# Structure of Neutron Stars in Massive Scalar-Tensor Gravity

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formalism

- (i)
- The boundary conditions are specified at different locations of the domain, so that we have a two-point-boundary-value problem.
- (ii)
- For realistic values of the polytropic exponent $\mathsf{\Gamma}$, the pressure will reach zero at a finite radius ${R}_{S}$; at this point, we need to match to an exterior solution with vanishing baryon density $\rho $.
- (iii)
- The asymptotic behaviour of the scalar field near infinity is determined by the scalar mass $\mu $ and is given by$$\underset{r\to \infty}{lim}\phi \sim {A}_{1}\frac{{e}^{-(\mu /\hslash )r}}{r}+{A}_{2}\frac{{e}^{(\mu /\hslash )r}}{r}\phantom{\rule{0.166667em}{0ex}},$$

## 3. Numerical Framework

## 4. Results

#### 4.1. Overall Phenomenology

#### 4.2. Dependence on $\mu $

#### 4.3. Dependence on ${\alpha}_{0}$

#### 4.4. Dependence on ${\beta}_{0}$

#### 4.5. Stability of Models

## 5. Conclusions

- In agreement with previous literature studies of NS equilibrium models in massive and massless ST gravity, we find larger values of ${\alpha}_{0}$ and ${\beta}_{0}$ to result in larger deviations from the NS solutions in GR, whereas larger values of the scalar mass tend to reduce these deviations; cf. Figure 1 and Figure 3.
- For ${\alpha}_{0}=0$, the NS models of GR are also solutions of the field Equations of massive ST gravity. For $\beta \lesssim -4.35$, we find, additionally to the GR branch, the spontaneously scalarized class of NS solutions that Damour and Esposito-Farèse discovered in their original exploration of massless ST theory [18] and that were also identified in massive ST theory in [21]. These solutions are invariant under the scalar field transformation $\phi \to -\phi $.
- A non-zero ${\alpha}_{0}$ breaks this degeneracy and results in a dissection of the branches around the branch points; instead of the two connected branches of scalarized and non-scalarized solutions for ${\alpha}_{0}=0$, we now find a main branch I and a smaller loop of branch $II$ solutions; cf. Figure 2. The solutions on branches I and $II$ are characterized by different signs of the central scalar-field value ${\phi}_{c}$; cf. Figure 4.
- For sufficiently negative ${\beta}_{0}$, roughly ${\beta}_{0}\lesssim -15$, we observe a qualitative change in the strongly scalarized branch S of solutions. Instead of smoothly approaching the weakly scalarized branch W as happens for milder ${\beta}_{0}$, its upper (in the sense of increasing central baryon density) tail now either crosses or completely detaches from the W branch.
- For highly negative values of ${\beta}_{0}$, we furthermore encounter a new type of strongly scalarized solutions at this upper end of the S branch: the maximum of the scalar field is located away from the stellar center; cf. Figure 6 and Figure 7. In its most extreme form, these solutions are composed of highly compact NS models surrounded by a scalar shell; see, e.g., [77,78] for similar “gravitational atom” like configurations in other theories of gravity.
- Whenever multiple NS models with equal baryon mass exist, we find the scalarized model to be the stable configurations in the sense of minimal binding energy. Typically, though not always, this is the model with the largest radius; cf. Figure 8 and Figure 9. We also observe that the stable configurations agree in the sign of the central scalar field value, ${\phi}_{c}<0$ in our convention.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Will, C.M. The Confrontation between General Relativity and Experiment. Living Rev. Rel.
