# Stability of Accelerating Cosmology in Two Scalar-Tensor Theory: Little Rip versus de Sitter

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## Abstract

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**PACS**95.36.+x; 98.80.Cq

## 1. Introduction

- Type I (“Big Rip”) : For $t\to {t}_{s}$, $a\to \infty $, ${\rho}_{\text{DE}}\to \infty $ and $\left|{p}_{\text{DE}}\right|\to \infty $. This also includes the case of ${\rho}_{\text{DE}}$, ${p}_{\text{DE}}$ being finite at ${t}_{s}$.
- Type III : For $t\to {t}_{s}$, $a\to {a}_{s}$, ${\rho}_{\text{DE}}\to \infty $ and $\left|{p}_{\text{DE}}\right|\to \infty $.
- Type IV : For $t\to {t}_{s}$, $a\to {a}_{s}$, ${\rho}_{\text{DE}}\to 0$, $\left|{p}_{\text{DE}}\right|\to 0$ and higher derivatives of H diverge. This also includes the case in which ${p}_{\text{DE}}$ (${\rho}_{\text{DE}}$) or both of ${p}_{\text{DE}}$ and ${\rho}_{\text{DE}}$ tend to some finite values, whereas higher derivatives of H diverge.

## 2. Reconstruction of Scalar Model and (in)stability

#### 2.1. One Scalar Model

#### 2.2. Two Scalar Model

## 3. Reconstruction of Little Rip Cosmology

#### 3.1. A Model of Little Rip Cosmology

#### 3.2. Asymptotically de Sitter Phantom Model

#### 3.3. Asymptotically de Sitter Quintessence Dark Energy

#### 3.4. A Realistic Model Unifying Inflation with Little Rip Dark Energy Era

Models | Stability of the reconstructed solution | Existence of de Sitter solution | Stability of de Sitter solution |
---|---|---|---|

Equation (35) | stable | no | − |

Equation (51) | stable if $3>\frac{\lambda}{{H}_{0}^{\left(\text{II}\right)}}$ | yes if $12>\frac{\lambda}{{H}_{0}^{\left(\text{II}\right)}}$ | unstable |

Equation (59) | stable if $3>\frac{\lambda}{H}$ and $3>\frac{{H}_{1}^{\left(\text{III}\right)}\lambda {\mathrm{e}}^{-\lambda t}}{{H}^{2}}$ | yes if $12>\frac{\lambda}{{H}_{0}^{\left(\text{III}\right)}}$ | unstable |

Equation (68) | stable if $3>\frac{\lambda}{{H}_{0}^{\left(\text{IV}\right)}}$ | no | − |

## 4. Reconstruction in Terms of E-Foldings and Solution Flow

#### 4.1. Reconstruction of Two Scalar Model and (in)stability

#### 4.2. Fixed Points and Flow of General Solutions

**Point A**- $(X,Y,Z,W)=(1,1,1,0)$Here the solution is given by Equation (80).
**Point B**- $(X,Y,Z,W)=({\beta}_{1},{\beta}_{1},1,0)$Besides Point A $(X,Y,Z,W)=(1,1,1,0)$, there could be another solution for Equations (82–85). In order to show the existence of another solution, we now define $\beta \left(N\right)$ by$$\beta \left(N\right)\equiv {\beta}_{0}+{\beta}_{1}N\phantom{\rule{0.166667em}{0ex}}$$Here ${\beta}_{0}$ and ${\beta}_{1}$ are dimensionless constants. We now assume that the following equation could be satisfied,$$\frac{{\omega}_{,\varphi}(\beta \left(N\right)/\kappa )}{\kappa \phantom{\rule{0.166667em}{0ex}}\omega \left(\beta \right(N)/\kappa )}=\frac{{\eta}_{,\chi}(\beta \left(N\right)/\kappa )}{\kappa \phantom{\rule{0.166667em}{0ex}}\eta \left(\beta \right(N)/\kappa )}=-\frac{6}{1+{\beta}_{1}}\left[1+\frac{{f}^{\prime}\left(\beta \left(N\right)\right)}{3f\left(\beta \right(N\left)\right)}\right]\left[1-\frac{{f}^{\prime}\left(\beta \left(N\right)\right)}{3f\left(\beta \right(N\left)\right)}{\beta}_{1}\right]\phantom{\rule{0.166667em}{0ex}}$$Then if there exist ${\beta}_{0}$ and ${\beta}_{1}$ which satisfy Equation (89), we find Equations (82–85) is satisfied by the following solution:$$\varphi =\chi =\frac{\beta \left(N\right)}{\kappa}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}H=\frac{f\left(\beta \right(N\left)\right)}{\kappa}\sqrt{\frac{3f\left(\beta \left(N\right)\right)+{f}^{\prime}\left(\beta \left(N\right)\right)}{3f\left(\beta \left(N\right)\right)+{\beta}_{1}{}^{2}{f}^{\prime}\left(\beta \left(N\right)\right)}}\phantom{\rule{0.166667em}{0ex}}$$Especially when ${\beta}_{1}=0$, this point describes de Sitter space-time.

