Primordial black hole in Lorentz-violating theories: Insights from Bumblebee gravity

Abstract

The Bumblebee gravity (BG) model, featuring spontaneous Lorentz symmetry breaking via a vector field non-minimally coupled to curvature, has been widely used to explore Lorentz-violating effects in cosmology. We investigate primordial black hole (PBH) formation within this framework, deriving the complete set of modified perturbation equations. We demonstrate that BG, sourced by a timelike vector field, introduces three distinct enhancements to PBH abundance--modified expansion history, suppressed collapse threshold, and amplified power spectrum--which together render PBHs viable dark matter candidates across the asteroid-mass window for modest Lorentz-violating couplings. However, a systematic analysis of the quadratic action reveals that these phenomenological consequences emerge from a theoretically pathological foundation. The vector sector exhibits an intrinsic ghost instability, while the requirement of a stable symmetry-breaking minimum simultaneously induces a tachyonic instability on timescales far below cosmological scales. The model thus suffers from a fundamental inconsistency: the conditions for cosmological viability and spontaneous symmetry breaking are mutually exclusive within the minimal Bumblebee framework. Our results illustrate both the notable power of Lorentz violation to influence early Universe observables and the necessity of a consistent theoretical foundation for such predictions.