**2014**, 17, 4. [Google Scholar] [CrossRef] [PubMed][Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abernathy, M.R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; et al. Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett.
**2016**, 116, 061102. [Google Scholar] [CrossRef] [PubMed] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abernathy, M.R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; et al. Tests of general relativity with GW150914. Phys. Rev. Lett.
**2016**, 116, 221101, Erratum in**2018**, 121, 129902. [Google Scholar] [CrossRef] [PubMed] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A. Astrophys. J. Lett.
**2017**, 848, L13. [Google Scholar] [CrossRef] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. Tests of General Relativity with GW170817. Phys. Rev. Lett.
**2019**, 123, 011102. [Google Scholar] [CrossRef][Green Version] - Abbott, B.P.; Abbott, R.; Abbott, T.D.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; Adya, V.B.; et al. Tests of General Relativity with the Binary Black Hole Signals from the LIGO-Virgo Catalog GWTC-1. Phys. Rev.
**2019**, D100, 104036. [Google Scholar] [CrossRef][Green Version] - Yunes, N.; Yagi, K.; Pretorius, F. Theoretical Physics Implications of the Binary Black-Hole Mergers GW150914 and GW151226. Phys. Rev.
**2016**, D94, 084002. [Google Scholar] [CrossRef][Green Version] - Clifton, T.; Ferreira, P.G.; Padilla, A.; Skordis, C. Modified Gravity and Cosmology. Phys. Rept.
**2012**, 513, 1–189. [Google Scholar] [CrossRef][Green Version] - Berti, E.; Barausse, E.; Cardoso, V.; Gualtieri, L.; Pani, P.; Sperhake, U.; Stein, L.C.; Wex, N.; Yagi, K.; Baker, T.; et al. Testing General Relativity with Present and Future Astrophysical Observations. Class. Quant. Grav.
**2015**, 32, 243001. [Google Scholar] [CrossRef] - Barack, L.; Cardoso, V.; Nissanke, S.; Sotiriou, T.P.; Askar, A.; Belczynski, C.; Bertone, G.; Bon, E.; Blas, D.; Brito, R.; et al. Black holes, gravitational waves and fundamental physics: A roadmap. Class. Quant. Grav.
**2019**, 36, 143001. [Google Scholar] [CrossRef][Green Version] - Olmo, G.J.; Rubiera-Garcia, D.; Wojnar, A. Stellar structure models in modified theories of gravity: Lessons and challenges. arXiv
**2019**, arXiv:gr-qc/1912.05202. [Google Scholar] [CrossRef] - Capozziello, S.; D’Agostino, R.; Luongo, O. Extended Gravity Cosmography. Int. J. Mod. Phys.
**2019**, D28, 1930016. [Google Scholar] [CrossRef][Green Version] - Freese, K. Review of Observational Evidence for Dark Matter in the Universe and in upcoming searches for Dark Stars. EAS Publ. Ser.
**2009**, 36, 113–126. [Google Scholar] [CrossRef][Green Version] - Novosyadlyj, B.; Pelykh, V.; Shtanov, Y.; Zhuk, A. Dark Energy: Observational Evidence and Theoretical Models; Academperiodyka: Kyiv, Ukraine, 2013. [Google Scholar]
- Hamber, H.W. Quantum Gravitation: The Feynman Path Integral Approach; Springer: Berlin, Germany, 2009. [Google Scholar] [CrossRef]
- Lasky, P.D. Gravitational Waves from Neutron Stars: A Review. Publ. Astron. Soc. Austral.
**2015**, 32, e034. [Google Scholar] [CrossRef] - Özel, F.; Freire, P. Masses, Radii, and the Equation of State of Neutron Stars. Ann. Rev. Astron. Astrophys.
**2016**, 54, 401–440. [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, 2220–2223. [Google Scholar] [CrossRef] - Damour, T.; Esposito-Farese, G. Tensor–scalar gravity and binary pulsar experiments. Phys. Rev.
**1996**, D54, 1474–1491. [Google Scholar] [CrossRef][Green Version] - Harada, T. Neutron stars in scalar tensor theories of gravity and catastrophe theory. Phys. Rev.
**1998**, D57, 4802–4811. [Google Scholar] [CrossRef][Green Version] - Ramazanoğlu, F.M.; Pretorius, F. Spontaneous Scalarization with Massive Fields. Phys. Rev.
