Gravitational Wave Equation Research Articles

Gravitational wave probe of gravitational dark matter from preheating

We forecast high-frequency gravitational wave (GW) from preheating hosting gravitational dark matter (GDM) as the indirect probe of such GDM. We use proper lattice simulations to handle resonance, and to solve GW equation of motion with the resonance induced scalar field excitations as source term. Our numerical results show that Higgs scalar excitations in Higgs preheating model give rise to magnitudes of GW energy density spectra of order 10-10 at frequencies 10 – 103 MHz depending on the GDM mass of (6 – 9) × 1013 GeV, whereas inflaton fluctuation excitations in inflaton self-resonant preheating model yield magnitudes of GW energy density spectrum up to 10-9 (10-11) at frequencies near 30 (2) MHz for the index n=4 (6) with respect to the GDM mass of 1.04 (2.66) × 1014 GeV.

Parametric resonance of gravitational waves in general scalar-tensor theories

Gravitational waves offer a potent mean to test the underlying theory of gravity. In general theories of gravity, such as scalar-tensor theories, one expects modifications in the friction term and the sound speed in the gravitational wave equation. In that case, rapid oscillations in such coefficients, e.g. due to an oscillating scalar field, may lead to narrow parametric resonances in the gravitational wave strain. We perform a general analysis of such possibility within DHOST theories. We use disformal transformations to find the theory space with larger resonances, within an effective field theory approach. We then apply our formalism to a non-minimally coupled ultra-light dark matter scalar field, assuming the presence of a primordial gravitational wave background, e.g., from inflation. We find that the resonant peaks in the spectral density may be detectable by forthcoming detectors such as LISA, Taiji, Einstein Telescope and Cosmic Explorer.

Gauge-invariant formulation for the gravitational wave equations

A gauge-invariant formulation for the gravitational wave equations is presented. Using this approach, weak, plane wave solutions in a vacuum are derived in various theories. These include general relativity with two modes of polarization with helicity ±2, Yang’s theory with three modes of polarization with helicity ±2 and 0, and so-called ‘general metric theories’ with six modes of polarization with helicity ±2, ±1, and two 0s. To identify the polarizations of gravitational waves, it is explicitly demonstrated how the gauge-invariant approach reproduces the earlier results.

Gravitational radiation of a spherically symmetric source in f(R)-gravitation

It is shown that Birkhoff’s theorem for the general theory of relativity is overcome in the f(R)-theory of gravitation. That means, the f(R)-theory of gravitation, unlike Einstein’s general theory of relativity, does not forbid gravitational radiation from a spherically symmetric source (whether stationary or non-stationary). As a consequence, in the f(R)-theory a spherically symmetric gravitational deformation (e.g., collapse/expansion or pulsation) could emit gravitational waves (of tensor- and scalar polarization modes), a phenomenon impossible in the general relativity. A test model is examined and it turns out that the gravitational radiation is strongest when the surface of the deforming object is in the vicinity of the (modified) event horizon, even suddenly flares up just outside the latter. In this letter, within the f(R)-theory of gravitation, a gravitational wave equation and a formula for the gravitational emission power are derived. These formulae, along with searching for signals, can be used for the experimental test of the f(R)-theory. In general, including the spherically symmetry case, gravitational radiation of both tensor- and scalar polarization modes are allowed, although under some circumstance the contribution of scalar modes is strongly suppressed.

Teukolsky-like equations with various spins in spherically symmetric spacetime* *Supported in part by the National Natural Science Foundation of China (11973025)

We study wave equations with various spins on the background of a general spherically symmetric spacetime. We obtain the unified expression of the Teukolsky-like master equations and the corresponding radial equations with the general spins. We also discuss the gauge dependence in the gravitational-wave equations, which have appeared in previous studies.

