# Inflation inside Non-Topological Defects and Scalar Black Holes

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Results

#### 3.1. Black Hole Solutions

#### 3.1.1. Solutions without Scalar Fields

#### 3.1.2. Q-Clouds on Schwarzschild Black Holes

#### 3.1.3. Backreaction of Q-Clouds

#### 3.2. Globally Regular Solutions

#### 3.2.1. (Un)Charged Q-Balls

#### 3.2.2. Charged Boson Stars

## 4. Discussion

## 5. Materials and Methods

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BH | Black Hole |

BS | Boson Star |

NS | Neutron Star |

RN | Reissner–Nordström |

## References

- Ruffini, R.; Wheeler, J.A. Introducing the black hole. Phys. Today
**1971**, 24, 30. [Google Scholar] [CrossRef] [Green Version] - Herdeiro, C.A.R.; Radu, E. Asymptotically flat black holes with scalar hair: A review. Int. J. Mod. Phys. D
**2015**, 24, 1542014. [Google Scholar] [CrossRef] [Green Version] - Damour, T.; Esposito-Farese, G. Nonperturbative strong field effects in tensor-scalar theories of gravitation. Phys. Rev. Lett.
**1993**, 70, 2220. [Google Scholar] [CrossRef] [PubMed] - Silva, H.O.; Sakstein, J.; Gualtieri, L.; Sotiriou, T.P.; Berti, E. Spontaneous scalarization of black holes and compact stars from a Gauss-Bonnet coupling. Phys. Rev. Lett.
**2018**, 120, 131104. [Google Scholar] [CrossRef] [Green Version] - Hod, S. Stationary Scalar Clouds Around Rotating Black Holes. Phys. Rev. D
**2012**, 86, 104026. [Google Scholar] [CrossRef] [Green Version] - Herdeiro, C.A.R.; Radu, E. Kerr black holes with scalar hair. Phys. Rev. Lett.
**2014**, 112, 221101. [Google Scholar] [CrossRef] [Green Version] - Coleman, S.R. Q-Balls. Nucl. Phys. B
**1985**, 262, 263. [Google Scholar] [CrossRef] - Schunck, F.E.; Mielke, E.W. General relativistic boson stars. Class. Quant. Grav.
**2003**, 20, R301. [Google Scholar] [CrossRef] [Green Version] - Eby, J.; Kouvaris, C.; Nielsen, N.G.; Wijewardhana, L.C.R. Boson Stars from Self-Interacting Dark Matter. JHEP
**2016**, 02, 028. [Google Scholar] [CrossRef] [Green Version] - Guzman, F.S.; Rueda-Becerril, J.M. Spherical boson stars as black hole mimickers. Phys. Rev. D
**2009**, 80, 084023. [Google Scholar] [CrossRef] [Green Version] - Macedo, C.F.B.; Pani, P.; Cardoso, V.; Crispino, L.C.B. Astrophysical signatures of boson stars: Quasinormal modes and inspiral resonances. Phys. Rev. D
**2013**, 88, 064046. [Google Scholar] [CrossRef] [Green Version] - Brihaye, Y.; Diemer, V.; Hartmann, B. Charged Q-balls and boson stars and dynamics of charged test particles. Phys. Rev. D
**2014**, 89, 084048. [Google Scholar] [CrossRef] [Green Version] - Campanelli, L.; Ruggieri, M. Supersymmetric Q-balls: A Numerical study. Phys. Rev. D
**2008**, 77, 043504. [Google Scholar] [CrossRef] [Green Version] - Copeland, E.J.; Tsumagari, M.I. Q-balls in flat potentials. Phys. Rev. D
**2009**, 80, 025016. [Google Scholar] [CrossRef] - Hartmann, B.; Riedel, J. Supersymmetric Q-balls and boson stars in (d+1) dimensions. Phys. Rev. D
**2013**, 87, 044003. [Google Scholar] [CrossRef] [Green Version] - Hartmann, B.; Kleihaus, B.; Kunz, J.; Schaffer, I. Compact Boson Stars. Phys. Lett. B
**2012**, 714, 120. [Google Scholar] [CrossRef] [Green Version] - Herdeiro, C.A.R.; Radu, E. Spherical electro-vacuum black holes with resonant, scalar Q-hair. Eur. Phys. J. C
**2020**, 80, 390. [Google Scholar] [CrossRef] - Brihaye, Y.; Hartmann, B. Strong gravity effects of charged Q-clouds and inflating black holes. arXiv
**2020**, arXiv:2009.08293. [Google Scholar] - Breitenlohner, P.; Forgacs, P.; Maison, D. Gravitating monopole solutions. Nucl. Phys. B
**1992**, 383, 357. [Google Scholar] [CrossRef] [Green Version] - Breitenlohner, P.; Forgacs, P.; Maison, D. Gravitating monopole solutions 2. Nucl. Phys. B
**1995**, 442, 126. [Google Scholar] [CrossRef] [Green Version] - Blázquez-Salcedo, J.L.; Kahlen, S.; Kunz, J. Critical solutions in scalarized black holes. arXiv
**2020**, arXiv:2011.01326. [Google Scholar] - Vilenkin, A. Topological inflation. Phys. Rev. Lett.
**1994**, 72, 3137. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Linde, A.D. Monopoles as big as a universe. Phys. Lett. B
**1994**, 327, 208. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**