**2016**, D93, 064005. [Google Scholar] [CrossRef][Green Version] - Bergmann, P.G. Comments on the scalar tensor theory. Int. J. Theor. Phys.
**1968**, 1, 25–36. [Google Scholar] [CrossRef] - Wagoner, R.V. Scalar tensor theory and gravitational waves. Phys. Rev.
**1970**, D1, 3209–3216. [Google Scholar] [CrossRef] - Andreou, N.; Franchini, N.; Ventagli, G.; Sotiriou, T.P. Spontaneous scalarization in generalised scalar-tensor theory. Phys. Rev.
**2019**, D99, 124022, Erratum in**2020**, 101, 109903. [Google Scholar] - Ventagli, G.; Lehébel, A.; Sotiriou, T.P. The onset of spontaneous scalarization in generalised scalar-tensor theories. arXiv
**2020**, arXiv:gr-qc/2006.01153. [Google Scholar] - 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] - Doneva, D.D.; Yazadjiev, S.S. New Gauss-Bonnet Black Holes with Curvature-Induced Scalarization in Extended Scalar-Tensor Theories. Phys. Rev. Lett.
**2018**, 120, 131103. [Google Scholar] [CrossRef][Green Version] - Barausse, E.; Jacobson, T.; Sotiriou, T.P. Black holes in Einstein-aether and Horava-Lifshitz gravity. Phys. Rev.
**2011**, D83, 124043. [Google Scholar] [CrossRef][Green Version] - Ramazanoğlu, F.M. Spontaneous growth of vector fields in gravity. Phys. Rev.
**2017**, D96, 064009. [Google Scholar] [CrossRef][Green Version] - Annulli, L.; Cardoso, V.; Gualtieri, L. Electromagnetism and hidden vector fields in modified gravity theories: Spontaneous and induced vectorization. Phys. Rev.
**2019**, D99, 044038. [Google Scholar] [CrossRef][Green Version] - Ramazanoğlu, F.M. Spontaneous tensorization from curvature coupling and beyond. Phys. Rev.
**2019**, D99, 084015. [Google Scholar] [CrossRef][Green Version] - Sotani, H. Slowly Rotating Relativistic Stars in Scalar-Tensor Gravity. Phys. Rev.
**2012**, D86, 124036. [Google Scholar] [CrossRef][Green Version] - Pani, P.; Berti, E. Slowly rotating neutron stars in scalar-tensor theories. Phys. Rev.
**2014**, D90, 024025. [Google Scholar] [CrossRef][Green Version] - 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] - Yazadjiev, S.S.; Doneva, D.D.; Popchev, D. Slowly rotating neutron stars in scalar-tensor theories with a massive scalar field. Phys. Rev.
**2016**, D93, 084038. [Google Scholar] [CrossRef][Green Version] - Altaha Motahar, Z.; Blázquez-Salcedo, J.L.; Kleihaus, B.; Kunz, J. Scalarization of neutron stars with realistic equations of state. Phys. Rev.
**2017**, D96, 064046. [Google Scholar] [CrossRef][Green Version] - Staykov, K.V.; Popchev, D.; Doneva, D.D.; Yazadjiev, S.S. Static and slowly rotating neutron stars in scalar–tensor theory with self-interacting massive scalar field. Eur. Phys. J.
**2018**, C78, 586. [Google Scholar] [CrossRef] [PubMed][Green Version] - Doneva, D.D.; Yazadjiev, S.S.; Stergioulas, N.; Kokkotas, K.D. Rapidly rotating neutron stars in scalar-tensor theories of gravity. Phys. Rev.
**2013**, D88, 084060. [Google Scholar] [CrossRef][Green Version] - Doneva, D.D.; Yazadjiev, S.S.; Staykov, K.V.; Kokkotas, K.D. Universal I-Q relations for rapidly rotating neutron and strange stars in scalar-tensor theories. Phys. Rev.
**2014**, D90, 104021. [Google Scholar] [CrossRef][Green Version] - Doneva, D.D.; Yazadjiev, S.S.; Stergioulas, N.; Kokkotas, K.D. Differentially rotating neutron stars in scalar-tensor theories of gravity. Phys. Rev.
**2018**, D98, 104039. [Google Scholar] [CrossRef][Green Version] - Sotani, H.; Kokkotas, K.D. Maximum mass limit of neutron stars in scalar-tensor gravity. Phys. Rev.
