Counting the degrees of freedom of generalized Galileons

TL;DR

This paper proves that generalized Galileon models on curved spacetime have only three degrees of freedom, using gauge-invariant methods and Hamiltonian analysis, confirming their second-order nature and clarifying their dynamical content.

Contribution

It provides a gauge-invariant proof that these models have at most three degrees of freedom and performs a Hamiltonian analysis to show the count is strictly less than four in a specific case.

Findings

All third-order derivatives can be eliminated to yield second-order equations.

The models have only three degrees of freedom, two for the graviton and one for the scalar.

In a specific model, the degrees of freedom are strictly less than four.

Abstract

We consider Galileon models on curved spacetime, as well as the counterterms introduced to maintain the second-order nature of the field equations of these models when both the metric and the scalar are made dynamical. Working in a gauge invariant framework, we first show how all the third-order time derivatives appearing in the field equations -- both metric and scalar -- of a Galileon model or one defined by a given counterterm can be eliminated to leave field equations which contain at most second-order time derivatives of the metric and of the scalar. The same is shown to hold for arbitrary linear combinations of such models, as well as their k-essence-like/Horndeski generalizations. This supports the claim that the number of degrees of freedom in these models is only 3, counting 2 for the graviton and 1 for the scalar. We comment on the arguments given previously in support of this claim. We then prove that this number of degrees of freedom is strictly less that 4 in one particular such model by carrying out a full-fledged Hamiltonian analysis. In contrast to previous results, our analyses do not assume any particular gauge choice of restricted applicability.