Boson stars with negative Gauss–Bonnet coupling

In this paper, we discuss asymptotically flat and anti-de Sitter (AdS) boson star in (4+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(4+1)$$\end{document}-dimensional Gauss–Bonnet gravity. We describe the dependence of the mass, the charge and the radius of the boson star on the model parameters, such as Gauss–Bonnet coupling α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}, cosmological constant Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} and gravitational constant κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}. The basic properties of the solutions of boson stars have been studied for the different negative values of Gauss–Bonnet coupling. We found that when κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} is large and α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is negative enough, the spiral shrinks and pulls back to the larger internal frequency ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}, and there is only one branch exists. We have also observed that when κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} is small enough and if α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is close to zero, the spiral will unfold.

In this paper, we study the Gauss-Bonnet boson stars in asymptotically flat and AdS space-time. We construct the solutions numerically using COLSYS [3,4] (Fortran ODE solver package). We describe the dependence of the mass M, charge Q and radius R of the boson stars on the model parameters such as the Gauss-Bonnet coupling α, the cosmological constant and the gravitational constant κ. Previously, this subject has been studied in flat space-time in [28], but for positive α. Here we pay our attention to the negative case of this coupling parameter and also extended it studying in AdS space-time.
The paper is organized as follows: in Sect. 2, we introduce the basic model for boson star and derive the field equations using appropriate boundary conditions. Next, in Sect. 3, we present and discuss our numerical results for different values of model parameters. And finally, in Sect. 4, some concluding remarks are given.
Throughout the paper, we use a space-like signature as (−, +, +, +) and a system of units c = 1.

The model
In this section we construct asymptotically flat and anti-de Sitter (AdS) boson stars in (4 + 1)-dimensional Gauss-Bonnet gravity. We consider standard Einstein-Gauss-Bonnet theory minimally coupled to a complex valued and self-interacting scalar field. The action for boson star model in five-dimensional anti-de Sitter space-time in Gauss-Bonnet gravity reads: where = −6/ 2 is the cosmological constant, α is the Gauss-Bonnet coupling and G 5 is Newton's constant in 5 dimensions. L matter is the matter Lagrangian for the complex scalar field ψ and reads : where U (ψ) is the scalar field potential that arises in gauge-mediated supersymmetric breaking in the Minimal Supersymmetric extension of the Standard Model (MSSM) and it is given by the expression where σ corresponds to the scale below which super-symmetry is broken, while m denotes the scalar boson mass. This potential is not differentiable at |ψ| = σ . Therefore the following approximation of the above potential has been suggested [20]: For simplicity we develop this potential into a series and keep the terms only up to 6th order in ψ Using the variation principle we can derive the gravity and Klein-Gordon equations as follows: − ∂U ∂|ψ| 2 ψ = 0, where the tensor H M N is given by and T M N is the energy-momentum tensor Since the matter Lagrangian is invariant under the global U(1) transformation the system posses the locally conserved Noether current j M and the globally conserved Noether charge Q. The symmetry for this transformation is given by with a conserved current: and a conserved charge, namely, the number of scalar particles:

Ansatz, field equations and boundary conditions
We choose the following Ansatz for the metric: where N and A are functions of r only. We further choose such that n(∞) will determine the gravitational mass of the solution at infinity. For the scalar field, we choose the following stationary Ansatz where ω is the internal frequency and φ is function of r only. Imposing the Ansatz (13) and (15) into the field equations (6), (7) we can derive the equations of motion as follows: Here the prime denotes the derivative with respect to r . These equations possess the following scaling symmetries:  (20) where M pl is the Planck mass.
To solve these equations, we have to set the appropriate boundary conditions at the origin r = 0 and as well as at infinity. At the origin, we require the regularity conditions and at infinity Since above system does not have any analytic solution we solve it numerically.

Definition of mass, charge and radius
As shown before in [1,2,5,27,30,31], we follow [28] and use the same definitions for the radius R, charge Q and mass M as follows:  Since A ≡ 1 and n ≡ 0 when κ = 0, we can use the following definition of mass and for κ = 0 case we can use the asymptotic behaviour of the metric function at infinity and the mass can be read off as where M n(∞) and n 1 is a constant that depends on AdS radius .