**2017**, D95, 044032. [Google Scholar] [CrossRef][Green Version] - Rezaei, Z.; Dezdarani, H.Y. Ferromagnetic Neutron Stars in Scalar-Tensor Theories of Gravity. JCAP
**2019**, 1903, 013. [Google Scholar] [CrossRef][Green Version] - Anderson, D.; Yunes, N. Scalar charges and scaling relations in massless scalar–tensor theories. Class. Quant. Grav.
**2019**, 36, 165003. [Google Scholar] [CrossRef][Green Version] - Morisaki, S.; Suyama, T. Spontaneous scalarization with an extremely massive field and heavy neutron stars. Phys. Rev.
**2017**, D96, 084026. [Google Scholar] [CrossRef][Green Version] - Sotani, H.; Kokkotas, K.D. Compactness of neutron stars and Tolman VII solutions in scalar-tensor gravity. Phys. Rev.
**2018**, D97, 124034. [Google Scholar] [CrossRef][Green Version] - Tsuchida, T.; Kawamura, G.; Watanabe, K. A Maximum mass-to-size ratio in scalar tensor theories of gravity. Prog. Theor. Phys.
**1998**, 100, 291–313. [Google Scholar] [CrossRef] - Abbott, R.; Abbott, T.D.; Abraham, S.; Acernese, F.; Ackley, K.; Adams, C.; Adhikari, R.X.; Adya, V.B.; Affeldt, C.; Agathos, M. GW190814: Gravitational Waves from the Coalescence of a 23 Solar Mass Black Hole with a 2.6 Solar Mass Compact Object. Astrophys. J.
**2020**, 896, L44. [Google Scholar] [CrossRef] - Novak, J. Neutron star transition to strong scalar field state in tensor scalar gravity. Phys. Rev.
**1998**, D58, 064019. [Google Scholar] [CrossRef][Green Version] - Mendes, R.F.P.; Matsas, G.E.A.; Vanzella, D.A.T. Instability of nonminimally coupled scalar fields in the spacetime of slowly rotating compact objects. Phys. Rev.
**2014**, D90, 044053. [Google Scholar] [CrossRef][Green Version] - Mendes, R.F.P. Possibility of setting a new constraint to scalar-tensor theories. Phys. Rev.
**2015**, D91, 064024. [Google Scholar] [CrossRef][Green Version] - Mendes, R.F.P.; Ortiz, N. Highly compact neutron stars in scalar-tensor theories of gravity: Spontaneous scalarization versus gravitational collapse. Phys. Rev.
**2016**, D93, 124035. [Google Scholar] [CrossRef][Green Version] - Palenzuela, C.; Liebling, S.L. Constraining scalar-tensor theories of gravity from the most massive neutron stars. Phys. Rev.
**2016**, D93, 044009. [Google Scholar] [CrossRef][Green Version] - Bertotti, B.; Iess, L.; Tortora, P. A test of general relativity using radio links with the Cassini spacecraft. Nature
**2003**, 425, 374–376. [Google Scholar] [CrossRef] [PubMed] - Williams, J.G.; Turyshev, S.G.; Boggs, D.H. Lunar laser ranging tests of the equivalence principle with the earth and moon. Int. J. Mod. Phys.
**2009**, D18, 1129–1175. [Google Scholar] [CrossRef][Green Version] - Wex, N. Testing Relativistic Gravity with Radio Pulsars. arXiv
**2014**, arXiv:gr-qc/1402.5594. [Google Scholar] - Novak, J. Spherical neutron star collapse in tensor–scalar theory of gravity. Phys. Rev.
**1998**, D57, 4789–4801. [Google Scholar] [CrossRef][Green Version] - Novak, J.; Ibanez, J.M. Gravitational waves from the collapse and bounce of a stellar core in tensor scalar gravity. Astrophys. J.
**2000**, 533, 392–405. [Google Scholar] [CrossRef][Green Version] - Gerosa, D.; Sperhake, U.; Ott, C.D. Numerical simulations of stellar collapse in scalar-tensor theories of gravity. Class. Quant. Grav.
**2016**, 33, 135002. [Google Scholar] [CrossRef][Green Version] - Barausse, E.; Palenzuela, C.; Ponce, M.; Lehner, L. Neutron-star mergers in scalar-tensor theories of gravity. Phys. Rev.
**2013**, D87, 081506. [Google Scholar] [CrossRef][Green Version] - Palenzuela, C.; Barausse, E.; Ponce, M.; Lehner, L. Dynamical scalarization of neutron stars in scalar-tensor gravity theories. Phys. Rev.
**2014**, D89, 044024. [Google Scholar] [CrossRef][Green Version] - Shibata, M.; Taniguchi, K.; Okawa, H.; Buonanno, A. Coalescence of binary neutron stars in a scalar-tensor theory of gravity. Phys. Rev.
