# Analytical study of charged boson stars with large scalar self-couplings

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## Abstract

We give good approximate analytic solutions for spherical charged boson stars in the large scalar-self-coupling limit in general relativity. We show that if the charge *e* and mass *m* of the scalar field nearly satisfy the critical relation \(e^2\approx Gm^2\) (where *G* is the Newton constant), our analytic expressions for stable solutions agree well with the numerical solutions.

## 1 Introduction

One of the great problems in astroparticle physics is the dark matter problem [1, 2]. Many non-baryonic dark matter candidates have been supposed in the last several decades; for instance, weakly interacting massive particles (WIMPs) have been studied along with development of phenomenological supersymmetric particle theory.

There is a novel idea that condensation of unknown scalar bosons as a compact object may play a role in dark matter. Such a gravitating configuration is called a boson star [3, 4, 5, 6] and serves as a simple model to solve some problems arising in astrophysics, such as galactic dynamics and stellar structure, avoiding restrictions on WIMPs and other models.

Many authors have studied various models for boson stars so far, and the studies on boson stars can yield not only a clue to astrophysical problems but also new insights into compact configurations in general relativity and in modified gravity theories on theoretical grounds.

As a specific example, stable boson stars in scalar theory with a large quartic self-coupling, first studied by Colpi et al. [7], typically have a large length scale, and thus the idea of boson stars with a galactic size naturally arises as an explanation of the flat rotation curves of galaxies [8, 9, 10, 11, 12, 13, 14].^{1}

Another interesting object is a charged boson star [3, 17, 18, 19, 20]. The typical size of charged boson stars is larger than that of neutral boson stars, because of partial subtraction of the magnitude of the attractive force by the “electric” force.

Consider a system of particles with mass *m* and charge *e*. In the limit of \(e^2\rightarrow Gm^2\), where *G* is the Newton constant, the long-range forces are mutually canceled. This fact raises a question: near the critical point \(e^2=Gm^2\), can one find some simple (or peculiar) behavior in the system?

A few decades ago, Jetzer et al. found the critical behavior in the mass of a time-independent, spherical charged boson star [3, 17]. Pugliese et al. recently investigated such behavior for gravitating charged scalar theory without scalar self-interactions [20]. Both analyses relied on numerical methods. In the present paper, we will study critical behaviors in stationary spherical charged boson stars with a large scalar self-coupling, using analytical approximations.

If the charge is near critical, i.e., \(e^2\approx Gm^2\), the equilibrium density distribution is expected to be dilute as well as large scale. The value of the central density goes to zero as the total mass becomes infinite. Because the pressure in the center of the almost critical charged boson star is small compared to the energy density, the configuration approaches a Newtonian boson star in the critical limit. Therefore, we first arrive at the idea of obtaining solutions for boson stars with a small \(\epsilon \), which represents the (appropriately normalized) central value of a square of the scalar field.

It should be noted that it is necessary to find solutions of the next order in \(\epsilon \), because the post-Newtonian effect determines the stability of a definite mass and radius. The boundary between a stable and an unstable star is given by the maximum mass. It is reported [3, 17] that the maximum mass increases with increasing gauge coupling constant. An important aim of the present paper is to reproduce this behavior semi-quantitatively in our approximation.

The present paper is organized as follows. In Sect. 2, we give the field equations for a boson star in general relativity and their large coupling limit. In Sect. 3, in order to generate simple approximate solutions, we construct an approximate differential equation for the square of the scalar field. Linearizing the equation, we obtain analytical approximate solutions expressed by trigonometric functions. The critical behavior in the mass of the boson star is qualitatively confirmed by using the approximate solutions. In Sect. 4, we reconsider the energy density of the electric field, which is ignored in Sect. 3. After including the electromagnetic contribution to the total mass, we again compare our approximate analysis with numerical results. Finally, we summarize and discuss our results in Sect. 5.

In Appendix A, a naive perturbative treatment of the equation as a power expansion of \(\epsilon \) is given. In Appendix B, we present the method which relies on the Taylor expansion in radius coordinates and estimates the mass of the stable boson star in the critical limit.

