# The Singularity Problem in Brane Cosmology

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 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

## References

- Arcani-Hamed, N.; Dimopoulos, S.; Dvali, G.R. The hierarchy problem and New Dimensions at a millimeter. Phys. Lett. B
**1998**, 429, 263–272. [Google Scholar] [CrossRef] - Arcani-Hamed, N.; Antoniadis, I.; Dimopoulos, S.; Dvali, G.R. New dimensions at a millimeter to a fermi and superstrings at a TeV. Phys. Lett. B
**1998**, 436, 257–263. [Google Scholar] - Antoniadis, I. A new approach to supersymmetry breaking in superstring models. Phys. Lett. B
**1990**, 246, 377–384. [Google Scholar] [CrossRef] - Arkani-Hamed, N.; Dimopoulos, S.; Kaloper, N.; Sundrum, R. A small cosmological constant from a large extra dimension. Phys. Lett. B
**2000**, 480, 193–199. [Google Scholar] [CrossRef] - Kachru, S.; Schulz, M.; Silverstein, E. Bounds on curved domain walls in 5d gravity. Phys. Rev. D
**2000**, 62, 085003. [Google Scholar] [CrossRef] - Antoniadis, I.; Cotsakis, S.; Klaoudatou, I. Brane singularities and their avoidance. Class. Quantum Gravity
**2010**, 27, 235018. [Google Scholar] [CrossRef] - Antoniadis, I.; Cotsakis, S.; Klaoudatou, I. Enveloping branes and braneworld singularities. Eur. Phys. J. C
**2014**, 74, 3192. [Google Scholar] [CrossRef] [PubMed] - Antoniadis, I.; Cotsakis, S.; Klaoudatou, I. Brane singularities with mixtures in the bulk. Fortschr. Phys.
**2013**, 61, 20–49. [Google Scholar] [CrossRef] - Antoniadis, I.; Cotsakis, S.; Klaoudatou, I. Curved branes with regular support. Eur. Phys. J. C
**2016**, 76, 511. [Google Scholar] [CrossRef] - Gubser, S.S. Curvature singularities: The good, the bad, and the naked. Adv. Theor. Math. Phys.
**2000**, 4, 679–745. [Google Scholar] [CrossRef] - Gasperini, M. Elements of String Cosmology; Cambridge University Press: Cambridge, UK, 2007; Chapter 10. [Google Scholar]
- Peterson, P. Riemannian Geometry; Springer: New York, NY, USA, 2006. [Google Scholar]
- O’Neill, B. Semi-Riemannian Geometry with Applications to Relativity; Academic Press: New York, NY, USA, 1983. [Google Scholar]
- Barrow, J.D.; Yamamoto, K. Anisotropic Pressures at Ultra-stiff Singularities and the Stability of Cyclic Universes. Phys. Rev. D
**2010**, 82, 063516. [Google Scholar] [CrossRef] - Randall, L.; Sundrum, S. An alternative to compactification. Phys. Rev. Lett.
**1999**, 83, 4960. [Google Scholar] [CrossRef] - Antoniadis, I.S.; Cotsakis, S. The large-scale structure of the ambient boundary. In Proceedings of the Fourteenth Marcel Grossman Meeting on General Relativity, Rome, Italy, 12–18 July 2015; Bianchi, M., Jantzen, R.T., Ruffini, R., Eds.; World Scientific: Singapore, 2017. [Google Scholar]

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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