**2014**, D89, 084005. [Google Scholar] [CrossRef][Green Version] - Chen, P.; Suyama, T.; Yokoyama, J. Spontaneous scalarization: Asymmetron as dark matter. Phys. Rev.
**2015**, D92, 124016. [Google Scholar] [CrossRef][Green Version] - Doneva, D.D.; Yazadjiev, S.S. Rapidly rotating neutron stars with a massive scalar field—structure and universal relations. JCAP
**2016**, 1611, 019. [Google Scholar] [CrossRef] - Sperhake, U.; Moore, C.J.; Rosca, R.; Agathos, M.; Gerosa, D.; Ott, C.D. Long-lived inverse chirp signals from core collapse in massive scalar-tensor gravity. Phys. Rev. Lett.
**2017**, 119, 201103. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rosca-Mead, R.; Sperhake, U.; Moore, C.J.; Agathos, M.; Gerosa, D.; Ott, C.D. Core collapse in massive scalar-tensor gravity. arXiv
**2020**, arXiv:gr-qc/2005.09728. [Google Scholar] [CrossRef] - Geng, C.Q.; Kuan, H.J.; Luo, L.W. Inverse-Chirp Imprint of Gravitational Wave Signals in Scalar Tensor Theory. arXiv
**2020**, arXiv:gr-qc/2005.11629. [Google Scholar] - Cheong, P.C.K.; Li, T.G.F. Numerical studies on core collapse supernova in self-interacting massive scalar-tensor gravity. Phys. Rev.
**2019**, D100, 024027. [Google Scholar] [CrossRef][Green Version] - Rosca-Mead, R.; Moore, C.J.; Agathos, M.; Sperhake, U. Inverse-chirp signals and spontaneous scalarisation with self-interacting potentials in stellar collapse. Class. Quant. Grav.
**2019**, 36, 134003. [Google Scholar] [CrossRef][Green Version] - Fujii, Y.; Maeda, K. The Scalar-Tensor Theory of Gravitation; Cambridge Monographs on Mathematical Physics; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar] [CrossRef]
- Salgado, M. The Cauchy problem of scalar tensor theories of gravity. Class. Quant. Grav.
**2006**, 23, 4719–4742. [Google Scholar] [CrossRef][Green Version] - Faraoni, V.; Gunzig, E. Einstein frame or Jordan frame? Int. J. Theor. Phys.
**1999**, 38, 217–225. [Google Scholar] [CrossRef] - Salgado, M.; Martinez-del Rio, D.; Alcubierre, M.; Nunez, D. Hyperbolicity of scalar-tensor theories of gravity. Phys. Rev.
**2008**, D77, 104010. [Google Scholar] [CrossRef][Green Version] - Geng, C.Q.; Kuan, H.J.; Luo, L.W. Viable Constraint on Scalar Field in Scalar-Tensor Theory. Class. Quant. Grav.
**2020**, 37, 115001. [Google Scholar] [CrossRef][Green Version] - Press, W.H.; Teukolsky, S.A.; Flannery, B.P.; Vetterling, W.T. Numerical Recipes in C (2nd ed.): The Art of Scientific Computing; Cambridge University Press: New York, NY, USA, 1992. [Google Scholar]
- Shapiro, S.L.; Teukolsky, S.A. Black Holes, White Dwarfs, and Neutron Stars; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 1983. [Google Scholar]
- Arnowitt, R.L.; Deser, S.; Misner, C.W. The Dynamics of general relativity. Gen. Rel. Grav.
**2008**, 40, 1997–2027. [Google Scholar] [CrossRef][Green Version] - Brito, R.; Cardoso, V.; Macedo, C.F.B.; Okawa, H.; Palenzuela, C. Interaction between bosonic dark matter and stars. Phys. Rev.
**2016**, D93, 044045. [Google Scholar] [CrossRef][Green Version] - Baumann, D.; Chia, H.S.; Stout, J.; ter Haar, L. The Spectra of Gravitational Atoms. JCAP
**2019**, 1912, 006. [Google Scholar] [CrossRef][Green Version] - Akmal, A.; Pandharipande, V.; Ravenhall, D. The Equation of state of nucleon matter and neutron star structure. Phys. Rev.
**1998**, C58, 1804–1828. [Google Scholar] [CrossRef][Green Version] - Read, J.S.; Markakis, C.; Shibata, M.; Uryu, K.; Creighton, J.D.; Friedman, J.L. Measuring the neutron star equation of state with gravitational wave observations. Phys. Rev.
**2009**, D79, 124033. [Google Scholar] [CrossRef]

