Gravitational Wave Flux Research Articles

Determination of new coefficients in the angular momentum and energy fluxes at infinity to 9PN order for eccentric Schwarzschild extreme-mass-ratio inspirals using mode-by-mode fitting

We present an extension of work in an earlier paper showing high precision comparisons between black hole perturbation theory and post-Newtonian (PN) theory in their region of overlapping validity for bound, eccentric-orbit, Schwarzschild extreme-mass-ratio inspirals. As before we apply a numerical fitting scheme to extract eccentricity coefficients in the PN expansion of the gravitational wave fluxes, which are then converted to exact analytic form using an integer-relation algorithm. In this work, however, we fit to individual $lmn$ modes to exploit simplifying factorizations that lie therein. Since the previous paper focused solely on the energy flux, here we concentrate initially on analyzing the angular momentum flux to infinity. A first step involves finding convenient forms for hereditary contributions to the flux at low-PN order, analogous to similar terms worked out previously for the energy flux. We then apply the upgraded techniques to find new PN terms through 9PN order and (at many PN orders) to $e^{30}$ in the power series in eccentricity. With the new approach applied to angular momentum fluxes, we return to the energy fluxes at infinity to extend those previous results. Like before, the underlying method uses a \textsc{Mathematica} code based on use of the Mano-Suzuki-Takasugi (MST) function expansion formalism to represent gravitational perturbations and spectral source integration (SSI) to find numerical results at arbitrarily high precision.

Extreme mass ratio inspirals with spinning secondary: A detailed study of equatorial circular motion

Extreme mass-ratio inspirals detectable by the future Laser Interferometer Space Antenna provide a unique way to test general relativity and fundamental physics. Motivated by this possibility, here we study in detail the EMRI dynamics in the presence of a spinning secondary, collecting and extending various results that appeared in previous work and also providing useful intermediate steps and new relations for the first time. We present the results of a frequency-domain code that computes gravitational-wave fluxes and the adiabatic orbital evolution for the case of circular, equatorial orbits with (anti)aligned spins. The spin of the secondary starts affecting the gravitational-wave phase at the same post-adiabatic order as the leading-order self-force terms and introduces a detectable dephasing, which can be used to measure it at $5-25\%$ level, depending on individual spins. In a companion paper we discuss the implication of this effect for tests of the Kerr bound.

Tidal effects in the equations of motion of compact binary systems to next-to-next-to-leading post-Newtonian order

As a first step in the computation of the orbital phase evolution of spinless compact binaries including tidal effects up to the next-to-next-to-leading (NNL) order, we obtain the equations of motion of those systems and the associated conserved integrals in harmonic coordinates. The internal structure and finite size effects of the compact objects are described by means of an effective Fokker-type action. Our results, complete to the NNL order, correspond to the second-post-Newtonian (2PN) approximation beyond the leading tidal effect itself, already occurring at the 5PN order. They are parametrized by three polarizability (or deformability) coefficients describing the mass quadrupolar, mass octupolar and current quadrupolar deformations of the objects through tidal interactions. Up to the next-to-leading (NL) order, we recover previous results in the literature; up to the NNL order for quasi-circular orbits, we confirm the known tidal effects in the (PN re-expansion of the) effective-one-body (EOB) Hamiltonian. In a future work, we shall derive the tidal contributions to the gravitational-wave flux up to the NNL order, which is the second step required to find the orbital phase evolution.

Publisher’s Note: Asymptotic gravitational wave fluxes from a spinning particle in circular equatorial orbits around a rotating black hole [Phys. Rev. D 93, 044015 (2016)

Publisher’s Note: Asymptotic gravitational wave fluxes from a spinning particle in circular equatorial orbits around a rotating black hole [Phys. Rev. D 93, 044015 (2016)

Eccentric-orbit extreme-mass-ratio-inspiral radiation: Analytic forms of leading-logarithm and subleading-logarithm flux terms at high PN orders

