are real, either

[TABLE]

or

[TABLE]

In order to find out the stable de Sitter solutions, we choose $F=F_{0}/\phi^{8}$ and obtain $V_{eff}$, which has minima, see the red solid curve in Fig. 5.

V Conclusion

In the present paper, we investigate a homogeneous cosmological dynamics of a nonminimally coupled scalar field both with the curvature and with the Gauss–Bonnet term. We are mainly interested in looking for the fixed points of scalar field dynamics which correspond to de Sitter solutions. We show that, in the case of a positive coupling function $U(\phi)$, it is possible to introduce an effective potential $V_{eff}$ which can be expressed through curvature $U$, the scalar field potential $V$, and the coupling function with the Gauss–Bonnet term denoted by $F$. We show that it is convenient to investigate the structure of fixed points using the effective potential, indeed, the stable de Sitter solutions correspond to minima of the effective potential. The latter implies that the existence and stability of de Sitter solutions in the system under consideration can be studied with the help of function $V_{eff}$, since the stability of de Sitter solutions is analogous to the stability of a classical mechanical system moving in the potential field $V_{eff}$. One can get the structure and stability properties of de Sitter solutions using only graphical representation of the effective potential.

In the model under consideration, the effective potential is a sum of two terms. The first one includes the contributions from $U$ and $V$ — the effective potential for models in absence of the Gauss–Bonnet term, proposed in Skugoreva:2014gka ; the second term includes the function $F$ or the contribution from the Gauss–Bonnet term. Thus, the use of the effective potential is the simplest way to compare the results on existence and stability of de Sitter solutions in models with and without the Gauss–Bonnet term. We in general derive the structure of the effective potential and conditions for existence and stability of de Sitter solutions. Using this approach, we have studied concrete models with the Gauss–Bonnet term and described a number of situations where de Sitter solutions exist due to the presence of the Gauss–Bonnet term and disappear otherwise.

Let us note that the effective potential for models without the Gauss–Bonnet term has a simple physical meaning: it is an invariant under the conformal rescaling of the metric that coincides with the potential in the Einstein frame Jarv:2015kga . It would be interesting to get a similar interpretation for the effective potential proposed in this paper.

Let us also emphasize that the knowledge of unstable de Sitter solutions can be useful to describe inflation (see, for example, Elizalde:2014xva for details), whereas stable de Sitter solutions are often used in models of late time acceleration of the Universe. We should, however, stress that the effective potential formalism in the Gauss–Bonnet cosmology works only for the study of an exact de Sitter solution. When we deviate from an exact de Sitter solution to a quasi–de Sitter one, the effective potential as a single construction from three functions, entering the action of the theory, may not be enough. We leave the detailed study of quasi–de Sitter solutions in the considered models for our future investigations.

Acknowledgement

This work is partially supported by the Indo–Russia Project; E.P., A.T., and S.V. are supported by RFBR Grant No. 18-52-45016, and M.S. is supported by Grant No. INT/RUS/RFBR/P-315 of the Department of science and technology of India government. A.T. is supported by the Russian Government Program of Competitive Growth of Kazan Federal University.