post-Newtonian Templates Research Articles

Post-Newtonian templates for binary black-hole inspirals: the effect of the horizon fluxes and the secular change in the black-hole masses and spins

Black holes (BHs) in an inspiraling compact binary system absorb the gravitational-wave (GW) energy and angular-momentum fluxes across their event horizons and this leads to the secular change in their masses and spins during the inspiral phase. The goal of this paper is to present ready-to-use, post-Newtonian (PN) template families for spinning, non-precessing, binary BH inspirals in quasicircular orbits, including the PN and PN horizon-flux contributions as well as the correction due to the secular change in the BH masses and spins through PN order, respectively, in phase. We show that, for binary BHs observable by Advanced LIGO with high mass ratios (larger than ∼10) and large aligned-spins (larger than ), the mismatch between the frequency-domain template with and without the horizon-flux contribution is typically above the mark. For (supermassive) binary BHs observed by LISA, even a moderate mass-ratios and spins can produce a similar level of the mismatch. Meanwhile, the mismatch due to the secular time variations of the BH masses and spins is well below the mark in both cases, hence this is truly negligible. We also point out that neglecting the cubic-in-spin, point-particle phase term at PN order would deteriorate the effect of BH absorption in the template.

Next-to-leading tail-induced spin–orbit effects in the gravitational radiation flux of compact binaries

The imprint of nonlinearities in the propagation of gravitational waves—the tail effect—is responsible for new spin contributions to the energy flux and orbital phasing of spinning black hole binaries. The spin–orbit (linear in spin) contribution to this effect is currently known at leading post-Newtonian (PN) order, namely 3PN for maximally spinning black holes on quasi-circular orbits. In the present work, we generalize these tail-originated spin–orbit terms to the next-to-leading 4PN order. This requires in particular extending previous results on the dynamical evolution of precessing compact binaries. We show that the tails represent the only spin–orbit terms at that order for quasi-circular orbits, and we find perfect agreement with the known result for a test particle around a Kerr black hole, computed by perturbation theory. The BH-horizon absorption terms have to be added to the PN result computed here. Our work completes the knowledge of the spin–orbit effects to the phasing of compact binaries up to the 4PN order, and will allow the building of more faithful PN templates for the inspiral phase of black hole binaries, improving the capabilities of ground-based and space-based gravitational wave detectors.

Constraining the equation of state of nuclear matter with gravitational wave observations: Tidal deformability and tidal disruption

We study how to extract information on the neutron star equation of state from the gravitational wave signal emitted during the coalescence of a binary system composed of two neutron stars or a neutron star and a black hole. We use post-Newtonian templates which include the tidal deformability parameter and, when tidal disruption occurs before merger, a frequency cut-off. Assuming that this signal is detected by Advanced LIGO/Virgo or ET, we evaluate the uncertainties on these parameters using different data analysis strategies based on the Fisher matrix approach, and on recently obtained analytical fits of the relevant quantities. We find that the tidal deformability is more effective than the stellar compactness to discriminate among different possible equations of state.

Repeated bursts from relativistic scattering of compact objects in galactic nuclei

Galactic nuclei are densely populated by stellar mass compact objects such as black holes and neutron stars. Bound, highly eccentric binaries form as a result of gravitational wave (GW) losses during close flybys between these objects. We study the evolution of these systems using 2.5 and 3.5 order post-Newtonian (PN) equations of motion. The GW signal consists of many thousand repeated bursts (RB) for minutes to days (depending on the impact parameter and masses), followed by a powerful GW chirp and an eccentric merger. We show that a significant signal-to-noise ratio accumulates already in the RB phase, corresponding to a detection limit around 200--300 Mpc and 300--600 Mpc for advanced LIGO for an average orientation black hole/neutron star or black hole/black hole binary, respectively. The theoretical errors introduced by the inaccuracy of the PN templates are typically much less severe for the RB phase than in the following eccentric merger. The GW signal in the RB phase is broadband; we show that encounters involving intermediate mass black holes are detectable in multiple frequency bands coincidentally using LIGO and LISA.

