There is a host of alternative theories of gravitation in the literature, among them the $f(R,T)$ and $f(R,T^{\varphi })$ theories recently elaborated by Harko et al. In these theories, $R$, $T$ and $T^{\varphi }$ are respectively the Ricci scalar and the traces of the energy-momentum tensors of matter and of a scalar field. There is already in the literature a series of studies of different forms of the $f(R,T)$ and $f(R,T^{\varphi })$ functions as well as their cosmological consequences. However, there have been no studies so far related to gravitational waves. Here we consider such an issue, in particular, studying the putative extra polarization modes that can appear in the scope of such theories. Different functional forms of $f(R,T^{\varphi })$ are considered and the gravitational waveforms are found for the extra polarization modes in the cases in which they are present.

The direct observation of gravitational waves (GW) in the near future, and the corresponding determination of the number of independent polarizations, is a powerful tool to test general relativity and alternative theories of gravity. In the present work we use the Newman–Penrose formalism to characterize GWs in quadratic gravity and in a particular class of f(R) Lagrangians. We find that both quadratic gravity and the f(R) theory belong to the most general invariant class of GWs, i.e., they can present up to six independent polarizations of GWs. For a particular combination of the parameters, we find that quadratic gravity can present up to five polarizations states. On the other hand, if we use the Palatini approach for f(R) theories, GWs present only the usual two transverse-traceless polarizations such as in general relativity. Thus, we conclude that the observation of GWs can strongly constrain the suitable formalism for these theories

The direct observation of gravitational waves will provide a unique tool for probing the dynamical properties of highly compact astrophysical objects, mapping ultrarelativistic regions of space-time, and testing Einstein's general theory of relativity. LISA (Laser Interferometer Space Antenna), a joint National Aeronautics and Space Administration and European Space Agency mission to be launched in the next decade, will perform these scientific tasks by detecting and studying low-frequency cosmic gravitational waves through their influence on the phases of six modulated laser beams exchanged between three remote spacecraft. By directly measuring the polarization components of the waves LISA will detect, we will be able to test Einstein's theory of relativity with good sensitivity. Since a gravitational wave signal predicted by the most general relativistic metric theory of gravity accounts for six polarization modes (the usual two Einstein's tensor polarizations as well as two vector and two scalar wave components), we have derived the LISA time-delay interferometric responses and estimated their sensitivities to vector- and scalar-type waves. We find that (i)~at frequencies larger than roughly the inverse of the one-way light time ($\approx${}6\ifmmode\times\else\texttimes\fi{}10-2\,\,Hz), LISA is more than ten times sensitive to scalar-longitudinal and vector signals than to tensor and scalar-transverse waves, and (ii)~in the low part of its frequency band is equally sensitive to tensor and vector waves and somewhat less sensitive to scalar signals.

The direct detection of gravitational waves (GWs) opened a new chapter in the modern cosmology to probe possible deviations from the general relativity (GR) theory. In the present work, we investigate for the first time the modified GW form propagation from the inspiraling of compact binary systems within the context of $f(T)$ gravity in order to obtain new forecasts/constraints on the free parameter of the theory. First, we show that the modified waveform differs from the GR waveform essentially due to induced corrections on the GWs amplitude. Then, we discuss the forecasts on the $f(T)$ gravity assuming simulated sources of GWs as black hole binaries, neutron star binaries and black hole–neutron star binary systems, which emit GWs in the frequency band of the Advanced LIGO (aLIGO) interferometer and of the third generation Einstein Telescope (ET). We show that GW sources detected within the aLIGO sensitivity can return estimates of the same order of magnitude of the current cosmological observations. On the other hand, detection within the ET sensitivity can improve by up to 2 orders of magnitude the current bound on the $f(T)$ gravity. Therefore, the statistical accuracy that can be achieved by future ground based GW observations, mainly with the ET detector (and planed detectors with a similar sensitivity), can allow strong bounds on the free parameter of the theory, and can be decisive to test the theory of gravitation.

We investigate the propagation of primordial gravitational waves within the context of the Horndeski theories; for this, we present a generalized transfer function quantifying the subhorizon evolution of gravitational wave modes after they enter the horizon. We compare the theoretical prediction of the modified primordial gravitational wave spectral density with the aLIGO, Einstein telescope, LISA, gLISA and DECIGO sensitivity curves. Assuming reasonable and different values for the free parameters of the theory (in agreement with the event GW170817 and stability conditions of the theory), we note that the gravitational wave amplitude can vary significantly in comparison with general relativity. We find that in some cases the gravitational primordial spectrum can cross the sensitivity curves for the DECIGO detector with the maximum frequency sensitivity to the theoretical predictions around 0.05–0.30 Hz. From our results, it is clear that the future generations of space based interferometers can bring new perspectives to probing modifications in general relativity.