Gravitational waveform of constantly accelerating sources

Gravitationally bound hierarchies containing three or more components are very common in our Universe. In this paper we study {\em periodic} gravitational wave (GW) form, their polarizations, response function, its Fourier transform, and energy loss rate of a triple system through three different channels of radiation, the scalar, vector and tensor modes, in Einstein-aether theory of gravity. In the weak-field approximations and with the recently obtained constraints of the theory, we first analyze the energy loss rate of a binary system, and find that the dipole contributions from the scalar and vector modes could be of the order of ${\cal{O}}\left(c_{14}\right){\cal{O}}\left(G_Nm/d\right)^2$, where $c_{14} \; (\equiv c_{1} + c_{4})$ is constrained to $c_{14} \lesssim {\cal{O}}\left(10^{-5}\right)$ by current observations, where $c_i$'s are the four coupling constants of the theory. On the other hand, the "strong-field" effects for a binary system of neutron stars are about six orders lower than that of GR. So, in this paper we ignore these "strong-field" effects and first develop the general formulas to the lowest post-Newtonian order, by taking the coupling of the aether field with matter into account. Within this approximation, we find that the scalar breather mode and the scalar longitudinal mode are all suppressed by a factor of ${\cal{O}}\left(c_{14}\right)$ with respect to the transverse-traceless modes ($h_{+}$ and $h_{\times}$), while the vectorial modes $(h_{X}$ and $h_{Y}$) are suppressed by a factor of $c_{13} \lesssim {\cal{O}}\left(10^{-15}\right)$. Applying the general formulas to a triple system with periodic orbits, we find that the corresponding GW form, response function, and its Fourier transform depend sensitively on both the configuration of the triple system and their orientations with respect to the detectors.

The purpose of this paper is to investigate the effects of the dynamical parameters, the spin-orbit coupling, the spin-spin coupling and the classification of orbits on the gravitational waveforms emitted by the conservative post-Newtonian Hamiltonian system for the spinning compact binaries by means of symplectic integrator and power spectrum. The numerical results show that the gravitational waveforms of ordered orbits vary periodically with time, while those of chaotic orbits are stochastic under the radiation--reaction force turning off. In particular, the chaotic dynamic behavior can enhance the strength of emission power. In addition, the magnitude of the spin parameter does exert a significant influence on the gravitational waveforms.

Classical soft photon and soft graviton theorems determine long wavelength electromagnetic and gravitational waveforms for a general classical scattering process in terms of the electric charges and asymptotic momenta of the ingoing and outgoing macroscopic objects. Performing Fourier transformation of the electromagnetic and gravitational waveforms in the frequency variable one finds electromagnetic and gravitational waveforms at late and early retarded time. Here extending the formalism developed in [1], we derive sub-subleading electromagnetic and gravitational waveforms which behave like u−2(ln u) at early and late retarded time u in four spacetime dimensions. We also have derived the sub-subleading soft photon theorem analyzing two loop amplitudes in scalar QED. Finally, we conjectured the structure of leading non-analytic contribution to (sub)n-leading classical soft photon and graviton theorems which behave like u−n(ln u)n−1 for early and late retarded time u.

We provide ready-to-use time-domain gravitational waveforms for spinning compact binaries with precession effects through 1.5 post-Newtonian (PN) order in amplitude, and compute their mode decomposition using spin-weighted $\ensuremath{-}2$ spherical harmonics. In the presence of precession, the gravitational-wave modes $(\ensuremath{\ell},m)$ contain harmonics originating from combinations of the orbital frequency and precession frequencies. We find that the gravitational radiation from binary systems with large mass asymmetry and large inclination angle can be distributed among several modes. For example, during the last stages of inspiral, for some maximally spinning configurations, the amplitude of the (2, 0) and (2, 1) modes can be comparable to the amplitude of the (2, 2) mode. If the mass ratio is not too extreme, the $\ensuremath{\ell}=3$ and $\ensuremath{\ell}=4$ modes are generally 1 or 2 orders of magnitude smaller than the $\ensuremath{\ell}=2$ modes. Restricting ourselves to spinning, nonprecessing compact binaries, we apply the stationary-phase approximation and derive the frequency-domain gravitational waveforms including spin-orbit and spin(1)-spin(2) effects through 1.5PN and 2PN order, respectively, in amplitude, and 2.5PN order in phase. Since spin effects in the amplitude through 2PN order affect only the first and second harmonics of the orbital phase, they do not extend the mass reach of gravitational-wave detectors. However, they can interfere with other harmonics and lower or raise the signal-to-noise ratio depending on the spin orientation. These ready-to-use waveforms could be employed in the data analysis of the spinning, inspiraling binaries as well as in comparison studies at the interface between analytical and numerical relativity.

