Ready-to-use Fourier domain templates for compact binaries inspiraling along moderately eccentric orbits
Srishti Tiwari, Gopakumar Achamveedu, Maria Haney, Phurailatapam, Hemantakumar
TL;DR
This paper develops accurate, ready-to-use Fourier domain gravitational wave templates for inspiraling compact binaries with moderate eccentricity, incorporating amplitude and phase corrections up to high post-Newtonian orders.
Contribution
It introduces a novel analytic template family for eccentric binary inspirals, including amplitude and phase corrections up to 3PN order, with a prescription for periastron advance effects.
Findings
Templates are accurate to 1PN amplitude order and 3PN phase order.
Inclusion of eccentricity effects up to sixth order in phase evolution.
Preliminary data analysis indicates these templates are essential for modeling eccentric binary coalescence.
Abstract
We derive analytic expressions that provide Fourier domain gravitational wave (GW) response function for compact binaries inspiraling along moderately eccentric orbits. These expressions include amplitude corrections to the two GW polarization states that are accurate to the first post-Newtonian (PN) order. Additionally, our fully 3PN accurate GW phase evolution incorporates eccentricity effects up to sixth order at each PN order. Further, we develop a prescription to incorporate analytically the effects of 3PN accurate periastron advance in the GW phase evolution. This is how we provide a ready-to-use and efficient inspiral template family for compact binaries in moderately eccentric orbits. Preliminary GW data analysis explorations suggest that our template family should be required to construct analytic inspiral-merger-ringdown templates to model moderately eccentric compact binary…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Ready-to-use Fourier domain templates for compact binaries inspiraling along moderately eccentric orbits
Srishti Tiwari 1 [email protected], Achamveedu Gopakumar 1, Maria Haney 2, and Phurailatapam Hemantakumar 3
1Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Mumbai 400005, India
2Physik-Institut, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
3Indian Institute of Technology, Mumbai 400076, India
Abstract
We derive analytic expressions that provide Fourier domain gravitational wave (GW) response function for compact binaries inspiraling along moderately eccentric orbits. These expressions include amplitude corrections to the two GW polarization states that are accurate to the first post-Newtonian (PN) order. Additionally, our fully 3PN accurate GW phase evolution incorporates eccentricity effects up to sixth order at each PN order. Further, we develop a prescription to incorporate analytically the effects of 3PN accurate periastron advance in the GW phase evolution. This is how we provide a ready-to-use and efficient inspiral template family for compact binaries in moderately eccentric orbits. Preliminary GW data analysis explorations suggest that our template family should be required to construct analytic inspiral-merger-ringdown templates to model moderately eccentric compact binary coalescence.
pacs:
04.30.-w, 04.30.Tv
††preprint: APS/123-QED
I Introduction
Observations of GW events by the advanced LIGO and VIRGO GW interferometers are ushering in the era of GW astronomy Abbott et al. (2016a); Acernese et al. (2015). These GW events include merging black hole (BH) binaries and an inspiraling neutron star (NS) binary Abbott et al. (2016b, c, 2017a, 2017b, 2017c, 2017d); The LIGO Scientific Collaboration and the Virgo Collaboration (2018). Several scenarios that include long-lived (galactic) field binaries, star clusters, galactic nuclei and active galactic nuclei can produce these observed GW events Belczynski et al. (2016); Park et al. (2017); Hoang et al. (2018); McKernan et al. (2018); Kremer et al. (2018). Fortunately, it may be possible to extract valuable information about the astrophysical origins of GW events in the near future. This requires accurate GW measurements of the spin-orbit misalignment or the orbital eccentricities of these GW events Rodriguez et al. (2016a); Chen and Amaro-Seoane (2017); Nishizawa et al. (2017). Using both frequency and time domain inspiral-merger-ringdown (IMR) waveforms, residual orbital eccentricities of the first two GW events were restricted to be below when these binaries entered aLIGO frequency windowAbbott et al. (2016d); Huerta et al. (2017). Strictly speaking, the so far detected GW events do not exhibit any observational signatures of residual orbital eccentricities and are faithfully captured by IMR templates associated with compact binaries merging along quasi-circular orbits.
