Perturbations of Solitonic Boson Stars: Nonlinear Radial Stability and Binding Energy

Technical Summary: Perturbations of Solitonic Boson Stars

Problem Statement
The stability of Boson Stars (BSs), particularly those modeled with a solitonic potential, remains a critical question for their viability as astrophysical objects. While linear radial perturbation theory has historically been sufficient to predict the nonlinear stability of "mini" boson stars (those with a simple massive Klein-Gordon potential), the behavior of solitonic BSs is less understood. These models support ultracompact objects (UCOs) with high compactness and light rings, raising questions about whether nonlinear effects or specific energy conditions alter stability outcomes.

A prevailing heuristic in the literature suggests that negative binding energy ($E_B < 0$) is a necessary condition for the dynamical stability of compact objects. Conversely, models with positive binding energy ($E_B > 0$) are often assumed to be unstable, expected to disperse or undergo fission under perturbation. This paper investigates whether this heuristic holds for solitonic BSs, specifically testing if perturbatively stable models with positive binding energy can remain dynamically stable against explicit radial perturbations.

Methodology
The study employs a numerical relativity approach to evolve solitonic boson star models in spherical symmetry.

• Theoretical Framework: The models are derived from Einstein's general relativity coupled to a massive complex scalar field with a solitonic potential $V(\phi) = \mu^2|\phi|^2(1 - 2|\phi|^2/\sigma_0^2)^2$. The system is solved using the Einstein-Klein-Gordon equations with an ansatz $\phi(t, r) = A(r)e^{i\omega t}$.
• Numerical Scheme: The authors utilize sphericalbsevolver, a code implementing dimensional reductions of the CCZ4 and BSSN formulations of numerical relativity.
• Perturbation Strategy: To test stability, explicit perturbations are introduced to the real part of the scalar field ($\phi$) and the imaginary part of the field momentum ($\Pi_\phi$). These perturbations follow a Gaussian profile $\delta\phi = a \exp(-(r-r_0)^2/k^2)$ and are designed such that the Noether charge $N$ remains constant to first order.
• Parameter Space: The study scans families of solutions across varying central amplitudes ($A_0$) and solitonic parameters ($\sigma_0$). Special attention is paid to:
  1. Models on both stable and unstable branches (with both signs of $E_B$).
  2. Linearly stable models with $E_B > 0$.
  3. Ultracompact models (UCOs) located near the ends of the stable branches.

Key Results
The authors performed long-term dynamic evolutions (up to $t = 10^5 M$) for various configurations, categorized by their linear stability and binding energy:

1. Validation of Linear Theory: For all tested cases, the nonlinear dynamical evolution aligned with predictions from linear radial perturbation theory.

   • Unstable Models: Linearly unstable models either collapsed to black holes (if $E_B < 0$) or dispersed to spatial infinity (if $E_B > 0$).
   • Stable Models: Linearly stable models remained stable under perturbation, provided the perturbation did not alter the Noether charge sufficiently to displace the system onto an unstable branch.

2. Stability of Positive Binding Energy Models: A significant finding is that linearly stable models with positive binding energy ($E_B > 0$) did not exhibit instability or fission. Even when perturbations removed scalar matter (further increasing $E_B$), these models settled into stationary solutions rather than dispersing. This directly challenges the heuristic that $E_B < 0$ is a strict requirement for stability.

3. Ultracompact Objects (UCOs): The stability of UCOs, which possess light rings and exist near the boundaries of the stable branch, was confirmed. Despite expectations that nonlinear effects might become significant in such highly compact configurations, the simulations showed no deviation from linear predictions. No evidence of a light-ring instability was found.

4. Oscillation Spectra: Power spectrum analysis of the central amplitude (e.g., in run B1) confirmed that the fundamental and first excited radial oscillation frequencies matched the peaks observed in the time-domain evolution, further underscoring the accuracy of linear theory in describing these dynamics.

Significance and Claims
The paper concludes that, at least within spherical symmetry, the radial stability of solitonic boson stars is robust and does not depend significantly on nonlinear effects, even for ultracompact configurations.

The primary contribution is the demonstration that positive binding energy is not an effective instability trigger for perturbatively stable boson stars. The authors argue that the condition $E_B < 0$ should be regarded as a rule of thumb rather than a strict necessity for dynamical stability. They conjecture that while such positive-energy models may not generically form via gravitational collapse, the solitonic potential creates an obstruction preventing the transition to the energetically preferred state of dispersion.

The authors remain modest regarding the scope of these findings, noting that the results are specific to spherical symmetry. They suggest that future work is required to determine if non-radial modes could introduce dispersive instabilities in non-spherical configurations and to explore whether similar behaviors exist in other scalar potentials with degenerate vacuua (e.g., axionic potentials). Additionally, the formation mechanisms of these solitonic models, particularly UCOs, remain an open area for investigation.