First order post-Newtonian gravitational waveforms of binaries on eccentric orbits with Hansen coefficients

Mikóczi, Balázs; Forgács, Péter; Vasúth, M.

doi:10.1103/physrevd.92.044038

Cited by 17 publications

(18 citation statements)

References 54 publications

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“…In order to explicitly find l max and t max , we begin by associating t max with the maximum value of the sum of the absolute values of the inverse Fourier transforms appearing in Eq. (37). Inspection of Eq.…”

Section: B Eccentric Match Maximizationmentioning

confidence: 99%

Towards a Fourier domain waveform for non-spinning binaries with arbitrary eccentricity

Moore

Robson

Loutrel

et al. 2018

Class. Quantum Grav.

105

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Although the gravitational waves observed by advanced LIGO and Virgo are consistent with compact binaries in a quasi-circular inspiral prior to coalescence, eccentric inspirals are also expected to occur in Nature. Due to their complexity, we currently lack ready-to-use, analytic waveforms in the Fourier domain valid for sufficiently high eccentricity, and such models are crucial to coherently extract weak signals from the noise. We here take the first steps to derive and properly validate an analytic waveform model in the Fourier domain that is valid for inspirals of arbitrary orbital eccentricity. As a proof-of-concept, we build this model to leading post-Newtonian order by combining the stationary phase approximation, a truncated sum of harmonics, and an analytic representation of hypergeometric functions. Through comparisons with numerical post-Newtonian waveforms, we determine how many harmonics are required for a faithful (matches above 99%) representation of the signal up to orbital eccentricities as large as 0.9. As a first byproduct of this analysis, we present a novel technique to maximize the match of eccentric signals over time of coalescence and phase at coalescence. As a second byproduct, we determine which of the different approximations we employ leads to the largest loss in match, which could be used to systematically improve the model because of our analytic control. The future extension of this model to higher post-Newtonian order will allow for an accurate and fast phenomenological hybrid that can account for arbitrary eccentricity inspirals and mergers.PACS numbers: 04.30.-w,04.25.-g,04.25.Nx 1 The post-Newtonian approximation is one in which the field equations are solved assuming small velocities and weak gravitational fields in an expansion in powers of ( v c ), where v is the orbital velocity and c is the speed of light [20]. By nPN order we mean an expansion to order (v/c) 2n . arXiv:1807.07163v1 [gr-qc]

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Section: B Eccentric Match Maximizationmentioning

confidence: 99%

Towards a Fourier domain waveform for non-spinning binaries with arbitrary eccentricity

Moore

Robson

Loutrel

et al. 2018

Class. Quantum Grav.

105

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“…Especially useful for GW data analysis applications is the development of frequency-domain eccentric waveforms. References [97,98] express the waveform amplitude to arbitrary eccentricity at 1PN order in the frequency domain but do not express the phasing explicitly as a function of frequency. This is extended to 2PN order in Ref.…”

Section: B Previous Work On Eccentric Waveform Modelsmentioning

confidence: 99%

Gravitational-wave phasing for low-eccentricity inspiralling compact binaries to 3PN order

et al. 2016

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Although gravitational radiation causes inspiralling compact binaries to circularize, a variety of astrophysical scenarios suggest that binaries might have small but nonnegligible orbital eccentricities when they enter the low-frequency bands of ground-and space-based gravitational-wave detectors. If not accounted for, even a small orbital eccentricity can cause a potentially significant systematic error in the mass parameters of an inspiralling binary [M. Favata, Phys. Rev. Lett. 112, 101101 (2014)]. Gravitational-wave search templates typically rely on the quasi-circular approximation, which provides relatively simple expressions for the gravitational-wave phase to 3.5 post-Newtonian (PN) order. Damour, Gopakumar, Iyer, and others have developed an elegant but complex quasiKeplerian formalism for describing the post-Newtonian corrections to the orbits and waveforms of inspiralling binaries with any eccentricity. Here we specialize the quasi-Keplerian formalism to binaries with low eccentricity. In this limit the non-periodic contribution to the gravitational-wave phasing can be expressed explicitly as simple functions of frequency or time, with little additional complexity beyond the well-known formulas for circular binaries. These eccentric phase corrections are computed to 3PN order and to leading order in the eccentricity for the standard PN approximants. For a variety of systems these eccentricity corrections cause significant corrections to the number of gravitational wave cycles that sweep through a detector's frequency band. This is evaluated using several measures, including a modification of the useful cycles. By comparing to numerical solutions valid for any eccentricity, we find that our analytic solutions are valid up to e0 0.1 for comparable mass systems, where e0 is the eccentricity when the source enters the detector band. We also evaluate the role of periodic terms that enter the phasing and discuss how they can be incorporated into some of the PN approximants. While the eccentric extension of the PN approximants is our main objective, this work collects a variety of results that may be of interest to others modeling eccentric relativistic binaries. This includes a consistent eccentricity expansion of the Newtonian-order polarizations and a comparison of quasi-Keplerian results with numerical simulations. In addition to applications in gravitational wave data analysis, the formulas derived here could be of use in comparing PN theory with numerical relativity or self-force calculations of eccentric binaries. They could also be useful in the construction of phenomenological inspiral-merger-ringdown waveforms that include eccentricity effects.