Current gravitational waveform models used by detection data analysis have not included the effect of the moving speed of the sources relative to detector. Few works about the gravitational waveform problem for moving sources exist in the literature. And all of them have taken some approximations. Some of them just considered frequency shift. Some of them used slow speed approximation. In the current paper, we apply the precise Lorentz transformation of gravitational wave to generate the gravitational waveform for moving sources with any high speed. Our calculation is straightforward but without any approximation. That is to say our result is valid in the whole velocity range 0 ⩽ v < c which is only limited by the speed of light coming from general relativity itself. If only a waveform model of a rest source is given, our result can be applied directly to generate the waveform for the corresponding moving source with any high speed. Such waveform is valid for binary black hole coalescence happening extremely close to a super-massive black hole. As an example, we apply our method to the effect of kick velocity of binary black hole. The adjusted waveform by the kick velocity is presented in the current paper.

We calculate the gravitational waveform radiated from spinning black holes (BHs) binary in dynamical Chern-Simons (dCS) gravity. The equation of motion (EOM) of the spinning binary BHs is derived based on the modified Mathisson-Papapetrou-Dixon equation for the spin-aligned circular orbits. The leading-order effects induced by the dCS theory contain spin-spin interaction and monopole-quadrupole interaction, which influences both the EOM of the binary system and corresponding gravitational waveform at the second post-Newtonian (PN) order (i.e., 2PN order). After reporting the waveforms, we investigate the polarization modes of gravitational waves (GWs) in dCS theory. None of the extra modes appears in this theory up to the considered PN order. Moreover, since the time scale of the binary merger is much smaller than that of the cosmological expansion, the parity-violating effect of the dCS theory does not appear in the process of GW generation. However, during the process of GW propagation, amplitude birefringence, a typical parity-violating effect, makes plus and cross modes convert to each other, which modifies the gravitational waveform at 1.5PN order.

Extreme mass-ratio inspirals are crucial sources for future space-based gravitational wave detections. Gravitational waveforms emitted by extreme mass-ratio inspirals are closely related to the orbital dynamics of small celestial objects, which vary with the underlying spacetime geometry. Despite the tremendous success of general relativity, there are unsolved issues such as singularities in both black holes and cosmology. Loop quantum gravity, a theory addressing these singularity problems, offers a framework for regular black holes. In this paper, we focus on periodic orbits of a small celestial object around a supermassive quantum-corrected black hole in loop quantum gravity and compute the corresponding gravitational waveforms. We view the small celestial object as a massive test particle and obtain its four-velocity and effective potential. We explore the effects of quantum corrections on marginally bound orbits, innermost stable circular orbits, and other periodic orbits. Using the numerical kludge scheme, we further explore the gravitational waveforms of the small celestial object along different periodic orbits. The waveforms exhibit distinct zoom and whirl phases in a complete orbital period, closely tied to the quantum parameter α̂. We also perform a spectral analysis of the gravitational waves from these periodic orbits and assess their detectability. With the steady progress of space-based gravitational wave detection programs, our findings will contribute to utilizing extreme mass-ratio inspirals to test and understand the properties of quantum-corrected black holes.

We provide ready-to-use time-domain gravitational waveforms for spinning compact binaries with precession effects through 1.5PN order in amplitude and compute their mode decomposition using spin-weighted -2 spherical harmonics. In the presence of precession, the gravitational-wave modes (l,m) contain harmonics originating from combinations of the orbital frequency and precession frequencies. We find that the gravitational radiation from binary systems with large mass asymmetry and large inclination angle can be distributed among several modes. For example, during the last stages of inspiral, for some maximally spinning configurations, the amplitude of the (2,0) and (2,1) modes can be comparable to the amplitude of the (2,2) mode. If the mass ratio is not too extreme, the l=3 and l=4 modes are generally one or two orders of magnitude smaller than the l = 2 modes. Restricting ourselves to spinning, non-precessing compact binaries, we apply the stationary-phase approximation and derive the frequency-domain gravitational waveforms including spin-orbit and spin(1)- spin(2) effects through 1.5PN and 2PN order respectively in amplitude, and 2.5PN order in phase. Since spin effects in the amplitude through 2PN order affect only the first and second harmonics of the orbital phase, they do not extend the mass reach of gravitational-wave detectors. However, they can interfere with other harmonics and lower or raise the signal-to-noise ratio depending on the spin orientation. These ready-to-use waveforms could be employed in the data-analysis of the spinning, inspiraling binaries as well as in comparison studies at the interface between analytical and numerical relativity.

