# The Dark Energy Properties of the Dirac–Born–Infeld Action

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## Abstract

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**PACS**Nos.: 98.80.Cq.

## 1. Introduction

## 2. The New General Lagrangian of DBI Inflation

## 3. Energy-Momentum Tensor and Sound Speed of the New DBI Action

## 4. Dark Energy with DBI Inflation

- Null Energy Condition (NEC): $\rho +P\ge 0$ ;
- Weak Energy Condition (WEC): $\rho \ge 0$ and $\rho +P\ge 0$ ;
- Strong Energy Condition (SEC): $\rho +3P\ge 0$ and $\rho +P\ge 0$ ;
- Dominant Energy Condition (DEC): $\rho \ge 0$ and $\rho \pm P\ge 0$ .

- (i) Let us first consider the scenario where no scalar potential is present, i.e., $U(\varphi )=1$, $\xi V(\varphi )=0$, $\xi V(\varphi )/U(\varphi )=0$, that is to study solely the brane action. In this case, the equation of state is rewritten in$$\begin{array}{c}\hfill \omega =\frac{{\dot{\varphi}}^{2}}{-2-F(\varphi ){\dot{\varphi}}^{4}-{\dot{\varphi}}^{2}}\end{array}$$$$\begin{array}{c}\hfill \omega =\frac{{\dot{\varphi}}^{2}}{-2-F(\varphi ){\dot{\varphi}}^{4}-{\dot{\varphi}}^{2}}<-1\end{array}$$$$\begin{array}{c}\hfill -2>F(\varphi ){\dot{\varphi}}^{4}\end{array}$$
- (ii) Let us now turn on the scalar potential term $V(\varphi )$. A first simple solution subclass would be to consider $F(\varphi )=0$, where we obtain $\omega =-1$ that recovers the case of pure de Sitter expansion.
- (iii) In the general case of nonzero potential $V(\varphi )$ and tension terms $f(\varphi )$, but with $V(\varphi )\gg f(\varphi )$ (corresponds to $U(\varphi )\gg F(\varphi )$ in the new DBI Lagrangian), and then we have$$\begin{array}{c}\hfill \omega =\frac{-1+\frac{1}{U(\varphi )}}{F(\varphi ){\dot{\varphi}}^{2}+1-\frac{1}{U(\varphi )}}<-1\end{array}$$$$\begin{array}{c}\hfill {\dot{\varphi}}^{2}>0\end{array}$$
- (iv) Another class of solutions will occur when we have nonzero potential $V(\varphi )$ and tension terms $f(\varphi )$, but with $V(\varphi )\ll f(\varphi )$(corresponds to $U(\varphi )\ll F(\varphi )$ in the new DBI Lagrangian), and then we have$$\begin{array}{c}\hfill \omega =\frac{{\dot{\varphi}}^{2}+\frac{1}{F}}{-2-F(\varphi ){\dot{\varphi}}^{4}+(U(\varphi )-2){\dot{\varphi}}^{2}-\frac{1}{F}}<-1\end{array}$$$$\begin{array}{c}\hfill 3U(\varphi )+F(\varphi )U(\varphi ){\dot{\varphi}}^{2}-U{(\varphi )}^{2}\ll 0\end{array}$$
- (v) The last class of solutions will occur when we have nonzero potential $V(\varphi )$ and tension terms $f(\varphi )$, but with $V(\varphi )\approx f(\varphi )$ ($U(\varphi )\approx F(\varphi )$), and then we have$$\begin{array}{c}\hfill \omega =\frac{{\dot{\varphi}}^{2}-1+\frac{1}{U(\varphi )}}{-1-\frac{1}{U(\varphi )}-F(\varphi ){\dot{\varphi}}^{4}+(U(\varphi )-2){\dot{\varphi}}^{2}}<-1\end{array}$$$$\begin{array}{c}\hfill \gamma (\varphi )=1-\frac{1}{2}F(\varphi )(-{\dot{\varphi}}^{2}+\frac{\xi V(\varphi )}{U(\varphi )})\approx 1+\frac{1}{2}F(\varphi )({\dot{\varphi}}^{2}-1)\end{array}$$

$V,U,F$ | quintessence $\omega >-1$ | de-Sitter $\omega =-1$ | phantom $\omega <-1$ |
---|---|---|---|

(i) $V=0$, $U=1$ | $-2<F(\varphi ){\dot{\varphi}}^{4}$ | $-2=F(\varphi ){\dot{\varphi}}^{4}$ | $-2>F(\varphi ){\dot{\varphi}}^{4}$ |

(ii) $V\ne 0$, $F=0$ | − | $\omega (\varphi )=-1$ | − |

(iii) $V\ne 0$, $U\gg F$ | $F(\varphi ){\dot{\varphi}}^{2}+1-\frac{1}{\mathrm{U}(\varphi )}<0$ | − | $F(\varphi ){\dot{\varphi}}^{2}+1-\frac{1}{\mathrm{U}(\varphi )}>0$ |

(iv) $V\ne 0$, $U\ll F$ | $U(\varphi ){\dot{\varphi}}^{2}>2+F(\varphi ){\dot{\varphi}}^{4}+{\dot{\varphi}}^{2}$ | − | $U(\varphi ){\dot{\varphi}}^{2}<2+F(\varphi ){\dot{\varphi}}^{4}+{\dot{\varphi}}^{2}$ |

(v) $V\ne 0$, $U\approx F$ | $F(\varphi )<0$ | − | $-U(\varphi )+1>2\sqrt{2F(\varphi )}>0$ |

## 5. Discussion and Conclusion

## Acknowledgments

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Zhang, X.; Zhang, Q.; Huang, Y.
The Dark Energy Properties of the Dirac–Born–Infeld Action. *Entropy* **2012**, *14*, 1203-1220.
https://doi.org/10.3390/e14071203

**AMA Style**

Zhang X, Zhang Q, Huang Y.
The Dark Energy Properties of the Dirac–Born–Infeld Action. *Entropy*. 2012; 14(7):1203-1220.
https://doi.org/10.3390/e14071203

**Chicago/Turabian Style**

Zhang, Xinyou, Qing Zhang, and Yongchang Huang.
2012. "The Dark Energy Properties of the Dirac–Born–Infeld Action" *Entropy* 14, no. 7: 1203-1220.
https://doi.org/10.3390/e14071203