We present new results on the analytic eccentricity dependence of several sequences of gravitational wave flux terms at high post-Newtonian (PN) order for extreme-mass-ratio inspirals. These sequences are the leading logarithms, which appear at PN orders $x^{3k} \log^k(x)$ and $x^{3k+3/2} \log^k(x)$ for integers $k\ge 0$ ($x$ a PN compactness parameter), and the subleading logarithms, which appear at orders $x^{3k} \log^{k-1}(x)$ and $x^{3k+3/2} \log^{k-1}(x)$ ($k\ge 1$), in both the energy and angular momentum radiated to infinity. For the energy flux leading logarithms, we show that to arbitrarily high PN order their eccentricity dependence is determined by particular sums over the function $g(n,e_t)$, derived from the Newtonian mass quadrupole moment, that normally gives the spectral content of the Peters-Mathews flux as a function of radial harmonic $n$. An analogous power spectrum $\tilde{g}(n,e_t)$ determines the leading logarithms of the angular momentum flux. For subleading logs, the quadrupole power spectra are again shown to play a role, providing a distinguishable part of the eccentricity dependence of these flux terms to high PN order. With the quadrupole contribution understood, the remaining analytic eccentricity dependence of the subleading logs can in principle be determined more easily using black hole perturbation theory. We show this procedure in action, deriving the complete analytic structure of the $x^6 \log(x)$ subleading-log term and an analytic expansion of the $x^{9/2}$ subleading log to high order in a power series in eccentricity. We discuss how these methods might be extended to other sequences of terms in the PN expansion involving logarithms.

Neutron star sensitivities in Hořava gravity after GW170817

Ho\v{r}ava gravity breaks boost invariance in the gravitational sector by introducing a preferred time foliation. The dynamics of this preferred slicing is governed, in the low-energy limit suitable for most astrophysical applications, by three dimensionless parameters $\alpha$, $\beta$ and $\lambda$. The first two of these parameters are tightly bound by solar system and gravitational wave propagation experiments, but $\lambda$ remains relatively unconstrained ($0\leq\lambda\lesssim 0.01-0.1$). We restrict here to the parameter space region defined by $\alpha=\beta=0$ (with $\lambda$ kept generic), which in a previous paper we showed to be the only one where black hole solutions are non-pathological at the universal horizon, and we focus on possible violations of the strong equivalence principle in systems involving neutron stars. We compute neutron star 'sensitivities', which parametrize violations of the strong equivalence principle at the leading post-Newtonian order, and find that they vanish identically, like in the black hole case, for $\alpha=\beta=0$ and generic $\lambda\neq0$. This implies that no violations of the strong equivalence principle (neither in the conservative sector nor in gravitational wave fluxes) can occur at the leading post-Newtonian order in binaries of compact objects, and that data from binary pulsars and gravitational interferometers are unlikely to further constrain $\lambda$.

Flux-balance equations for linear momentum and center-of-mass position of self-gravitating post-Newtonian systems

We revisit the problem of flux-balance equations for isolated post-Newtonian matter systems due to the emission of gravitational waves. In particular we show by a local derivation confined to the system, using the expression of radiation-reaction forces up to the 3.5 PN order, that not only the energy, angular momentum and linear momentum of the system, but also the position of its center of mass, obey some (non-trivial) flux-balance equations. The balance equation for the center-of-mass position completes the description of the secular evolution by gravitational waves of relativistic post-Newtonian isolated matter systems. We then confirm this result by a direct computation of the gravitational-wave fluxes at future null infinity, obtaining the full multipole-moment expansion of the flux associated with the center-of-mass position (probably new with this paper), and rederiving as well the known multipole-moment expansions of the fluxes of energy, angular momentum and linear momentum. We also check our analysis by a direct calculation of radiation-reaction effects in the case of compact binary systems.