In a subclass of Horndeski theories with the speed of gravity equivalent to that of light, we study gravitational radiation emitted during the inspiral phase of compact binary systems. We compute the waveform of scalar perturbations under a post-Newtonian expansion of energy-momentum tensors of pointlike particles that depend on a scalar field. This scalar mode not only gives rise to breathing and longitudinal polarizations of gravitational waves, but it is also responsible for scalar gravitational radiation in addition to energy loss associated with transverse and traceless tensor polarizations. We calculate the Fourier-transformed gravitational waveform of two tensor polarizations under a stationary phase approximation and show that the resulting waveform reduces to the one in a parametrized post-Einsteinian (ppE) formalism. The ppE parameters are directly related to a scalar charge in the Einstein frame, whose existence is crucial to allow the deviation from general relativity (GR). We apply our general framework to several concrete theories and show that a new theory of spontaneous scalarization with a higher-order scalar kinetic term leaves interesting deviations from GR that can be probed by the observations of gravitational waves emitted from neutron star--black hole binaries. If the scalar mass exceeds the order of typical orbital frequencies $\ensuremath{\omega}\ensuremath{\simeq}{10}^{\ensuremath{-}13}\text{ }\text{ }\mathrm{eV}$, which is the case for a recently proposed scalarized neutron star with a self-interacting potential, the gravitational waveform practically reduces to that in GR.

This paper mainly deals with the effects of eccentricity on the gravitational waveforms emitted by the non-conservative post-Newtonian (PN) Hamiltonian formulations of the spinning compact binaries. The numerical results show that the change of eccentricity has a slight influence on the time-domain gravitational waveforms from the conservative spinning compact binaries, but the frequency bands of gravitational waveforms is broadened with increasing eccentricity. Owing to the effects of dissipation from the gravitational radiation reaction, the separation and the eccentricity decrease gradually with time, and the gravitational waveforms emitted from the non-conservative PN spinning compact binaries are modulated by the eccentricity, meanwhile the amplitude of the waveforms enhances with the increase of eccentricity; the duration of the waveforms decreases.

The future LISA detector will constitute the prime instrument for high-precision gravitational wave observations. Among other goals, LISA is expected to materialize a ‘spacetime-mapping’ program that is to provide information for the properties of spacetime in the vicinity of supermassive black holes which reside in the majority of galactic nuclei. Such black holes can capture stellar-mass compact objects, which afterwards slowly inspiral under the emission of gravitational radiation. The small body's orbital motion and the associated waveform observed at infinity carry information about the spacetime metric of the massive black hole, and in principle it is possible to extract this information and experimentally identify (or not!) a Kerr black hole. In this paper we lay the foundations for a practical spacetime-mapping framework. Our work is based on the assumption that the massive body is not necessarily a Kerr black hole, and that the vacuum exterior spacetime is stationary axisymmetric, described by a metric which deviates slightly from the known Kerr metric. We first provide a simple recipe for building such a ‘quasi-Kerr’ metric by adding to the Kerr metric the leading order deviation which appears in the value of the spacetime's quadrupole moment. We then study geodesic motion of a test body in this metric, mainly focusing on equatorial orbits, but also providing equations describing generic orbits formulated by means of canonical perturbation theory techniques. We proceed by computing approximate ‘kludge’ gravitational waveforms which we compare with their Kerr counterparts. We find that a modest deviation from the Kerr metric is sufficient for producing a significant mismatch between the waveforms, provided we fix the orbital parameters. This result suggests that an attempt to use Kerr waveform templates for studying extreme mass ratio inspirals around a non-Kerr object might result in serious loss of signal-to-noise ratio and total number of detected events. The waveform comparisons also unveil a ‘confusion’ problem, that is the possibility of matching a true non-Kerr waveform with a Kerr template of different orbital parameters.

