De Sitter Solutions in Models with the Gauss-Bonnet Term ^†

Abstract

De Sitter solutions play an important role in cosmology because the knowledge of unstable de Sitter solutions can be useful in describing inflation, whereas stable de Sitter solutions are often used in models of the late-time acceleration of the Universe. Einstein–Gauss–Bonnet models are actively used as both inflationary models and dark energy models. To modify the Einstein equations, one can add a nonlinear function of the Gauss–Bonnet term or a function of the scalar field multiplied on the Gauss–Bonnet term. The effective potential method essentially simplifies the search and stability analysis of de Sitter solutions, because the stable de Sitter solutions correspond to the minima of the effective potential.

1. Introduction

It is well-known that one can add the Gauss–Bonnet term to the Hilbert–Einstein Lagrangian of General Relativity, and this does not change the equations of motion. On the other hand, this term, when multiplied by some nonconstant function of a scalar field, modifies the equations of motion. Additionally, models with a non-linear function of the Gauss–Bonnet term can be rewritten in the equivalent form, which includes a scalar field without a kinetic term.

The cosmological models with the Gauss–Bonnet term are motivated by string theory [1,2,3,4,5,6,7,8] and are actively used for describing of both the early Universe evolution [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] and the current dark energy dominated epoch [5,6,7,25,26,27,28,29,30].

Note that these studies of the Universe’s evolution are characterized by the quasi de Sitter accelerated expansion of the Universe. Therefore, it is important to have an effective method to sear the de Sitter solutions and for the study of their stability. For the Gauss–Bonnet model with the standard scalar field, such a method has been proposed in [31]. This is a generalization of the effective potential method [32,33]. In this paper, we generalize this method on a model with nonlinear functions of the Gauss–Bonnet term. We also consider the case of a phantom scalar field and show that, in this case, the situation is more difficult.

2. Models the Gauss-Bonnet Term

Let us consider the model with the Gauss–Bonnet term, described by the following action:

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

(1)

where the functions $U\left(\varphi \right)$, $V\left(\varphi \right)$, and $F\left(\varphi \right)$ are double differentiable ones, c is a constant, R is the Ricci scalar and $\mathcal{G}$ is the Gauss–Bonnet term,

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

Note that the action

$$S=\int d^{4}x\sqrt{-g}W\left(\mathcal{G}\right),$$

(2)

where $W\left(\mathcal{G}\right)$ is a double differentiable function, can be rewritten in the following form [8,27]:

$$S=\int d^{4}x\sqrt{-g}\left[W^{\prime }\left(\varphi \right)(\mathcal{G}-\varphi )+W\left(\varphi \right)\right],$$

(3)

where a prime denotes the derivatives with respect to $\varphi$. Varying action (3) over $\varphi$, one gets $\varphi =\mathcal{G}$ and the initial $W\left(\mathcal{G}\right)$ model. Therefore, action (1) with $c=0$ describes $W\left(\mathcal{G}\right)$ models.

In the spatially flat Friedmann–Lemaître–Robertson–Walker metric with

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

(4)

one obtains the following evolution equations:

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

(5)

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

(6)

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

(7)

where $H=\dot{a}/a$ is the Hubble parameter; dots and primes denote the derivatives with respect to the cosmic time and the scalar field $\varphi$, respectively. At $c=1$, these equations have been investigated in many papers (see, for example, [11,31]).

To find de Sitter solutions with a constant $\varphi$ in the model (2), we substitute $\varphi ={\varphi }_{dS}$ and $H=H_{dS}$ into Equations (5) and (7). A de Sitter solution does not depend on the value of c, so we obtain the same results as in the case $c=1$ considered in [31]:

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

(8)

and

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

(9)

where $A_{dS}\equiv A\left({\varphi }_{dS}\right)$ for any function A. Therefore, for arbitrary functions $U\left(\varphi \right)$ and $V\left(\varphi \right)$ with $V_{dS}{U}_{dS}>0$, we can choose $F\left(\varphi \right)$ such that the corresponding point becomes a de Sitter solution, with the Hubble parameter defined by Equation (8). We always choose that $H_{dS}>0$.

