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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Modified gravity theories with an effective Newton constant that varies over cosmological timescales generally predict a different gravitational wave luminosity distance than General Relativity. While this holds for a uniform variation, we show that if locally screened at the source and at the observer as required to pass stringent astrophysical tests of gravity, the General Relativistic distance is restored. In the absence of such a screening, the same effect must modify electromagnetic luminosity distances inferred from supernovae Type Ia, to the extent that the effects can cancel in the comparison. Hence, either the modifications considered employ screening, which leaves no signature in Standard Sirens of a cosmological modification of gravity, or screening does not operate, in which case there can be a signal that is however well below the forseeable sensitivity of the probe when astrophysical bounds are employed. We recover these results both in the Jordan and Einstein frames, paying acute attention to pecularities of each frame such as the notion of redshift or geodesic motions. We emphasise that despite these limitations, Standard Sirens provide valuable independent tests of gravity that differ fundamentally from other probes, a circumstance that is generally important for the wider scope of gravitational modifications and related scenarios. Finally, we use our results to show that the gravitational wave propagation is not affected by dark sector interactions, which restores a dark degeneracy between conformal and disformal couplings that enables observationally viable cosmic self-acceleration to emenate from those.

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In dark-energy models where a scalar field is nonminimally coupled to the spacetime geometry, gravitational waves are expected to be supplemented with a scalar mode. Such scalar waves may interact with the standard tensor waves, thereby affecting their observed amplitude and polarization. Understanding the role of scalar waves is thus essential in order to design reliable gravitational-wave probes of dark energy and gravity beyond general relativity. In this article, we thoroughly investigate the propagation of scalar and tensor waves in the subset of Horndeski theories in which tensor waves propagate at the speed of light. We work at linear order in scalar and metric perturbations, in the eikonal regime, and for arbitrary scalar and spacetime backgrounds. We diagonalize the system of equations of motion and identify the physical tensor mode, which differs from the metric perturbation. We find that interactions between scalar and tensor waves generally depend on the scalar propagation speed. If the scalar waves are luminal or quasiluminal, then interactions are negligible. In the subluminal case, scalar-tensor interactions are effectively suppressed due to the incoherence of the wave's phases.