Covariant Galileon

Abstract: We consider the recently introduced "galileon" field in a dynamical spacetime. When the galileon is assumed to be minimally coupled to the metric, we underline that both field equations of the galileon and the metric involve up to third-order derivatives. We show that a unique nonminimal coupling of the galileon to curvature eliminates all higher derivatives in all field equations, hence yielding second-order equations, without any extra propagating degree of freedom. The resulting theory breaks the generalized "Galilean" invariance of the original model.

• The paper introduces a nonminimal curvature coupling that reduces problematic third-order derivatives to a consistent second-order formulation in curved spacetime.
• It demonstrates that the modified field equations maintain the original degrees of freedom, thereby avoiding instabilities linked with higher-order derivatives.
• The study paves the way for alternative dark energy-free cosmological models through a refined, covariant framework for modified gravity.

Overview of Covariant Galileon

The paper "Covariant Galileon" by C. Deffayet, G. Esposito-Farese, and A. Vikman focuses on the theoretical development and refinement of the galileon field in a dynamical spacetime context. Galileon theories, initially inspired by the Dvali-Gabadadze-Porrati (DGP) model, are scalar field theories notable for their capability to explain cosmic acceleration without the need for dark energy or a cosmological constant. This paper critically analyzes the covariantization of galileon theories and introduces a unique modification that ensures second-order field equations across different spacetime backgrounds.

Key Contributions

1. Third-Order Derivative Elimination: The authors identify that the minimally-coupled galileon field equations, when generalized to curved spacetime, inherently involve higher derivatives up to third order. To address this, they present a singular nonminimal coupling of the galileon to curvature that effectively reduces these derivatives to second order in both the scalar field and metric equations.
2. Preservation of Degrees of Freedom: Importantly, the proposed modification succeeds without introducing any additional propagating degrees of freedom, ensuring the theoretical model remains consistent with known physics frameworks. This is primarily achieved by strategically manipulating the coupling to the curvature, thus simplifying the field dynamics.
3. Symmetry Considerations: In Minkowski spacetime, the galileon field enjoys specific symmetries, including invariance under constant shifts of both the field and its derivatives. The introduction of nonminimal coupling necessarily breaks some of these symmetries in curved spacetime, but it aligns the theory more closely with general theories of modification in dynamical settings.
4. Nonminimal Couplings: The paper describes a detailed construction of nonminimal coupling terms for specific Lagrangians—particularly L4 and L5—which deal explicitly with the problematic higher-order derivatives. This construction is unique and assures the absence of third- or higher-order terms in both the scalar field equations and the associated energy-momentum tensor expressions.

Theoretical and Practical Implications

The authors' exploration into the covariantized galileon theory has significant implications for both fundamental theoretical physics and potential observational impacts:

• Theoretical Implications: By ensuring equations of motion remain of second order, the issues linked to Ostrogradski’s instability, which are prevalent in higher-derivative theories, are avoided. This contributes to the development of tractable and potentially stable modifications of General Relativity that might be probed in cosmological contexts.
• Cosmological Applications: The galileon model, particularly in explaining cosmic acceleration sans dark energy, aligns with the attempts to construct alternative theories of gravity. The refined covariant galileon model could serve as a robust basis for exploring such cosmological phenomena. Further investigation into the phenomenological predictions associated with this theory is suggested.

Future Exploration

The absence of exotic energies and instabilities makes this adapted covariant framework promising for future exploration. Prospective studies might focus on the following:

• Behavior in Higher Dimensions: Extending this model to curved and higher-dimensional backgrounds could reveal new properties or constraints, lending insight into the behavior of scalar fields in general relativity and its modifications.
• Phenomenological Predictions: The modified galileon theories can be detailed into specific cosmological or astrophysical predictions that could contrast with standard ΛCDM models.

In conclusion, this paper contributes a crucial refinement to the existing framework of galileon theories, particularly in dynamic and curved spacetimes. The proposed nonminimal coupling and the resultant simplification of field equations present a step forward in the quest to integrate theoretical elegance with observational veracity in modified gravity models.