Some New Results on Charged Compact Boson Stars

In this work we present some new results obtained in a study of the phase diagram of charged compact boson stars in a theory involving a complex scalar field with a conical potential coupled to a U(1) gauge field and gravity. We here obtain new bifurcation points in this model. We present a detailed discussion of the various regions of the phase diagram with respect to the bifurcation points. The theory is seen to contain rich physics in a particular domain of the phase diagram.

for a complex scalar field with only a conical potential, i.e., the scalar field is considered to be massless. Such a choice is possible for boson stars in a theory with a conical potential, since this potential yields compact boson star solutions with sharp boundaries, where the scalar field vanishes. This is in contrast to the case of non-compact boson stars, where the the mass of the scalar field is a basic ingredient for the asymptotic exponential fall-off of the solutions.
We construct the boson star solutions of this theory numerically. Our numerical method is based on the Newton-Raphson scheme with an adaptive stepsize Runge-Kutta method of order 4. We have calibrated our numerical techniques by reproducing the work of Refs. [1,2] and [14,15,16,17].
We consider the theory defined by the following action (with V (|Φ|) := λ|Φ|, where λ is a constant parameter): tional constant. Also, g = det(g µν ), where g µν is the metric tensor, and the asterisk in the above equation denotes complex conjugation. Using the variational principle, the equations of motion are obtained as: The energy-momentum tensor T µν is given by −g µν ((D α Φ) * (D β Φ)) g αβ − g µν λ(|Φ|) .
To construct spherically symmetric solutions we adopt the static spherically symmetric metric with Schwarzschildlike coordinates This leads to the components of Einstein tensor (G µν ) Here the arguments of the functions A(r) and N (r) have been suppressed. For solutions with a vanishing magnetic field, the Ansätze for the matter fields have the form: We introduce new constant parameters: Here a := α 2 is dimensionless. We then redefine φ(r) and A t (r): Introducing a dimensionless coordinater defined byr := The equations of motion in terms of h(r) and b(r) (where the primes denote differentiation with respect tô r, and sign(h) denotes the usual signature function) read: We thus obtain the set of equations: For the metric function A(r) we choose the boundary condition A(r o ) = 1, wherer o is the outer radius of the star. For constructing globally regular ball-like boson star solutions, we choose: In the exterior regionr >r o we match the Reissner-Nordström solution.
The theory has a conserved Noether current: The charge Q of the boson star is given by For all boson star solutions we obtain the mass M (in the units employed): We now study the numerical solutions of Eqs. IIA and IIB (as seen in Fig. 1(a)). The asterisks seen in The regions IA, IIA and IIB do not have any further bifurcation points. However, the region IB is seen to contain rich physics as evidenced by the occurrence of more bifurcation points in this region. For better detail, the re-       Fig. 1). The region IB shown in Fig. (a) is separately depicted in detail in Fig. (b) and similarly a part of the region shown in Fig. (b) is separately depicted in detail in Fig. (c). The asterisks shown in gion IB is magnified in Fig. 1(b). The region IB is then further divided into the regions IB1, IB2 and IB3 in the vicinity of B 2 , as seen in Fig. 1(b).
The region IB3 finally is seen to have the further bifurcation point B 3 . In the vicinity of B 3 we therefore further subdivide the phase diagram into the regions IB3a, IB3b and IB3c, as seen in Fig. 1(c). The region IB3b is seen to have closed loops and the behaviour of the phase diagram in this region is akin to the one of the region IB2. Also, the insets shown in Figs. 1(b) and 1(c) represent parts of the phase diagram with higher resolution.
The figures demonstrate, that as we change the value of a from a = 0.225 to a = 0, we observe a lot of new rich physics. While going from a = 0.225 to the critical value a = a c1 , we observe that the solutions exist in two separate domains, IIA and IIB (as seen in Fig. 1(a)). However, as As we decrease the value of a from the first critical value a = a c1 to the next critical value a = a c2 , we notice that the region IA in the phase diagram shows a continuous deformation of the curves, and the region IB is seen to have its own rich physics as explained in the foregoing.
As we decrease a below a c2 , we observe that in the A plot of the radiusr o of the solutions versus the vector field at the center of the star b(0) is depicted in Fig. 2(a).
As before, the point B 1 corresponds to the first bifurcation point, and the four regions IA, IB and IIA, IIB in the vicinity of the bifurcation point are indicated. Again, the region IB shown in Fig. 2(a) is enlarged and shown in Fig. 2(b), with the region IB3 being enlarged further and depicted in Fig. 2(c). The asterisks shown in Fig. 2(a)   (c) Figure 3: Fig. (a) depicts the mass M versus the radius of the star ro for the same sequence of values of the parameter a. As before, the asterisks represent the transition points from the boson stars to boson shells, and the insets magnify parts of the diagram. Fig. (b) zooms into the region of the bifurcations, with the inset giving a magnified view of the bifurcation B3. Fig. (c) is the analog of Fig. (b) for the charge Q.  In conclusion, we have studied in this work a theory of a complex scalar field with a conical potential, coupled to a U(1) gauge field and gravity [1,2].