However, there exists a number of astrophysical scenarios that can produce GW events with non-negligible eccentricities in the frequency windows of ground-based GW detectors. Dense star clusters like the ubiquitous globular clusters are the most promising sites to form aLIGO relevant compact binaries with non-negligible orbital eccentricities Rodriguez et al. (2016b). A recent realistic modeling of globular clusters that involve general relativistic few body interactions provided non-negligible fraction of BH binaries with eccentricities as they enter the aLIGO frequency window Samsing et al. (2014); Samsing and Ramirez-Ruiz (2017); Rodriguez et al. (2018a); Samsing (2018); Rodriguez et al. (2018b); Kremer et al. (2018). Additionally, there exists a number of other astrophysical scenarios that can force stellar mass compact binaries to merge with orbital eccentricities. This include GW induced merger during hyperbolic encounters between BHs in dense clusters O’Leary et al. (2016) and mergers influenced by Kozai effect in few body systems as explored in many detailed investigations (see Ref. Randall and Xianyu (2018) and references therein). Further, a very recent investigation pointed out that less frequent binary-binary encounters in dense star clusters can easily produce eccentric compact binary coalescence Zevin et al. (2018). These detailed investigations suggest that it may be reasonable to expect GW events with non-negligible orbital eccentricities in the coming years. Non-negligible orbital eccentricities may be helpful to improve the accuracy with a network of GW interferometers to constrain parameters of compact binary mergers Gondán et al. (2018); Gondán and Kocsis (2018). Moreover, massive BH binaries in eccentric orbits are of definite interest to maturing Pulsar Timing Arrays and the planned Laser Interferometer Space Antenna (LISA) Burke-Spolaor et al. (2018); Bonetti et al. (2018).
There are different on-going investigations to model eccentric compact binary coalescence. These efforts aim to provide template families that model GWs from IMR phases of eccentric coalescence. The initial effort, detailed in Ref. Huerta et al. (2017), provided a time-domain IMR family that requires orbital eccentricity to be negligible during the merger phase. The inspiral part of the above waveform family was based on certain -model, introduced in Ref. Hinder et al. (2010), that adapted GW phasing formalism of Refs. Damour et al. (2004); Königsdörffer and Gopakumar (2006). Additionally, a preliminary comparison with two numerical relativity (NR) waveforms was also pursued in Ref. Huerta et al. (2017). An improved version of the above family was presented in Ref. Hinder et al. (2018) that employed certain quasi-circular merger waveform and which can reproduce their NR simulations for any mass ratio below . These waveform families are expected to model GWs from eccentric coalescence when initial eccentricities were usually below . Very recently, another time domain IMR family was introduced in Ref. Huerta et al. (2018). This detailed effort combined various elements from post-Newtonian, self-force and black hole perturbation approaches in tandem with NR simulations to model GWs from moderately eccentric non-spinning BH binary coalescence. The resulting IMR waveforms were validated with many NR simulations for eccentric binary BH mergers lasting around ten orbits with mass ratios below and initial eccentricities below . The eccentric binary BH coalescence is also explored in the framework of the Effective-One-Body (EOB) approach Damour and Nagar (2016). A formalism to incorporate orbital eccentricity in the existing EOB approach to model quasi-circular compact binary coalescence is presented in Ref. Hinderer and Babak (2017). Additionally, Ref. Cao and Han (2017) presented an EOB waveform family that incorporated elements of 2PN accurate eccentric orbital description while comparing with few NR simulations for eccentric binary BH coalescence. In contrast, the LIGO Scientific Collaboration (LSC) adapted Ref. East et al. (2013) that provided a crude IMR prescription to model GW signals from merging highly eccentric compact binaries. This was employed to probe the ability of few LSC algorithms to extract burst-like signals in the LIGO data Tiwari et al. (2016). Further, some of us developed a ready-to-use ‘effective eccentric variant’ of IMRPhenomD waveform to constrain the initial orbital eccentricity of the GW150914 black hole binary. This was pursued to justify the assumption of binary evolution along circular orbits for the event Abbott et al. (2016d). A crucial ingredient of the above IMR waveform family involved an eccentric version of TaylorF2 approximant that incorporated in its Fourier phase the leading-order eccentricity corrections up to 3PN order. The present paper provides fully analytic frequency domain interferometric response function relevant for GW data analysis of nonspinning compact binaries inspiraling along moderately eccentric PN-accurate orbits.
