Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian

Abstract: We study the structure of scalar-tensor theories of gravity based on derivative couplings between the scalar and the matter degrees of freedom introduced through an effective metric. Such interactions are classified by their tensor structure into conformal (scalar), disformal (vector) and extended disformal (traceless tensor), as well as by the derivative order of the scalar field. Relations limited to first derivatives of the field ensure second order equations of motion in the Einstein frame and hence the absence of Ostrogradski ghost degrees of freedom. The existence of a mapping to the Jordan frame is not trivial in the general case, and can be addressed using the Jacobian of the frame transformation through its eigenvalues and eigentensors. These objects also appear in the study of different aspects of such theories, including the metric and field redefinition transformation of the path integral in the quantum mechanical description. Although sane in the Einstein frame, generic disformally coupled theories are described by higher order equations of motion in the Jordan frame. This apparent contradiction is solved by the use of a hidden constraint: the contraction of the metric equations with a Jacobian eigentensor provides a constraint relation for the higher field derivatives, which allows one to express the dynamical equations in a second order form. This signals a loophole in Horndeski's theorem and allows one to enlarge the set of scalar-tensor theories which are Ostrogradski-stable. The transformed Gauss-Bonnet terms are also discussed for the simplest conformal and disformal relations.

• The paper introduces a new class of scalar-tensor theories that extend Horndeski by employing hidden constraints to retain second-order dynamics.
• It classifies derivative couplings into conformal, disformal, and extended disformal types while using Jacobian analysis for robust frame transformations.
• The work challenges conventional gravitational theory limits and offers a framework with practical implications for explaining cosmic acceleration without dark energy.

Overview of Scalar-Tensor Theories Beyond the Horndeski Framework

The paper focuses on the exploration of scalar-tensor theories of gravity, particularly those characterized by derivative couplings to matter via an effective metric. These derivative couplings are classified by their tensor structure into conformal (scalar), disformal (vector), and extended disformal (traceless tensor) forms. This classification leads to a framework where interactions involve specific derivative orders of the scalar field, offering a broader spectrum beyond the classical Horndeski theory.

Main Contributions

The research primarily investigates the transformation and stability of these derivative-coupled scalar-tensor theories:

1. Classification and Metrics:
   • The interaction frameworks are classified based on their tensor structures and derivative orders from the scalar field.
   • They propose derivative couplings through Bekenstein's disformal relation and potential generalized forms, with primary focus on maintaining second-order equations of motion to avoid Ostrogradski instability.
2. Frame Transformation and Jacobian Analysis:
   • The study emphasizes the importance of mapping to the Jordan frame (minimal couplings to matter).
   • It employs the Jacobian determinant criteria to ensure frame transformation’s validity, discussing circumstances where inverse transformations are feasible.
   • The eigenvalues of the Jacobian provide insights into potential singularities and continuity of the inverse map, enhancing understanding of the theoretical landscape provided by derivative couplings.
3. Jordan Frame and Ostrogradski Stability:
   • By examining the gravitational Lagrangian in the Jordan frame, the paper reveals implicit strategies to ensure second-order dynamics despite apparent higher derivative terms.
   • The work identifies a "loophole" in Horndeski's theorem. It establishes that certain theories beyond the traditional Horndeski Lagrangian can also retain second-order equations, hence avoiding Ostrogradski instability via hidden constraints.
4. Horndeski Theorem Extension:
   • The paper suggests that the current understanding of stable scalar-tensor theories may be extended using this new class of derivative-coupled frameworks that exploit field redefinitions, implicit constraints, and innovative utilization of degenerate structures.

Practical and Theoretical Implications

The implications of these findings are twofold:

• Theoretical Implications: This work broadens theoretical possibilities, suggesting that the landscape of viable gravitational theories is larger than previously considered. By avoiding instability through implicit constraints, these models extend beyond the established Horndeski framework, offering new opportunities for theoretical exploration and model-building.
• Practical Implications: Practically, the development of stable alternative theories of gravity can potentially offer explanations for cosmic phenomena that are not adequately addressed by standard models or classical Horndeski theories, such as cosmic acceleration without dark energy or modified gravity frameworks.

Speculation on Future Directions

This research opens several potential avenues for ongoing work:

• Exploration of Non-Horndeski Terms: Future research might explore how terms outside the Horndeski Lagrangian contribute to the second-order nature of these models, elucidating new constraints that might emerge.
• Quantum Mechanical Considerations: The role of Jacobians and potential quantum anomalies may demand further exploration, especially regarding their classical and quantum mechanical equivalence.
• Phenomenological Applications: Developing models to test these theories against cosmological and astrophysical data could validate the theoretical extensions proposed, leading to better constraints on scalar-tensor theories of gravity.

This paper thus serves as a pivotal exploration of extending gravitational theory beyond established norms, suggesting a richer framework for understanding fundamental interactions in the cosmological context.