“Flux-balance formulae” for extreme mass-ratio inspirals

The "flux-balance formulae" that determine the averaged evolution of energy, azimuthal angular momentum, and Carter constant in terms of the averaged asymptotic gravitational-wave fluxes for inspirals of small bodies into Kerr black holes were first derived about 15 years ago. However, this derivation is restricted to the case that the background Kerr geodesics are non-resonant (i.e., the radial and angular motions are always incommensurate), and excludes the resonant case that can be important for the radiative dynamics of extreme mass-ratio inspirals. We give here a new derivation of the flux formulae based on Hamiltonian dynamics of a self-forced particle motion, which is a valuable tool for analyzing self-force effects on generic (eccentric, inclined) bound orbits in the Kerr spacetime. This Hamiltonian derivation using action-angle variables is much simpler than the previous one, applies to resonant inspirals without any complication, and can be straightforwardly implemented by using analytical/numerical Teukolsky-based flux codes.

Gravitational self-force on generic bound geodesics in Kerr spacetime

In this work we present the first calculation of the gravitational self-force on generic bound geodesics in Kerr spacetime to first order in the mass-ratio. That is, the local correction to equations of motion for a compact object orbiting a larger rotating black hole due to its own impact on the gravitational field. This includes both dissipative and conservative effects. Our method builds on and extends earlier methods for calculating the gravitational self-force on equatorial orbits. In particular we reconstruct the local metric perturbation in the outgoing radiation gauge from the Weyl scalar $\psi_4$, which in turn is obtained by solving the Teukolsky equation using semi-analytical frequency domain methods. The gravitational self-force is subsequently obtained using (spherical) $l$-mode regularization. We test our implementation by comparing the large $l$-behaviour against the analytically known regularization parameters. In addition we validate our results be comparing the long-term average changes to the energy, angular momentum, and Carter constant to changes to these constants of motion inferred from the gravitational wave flux to infinity and down the horizon.

Dynamics of marginally trapped surfaces in a binary black hole merger: Growth and approach to equilibrium

The behavior of quasi-local black hole horizons in a binary black hole merger is studied numerically. We compute the horizon multipole moments, fluxes and other quantities on black hole horizons throughout the merger. These lead to a better qualitative and quantitative understanding of the coalescence of two black holes; how the final black hole is formed, initially grows and then settles down to a Kerr black hole. We calculate the rate at which the final black hole approaches equilibrium in a fully non-perturbative situation and identify a time at which the linear ringdown phase begins. Finally, we provide additional support for the conjecture that fields at the horizon are correlated with fields in the wave-zone by comparing the in-falling gravitational wave flux at the horizon to the outgoing flux as estimated from the gravitational waveform.

Spinning test body orbiting around a Kerr black hole: Circular dynamics and gravitational-wave fluxes

In a recent work, [Phys. Rev. D. 94, 104010 (2016)], hereafter Paper I, we have numerically studied different prescriptions for the dynamics of a spinning particle in circular motion around a Schwarzschild black hole. In the present work, we continue this line of investigation to the rotating Kerr black hole. We consider the Mathisson-Papapetrou formalism under three different spin-supplementary-conditions (SSC), the Tulczyjew SSC, the Pirani SSC and the Ohashi-Kyrian-Semerak SSC, and analyze the different circular dynamics in terms of the ISCO shifts and the frequency parameter ${x \equiv (M \Omega)^{2/3}}$, where $\Omega$ is the orbital frequency and $M$ is the Kerr black hole mass. Then, we solve numerically the inhomogeneous $(2+1)D$ Teukolsky equation to contrast the asymptotic gravitational wave fluxes for the three cases. Our central observation made in Paper I for the Schwarzschild limit is found to hold true for the Kerr background: the three SSCs reduce to the same circular dynamics and the same radiation fluxes for small frequency parameters but differences arise as $x$ grows close to the ISCO. For a positive Kerr parameter $a=0.9$ the energy fluxes mutually agree with each other within a $0.2\%$ uncertainty up to $x<0.14$, while for $a=-0.9$ this level of agreement is preserved up to $x<0.1$. For large frequencies ($x \gtrsim 0.1$), however, the spin coupling of the Kerr black hole and the spinning body results in significant differences of the circular orbit parameters and the fluxes, especially for the $a=-0.9$ case. Instead, in the study of ISCO the negative Kerr parameter $a=-0.9$ results in less discrepancies in comparison with the positive Kerr parameter $a=0.9$.

