Nonlinear Radial Stability of Boson Stars

• The paper reveals that turning points identified via catastrophe theory mark the onset and loss of radial stability in boson stars.
• It demonstrates how scalar field self-interaction creates two stable phases analogous to white dwarfs and neutron stars.
• Nonlinear simulations and perturbative analyses map the multi-branch structure, highlighting spiraling instabilities near the black hole limit.

Boson stars are gravitating configurations of a complex scalar field exhibiting macroscopic quantum coherence due to a global U(1) symmetry and corresponding Noether charge. The concept of nonlinear radial stability in boson stars refers to the dynamical robustness of a given equilibrium configuration against finite, spherically symmetric perturbations—extending beyond standard linear stability analysis and accounting for generic or strong deviations from exact equilibrium. Such an analysis explores the existence, extent, and delineation of stable solution branches, the onset of unstable regimes, and their precise connection to physical analogs such as white dwarfs, neutron stars, and black holes. Theoretical frameworks making essential use of catastrophe theory, full nonlinear simulations, and perturbative expansions reveal a rich, multi-branch phase structure governed by gravitational coupling, global/Noether charge, field self-interaction, and the underlying scalar potential.

1. Catastrophe Theory and Equilibrium Space

Nonlinear radial stability of boson stars is fundamentally determined by the topology of the solution manifold in the so-called “equilibrium space”: $\mathcal{M} = \{\omega_s,\;\kappa,\; Q\}$ where $\omega_s$ is the scalar field oscillation frequency, $\kappa$ is the dimensionless gravitational coupling constant, and $Q$ denotes the conserved Noether charge. Catastrophe theory provides a rigorous criterion: changes in stability occur exclusively at turning points (cusps or folds) where $(\partial Q/\partial \omega_s)_{\kappa}=0$. Each such turning point marks the onset or loss of stability for a given mode of the system. Evolutionary branches in this equilibrium space are segmented by these points, which correspond in parameter space (such as the plane of $(\kappa,Q)$ or $(M,Q)$) to changes in qualitative solution behavior—mirrored in the structure of the nonlinear equations’ solution curves.

As one increases the central scalar field amplitude starting from vacuum ($Q=0$), the solution curve proceeds through a succession of turning points (labeled A, B, C, …). Stability changes occur only at these points, and the sign of $(\partial Q/\partial \omega_s)_{\kappa}$ determines the character (stable vs. unstable) of each branch.

2. Structure and Significance of the Stable and Unstable Regions

Projecting the equilibrium space onto the $(\kappa,Q)$ control plane uncovers a multi-branch structure:

Branch	Parameter Range	Stability	Physical Analogy
Vacuum → A	$Q:0\to Q_A$	Stable	White dwarf–like (low density)
A → B	$Q:Q_A\to Q_B$	Unstable	Unstable region
B → C	$Q:Q_B\to Q_C$	Stable	Neutron star–like (high density)
Beyond C (spiral)	$Q:Q_C\to$ spiral	Unstable	Black hole limit, multiple modes

• First stable branch (0 to A): Analogous to white dwarfs; low central densities, stability derived from pressure-like effects of the self-interacting scalar field and gravity.
• Unstable branch (A to B): Characterized by the turning on of an unstable mode; small perturbations in this region will either collapse the star or cause migration toward one of the neighboring stable branches.
• Second stable branch (B to C): Analogous to neutron stars; higher densities, stability supported by more strongly self-interacting scalar configurations.
• Beyond C, spiral region: Features a series of extrema corresponding to further changes in the number of unstable modes, revealed as damped oscillations in $M(R)$ or $M(Q)$ diagrams. As the “black hole limit” is approached, solutions enter a regime of uncontrolled instability (“spiral instability"), akin to the termination of the neutron star branch.

3. The Spiraling Phenomenon Near the Black Hole Limit

Upon reaching the high density maxima (point C), the sequence of solutions exhibits spiral behavior in the $M(Q)$ or $M(R)$ diagram, with successive turning points reflecting the advent of new unstable (radial) modes. Mathematically, this is seen as the solution curve winding around a limiting configuration—a phenomenon common in compact-object physics and understood through the accumulation of turning points. Each local extremum marks the loss of stability of an additional mode. This is the direct analog of the instability spiraling seen in the mass-radius relation of neutron stars as they approach collapse.

Physically, in this region, configurations become increasingly compact and susceptible to dynamic collapse or nonuniqueness. Catastrophe theory mandates that only at these turning points (cusps of the spiral) does the number of unstable modes increase or decrease, and the direction of the curve’s flow through the control parameter space determines the sign.