## 2 Charged boson stars in large coupling limit

*m*and charge

*e*, governed by the following action (where \(\hbar =c=1\)):

*G*is the Newton constant,

*R*is the scalar curvature, and \(F^2=g^{\mu \rho }g^{\nu \sigma }F_{\mu \nu }F_{\rho \sigma }\). The field strength is defined as \(F_{\mu \nu }=\partial _\mu A_\nu -\partial _\nu A_\mu \), where \(A_\mu \) is a

*U*(1) gauge field. The gauge field also appears in \(|D_\mu \phi |^2\equiv g^{\mu \nu }(D_\mu \phi _i)^*(D_\nu \phi _i)\), where the covariant derivative is \(D_{\mu }=\partial _\mu +ie A_\mu \). The scalar self-coupling constant \({\lambda }\) is assumed to be positive.

*r*.

*t*.

*A*(

*x*), \(\mu (x)\), and \(\delta (x)\). The surface of the spherical boson star is defined by the radius \(x=x_*\) at which \(\sigma (x_*)=\Phi ^2(x_*)=0\). Outside the boson star, it is thought that \(\Phi (x)\) vanishes for \(x>x_*\).

The numerical solutions for the system in the large coupling limit have been investigated, for example, in Refs. [3, 17]. We will consider an approximation that leads to analytic solutions for the system in the next section.

## 3 Approximate equation for square of the scalar field

When solving the field equations mathematically, we initially regard the region of definition for *A*(*x*), \(\mu (x)\), and \(\delta (x)\) as \([0,\infty )\), though physical meanings of the solutions hold only in the region of positive \(\sigma (x)=\Phi ^2(x)\); i.e., \([0, x_*]\).

^{2}) for a stable boson star.

Therefore, we can examine the expansion in terms of a “small” parameter \(\epsilon \) defined by \(\epsilon \equiv {A_0^2-1}=\sigma (0)\) for solving the differential equations. Although the equations at the lowest order of \(\epsilon \) become very simple, those at the next order are very complicated to analyze. This approach is described in Appendix A. Thus, in the present section, we consider the other approach.

Now, we investigate the relationship in terms of derivatives of \(\sigma (x)\) by using the following approximation. Because we wish to consider a stable dilute boson star, we first assume \(\frac{2\mu (x)}{x}\ll 1\) and \(\delta (x)\ll 1\), which are near-vacuum values of the variables. We will, however, take care of their derivative, which is expressed by \(\sigma (x)\) and its derivative.

We next assume \((\alpha '(x))^2\ll \sigma (x)/g^2\). This assumption implies that the energy density of the electric field is negligible compared with the energy density of the scalar field and can therefore be omitted, because the right-hand side of Eq. (3.2) is proportional to the total energy density. For finite values of \(\sigma (x)\) and small \(g^2\), this approximation is reasonable. Because we are now going to study the behavior of \(\sigma (x)\) in the nearly flat background, we take only the first order in the electric field in the present approximation. It is noteworthy that we do not assume \(\sigma (x)\ll 1\) at the first time, whose value should be small for a stable dilute boson star in the critical limit \(g\approx 0\).

^{3}

^{4}That is

In the general relativistic system, it is known that the mass of the star increases monotonically up to a maximum as the central density increases. The maximum mass defines the border between the stable and unstable configurations.

*g*,

^{5}Therefore, the deviation from the precise value is \(\sim \! 33\%\) for \(M_{*max}\) and \(\sim \! 20\%\) for \(\Phi (0)_{*max}\). The definition of the radius of the boson star in Ref. [3] is the average of \(r_*\) over the particle density. In the present approximation, the function \(\sigma (x)\) represents both the particle and the energy density. Thus, their definition of the radius should be recognized as \(R_*\approx 0.344~r_{*max}\), because the solution for \(\sigma (x)\) is proportional to \(\sin kx/kx\), and \(\{\int _0^\pi x(\sin x/x) dx\}/\{\pi \int _0^\pi (\sin x/x) dx\}\approx 0.344\). Our approximate value is \(R_*=0.415\times 2^{3/4}\times \sqrt{2}\frac{1}{\sqrt{1-q^2}}=0.985\frac{1}{\sqrt{1-q^2}} =0.344\times 2.86\frac{1}{\sqrt{1-q^2}}\). The deviation of \(r_{*max}\) is considered to be \(\sim \! 58\%\).

Finally, in this section, we mention that it is also possible to show the qualitative \(M_*\)-\(\sqrt{\epsilon }\) relation in the other approximation. The approximation, which utilizes the approximate functions of the order \(O(x^2)\), is shown in Appendix B.