**Figure 1.**Branches of NS models are shown in the form of baryon-mass vs. radius (${M}_{b}-{R}_{S}$) diagrams. (

**Left**): For fixed values ${\alpha}_{0}={10}^{-4}$ and $\mu =4.8\times {10}^{-13}\phantom{\rule{0.166667em}{0ex}}\mathrm{eV}$, we plot the strongly scalarized branches obtained for selected values of ${\beta}_{0}$. For reference, the dashed black curve displays the solutions obtained in GR with ${\alpha}_{0}={\beta}_{0}=0$. (

**Right**): Here we fix ${\alpha}_{0}={10}^{-4}$ and a more extreme value of ${\beta}_{0}=-15$ and vary the scalar mass $\mu $; larger deviations from the GR structure are clearly visible in this case. In both panels the color scale measures the central value of $\left|\phi \right|$ and the “S” and “W” label the strongly and weakly scalarized branches described in the text.

**Figure 2.**${M}_{b}-{R}_{S}$ diagrams are shown for $\mu =4.8\times {10}^{-13}\phantom{\rule{0.166667em}{0ex}}\mathrm{eV}$ and ${\beta}_{0}=-4.5$, as well as ${\alpha}_{0}=0$ (

**top left**), ${\alpha}_{0}={10}^{-4}$ (

**top right**) and ${\alpha}_{0}={10}^{-3}$ (

**bottom**). The color scale measures the central value of $\left|\phi \right|$. Whereas the S and W branches connect at two points when ${\alpha}_{0}=0$, the S branch splits in two for nonzero ${\alpha}_{0}$ with each part connecting to GR-like models in such a way that we obtain a “loop” of models separate from the main branch of solutions. We refer to the main branch as branch I and to the loop as branch $II$.

**Figure 3.**${M}_{b}-{R}_{S}$ diagrams are shown for a scalar mass $\mu =4.8\times {10}^{-13}$ with ${\beta}_{0}=-4.5$ (

**left**) and $-5$ (

**right**). As we increase ${\alpha}_{0}$, the loop of branch $II$ solutions shrinks in size and eventually disappears. For reference, we include the GR branch in the right panel.

**Figure 4.**Distribution of scalarized NS models based on the sign of ${\phi}_{c}$ in the ${M}_{b}-{R}_{S}$ plane for $\mu =4.8\times {10}^{-13}\phantom{\rule{0.166667em}{0ex}}\mathrm{eV}$ and, from top-left to bottom-right, $({\alpha}_{0},{\beta}_{0})=({10}^{-1},-4.5)$, $({10}^{-2},-5.5)$, $({10}^{-2},-5)$, $(3\times {10}^{-2},-5)$. The orange points represent models with ${\phi}_{c}<0$ whereas the black ones have ${\phi}_{c}>0$. The type $II$ models on the loop differ in the sign of ${\phi}_{c}$ from the nearby main branch I models. Furthermore, we always observe a sign flip at the high-density end of branch I (around ${R}_{S}\approx 8\phantom{\rule{0.166667em}{0ex}}\mathrm{km}$ in the figure) but these NS models are unstable; cf. Section 4.5.