Minimum energy and the end of the inspiral in the post-Newtonian approximation

The early inspiral phase of a compact binary coalescence is well modelled by the post-Newtonian (PN) approximation to the orbital energy and gravitational wave flux. The transition from the inspiral phase to the plunge can be defined by the minimum energy circular orbit (MECO). In the extreme mass-ratio limit, up to the highest PN order known, the PN energy equals the energy of the exact Kerr solution. However, for comparable-mass systems the MECO of the PN energy does not exist when bodies have large spins. By including the exact Kerr limit and recently published post-Newtonian terms we extract a well-defined minimum of the orbital energy beyond which the plunge or merger occurs. We study the hybrid condition for a number of cases of both black hole and neutron stars and compare to other commonly employed definitions. Our method can be used for any known order of the post-Newtonian series and enables the MECO condition to be used to define the end of the inspiral phase for highly spinning, comparable mass systems.

Importance of transient resonances in extreme-mass-ratio inspirals

The inspiral of stellar-mass compact objects, like neutron stars or stellar-mass black holes, into supermassive black holes provides a wealth of information about the strong gravitational-field regime via the emission of gravitational waves. In order to detect and analyse these signals, accurate waveform templates which include the effects of the compact object's gravitational self-force are required. For computational efficiency, adiabatic templates are often used. These accurately reproduce orbit-averaged trajectories arising from the first-order self-force, but neglect other effects, such as transient resonances, where the radial and poloidal fundamental frequencies become commensurate. During such resonances the flux of gravitational waves can be diminished or enhanced, leading to a shift in the compact object's trajectory and the phase of the waveform. We present an evolution scheme for studying the effects of transient resonances and apply this to an astrophysically motivated population. We find that a large proportion of systems encounter a low-order resonance in the later stages of inspiral; however, the resulting effect on signal-to-noise recovery is small as a consequence of the low eccentricity of the inspirals. Neglecting the effects of transient resonances leads to a loss of 4% of detectable signals.

Spinning test body orbiting around a Schwarzschild black hole: Circular dynamics and gravitational-wave fluxes

We consider a spinning test-body in circular motion around a nonrotating black hole and analyze different prescriptions for the body's dynamics. We compare, for the first time, the Mathisson-Papapetrou formalism under the Tulczyjew spin-supplementary-condition (SSC), the Pirani SSC and the Ohashi-Kyrian-Semerak SSC, and the spinning particle limit of the effective-one-body Hamiltonian of [Phys.~Rev.~D.90,~044018(2014)]. We analyze the four different dynamics in terms of the ISCO shifts and in terms of the coordinate invariant binding energies, separating higher-order spin contributions from spin-orbit contributions. The asymptotic gravitational wave fluxes produced by the spinning body are computed by solving the inhomogeneous $(2+1)D$ Teukolsky equation and contrasted for the different cases. For small orbital frequencies $\Omega$, all the prescriptions reduce to the same dynamics and the same radiation fluxes. For large frequencies, ${x \equiv (M \Omega)^{2/3} >0.1 }$, where $M$ is the black hole mass, and especially for positive spins (aligned with orbital angular momentum) a significant disagreement between the different dynamics is observed. The ISCO shifts can differ up to a factor two for large positive spins; for the Ohashi-Kyrian-Semerak and the Pirani SSC the ISCO diverges around dimensionless spins $\sim0.52$ and $\sim0.94$ respectively. In the spin-orbit part of the energetics the deviation from the Hamiltonian dynamics is largest for the Ohashi-Kyrian-Semerak SSC; it exceeds $10\%$ for $x>0.17$. The Tulczyjew and the Pirani SSCs behave compatible across almost the whole spin and frequency range. Our results will have direct application in including spin effects to effective-one-body waveform models for circularized binaries in the extreme-mass-ratio limit.

