4.1 Gravitational stability

A linear stability analysis consists of studying the time evolution of infinitesimal perturbations about an equilibrium configuration, usually with the additional constraint that the total number of particles must be conserved. In the case of spherically symmetric, fermionic stars described by perfect fluid, it is possible to find an eigenvalue equation for the perturbations that determines the normal modes and frequencies of the radial oscillations (see, for example, Ref. [86]). Stability theorems also allow for a direct characterization of the stability branches of the equilibrium solutions [91Jump To The Next Citation Point, 60Jump To The Next Citation Point]. Analogously, one can write a similar eigenvalue equation for boson stars and show the validity of similar stability theorems. In addition to these methods, the stability of boson stars has also been studied using two other, independent methods: by applying catastrophe theory and by solving numerically the time dependent Einstein–Klein–Gordon equations. All these methods agree with the results obtained in the linear stability analysis.

4.1.1 Linear stability analysis

Assume that a spherically symmetric boson star in an equilibrium configuration is perturbed only in the radial direction. The equations governing these small radial perturbations are obtained by linearizing the system of equations in the standard way; expand the metric and the scalar field functions to first order in the perturbation and neglect higher order terms in the equations [93, 118]. Considering the collection of fields for the system fi, one expands them in terms of the background solution 0 fi and perturbation as

f (r,t) = 0f (r) + 1f(r)eiσt, (79 ) i i i
which assumes harmonic time dependence for the perturbation. Substitution of this expansion into the system of equations then provides a linearized system, which reduces to a set of coupled equations that determines the spectrum of modes 1fi and eigenvalues σ2
1 2 1 Lij fi = σ Mij fi , (80 )
where Lij is a differential operator containing partial derivatives and Mij is a matrix depending on the background equilibrium fields 0f i. Solving this system, known as the pulsation equation, produces the spectrum of eigenmodes and their eigenvalues σ.

The stability of the star depends crucially on the sign of the smallest eigenvalue. Because of time reversal symmetry, only σ2 enters the equations [148], and we label the smallest eigenvalue σ20. If it is negative, the eigenmode grows exponentially with time and the star is unstable. On the other hand, for positive eigenvalues the configuration has no unstable modes and is therefore stable. The critical point at which the stability transitions from stable to unstable, therefore, occurs when the smallest eigenvalue vanishes, σ = 0 0.

Equilibrium solutions can be parametrized with a single variable, such as the central value of the scalar field ϕc, and so we can write M = M (ϕc) and N = N (ϕc). Stability theorems then indicate that transitions between stable and unstable configurations occur only at critical points of the parameterization (M ′(ϕ ) = N ′(ϕ ) = 0 c c[60, 91, 107, 207]. Linear perturbation analysis provides a more detailed picture such as the growth rates and the eigenmodes of the perturbations.

Ref. [94] carries out such an analysis for perturbations that conserve mass and charge. They find the first three perturbative modes and their growth rates, and they identify at which precise values of ϕc these modes become unstable. Starting from small values, they find that ground state BSs are stable up to the critical point of maximum mass. Further increases in the central value subsequently encounter additional unstable modes. This same type of analysis applied to excited state BSs showed that the same stability criterion applies for perturbations that conserve the total particle number [120]. For more general perturbations that do not conserve particle number, excited states are generally unstable to decaying to the ground state.

A more involved analysis by [148] uses a Hamiltonian formalism to study BS stability. Considering first order perturbations that conserve mass and charge (δN = 0), their results agree with those of [94, 120]. However, they extend their approach to consider more general perturbations, which do not conserve the total number of particles (i.e., δN ⁄= 0). To do so, they must work with the second order quantities. They found complex eigenvalues for the excited states that indicate that excited state boson stars are unstable. More detail and discussion on the different stability analysis can be found in Ref. [122].