Our computation is aimed at extending the widely used TaylorF2 approximant that provides analytic frequency domain GW templates for compact binaries inspiraling along quasi-circular orbits Buonanno et al. (2009). This waveform family employs the method of stationary phase approximation (SPA) to compute analytically Fourier transform of temporally evolving GW polarization states, and , for quasi-circular inspirals. The popular LSC approximant provides fully analytic Fourier domain GW response function that incorporates 3.5PN-accurate Fourier phase Buonanno et al. (2009).In other words, this approximant provides general relativistic corrections to GW phase evolution that are accurate to order beyond the dominant quadrupolar order, where is the orbital velocity. The present manuscript details our derivation of a fully analytic with 3PN-accurate Fourier phase with sixth order eccentricity contributions in terms of certain initial eccentricity at each PN order. Additionally, we include 1PN-accurate amplitude corrections and the effect of 3PN-accurate periastron advance on the Fourier phases.
To derive our eccentric approximant, we extend the post-circular scheme of Ref. Yunes et al. (2009) to higher PN orders. This scheme involves expanding the Newtonian accurate and as a power series in orbital eccentricity that requires analytic solution to the classic Kepler equation. We extend such a Newtonian approach by invoking a recent effort to solve analytically PN-accurate Kepler equation in the small eccentricity limit Boetzel et al. (2017). This detailed computation also provided analytic 1PN-accurate amplitude corrected expressions for and as a sum over harmonics in certain mean anomaly of PN-accurate Keplerian type parametric solution Boetzel et al. (2017). Additionally, the above PN-accurate decomposition explicitly incorporated the effect of periastron advance on individual harmonics, numerically explored using PN description in Ref. Tessmer and Gopakumar (2007). We combine such 1PN-accurate amplitude corrected and expressions that incorporated eccentricity contributions to sixth order at each PN order with the two beam pattern functions, and , to obtain fully analytic time domain GW response function . Our eccentric TaylorF2 approximant is obtained by applying the method of stationary phase approximation to such an analytic expression.
To obtain analytic expressions for several Fourier phases at their associated stationary points of , we require additional PN-accurate expressions. This involves deriving 3PN-accurate expression for the time eccentricity , present in the 3PN-accurate Kepler Equation Memmesheimer et al. (2004), as a bivariate expansion in terms of orbital angular frequency , its initial value and , the value of at . This lengthy computation extends to 3PN order, the idea of certain asymptotic eccentricity invariant at the quadrupolar order, introduced in Ref. Królak et al. (1995), and extended to 2PN in Ref. Tanay et al. (2016). In fact, we adapted the approach of Ref. Tanay et al. (2016) by employing the appropriately modified 3PN-accurate and expressions of Refs. Arun et al. (2009); Klein et al. (2018) to obtain 3PN-accurate bivariate expression for . A careful synthesis of the above listed PN-accurate expressions lead to a fully analytic frequency domain TaylorF2 approximant that included 1PN-accurate amplitude corrections and 3PN-accurate Fourier phases. An additional feature of our approximant is the inclusion of periastron advance effects to 3PN order. To explore GW data analysis implications of these features, we perform preliminary match computations Damour et al. (1998). We conclude that the influences of periastron advance are non-negligible for moderately eccentric binaries, especially in the aLIGO frequency window. This observation should be relevant while constructing IMR waveform family for compact binaries merging along moderate eccentric orbits.
This paper is structured as follows. In Sec. II, we summarize the efforts of Refs. Yunes et al. (2009); Tanay et al. (2016) to obtain analytic with PN-accurate Fourier phase. The crucial inputs to construct our eccentric TaylorF2 approximant is also listed in this section. Our approach and crucial expressions to implement our eccentric approximant that incorporates eccentricity contributions up to to 3PN are presented in Sec. III. A brief summary and possible extensions are listed in Sec. IV while detailed expressions, accurate to are given in Appendix C
II Post-circular extensions to circular inspiral templates
We begin by reviewing two key efforts to include the effects of orbital eccentricity onto the circular inspiral templates Królak et al. (1995); Yunes et al. (2009). This involves listing in Sec. II.1 the steps that are crucial to compute analytic frequency domain GW response function with quadrupolar amplitudes and PN-accurate Fourier phase in some detail. Various lengthy expressions, extracted from Refs. Boetzel et al. (2017); Arun et al. (2009); Klein et al. (2018), are listed in Sec. II.2 that will be crucial to compute the time domain response function for eccentric binaries while incorporating effects of periastron advance, higher order radiation reaction and amplitude corrections.
