Stability and Formation of Bosonic Stars

An equilibrium bosonic star can exist yet be linearly unstable, fail under a finite perturbation, or be unreachable from plausible initial data. Stability and formation must therefore be tested with distinct evidence: turning points on constrained equilibrium families, perturbation spectra, converged nonlinear evolutions, and surveys over initial conditions. Results do not transfer automatically between scalar, vector, rotating, excited, or self-interacting branches. This comparison is current through 2026-08-18.

Required background. The Einstein equations supply the coupled geometry and field evolution.

Helpful background. Stress–Energy and Conservation supplies mass and charge budgets; Relativistic Matter and Compact Objects supplies equilibrium and stability comparisons; Black-Hole Geometry supplies horizon diagnostics; and Computational Relativity supplies perturbative and nonlinear validation.

Four different questions

Existence asks whether regular, asymptotically flat boundary data solve the stationary Einstein–matter equations. Linear stability asks whether a small perturbation contains a mode with growth rate $\operatorname{Im}\sigma>0$ under $e^{-i\sigma t}$. Nonlinear survival asks whether a finite perturbation migrates to another star, disperses, or collapses. Formation asks whether an open family of physically motivated initial data reaches the stable branch with acceptable efficiency.

For a complex scalar sequence labeled by central amplitude or field frequency, mass $M$ and Noether charge $Q$ obey

$$dM=\omega\,dQ$$

along a regular one-parameter family. Coincident extrema of $M$ and $Q$ can mark a change in the number of unstable radial modes when the constrained turning-point assumptions hold. They do not reveal nonradial modes, prove nonlinear stability, or classify a different potential. Liebling and Palenzuela review equilibrium branches, perturbations, evolutions, and formation scenarios in Liebling and Palenzuela 2023, §§ 4.1.1–4.1.3 and 4.2.

The role of the conserved charge is easiest to see variationally. A stationary star is a constrained critical point of mass: at a given background frequency, $\delta(M-\omega Q)=0$. Consequently a nearby equilibrium satisfies the first law above. Whether perturbations increase or decrease the constrained energy depends on its second variation, after solving the gravitational constraints and removing gauge and symmetry directions. An extremum along one plotted curve supplies only one direction through this larger space.

There is also a useful energetic check. For a massive complex field with $Q\gt0$, separated free bosons carrying that charge have energy at least $\mu Q$. A configuration with $M\lt\mu Q$ cannot completely disperse into those particles while conserving total energy and charge. This excludes one final state, not every instability: collapse, fragmentation, or migration to a lower-mass bound configuration may still be possible. The sign of binding energy and the signs of perturbation eigenvalues answer different questions.

Linear modes and nonlinear fates

Radial perturbations form an eigenvalue problem after the constraint variables are consistently eliminated. A fundamental eigenvalue crossing $\sigma_0^2=0$ provides stronger local evidence than a mass maximum alone. The perturbation calculation must state what is held fixed, how gauge and constraints are treated, and which boundary conditions enforce regularity and decay.

Nonlinear evolution supplies a different test. An unstable star with slightly too little binding can disperse; one with suitable mass and charge can migrate toward a stable branch by ejecting scalar radiation; a more compact configuration can form an apparent horizon. Mass, charge, constraints, and boundary fluxes must close before the outcome is interpreted. A simulation that ends before secular or nonaxisymmetric growth cannot establish long-term survival.

Rotation opens additional azimuthal modes, and excited states with radial or angular nodes need separate spectra. A nodeless spherical scalar branch may have a familiar first turning point, while a rotating vector star or a solitonic thin-wall branch can have different instability structure. Competing positions based on “before the mass maximum is stable” and on direct mode/evolution evidence agree only inside the former statement’s assumptions.

Comparison of methods and results

The matrix distinguishes methods and model classes as of 2026-08-18.

