Boson Stars in $D \ge 4$ Dimensions: Stability, Oscillation Frequencies, and Dynamical Evolutions

Technical Summary: Boson Stars in D ≥ 4 Dimensions: Stability, Oscillation Frequencies, and Dynamical Evolutions

Problem Statement
The paper addresses the existence and stability of spherically symmetric boson star (BS) solutions in spacetime dimensions $D \geq 4$. While BSs are well-studied in four dimensions ($D=4$) as horizonless, self-gravitating configurations of a massive complex scalar field, their stability in higher dimensions ($D > 4$) has been an open question. Previous work suggested that the characteristic $1/r^{D-3}$ decay of the gravitational field in $D > 4$ might prevent the balance between gravitational attraction and scalar kinetic energy required for stability, potentially rendering higher-dimensional BSs generically unstable. Furthermore, while some higher-dimensional models (mini, Dirac, Proca) had been constructed, rigorous evidence for their dynamical stability was lacking. The authors note that linear stability analysis alone is insufficient to guarantee nonlinear stability, as seen in $D=4$ where radially stable rotating or excited BSs can suffer nonlinear instabilities.

Methodology
The authors employ a dual approach combining perturbative linear analysis and nonlinear dynamical evolution:

1. Model Construction: They construct static, spherically symmetric BS solutions in $D \in \{4, 5, 6\}$ using the Einstein-Klein-Gordon equations. Two scalar potentials are considered:

   • A quartic self-interaction potential ($V \propto \mu^2|\phi|^2 + \frac{\lambda}{2}|\phi|^4$), covering both mini-BSs ($\lambda=0$) and massive BSs ($\lambda > 0$).
   • A solitonic potential ($V \propto \mu^2|\phi|^2(1 - 2|\phi|^2/\sigma_0^2)^2$), which supports degenerate vacuum states.
     Solutions are obtained via a two-way shooting method, utilizing asymptotic expansions for the exterior region to handle compact solutions accurately.

2. Perturbative Analysis: The authors generalize the pulsation equations to arbitrary dimensions and arbitrary potentials. By linearizing the Einstein-Klein-Gordon system around a stationary background, they derive a system of ordinary differential equations (ODEs) governing radial perturbations. They solve for the eigenfrequencies $\chi$ of these modes. A solution is considered linearly stable if the fundamental mode frequency satisfies $\chi_0^2 > 0$; $\chi_0^2 < 0$ indicates a radial instability.

3. Nonlinear Dynamical Evolutions: To test linear predictions against nonlinear dynamics, the authors perform numerical evolutions in spherical symmetry using the sphericalbsevolver (SBSE) code. This code employs a "modified cartoon" method to reduce the $D$-dimensional BSSN and CCZ4 formulations of numerical relativity to a 1+1 dimensional grid while preserving full gauge freedom. They evolve both unperturbed solutions (to check for numerical drift) and explicitly perturbed solutions (using internal and external Gaussian perturbations) to observe the long-term fate of the stars (stability, dispersion, collapse, or migration).

Key Contributions and Results

• Existence of Stable Higher-Dimensional BSs: The primary finding is that stable higher-dimensional boson stars do exist, contrary to the expectation that $D>4$ implies generic instability.

  • Massive BSs: For $D=5$ and $D=6$, stable branches emerge when the self-interaction strength $\hat{\lambda} = \lambda/\mu^2$ exceeds a critical value ($\hat{\lambda} \approx 63.4$ for $D=5$ and $\hat{\lambda} \approx 416$ for $D=6$).
  • Solitonic BSs: For $D=5$, stable branches exist for sufficiently small solitonic parameters ($\sigma_0 \lesssim 0.236$). However, for $D=6$, solitonic models exhibit a pathological divergence in mass and radius at finite central amplitude for small $\sigma_0$, and no stable solitonic branches were found.
  • Mini BSs: Consistent with previous work, mini-BSs (without self-interaction) are found to be radially unstable in $D=5$ and $D=6$.

• Dimensional Differences in Stability Structure:

  • In $D=4$, stable massive BS branches connect to the vacuum limit ($A_0 \to 0$).
  • In $D=6$, the stable branch of massive BSs (for $\hat{\lambda} > \hat{\lambda}_{crit}$) does not include the $A_0 \to 0$ limit; instead, it is an isolated branch separated from the vacuum by a region of instability. This implies that arbitrarily low compactness stable models cannot be formed in $D=6$ for a fixed potential, though the gap can be made arbitrarily small by increasing $\hat{\lambda}$.

• Linear vs. Nonlinear Agreement: The dynamical evolutions confirm the predictions of linear theory. Models identified as perturbatively stable ($\chi_0^2 > 0$) remain stable under both unperturbed and explicitly perturbed nonlinear evolution. Conversely, models with $\chi_0^2 < 0$ evolve into black holes, disperse, or migrate to a stable branch.

• Role of Binding Energy: The sign of the binding energy ($E_B = M - \mu N$) is shown to be neither necessary nor sufficient for stability. The authors identify stable models with positive binding energy in $D=5$ (solitonic) and $D=6$ (massive). These models appear "metastable": they are dynamically stable once formed but unlikely to form generically via gravitational collapse of scalar clouds.

• Numerical Observations:

  • The authors observe that evolving $D=6$ spacetimes requires a smaller Courant factor ($C=0.25$) compared to $D=4,5$ ($C=0.4$) to avoid numerical instabilities, particularly for strong self-interactions.
  • Comparisons between BSSN and CCZ4 formulations show that while BSSN exhibits long-term linear growth in Hamiltonian constraint violations, CCZ4 allows constraints to settle to finite values, suggesting CCZ4 may be superior for long-term stability studies.

Significance and Claims
The paper claims to provide the first convincing evidence of asymptotically flat, higher-dimensional boson star solutions that are dynamically stable to spherical dynamics. By demonstrating that self-interactions (quartic or solitonic) can counteract the destabilizing effects of higher-dimensional gravity, the work extends the body of knowledge regarding exotic compact objects in $D \geq 4$.

The authors emphasize that their results validate the use of these models for future studies of higher-dimensional critical phenomena and the hoop conjecture, where such stable, horizonless objects are required as initial data or collision targets. They explicitly state that while their results confirm stability under spherical symmetry, the question of non-radial (nonspherical) stability remains open and is the most obvious avenue for future work. They do not claim these objects are confirmed astrophysical candidates but rather robust theoretical models for numerical relativity applications in higher dimensions.