Hamiltonian analysis of higher derivative scalar-tensor theories (1512.06820v1)

Abstract: We perform a Hamiltonian analysis of a large class of scalar-tensor Lagrangians which depend quadratically on the second derivatives of a scalar field. By resorting to a convenient choice of dynamical variables, we show that the Hamiltonian can be written in a very simple form, where the Hamiltonian and the momentum constraints are easily identified. In the case of degenerate Lagrangians, which include the Horndeski and beyond Horndeski quartic Lagrangians, our analysis confirms that the dimension of the physical phase space is reduced by the primary and secondary constraints due to the degeneracy, thus leading to the elimination of the dangerous Ostrogradski ghost. We also present the Hamiltonian formulation for nondegenerate theories and find that they contain four degrees of freedom, as expected. We finally discuss the status of the unitary gauge from the Hamiltonian perspective.

• The paper demonstrates that degenerate scalar-tensor Lagrangians yield three degrees of freedom through primary and secondary constraints, eliminating Ostrogradski ghosts.
• It employs a Hamiltonian framework to clearly distinguish between degenerate and nondegenerate formulations, with nondegenerate theories exhibiting four degrees of freedom.
• The study advances modified gravity research by providing a robust methodology for constructing ghost-free models applicable to cosmological scenarios, particularly for dark energy exploration.

Hamiltonian Analysis of Higher Derivative Scalar-Tensor Theories

The research conducted by Langlois and Noui presents a comprehensive Hamiltonian analysis of scalar-tensor Lagrangians encompassing dependencies on second derivatives of a scalar field. In particular, this paper addresses the degeneracy characteristics of such theories, shedding light on their potential to resolve the problematic Ostrogradski instability, often identified with nonphysical, ghost-like degrees of freedom. The paper effectively distinguishes between degenerate and nondegenerate Lagrangian forms, providing a detailed exploration of their Hamiltonian formulations.

Considering the class of degenerate Lagrangians, which includes theories like Horndeski and beyond Horndeski quartic forms, the paper verifies that the dimensionality of the physical phase space diminishes due to primary and secondary constraints attributed to degeneracy. This reduction results in the exclusion of the Ostrogradski ghost, thereby circumventing an issue prevalent in higher derivative theories. For nondegenerate theories, the analysis identifies the presence of four degrees of freedom, aligning with theoretical predictions. The work also revisits the principles of unitary gauge evaluation through the Hamiltonian framework.

Theoretical implications of this exploration are significant. In degenerate theories, the work asserts the presence of three degrees of freedom, a configuration absent of additional ghost states, confirming prior conjectures. The nondegenerate configurations indicate an additional degree of freedom, implying a further understanding of instability mechanisms. The distinction between these outcomes offers crucial insights into constructing ghost-free scalar-tensor models in modified gravitational frameworks.

Practically, this research favors advancements in cosmological modeling, particularly where modified gravity theories are employed to explain phenomena like dark energy or cosmic acceleration, without incurring the pitfalls associated with higher order derivatives. Scalar-tensor theories, evolving beyond Horndeski's original formulation, receive validation through this detailed Hamiltonian analysis, supporting their applicability in describing our universe's dynamics.

Future developments in this field might consider extending the method presented in this paper to encompass a broader class of theories, particularly those featuring cubic-order derivative terms in their Lagrangians. Such extensions could markedly broaden the field's understanding of the dynamics in scalar-tensor models and their implications for both cosmological and theoretical physics. Further exploration of the linkage between the equations of motion's order and the number of physical degrees of freedom could also advance fundamental knowledge in this domain. Langlois and Noui's analytical method sets a precedent for future works aiming to elucidate the complex nature of higher derivative scalar-tensor frameworks.