We discuss the scalar mode of gravitational waves emerging in the context of $F(R)$ gravity by taking into account the chameleon mechanism. Assuming a toy model with a specific matter distribution to reproduce the environment of detection experiment by a ground-based gravitational wave observatory, we find that chameleon mechanism remarkably suppresses the scalar wave in the atmosphere of Earth, compared with the tensor modes of the gravitational waves. We also discuss the possibility to detect and constrain scalar waves by the current gravitational observatories and advocate a necessity of the future space-based observations.
We point out that there are only three polarizations for gravitational waves in $f(R)$ gravity, and the polarization due to the massive scalar mode is a mix of the pure longitudinal and transverse breathing polarization. The classification of the six polarizations by the Newman-Penrose quantities is based on weak, plane and null gravitational waves, so it is not applicable to the massive mode.
In this work we shall develop a quantitative approach for extracting predictions on the primordial gravitational waves energy spectrum for $f(R)$ gravity. We shall consider two distinct models which yield different phenomenology, one pure $f(R)$ gravity model and one Chern-Simons corrected potential-less $k$-essence $f(R)$ gravity model in the presence of radiation and non-relativistic perfect matter fluids. The two $f(R)$ gravity models were carefully chosen in order for them to describe in a unified way inflation and the dark energy era, in both cases viable and compatible with the latest Planck data. Also both models mimic the $Lambda$-Cold-Dark-Matter model and specifically the pure $f(R)$ model only at late times, but the Chern-Simons $k$-essence model during the whole evolution of the model up to the radiation domination era. In addition they guarantee a smooth transition from the inflationary era to the radiation, matter domination and subsequently to the dark energy era. Using a WKB approach introduced in the relevant literature by Nishizawa, we derive formulas depending on the redshift that yield the modified gravity effect, quantified by a multiplicative factor, a ``damping in front of the General Relativistic waveform. In order to calculate the effect of the modified gravity, which is the ``damping factor, we solve numerically the Friedmann equations using appropriate initial conditions and by introducing specific statefinder quantities. As we show, the pure $f(R)$ gravity gravitational wave energy spectrum is slightly enhanced, but it remains well below the sensitivity curves of future gravitational waves experiments. In contrast, the Chern-Simons $k$-essence $f(R)$ gravity model gravitational wave energy spectrum is significantly enhanced and two signals are predicted which can be verified by future gravitational wave experiments.
The detection of gravitational waves by the Laser Interferometer Gravitational-Wave Observatory opens a new era to use gravitational waves to test alternative theories of gravity. We investigate the polarizations of gravitational waves in $f(R)$ gravity and Horndeski theory, both containing scalar modes. These theories predict that in addition to the familiar $+$ and $times$ polarizations, there are transverse breathing and longitudinal polarizations excited by the massive scalar mode and the new polarization is a single mixed state. It would be very difficult to detect the longitudinal polarization by interferometers, while pulsar timing array may be the better tool to detect the longitudinal polarization.
The direct detection of gravitational waves (GWs) is an invaluable new tool to probe gravity and the nature of cosmic acceleration. A large class of scalar-tensor theories predict that GWs propagate with velocity different than the speed of light, a difference that can be $mathcal{O}(1)$ for many models of dark energy. We determine the conditions behind the anomalous GW speed, namely that the scalar field spontaneously breaks Lorentz invariance and couples to the metric perturbations via the Weyl tensor. If these conditions are realized in nature, the delay between GW and electromagnetic (EM) signals from distant events will run beyond human timescales, making it impossible to measure the speed of GWs using neutron star mergers or other violent events. We present a robust strategy to exclude or confirm an anomalous speed of GWs using eclipsing binary systems, whose EM phase can be exquisitely determined. he white dwarf binary J0651+2844 is a known example of such system that can be used to probe deviations in the GW speed as small as $c_g/c-1gtrsim 2cdot 10^{-12}$ when LISA comes online. This test will either eliminate many contender models for cosmic acceleration or wreck a fundamental pillar of general relativity.
We consider the gravitational radiation in conformal gravity theory. We perturb the metric from flat Mikowski space and obtain the wave equation after introducing the appropriate transformation for perturbation. We derive the effective energy-momentum tensor for the gravitational radiation, which can be used to determine the energy carried by gravitational waves.