Ghost instabilities of cosmological models with vector fields nonminimally coupled to the curvature

Abstract: We prove that many cosmological models characterized by vectors nonminimally coupled to the curvature (such as the Turner-Widrow mechanism for the production of magnetic fields during inflation, and models of vector inflation or vector curvaton) contain ghosts. The ghosts are associated with the longitudinal vector polarization present in these models, and are found from studying the sign of the eigenvalues of the kinetic matrix for the physical perturbations. Ghosts introduce two main problems: (1) they make the theories ill-defined at the quantum level in the high energy/sub horizon regime (and create serious problems for finding a well behaved UV completion); (2) they create an instability already at the linearized level. This happens because the eigenvalue corresponding to the ghost crosses zero during the cosmological evolution. At this point the linearized equations for the perturbations become singular (we show that this happens for all the models mentioned above). We explicitly solve the equations in the simplest cases of a vector without vev in a FRW geometry, and of a vector with vev plus a cosmological constant, and we show that indeed the solutions of the linearized equations diverge when these equations become singular.

• The paper demonstrates that vector fields nonminimally coupled to curvature intrinsically develop ghost instabilities, undermining model consistency.
• The analysis of kinetic matrix eigenvalues uncovers negative energy modes, indicating vacuum decay during key cosmological epochs.
• By examining both classical and quantum perturbations, the study constrains vector field models and suggests pathways for ghost-free UV completions.

Ghost Instabilities in Cosmological Models with Nonminimally Coupled Vector Fields

This paper investigates the stability of cosmological models characterized by vector fields that are nonminimally coupled to curvature, focusing on the presence of ghost instabilities in such systems. Nonminimal couplings to curvature in vector field models present an intriguing possibility for understanding the early universe's evolution and the generation of cosmological anisotropies. However, the emergence of ghosts presents a fundamental theoretical challenge that jeopardizes these models' consistency.

Main Findings

1. Presence of Ghosts: The authors rigorously demonstrate that the ghost instabilities are intrinsic to a class of cosmological models where vector fields have nonminimal coupling to curvature, as seen in the Turner-Widrow mechanism for magnetic field production, and in models such as vector inflation or vector curvaton. The analysis reveals that the longitudinal polarization mode of the vector field evolves into a ghost under such couplings.
2. Eigenvalues of the Kinetic Matrix: By examining the eigenvalues of the kinetic matrix for physical perturbations, the paper shows that these models possess negative energy excitations (ghosts) due to the evolution of these eigenvalues, which become negative at certain cosmological epochs. This indicates vacuum instability and the potential for the decay of the vacuum into ghost-nonghost pairs, posing significant issues for a well-behaved quantum theory.
3. Implications for Cosmological Evolution: The study covers illustrative cases like a single vector field in an FRW universe and models including a cosmological constant or a scalar inflaton. It demonstrates how ghost instabilities manifest during the early universe's sub-horizon evolution and post-inflationary stages, especially around the horizon crossing and when vector mass terms vanish.
4. Instabilities at the Linearized Level: The analysis rigorously shows that the system of linearized equations for perturbations becomes singular at the moment an eigenvalue of the kinetic matrix crosses zero. This singularity typically leads to a divergence in the solutions, suggesting that even classical analyses of these models without quantum considerations face significant robustness issues.

Implications and Future Directions

The findings of this paper have profound implications for the theoretical construction of cosmological models:

• Theoretical Consistency: Models with ghosts can only function as effective theories up to a certain energy scale, necessitating a UV completion that eliminates these negative energy artifacts. The challenge remains to construct a higher-order theory that incorporates the intriguing features of these vector field models without the associated ghosts.
• Constraints on Vector Models: The detailed calculations impose strict constraints on the realistic parameter spaces of models employing vector fields in early universe cosmology. Future theoretical developments must reconcile the benefits of nonminimal coupling (such as addressing isotropy anomalies) with the need for a stable, ghost-free formulation.
• Quantum and Classical Analyses: This work underscores the importance of considering both quantum and classical instabilities in cosmological models. The synchronization of results from quantized linear perturbations with classical dynamics is essential for accurate phenomenology.
• Exploration of Alternatives: The issues raised by this study encourage further research into alternative frameworks, such as models with modified kinetic terms or those involving non-Abelian structures, which might circumvent these challenges.

This research has advanced the understanding of ghost instabilities within cosmological models featuring vector fields nonminimally coupled to curvature, setting a foundation for future explorations toward more stable and theoretically robust cosmological theories.