#### 4.2.1. Model with Exponential Growth

**Figure 1.**Each vector denotes $({X}^{\prime}/50,{Y}^{\prime}/50)$, which is independent of Z and W. The parameters are $\tilde{\lambda}=3/2$ and ${\tilde{\alpha}}_{0}=1$. The point A is located in $(1,1)$, where the EoS parameter is $-2$. The point B is located in $(2,2)$, where the EoS parameter is $-3$.

**Figure 2.**Each vector denotes $({X}^{\prime}/20,{Y}^{\prime}/20)$, which is independent of Z and W. The parameters are $\tilde{\lambda}=-3/2$ and ${\tilde{\alpha}}_{0}=1$. The point A is located in $(1,1)$, where the EoS parameter is 0.

#### 4.2.2. Little Rip Model

**Figure 3.**Each vector denotes $({X}^{\prime}/50,{Y}^{\prime}/50)$ with $Z=1$ and $W=0$ ($\varphi =\chi =1/\kappa $). The parameters are $\gamma =1$ and ${\tilde{\alpha}}_{0}=1$. The point A is located in $(1,1)$, which corresponds to the Little Rip universe.

**Figure 4.**Each vector denotes $({X}^{\prime}/20,{Y}^{\prime}/20)$ with $Z=1$ and $W=0$ ($\varphi =\chi =1/2\kappa $). The parameters are $\gamma =-1$ and ${\tilde{\alpha}}_{0}=1$. The point A is located in $(1,1)$. The point B is located in $(0,0)$, which corresponds to the de Sitter universe.

**Figure 5.**Each vector denotes $({Z}^{\prime}/20,{W}^{\prime}/20)$, which is independent of the form of $\omega \left(\varphi \right)$, $\eta \left(\chi \right)$ and $V(\varphi ,\chi )$. The dynamics of Z and W are classified into four types according to the values of X and Y as (

**a**) $0<Y$, $X<Y$; (

**b**) $0<Y<X$; (

**c**) $0>Y$, $X>Y$ and (

**d**) $0>Y>X$. The fixed points are located in $(1,0)$.

#### 4.3. The Potential and the (in)stability

## 5. Discussion

- (1)
- The potential does not have maximum and it goes to infinity.
- (2)
- There is a path in the potential that the potential becomes infinite but the kinetic energy of the canonical scalar field is vanishing.

## Acknowledgments

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**MDPI and ACS Style**

Ito, Y.; Nojiri, S.; Odintsov, S.D.
Stability of Accelerating Cosmology in Two Scalar-Tensor Theory: Little Rip *versus* de Sitter. *Entropy* **2012**, *14*, 1578-1605.
https://doi.org/10.3390/e14081578

**AMA Style**

Ito Y, Nojiri S, Odintsov SD.
Stability of Accelerating Cosmology in Two Scalar-Tensor Theory: Little Rip *versus* de Sitter. *Entropy*. 2012; 14(8):1578-1605.
https://doi.org/10.3390/e14081578

**Chicago/Turabian Style**

Ito, Yusaku, Shin’ichi Nojiri, and Sergei D. Odintsov.
2012. "Stability of Accelerating Cosmology in Two Scalar-Tensor Theory: Little Rip *versus* de Sitter" *Entropy* 14, no. 8: 1578-1605.
https://doi.org/10.3390/e14081578