A bstract We develop a formalism to compute non-perturbative 5-point scattering amplitudes and apply it to gravitational waveforms in the two-body problem for arbitrary trajectories. Drawing inspiration from Feshbach’s projector formalism in nuclear physics, we introduce effective potentials governing graviton emission and relate them to perturbative scattering amplitudes at arbitrary order in the gravitational coupling and mass ratio. Once these potentials are determined, the corresponding non-perturbative amplitudes in the classical limit are obtained by iterative insertions and subsequently translated into gravitational waveforms using the KMOC formalism. As an application, we compute the gravitational waveform emitted by a conservative two-body dynamics moving along a generic, potentially highly bent, trajectory. Our formalism extends effective field theory matching of the gravitational two-body potential to radiative phenomena, enabling the extraction of gravitational-wave source terms directly from perturbative on-shell amplitudes.

Extreme mass-ratio inspirals (EMRIs) are among the key targets for future space-based gravitational wave detectors. The gravitational waveforms emitted by EMRIs are highly sensitive to the orbital dynamics of the small compact object, which in turn are determined by the geometry of the underlying spacetime. In this paper, we explore the detectability of regular black holes with sub-Planckian curvature, which can be interpreted as regularized versions of the Schwarzschild black hole (RSBH). To do so, we begin by analyzing the metric and geodesics, determining the effective potential, and investigating the marginally bound orbits and the innermost stable circular orbits for timelike particles. Our analysis reveals that orbital radius, angular momentum, and energy significantly depend on the model parameter α for both orbits. In addition, we study how variations in α influence the photon sphere and the corresponding shadow silhouette. Observations of M87* and Sgr A* motivate testing whether α falls within an observationally constrained range, narrower than the theoretical bound ensuring a singularity-free BH structure. Our main aim is to focus on the influence of the model parameter on a specific kind of orbit, the periodic orbit, surrounding a supermassive RSBH. The findings show that, for a constant rational integer, α has a significant impact on the energy and angular momentum of the periodic orbit. Utilising the numerical kludge method, we further investigate the gravitational waveforms of the small celestial body over various periodic orbits. The waveforms display discrete zoom and spin phases within a complete orbital period, influenced by the RSBH parameter α. As the system evolves, the phase shift in the gravitational waveforms grows progressively more pronounced, with cumulative deviations amplifying over time. With the ongoing advancements in space-based gravitational wave detection systems, our results will aid in leveraging EMRIs to probe and characterize the RSBH properties.

In the broad subclass of Horndeski theories with a luminal speed of gravitational waves, we derive gravitational waveforms emitted from a compact binary by considering the wave propagation on a spatially flat cosmological background. A scalar field nonminimally coupled to gravity gives rise to hairy neutron star (NS) solutions with a nonvanishing scalar charge, whereas black holes (BHs) do not have scalar hairs in such theories. A binary system containing at least one hairy neutron star modifies the gravitational waveforms in comparison to those of the BH-BH binary. Using the tensor gravitational waveforms, we forecast the constraints on a parameter characterizing the difference of scalar charges of NS-BH or NS-NS binaries for Advanced LIGO and Einstein Telescope. We illustrate how these constraints depend on redshift and signal-to-noise ratio, and on different possible priors. We show that in any case it is possible to constrain the scalar charge precisely, so that some scalarized NS solutions known in the literature can be excluded.

Core collapse supernovae are a promising source of detectable gravitational waves. Most of the existing (multidimensional) numerical simulations of core collapse in general relativity have been done using approximations of the Einstein field equations. As recently shown by Dimmelmeier et al (2002a,b), one of the most interesting such approximation is the so-called conformal flatness condition (CFC) of Isenberg, Wilson and Mathews. Building on this previous work we present here new results from numerical simulations of relativistic rotational core collapse in axisymmetry, aiming at improving the dynamics and the gravitational waveforms. The computer code used for these simulations evolves the coupled system of metric and fluid equations using the 3+1 formalism, specialized to a new framework for the gravitational field equations which we call CFC+. In this approach we add new degrees of freedom to the original CFC equations, which extend them by terms of second post-Newtonian order. The corrections for CFC+ are computed solving a system of elliptic linear equations. The new formalism is assessed with time evolutions of both rotating neutron stars in equilibrium and gravitational core collapse of rotating polytropes. Gravitational wave signals for a comprehensive sample of collapse models are extracted using either the quadrupole formula or directly from the metric. We discuss our results on the dynamics and the gravitational wave emission through a detailed comparison between CFC and CFC+ simulations. The main conclusion is that, for the neutron star spacetimes analyzed in the present work, no significant differences are found among CFC, CFC+, and full general relativity, which highlights the suitability of the former.