The gauge problem and polarization modes of gravitational waves in anisotropic Bianchi type I cosmological models

Rastall theory, originally introduced in 1972, suggests a violation of the usual conservation law. We consider two generalizations of Rastall theory: Brans–Dicke–Rastall theory and the newly established scalar-tensor-Rastall theory, the latter being a further generalization of the former. The field equations in these two generalized theories are studied across different parameter spaces, and the polarization modes of gravitational waves, as a key focus, are subsequently investigated. The results show that the polarization modes of gravitational waves in Brans–Dicke–Rastall theory are the same as those in Brans–Dicke theory; specifically, both theories exhibit the plus, cross, and breathing modes. However, in scalar-tensor-Rastall theory, the polarization modes of gravitational waves depend on the parameter space of the theory. Particularly, over a broad range of the parameter space, regardless of some special values of the parameters, it allows only two tensor modes, just as in general relativity, without introducing any additional degrees of freedom. This indicates that Rastall theory offers a novel approach to constructing modified gravity theories that propagate only two tensor degrees of freedom. In the remaining regions of the parameter space, there is also one scalar mode in addition to the two tensor modes. The scalar mode can be either a mixture of the breathing and longitudinal modes or just a pure breathing mode, depending on the parameter space. These results will play a crucial role in constraining the theoretical parameters through future gravitational wave detection projects, such as LISA, Taiji, and TianQin.

In this paper, the polarization modes of gravitational waves in Horndeski gravity are studied under the Palatini formalism. After obtaining the linearized equation of perturbations in Minkowski background, we find that the polarization modes of gravitational waves depend on the selection of the theoretical parameters. The polarization modes can be divided into quite rich cases by parameters. In all cases of parameter selection, there are $+$ and $\times$ modes propagating at the speed of light but no vector modes. The only difference from general relativity is scalar modes, especially the scalar degrees of freedom can be 0, 1 or 2 in different cases. The appropriate parameter cases can be expected to be selected in the detection of gravitational wave polarization modes by Lisa, Taiji and TianQin in the future.

We find polarization modes of gravitational waves in topologically massive and new massive gravities by using the Newman-Penrose formalism where the null real tetrad is necessary to specify gravitational waves. The number of polarization modes is two for the new massive gravity and one for the topologically massive gravity, which is consistent with the metric-perturbation approach.

The observation of the inspiral and merger of compact binaries by the LIGO-Virgo collaboration has allowed for new tests of Einstein's theory in the extreme gravity regime, where gravitational interactions are simultaneously strong, non-linear, and dynamical. Theories beyond Einstein's can also be constrained by detecting the polarization modes of gravitational waves. In this paper, we show that dynamical Chern-Simons and Einstein-dilaton-Gauss-Bonnet gravity cannot be differentiated from general relativity based on the detection of polarization modes alone. To prove this result, we use the Newman-Penrose method and an irreducible decomposition method to find that only the tensorial modes can be detected in both these theories.

General Relativity predicts only two tensor polarization modes for gravitational waves while at most six possible polarization modes are allowed in the general metric theory of gravity. The number of polarization modes is determined by the specific modified theory of gravity. Therefore, the determination of polarization modes can be used to test gravitational theory. We introduce a concrete data analysis pipeline for a space-based detector such as LISA to detect the polarization modes of gravitational waves. This method can be used for monochromatic gravitational waves emitted from any compact binary system with a known sky position and frequency to detect mixtures of tensor and extra polarization modes. We use the source $\mathrm{J}0806.3+1527$ with one year of simulation data as an example to show that this approach is capable of probing pure and mixed polarizations without knowing the exact polarization modes. We also find that the ability of detection of extra polarization depends on the gravitational-wave source location and the amplitude of nontensorial components.

