# Chandrasekhar Mass Limit of White Dwarfs in Modified Gravity

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Tolman–Oppenheimer–Volkoff Equations in GR

## 3. Spherically Symmetric Stars in f(R)-Gravity

## 4. Simple Model of R^{2}-Gravity: Perturbative Approach and Numerical Integration of Reduced System

## 5. Realistic Equation of State

## 6. Chandrasekhar Limit of Mass in Another Model of Modified Gravity

## 7. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Capozziello, S.; Laurentis, M.D. Extended Theories of Gravity. Phys. Rept.
**2011**, 509, 167. [Google Scholar] [CrossRef] - Capozziello, S.; Faraoni, V. Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics; Springer: Dordrecht, The Netherlands, 2011; Volume 170. [Google Scholar]
- Nojiri, S.; Odintsov, S.D.; Oikonomou, V.K. Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution. Phys. Rep.
**2017**, 692, 1–104. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Unified cosmic history in modified gravity: From F(R) theory to Lorentz non-invariant models. Phys. Rep.
**2011**, 505, 59. [Google Scholar] [CrossRef] - Cruz-Dombriz, A.d.; Saez-Gomez, D. Black holes, cosmological solutions, future singularities, and their thermodynamical properties in modified gravity theories. Entropy
**2012**, 14, 1717. [Google Scholar] [CrossRef] - Olmo, G.J. Palatini Approach to Modified Gravity: F(R) Theories and Beyond. Int. J. Mod. Phys. D
**2011**, 20, 413. [Google Scholar] [CrossRef] - Dimopoulos, K. Introduction to Cosmic Inflation and Dark Energy; CRC Press: Boca Raton, FL, USA, 2021. [Google Scholar]
- Perlmutter, S.; Aldering, G.; Goldhaber, G.; Knop, R.A.; Nugent, P.; Castro, P.G.; Deustua, S.; Fabbro, S.; Goobar, A.; Groom, D.E.; et al. Measurements of ω and ∧ from 42 High-Redshift Supernovae. Astrophys. J.
**1999**, 517, 565. [Google Scholar] [CrossRef] - Riess, A.G.; Filippenko, A.V.; Challis, P.; Clocchiatti, A.; Diercks, A.; Garnavich, P.M.; Gilliland, R.L.; Hogan, C.J.; Jha, S.; Kirshner, P.R.; et al. Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. Astron. J.
**1998**, 116, 1009. [Google Scholar] [CrossRef] - Riess, A.G.; Strolger, L.; Tonry, J.; Casertano, S.; Ferguson, H.C.; Mobasher, B.; Challis, P.; Filippenko, A.V.; Jha, S.; Li, W.; et al. Type Ia Supernova Discoveries at z > 1 from the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution. Astrophys. J.
**2004**, 607, 665. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Modified gravity with negative and positive powers of the curvature: Unification of the inflation and of the cosmic acceleration. Phys. Rev. D
**2003**, 68, 123512. [Google Scholar] [CrossRef] - Weinberg, S. The cosmological constant problem. Rev. Mod. Phys.
**1989**, 61, 1. [Google Scholar] [CrossRef] - Spergel, D.N.; Verde, L.; Peiris, H.V.; Komatsu, E.; Nolta, M.R.; Bennett, C.L.; Halpern, M.; Hinshaw, G.; Jarosik, N.; Kogut, A.; et al. First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters. Astrophys. J. Suppl.
**2003**, 148, 175. [Google Scholar] [CrossRef] - Schimdt, C.; Tereno, I.; Uzan, J.-P.; Mellier, Y.; van Waerbeke, L.; Semboloni, E.; Hoekstra, H.; Fu, L.; Riazuelo, A. Tracking quintessence by cosmic shear. Astron. Astrophys.
