We present all scalar-tensor Lagrangians that are cubic in second derivatives of a scalar field, and that are degenerate, hence avoiding Ostrogradsky instabilities. Thanks to the existence of constraints, they propagate no more than three degrees of freedom, despite having higher order equations of motion. We also determine the viable combinations of previously identified quadratic degenerate Lagrangians and the newly established cubic ones. Finally, we study whether the new theories are connected to known scalartensor theories such as Horndeski and beyond Horndeski, through conformal or disformal transformations.

We study new consistent scalar-tensor theories of gravity recently introduced by Langlois and Noui with potentially interesting cosmological applications. We derive the conditions for the existence of a primary constraint that prevents the propagation of an additional dangerous mode associated with higher order equations of motion. We then classify the most general, consistent scalar-tensor theories that are at most quadratic in the second derivatives of the scalar field. In addition, we investigate the possible connection between these theories and (beyond) Horndeski through conformal and disformal transformations. Finally, we point out that these theories can be associated with new operators in the effective field theory of dark energy, which might open up new possibilities to test dark energy models in future surveys. I. INTRODUCTIONScalar tensor theories of gravity play an essential role for building models of inflation and dark energy. The most general scalar-tensor theory leading to second order equations of motion, the theory of Horndeski [1, 2], propagates three degrees of freedom (dof) in the vacuum. When coupled with matter, this set-up provides a convenient framework for parametrising linear cosmological perturbations and for developing tests of General Relativity using cosmological observations (see e.g. [3] for a review). On the other hand, it has been recently pointed out [4-6] that there are scalar-tensor theories more general than Horndeski, which are characterised by higher order equations of motion, but which nevertheless propagate three degrees of freedom in the vacuum. This property is ensured by the existence of constraints that forbid the propagation of additional, dangerous modes. Such theories are dubbed 'beyond Horndeski' [4,5].The existence of these theories raises various broad questions that we consider in this paper:• What is the most general, consistent theory of gravity coupled with a scalar field? Beyond Horndeski is an explicit example of a consistent generalisation of Horndenski, but it is not necessarily the most general one. In fact, [7,8] provided explicit examples of theories more general than beyond Horndeski. They study degeneracy conditions for the kinetic matrix associated with Lagrangians that are at most quadratic in the scalar second derivatives. In the present work, we re-derive the results of [7] using a method that provides the conditions for the existence of a primary constraint starting from the conjugate momenta of the scalar and tensor fields. Our method is equivalent to the one developed in [7] to impose the degeneracy conditions for the kinetic matrix. In the present work we limit our attention to theories quadratic in second derivatives of the scalar and we dub them extended scalar-tensor theories of gravity, in brief EST theories, and classify them carefully.• Is there any connection between EST theories and the original Horndeski, or beyond Horndeski actions? It is known that generalised disformal transformations allow to generate beyond Horndeski Lagrangia...

Modifications of General Relativity leave their imprint both on the cosmic ex-Contents A GW luminosity distance and the flux-luminosity relation 53 B Technical details on bigravity 55 B.1 Hassan-Rosen theory of bigravity 55 B.2 Details on the WKB approximation for bigravity 56References 58 5. In the presence of anisotropic stress, or in theories where tensors couple with additional fields already at linearised level (as in theories breaking spatial diffeomorphisms), the tensor evolution equation contains a "source term" Π A in the right hand side of eq. (1.2). In absence of anisotropic stress, and in cosmological scenarios where spatial diffeomorphisms are preserved, we have Π A = 0.The physical consequences of these parameters have been discussed at length in the literature (see [18] for a review on their implications for GW astronomy). In this paper we investigate how they affect a specific observable, the GW luminosity distance, which can be probed by LISA standard sirens. The space-based interferometer LISA can qualitatively and quantitatively improve our tests on the propagation of gravitational waves in theories of modified gravity. LISA can probe signals from standard sirens of supermassive black hole mergers (MBHs) at redshifts z ∼ O(1 − 10), much larger than the redshifts z ∼ O(10 −1 ) of typical sources detectable from second-generation ground-based interferometers. This implies that LISA can test the possible time dependence of the parameters controlling deviations from GR or the standard ΛCDM model, since GWs travel large cosmological distances before reaching the observer. Moreover, as we will review in section 4, LISA can measure the luminosity distance to MBHs with remarkable precision, thereby reaching an accuracy not possible for second-generation ground-based detectors.It is also interesting to observe that LISA can probe GWs in the frequency range in the milli-Hz regime (more precisely, in the interval 10 −4 − 10 0 Hz), much smaller than the typical frequency interval of ground-based detectors, 10 1 − 10 3 Hz. This is a theoretically interesting range to explore since several theories of modified gravity designed to explain dark energy, such as Horndeski, degenerate higher order scalar-tensor (DHOST) theories or massive gravity, have a low UV cutoff, typically of order Λ cutoff ∼ H 2 0 M Pl 1/3 ∼ 10 2 Hz.This cutoff is within the frequency regime probed by LIGO, making a comparison between modified gravity predictions and GW observations delicate [19]. The frequency range tested by LISA, instead, is well below this cutoff, hence it lies within the range of validity of the theories under consideration. The paper is organized as follows. In section 2 we recall the notion of modified GW propagation and GW luminosity distance, that emerges generically in modified theories of gravity. In section 3 we discuss the prediction on modified GW propagation of some of the best studied modified-gravity theories: scalar-tensor theories (with particular emphasis on Horndeski and DHOST theories), infrared non-l...