# Structure of Neutron Stars in Massive Scalar-Tensor Gravity

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## 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

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**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, $({\alpha}_{0},{\beta}_{0})=({10}^{-3},-4.5)$, $({10}^{-2},-5.5)$, $(3\times {10}^{-2},-5)$.

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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