# 3-Form Cosmology: Phantom Behaviour, Singularities and Interactions

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## Abstract

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## 1. Introduction

## 2. 3-Form Cosmology and the Little Sibling of the Big Rip

## 3. Removing the LSBR

- Fixed points verifying ${z}_{fp}=0$ for finite values of the 3-form field χ (${u}_{fp}\ne \pm 1$). This category includes the fixed points that correspond to the future LSBR event: $(\pm 1/2,\phantom{\rule{0.166667em}{0ex}}\pm 1,\phantom{\rule{0.166667em}{0ex}}0)$.
- Fixed points verifying ${z}_{fp}\ne 0$ for finite values of the 3-form field χ (${u}_{fp}\ne \pm 1$). In the non-interacting case, the fixed points of Type II correspond to extrema of the potential in the interval $]-{\chi}_{\mathrm{c}},{\chi}_{\mathrm{c}}[$.
- Fixed points that are characterized by $u=\pm 1$; i.e., infinite values of the 3-form field χ. The fixed points of Type III with a repulsive nature represent the asymptotic past of the system [18]. The position of these fixed points needs to be analysed with care and must take into account the behaviour of V—and therefore of z—as $u\to \pm 1$ [18,26].

## 4. Distinguishing Interactions

## 5. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

LSBR | Little Sibling of the Big Rip |

DE | dark energy |

DM | dark matter |

CND | composite null diagnostic |

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**Figure 1.**Representation of the possible evolution of the field χ for the quadratic potential (

**left panel**) and the Gaussian potential (

**right panel**). Notice that—independently of the shape of the potential (blue)—the field χ decays monotonically until it enters the interval $[-{\chi}_{\mathrm{c}},{\chi}_{\mathrm{c}}]$. Once inside this interval, the field evolves towards a minimum of the effective potential (green), defined by ${V}_{,\chi}^{eff}=(1-{\chi}^{2}/{\chi}_{\mathrm{c}}^{2}){V}_{,\chi}$.

**Figure 2.**

**Left panel**: the position of the fixed points $({u}_{fp},\phantom{\rule{0.166667em}{0ex}}{y}_{fp},\phantom{\rule{0.166667em}{0ex}}{z}_{fp})$ for a Gaussian potential and no interaction between dark matter (DM) and the 3-form.

**Right panel**: the position of the fixed points changes when the interaction (9) is turned on with small coefficients ${\alpha}_{i}$. Depending on the value of the interaction coefficients, each of the fixed points can be decomposed in two new points which are shifted from the original position along the yellow curves.

**Figure 3.**The evolution of statefinder Hierarchy $\{{{S}_{3}}^{\left(1\right)},\phantom{\rule{0.166667em}{0ex}}{{S}_{4}}^{\left(1\right)}\}$ (

**left panel**), $\{{{S}_{3}}^{\left(1\right)},\phantom{\rule{0.166667em}{0ex}}{{S}_{5}}^{\left(1\right)}\}$ (

**middle panel**) and the CDN $\{\u03f5,\phantom{\rule{0.166667em}{0ex}}{{S}_{3}}^{\left(1\right)}\}$ (

**right panel**) for the non-interacting 3-form dark energy (DE) model (dotted line), and when interactions I (full line) and II (dashed line) are turned on. The point $\{1,\phantom{\rule{0.166667em}{0ex}}1\}$ indicates the ΛCDM model, the “×” symbols indicate the values of the statefinder parameters at the present time, and the arrows indicate the temporal direction.

© 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/).

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**MDPI and ACS Style**

Morais, J.; Bouhmadi-López, M.; Marto, J.
3-Form Cosmology: Phantom Behaviour, Singularities and Interactions. *Universe* **2017**, *3*, 21.
https://doi.org/10.3390/universe3010021

**AMA Style**

Morais J, Bouhmadi-López M, Marto J.
3-Form Cosmology: Phantom Behaviour, Singularities and Interactions. *Universe*. 2017; 3(1):21.
https://doi.org/10.3390/universe3010021

**Chicago/Turabian Style**

Morais, João, Mariam Bouhmadi-López, and João Marto.
2017. "3-Form Cosmology: Phantom Behaviour, Singularities and Interactions" *Universe* 3, no. 1: 21.
https://doi.org/10.3390/universe3010021