**2007**, 463, 405–421. [Google Scholar] [CrossRef] - McDonald, P.; Seljak, U.; Burles, S.; Schlegel, D.J.; Weinberg, D.H.; Cen, R.; Shih, D.; Schaye, J.; Schneider, D.P.; Bahcall, N.A.; et al. The Lyα Forest Power Spectrum from the Sloan Digital Sky Survey. Astrophys. J. Suppl.
**2006**, 163, 80. [Google Scholar] [CrossRef] - Sarmah, L.; Kalita, S.; Wojnar, A. Stability criterion for white dwarfs in Palatini f(R) gravity. Phys. Rev. D
**2022**, 105, 024028. [Google Scholar] [CrossRef] - Wojnar, A. White dwarf stars in modified gravity. Int. J. Geom. Meth. Mod. Phys.
**2021**, 18, 2140006. [Google Scholar] [CrossRef] - Upadhye, A.; Hu, W. Existence of relativistic stars in f(R) gravity. Phys. Rev. D
**2009**, 80, 064002. [Google Scholar] [CrossRef] - Babichev, E.; Langlois, D. Relativistic stars in f(R) gravity. Phys. Rev. D
**2009**, 80, 121501. [Google Scholar] [CrossRef] - Arapoglu, A.S.; Deliduman, C.; Eksi, K.Y. Constraints on Perturbative f(R) Gravity via Neutron Stars. J. Cosmol. Astropart. Phys.
**2011**, 7, 20. [Google Scholar] [CrossRef] - Astashenok, A.V.; Capozziello, S.; Odintsov, S.D. Extreme neutron stars from Extended Theories of Gravity. J. Cosmol. Astropart. Phys.
**2015**, 1, 1. [Google Scholar] [CrossRef] - Capozziello, S.; Laurentis, M.D.; Farinelli, R.; Odintsov, S.D. Mass-radius relation for neutron stars in f(R) gravity. Phys. Rev. D
**2016**, 93, 023501. [Google Scholar] [CrossRef] - Resco, M.A.; Cruz-Dombriz, A.D.; Llanes-Estrada, F.J.; Castrillo, V.Z. On neutron stars in f(R) theories: Small radii, large masses and large energy emitted in a merger. Phys. Dark Univ.
**2016**, 13, 147–161. [Google Scholar] [CrossRef] - Zubair, M.; Abbas, G. Some interior models of compact stars in f(R) gravity. Astrophys. Space Sci.
**2016**, 361, 342. [Google Scholar] [CrossRef] - Das, A.; Rahaman, F.; Guha, B.K.; Ray, S. Compact stars in f(R,τ) gravity. Eur. Phys. J. C
**2016**, 76, 654. [Google Scholar] [CrossRef] - Yazadjiev, S.S.; Doneva, D.D.; Kokkotas, K.D. Tidal Love numbers of neutron stars in f(R) gravity. Eur. Phys. J. C
**2018**, 78, 818. [Google Scholar] [CrossRef] [PubMed] - Kase, R.; Tsujikawa, S. Neutron stars in f(R) gravity and scalar-tensor theories. J. Cosmol. Astropart. Phys.
**2019**, 1909, 54. [Google Scholar] [CrossRef] - Blazquez-Salcedo, J.L.; Khoo, F.S.; Kunz, J. Ultra-long-lived quasi-normal modes of neutron stars in massive scalar-tensor gravity. Europhys. Lett.-EPL
**2020**, 130, 50002. [Google Scholar] [CrossRef] - Astashenok, A.V.; Capozziello, S.; Odintsov, S.D.; Oikonomou, V.K. Extended Gravity Description for the GW190814 Supermassive Neutron Star. Phys. Lett. B
**2020**, 811, 135910. [Google Scholar] [CrossRef] - Astashenok, A.V.; Odintsov, S.D. Supermassive Neutron Stars in Axion F(R) Gravity. Mon. Not. R. Astron. Soc.
**2020**, 493, 78–86. [Google Scholar] [CrossRef] - Lobato, R.; Lourenço, O.; Moraes, P.H.R.S.; Lenzi, C.H.; de Avellar, M.; de Paula, W.; Dutra, M.; Malheiro, M. Neutron stars in f(R,T)) gravity using realistic equations of state in the light of massive pulsars and GW170817. J. Cosmol. Astropart. Phys.
**2020**, 12, 039. [Google Scholar] [CrossRef] - Astashenok, A.V.; Capozziello, S.; Odintsov, S.D.; Oikonomou, V.K. Causal Limit of Neutron Star Maximal Mass in f(R) Gravity in View of GW190814. Phys. Lett. B
**2021**, 816, 136222. [Google Scholar] [CrossRef] - Panotopoulos, G.; Tangphati, T.; Banerjee, A.; Jasim, M.K. Anisotropic quark stars in R
^{2}gravity. Phys. Lett. B**2021**, 817, 136330. [Google Scholar] [CrossRef] - Oikonomou, V.K. Universal inflationary attractors implications on static neutron stars. Class. Quant. Grav.