II.1 Quadrupolar order with PN-accurate Fourier phase
Following Thorne (1987), we may express the GW interferometric response function as
[TABLE]
where are the two detector antenna patterns. These quantities depend on , the right ascension and declination of the source, and certain polarization angle Thorne (1987). For eccentric inspirals, the explicit expressions for the quadrupolar order GW polarization states, and , are given by Eqs. (3.1) of Ref. Yunes et al. (2009). It is rather straightforward to express these Newtonian accurate expressions as a sum over harmonics in terms of the mean anomaly . The resulting expressions read
[TABLE]
where denotes the luminosity distance while the symmetric mass ratio of a binary consisting of individual masses and is defined to be while the total mass . Further, we use the commonly used dimensionless PN expansion parameter where , and are the gravitational constant, the speed of light in vacuum and the orbital angular frequency, respectively. The Newtonian accurate amplitudes, and , are written as power series in orbital eccentricity whose coefficients involve trigonometric functions of the two angles that specify the line of sight vector in a certain inertial frame. The derivation of these expressions is detailed in Ref. Yunes et al. (2009) and the required inputs are obtained by adapting a standard analytic approach to solve the classical Kepler equation in terms of the Bessel functionsColwell (1993).
With the help of Eqs. (1) and (2), we obtain interferometric strain for GWs from eccentric binaries as
[TABLE]
where and . The two new functions, and , are defined as and , respectively as in Ref. Yunes et al. (2009). We impose the effects of GW emission on the above strain by specifying how and , being the orbital frequency, vary in time. In Ref. Yunes et al. (2009), the temporal evolutions of and are governed by the following Newtonian (or quadrupolar) equations that were adapted from Refs. Peters and Mathews (1963); Peters (1964); Junker and Schaefer (1992).
[TABLE]
It is customary to solve these two coupled differential equations numerically to obtain and and hence temporally evolving . Interestingly, earlier efforts provided certain analytic way for obtaining temporal evolution for and that mainly involves the usage of hypergeometric functions Moore et al. (2018); Mikóczi et al. (2012); Pierro et al. (2002, 2001)
However, it is possible to obtain analytic frequency domain counterpart of the above as demonstrated in Ref. Królak et al. (1995); Yunes et al. (2009). This traditional approach involves the method of SPA, detailed in Ref. Bender and Orszag (1999), to compute analytically the Fourier Transform of . This was essentially demonstrated at the leading order in initial eccentricity in Ref. Królak et al. (1995) and later extended to in Ref. Yunes et al. (2009). Following Refs. Królak et al. (1995); Yunes et al. (2009), we write
[TABLE]
where the overall amplitude and the amplitudes of Fourier coefficients are given by
[TABLE]
In the approach of stationary phase approximation, the crucial Fourier phase is given by
[TABLE]
where stands for . Note that one needs to evaluate the above integrals at appropriate stationary points , defined by .
To obtain a fully analytic ready-to-use expression for , we need to follow few additional steps. Clearly, we require to specify the frequency evolution of with the help of above Eqs. (4). The structure of these equations for and allows us to write and it turns out that depends only on . This allows to integrate analytically the resulting equation. The resulting expression can be written symbolically as where is the value of at the initial value, namely (see Eq. (62) in Ref. Damour et al. (2004) for the explicit form for ). Interestingly, one may invert such an expression in the limit to obtain in terms of and and it reads
[TABLE]
where is defined as . We note that the above result was first obtained in Ref. Królak et al. (1995) which influenced them to introduce the idea of an *asymptotic eccentric invariant *. This relation allows us to write in terms of and as
[TABLE]
It is now straightforward to compute analytically the indefinite integral for , namely
[TABLE]
that appears in Eq. (7) for . This leads to the following expression for , accurate to corrections:
[TABLE]
where and are the orbital phase at coalescence and the time of coalescence, respectively. Note that now stands for due to the use of the stationary phase condition. Additionally, we have re-scaled to ensure that while employing the above expression for , given by Eq. (8). Indeed, our expression is consistent with Eq. (4.28) of Ref. Yunes et al. (2009) that employs the chirp mass to characterize the binary. A number of extensions to the above result is available in the literature. In fact, Ref. Yunes et al. (2009) computed the higher order corrections to in terms of up to and extended to . Its PN extension, available in Ref. Tanay et al. (2016), provided 2PN corrections for that incorporated eccentricity corrections, accurate to at every PN order while Ref. Moore et al. (2016) computed 3PN-accurate that included leading order contributions.
