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...

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We perform the Hamiltonian analysis of several mimetic gravity models and compare our results with those obtained previously by different authors. We verify that for healthy mimetic scalartensor theories the condition for the corresponding part of the Hamiltonian to be bounded from below is the positive value of the mimetic field energy density λ. We show that for mimetic dark matter possessing a shift symmetry the mimetic energy density remains positive in time, provided appropriate boundary conditions are imposed on its initial value, while in models without shift symmetry the positive energy density can be maintained by simply replacing λ → e λ . The same result also applies to mimetic f (R) gravity, which is healthy if the usual stability conditions of the standard f (R) gravity are assumed and λ > 0. In contrast, if we add mimetic matter to an unhealthy seed action, the resulting mimetic gravity theory remains, in general, unstable. As an example, we consider a scalar-tensor theory with the higher-derivative term ( ϕ) 2 , which contains an Ostrogradski ghost. We also revisit results regarding stability issues of linear perturbations around the FLRW background of the mimetic dark matter in the presence of ordinary scalar matter.We find that the presence of conventional matter does not revive dynamical ghost modes (at least in the UV limit). The modes, whose Hamiltonian is not positive definite, are non-propagating (have zero sound speed) and are associated with the mimetic matter itself. They are already present in the case in which the ordinary scalar fluid is absent, causing a growth of dust overdensity.