# Compact Objects in Alternative Gravities

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Neutron Stars

#### 2.1. Neutron Stars in General Relativity

#### 2.2. Neutron Stars beyond General Relativity

## 3. Black Holes

#### 3.1. Black Holes in General Relativity

#### 3.2. Black Holes beyond General Relativity

## 4. Conclusions

**ALTECOSMOFUN’21**by the schedule of the meeting.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

**Alternative Gravities and Fundamental Cosmology—ALTECOSMOFUN’21**. We would also like to thank our collaborators: Emanuele Berti, Vitor Cardoso, Lucas G. Collodel, Daniela D. Doneva, Valeria Ferrari, Luis M. González-Romero, Leonardo Gualtieri, Sarah Kahlen, Panagiota Kanti, Fech Scen Khoo, Caio F. B. Macedo, Sindy Mojica, Zahra A. Motahar, Francisco Navarro-Lérrida, Petya Nedkova, Paolo Pani, Vincent Preut, Eugen Radu, Kalin V. Staykov, Stoytcho S. Yazadjiev. We gratefully acknowledge support by the DFG Research Training Group 1620 Models of Gravity and the COST Actions CA15117 and CA16104. JLBS would like to acknowledge support from FCT project PTDC/FIS-AST/3041/2020.

## Conflicts of Interest

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**Figure 2.**$l=2$ polar QNMs: (

**a**) frequency (in kHz) vs. compactness $M/R$ for the fundamental f-mode, the pressure p-mode, and the space-time w-mode for several equations of state; (

**b**) universal relation for the scaled frequency $\omega R$ vs. the compactness $M/R$ for the w-mode [43].

**Figure 3.**Spontaneous scalarization of neutron stars: (

**a**) mass-radius relation for several equations of state for two values of ${\beta}_{0}$, also shown is the GR relation; (

**b**) universal relation for the value of the scalar field at the center $\phi \left(0\right)$ vs. the $tt$ component of the metric at the center ${g}_{tt}\left(0\right)$. ${\beta}_{\mathrm{cr}}$ denotes the onset of scalarization [46].

**Figure 4.**Universal relations for axial $l=2$ QNMs of neutron stars in ${R}^{2}$ gravity: (

**a**) scaled frequency ${\omega}_{R}R$ vs. compactness $M/R$ for $a=10$ (red) and $a={10}^{5}$ (blue); (

**b**) scaled damping time $M/\tau $ vs. $M/R$. Additionally shown are the deviations from the respective best fits, as well as the GR modes (green) for comparison [57].

**Figure 5.**Polar QNMs of neutron stars in ${R}^{2}$ gravity: (

**a**) frequency ${\omega}_{R}$ vs. mass M of the $l=0$ F mode for several values of a, also shown are the GR values and the massless case; (

**b**) frequency ${\omega}_{R}$ vs. scalar Compton wavelength ${L}_{\phi}$ for the F mode, H modes and $\phi $ mode for a given neutron star central density. The equation of state is fixed [58].

**Figure 6.**Dilatonic GB black holes: (

**a**) scaled event horizon radius ${r}_{H}/\sqrt{\alpha}$ vs. scaled mass $M/\sqrt{\alpha}$ of static black holes, compared to Schwarzschild; (

**b**) scaled horizon area ${a}_{H}={A}_{H}/16\pi {M}^{2}$ vs. scaled angular momentum $j=J/{M}^{2}$ of rotating black holes, delimited by static and Kerr black holes, and critical solutions. Additionally shown are curves of fixed scaled horizon angular velocity ${\Omega}_{H}{\alpha}^{1/2}$ [84].

**Figure 7.**Polar $l=2$ QNMs of dilaton GB black holes: (

**a**) scaled frequency ${\omega}_{R}/{\omega}_{R}^{S}$ vs. dimensionless coupling constant $\zeta =\alpha /{M}^{2}$; (

**b**) scaled decay rate ${\omega}_{I}/{\omega}_{I}^{S}$ vs. $\zeta =\alpha /{M}^{2}$. Shown are the scalar-led and grav-led modes. The scaling is w.r.t. Schwarzschild [92].

**Figure 8.**Domain of existence of static spontaneously scalarized EsGB black holes: (

**a**) horizon radius ${r}_{H}$ vs. horizon scalar field ${\phi}_{H}$ at fixed GB coupling $\lambda =1$ for exponential coupling function; (

**b**) same for quadratic coupling function. Shown are the fundamental mode $n=0$, and the lowest radially excited modes $n=1$ and 2. The colors indicate forbidden regions [58].

**Figure 9.**Polar and axial $l=2$ QNMs of static spontaneously scalarized EsGB black holes with exponential coupling: (

**a**) scaled frequency $\lambda {\omega}_{R}$ vs. scaled mass $M/\lambda $; (

**b**) scaled decay rate $\lambda {\omega}_{I}$ vs. $M/\lambda $. Additionally shown are the Schwarzschild modes, the bifurcation point from Schwarzschild ${r}_{\mathrm{B}}$, and the point ${r}_{\mathrm{S}2}$, where hyperbolicity is lost for the axial modes [100,101].

**Figure 10.**Domain of existence of rotating spontaneously scalarized GB black holes with quadratic coupling function: (

**a**) scaled horizon area ${a}_{H}={A}_{H}/16\pi {M}^{2}$ vs. scaled angular momentum $j=J/{M}^{2}$ for positive coupling constant; (

**b**) ${a}_{H}={A}_{H}/16\pi {M}^{2}$ vs. $j=J/{M}^{2}$ for spin-induced black holes for negative coupling constant, both for even and odd scalar field. The domains are delimited by Kerr black holes, and critical solutions. Additionally shown are curves of fixed horizon angular velocity ${\Omega}_{H}$ [104,108].

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**MDPI and ACS Style**

Blázquez-Salcedo, J.L.; Kleihaus, B.; Kunz, J.
Compact Objects in Alternative Gravities. *Universe* **2022**, *8*, 153.
https://doi.org/10.3390/universe8030153

**AMA Style**

Blázquez-Salcedo JL, Kleihaus B, Kunz J.
Compact Objects in Alternative Gravities. *Universe*. 2022; 8(3):153.
https://doi.org/10.3390/universe8030153

**Chicago/Turabian Style**

Blázquez-Salcedo, Jose Luis, Burkhard Kleihaus, and Jutta Kunz.
2022. "Compact Objects in Alternative Gravities" *Universe* 8, no. 3: 153.
https://doi.org/10.3390/universe8030153