Assessing the stability of ultracompact spinning boson stars with nonlinear evolutions

Abstract

We reinvestigate the stability properties of ultracompact spinning boson stars with a stable light ring using fully nonlinear 3+1 and 2+1 numerical relativity simulations and two different formulations of the Einstein equations. We find no evidence of an instability on timescales of $t μ\sim 10^4$ (in units of the scalar mass), when allowing the star to be perturbed either solely by discretization error or by imposing various types of perturbations to our initial data. We find that the initially imposed perturbations exhibit slow decay, even for magnitudes just below the order where immediate collapse is induced.

Figures (9)

• Figure 1: Illustration of the stationarity of the ultracompact spinning BS solution S005 (perturbed only by numerical discretization error) using measures $\Phi$ and $G$, defined in Eqs. \ref{['eq:phi_diagnostics']}--\ref{['eq:gtt_diagnostics']}. We also show the first few and higher-order $\tilde{m}$-modes of $\Phi$, normalized by $\Phi_0$, so that $|\tilde{\Phi}_{\tilde{m}}| = |\Phi_{\tilde{m}}|/|\Phi_0|$. The magnitude of the $\tilde{m}=4$ mode is comparable to the magnitude of the $\tilde{m}=1$, and exhibits a slight growth, which is discussed in the main text.
• Figure 2: Left: Behavior of $\Phi$ throughout the evolution of star S005 with perturbation type \ref{['eq:perturbation']} varying both amplitude $\epsilon$ and azimuthal dependence characterized by $n$. The perturbation with $\epsilon = 0.1$ results in BH formation, signaled by a sudden drop of $\Phi$. Right: Same as the left panel but with perturbation of type \ref{['eq:perturbation_lr']}; BH formation is now triggered for $\epsilon = 0.01$. Black dashed lines indicate the logarithmic-in-time decay of $\Phi$.
• Figure 3: Evolution of the ultracompact spinning boson star S02 that exhibits a gauge instability in stationary gauge using the GHC. The solid lines are moving averages of each diagnostic (shaded lines). Top panel: Maximum of the derivative of the scalar field amplitude, as defined in Eq. \ref{['eq:phi_diagnostics']}; $\mu \mathrm{d} x$ is the grid spacing of the reference resolution quoted in the main text. Middle panel: The ratio of the maximum of the scalar field amplitude to its value at initial time $t=0$, as defined in Eq. \ref{['eq:phi_max']}. Bottom panel: Integrated constraint violations, defined in Eq. , and normalized by $\Phi$.
• Figure 4: A zoom-in on volume-weighted $L^2$-norms of Hamiltonian constraint for the S005 BS model evolved with $\mu \mathrm{d} x \approx 0.083$. We compare the evolution of constraints using CCZ4 and BSSNOK formulations, where the latter evidently exhibits problematic growth.
• Figure 5: Convergence of the volume-weighted $L^2$-norms of the Hamiltonian (left) and momentum (right) constraints in ExoZvezda. Medium (low) resolutions use $\mu \mathrm{d} x \approx 0.167$ ($\mu \mathrm{d} x \approx 0.25$) on the finest levels. We find consistent convergent behavior in the constraints, sandwiched between third and fourth orders.