[Paper Review] Gravitational Wave Propagation and Polarizations in the Teleparallel analog of Horndeski Gravity

Sebastián Bahamonde, Maria Caruana+6 more|arXiv (Cornell University)|May 27, 2021

Cosmology and Gravitation TheoriesPhysics and Astronomy96 references43 citations

TL;DR

This paper investigates gravitational wave (GW) polarization modes in the teleparallel analog of Horndeski gravity, a higher-order, torsion-based theory of gravity. By perturbing around a Minkowski background and decomposing modes into scalar–vector–tensor (SVT) components, the authors find up to seven propagating degrees of freedom, including massive and massless scalar and tensor modes, but no vector polarizations—revealing a richer polarization structure than in standard Horndeski gravity.

ABSTRACT

Gravitational waves (GWs) have opened a new window on fundamental physics in a number of important ways. The next generation of GW detectors may reveal more information about the polarization structure of GWs. Additionally, there is growing interest in theories of gravity beyond GR. One such theory which remains viable within the context of recent measurements of the speed of propagation of GWs is the teleparallel analogue of Horndeski gravity. In this work, we explore the polarization structure of this newly proposed formulation of Horndeski theory. In curvature-based gravity, Horndeski theory is almost synonymous with extensions to GR since it spans a large portion of these possible extensions. We perform this calculation by taking perturbations about a Minkowski background and consider which mode propagates. The result is that the polarization structure depends on the choice of model parameters in the teleparallel Horndeski Lagrangian with a maximum of seven propagating degrees of freedom. While the curvature-based Horndeski results follows as a particular limit within this setup, we find a much richer structure of both massive and massless cases which produce scalar--vector--tensor propagating degrees of freedom. We also find that the GW polarization that emerges from the teleparallel analogue of Horndeski gravity results in analogous massive and massless modes which take on at most four polarizations in the massless sector and two scalar ones in the massive sector. In none of the cases do we find vector polarizations.

Motivation & Objective

• To investigate the polarization structure of gravitational waves in the teleparallel analog of Horndeski gravity, a recently proposed extension of Horndeski theory.
• To determine how the choice of model parameters in the teleparallel Horndeski Lagrangian affects the number and type of propagating degrees of freedom (DoF).
• To compare the polarization spectrum of this theory with that of standard curvature-based Horndeski gravity and GR.
• To assess the viability of this theory in light of recent GW speed constraints, particularly through its polarization content.
• To explore whether the theory supports additional polarizations beyond the two tensor modes of general relativity, especially in the context of next-generation GW detectors.

Proposed method

• Perturbing the field equations of the teleparallel Horndeski Lagrangian around a Minkowski background to linear order.
• Decomposing the perturbations into scalar, vector, and tensor (SVT) modes to analyze their propagation behavior.
• Applying the formalism of linearized gravity in the tetrad formalism, using the Weitzenb"ock connection to describe torsion-based gravity.
• Deriving the effective wave equations for each SVT mode to determine which degrees of freedom propagate and their dispersion relations.
• Identifying the number of propagating degrees of freedom by analyzing the rank of the field equations' principal symbol using H"ormander's criterion.
• Taking the curvature-based Horndeski limit to confirm consistency with known results and to highlight the richer structure in the teleparallel formulation.

Experimental results

Research questions

• RQ1 How many propagating degrees of freedom does the teleparallel analog of Horndeski gravity support, and what types (scalar, vector, tensor) are present?
• RQ2 Does the teleparallel Horndeski theory admit vector polarization modes, and if not, why do they decouple?
• RQ3 How does the polarization structure in the teleparallel formulation compare to that of standard curvature-based Horndeski gravity?
• RQ4 What is the role of model parameters in determining the number and nature (massive or massless) of propagating modes?
• RQ5 Can the theory support a richer polarization spectrum than GR, and if so, how does this affect its detectability with future GW observatories?

Key findings

• The teleparallel analog of Horndeski gravity supports up to seven propagating degrees of freedom, significantly more than the two tensor modes of general relativity.
• The polarization spectrum includes at most four massless modes—two scalar and two tensor—along with two massive scalar modes, but no vector polarizations.
• The absence of vector modes is consistent across all parameter choices, indicating a fundamental suppression of vector degrees of freedom in this formulation.
• The massive scalar modes arise from non-trivial couplings in the Lagrangian and are distinct from the massless breathing and longitudinal modes found in other scalar-tensor theories.
• The curvature-based Horndeski theory is recovered as a particular limit of the teleparallel formulation, confirming consistency with known results.
• The theory's richer structure, including multiple scalar and tensor polarizations, makes it a promising candidate for testing beyond-GR gravity with next-generation gravitational wave detectors like LISA and the Einstein Telescope.