Cosmology in generalized Horndeski theories with second-order equations of motion

TL;DR

This paper explores extended Horndeski theories with multiple scalar fields in cosmology, analyzing their perturbation speeds and stability, and distinguishes between different Galileon models based on their perturbation behavior.

Contribution

It extends Horndeski theories to include multiple scalar fields, deriving their perturbation properties and stability conditions, and compares covariantized Galileon models in a cosmological context.

Findings

Propagation speeds of scalar and tensor perturbations are derived.

Modifications to matter field speeds are generally suppressed.

The scalar speed squared for the dark energy field can become negative during matter era.

Abstract

We study the cosmology of an extended version of Horndeski theories with second-order equations of motion on the flat Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) background. In addition to a dark energy field $\chi$ associated with the gravitational sector, we take into account multiple scalar fields $\phi_I$ ($I=1,2\cdots,N-1$) characterized by the Lagrangians $P^{(I)}(X_I)$ with $X_I=\partial_{\mu}\phi_I\partial^{\mu}\phi_I$. These additional scalar fields can model the perfect fluids of radiation and non-relativistic matter. We derive propagation speeds of scalar and tensor perturbations as well as conditions for the absence of ghosts. The theories beyond Horndeski induce non-trivial modifications to all the propagation speeds of $N$ scalar fields, but the modifications to those for the matter fields $\phi_I$ are generally suppressed relative to that for the dark energy field $\chi$. We apply our results to the covariantized Galileon with an Einstein-Hilbert term in which partial derivatives of the Minkowski Galileon are replaced by covariant derivatives. Unlike the covariant Galileon with second-order equations of motion in general space-time, the scalar propagation speed square $c_{s1}^2$ associated with the field $\chi$ becomes negative during the matter era for late-time tracking solutions, so the two Galileon theories can be clearly distinguished at the level of linear cosmological perturbations.