Asymptotic gravitational wave fluxes from a spinning particle in circular equatorial orbits around a rotating black hole

We present a new computation of the asymptotic gravitational wave energy fluxes emitted by a {\it spinning} particle in circular equatorial orbits about a Kerr black hole. The particle dynamics is computed in the pole-dipole approximation, solving the Mathisson-Papapetrou equations with the Tulczyjew spin-supplementary-condition. The fluxes are computed, for the first time, by solving the 2+1 Teukolsky equation in the time-domain using hyperboloidal and horizon-penetrating coordinates. Denoting by $M$ the black hole mass and by $\mu$ the particle mass, we cover dimensionless background spins $a/M=(0,\pm0.9)$ and dimensionless particle spins $-0.9\leq S/\mu^2 \leq +0.9$. Our results span orbits of Boyer-Lindquist coordinate radii $4\leq r/M \leq 30$; notably, we investigate the strong-field regime, in some cases even beyond the last-stable-orbit. We confirm, numerically, the Tanaka {\it et al.} [Phys.\ Rev.\ D 54, 3762] 2.5th order accurate Post-Newtonian (PN) predictions for the gravitational wave fluxes of a spinning particle in circular equatorial orbit: we find an unambiguos trend of the PN-prediction towards the numerical results when $r$ is large. At $r/M=30$ the fractional agreement between the full numerical flux, approximated as the sum over the modes $m=1,2,3$, and the PN prediction is $\lesssim0.5\%$ in all cases tested. This is close to our fractional numerical accuracy ($\sim 0.2\%$). For smaller radii, the agreement between the 2.5PN prediction and the numerical result progressively deteriorates. Our numerical data will be essential to develop suitably resummed expressions of the 2.5PN analytical fluxes in order to improve their accuracy in the strong-field regime.

Estimating effective higher order terms in the post-Newtonian binding energy and gravitational-wave flux: Nonspinning compact binary inspiral

In the adiabatic post-Newtonian (PN) approximation, the phase evolution of gravitational waves (GWs) from inspiralling compact binaries in quasicircular orbits is computed by equating the change in binding energy with the GW flux. This energy balance equation can be solved in different ways, which result in multiple approximants of the PN waveforms. Due to the poor convergence of the PN expansion, these approximants tend to differ from each other during the late inspiral. Which of these approximants should be chosen as templates for detection and parameter estimation of GWs from inspiraling compact binaries is not obvious. In this paper, we present estimates of the effective higher order (beyond the currently available 4PN and 3.5PN) non-spinning terms in the PN expansion of the binding energy and the GW flux that minimize the difference of multiple PN approximants (TaylorT1, TaylorT2, TaylorT4, TaylorF2) with effective one body waveforms calibrated to numerical relativity (EOBNR). We show that PN approximants constructed using the effective higher order terms show significantly better agreement (as compared to 3.5PN) with the inspiral part of the EOBNR. For non-spinning binaries with component masses $m_{1,2} \in [1.4 M_\odot, 15 M_\odot]$, most of the approximants have a match (faithfulness) of better than 99% with both EOBNR and each other.

A new gravitational wave generation algorithm for particle perturbations of the Kerr spacetime

We present a new approach to solve the 2+1 Teukolsky equation for gravitational perturbations of a Kerr black hole. Our approach relies on a new horizon penetrating, hyperboloidal foliation of Kerr spacetime and spatial compactification. In particular, we present a framework for waveform generation from point-particle perturbations. Extensive tests of a time domain implementation in the Teukode code are presented. The code can efficiently deliver waveforms at future null infinity. The accuracy and convergence of the waveforms’ phase and amplitude is demonstrated. As a first application of the method, we compute the gravitational waveforms from inspiraling and coalescing black-hole binaries in the large-mass-ratio limit. The smaller mass black hole is modeled as a point particle whose dynamics is driven by an effective-one-body-resummed analytical radiation reaction force. We compare the analytical, mechanical angular momentum loss (computed using two different prescriptions) to the gravitational wave angular momentum flux. We find that higher-order post-Newtonian corrections are needed to improve the consistency for rapidly spinning binaries. We characterize the multipolar waveform as a function of the black-hole spin. Close to merger, the subdominant multipolar amplitudes (notably the m = 0 ones) are enhanced for retrograde orbits with respect to prograde ones. We argue that this effect mirrors nonnegligible deviations from the circularity of the dynamics during the late-plunge and merger phase. For the first time, we compute the gravitational wave energy flux flowing into the black hole during the inspiral using a time-domain formalism proposed by Poisson. Finally, a self-consistent, iterative method to compute the gravitational wave fluxes at leading-order in the mass of the particle is developed. The method can be used alternatively to the analytical radiation reaction in cases where the analytical information is poor or not sufficient. Specifically, we apply it to compute dynamics and waveforms for a rapidly rotating black hole with dimensionless spin parameter . For this case, the simulation with the consistent flux differs from the one with the analytical flux by gravitational wave cycles over a total of about 250 cycles. In this simulation the horizon absorption accounts for about gravitational wave cycles.

