Stability of Horndeski vector-tensor interactions

TL;DR

This paper analyzes the stability and dynamics of a simple Horndeski vector-tensor theory with a non-minimal coupling, exploring its behavior in various spacetimes and identifying conditions for stability and the presence of superluminal propagation.

Contribution

It provides a comprehensive stability analysis of a minimal Horndeski vector-tensor model, including cosmological and static backgrounds, highlighting conditions for ghost-free and Laplacian-stable configurations.

Findings

The theory is an attractor in de Sitter spacetime.

Positive M^2 ensures absence of ghosts in FLRW backgrounds.

Superluminal propagation is common in regions dominated by non-minimal interactions.

Abstract

We study the Horndeski vector-tensor theory that leads to second order equations of motion and contains a non-minimally coupled abelian gauge vector field. This theory is remarkably simple and consists of only 2 terms for the vector field, namely: the standard Maxwell kinetic term and a coupling to the dual Riemann tensor. Furthermore, the vector sector respects the U(1) gauge symmetry and the theory contains only one free parameter, M^2, that controls the strength of the non-minimal coupling. We explore the theory in a de Sitter spacetime and study the presence of instabilities and show that it corresponds to an attractor solution in the presence of the vector field. We also investigate the cosmological evolution and stability of perturbations in a general FLRW spacetime. We find that a sufficient condition for the absence of ghosts is M^2>0. Moreover, we study further constraints coming from imposing the absence of Laplacian instabilities. Finally, we study the stability of the theory in static and spherically symmetric backgrounds (in particular, Schwarzschild and Reissner-Nordstr\"om-de Sitter). We find that the theory, quite generally, do have ghosts or Laplacian instabilities in regions of spacetime where the non-minimal interaction dominates over the Maxwell term. We also calculate the propagation speed in these spacetimes and show that superluminality is a quite generic phenomenon in this theory.