On graviton–photon conversions in magnetic environments

Graviton–photon conversions in a given external electric or magnetic field, known as the Gertsenshtein mechanism, are usually treated using the four-potential for photons. In terms of the electric and magnetic (EM) fields, however, proper identification of the fields in curved spacetime is important. By misidentifying the fields in Minkowski form, as is often practiced in the literature, we show that the final equation for photon conversion is correct in transverse-tracefree gauge only for planar gravitational waves in a uniform and constant external field. Even in the former method, to recover the EM fields from the four-potential in curved spacetime, one should properly take into account the metric involved in the relation. By including the metric perturbation in the graviton conversion equation, we show that a magnetic environment can cause tachyonic instability term in gravitational wave equation.

Constrain the time variation of the gravitational constant via the propagation of gravitational waves

The gravitational constant variation means the breakdown of the strong equivalence principle. As the cornerstone of general relativity, the validity of general relativity can be examined by studying the gravitational constant variation. Such variations have the potential to affect both the generation and propagation of gravitational waves. Here, our focus lies on the effect of gravitational constant variation specifically on the propagation of gravitational waves. In a previous paper (An et al., 2023 [13]) we have studied such effect based on a Maxwell-like equation. That work admits two drawbacks. The assumption that describing gravitational waves with Maxwell-like equation is not solid. And more such treatment can only deal with space dependent gravitational constant. In the current paper we will remedy these two issues. We employ two analytical methods, namely based on the Fierz-Pauli action and the perturbation of Einstein-Hilbert action around Minkowski spacetime, both leading to the same gravitational wave equation. By solving this equation, we find the effects of gravitational constant variation on gravitational wave propagation. The result is consistent with previous investigations based on Maxwell-like equations for gravitational waves for space dependent gravitational constant cases. In addition we can constrain the time variation of the gravitational constant via the propagation of gravitational waves based on GW170817 data. And more, we find that small variations in the gravitational constant result in an amplitude correction at the leading order and a phase correction at the sub-leading order for gravitational waves. These results provide valuable insights for probing gravitational constant variation and can be directly applied to gravitational wave data analysis.

Evolution of gravitational waves in non-minimal coupling between geometry and matter theories of gravity

We consider some specific models of non-minimal matter–geometry coupling theories and investigate the propagation of the gravitational waves in them. Extracting the temporal evolution of the gravitational wave equation within the framework of a flat FRW universe with a perfect fluid distribution, we analyze the waveforms traveling during the time. We find that while both the amplitude and frequency of the GWs decay with time in all considered models, the rate of reduction is highly sensitive to the values of the equation of state parameter and input parameters of the considered models.

Gravitational-wave equation in effective one-body background for spinless binary

Gravitational-wave equation in effective one-body background for spinless binary

Propagation of scalar and tensor gravitational waves in Horndeski theory

Gravitational waves travel through the distributions of matter and dark energy during propagation. For this reason, gravitational waves emitted from binary compact objects serve as a useful tool especially to probe the nature of dark energy. The geometrical optics approximation is a conventional way of investigating wave propagation. However, the approximation becomes less accurate as the wavelength approaches the curvature radius of the background, which can occur in generic situations. In this paper, we suggest a formulation for higher-order corrections of the geometrical optics expansion, applied to Horndeski theory which accommodates many dark energy models. At the level of the background, assuming that the derivative of the scalar field is non-vanishing and timelike, we choose the time slices to coincide with the contours of the scalar field. This choice of the background time slices is advantageous as the sound cones of both scalar and tensor gravitational waves are upright with respect to the background time slices whenever the scalar field behaves as a perfect fluid. We then analyze the equations of motion for scalar and tensor components of gravitational waves at the leading and next-to-leading order in the geometrical optics expansion, deriving the evolution equations for their amplitudes under certain conditions. In particular, for Generalized Brans-Dicke theories, we find a simple description of equations for gravitational waves in terms of an effective metric.

Gravitational wave propagation on the brane in different braneworld scenarios

In this paper, we consider several well-known braneworld gravity scenarios such as the Rundall-Sundrum type II ( RSII ) theory, the Dvali, Gabadadze, and Porrati (DGP) braneworld, and the braneworld mimetic gravity. We then focus on studying the behavior of gravitational wave propagation on the brane in each setup. Using a specific approach and considering each setup’s brane separately, which requires distinct cosmological background, we show that different braneworld scenarios imply various amplitude and frequency reduction rates. By plotting the temporal part of the gravitational wave equation and comparing results, we deduce that the RSII braneworld scenario implies a more stable wave on its brane than the standard model. In the case of DGP gravity, we investigate the two branches separately and show that in the normal branch of this model, the amplitude and frequency of brane gravitational waves decay extremely fast.

