# The Singularity Problem in Brane Cosmology

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## Abstract

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## 1. Introduction

## 2. Flat Branes

**Theorem**

**1**(Minkowski brane-massless dilaton).

- Add self-interaction to the dilaton
- Add a perfect fluid in the bulk
- Add a mixture of a fluid and a (possibly interacting) dilaton field.

**Theorem**

**2**(Minkowski brane-Single bulk fluid).

- Collapse-type I: $\gamma >-1/2$$$a\to 0,\phantom{\rule{1.em}{0ex}}{a}^{\prime}\to \infty ,\phantom{\rule{1.em}{0ex}}\rho \to \infty ,$$
- Collapse-type II: $\gamma =-1/2$$$a\to 0,\phantom{\rule{1.em}{0ex}}{a}^{\prime}\to \mathit{const}.,\phantom{\rule{1.em}{0ex}}\rho \to \infty ,$$
- Big rip: $\gamma <-1$$$a\to \infty ,\phantom{\rule{1.em}{0ex}}{a}^{\prime}\to -\infty ,\phantom{\rule{1.em}{0ex}}\rho \to \infty ,$$
- At envelope: $\gamma \in (-1,-1/2)$$$a\to 0,\phantom{\rule{1.em}{0ex}}{a}^{\prime}\to 0,\phantom{\rule{1.em}{0ex}}\rho \to \infty ,$$

**Theorem**

**3**(Minkowski brane: Non-interacting pair of massless dilaton–fluid).

- Collapse-type I: any γ$$a\to 0,\phantom{\rule{1.em}{0ex}}{a}^{\prime}\to \infty ,\phantom{\rule{1.em}{0ex}}{\varphi}^{\prime}\to \infty ,\phantom{\rule{1.em}{0ex}}{\rho}_{2}\to 0,{\rho}_{s},\infty ,$$
- Big rip: $\gamma <-1$$$a\to \infty ,\phantom{\rule{1.em}{0ex}}{a}^{\prime}\to -\infty ,\phantom{\rule{1.em}{0ex}}{\varphi}^{\prime}\to 0,\phantom{\rule{1.em}{0ex}}{\rho}_{2}\to \infty .$$

## 3. Curved Branes

**Theorem**

**4**(dS or AdS brane: Single bulk fluid with γ ≥ −1/2).

- $\gamma >-1/2$$$\begin{array}{ccc}\hfill x& =& \alpha \mathrm{Y}+{c}_{-1\phantom{\rule{0.166667em}{0ex}}1}-A\alpha /3{c}_{-2\phantom{\rule{0.166667em}{0ex}}3}{\mathrm{Y}}^{-1}+\cdots ,\hfill \end{array}$$$$\begin{array}{ccc}\hfill y& =& \alpha +A\alpha /3{c}_{-2\phantom{\rule{0.166667em}{0ex}}3}{\mathrm{Y}}^{-2}+\cdots ,\hfill \end{array}$$$$\begin{array}{ccc}\hfill w& =& {c}_{-2\phantom{\rule{0.166667em}{0ex}}3}{\mathrm{Y}}^{-4}+\cdots ,\hfill \end{array}$$
- $\gamma =-1/2$$$\begin{array}{ccc}\hfill x& =& \alpha \mathrm{Y}+{c}_{-1\phantom{\rule{0.166667em}{0ex}}1}\cdots ,\hfill \end{array}$$$$\begin{array}{ccc}\hfill y& =& \alpha \cdots ,\hfill \end{array}$$$$\begin{array}{ccc}\hfill w& =& {c}_{-2\phantom{\rule{0.166667em}{0ex}}3}{\mathrm{Y}}^{-2}+\cdots ,\hfill \end{array}$$

**Theorem**

**5.**

- a de Sitter brane in a single bulk fluid with negative energy density and $\gamma >-1/2$, or
- an Anti de Sitter brane in a single bulk fluid with positive energy density and $\gamma \in (-1,-1/2)$,

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Antoniadis, I.; Cotsakis, S.
The Singularity Problem in Brane Cosmology. *Universe* **2017**, *3*, 15.
https://doi.org/10.3390/universe3010015

**AMA Style**

Antoniadis I, Cotsakis S.
The Singularity Problem in Brane Cosmology. *Universe*. 2017; 3(1):15.
https://doi.org/10.3390/universe3010015

**Chicago/Turabian Style**

Antoniadis, Ignatios, and Spiros Cotsakis.
2017. "The Singularity Problem in Brane Cosmology" *Universe* 3, no. 1: 15.
https://doi.org/10.3390/universe3010015