Constraining the mass of the graviton using coalescing black-hole binaries

We study how well the mass of the graviton can be constrained from gravitational-wave (GW) observations of coalescing binary black holes. Whereas the previous investigations employed post-Newtonian (PN) templates describing only the inspiral part of the signal, the recent progress in analytical and numerical relativity has provided analytical waveform templates coherently describing the inspiral-merger-ringdown (IMR) signals. We show that a search for binary black holes employing IMR templates will be able to constrain the mass of the graviton much more accurately (about an order of magnitude) than a search employing PN templates. The best expected bound from GW observatories (lambda_g > 7.8 x 10^13 km from Adv. LIGO, lambda_g > 7.1 x 10^14 km from Einstein Telescope, and lambda_g > 5.9 x 10^17 km from LISA) are several orders-of-magnitude better than the best available model-independent bound (lambda_g > 2.8 x 10^12 km, from Solar system tests). Most importantly, GW observations will provide the first constraints from the highly dynamical, strong-field regime of gravity.

Comparison of post-Newtonian templates for compact binary inspiral signals in gravitational-wave detectors

The two-body dynamics in general relativity has been solved perturbatively using the post-Newtonian (PN) approximation. The evolution of the orbital phase and the emitted gravitational radiation are now known to a rather high order up to $\mathcal{O}({v}^{8})$, $v$ being the characteristic velocity of the binary. The orbital evolution, however, cannot be specified uniquely due to the inherent freedom in the choice of parameter used in the PN expansion, as well as the method pursued in solving the relevant differential equations. The goal of this paper is to determine the (dis)agreement between different PN waveform families in the context of initial and advanced gravitational-wave detectors. The waveforms employed in our analysis are those that are currently used by Initial LIGO/Virgo, that is, the time-domain PN models TaylorT1, TaylorT2, TaylorT3, the Fourier-domain representation TaylorF2 (or stationary phase approximant), and the effective-one-body model, and two more recent models, TaylorT4 and TaylorEt. For these models we examine their overlaps with one another for a number of different binaries at 2PN, 3PN, and 3.5PN orders to quantify their differences. We then study the overlaps of these families with the prototype effective-one-body family, currently used by Initial LIGO, calibrated to numerical-relativity simulations to help us decide whether there exist preferred families, in terms of detectability and computational cost, that are the most appropriate as search templates. We conclude that as long as the total mass remains less than a certain upper limit ${M}_{\mathrm{crit}}$, all template families at 3.5PN order (except TaylorT3 and TaylorEt) are equally good for the purpose of detection. The value of ${M}_{\mathrm{crit}}$ is found to be $\ensuremath{\sim}12{M}_{\ensuremath{\bigodot}}$ for Initial, Enhanced, and Advanced LIGO. From a purely computational point of view, we recommend that 3.5PN TaylorF2 be used below ${M}_{\mathrm{crit}}$ and that the effective-one-body model calibrated to numerical-relativity simulations be used for total binary mass $M>{M}_{\mathrm{crit}}$.

Comparison of high-accuracy numerical simulations of black-hole binaries with stationary-phase post-Newtonian template waveforms for initial and advanced LIGO

We study the effectiveness of stationary-phase approximated post-Newtonian waveforms currently used by ground-based gravitational-wave detectors to search for the coalescence of binary black holes by comparing them to an accurate waveform obtained from numerical simulation of an equal-mass non-spinning binary black hole inspiral, merger and ringdown. We perform this study for the initial- and advanced-LIGO detectors. We find that overlaps between the templates and signal can be improved by integrating the match filter to higher frequencies than used currently. We propose simple analytic frequency cutoffs for both initial and advanced LIGO, which achieve nearly optimal matches, and can easily be extended to unequal-mass, spinning systems. We also find that templates that include terms in the phase evolution up to 3.5 post-Newtonian (pN) order are nearly always better, and rarely significantly worse, than 2.0 pN templates currently in use. For initial LIGO we recommend a strategy using templates that include a recently introduced pseudo-4.0 pN term in the low-mass (M ⩽ 35 M⊙) region, and 3.5 pN templates allowing unphysical values of the symmetric reduced mass η above this. This strategy always achieves overlaps within 0.3% of the optimum, for the data used here. For advanced LIGO we recommend a strategy using 3.5 pN templates up to M = 12 M⊙, 2.0 pN templates up to M = 21 M⊙, pseudo-4.0 pN templates up to 65 M⊙, and 3.5 pN templates with unphysical η for higher masses. This strategy always achieves overlaps within 0.7% of the optimum for advanced LIGO.