A crucial ingredient to such PN extensions is the derivation of PN-accurate expression in terms of and . In what follows, we summarize the steps that are required to obtain 1PN-accurate expression for (see Ref. Tanay et al. (2016) for details). The starting point of such a derivation is the 1PN-accurate differential equations for and , obtainable from Eqs. (3.12) in Ref. Tanay et al. (2016). With these inputs, it is fairly straightforward to obtain the following 1PN accurate expression for that includes only the leading order contributions as
[TABLE]
The fact that term appears only at the 1PN order allows us to use the earlier derived Newtonian accurate relation to replace on the right hand side of the above equation. This leads to
[TABLE]
where . We can integrate this equation to obtain in terms of and . The exponential of the resulting expression and its bivariate expansion in terms of and result in
[TABLE]
We invert the above equation to obtain in terms of and after invoking the Newtonian accurate relation to replace the terms associated with the term. This inversion and the associated bivariate expansion in terms of and require that and . The resulting expression reads
[TABLE]
To obtain as a bivariate expansion in terms of the regular PN parameter and , we employ the fact that and this results in
[TABLE]
We are now in a position to obtain 1PN-accurate expression that includes contributions both at the Newtonian and 1PN orders with the help of 1PN-accurate expression that is accurate to terms. A straightforward computation leads to the desired expression which reads
[TABLE]
where the quantities and will have to be evaluated at the stationary point (see Ref. Tanay et al. (2016) for details). With the above equation, we explicitly listed our approach to compute PN-accurate that incorporates corrections at each PN order. In the present paper, we extend these computations to 3PN order while incorporating contributions at each PN order. These higher order corrections are included as we desire to to model GWs from moderately eccentric compact binary inspirals. In the next section, we provide crucial inputs that will be required to compute analytic 1PN-accurate amplitude corrected with 3PN-accurate Fourier phases.
II.2 Analytic PN-accurate amplitude corrected time domain eccentric GW templates
The previous section showed that we require analytic expressions for the two GW polarization states as a sum over harmonics to construct ready-to-use analytic . This influenced us to adapt Eqs. (44) and (45) in Ref. Boetzel et al. (2017) that provided analytic 1PN-accurate amplitude corrected which additionally included the effects of periastron advance on individual harmonics. This may be seen by a close inspection of appropriate terms in Eqs. (44),(45),(46) and (47) of Ref. Boetzel et al. (2017). To describe in detail how these improvements in GW polarization states change the harmonic structure of , we restrict our attention to quadrupolar order contributions to , given in Eqs. (44) and (45) of Ref. Boetzel et al. (2017). The explicit expressions for such ‘Newtonian‘ contributions to that include corrections read
[TABLE]
where , and provides the rate of periastron advance per orbit Damour et al. (2004). Further, we let , , and . Note that crucial ingredients to obtain above analytic expressions include developing approaches to solve PN-accurate Kepler equation and adapting them to derive PN-accurate relations to connect true and eccentric anomalies, detailed in Ref. Boetzel et al. (2017). A close inspection of the above two equations with Eqs. (3.3) and (3.4) of Ref. Yunes et al. (2009) reveals that the arguments of cosine and sine functions in above expressions involve and its multiples in addition to the usual orbital phase and its multiples. These additional contributions are clearly due to the periastron advance. It turns out that these additional angular contributions are sufficient to provide the numerically inferred side bands in the power spectrum of eccentric binaries due to the presence of Tessmer and Gopakumar (2007). This is why we explicitly included contributions to the above expressions as these contributions are required to reveal the underlying side band structure of waveforms due to the influence of periastron advance.
