Studies of dark energy at advanced gravitational-wave (GW) interferometers normally focus on the dark energy equation of state wDE(z). However, modified gravity theories that predict a nontrivial dark energy equation of state generically also predict deviations from general relativity in the propagation of GWs across cosmological distances, even in theories where the speed of gravity is equal to c. We find that, in generic modified gravity models, the effect of modified GW propagation dominates over that of wDE(z), making modified GW propagation a crucial observable for dark energy studies with standard sirens. We present a convenient parametrization of the effect in terms of two parameters (Ξ0, n), analogue to the (w0, wa) parametrization of the dark energy equation of state, and we give a limit from the LIGO/Virgo measurement of H0 with the neutron star binary GW170817. We then perform a Markov Chain Monte Carlo analysis to estimate the sensitivity of the Einstein Telescope (ET) to the cosmological parameters, including (Ξ0, n), both using only standard sirens, and combining them with other cosmological datasets. In particular, the Hubble parameter can be measured with an accuracy better than 1% already using only standard sirens while, when combining ET with current CMB+BAO+SNe data, Ξ0 can be measured to 0.8% . We discuss the predictions for modified GW propagation of a specific nonlocal modification of gravity, recently developed by our group, and we show that they are within the reach of ET. Modified GW propagation also affects the GW transfer function, and therefore the tensor contribution to the ISW effect.arXiv:1805.08731v3 [gr-qc]

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...

In modified gravity the propagation of gravitational waves (GWs) is in general different from that in general relativity. As a result, the luminosity distance for GWs can differ from that for electromagnetic signals, and is affected both by the dark energy equation of state wDE(z) and by a function δ(z) describing modified propagation. We show that the effect of modified propagation in general dominates over the effect of the dark energy equation of state, making it easier to distinguish a modified gravity model from ΛCDM. We illustrate this using a nonlocal modification of gravity that has been shown to fit remarkably well CMB, SNe, BAO and structure formation data, and we discuss the prospects for distinguishing nonlocal gravity from ΛCDM with the Einstein Telescope. We find that, depending on the exact sensitivity, a few tens of standard sirens with measured redshift at z ∼ 0.4, or a few hundreds at 1 < ∼ z < ∼ 2, could suffice.

We study the cosmological consequences of a recently proposed nonlocal modification of general relativity, obtained by adding a term m 2 R 2 −2 R to the Einstein-Hilbert action. The model has the same number of parameters as ΛCDM, with m replacing Ω Λ , and is very predictive. At the background level, after fixing m so as to reproduce the observed value of Ω M , we get a pure prediction for the equation of state of dark energy as a function of redshift, w DE (z), with w DE (0) in the range [−1.165, −1.135] as Ω M varies over the broad range Ω M ∈ [0.20, 0.36]. We find that the cosmological perturbations are well-behaved, and the model fully fixes the dark energy perturbations as a function of redshift z and wavenumber k. The nonlocal model provides a good fit to supernova data and predicts deviations from General Relativity in structure formation and in weak lensing at the level of 3-4%, therefore consistent with existing data but readily detectable by future surveys. For the logarithmic growth factor we obtain γ 0.53, to be compared with γ 0.55 in ΛCDM. For the Newtonian potential on subhorizon scales our results are well fitted by Ψ(a; k) = [1 + µ s a s ]Ψ GR (a; k) with a scale-independent µ s 0.09 and s 2, while the anisotropic stress is negligibly small.