1.1 The nature of a boson star

Boson stars (BS) are constructed with a complex scalar field coupled to gravity (as described in Section 2). A complex scalar field ϕ(t,r) can be decomposed into two real scalar fields ϕR and ϕI mapping every spacetime event to the complex plane
ϕ (t,r) ≡ ϕR (t,r) + iϕI(t,r). (1 )

Such a field possesses energy because of its spatial gradients and time derivatives and this energy gravitates holding the star together. Less clear is what supports the star against the force of gravity. Its constituent scalar field obeys a Klein–Gordon wave equation which tends to disperse fields. This is the same dispersion which underlies the Heisenberg uncertainty principle. Indeed, Kaup’s original work [126] found energy eigenstates for a semi-classical, complex scalar field, discovering that gravitational collapse was not inevitable. Ruffini and Bonazzola [188] followed up on this work by quantizing a real scalar field representing some number of bosons and they found the same field equations.

None of this guarantees that such solutions balancing dispersion against gravitational attraction exist. In fact, a widely known theorem, Derrick’s theorem [68] (see also [186]), uses a clever scaling argument to show that no regular, static, nontopological localized scalar field solutions are stable in three (spatial) dimensional flat space. This constraint is avoided by adopting a harmonic ansatz for the complex scalar field

iωt ϕ (r,t) = ϕ0 (r)e (2 )
and by working with gravity. Although the field is no longer static, as shown in Section 2 the spacetime remains static. The star itself is a stationary, soliton-like solution as demonstrated in Figure 1View Image.

There are, of course, many other soliton and soliton-like solutions in three dimensions finding a variety of ways to evade Derrick’s theorem. For example, the field-theory monopole of ’t Hooft and Polyakov is a localized solution of a properly gauged triplet scalar field. Such a solution is a topological soliton because the monopole possesses false vacuum energy which is topologically trapped. The monopole is one among a number of different topological defects that requires an infinite amount of energy to “unwind” the potential energy trapped within (see [218Jump To The Next Citation Point] for a general introduction to defects and the introduction of [189] for a discussion of relevant classical field theory concepts).

In Section 2, we present the underlying equations and mathematical solutions, but here we are concerned with the physical nature of these boson stars. When searching for an actual boson star, we look not for a quantized wave function, or even a semiclassical one. Instead, we search for a fundamental scalar, say the long-sought Higgs boson. The Large Hadron Collider (LHC) hopes to determine the existence and nature of the Higgs, with evidence at the time of writing suggesting a Higgs boson with mass 2 ≈ 125 GeV ∕c [184]. If the Higgs does not ultimately appear, there are other candidates such as an axion particle. Boson stars are then either a collection of stable fundamental bosonic particles bound by gravity, or else a collection of unstable particles that, with the gravitational binding, have an inverse process efficient enough to reach an equilibrium. They can thus be considered a Bose–Einstein condensate (BEC), although boson stars can also exist in an excited state as well.

Indeed, applying the uncertainty principle to a boson star by assuming it to be a macroscopic quantum state results in an excellent estimate for the maximum mass of a BS. One begins with the Heisenberg uncertainty principle of quantum mechanics

Δp Δx ≥ ℏ (3 )
and assumes the BS is confined within some radius Δx = R with a maximum momentum of Δp = mc where m is the mass of the constituent particle
mcR ≥ ℏ. (4 )
This inequality is consistent with the star being described by a Compton wavelength of λC = h∕(mc ). We look for the maximum possible mass M max for the boson star, which will saturate the uncertainty bound and drive the radius of the star towards its Schwarzschild radius 2 RS ≡ 2GM ∕c. Substituting yields
2Gm--Mmax-- c = ℏ , (5 )
which gives an expression for the maximum mass
1 ℏc Mmax = ------. (6 ) 2 Gm
Recognizing the Planck mass ∘ ------ MPlanck ≡ ℏc∕G, we obtain the estimate of 2 Mmax = 0.5M Planck∕m. This simple estimate indicates that the maximum mass of the BS is inversely related to the mass of the constituent scalar field. We will see below in Section 2 that this inverse relationship continues to hold with the explicit solution of the differential equations for a simple mass term in the potential, but can vary with the addition of self-interaction terms. Indeed depending on the strength of the coupling m and the other parameters of the self-interaction potential, the size and mass of the boson stars can vary from atomic to astrophysical scales. Despite their connection to fundamental physics, one can also view boson stars in analogy with models of neutron stars. In particular, as we discuss in the following sections, both types of stars demonstrate somewhat similar mass versus radius curves for their solutions with a transition in stability at local maxima of the mass. There is also a correspondence between (massless) scalar fields and a stiff, perfect fluid (see Section 2.1 and Appendix A of Ref. [35]), but the correspondence does not mean that the two are equivalent [78]. More than just an analogy, boson stars can serve as a very useful model of a compact star, having certain advantages over a fluid neutron star model: (i) the equations governing its dynamics avoid developing discontinuities, in particular there is no sharp stellar surface, (ii) there is no concern about resolving turbulence, and (iii) one avoids uncertainties in the equation of state.
View Image

Figure 1: Demonstration of the solitonic nature of the (mini-)boson star. Shown are snapshots of the magnitude squared of the complex scalar field for a head-on collision of two identical mini-boson stars. The interacting stars display an interference pattern as they pass through each other, recovering their individual identities after the collision. However, note that the BSs have a larger amplitude after their interaction and so are not true solitons. The collision can therefore be considered inelastic. Reprinted with permission from [49Jump To The Next Citation Point]. See also [141] (e.g., Figure 5.12).

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