Model or channel	Evidence type and result	Limitation or contrary evidence	Discriminating check
Nodeless spherical mini scalar	Turning point, radial spectrum, and nonlinear evolutions support a stable low-density branch and an unstable branch beyond the first relevant extremum.	The result assumes the canonical potential and radial family; it does not exclude nonradial or very slow modes in other models.	Compare the sign of $\sigma_0^2$ with both sides of the constrained turning point and evolve perturbed data at multiple resolutions.
Excited scalar states	Stationary solutions with nodes exist.	Many decay, migrate, or collapse under allowed perturbations; existence and node count do not imply viability.	Compute radial and nonradial spectra and test whether self-interaction or conserved quantities block the decay channel.
Rotating scalar or Proca stars	Conserved $J=mQ$ organizes stationary branches, and some evolutions find long-lived regions.	Nonaxisymmetric modes, mode competition, and symmetry-restricted simulations can change the conclusion.	Use unconstrained three-dimensional perturbations, vary azimuthal seeds, and evolve longer than the measured growth time.
Gravitational cooling	Diffuse scalar initial data can eject excess field energy and settle into a bound core in numerical evolutions.	Efficiency depends on mass, charge, phase coherence, initial profile, potential, and boundary loss; it does not predict cosmological abundance.	Survey an open initial-data family with converged mass/charge fluxes rather than tune one profile.
Bosonic-star mergers	Mergers can yield a remnant, disperse, or collapse depending on phase, boost, charge, compactness, and interaction.	Equal-mass or symmetry-restricted cases do not define a population formation rate.	Perform a three-dimensional parameter survey with consistent waveform, remnant, and conservation diagnostics.

Seidel and Suen demonstrated gravitational cooling in Seidel and Suen 1994, Eq. (1), Figures 1–3, pp. 2516–2518. That is evidence for formation from the studied scalar initial data, not evidence that generic cosmological collapse populates the full equilibrium family. A primary example of a radial-mode calculation is Gleiser and Watkins 1989, abstract, which describes the instability of sufficiently dense mini-boson stars. The perturbation equations and their interpretation are summarized explicitly in Liebling and Palenzuela 2023, § 4.1.1, Eqs. (86)–(88).

Formation is not abundance

Gravitational cooling is the emission of scalar waves that removes energy while leaving a bound core; it is not thermodynamic cooling. A complex field also carries charge, so the final mass and charge must be compatible with a stable equilibrium. A real oscillaton has no exact U(1) charge and slowly radiates. Self-interaction, expansion, ambient matter, or incoherent field phases can alter capture and relaxation.

Astrophysical viability needs an initial-data distribution and production environment, not just one successful evolution. The output must be folded into an abundance model without assuming every available boson condenses. Formation efficiency, mass function, spin distribution, merger history, and destructive encounters are independent of equilibrium existence.

The unresolved question is which stable branches are populated at appreciable rates in realistic environments. A result that could change the interpretation is a converged three-dimensional formation or merger survey, discovery of a new unstable mode, a counterexample to a constrained turning-point classification, or a quantified production channel whose abundance and observational limits are treated consistently.

Check a turning-point claim

Along a one-parameter scalar sequence, a calculation finds coincident extrema $dM/d\omega=0$ and $dQ/d\omega=0$. It reports no perturbation spectrum or time evolution. Does this prove that stability changes there?

Solution

No: the two extrema alone do not establish the hypotheses or the initial unstable-mode count. A valid constrained turning-point theorem can establish a radial stability change without a new time evolution, but one must verify its family, conserved quantities, regularity, nondegeneracy and mode-count assumptions. Alternatively, compute the radial spectrum and follow an eigenvalue through zero; it need not be the fundamental mode at every later turning point. Evolving perturbations on both sides, with mass, charge, constraint and boundary-flux checks, gives a separate dynamical test. Nonradial stability, nonlinear survival and formation remain distinct.

References

• Gleiser, Marcelo, and Richard Watkins. “Gravitational Stability of Scalar Matter.” Nuclear Physics B 319 (1989): 733–746. DOI.
• Liebling, Steven L., and Carlos Palenzuela. “Dynamical Boson Stars.” Living Reviews in Relativity 26 (2023): 1. DOI.
• Seidel, Edward, and Wai-Mo Suen. “Formation of Solitonic Stars through Gravitational Cooling.” Physical Review Letters 72 (1994): 2516–2519. DOI.