The continuum spectrum of nonrelativistic multi-frequency Proca stars

Abstract

Multi-frequency Proca stars are excited equilibrium configurations of a selfgravitating massive vector field that coexist with conventional stationary states in the solution space of the $s=1$ Schrödinger-Poisson system. In this paper, we present a systematic study of the continuum spectrum of spherical multi-frequency Proca stars and show that they interpolate between the discrete set of stationary states of constant polarization. Furthermore, we also analyze their stability and demonstrate that a subset of these multi-frequency configurations are linearly stable against general perturbations. We briefly discuss the potential implications of multi-frequency states for proving the particle spin in ultralight dark matter models.

Figures (7)

• Figure 1: $1$-component multi-frequency Proca stars (configurations). Real part of the vector field, $\vec{\psi}_R(t,\vec{x})$ [arrows], and particle number density, $n(t,\vec{x})$ [red shading], at $t=0$ for 1-component multi-frequency states belonging to the families $(n_x)=$ (0), (1) and (2), for $N=43.5$. Each family consists of a single, stationary configuration. The $(n_x)=(0)$ case corresponds to the ground state at fixed $N$. For $(n_x)=$ (1) and (2) the field orientation differs from one shell to another.
• Figure 2: $2$-component multi-frequency Proca stars. 2-component multi-frequency states belonging to the families $(n_x,n_y)=$ (0,1), (0,2), (0,3), (1,2), (1,3), (2,3), for $N=43.5$. Each family continuously connects the corresponding $(n_x)$ and $(n_y)$ 1-component states. Note that the ground state, $(n_x)=(0)$, is the lowest energy configuration. Figure \ref{['Fig.sol_2component2']} shows representative configurations from these families.
• Figure 3: $2$-component multi-frequency Proca stars (configurations). Same as Fig. \ref{['Fig.sol_1component']}, but for representative configurations from the families $(n_x,n_y)=$ (0,1), (0,2) and (0,3), for $N=43.5$. All these families start from the ground state $(n_x)=0$ and end at the excited $(n_y=1)$, $(n_y=2)$, and $(n_y=3)$ states, respectively. From left to right, particles are progressively transferred from the $x$ to the $y$ component. A movie illustrating the time evolution of a 2-component multi-frequency Proca star is provided in youtube.
• Figure 4: $3$-component multi-frequency Proca stars. 3-component multi-frequency states belonging to the families $(n_x,n_y,n_z)=$ (0,1,2), (0,1,3) and (0,2,3), for $N=43.5$. The planes $\sigma_{z0}=0$, $\sigma_{y0}=0$ and $\sigma_{x0}=0$ in these figures correspond to 2-component curves shown in Fig. \ref{['Fig.sol_space2']}. Note that the ground state, $(n_x)=(0)$, remains as the lowest energy configuration.
• Figure 5: Real eigenvalue spectra of 2-component multi-frequency Proca stars. Spectra of the 2-component families $(n_x,n_y)=$$(0,1)$ [left panel], $(0,2)$ [central panel], and $(0,3)$ [right panel] presented in Sec. \ref{['sec.2component']}, as function of the particle number ratio in the $y$-component, for perturbations with angular momentum number $J \le 6$. Modes with a positive real part of the eigenfrequency, $\lambda_R> 0$, signal the onset of a linear instability. The shaded regions indicate stability bands, where no unstable modes are observed. Families with $n_x\neq 0$ are unstable and are not shown here.