Non-linear stability analysis of $\ell$-Proca stars

TL;DR

This work investigates the non-linear stability of $oldsymbol{\\ell}$-Proca stars, a multi-field generalization of Proca stars formed from $oldsymbol{N=2\\u22121\\ell}$ complex Proca fields with angular dependence encoded by the spherical harmonics $Y^{\\ell m}$. Using long-term, fully three-dimensional numerical relativity simulations for the case $oldsymbol{\\ell=2}$, the authors show that these configurations are unstable across their domain of existence: radially unstable models collapse to Schwarzschild BHs, while radially stable models develop a non-axisymmetric $ ilde{m}=4$ mode that breaks spherical symmetry and drives a migration to a mixed multi-$\\ell$ configuration with $oldsymbol{\\ell=1}$ and $oldsymbol{\\ell=2}$ fields. Depending on compactness, the subsequent evolution leads either to BH formation or to further instabilities, including a non-axisymmetric $ ilde{m}=2$ (bar-mode) instability that can yield a non-rotating axially symmetric multi-$\\ell$ star or fragmentation into a binary Proca star. The results suggest that stable multi-field Proca stars may exist primarily for lower angular momenta (e.g., $oldsymbol{\\ell=1}$), with important implications for the astrophysical viability and gravitational-wave signatures of vector boson stars.

Abstract

Vector boson stars, also known as Proca stars, exhibit remarkable dynamical robustness, making them strong candidates for potential astrophysical exotic compact objects. In search of theoretically well-motivated Proca star models, we recently introduced the $\ell$-Proca star, a multi-field extension of the spherical Proca star, whose $(2\ell + 1)$ constitutive fields have the same time and radial dependence, and their angular structure is given by all the available spherical harmonics for a fixed angular momentum number $\ell$. In this work, we conduct a non-linear stability analysis of these stars by numerically solving the Einstein-(multi, complex) Proca system for the case of $\ell = 2$, which are formed by five constitutive independent, complex Proca fields with $m = 0, |1|$, and $|2|$. Our analysis is based on long-term, fully non-linear, 3-dimensional numerical-relativity simulations without imposing any symmetry. We find that ($\ell=2$)-Proca stars are unstable throughout their entire domain of existence. In particular, we highlight that less compact configurations dynamically lose their global spherical symmetry, developing a non-axisymmetric $\tilde{m}=4$ mode instability and a subsequent migration into a new kind of multi-field Proca star formed by fields with different angular momentum number, $\ell=1$ and $\ell=2$, that we identify as unstable multi-$\ell$ Proca stars.

Figures (18)

• Figure 1: Solution space of $\ell$-Proca stars for $\ell=0,2,3$. The specific configurations highlighted by dots correspond the ones specified in Table \ref{['table:2-Proca_configurations_initial']} in the domain of existence of the 2-Proca star. The black star corresponds the solution with maximum mass dividing this domain in two branches.
• Figure 2: Evolution of the minimum value of the lapse (top) and the 2-Proca star mass inside a sphere of radius $r_*=30$ (bottom) for the models $A$, $B_1$, $C$, and $D$ of Table \ref{['table:2-Proca_configurations_initial']}.
• Figure 3: Top: time evolution of the minimum value of the lapse for model $A$ and three different resolutions. Bottom: global convergence test for model $A$ using the $L_2$-norm of the Hamiltonian constraint. The curves are rescaled for an order of convergence between three and four.
• Figure 4: Same as Fig. \ref{['fig:convergence_test_A']} for model $B_1$ with a 4-refinement-level grid with the following resolutions for the innermost cube: low $m_V\{\Delta x, \Delta y, \Delta z\} =0.40$, medium $m_V\{\Delta x, \Delta y, \Delta z\} =0.36$, and high $m_V\{\Delta x, \Delta y, \Delta z\} =0.30$, and including the $L_2$-norm of the Gauss constraint $Re(Z_0)$. Unlike the model $A$, the timescale of the evolutions for this model is two orders of magnitude larger than the time offsets caused by the different-resolution truncation errors, making them unnoticeable. In consequence, we do not apply the time shifts in these plots.
• Figure 5: Local convergence test for model $B_1$ using different snapshots of the evolution of $|H|$ (left) and $|Re(Z_0)|$ (right) and applying a fourth-order rate rescaling to the lower-resolution constraints.