Nonlinear stability analysis of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mo>ℓ</mml:mo> </mml:math> -Proca stars

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 $\ensuremath{\ell}$-Proca star, a multifield extension of the spherical Proca star, whose $(2\ensuremath{\ell}+1)$ 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 $\ensuremath{\ell}$. In this work, we conduct a nonlinear stability analysis of these stars by numerically solving the Einstein- (multi-, complex) Proca system for the case of $\ensuremath{\ell}=2$, which are formed by five constitutive independent, complex Proca fields with $m=0,|1|$, and $|2|$. Our analysis is based on long-term, fully nonlinear, three-dimensional numerical-relativity simulations without imposing any symmetry. We find that ($\ensuremath{\ell}=2$)-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 nonaxisymmetric $\stackrel{\texttildelow{}}{m}=4$ mode instability and a subsequent migration into a new kind of multifield Proca star formed by fields with different angular momentum number, $\ensuremath{\ell}=1$ and $\ensuremath{\ell}=2$, that we identify as unstable multi-$\ensuremath{\ell}$-Proca stars.