Catastrophe theory is part of the study of dynamical systems that began in the 1960s and studies large changes in systems resulting from small changes to certain important parameters (for a physics-oriented review see [204]). Its use in the context of boson stars is to evaluate stability, and to do so one constructs a series of solutions in terms of a limited and appropriate set of parameters. Under certain conditions, such a series generates a curve smooth everywhere except for certain points. Within a given smooth expanse between such singular points, the solutions share the same stability properties. In other words, bifurcations occur at the singular points so that solutions after the singularity gain an additional, unstable mode. Much of the recent work in this area confirms the previous conclusions from linear perturbation analysis [209, 210, 211, 212] and from earlier work with catastrophe theory [140].

Another recent work using catastrophe theory finds that rotating stars share a similar stability picture as nonrotating solutions [135Jump To The Next Citation Point]. However, only fast spinning stars are subject to an ergoregion instability [44].

4.1.2 Non-linear stability of single boson stars

The dynamical evolution of spherically symmetric perturbations of boson stars has also been studied by solving numerically the Einstein–Klein–Gordon equations (Section 2.3), or its Newtonian limit (Section 3.2), the Schrödinger–Poisson system. The first such work was Ref. [196] in which the stability of the ground state was studied by considering finite perturbations, which may change the total mass and the particle number (i.e., δN ⁄= 0 and δM ⁄= 0). The results corroborated the linear stability analysis in the sense that they found a stable and an unstable branch with a transition between them at a critical value, ϕcrit, of the central scalar field corresponding to the maximal BS mass 2 Mmax = 0.633 M Planck∕m.

The perturbed configurations of the stable branch may oscillate and emit scalar radiation maintaining a characteristic frequency ν, eventually settling into some other stable state with less mass than the original. This characteristic frequency can be approximated in the non-relativistic limit as [196]

π mGM ν = -----2 − -------, (81 ) 4mR 2πR
where R is the effective radius of the star and M its total mass. Scalar radiation is the only damping mechanism available because spherical symmetry does not allow for gravitational radiation and because the Klein–Gordon equation has no viscous or dissipative terms. This process was named gravitational cooling, and it is extremely important in the context of formation of compact bosonic objects [198] (see below). The behavior of perturbed solutions can be represented on a plot of frequency versus effective mass as in Figure 9View Image. Perturbed stars will oscillate with a frequency below its corresponding solid line and they radiate scalar field to infinity. As they do so, they lose mass by oscillating at constant frequency, moving leftward on the plot until they settle on the stable branch of (unperturbed) solutions.
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Figure 9: Oscillation frequencies of various boson stars are plotted against their mass. Also shown are the oscillation frequencies of unstable BSs obtained from the fully nonlinear evolution of the dynamical system. Unstable BSs are observed maintaining a constant frequency as they approach a stable star configuration. Reprinted with permission from [196]; copyright by APS.

The perturbed unstable configurations will either collapse to a black hole or migrate to a stable configuration, depending on the nature of the initial perturbation. If the density of the star is increased, it will collapse to a black hole. On the other hand, if it is decreased, the star explodes, expanding quickly as it approaches the stable branch. Along with the expansion, energy in the form of scalar field is radiated away, leaving a very perturbed stable star, less massive than the original unstable one.

This analysis was extended to boson stars with self-interaction and to excited BSs in Ref. [15], showing that both branches of the excited states were intrinsically unstable under generic perturbations that do not preserve M and N. The low density excited stars, with masses close to the ground state configurations, will evolve to ground state boson stars when perturbed. The more massive configurations form a black hole if the binding energy EB = M − N m is negative, through a cascade of intermediate states. The kinetic energy of the stars increases as the configuration gets closer to EB = 0, so that for positive binding energies there is an excess of kinetic energy that tends to disperse the bosons to infinity. These results are summarized in Figure 10View Image, which shows the time scale of the excited star to decay to one of these states.

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Figure 10: The instability time scale of an excited boson star (the first excitation) to one of three end states: (i) decay to the ground state, (ii) collapse to a black hole, or (iii) dispersal. Reprinted with permission from [15]; copyright by APS.

More recently, the stability of the ground state was revisited with 3D simulations using a Cartesian grid [102]. The Einstein equations were written in terms of the BSSN formulation [200, 23], which is one of the most commonly used formulations in numerical relativity. Intrinsic numerical error from discretization served to perturb the ground state for both stable and unstable stars. It was found that unstable stars with negative binding energy would collapse and form a black hole, while ones with positive binding energy would suffer an excess of kinetic energy and disperse to infinity.