In the next section, we reconsider the definition of mass and inclusion of the energy density of the electric field.

## 4 The energy of the electric field

In the approximation scheme in the last section, we assumed \((\alpha ')^2\ll \sigma /g^2\). This corresponds to the omission of the energy density of the electric field as it is negligible compared with the energy density of the scalar field.

In the present section, we estimate the contribution of the electric energy density to the boson star mass. Although accounting for the electric energy in addition to the scalar energy sounds inconsistent judging from the ansatz, it can be considered that the configuration is the main source field of all the fields because the approximation for the scalar field configuration \(\sigma (x)\) fits numerical computations very well. Thus, we insist that the addition of the electric energy density has a physical meaning.

The definition of mass in Refs. [3, 17] includes the contribution of the electric field. It is pointed out [20] that \(M_*\) (in our notation) differs from the “actual” mass (which is proportional to the coefficient of the inverse of the distance from the origin in the asymptotic region). The difference is due to the electric contribution and has been ignored in the approximation scheme in Sect. 3.

*M*is attained if \(\frac{\partial M(\epsilon )}{\partial \epsilon }=0\), which reduces to

*M*from the values in Refs. [3, 17] is now approximately \(12\%\), while \(\Phi (0)_{max}\approx \sqrt{1.53}g=1.24\sqrt{1-q^2}\), and the deviation is \(14\%\). We have now obtained good approximate values by including the electric energy contribution.

## 5 Summary and discussion

In this paper, we presented approximate solutions for dilute charged boson stars with spherical symmetry in the large scalar self-coupling limit. An approximation scheme is presented in Sect. 3, where we first consider the approximate differential equation for the square of the scalar field \(\sigma (x)\). In this approximation, we assumed that the contribution of the energy density of the electric field is relatively small. A further linearized approximation yields a fully analytic approximation for a charged boson star. In Sect. 4, we improved the approximation by reconsidering the electric energy. Because it has been recognized that solutions with an \(\epsilon \) value that is smaller than the maximum \(\epsilon _m\) value are stable and the others are unstable, our approximation has a certain physical meaning for stable configurations of charged boson stars.

We confirmed that the maximum mass of the boson star increases with the gauge coupling constant as \((\sqrt{1-q^2})^{-1}\) for a charge close to the critical charge \(e^2\approx Gm^2\) in our approximation, whose deviation from the numerical result is on the order of a few ten percent.

It was pointed out that there is a localized configuration even if the charge of the scalar field is larger than the critical coupling for scalar theory without self-coupling [20]. The analysis of the critical behavior of the maximum mass in the large self-coupling limit under consideration is nevertheless valid, because the large coupling limit does not yield higher node solutions [3], whereas only solutions with nodes exist for over-critical cases as reported in Ref. [20] for scalar theory with no self-interaction.

Our analytically approximate solutions can be used to check the validity of numerical solutions generally. Analytic solutions can also be used as a background configuration in an investigation of the quantum vacuum around charged boson stars [18], as well as the seeds of an exact solution (for instance, nonspherical) in numerical computations.

We would like to improve the approximation for not so small \(\epsilon \). To this end, we have to try a basic approach such as the Padé approximation. In Sect. 3, a nonlinear equation for \(\sigma (x)\) has been derived. We wish to use some type of renormalization group methods [21, 22, 23, 24] to evaluate the solution, though it is difficult to directly apply the known methods to the present form of the equation.

Finally, we should consider the analysis of charged boson stars in scalar theory with an arbitrary self-coupling. We hope to return to these and other subjects in future work.

## Footnotes

- 1.
- 2.
Even if the self-interaction is absent, the density is expected to become low due to the uncertainty principle (“quantum force”).

- 3.
A comment is given in Sect. 5.

- 4.
- 5.
The factor \(2^{3/4}\) comes from the different definition of critical charge and its power, and the factor \(\sqrt{2}\) comes from the different definition of \(\lambda /G\).

- 6.
In some cases, however, quasi-stable configurations with a very long lifetime may be admitted as astrophysical objects.

- 7.
In this scheme, the approximate values may always be larger than the numerical values. Therefore, no purpose is served by making a further correction due to the electric field.

## Notes

### Acknowledgements

We thank Prof. Kenji Sakamoto for much inspiration from his master’s thesis submitted about two decades ago.

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