Dynamical Shortcomings in the Generalized SU(2) Proca Theory: Challenges for Cosmic Acceleration

The Generalized SU(2) Proca (GSU2P) theory has recently garnered attention for its potential to describe key phases of cosmic evolution, including primordial inflation and late-time accelerated expansion. However, its full cosmological implications remain unexplored. In this work, we perform a comprehensive analysis of the dynamical properties of the GSU2P theory in a flat, homogeneous, and isotropic spacetime, through a dynamical-system approach. Our analysis reveals the presence of three pairs of fixed points, one of them corresponding to de-Sitter expansion which may represent either a stable or unstable phase in the evolution of the universe. These points, nonetheless, give rise to an indeterminate or infinite Hubble parameter, which renders them cosmologically unviable. Additionally, we find two key pseudostationary states: the ``attractor lines'', along which the system exhibits constant-roll dynamics, and the ``central zone'', characterized by oscillatory radiation-like behaviour of the field. The dynamics within the central zone could represent a graceful exit from the primordial inflationary phase to a radiation dominated phase, or a state of the dark energy component prior to the late-time cosmic acceleration. However, within the central zone, the dynamics of the vector field leads to recurrent instances of a nonphysical expansion rate. The absence of a limit cycle in the central zone further exacerbates the issue, as the system may follow unbounded phase-space trajectories, and the expansion rate becomes complex once it escapes the region. Collectively, these challenges undermine the viability of the GSU2P theory as a cosmological model for cosmic acceleration.
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