That these unstable stars would disperse, instead of simply expanding into some less compact stable solution, disagrees with the previous results of Ref. [196], and was subsequently further analyzed in [104Jump To The Next Citation Point] in spherical symmetry with an explicit perturbation (i.e., a Gaussian shell of particles, which increases the mass of the star around 0.1%). The spherically symmetric results corroborated the previous 3D calculations, suggesting that the slightly perturbed configurations of the unstable branch have three possible endstates: (i) collapse to BH, (ii) migration to a less dense stable solution, or (iii) dispersal to infinity, dependent on the sign of the binding energy.

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Figure 11: Very long evolutions of a perturbed, slightly sub-critical, boson star with differing outer boundaries. The central magnitude of the scalar field is shown. At early times (t < 250 and the middle frame), the boson star demonstrates near-critical behavior with small-amplitude oscillations about an unstable solution. For late times (t > 250), the solution appears converged for the largest two outer boundaries and suggests that sub-critical boson stars are not dispersing. Instead, they execute large amplitude oscillations about low-density boson stars. Reprinted with permission from [142Jump To The Next Citation Point].

Closely related is the work of Lai and Choptuik [142Jump To The Next Citation Point] studying BS critical behavior (discussed in Section 6.1). They tune perturbations of boson stars so that dynamically the solution approaches some particular unstable solution for some finite time. They then study evolutions that ultimately do not collapse to BH, so-called sub-critical solutions, and find that they do not disperse to infinity, instead oscillating about some less compact, stable star. They show results with increasingly distant outer boundary that suggest that this behavior is not a finite-boundary-related effect (reproduced in Figure 11View Image). They use a different form of perturbation than Ref. [104], and, being only slightly subcritical, may be working in a regime with non-positive binding energy. However, it is interesting to consider that if indeed there are three distinct end-states, then one might expect critical behavior in the transition among the different pairings. Non-spherical perturbations of boson stars have been studied numerically in [14] with a 3D code to analyze the emitted gravitational waves.

Much less is known about rotating BSs, which are more difficult to construct and to evolve because they are, at most, axisymmetric, not spherically symmetric. However, as mentioned in Section 3.5, they appear to have both stable and unstable branches [135] and are subject to an ergoregion instability at high rotation rates [44]. To our knowledge, no one has evolved rotating BS initial data. However, as discussed in the next section, simulations of BS binaries [167Jump To The Next Citation Point, 173Jump To The Next Citation Point] have found rotating boson stars as a result of merger.

The issue of formation of boson stars has been addressed in [198] by performing numerical evolutions of the EKG system with different initial Gaussian distributions describing unbound states (i.e., the kinetic energy is larger than the potential energy). Quite independent of the initial condition, the scalar field collapses and settles down to a bound state by ejecting some of the scalar energy during each bounce. The ejected scalar field carries away excess ever-decreasing amounts of kinetic energy, as the system becomes bounded. After a few free-fall times of the initial configuration, the scalar field has settled into a perturbed boson star on the stable branch. This process is the already mentioned gravitational cooling, and allows for the formation of compact soliton stars (boson stars for complex scalar fields and oscillatons for real scalar fields). Although these evolutions assumed spherical symmetry, which does not include important processes such as fragmentation or the formation of pancakes, they demonstrate the feasibility of the formation mechanism; clouds of scalar field will collapse under their own self-gravity while shedding excess kinetic energy. The results also confirm the importance of the mass term in the potential. By removing the massive term in the simulations, the field collapses, rebounds and completely disperses to infinity, and no compact object forms. The evolution of the scalar field with and without the massive term is displayed in Figure 12View Image.

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Figure 12: The evolution of r2ρ (where ρ is the energy density of the complex scalar field) with massive field (left) and massless (right). In the massive case, much of the scalar field collapses and a perturbed boson star is formed at the center, settling down by gravitational cooling. In the massless case, the scalar field bounces through the origin and then disperses without forming any compact object. Reprinted with permission from [198]; copyright by APS.

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