Proca Stars: gravitating Bose-Einstein condensates of massive spin 1 particles

Abstract: We establish that massive complex Abelian vector fields (mass $\mu$) can form gravitating solitons, when minimally coupled to Einstein's gravity. Such Proca stars (PSs) have a stationary, everywhere regular and asymptotically flat geometry. The Proca field, however, possesses a harmonic time dependence (frequency $w$), realizing Wheeler's concept of geons for an Abelian spin 1 field. We obtain PSs with both a spherically symmetric (static) and an axially symmetric (stationary) line element. The latter form a countable number of families labelled by an integer $m\in \mathbb{Z}^+$. PSs, like (scalar) boson stars, carry a conserved Noether charge, and are akin to the latter in many ways. In particular, both types of stars exist for a limited range of frequencies and there is a maximal ADM mass, $M_{max}$, attained for an intermediate frequency. For spherically symmetric PSs (rotating PSs with $m=1,2,3$), $M_{max}\simeq 1.058 M_{Pl}^2/\mu$ ($M_{max}\simeq 1.568,\, 2.337, \, 3.247 \, M_{Pl}^2/\mu$), slightly larger values than those for (mini-)boson stars. We establish perturbative stability for a subset of solutions in the spherical case and anticipate a similar conclusion for fundamental modes in the rotating case. The discovery of PSs opens many avenues of research, reconsidering five decades of work on (scalar) boson stars, in particular as possible dark matter candidates.

• The paper introduces Proca stars as novel solutions in Einstein’s gravity, extending boson star models to massive spin 1 fields.
• It demonstrates precise mass-frequency relationships and angular momentum dynamics, with maximal masses exceeding those of mini-boson stars.
• The study assesses perturbative stability in both spherical and rotating configurations, suggesting these objects as viable dark matter candidates.

Proca Stars: Gravitating Bose-Einstein Condensates of Massive Spin 1 Particles

The paper by Richard Brito, Vitor Cardoso, Carlos A. R. Herdeiro, and Eugen Radu presents a comprehensive study of a novel class of astrophysical objects termed "Proca Stars" (PSs). These objects form as solutions within the framework of Einstein's gravity minimally coupled to a massive complex Abelian vector field. This study builds on the renowned concept of scalar boson stars (SBSs) but introduces an important distinction by considering massive spin 1 particles.

Introduction and Theoretical Background

The investigation is situated within the broader context of exploring alternative dark matter (DM) candidates. Proca stars emerge from complex vector fields and serve as solitonic solutions characterized by harmonic time dependence. Essential to the realization of these solutions is Wheeler’s initial proposition of geons adapted for Abelian spin 1 fields. Unlike traditional boson stars composed of scalar fields, the vector nature of Proca fields allows these stars to possess both spherically symmetric and axially symmetric configurations.

Numerical and Analytical Findings

The research yields several key findings pertinent to the structure and properties of Proca stars:

1. Mass and Frequency Relations: The study elucidates the existence of specific frequency bounds within which PSs manifest. A maximal Arnowitt-Deser-Misner (ADM) mass is achieved at an intermediate frequency, reminiscent of similar constraints observed in scalar boson stars. Notably, the maximal mass for spherically symmetric PSs slightly exceeds that of mini-boson stars, with $M_{max} \simeq 1.058 M_{Pl}^2/\mu$.
2. Rotational Aspects: The exploration extends to rotating Proca stars, elaborating on the solutions for different rotational quantum numbers $m$. The angular momentum is consistently shown to relate to the Noether charge through $J = m Q$, highlighting an intrinsic tie between rotational characteristics and conserved quantities. For modes with $m=1,2,3$, the study provides explicit values for maximal masses and highlights patterns similar to those found in rotating scalar boson stars.
3. Stability Considerations: A significant focus is directed towards the stability of Proca stars. The paper proposes that, analogous to SBSs, a subset of spherical solutions exhibit perturbative stability, specifically for those in the vicinity of the maximal ADM mass. This conclusion is extrapolated to some extent for rotating stars, based on theoretical and numerical parallels to known scalar field configurations.

Implications and Future Work

The theoretical and numerical evidence provided by the authors implies several notable consequences:

• Dark Matter Implications: Proca stars are proposed as potential dark matter constituents within the universe's energy budget. Given the increasing interest in non-scalar dark matter fields, the recognition of similar solitonic behaviors in vector fields could broaden the scope of dark matter modeling.
• Comprehensive Field Dynamics: The introduction of massive spin 1 fields necessitates revisiting the dynamics of self-gravitating systems. Future lines of inquiry may explore more intricate interactions, including those featuring black holes within such vector frameworks.
• Proca Oscillatons: The investigation taunts the potential existence of long-lived quasi-solitonic states termed Proca oscillatons, akin to the scalar counterparts known as oscillatons.

In summary, the investigation into Proca stars presented in the paper offers a rigorous extension of the analogous framework known in scalar boson stars, situating massive vector fields at center stage in theoretical astrophysics and cosmology. Such a paper paves the way for subsequent theoretical explorations and potentially observational endeavors to uncover signatures of these enigmatic objects.