Gravitational wave modes in matter

A general linear gauge-invariant equation for dispersive gravitational waves (GWs) propagating in matter is derived. This equation describes, on the same footing, both the usual tensor modes and the gravitational modes strongly coupled with matter. It is shown that the effect of matter on the former is comparable to diffraction and therefore negligible within the geometrical-optics approximation. However, this approximation is applicable to modes strongly coupled with matter due to their large refractive index. GWs in ideal gas are studied using the kinetic average-Lagrangian approach and the gravitational polarizability of matter that we have introduced earlier. In particular, we show that this formulation subsumes the kinetic Jeans instability as a collective GW mode with a peculiar polarization, which is derived from the dispersion matrix rather than assumed a priori. This forms a foundation for systematically extending GW theory to GW interactions with plasmas, where symmetry considerations alone are insufficient to predict the wave polarization.

On the \u201cEinstein\u2013Gauss\u2013Bonnet gravity in four dimension\u201d

To ensure the existence of a well defined linearized gravitational wave equation, we show that the spacetimes in the so-called “Einstein–Gauss–Bonnet gravity in four dimension” have to be locally conformally flat.

Gravitational multipole moments for asymptotically de Sitter spacetimes

We provide a prescription to compute the gravitational multipole moments of compact objects for asymptotically de Sitter spacetimes. Our prescription builds upon a recent definition of the gravitational multipole moments in terms of Noether charges associated to specific vector fields, within the residual harmonic gauge, dubbed multipole symmetries. We first derive the multipole symmetries for spacetimes which are asymptotically de Sitter; we also show that these symmetry vector fields eliminate the non-propagating degrees of freedom from the linearized gravitational wave equation in a suitable gauge. We then apply our prescription to the Kerr-de Sitter black hole and compute its multipole structure. Our result recovers the Geroch-Hansen moments of the Kerr black hole in the limit of vanishing cosmological constant.

Propagation of gravitational waves in various cosmological backgrounds

The present work investigates some exact solutions of the gravitational wave equation in some widely used cosmological spacetimes. The examples are taken from spatially flat and closed isotropic models as well as Kasner metric which is anisotropic. Various matter distributions are considered, from the standard dust or radiation distribution to exotic matters like that with an equation of state \(P = - \frac{1}{3} \rho \). In almost all cases the frequency and amplitude of the wave are found to be decaying with evolution, except for a closed radiation universe where there is a resurgence of the waveform, consistent with the recollapse of the universe into the big crunch singularity.

Gravitational waves for some dark energy models in FRW Universe

We study the gravitational waves phenomena for generalized Chaplygin gas, variable modified Chaplygin gas and some parametrizations of dark energy models. Motivated by their works, here we consider the Friedmann–Robertson–Walker (FRW) model of the non-flat Universe in the presence of dark matter with dark energy and we investigate the gravitational waves phenomena for some other candidates of dark energy like tachyonic field, generalized cosmic Chaplygin gas, new variable modified Chaplygin gas and Van der Waals fluid. By considering the individual conservation laws of dark matter and the dark energy components, we obtain the solutions of energy densities of the dark matter and dark energy components separately in terms of redshift. Next, we get the evolution equation for gravitational waves in FRW Universe. Then we analyze the natures of wave curves and their power spectrums for tachyonic field, generalized cosmic Chaplygin gas, new variable modified Chaplygin gas, and Van der Waals fluid.