**Figure 5.**${M}_{b}-{R}_{S}$ diagrams are shown for several values of ${\beta}_{0}$ in the regime of spontaneous scalarization ${\beta}_{0}<-4.35$. The other scalar field parameters are $\mu =-4.8\times {10}^{-13}$ eV, ${\alpha}_{0}={10}^{-2}$. For increasingly negative values of ${\beta}_{0}$, the S branch extends to larger values of the NS radius and baryon mass.

**Figure 6.**${M}_{b}-{R}_{S}$ diagrams are shown for $\mu =4.8\times {10}^{-13}\phantom{\rule{0.166667em}{0ex}}\mathrm{eV}$ and ${\alpha}_{0}={10}^{-4}$, as well as ${\beta}_{0}=-15$ (

**top left**), ${\beta}_{0}=-17$ (

**top right**), ${\beta}_{0}=-20$ (

**bottom left**) and ${\beta}_{0}=-25$ (

**bottom right panel**). The color scale measures the central value of $\left|\phi \right|$. This sequence of plots (from top left to bottom right) shows the upper end of the S branch disconnecting and separating from the W branch as ${\beta}_{0}$ becomes more negative.

**Figure 7.**The branches of NS models are shown in the ${M}_{b}-{R}_{S}$ plane in the bottom panels for $\mu =4.8\times {10}^{-13}\phantom{\rule{0.166667em}{0ex}}\mathrm{eV}$, ${\alpha}_{0}={10}^{-4}$ as well as ${\beta}_{0}=-6$ (

**left**) and ${\beta}_{0}=-17$ (

**right**). Several NS models are marked along the branches as colored circles. The top panels show the radial profiles of the baryon density $\rho \left(r\right)$ and the scalar field $\phi \left(r\right)$ for these NSs using their respective color. The density profile always reaches a maximum at the origin; however, the scalar field profile in some cases reaches a peak at a non-zero radius.

**Figure 8.**Plots showing the distribution of stable (green) and unstable (black) NS configurations in the ${M}_{b}-{R}_{S}$ plane. When two solutions with the same baryon mass ${M}_{b}$ exist, the one with the lower ADM mass is energetically favored. The scalar parameters are $\mu =4.8\times {10}^{-13}$ eV, ${\alpha}_{0}=-{10}^{-4}$ and, from top left to bottom right, ${\beta}_{0}=-5$, $-5.5$, $-6$ and $-10$.

**Figure 9.**Same as Figure 8 using scalar mass $\mu =4.8\times {10}^{-13}$ eV, and coupling parameters from top-left to bottom, $({\alpha}_{0},{\beta}_{0})=({10}^{-3},-4.5)$, $({10}^{-2},-5.5)$, $(3\times {10}^{-2},-5)$.

© 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**

Rosca-Mead, R.; Moore, C.J.; Sperhake, U.; Agathos, M.; Gerosa, D.
Structure of Neutron Stars in Massive Scalar-Tensor Gravity. *Symmetry* **2020**, *12*, 1384.
https://doi.org/10.3390/sym12091384

**AMA Style**

Rosca-Mead R, Moore CJ, Sperhake U, Agathos M, Gerosa D.
Structure of Neutron Stars in Massive Scalar-Tensor Gravity. *Symmetry*. 2020; 12(9):1384.
https://doi.org/10.3390/sym12091384

**Chicago/Turabian Style**

Rosca-Mead, Roxana, Christopher J. Moore, Ulrich Sperhake, Michalis Agathos, and Davide Gerosa.
2020. "Structure of Neutron Stars in Massive Scalar-Tensor Gravity" *Symmetry* 12, no. 9: 1384.
https://doi.org/10.3390/sym12091384