Gravitational waves have only two polarization modes in General Relativity.\nHowever, there are six possible modes of polarization in metric theory of\ngravity in general. The tests of gravitational waves polarization can be tools\nfor pursuing the nature of space-time structure. The observations of\ngravitational waves with a world-wide network of interferometric detectors such\nas Advanced LIGO, Advanced Virgo and KAGRA will make it possible to obtain the\ninformation of gravitational wave polarization from detector signals. We study\nthe separability of the polarization modes for the inspiral gravitational waves\nfrom the compact binary coalescences systematically. Unlike other waveforms\nsuch as burst, the binary parameters need to be properly considered. We show\nthat the three polarization modes of the gravitational waves would be separable\nwith the global network of three detectors to some extent, depending on\nsignal-to-noise ratio and the duration of the signal. We also show that with\nfour detectors the three polarization modes would be more easily distinguished\nby breaking a degeneracy of the polarization modes and even the four\npolarization modes would be separable.\n

There exist six possible polarization modes of gravitational waves in a general metric theory of gravity, while two tensor polarization modes are allowed in general relativity. The properties and number of polarization modes depend on gravity theories. For transient signals, the number of the detectors needs to be basically equal to the number of the gravitational-wave polarization modes for separation of polarizations. However, a single detector having great sensitivity at lower frequency could be effectively regarded as a virtual detector network including a set of detectors along its trajectory due to a long gravitational-wave signal from a compact binary and the Earth's rotation. Thus, time-varying antenna pattern functions can help test the polarizations of gravitational waves. We study the effects of the Earth's rotation on the polarization test and show a possibility to test the nontensorial polarization modes from future observations of compact binary mergers with ground-based gravitational detectors such as Einstein telescope and Cosmic Explorer.

A generalized Brans-Dicke (GBD) theory in the framework of Palatini formalism is proposed in this paper. We derive the field equations by using the variational approach and obtain the linearized equations by using the weak-field approximation method. We show various properties of the geometrical scalar field in the Palatini-formalism of GBD theory: it is massless and source-free, which are different from the results given in the metric-formalism of GBD theory. Also, we investigate the polarization modes of gravitational waves (GWs) by using the geodesic deviation method and the Newman-Penrose method in the Palatini-GBD theory. It is observed that there are three polarizations modes and four oscillations in the Palatini-GBD theory. Concretely, they are the two transverse tensor (+) and (×) standard polarization modes, and one breathing mode (with two oscillations). The results of GWs polarization in the Palatini-GBD theory are different from that in the metric-GBD theory, where there are four polarizations modes: the two standard tensorial modes (+ and ×), a scalar breathing mode, and a massive scalar mode that is a mix of the longitudinal and the breathing polarization. Comparing with the Palatini-f(R˜) theory and the General Relativity, we can see that the extra breathing mode of GWs polarization can be found in the Palatini-GBD theory. At last, the expression of the parameterized post Newton (PPN) parameter is derived, which could pass through the experimental test.

The detection of gravitational wave (GW) has opened a new window to test the theory of gravity in the strong field regime. In general relativity (GR), GW can only possess two tensor polarization modes, which are known as the $+$ and $\times$ modes. However, vector and scalar modes can exist in some modified theories of gravity, and we can test the gravitational theories by probing these extra polarization modes. As a space-borne GW detector, TianQin will be launched in the 2030s, and it is expected to observe plenty of GW signals, including those from nearly ten-thousands double white dwarfs (DWDs) in our galaxy. This offers an excellent chance for testing the existence of extra polarization modes. In this article, we analyze the capability of detecting the extra polarization modes with DWDs. For the extra modes, we consider both the leading order dipole radiation and the subleading quadrupole radiation. We find that the capability of TianQin has very strong dependency on the source orientation. We also analyze all the verification binaries with determined position and frequency, and find that ZTF J1539 can give the best constraint on the extra polarization modes. We have also considered the case of TianQin twin constellation, and the joint observation with LISA.

