A formal introduction to Horndeski and Galileon theories and their generalizations (1307.2450v1)

Abstract: We review different constructions of Galileon theories in both flat and curved space, and for both single scalar field models as well as multi-field models. Our main emphasis is on the formal mathematical properties of these theories and their construction.

Overview of Horndeski and Galileon Theories and Their Generalizations

The paper presents a comprehensive examination of Horndeski and Galileon theories, including their generalizations, within both flat and curved space-time contexts. The authors offer a detailed exploration of the mathematical properties and constructions of these theories, with particular emphasis on their applications to cosmology and scalar-tensor theories in physics.

Key Focus Areas

1. Galileon Theories in Flat Space-Time: Galileon theories are discussed as scalar field models characterized by field equations that are polynomial in second-order derivatives, involving no undifferentiated or singly differentiated terms. These models have historical roots in the work of Nicolis et al. (2008) and subsequent generalizations to curved space-time by Deffayet and collaborators. The formulations describe these theories in any arbitrary dimension, emphasizing their Lagrangian representations.
2. Field Equations: The paper explores the second-order nature of Galileon field equations. It is shown that the antisymmetric properties of certain tensors ensure the equations remain second order, which is crucial for avoiding higher-order derivatives that could lead to additional degrees of freedom and potentially destabilize the theory.
3. Extensions to Curved Space-Time: A significant portion of the discussion is devoted to extending Galileon theories to curved space-time. This process, known as "covariantization," modifies the Lagrangian to prevent the emergence of higher-than-second-order derivatives, thereby maintaining the desired characteristics of the theories.
4. Generalized Scalar Theories: The authors examine scalar theories that allow for more general constructs in flat space-time and seek the most inclusive scalar field theories that yield second-order equations when extended to curved space-time. This involves introducing arbitrary functions of the scalar field and its derivatives into the Lagrangian, while still ensuring the second-order nature of the field equations.
5. Multi-field Galileon Models: Theories incorporating multiple fields or $p$-forms explore models with several scalar fields, including investigations into how these can be implemented while maintaining second-order field equations.

Theoretical and Practical Implications

The insights presented in the paper have significant implications for cosmological applications, particularly in modeling cosmic acceleration and inflation scenarios without introducing ghost instabilities. The formalism of Galileons enables enhanced control over the scalar field interactions and supports the viable creation of cosmological scenarios where stability is maintained across both flat and curved space-time.

Speculation on Future Developments

Future work in this area may extend the understanding and application of Galileon theories to more complex cosmological models. The exploration of the UV completion of these theories remains a critical challenge, with ongoing research needed to confirm the potential for renormalization and effective field theory validity.

In conclusion, the paper serves as a robust and insightful paper of Horndeski and Galileon theories, providing a solid foundation for future theoretical developments and practical applications in advanced physics, particularly within the field of scalar-tensor and cosmological modeling.