We re-write the above expressions for in a more compact form to explicitly show how various harmonics are affected by the advance of periastron. The resulting expressions read
[TABLE]
where we denoted the coefficient of harmonic at the quadrupolar (Newtonian) order for the polarization by while the coefficient of is indicated by . We adopt a rather heavy notation as it is amenable to higher PN order contributions which will be tackled below. In this convention, we represent the coefficient of that appears in the 1PN contributions to polarization state by . It should be obvious that stands for the harmonic variable while provides a measure of the shift that each harmonic experiences due to periastron advance. A close comparison of Eqs. (18) and (19) reveals that these coefficients are functions of and contain powers of . Moreover, the arguments of cosine and sine functions clearly show that the eccentricity induced higher harmonics are not mere multiples of , where is the PN-accurate mean motion. Clearly, this is due to the presence of non-vanishing contributions due to periastron advance. Interestingly, the plus polarization state does provide harmonics which are integer multiples of . It is not difficult to show that these Newtonian like terms arise from specific cosine functions with arguments , as evident from Eqs. (19). Further, it is possible to show that these contributions arise from contributions to , given by Eq. (F2a) in Ref. Boetzel et al. (2017) and therefore not influenced by the periastron advance. Interestingly, similar conclusions were obtained in Ref. Tessmer and Gopakumar (2007).
With the above inputs, we write the time-domain GW detector response function for eccentric inspirals as
[TABLE]
where the amplitudes of the cosine and sine functions are denoted by rather complicated symbols and . The definition of ensures that while . We list in Appendix A, the lengthy expressions for these quantities in terms of and eccentricity contributions, accurate to . We display up to contributions to demonstrate the full harmonic structure of the quadrupolar order GW polarization states. It turns out that contributions are zero by construction. This is mainly because the un-shifted harmonics only appear with the cosine terms, present in the polarization state. Invoking familiar trigonometric identities, we simplify the above equation and obtain
[TABLE]
where we introduce two new multi-index symbols and to ensure that detector strain can be written in terms of only cosine functions. Influenced by Ref. Yunes et al. (2009), these symbols are defined as and . We do not list explicit expressions for these quantities that are accurate to in eccentricity corrections as they can be easily obtained from our Eqs. (59) and (60).
A close inspection of above equations reveal that they provide GW response function for compact binaries moving along precessing eccentric orbits. To obtain temporally evolving associated with compact binaries inspiraling along precessing eccentric orbits, we need to specify how and vary in time due to GW emission. We adapt the phasing formalism, detailed in Refs. Damour et al. (2004); Tanay et al. (2016), to provide differential equations for these variables. And, for the time being, we will concentrate on the secular evolution of these variables. In other words, we will neglect GW induced quasi-periodic variations to orbital elements and angles, detailed in Ref. Damour et al. (2004). The 3PN-accurate secular evolution to and in the modified harmonic gauge that are accurate to are given by
[TABLE]
The explicit and PN order contributions to and that incorporates all the corrections are provided in the Appendix B. The differential equations for and are extracted from expressions, available in Refs. Arun et al. (2009); Klein et al. (2018) and are in the modified harmonic gauge. These papers provided above 3PN accurate expressions as sum of certain ‘instantaneous’ and ‘tail’ contributions
[TABLE]
The 3PN-accurate instantaneous contributions that depend only on the binary dynamics at the usual retarded time while the hereditary contributions are sensitive to the binary dynamics at all epochs prior to the usual retarded time Blanchet et al. (1995). The instantaneous contributions to are extracted from Eqs. (6.14),(6.15a),(6.15),(C6) and (C7) of Ref. Arun et al. (2009) while for such contributions originate from Eqs. (6.16),(6.19a),(6.19b),(C10) and (C11) in Ref. Arun et al. (2009). It should be obvious that we have Taylor expanded these equations around to obtain eccentricity contributions accurate to . The hereditary contributions to and are adapted from Eqs. (6.24c) and (6.26) of Ref. Arun et al. (2009) and they depend on a number of eccentricity enhancement functions. We employ such enhancement functions provided in Ref. Klein et al. (2018) for our computations.