We consider the nondynamical Chern-Simons (nCS) modified gravity, which is regarded as a parity-odd theory of massive gravity in four dimensions. We first find polarization modes of gravitational waves for θ=x/μ in nCS modified gravity by using the Newman-Penrose formalism where the null complex tetrad is necessary to specify gravitational waves. We show that in the Newman–Penrose formalism, the number of polarization modes is one in addition to an unspecified Ψ4, implying three degrees of freedom for θ=x/μ. This compares with two for a canonical embedding of θ=t/μ. Also, if one introduces the Ricci tensor formalism to describe a massive graviton arising from the nCS modified gravity, one finds one massive mode after making second-order wave equations, which is compared to five found from the parity-even Einstein–Weyl gravity.

Scalar polarization modes of gravitational waves, which are often introduced in the context of the viable extension of gravity, have been actively searched for. However, couplings of the scalar modes to matter are strongly constrained by fifth-force experiments. Thus, the amplitude of scalar polarization in the observed gravitational-wave signal must be significantly suppressed compared to that of the tensor modes. Here, we discuss the implications of the experiments in the solar system on the detectability of scalar modes in gravitational waves from compact binary coalescences, taking into account the whole processes from the generation to the observation of gravitational waves. We first claim that the energy carried by the scalar modes at the generation is, at most, that of the tensor modes from the observed phase evolution of the inspiral gravitational waves. Next, we formulate general gravitational-wave propagation and point out that the energy flux hardly changes through propagation as long as the background changes slowly compared to the wavelength of the propagating waves. Finally, we show that the possible magnitude of scalar polarization modes detected by ground-based gravitational-wave telescopes is already severely constrained by the existing gravity tests in the solar system.

The formalism of the exact six polarization modes of gravitational waves is constructed in terms of both the small metric perturbations and the Newman-Penrose scalars. The obtained formulae are applicable to any metric-compatible gravity theories whose gravitational waves propagate along either the null or non-null geodesics. Once a gravity theory, specifically its linearized wave equation, is written, comparison to the observed data of the laser interferometer experiments is direct.

The vector and scalar polarization modes of gravitational waves do not exist in General Relativity, and their detection would have significant impacts on fundamental physics. In this paper, we explored the detectability of these anomalous polarization modes in a gravitational wave background around 1 mHz with the future LISA-Taiji network. The inherent geometrical symmetry of the network largely simplifies the correlation analysis. By taking a suitable linear combination of the correlated outputs, the contribution of the standard tensor modes can be canceled algebraically, and the anomalous modes can be exclusively examined. We provide concise expressions for the signal-to-noise ratios of the anomalous modes with this cancellation method. We also discuss the possibility of separately estimating the amplitudes of the vector and scalar modes, using the overall frequency dependence of the associated overlap reduction functions.

We investigate the different polarization modes of Gravitational Waves (GWs) in $f(R)$ gravity power law model in de Sitter space. It is seen that the massive scalar field polarization mode exists in this model. The mass of the scalar field depends highly on the background curvature and the power term $n$. However, we found that the model doesnot exhibit a massive scalar mode for $n=2$ and instead it shows a breathing mode in addition to the tensor plus and cross modes. Thus mass of the scalar field is found to vary with $n$ within the range $1 \leq n \leq 2$.

Many studies have been carried out in the literature to evaluate the number of polarization modes of gravitational waves in modified theories, in particular in $f(R)$ theories. In the latter ones, besides the usual two transverse-traceless tensor modes present in general relativity, there are two additional scalar ones: a massive longitudinal mode and a massless transverse mode (the so-called breathing mode). This last mode has often been overlooked in the literature, due to the assumption that the application of the Lorenz gauge implies transverse-traceless wave solutions. We however show that this is in general not possible and, in particular, that the traceless condition cannot be imposed due to the fact that we no longer have a Minkowski background metric. Our findings are in agreement with the results found using the Newman-Penrose formalism and thus clarify the inconsistencies found so far in the literature.