3. Stability of de Sitter Solutions

To analyze the stability of a de Sitter solution, we transform Equations (6) and (7) into the following dynamical system:

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

(10)

where

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

(11)

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

(12)

In the case $c=0$, the last equation is essentially simplified:

$$\dot{H}=\frac{24H^4{F}^{\prime }-12H^2{U}^{\prime }+V^{\prime }}{6\left(U^{\prime }-4H^2{F}^{\prime }\right)}.$$

(13)

At a de Sitter point system (10) is

$$\dot{\varphi }=0,\dot{\psi }=0,\dot{H}=0,$$

that corresponds to $X_{dS}=0$.

In Reference [31], the effective potential has been proposed for cosmological models with the Gauss–Bonnet term:

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

(14)

Using Equation (8), we obtain

$$X_{dS}=\frac{2}{3}{F}_{dS}^{\prime }-2\frac{U_{dS}^{\prime }{U}_{dS}}{V_{dS}}+\frac{V_{dS}^{\prime }{U}_{dS}^2}{V_{dS}^2}=V_{eff}^{\prime }\left({\varphi }_{dS}\right)=0,$$

(15)

therefore, de Sitter solutions correspond to extremum points of the effective potential $V_{eff}$.

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

$$H\left(t\right)=H_{dS}+\epsilon H_1\left(t\right)\varphi \left(t\right)={\varphi }_{dS}+\epsilon {\varphi }_1\left(t\right),\psi \left(t\right)=\epsilon {\psi }_1\left(t\right),$$

(16)

where $\epsilon$ is a small parameter. Therefore,

$$X=\epsilon (X_{,H}{H}_1+X_{,\varphi }{\varphi }_1)+\mathcal{O}\left({\epsilon }^2\right),$$

(17)

where

$$X_{,H}={\left.\frac{\partial X}{\partial H}\right|}_{\varphi ={\varphi }_{dS}}=\frac{4\sqrt{6}}{V_{dS}^{5/2}}{U}_{dS}^{3/2}\left(U_{dS}^{\prime }{V}_{dS}-V_{dS}^{\prime }{U}_{dS}\right),$$

$$X_{,\varphi }={\left.\frac{\partial X}{\partial \varphi }\right|}_{\varphi ={\varphi }_{dS}}=\frac{1}{V_{dS}^2}\left(\frac{2}{3}{F}_{dS}^{″}{V}_{dS}^2-2U_{dS}^{″}{U}_{dS}{V}_{dS}+V_{dS}^{″}{U}_{dS}^2\right).$$

The functions $H_1\left(t\right)$, ${\varphi }_1\left(t\right)$, and ${\psi }_1\left(t\right)$ are connected by Equation (5):

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

(18)

This expression does not depend on the value of c and coincides with the corresponding expression obtained in Reference [31].

Substituting (16)–(18) into Equation (10) in the first order of $\epsilon$, we obtain the following system of two linear differential equations:

$${\dot{\varphi }}_1={\psi }_1,$$

(19)

$${\dot{\psi }}_1=-\frac{\left[2F_{dS}^{″}{V}_{dS}^3-6U_{dS}^{″}{U}_{dS}{V}_{dS}^2+3V_{dS}^{″}{U}_{dS}^2{V}_{dS}-6{\left(U_{dS}^{\prime }{V}_{dS}-V_{dS}^{\prime }{U}_{dS}\right)}^2\right]}{3U_{dS}{V}_{dS}{B}_{dS}}{\varphi }_1-\frac{\sqrt{6U_{dS}{V}_{dS}}}{2U_{dS}}{\psi }_1,$$

(20)

where

$$B_{dS}=\frac{3}{V_{dS}^2}{\left(V_{dS}^{\prime }{U}_{dS}-U_{dS}^{\prime }{V}_{dS}\right)}^2+c{U}_{dS}.$$

(21)

