#
De Sitter Solutions in Einstein–Gauss–Bonnet Gravity^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Models the Gauss–Bonnet Term

## 3. Stability of de Sitter Solutions

## 4. Different Models with the Same Structure of de Sitter Solutions

## 5. Examples of $\mathcal{L}(\mathbf{R},\mathcal{G})$ Models

#### 5.1. Evolution Equations

#### 5.2. The Function F in a Role of the Effective Potential

#### 5.3. The Case of a Power Function $\mathcal{L}$ of the Gauss–Bonnet Term

- 1.
- At $\alpha =2$, we get de Sitter solution

- 2.
- At $\alpha =3$ we get$${\varphi}_{dS}=-{\left(\frac{3{U}_{0}^{2}}{8{C}^{2}}\right)}^{1/5}.$$

- 3.
- The case of $\alpha =1/3$. Similar models have been proposed in [31]. For$${\varphi}_{dS}=-\frac{512{C}^{6}}{19683{U}_{0}^{{6}^{\prime}}}$$$$V\left({\varphi}_{dS}\right)=\frac{16{C}^{3}}{81{U}_{0}^{2}}\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{V}_{eff}^{\u2033}\left({\varphi}_{dS}\right)=-\frac{3486784401{U}_{0}^{16}}{4194304{C}^{15}}.$$

#### 5.4. The Case of a Quadratic Polynomial $\mathcal{L}$

#### 5.4.1. Equation for ${\varphi}_{dS}$

#### 5.4.2. The Case $\mathcal{L}\left(R\right)$ Gravity

#### 5.4.3. The Case $\mathcal{L}\left(\mathcal{G}\right)$ Gravity

#### 5.4.4. The Case of the Absence of the Cosmological Constant

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The function $J\left(\varphi \right)$ for different values of parameters: ${B}_{2}=1$, ${B}_{1}=-2$ (blue solid curve) and ${B}_{1}=1$ (green dash-dot curve) (

**left**); ${B}_{2}=-1$, ${B}_{1}=-1/2$ (blue solid curve) and ${B}_{1}=2$ (green dash-dot curve) (

**right**).

**Figure 2.**The functions $J\left(\varphi \right)$ and $U\left(\varphi \right)$ for ${B}_{2}=1$ (

**left**) and ${B}_{1}=-1$ (

**right**). Red solid curves show the function $J\left(\varphi \right)$. Black dash curves show the function $U\left(\varphi \right)$ at ${Q}_{0}=-1$ and grey dash-dot curves show the function $U\left(\varphi \right)$ at ${Q}_{0}=1$.

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Vernov, S.; Pozdeeva, E.
De Sitter Solutions in Einstein–Gauss–Bonnet Gravity. *Universe* **2021**, *7*, 149.
https://doi.org/10.3390/universe7050149

**AMA Style**

Vernov S, Pozdeeva E.
De Sitter Solutions in Einstein–Gauss–Bonnet Gravity. *Universe*. 2021; 7(5):149.
https://doi.org/10.3390/universe7050149

**Chicago/Turabian Style**

Vernov, Sergey, and Ekaterina Pozdeeva.
2021. "De Sitter Solutions in Einstein–Gauss–Bonnet Gravity" *Universe* 7, no. 5: 149.
https://doi.org/10.3390/universe7050149