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“…Previous work related to this particular subject includes the following: (i) frequency domain inspiral-only waveforms that include leading-order post-Newtonian (PN 1 ) corrections in a post-circular or small eccentricity approximation [44,45]; (ii) frequency and time domain waveforms that reduce to the PN-based approximants TaylorF2 and TaylorT4 at 2PN in the quasicircular limit [46]; (iii) inspiralonly waveforms that include 2PN and 3PN corrections to the radiative and conservative pieces of the dynamics, respectively [47]; (iv) inspiral-only waveforms that include 3PN corrections to the radiative and conservative pieces of the dynamics [31,[48][49][50]; (v) inspiral-only frequency domain waveforms that reduce to the PN-based approximant TaylorF2 3.5PN at zero eccentricity, and to the post-circular approximation of Ref. [44] at small eccentricity [34]; (vi) hybrid waveforms that describe highly eccentric systems: these waveforms describe the inspiral evolution using geodesic equations of motion, and the merger phase is modeled using a semianalytical prescription that captures the features of NR simulations [51]; (vii) self-force calculations for nonspinning BHs along eccentric orbits [52][53][54][55][56][57][58][59]; and (viii) NR simulations that explore the dynamics of eccentric binary systems [47,[60][61][62][63][64][65][66][67][68][69].…”

Section: Introductionmentioning

confidence: 99%

Complete waveform model for compact binaries on eccentric orbits

et al. 2017

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We present a time domain waveform model that describes the inspiral, merger and ringdown of compact binary systems whose components are nonspinning, and which evolve on orbits with low to moderate eccentricity. The inspiral evolution is described using third-order post-Newtonian equations both for the equations of motion of the binary, and its far-zone radiation field. This latter component also includes instantaneous, tails and tails-of-tails contributions, and a contribution due to nonlinear memory. This framework reduces to the post-Newtonian approximant TaylorT4 at third postNewtonian order in the zero-eccentricity limit. To improve phase accuracy, we also incorporate higher-order post-Newtonian corrections for the energy flux of quasicircular binaries and gravitational self-force corrections to the binding energy of compact binaries. This enhanced prescription for the inspiral evolution is combined with a fully analytical prescription for the merger-ringdown evolution constructed using a catalog of numerical relativity simulations. We show that this inspiral-mergerringdown waveform model reproduces the effective-one-body model of Ref. [Y. Pan et al., Phys. Rev. D 89, 061501 (2014).] for quasicircular black hole binaries with mass ratios between 1 to 15 in the zero-eccentricity limit over a wide range of the parameter space under consideration. Using a set of eccentric numerical relativity simulations, not used during calibration, we show that our new eccentric model reproduces the true features of eccentric compact binary coalescence throughout merger. We use this model to show that the gravitational-wave transients GW150914 and GW151226 can be effectively recovered with template banks of quasicircular, spin-aligned waveforms if the eccentricity e 0 of these systems when they enter the aLIGO band at a gravitational-wave frequency of 14 Hz satisfies e GW150914 0 ≤ 0.15 and e GW151226 0 ≤ 0.1. We also find that varying the spin combinations of the quasicircular, spin-aligned template waveforms does not improve the recovery of nonspinning, eccentric signals when e 0 ≥ 0.1. This suggests that these two signal manifolds are predominantly orthogonal.

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First order post-Newtonian gravitational waveforms of binaries on eccentric orbits with Hansen coefficients

Cited by 17 publications

References 54 publications

Towards a Fourier domain waveform for non-spinning binaries with arbitrary eccentricity

Towards a Fourier domain waveform for non-spinning binaries with arbitrary eccentricity

Gravitational-wave phasing for low-eccentricity inspiralling compact binaries to 3PN order

Complete waveform model for compact binaries on eccentric orbits

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