**2021**, 38, 175005. [Google Scholar] [CrossRef] - Odintsov, S.D.; Oikonomou, V.K. Neutron stars phenomenology with scalar–tensor inflationary attractors. Phys. Dark Univ.
**2021**, 32, 100805. [Google Scholar] [CrossRef] - Niu, R.; Zhang, X.; Wang, B.; Zhao, W. Constraining Scalar-tensor Theories Using Neutron Star—Black Hole Gravitational Wave Events. Astrophys. J.
**2021**, 921, 149. [Google Scholar] [CrossRef] - Numajiri, K.; Katsuragawa, T.; Nojiri, S. Compact star in general F(R) gravity: Inevitable degeneracy problem and non-integer power correction. Phys. Lett. B
**2022**, 826, 136929. [Google Scholar] [CrossRef] - Olmo, G.J.; Rubiera-Garcia, D.; Wojnar, A. Stellar structure models in modified theories of gravity: Lessons and challenges. Phys. Rep.
**2020**, 876, 1–75. [Google Scholar] [CrossRef] - Pani, P.; Berti, E. Slowly rotating neutron stars in scalar-tensor theories. Phys. Rev. D
**2014**, 90, 024025. [Google Scholar] [CrossRef] - Doneva, D.D.; Yazadjiev, S.S.; Stergioulas, N.; Kokkotas, K.D. Rapidly rotating neutron stars in scalar-tensor theories of gravity. Phys. Rev. D
**2013**, 88, 084060. [Google Scholar] [CrossRef] - Horbatsch, M.; Silva, H.O.; Gerosa, D.; Pani, P.; Berti, E.; Gualtieri, L.; Sperhake, U. Tensor-multi-scalar theories: Relativistic stars and 3 + 1 decomposition. Class. Quant. Grav.
**2015**, 32, 204001. [Google Scholar] [CrossRef] - Silva, H.O.; Macedo, C.F.B.; Berti, E.; Crispino, L.C.B. Slowly rotating anisotropic neutron stars in general relativity and scalar–tensor theory. Class. Quant. Grav.
**2015**, 32, 145008. [Google Scholar] [CrossRef] - Chew, X.Y.; Kleihaus, B.; Kunz, J.; Dzhunushaliev, V.; Folomeev, V. Rotating wormhole solutions with a complex phantom scalar field. Phys. Rev. D
**2019**, 100, 044019. [Google Scholar] [CrossRef] - Motahar, Z.A.; Blázquez-Salcedo, J.L.; Kleihaus, B.; Kunz, J. Scalarization of neutron stars with realistic equations of state. Phys. Rev. D
**2017**, 96, 064046. [Google Scholar] [CrossRef] - Astashenok, A.V.; Odintsov, S.D.; Oikonomou, V. Maximal Masses of White Dwarfs for Polytropes in R2 Gravity and Theoretical Constraints. Phys. Rev. D
**2022**, 106, 124010. [Google Scholar] [CrossRef] - Astashenok, A.V.; Capozziello, S.; Odintsov, S.D. Nonperturbative models of quark stars in f(R) gravity. Phys. Lett. B
**2015**, 742, 160. [Google Scholar] [CrossRef] - Astashenok, A.V.; Odintsov, S.D.; Cruz-Dombriz, A.D. The realistic models of relativistic stars in f(R) = R + alpha R
^{2}gravity. Class. Quant. Grav.**2017**, 34, 205008. [Google Scholar] [CrossRef] - Arnowitt, R.; Deser, S.; Misner, C. Dynamical Structure and Definition of Energy in General Relativity. Phys. Rev.
**1959**, 116, 1322. [Google Scholar] [CrossRef] - Naf, J.; Jetzer, P. On the 1/c expansion of f(R) gravity. Phys. Rev. D
**2010**, 81, 104003. [Google Scholar] [CrossRef] - Kilic, M.; Bergeron, P.; Blouin, S.; Bedard, A. The most massive white dwarfs in the solar neighbourhood. Mon. Not. R. Astron. Soc.
**2021**, 503, 5397–5408. [Google Scholar] [CrossRef]

