Nonrelativistic Proca Stars

• Nonrelativistic Proca stars are self-gravitating configurations of a massive, complex vector field stabilized by particle number and self-interactions.
• They are modeled using a Gross–Pitaevskii–Poisson system derived from a nonrelativistic expansion of the Proca Lagrangian, capturing density and spin-spin effects.
• The structures feature both stationary and multi-frequency states, with rich stability properties that have implications for vector bosonic dark matter and astrophysical phenomena.

Nonrelativistic Proca stars are self-gravitating, localized configurations of a massive, complex spin-1 (vector) field in the nonrelativistic (Newtonian) regime, stabilized by particle number and self-interactions. They generalize the concept of boson stars to spin-1 fields, which are described by nonrelativistic effective theories derived from the weak-field, slow-mode limit of relativistic Proca field theory. These objects exist as stationary and, in certain sectors, genuinely multi-frequency bound states, with structure and stability properties determined by both scalar (density) and spin-spin self-interactions.

1. Effective Theory Formulation and Field Equations

The nonrelativistic regime is obtained by expanding the relativistic Proca Lagrangian for a complex vector field $A_\mu$ of mass $m_0$ with quartic self-interactions in the presence of Newtonian gravity. After integrating out the non-dynamical temporal component and keeping operators up to dimension six, the effective Lagrangian is

$$\mathcal L = \frac{1}{8\pi G}\, \mathcal U\, \Delta\, \mathcal U + \psi_i^*\Big(i\partial_t + \frac{\Delta}{2m_0}\Big)\psi^i - \frac{\lambda_n}{4m_0^2} (\psi_i^*\psi^i)^2 - \frac{\lambda_s}{4m_0^2}\left(-i\,\psi^*\!\times\psi\right)^2 - m_0\,\mathcal U(\psi_i^*\psi^i),$$

where $\psi_i$ is a slowly varying three-component complex vector field, $\mathcal U$ is the Newtonian gravitational potential, and the couplings $\lambda_n$, $\lambda_s$ encode density and spin-spin self-interactions. Particle density is $n = \psi_i^*\psi^i$, spin density is $\mathbf s = -i\,\psi^*\!\times\psi$.

The Euler–Lagrange equations yield a Gross–Pitaevskii–Poisson (GPP) system for vectors: $$i\,\partial_t\psi = -\frac{1}{2m_0}\Delta\psi + \frac{\lambda_n}{2m_0^2} n\psi + i\frac{\lambda_s}{2m_0^2}(\mathbf s \times \psi) + m_0\,\mathcal U\,\psi, \quad \Delta\mathcal U = 4\pi G m_0 n.$$ This system admits both stationary ($\psi \sim e^{-iEt}$) and, in special cases, multi-frequency ($\psi \sim \sum_i e^{-iE_i t}$) solutions (Nambo et al., 2024).

2. Classification and Structure of Solutions

Equilibrium Proca star configurations are classified into two sectors, governed by the spin-spin coupling $\lambda_s$:

• Generic sector ($\lambda_s \neq 0$): The only spherically symmetric, finite energy solutions are stationary single-frequency states, with possible polarizations: linear, circular, or radial. Stationary states solve the nonlinear vector eigenvalue problem:

$$E\sigma = -\frac{1}{2m_0}\Delta\sigma + \frac{\lambda_n}{2m_0^2}n\sigma + i\frac{\lambda_s}{2m_0^2}(\mathbf s\times\sigma) + m_0\,\mathcal U\sigma.$$

• Symmetry-enhanced sector ($\lambda_s = 0$): The theory acquires a global $U(3)$ symmetry. In addition to stationary solutions, there exist multi-frequency states where each vector component oscillates with an independent frequency:

$$\psi(t,\mathbf x) = \sum_{a=1}^3 e^{-iE_a t}\,\sigma_a(\mathbf x)\mathbf e_a,$$

with coupled equations for each $\sigma_a$. These multi-frequency families interpolate continuously between pure stationary solutions of distinct constant polarizations (Nambo et al., 2024, Nambo et al., 4 Dec 2025).

The spatial structure is obtained via a radial ansatz, with boundary conditions for regularity at the origin and decay at infinity. States are further labeled by node numbers $(n_x, n_y, n_z)$, corresponding to the zeros in each component.