Relativistic equations of motion of massive bodies

AbstractHighly relativistic equations of motions will play a crucial role for the detection and analysis of gravitational waves emitted by inspiralling compact binaries in detectors LIGO/VIRGO on ground and LISA in space. Indeed these very relativistic systems (with orbital velocities of the order of half the speed of light in the last orbital rotations) require the application of a high-order post-Newtonian formalism in general relativity for accurate description of their motion and gravitational radiation [1]. In this contribution the current state of the art which has reached the third post-Newtonian approximation for the equations of motion [2–6] and gravitational waveform [7–9] has been described (see [10] for an exhaustive review). We have also emphasized the successful matching of the post-Newtonian templates to numerically generated predictions for the merger and ring-down in the case of black-hole binaries [11].

Data-analysis driven comparison of analytic and numerical coalescing binary waveforms: Nonspinning case

We compare waveforms obtained by numerically evolving nonspinning binary black holes to postNewtonian (PN) template families currently used in the search for gravitational waves by groundbased detectors. We find that the time-domain 3.5PN template family, which includes the inspiral phase, has fitting factors (FFs) ≥ 0.96 for binary systems with total mass M = 10–20M⊙. The timedomain 3.5PN effective-one-body template family, which includes the inspiral, merger and ring-down phases, gives satisfactory signal-matching performance with FFs ≥ 0.96 for binary systems with total mass M = 10–120M⊙. If we introduce a cutoff frequency properly adjusted to the final black-hole ring-down frequency, we find that the frequency-domain stationary-phase-approximated template family at 3.5PN order has FFs ≥ 0.96 for binary systems with total mass M = 10–20M⊙. However, to obtain high matching performances for larger binary masses, we need to either extend this family to unphysical regions of the parameter space or introduce a 4PN order coefficient in the frequencydomain GW phase. Finally, we find that the phenomenological Buonanno-Chen-Vallisneri family has FFs ≥ 0.97 with total mass M = 10–120M⊙. The main analyses use the noise spectral-density of LIGO, but several tests are extended to VIRGO and advanced LIGO noise-spectral densities.

New template family for the detection of gravitational waves from comparable-mass black hole binaries

In order to improve the phasing of the comparable-mass waveform as we approach the last stable orbit for a system, various resummation methods have been used to improve the standard post-Newtonian waveforms. In this work we present a new family of templates for the detection of gravitational waves from the inspiral of two comparable-mass black hole binaries. These new adiabatic templates are based on reexpressing the derivative of the binding energy and the gravitational wave flux functions in terms of shifted Chebyshev polynomials. The Chebyshev polynomials are a useful tool in numerical methods as they display the fastest convergence of any of the orthogonal polynomials. In this case they are also particularly useful as they eliminate one of the features that plagues the post-Newtonian expansion. The Chebyshev binding energy now has information at all post-Newtonian orders, compared to the post-Newtonian templates which only have information at full integer orders. In this work, we compare both the post-Newtonian and Chebyshev templates against a fiducially exact waveform. This waveform is constructed from a hybrid method of using the test-mass results combined with the mass dependent parts of the post-Newtonian expansions for the binding energy and flux functions. Our results show that the Chebyshev templates achieve extremely high fitting factors at all post-Newtonian orders and provide excellent parameter extraction. We also show that this new template family has a faster Cauchy convergence, gives a better prediction of the position of the last stable orbit and in general recovers higher Signal-to-Noise ratios than the post-Newtonian templates.

