The Horndeski action is the most general one involving a metric and a scalar field that leads to second order field equations in four dimensions. Being the natural extension of the well known ScalarTensor theories, its structure and properties are worth analyzing along the experience accumulated in the latter context. Here we argue that disformal transformations play, for the Horndeski theory, a similar role to that of conformal transformations for Scalar-Tensor theories a là Brans-Dicke. We identify the most general transformation preserving second order field equations and discuss the issue of viable frames for this kind of theories, in particular the possibility to cast the action in the so called Einstein frame. Interestingly, we find that only for a subset of Horndeski Lagrangian such frame exists. Finally, we investigate the transformation properties of such frames under field redefinitions and frame transformations and their reciprocal relationship.

The direct detection of gravitational waves (GWs) is an invaluable new tool to probe gravity and the nature of cosmic acceleration. A large class of scalar-tensor theories predict that GWs propagate with velocity different than the speed of light, a difference that can be O(1) for many models of dark energy. We determine the conditions behind the anomalous GW speed, namely that the scalar field spontaneously breaks Lorentz invariance and couples to the metric perturbations via the Weyl tensor. If these conditions are realized in nature, the delay between GW and electromagnetic (EM) signals from distant events will run beyond human timescales, making it impossible to measure the speed of GWs using neutron star mergers or other violent events. We present a robust strategy to exclude or confirm an anomalous speed of GWs using eclipsing binary systems, whose EM phase can be exquisitely determined. he white dwarf binary J0651+2844 is a known example of such system that can be used to probe deviations in the GW speed as small as cg/c − 1 2 · 10 −12 when LISA comes online. This test will either eliminate many contender models for cosmic acceleration or wreck a fundamental pillar of general relativity.PACS numbers: 04.30.Nk 04.50. Kd, 95.36.+x, Introduction and summary. The direct detection of gravitational radiation [1,2] has initiated a new era for astronomy, astrophysics and fundamental physics. The observed gravitational wave (GW) events and the ones to come will usher in novel ways to test the nature of gravity [3]. Here, we will argue that probing the speed of GWs will be a decisive test for gravity and dark energy models.