Exploring gravitational theories beyond Horndeski (1408.1952v2)

Abstract: We have recently proposed a new class of gravitational scalar-tensor theories free from Ostrogradski instabilities, in arXiv:1404.6495. As they generalize Horndeski theories, or "generalized" galileons, we call them G$^3$. These theories possess a simple formulation when the time hypersurfaces are chosen to coincide with the uniform scalar field hypersurfaces. We confirm that they contain only three propagating degrees of freedom by presenting the details of the Hamiltonian formulation. We examine the coupling between these theories and matter. Moreover, we investigate how they transform under a disformal redefinition of the metric. Remarkably, these theories are preserved by disformal transformations that depend on the scalar field gradient, which also allow to map subfamilies of G$^3$ into Horndeski theories.

• The paper introduces G3 theories as extensions beyond Horndeski that incorporate additional scalar-tensor interactions while avoiding Ostrogradski instabilities.
• The paper employs Hamiltonian analysis and the unitary gauge within the ADM formalism to confirm the preservation of three degrees of freedom.
• The paper discusses disformal transformations and unique matter couplings, providing novel insights into cosmic acceleration and structure formation.

Beyond Horndeski: Expanding the Landscape of Gravitational Theories

The exploration of gravitational theories has increasingly turned toward extensions beyond the well-studied Horndeski theories, responding to the complex demands posed by cosmological observations, notably the late-time accelerated expansion of the universe. The paper "Exploring gravitational theories beyond Horndeski" by Jérôme Gleyzes et al. explores one such class of theories, referred to as "Generalized Generalized Galileons," or G$_3$.

Key Theoretical Advances

The authors build upon the foundational framework of scalar-tensor theories, which incorporate a scalar field alongside the tensorial degrees of freedom from general relativity. A significant focus is given to overcoming Ostrogradski instabilities, which are typically induced by higher-order derivatives in the equations of motion. The proposed G$_3$ theories extend Horndeski theories by integrating additional terms that preserve the desired second-order nature of the equations of motion, thereby circumventing ghost-like instabilities.

Methodological Insights

The Hamiltonian analysis is central to this research, confirming that the proposed theories maintain three degrees of freedom, consistent with absence of Ostrogradski instabilities. This analysis effectively isolates the viable regions of the phase space for these theories. Notably, the unitary gauge within the ADM formalism demonstrates the theoretical robustness of G$_3$ theories by encoding the scalar degrees of freedom within the metric components, simplifying stability analysis.

Disformal Transformations and Theoretical Implications

The work explores the impact of disformal transformations, beyond the usual conformal transformations, as a method for mapping G$_3$ theories into the Horndeski framework. The authors illustrate that while a complete conversion is constrained, individual components of the G$_3$ Lagrangian can be related to Horndeski theories under specific conditions. This elucidates the underlying geometric structures and emphasizes the potential for novel phenomenology in cosmic dynamics and gravitational interactions.

Cosmological Perturbations and Matter Couplings

The paper addresses the behavior of linear cosmological perturbations in G$_3$ theories, both in the presence and absence of matter. The analysis shows no additional ghost modes, laying the ground for further exploration of their cosmological viability. These theories exhibit unique matter couplings, suggesting intriguing modifications to the standard Jeans instability that governs structure formation.

Numerical and Theoretical Implications

Theoretical explorations in this paper set the stage for vast implications in cosmology, especially in understanding the complex gravitational dynamics that may underlie accelerated cosmic expansion. Future studies could further examine these extensions' implications for cosmological observations, distinct signatures in gravitational waves, and potential interactions with dark matter.

This paper, through its detailed expansion of the Horndeski framework and careful analysis of stability, contributes significantly to the ongoing development of viable gravitational theories that go beyond traditional paradigms. As such, it represents a critical step in broadening our understanding of ultimate gravitational law, with promising directions for addressing existing cosmological paradoxes.