3. Existence, Symmetry, and Energy Minimization

The existence of equilibrium states at fixed particle number $N$ is determined by boundedness of the energy functional

$$E[\psi] = \int d^3x\,\left[ \frac{1}{2m_0}|\nabla\psi|^2 + \frac{\lambda_n}{4m_0^2}n^2 + \frac{\lambda_s}{4m_0^2}s^2 + \frac{m_0}{2}\,n\mathcal U \right].$$

Defining

$$\lambda_0 = \begin{cases} \lambda_n, & \lambda_s \geq 0\ \lambda_n - |\lambda_s|, & \lambda_s < 0 \end{cases},$$

the energy is bounded below for $\lambda_0 \geq 0$, guaranteeing the existence of a global minimizer. The minimizer is always spherically symmetric and of constant polarization. The minimal state is linearly polarized ($\mathbf s = 0$) for $\lambda_s > 0$, and circularly polarized ($|\mathbf s| = n$) for $\lambda_s < 0$ (Nambo et al., 2024). If $\lambda_0 < 0$, the energy is unbounded below and solutions are unstable to collapse.

4. Numerical Construction and Physical Properties

The nonlinear eigenvalue problems for stationary and multi-frequency Proca stars are solved using shooting methods (e.g., Runge–Kutta with bisection), typically after reduction to a dimensionless form by setting $4\pi G m_0^3/\lambda_* = 1$, where $\lambda_*$ depends on the dominant self-interaction. Key numerical results include:

• Stationary states resemble scalar boson stars in their mass–radius and eigenfrequency–particle number curves. Repulsive ($\lambda_n > 0$) interactions increase maximal mass; attractive interactions decrease it.
• Radial polarization ($\gamma=1$) produces “$\ell=1$”–like profiles with a central hole and matching the mass–radius relations of “$\ell=1$” scalar boson stars in the free limit.
• Multi-frequency families fill 2D regions in parameter space, bounded by the single-frequency ground and first-excited branches, allowing continuous interpolation between linearly and circularly polarized states when $\lambda_s=0$ (Nambo et al., 2024).

A summary table of key solution types is below:

Sector	Solution Types	Polarizations Allowed
Generic ($\lambda_s\neq0$)	Stationary only	Linear, circular, radial
Symmetry-enhanced ($\lambda_s=0$)	Stationary, multi-frequency	Linear, circular, arbitrary mix

5. Linear Stability and Mode Spectrum

The mode stability of nonrelativistic Proca stars is determined by linear perturbation analysis. For ground state equilibria ($n=0$) with $\lambda_0 \geq 0$, analytic arguments and full numerical eigenvalue calculations confirm that all eigenmodes are either purely oscillatory or bounded ($\Re\lambda = 0$), ensuring mode-stability (Nambo et al., 4 Dec 2025).

Notably, Proca stars admit a richer spectrum of stable states than scalar boson stars:

• Excited stationary states ($n=1$) with constant polarization develop mode instabilities for small amplitudes, but, with sufficient repulsive self-interaction, stability bands can appear at larger amplitudes.
• Radially polarized ground states are stable in the free and repulsive cases, but are destabilized by even small nonzero spin-spin coupling.
• Multi-frequency solutions exhibit stability for configurations with sufficiently small admixture of higher-frequency components; explicit bounds are obtained numerically (e.g., for the fundamental $(0,1)$ family, $0 \lesssim N_y/N \lesssim 0.55$, above which instabilities emerge).
• For attractive self-interactions, the region of stability shrinks and is limited to amplitudes below the maximum-mass point.

This behavior is distinct from the scalar case, where typically only the nodeless ground state is stable under perturbations (Nambo et al., 4 Dec 2025).

6. Astrophysical and Cosmological Relevance

Nonrelativistic Proca stars act as theoretical models for self-gravitating condensates of ultralight spin-1 particles—vector bosonic dark matter. The existence of both stationary and multi-frequency local minima (or long-lived excited states) has implications for halo structure, small-scale galactic substructure, and gravitational wave phenomenology:

• Stable excited and multi-frequency states can serve as long-lived, coherent dark matter overdensities.
• Multiple quasi-stable halos may coexist, altering structure formation scenarios relative to scalar field dark matter models.
• The distinct stability bands and transition mechanisms between solution types offer phenomenological “islands” that are tunable via self-coupling.
• Spherical, nonrelativistic Proca stars provide precise initial data for fully relativistic evolutions relevant to gravitational-wave signatures from mergers or oscillatory dynamics (Nambo et al., 2024, Nambo et al., 4 Dec 2025).

7. Outlook and Open Problems

Recent advances establish that nonrelativistic Proca stars exhibit a wide range of stable equilibrium structures—single-frequency, radial, and multi-frequency—depending on the nature and sign of self-interactions. This suggests broader diversity in the phenomenology of vector field dark matter compared to the scalar case. The stability of excited and multi-frequency configurations opens questions about their formation, merger dynamics, and observational signatures in astrophysical contexts.

A plausible implication is that further exploration of Proca star mergers, fully relativistic simulations with these nonrelativistic states as initial data, and models incorporating additional interactions (e.g., electromagnetic or anomaly-induced) are likely to provide crucial insights into the role of spin-1 fields in cosmology and gravitational physics.