Small mass plunging into a Kerr black hole: Anatomy of the inspiral-merger-ringdown waveforms

We numerically solve the Teukolsky equation in the time domain to obtain the gravitational-wave emission of a small mass inspiraling and plunging into the equatorial plane of a Kerr black hole. We account for the dissipation of orbital energy using the Teukolsky frequency-domain gravitational-wave fluxes for circular, equatorial orbits, down to the light-ring. We consider Kerr spins $\ensuremath{-}0.99\ensuremath{\le}q\ensuremath{\le}0.99$, and compute the inspiral-merger-ringdown (2,2), (2,1), (3,3), (3,2), (4,4), and (5,5) modes. We study the large-spin regime, and find a great simplicity in the merger waveforms, thanks to the extremely circular character of the plunging orbits. We also quantitatively examine the mixing of quasinormal modes during the ringdown, which induces complicated amplitude and frequency modulations in the waveforms. Finally, we explain how the study of small mass-ratio black-hole binaries helps extending effective-one-body models for comparable-mass, spinning black-hole binaries to any mass ratio and spin magnitude.

Gravitational wave recoils in non-axisymmetric Robinson–Trautman spacetimes

We examine the gravitational wave recoil waves and the associated net kick velocities in non-axisymmetric Robinson-Trautman spacetimes. We use characteristic initial data for the dynamics corresponding to non-head-on collisions of black holes. We make a parameter study of the kick distributions, corresponding to an extended range of the incidence angle $\rho_0$ in the initial data. For the range of $\rho_0$ examined ($3^{\circ} \leq \rho_0 \leq 110^{\circ}$) the kick distributions as a function of the symmetric mass parameter $\eta$ satisfy a law obtained from an empirical modification of the Fitchett law, with a parameter $C$ that accounts for the non-zero net gravitational momentum wave fluxes for the equal mass case. The law fits accurately the kick distributions for the range of $\rho_0$ examined, with a rms normalized error of the order of $5 \%$. For the equal mass case the nonzero net gravitational wave momentum flux increases as $\rho_0$ increases, up to $\rho_0 \simeq 55^{\circ}$ beyond which it decreases. The maximum net kick velocity is about $190 {\rm km/s}$ for for the boost parameter considered. For $\rho_0 \geq 50^{\circ}$ the distribution is a monotonous function of $\eta$. The angular patterns of the gravitational waves emitted are examined. Our analysis includes the two polarization modes present in wave zone curvature.

Gravitational-wave flux for a particle orbiting a Kerr black hole to 20th post-Newtonian order: A numerical approach

In this article we present the post-Newtonian (pN) coefficients of the energy flux (and angular momentum flux) at infinity and event horizon for a particle in circular, equatorial orbits about a Kerr black hole (of mass $M$ and spin-parameter $a$) up to 20-pN order. When a pN term is not a polynomial in $a/M$ and includes irrational functions (like polygamma functions), it is written as a power series of $a/M$. This is achieved by calculating the fluxes numerically with an accuracy greater than 1 part in $10^{600}$. Such high accuracy allows us to extract analytical values of pN coefficients that are linear combinations of transcendentals like the Euler constant, logarithms of prime numbers and powers of $\pi$. We also present the 22-pN expansion (spin-independent pN expansion) of the ingoing energy flux at the event horizon for a particle in circular orbit about a Schwarzschild black hole.