**Figure 1.**Profile of scalar field (solid lines) as function of dimensionless variable x in comparison with approximation (25) (black dotted line) for some central densities. Parameter $\alpha ={10}^{14}$ cm${}^{2}$; ${\varphi}_{0}$ means $\varphi \left(0\right)$. For the exact solution, the scalar field starts from smaller values ($\varphi \left(0\right)/{\varphi}_{0}<1$).

**Figure 2.**Mass–density relation in ${R}^{2}$ gravity for some $\alpha $ in comparison with GR. The dotted lines correspond to results obtained with simple approximations of the scalar field; ${\alpha}_{13}$ means that the value of $\alpha $ is given in units of ${10}^{13}$ cm${}^{2}$.

**Figure 3.**Profile of the scalar field (solid lines) as a function of radial coordinates in comparison with the approximation (25) (dotted lines) for ${\rho}_{c}={10}^{9}$ g/cm${}^{3}$ and Chandrasekhar EoS.

**Figure 4.**Mass–density relation for polytrope with $n=3$ in $R+{\alpha}^{l-1}{R}^{l}$ gravity for ($\alpha ={10}^{14}$ cm${}^{2}$, $l=2.1$, green lines) and ($\alpha =5\times {10}^{14}$ cm${}^{2}$, $l=2.4$, black lines). The dotted lines correspond to results obtained with simple approximation of the scalar field.

**Table 1.**Differences between results for stellar masses from exact solutions of reduced system (M) and perturbative solution (${M}_{p}$) for $\alpha ={10}^{14}$ cm${}^{2}$. We also provide corresponding values of dimensionless parameter $\tilde{\alpha}$ and the relation ${\varphi}_{c}/{\varphi}_{p}\left(0\right)$. Here, ${\varphi}_{c}$ is the value of $\varphi $ at $r=0$, and ${\varphi}_{p}\left(0\right)$ is the value of the scalar field from the perturbative approximation.

$ln{\mathit{\rho}}_{\mathit{c}}$ | $\tilde{\mathit{\alpha}},$ | ${\mathit{\varphi}}_{\mathit{c}}/{\mathit{\varphi}}_{\mathit{p}}\left(0\right)$ | $\mathit{M},$ | ${\mathit{M}}_{\mathit{p}},$ |
---|---|---|---|---|

${\mathbf{10}}^{-\mathbf{3}}$ | ${\mathit{M}}_{\odot}$ | ${\mathit{M}}_{\odot}$ | ||

7 | 1.97 | ∼1 | 1.448 | 1.448 |

7.5 | 4.25 | ∼1 | 1.439 | 1.439 |

8 | 9.15 | 0.92203 | 1.419 | 1.418 |

8.5 | 19.71 | 0.76707 | 1.389 | 1.377 |

9 | 42.48 | 0.62547 | 1.334 | 1.295 |

9.5 | 91.52 | 0.46316 | 1.260 | 1.162 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Astashenok, A.V.; Odintsov, S.D.; Oikonomou, V.K.
Chandrasekhar Mass Limit of White Dwarfs in Modified Gravity. *Symmetry* **2023**, *15*, 1141.
https://doi.org/10.3390/sym15061141

**AMA Style**

Astashenok AV, Odintsov SD, Oikonomou VK.
Chandrasekhar Mass Limit of White Dwarfs in Modified Gravity. *Symmetry*. 2023; 15(6):1141.
https://doi.org/10.3390/sym15061141

**Chicago/Turabian Style**

Astashenok, Artyom V., Sergey D. Odintsov, and Vasilis K. Oikonomou.
2023. "Chandrasekhar Mass Limit of White Dwarfs in Modified Gravity" *Symmetry* 15, no. 6: 1141.
https://doi.org/10.3390/sym15061141