Parameter estimation of coalescing supermassive black hole binaries with LISA

Laser Interferometer Space Antenna (LISA) will routinely observe coalescences of supermassive black hole (BH) binaries up to very high redshifts. LISA can measure mass parameters of such coalescences to a relative accuracy of $10^{-4}-10^{-6}$, for sources at a distance of 3 Gpc. The problem of parameter estimation of massive nonspinning binary black holes using post-Newtonian (PN) phasing formula is studied in the context of LISA. Specifically, the performance of the 3.5PN templates is contrasted against its 2PN counterpart using a waveform which is averaged over the LISA pattern functions. The improvement due to the higher order corrections to the phasing formula is examined by calculating the errors in the estimation of mass parameters at each order. The estimation of the mass parameters ${\cal M}$ and $\eta$ are significantly enhanced by using the 3.5PN waveform instead of the 2PN one. For an equal mass binary of $2\times10^6M_\odot$ at a luminosity distance of 3 Gpc, the improvement in chirp mass is $\sim 11%$ and that of $\eta$ is $\sim 39%$. Estimation of coalescence time $t_c$ worsens by 43%. The improvement is larger for the unequal mass binary mergers. These results are compared to the ones obtained using a non-pattern averaged waveform. The errors depend very much on the location and orientation of the source and general conclusions cannot be drawn without performing Monte Carlo simulations. Finally the effect of the choice of the lower frequency cut-off for LISA on the parameter estimation is studied.

The Cauchy convergence of T and P-approximant templates for test-mass Kerr binary systems

In this work, we examine the Cauchy convergence of both post-Newtonian (T-approximant) and re-summed post-Newtonian (P-approximant) templates for the case of a test-mass orbiting a Kerr black hole along a circular equatorial orbit. The Cauchy criterion demands that the inner product between the n and n + 1 order approximation approaches unity, as we increase the order of approximation. In previous works, it has been shown that we achieve greater fitting factors and better parameter estimation using the P-approximant templates for both Schwarzschild and Kerr black holes. In this work, we show that the P-approximant templates also display a faster Cauchy convergence making them a superior template to the standard post-Newtonian templates.

Erratum: Comparison of search templates for gravitational waves from binary inspiral [Phys. Rev. D63, 044023 (2001)

We compare the performances of the templates defined by three different types of approaches: traditional post-Newtonian templates (Taylor-approximants), ``resummed'' post-Newtonian templates assuming the adiabatic approximation and stopping before the plunge (P-approximants), and further ``resummed'' post-Newtonian templates going beyond the adiabatic approximation and incorporating the plunge with its transition from the inspiral (Effective-one-body approximants). The signal to noise ratio is significantly enhanced (mainly because of the inclusion of the plunge signal) by using these new effective-one-body templates relative to the usual post-Newtonian ones for binary masses greater than $ 30 M_\odot$, the most likely sources for initial laser interferometers. Independently of the question of the plunge signal, the comparison of the various templates confirms the usefulness of using resummation methods. The paper also summarizes the key elements of the construction of various templates and thus can serve as a resource for those involved in writing inspiral search software.

Are post-Newtonian templates faithful and effectual in detecting gravitational signals from neutron star binaries?

We compute the overlap function between Post-Newtonian (PN) templates and gravitational signals emitted by binary systems composed of one neutron star and one point mass, obtained by a perturbative approach. The calculations are performed for different stellar models and for different detectors, to estimate how effectual and faithful the PN templates are, and to establish whether effects related to the internal structure of neutron stars may possibly be extracted by the matched filtering technique.

Comparison of search templates for gravitational waves from binary inspiral

We compare the performances of the templates defined by three different types of approaches: traditional post-Newtonian templates (Taylor approximants), “resummed” post-Newtonian templates assuming the adiabatic approximation and stopping before the plunge (P approximants), and further “resummed” post-Newtonian templates going beyond the adiabatic approximation and incorporating the plunge with its transition from the inspiral (effective-one-body approximants). The signal to noise ratio is significantly enhanced (mainly because of the inclusion of the plunge signal) by using these new effective-one-body templates relative to the usual post-Newtonian ones for a total binary mass m≳30M⊙, and reaches a maximum around m∼80M⊙. Independently of the question of the plunge signal, the comparison of the various templates confirms the usefulness of using resummation methods. The paper also summarizes the key elements of the construction of various templates and thus can serve as a resource for those involved in writing inspiral search software. © 2001 The American Physical Society

Gravitational waveforms from spinning objects

According to general relativity, a binary consisting of spinning bodies will precess due to spin-orbit and spin-spin coupling. The corresponding modulation of its gravitational waves might be a serious problem for detecting such waves with simple post-Newtonian templates. A new family of templates that takes into account the complications arising from the orbital precession is proposed and its application and performance are discussed.