**Left**: We show the value of the scalar field $\psi $ (dashed) and the radial electric field ${E}_{x}$ (solid) on the horizon dependent on $e{V}_{\infty}\equiv \mathsf{\Omega}$ for ${x}_{h}=0.15$ and two different values of e.

**Right**: We show the values of the electric charge Q (solid) and the charge contained in ${Q}_{N}$ charges e, $e{Q}_{N}$, (dashed) dependent on $e{V}_{\infty}\equiv \mathsf{\Omega}$ for the same solutions.

**Figure 2.**

**Left**: Profiles of the scalar field $\psi \left(x\right)$ and the gauge potential $V\left(x\right)$ for the two possible Q-cloud solutions on Schwarzschild black holes with ${x}_{h}=0.15$, $e=0.08$, and $e{V}_{\infty}=\mathsf{\Omega}=0.94$.

**Right**: Profiles of the scalar field $\psi \left(x\right)$ and the gauge potential $V\left(x\right)$ for the two possible Q-ball solutions for $e=0.08$ and $\mathsf{\Omega}=0.94$.

**Figure 3.**We show the electric charge Q and the electric charge contained in ${Q}_{N}$ charges e of Q-clouds on Schwarschild black holes with ${x}_{h}=0.15$ and for $e=0.08$, ${V}_{\infty}=11.75$, i.e., $\mathsf{\Omega}=e{V}_{\infty}=0.94$.

**Figure 4.**We show the approach to critically for a back hole with charged scalar hair for $\mathsf{\Omega}=0.6$ and ${x}_{h}=0.15$.

**Figure 5.**

**Left**: We show the profiles of the energy density components (see (17)) for ${x}_{h}=0.15$, $\mathsf{\Omega}=0.6$ and close to the critical value of $\alpha $, ${\alpha}_{cr}\approx 0.000113$.

**Right**: We show the effective potential ${U}_{\mathrm{eff}}=U-\frac{{(\omega -eV)}^{2}}{N{\sigma}^{2}}{\psi}^{2}$ and the first (${\partial}_{\psi}{U}_{\mathrm{eff}}$) and second derivative (${\partial}_{\psi}^{2}{U}_{\mathrm{eff}}$) with respect to $\psi $ at the approach to criticality for ${x}_{h}=0.15$ and $\mathsf{\Omega}=0.6$.

**Figure 6.**

**Left**: We show the Noether charge ${Q}_{N}$ of (un)charged Q-balls dependent on $\mathsf{\Omega}$ for four different values of e.

**Right**: We show the central value of the scalar field, $\psi \left(0\right)$, of (un)charged Q-balls dependent on $\mathsf{\Omega}$ for four different values of e.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 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**

Brihaye, Y.; Console, F.; Hartmann, B.
Inflation inside Non-Topological Defects and Scalar Black Holes. *Symmetry* **2021**, *13*, 2.
https://doi.org/10.3390/sym13010002

**AMA Style**

Brihaye Y, Console F, Hartmann B.
Inflation inside Non-Topological Defects and Scalar Black Holes. *Symmetry*. 2021; 13(1):2.
https://doi.org/10.3390/sym13010002

**Chicago/Turabian Style**

Brihaye, Yves, Felipe Console, and Betti Hartmann.
2021. "Inflation inside Non-Topological Defects and Scalar Black Holes" *Symmetry* 13, no. 1: 2.
https://doi.org/10.3390/sym13010002