This system can be rewritten in the matrix form:

$$\left(\begin{array}{c}{\dot{\varphi }}_1\\ {\dot{\psi }}_1\end{array}\right)=\left(\begin{array}{c}{\tilde{A}}_{11}{\tilde{A}}_{21}\\ {\tilde{A}}_{12}{\tilde{A}}_{22}\end{array}\right)\left(\begin{array}{c}{\varphi }_1\\ {\psi }_1\end{array}\right)$$

(22)

where

$$\tilde{A}=\Vert \begin{array}{cc}0,& 1\\ -\frac{V_{dS}^2{V}_{eff}^{″}\left({\varphi }_{dS}\right)}{U_{dS}{B}_{dS}},& -3H_{dS}\end{array}\Vert$$

The general solution of system (22) has the following form

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

(23)

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

(24)

where $c_{ij}$ are some constants. Solving the characteristic equation:

$$det(\tilde{A}-\lambda ·I)={\lambda }^2-3H_{dS}\lambda +\frac{V_{dS}^2{V}_{eff}^{″}\left({\varphi }_{dS}\right)}{U_{dS}{B}_{dS}}=0,$$

(25)

we obtain the following roots:

$${\lambda }_{\pm }=-\frac{3}{2}{H}_{dS}\pm \sqrt{\frac{9}{4}{H}_{dS}^2-\frac{V_{dS}^2}{U_{dS}{B}_{dS}}{V}_{eff}^{″}\left({\varphi }_{dS}\right)}.$$

(26)

A de Sitter solution is stable if the real parts of both ${\lambda }_{-}$ and ${\lambda }_{+}$ are negative. We consider the case $H_{dS}=\sqrt{\frac{V}{6U}}>0$, hence, $\Re e\left({\lambda }_{-}\right)<0$.

In the case of a positive $U_{dS}$, we see that $B_{dS}>0$ for $c⩾0$ and the condition $\Re e\left({\lambda }_{+}\right)<0$ is equivalent to $V_{eff}^{″}\left({\varphi }_{dS}\right)>0$. In the cases $c>0$ and $c=0$, a de Sitter solution is stable if $V_{eff}^{″}\left({\varphi }_{dS}\right)>0$ and unstable if $V_{eff}^{″}\left({\varphi }_{dS}\right)<0$.

In the case $c<0$, we see that $B_{dS}$ can be negative. Therefore, in this case, the de Sitter solution is stable if the $V_{eff}^{″}\left({\varphi }_{dS}\right)B_{dS}>0$. Therefore, the main result of Reference [31] can be generalized on the case $c=0$ without any correction, whereas the condition should be changed to $V_{eff}^{″}\left({\varphi }_{dS}\right)B_{dS}>0$ in the case of $c<0$, which corresponds to a phantom scalar field $\varphi$.

4. Conclusions

In this paper, we consider de Sitter solutions in models with the Gauss–Bonnet term, including $W\left(\mathcal{G}\right)$ models. We show that the effective potential proposed [31] for model with the Gauss–Bonnet term, multiplied on a function of the scalar field, can be used in $W\left(\mathcal{G}\right)$ models as well. To find de Sitter solutions in a $W\left(\mathcal{G}\right)$ model, we rewrite the action of this model in the form (3) and construct the corresponding effective potential $V_{eff}$. A stable de Sitter solution corresponds to $V_{eff}^{″}\left({\varphi }_{dS}\right)>0$, where the values of the scalar field at the de Sitter point ${\varphi }_{dS}$ are determined by the condition $V_{eff}^{\prime }\left({\varphi }_{dS}\right)=0$. Examples of de Sitter solutions in $W\left(\mathcal{G}\right)$ gravity models and in more complicated models with the $\mathcal{L}(R+\mathcal{G})$ function in the action are given in [34].

Note that the effective potential provides a useful tool for the construction of inflationary scenarios in models where the Gauss–Bonnet term is multiplied to a function of the scalar field [24,35,36]. We plan to generalize this approach to inflationary scenarios in $W\left(\mathcal{G}\right)$ models.

The Supplementary Materials include the presentation on the first Electronic Conference on Universe.