Dynamics of primordial fields in quantum cosmological spacetimes

Quantum cosmological models are commonly described by means of semiclassical approximations in which a smooth evolution of the expectation values of elementary geometry operators replaces the classical and singular dynamics. The advantage of such descriptions is that they are relatively simple and display the classical behavior for large universes. However, they may smooth out an important inner structure and to include it a more detailed treatment is needed. The purpose of the present work is to investigate quantum uncertainty in the basic background variables and its influence on primordial gravitational waves. To this end we quantize a model of the Friedmann-Lemaitre-Robertson-Walker universe filled with a linear barotropic cosmological fluid and with gravitational waves. We carefully derive the dynamical equations for the perturbations in quantum spacetime. The quantization yields an equation of motion for the Fourier modes of gravitational radiation, which is a quantum extension to the usual parametric oscillator equation for gravitational waves propagating in an expanding universe. The two quantum effects from the cosmological background that enter the enhanced equation of motion are (i) a repulsive potential resolving the big bang singularity and replacing it with a big bounce and (ii) uncertainties in the numerical values for the background spacetime dynamical variables. First we study the former effect and its consequences for the primordial amplitude spectrum and carefully discuss the relation between the bounce scale and the physical predictions of the model. Next we investigate the latter effect, in particular the extent to which it may affect the primordial amplitude of gravitational waves. Making use of the WKB approximation we find an analytical formula for the amplitude spectrum as a function of the quantum dispersion of the background spacetime.

Gravitational Waves in a Universe with Time-Varying Curvature

In this paper, we present a complete solution of Einstein’s equations for the gravitational wave (GW) problem. The full metric is taken in the usual way to be the sum of a background vacuum metric plus a perturbation metric describing the GW. The background metric used is characterized by time-varying curvature as described in a recent paper. The solution we develop here does exhibit some features found in the standard model but it also contains others that are not found in the standard model. One difference is that the solution with time-varying curvature only allows for outward-directed waves. While this might seem a minor point regarding the GW equations, it is actually a significant verification of the solution presented in our earlier paper. A more obvious difference is that the solution demands that the vacuum along with all matter must experience transverse motion with the passing of the waves. This fact leads to the idea that a new approach to the detection problem based on the Doppler effect could well be practical. Such an approach, if feasible, would be much simpler and less costly to implement than the large-scale interferometer system currently under development.

Primordial gravitational waves spectrum in the Coupled-Scalar-Tachyon Bounce Universe

We extend our study on the Coupled-Scalar-Tachyon Bounce Universe to obtain its gravitational waves spectrum. We derive in detail the equations of motion for the tensorial modes of primordial metric perturbations in the Coupled-Scalar-Tachyon Bounce Universe. We solve for the gravitational wave equations in the pre-bounce contraction and the post-bounce expansion epochs. To match the solutions of the tensor perturbations, we idealise the bounce process yet retaining the essential physical properties of the bounce universe. We put forward two matching conditions: one ensures the continuity of the gravitational wave functions and the other respects the symmetric nature of the bounce dynamics. The matching conditions connect the two independent modes of gravitational waves solutions before and after the bounce. We further analyze the scale dependence and time dependence of the gravitational waves spectra in the bounce universe and compare them with the primordial spectrum in the single field inflation scenario. We discuss the implications to early universe physics and present model independent observational signatures extracted from the bounce universe.

The timestep constraint in solving the gravitational wave equations sourced by hydromagnetic turbulence

ABSTRACTHydromagnetic turbulence produced during phase transitions in the early universe can be a powerful source of stochastic gravitational waves (GWs). GWs can be modelled by the linearised spatial part of the Einstein equations sourced by the Reynolds and Maxwell stresses. We have implemented two different GW solvers into the Pencil Code – a code which uses a third order timestep and sixth order finite differences. Using direct numerical integration of the GW equations, we study the appearance of a numerical degradation of the GW amplitude at the highest wavenumbers, which depends on the length of the timestep – even when the Courant–Friedrichs–Lewy condition is ten times below the stability limit. This degradation leads to a numerical error, which is found to scale with the third power of the timestep. A similar degradation is not seen in the magnetic and velocity fields. To mitigate numerical degradation effects, we alternatively use the exact solution of the GW equations under the assumption that the source is constant between subsequent timesteps. This allows us to use a much longer timestep, which cuts the computational cost by a factor of about ten.