Linear stability of nonrelativistic Proca stars

TL;DR

This work analyzes the linear stability of nonrelativistic Proca stars within the $s=1$ Gross-Pitaevskii-Poisson framework, identifying the ground state as always mode-stable and revealing several vector-specific stable excited configurations. By combining analytic perturbation theory with a comprehensive numerical eigenvalue study across sectors defined by selfinteractions, the authors classify stationary and multi-frequency spherical equilibria by their stability under angular-momentum-resolved perturbations. Key findings include stability bands for certain excited states in the presence of repulsive particle-particle selfinteractions, and strong destabilization of radial polarization states when spin-spin selfinteractions are nonzero; the ground state remains robust in all regimes with $oldsymbol{oldsymbol{ ext{lambda_0}}}oldsymbol{oldsymbol{≥0}}$. These results imply a richer phenomenology for spin-1 ultralight dark matter than for scalar cases and offer a predictive map for which configurations are dynamically viable, with nonlinear evolutions to be explored in follow-up work.

Abstract

We study the linear stability of nonrelativistic Proca stars under generic perturbations. Using a combination of analytic and numerical methods, we demonstrate that, as expected, the ground state is always mode-stable. Additionally, we identify several mode-stable spherically symmetric excited states, including stationary states of constant and radial polarization, as well as multi-frequency states in case that the spin-spin selfinteraction vanishes. The existence of stable excited states may have implications for spin-$1$ ultralight dark matter models.

Figures (7)

• Figure 1: Eigenvalue spectra of stationary Proca stars with constant polarization (free theory): Spectra corresponding to the ground state ($n=0$, left panel) and two excited states ($n=1$, center; $n=2$, right) for configurations of constant polarization, unit amplitude $\sigma_0=1$, and perturbations up to $J\le 5$. Eigenvalues with $(\lambda_R, \lambda_I) \neq 0$ appear in quadruples ${\lambda, -\lambda, \lambda^*, -\lambda^*}$, while purely real and imaginary ones occur in pairs. Only the nodeless configuration ($n=0$) exhibits a purely imaginary spectrum and is therefore linearly stable. In contrast, excited states ($n\ge 1$) possess eigenvalues with nonzero real parts, implying linear instability.
• Figure 2: Eigenvalue spectra of stationary Proca stars with radial polarization (free theory): Same as in Fig. \ref{['FigmodosSCPLinear']} but for radially polarized configurations. As for the constant polarization case, only the nodeless state is stable, although in this case it does not represent a ground state.
• Figure 3: Phase diagrams of multi-frequency states: A family of two-component multi-frequency states with $(n_x,n_y)=(0,1)$ and $N=25.4$ in the attractive ($\lambda_n=-1$), free ($\lambda_n=0$), and repulsive ($\lambda_n=+1$) cases of the symmetry-enhanced sector of the theory ($\lambda_s=0$).
• Figure 4: Real eigenvalue spectra of two-component multi-frequency Proca stars: Spectra corresponding to the family presented in Fig. \ref{['FigProfMultifrequency']} (left: $\lambda_n=+1$, center: $\lambda_n=0$, right: $\lambda_n=-1$), for perturbations with $J\le 5$. The shaded regions indicate stability bands, where no unstable modes are observed. These configurations correspond to stable excited states.
• Figure 5: Real eigenvalue spectra of stationary Proca stars with constant polarization ($\lambda_n\neq0$, $\lambda_s=0$): Spectra corresponding to selfinteracting stationary states of constant polarization ($\lambda_n = +1$, left panel; $\lambda_n=-1$, right panel), for perturbations with $J\le 5$. The shaded regions indicate stability bands where no unstable modes are observed.