3.1 Self-interaction potentials

Originally, boson stars were constructed with a free-field potential without any kind of self-interaction, obtaining a maximum mass with a dependence 2 M ≈ M Planck∕m. This mass, for typical masses of bosonic particle candidates, is much smaller than the Chandrasekhar mass MCh ≈ M 3Planck∕m2 obtained for fermionic stars, and so they were known as mini-boson stars. In order to extend this limit and reach astrophysical masses comparable to the Chandrasekhar mass, the potential was generalized to include a self-interaction term that provided an extra pressure against gravitational collapse.

Although the first expansion to nonlinear potentials was considered in [161] including fourth and sixth power |ϕ |-terms, a deeper analysis was performed later considering a potential with only the quartic term [57]

( 2) 2 2 λ 4 V |ϕ| = m |ϕ| + --|ϕ| , (51 ) 2
with λ a dimensionless coupling constant. Written in terms of a general potential, the EKG equations remain the same. The families of gravitational equilibrium can be parametrized by the single dimensionless quantity Λ ≡ λ∕ (4 πGm2 ). The potential of Eq. (51View Equation) results in a maximum boson-star mass that now scales as
M ≈ 0.22Λ1 ∕2M ∕m = (0.1 GeV2 ) M λ1∕2∕m2 (52 ) max Planck ⊙
which is comparable to the Chandrasekhar mass for fermions with mass m ∕λ1∕4 [57]. This self-interaction, therefore, allows much larger masses than the mini-boson stars as long as Λ ≫ 1, an inequality that may be satisfied even when λ ≪ 1 for reasonable scalar boson masses. The maximum mass as a function of the central value of the scalar field is shown in Figure 3View Image for different values of Λ. The compactness of the most massive stable stars was studied in [8], finding an upper bound M ∕R ≲ 0.16 for Λ ≫ 1. Figure 4View Image displays this compactness as a function of Λ along with the compactness of a Schwarzschild BH (black hole) and non-spinning neutron star for comparison.
View Image

Figure 3: Left: The mass of the boson star as a function of the central value of the scalar field in adimensional units √ ----- σc = 4π G ϕc. Right: Maximum mass as a function of Λ (squares) and the asymptotic Λ → ∞ relation of Eq. (52View Equation) (solid curve). Reprinted with permission from [57]; copyright by APS.
View Image

Figure 4: The compactness of a stable boson star (black solid line) as a function of the adimensional self-interaction parameter Λ ≡ λ ∕(4π Gm2 ). The compactness is shown for the most massive stable star (the most compact BS is unstable). This compactness asymptotes for Λ → ∞ to the value indicated by the red, dashed line. Also shown for comparison is the compactness of a Schwarzschild BH (green dot-dashed line), and the maximum compactness of a non-spinning neutron star (blue dotted line). Reprinted with permission from [8]; copyright by IOP.

Many subsequent papers further analyze the EKG solutions with polynomial, or even more general non-polynomial, potentials. One work in particular [195] studied the properties of the galactic dark matter halos modeled with these boson stars. They found that a necessary condition to obtain stable, compact solutions with an exponential decrease of the scalar field, the series expansion of these potentials must contain the usual mass term 2 2 m |ϕ |.

More exotic ideas similarly try to include a pressure to increase the mass of BSs. Ref. [2] considers a form of repulsive self-interaction mediated by vector mesons within the mean-field approximation. However, the authors leave the solution of the fully nonlinear system of the Klein–Gordon and Proca equations to future work. Ref. [18] models stars made from the condensation of axions, using the semi-relativistic approach with two different potentials. Mathematically this approach involves averages such that the equations are equivalent to assuming the axion is constituted by a complex scalar field with harmonic time dependence.

Other generalizations of the potential allow for the presence of nontopological soliton solutions even in the absence of gravity, with characteristics quite different than those of the mini-boson stars. In order to obtain these solutions the potential must satisfy two conditions. First, it must be a function of |ϕ|2 to preserve the global U(1) invariance. Second, the potential should have an attractive term, bounded from below and positive for |ϕ| → ∞. These conditions imply a potential of at least sixth order, a condition that is satisfied by the typical degenerate vacuum form [147, 89, 90]

( )2 ( 2) 2 2 |ϕ|2 V |ϕ | = m |ϕ| 1 − --2- , (53 ) ϕ0
for which the potential has two degenerate minima at ± ϕ 0. The case |ϕ| = 0 corresponds to the true vacuum state, while |ϕ| = ϕ0 represents the degenerate vacuum state.

The resulting soliton solution can be split in three different regions. When gravity is negligible, the interior solution satisfies ϕ ≈ ϕ0, followed by a shell of width 1∕m over which ϕ changes from ϕ0 to zero, and an exterior that is essentially vacuum. This potential leads to a different scaling of the mass and the radius than that of the ground state [149]

4 2 2 2 Mmax ≈ M Planck∕ (m ϕ0), Rmax ≈ M Planck∕(m ϕ0). (54 )

There is another type of non-topological soliton star, called Q-stars [155], which also admits soliton solutions in the absence of gravity (i.e., Q-balls [56, 149]). The potential, besides being also a function of 2 |ϕ|, must satisfy the following conditions: it must behave like 2 ≈ ϕ near ϕ = 0, it has to be bounded < ϕ2 in an intermediate region and must be larger > ϕ2 for |ϕ| → ∞. The Q-stars also have three regions; an interior solution of radius R ≈ MPlanck∕ϕ20, (i.e., ϕ0 ≈ m is the free particle inverse Compton wavelength) a very thin surface region of thickness 1∕ ϕ 0, and finally the exterior solution without matter, which reduces to Schwarzschild in spherical symmetry. The mass of these Q-stars scales now as 3 2 M Planck∕ϕ 0. The stability of these Q-stars has been studied recently using catastrophe theory, such as [209, 135Jump To The Next Citation Point]. Rotating, axisymmetric Q-balls were constructed in [133, 134]. Related, rotating solutions in 2+1 with the signum-Gordon equation instead of the KG equation are found in [10].

Other interesting works have studied the formation of Q-balls by the Affleck–Dine mechanism [125], their dynamics in one, two and three spatial dimensions [22], and their viability as a self-interacting dark matter candidate [139].

Ref. [29] considers a chemical potential to construct BSs, arguing that the effect of the chemical potential is to reduce the parameter space of stable solutions. Related work modifies the kinetic term of the action instead of the potential. Ref. [1] studies the resulting BSs for a class of K field theories, finding solutions of two types: (i) compact balls possessing a naked singularity at their center and (ii) compact shells with a singular inner boundary which resemble black holes. Ref. [3] considers coherent states of a scalar field instead of a BS within k-essence in the context of explaining dark matter. Ref. [72] modifies the kinetic term with just a minus sign to convert the scalar field to a phantom field. Although, a regular real scalar field has no spherically symmetric, local static solutions, they find such solutions with a real phantom scalar field.

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