4. Comparison with White Dwarfs and Neutron Stars

The radial stability diagram for boson stars replicates, at a formal level, the well-known Chandrasekhar–Landau stability structure of fermionic compact stars:

• White dwarfs: Supported by electron degeneracy pressure, stable up to a critical mass; beyond this point, gravitational collapse ensues.
• Neutron stars: Stable up to the “Tolman–Oppenheimer–Volkoff limit”, above which instability to collapse or dispersal dominates.
• Spiral instability: For both boson stars and neutron stars, the solution curve near the maximum mass exhibits the same “spiral” structure, indicating the presence of multiple unstable radial modes and the breakdown of equilibrium prior to black-hole formation.

In the context of boson stars, this analogy is enriched by the role of the self-interaction in the scalar sector, which can mimic the stiff equation of state of nuclear matter in neutron stars and yield a second region of stability, otherwise absent in the simpler “mini-boson star” models.

5. Role of Scalar Field Self-Interaction and Mini Boson Stars

The specific form of the scalar potential is crucial. The model considered features a non-topological soliton (Q-ball)–type potential: $$U(|\Phi|) = \lambda|\Phi|^2(|\Phi|^4 - a|\Phi|^2 + b)$$ with judiciously chosen parameters to allow for both attractive and repulsive nonlinearity. Significant self-interaction (quartic and/or sextic terms) generates an additional (higher density) stable branch. In the limit where only a mass term remains ($v(|\Phi|)\to \mu^2|\Phi|^2$), i.e., self-interaction is negligible, the branch structure reduces to a single stable region: mini boson stars only exhibit one stable phase, without the higher-density analogy to neutron stars.

This dichotomy is summarized:

Model Type	Potential	Number of Stable Branches
Mini boson star	$\mu^2|\Phi|^2$	One
Solitonic/self-interacting	Quartic/sextic	Two

Thus, self-interactions allow for richer equilibrium structures and are necessary for the existence of the upper-density, neutron-star–like branch.

6. Mathematical Structure of the Stability Criteria

The primary stability criterion in equilibrium space is the vanishing of the derivative: $$\left.\frac{\partial Q}{\partial \omega_s}\right|_{\kappa}=0$$ This defines the locus of turning points where the stability of a given branch changes. The equilibrium structure is specified by: $\mathcal{M} = \{ \omega_s,\, \kappa,\, Q\}$ The corresponding control space (projection onto $\{\kappa,Q\}$) subdivides the sets of equilibrium solutions into regions characterized by the number of unstable modes—single-mode ($S_1$), unstable ($U$), and ultimately the spiral region with ever-increasing instability.

The sequence of solution branches and the role of each segment can be charted as follows:

Segment	$(\partial Q/\partial \omega_s)_\kappa$	Stability
Pre-A	$>0$	Stable
A-B	$<0$	Unstable
B-C	$>0$	Stable
Spiral	Oscillatory sign; new modes turn unstable	Increasing instability

The change in sign signals both the change in the number of unstable modes and the boundaries of stable phases.

7. Broader Implications and Physical Insights

The catastrophe-theoretic nonlinear analysis of radial stability in boson stars leads to several key insights:

• The existence of two stable regions for self-interacting boson stars draws a precise analogy with the two phases of compact fermion stars, mapping scalar field self-interaction to the effective equation of state in the latter.
• The structure and extent of the stable/unstable branches are determined purely by the local geometry of the solution manifold in equilibrium space—unlike, e.g., dynamical stability to generic (non-radial) perturbations, which may require separate analysis.
• Mini boson stars, lacking appreciable self-interaction, are structurally more limited, possessing only a single stable phase, thereby lacking any neutron-star–like high-density analog.
• The spiral instability marks the endpoint of stable equilibrium and delineates the parameter space relevant for “compact object mimickers,” i.e., non-black hole objects exhibiting features akin to neutron stars or black holes.

These findings motivate further exploration of the stability properties under additional degrees of freedom (rotation, charge, higher-dimensional analogs) and imply that, in practice, only those configurations residing within the first and second stable branches (as defined above) can serve as dynamically viable astrophysical objects.

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In summary, catastrophe theory provides a comprehensive, mathematically rigorous classification of nonlinear radial stability in boson stars, with two key stable phases—one “white dwarf–like” and one “neutron star–like”—arising in self-interacting models. Mini boson stars lack the high-density stable branch. The transition to instability is precisely governed by turning points in equilibrium space, and the spiral regime signals the approach to collapse or dispersal. The interplay between gravitational coupling and scalar field self-interaction underpins the entire stable/unstable phase structure. This analysis situates boson stars within the broader landscape of compact-object physics and provides a robust framework for future model-building and astrophysical application (Kleihaus et al., 2011).