Numerical results
The goal of the paper is to study the basic properties of the boson stars in the presence of negative Gauss-Bonnet coupling in 5-dimensional asymptotically flat and AdS space-time. Let us first start with a flat space-time.   But it is very difficult to find the exact value for α where the 3rd branch disappears. If one compare the 3rd and 2nd branches we can easily see from Figs. 1, 2 and 3 that the mass M, charge Q and radius R always take the higher values on the 3rd branch than the value of 2nd branch at ω cr . The values of the 2nd branch are lower than the values of the 1st branch at ω min . If we continue decreasing α further the 2nd branch also disappears. As a result we end up only with one fundamental branch. In [28], they observed that for large enough positive α the spiral unfolds. We also observed similar effect in Figs. 1a, 2a and 3a for the negative case if the values of κ and α are small enough (say κ = 0.0015 and α < −2.0). If we keep κ fixed and decrease α further we again observe the spiralling behaviour for large negative values of α (see Fig. 5). Similarly, fixing α and increasing κ lead the solutions spiralling (see Figs. 6, 7).
We have also studied the behaviour of the scalar field function at the origin. It is plotted in Fig. 4. In this figure, we give the values of the scalar field function at the origin, φ(0), as function of ω for different values of α and κ. As shown from Fig. 4, we find that the range of values of φ(0) is limited and the maximal value for φ(0) decreases with decreasing α. In the positive α case (see [28]) the ω min decreases with increasing α and ω cr takes oscillating behaviour. But in the case when α is negative, the values of ω min increase with decreasing α and ω cr first decreases until some critical α = α cr and then it starts to increase further with decreasing α. It is shown more clearly in Fig. 8. At the critical point (the point which two solutions join) numerics become very difficult. In this point, the tip of the metric function N (r ) at some r = r cr and as well the central value of the metric function A(0) seems to drop forward to zero. To understand the behaviour of these solutions in more detail we plotted the profiles of metric function in Figs. 9, 10 and 11. Our observations show that the value of metric function A(r ) at the origin and the tip of the metric function N (r ) decrease with increasing φ(0) for fixed α and κ. Numerically, it is very difficult to reach A(0) = 0 and N (r cr ) = 0 limit which corresponds to φ(0) → ∞. It would be interesting to see wether A(0) ≡ 0 and N (r cr ) ≡ 0 at the critical point. Since our numerical code does not converge near φ(0) → ∞, we could not reach this point.
As we can see from Fig. 12, the minima of the metric function N (r ) increases with decreasing α for fixed value of φ(0). Hence it takes opposite character for the metric function A(r). The value of metric function A(r ) at the origin decreases with decreasing α.   Fig. 9. Here = 0 the opposite, the values of the 3nd branch are lower than the values of the 2nd branch at ω cr . If we continue decreasing α further, 2nd branch also disappears. As a result we end up only with one fundamental branch. Similarly, as in the flat case, when α is negative, the values of ω min increase with decreasing α and ω cr first decreases until some critical α = α cr and then it starts to increase further with decreasing α. It is shown more clearly in Fig. 22 for = −0.02 and κ = 0.01.
Next we discuss the properties of solutions for different values of cosmological constant for fixed Gauss-Bonnet coupling α and the gravitational constant κ. If one compare the maximal mass of boson stars for < 0 with = 0 case (see Figs. 23,24), it is shown clearly from the figures that in the flat case it is always higher than AdS case. The same feature occurs for maximal charge Q max and maximal radius R max . In the flat case, the maximal charge and the maximal radius is always bigger than the case in AdS. Comparing the flat case with the AdS space time the small decrease of lowers the maximal mass M max suddenly. These can be seen in Fig. 23a. But with decreasing , the maximal frequency ω max increases being ω max > 1.

Conclusion
In this paper, we have studied the properties of asymptotically flat and anti-de Sitter boson stars in fivedimensional Gauss-Bonnet gravity in more detail for different values of the cosmological constant , Gauss-Bonnet coupling α and the gravitational constant κ. First, we studied the flat case. In [28], the authors showed that the spiralling behaviour characteristic for boson stars is observed for α = 0. But our analysis show that this is valid only if κ is large enough. We find that the spiralling behaviour disappears for small enough κ even when α = 0. We also observed that the maximal mass M, the maximal charge Q and the maximal frequency is always larger in excited solutions than the ground solutions. Comparing the flat case with the AdS space time the small decrease of lowers the maximal mass M max , maximal charge Q max and minimal frequency ω min . On the basis of our numerical analysis, we can state: • the maximal mass M max , the maximal charge Q max , the maximal radius R max and the minimal radius R min of the boson star decreases with decreasing cosmological constant ; • the minimal and the maximal internal frequency increases with decreasing cosmological constant ; • the ω min increases with decreasing Gauss-Bonnet coupling α in both flat and AdS space-time; • the ω cr decreases with decreasing Gauss-Bonnet coupling α in both flat and AdS space-time; • the ω min and ω cr increases with increasing gravitation constant κ; • the maximal mass M max , the maximal charge Q max , the maximal radius R max and the minimal radius R min decreases with increasing κ in both flat and AdS space-time.
It has been previously shown [25] that the boson star solutions exist only in a limited parameter range of ω. This parameter range depends on the choice of potential and the cosmological constant . In a flat space-time for our potential (5) it obeys ω ∈ [0 : 1]. However, we observed that the maximal frequency increases with decreasing and it is always bigger than one (ω max > 1) in the AdS space-time. We find that changing the Gauss-Bonnet coupling α or the gravitational constant κ does not make any influence on the maximal frequency ω max in both flat and AdS space-time.