We now have all the inputs to obtain the restricted time-domain to model GWs from non-spinning compact binaries inspiraling along precessing moderately eccentric orbits. To obtain such time domain templates, we numerically solve the above listed differential equations for and and impose their temporal evolution in the quadrupolar order GW response function, given by Eq. (II.2). We now move onto describe how we extend the quadrupolar order GW response function.
It should be obvious that we require a prescription to compute analytically PN-accurate amplitude corrected GW polarization states to improve the above listed quadrupolar order GW response function. Therefore, we adapt 1PN-accurate amplitude corrected and fully analytic expressions for , available in Ref. Boetzel et al. (2017), to compute GW response function for eccentric inspirals that incorporates PN contributions even to its amplitudes. We list below certain ingredients that will be crucial to write down analytic that incorporates 1PN-accurate amplitude corrections to while consistently keeping eccentricity contributions up to . We begin by displaying Eqs. (44) and (45) of Ref. Boetzel et al. (2017) as a single sum which reads
[TABLE]
Various PN order amplitude contributions take the following form
[TABLE]
where and we let to be the heavier of the two binary components. We do not list explicitly very lengthy expressions for these amplitudes. However, they can be easily extracted from the attached Mathematica notebook. The derivation of above lengthy expressions include developing analytic approaches to solve PN-accurate Kepler equation and PN-accurate relations connecting true and eccentric anomalies, detailed in Ref. Boetzel et al. (2017). Indeed, we have verified that these expressions reduce to their circular counterparts, provided in Ref. Blanchet et al. (1996).
The associated GW detector strain for eccentric binaries is given by
[TABLE]
where as expected, we have defined
[TABLE]
A further simplification is possible which requires, as expected, additional multi-index functions
[TABLE]
such that
[TABLE]
A cursory look at the above equation may give the impression that the summation indices in various sums are terminated in an arbitrary manner. Interestingly, we find a possible way to predict the maximum value that j index can take in each of the above summations. This is related to the argument of in each of these cosine series. We infer that the argument of can take a maximum value of six as we are restricting eccentricity contributions to sixth order in . This ensures that index can take maximum values of and at the Newtonian order in the above expression. In other words, in the above expression is given such that where value arises from the the argument of variable in various summations. It is easy to see that the above relation holds true even at 0.5 and 1PN orders and it provides a natural check on the structure of these higher order PN contributions to .
To obtain GW response function for eccentric inspirals, we need to incorporate temporal evolution in and , given by our earlier listed 3PN-accurate differential equations. The fact that we require to solve the above four coupled differential equations numerically ensures that our approach to obtain ready-to-use will be computationally expensive. This is clearly one of the motivation to obtain fully analytic for compact binaries inspiraling along moderately eccentric orbits. Fortunately, we are in a position to compute analytic amplitude corrected that incorporates 3PN-accurate Fourier phase while keeping eccentricity contributions accurate to sixth order in at every PN order.
III Analytic for eccentric inspirals with 1PN amplitude corrections
We first provide a detailed description of our approach to compute analytic Fourier transform of the restricted time domain inspiral family, given by Eq. (II.2). This will be followed by computing associated with Eq. (II.2). Preliminary data analysis implications of our analytic are probed in Sec. III.2.
III.1 Approach to compute Fourier transform of for compact binaries inspiraling along precessing eccentric orbits
We begin by listing the expanded version of our quadrupolar order , namely Eq. (II.2) with eccentricity contributions as
[TABLE]
Clearly, we see three distinct square brackets that contain three cosine functions with explicitly time dependent arguments, namely , and . Note that and experience implicit temporal evolution due to the GW emission induced variations to and . The main reason for displaying the above equation is to show explicitly how the periastron advance, defined by , influences the harmonic structure of in comparison with Eq. (4.21) of Ref. Yunes et al. (2009) or our Eq. (3).
We obtain an analytic Fourier domain version of the above equation with the help of the Stationary Phase Approximation, detailed in Bender and Orszag (1999). How this approach can be employed to compute for compact binaries spiraling along Keplerian eccentric orbits can be found in Sec. IV of Ref. Yunes et al. (2009). This approximation is quite appropriate for us as it provides a prescription to compute the asymptotic behavior of generalized cosine time series, as given by our Eq. (III.1). Without loss of any generality, we may write such a time series as
[TABLE]
where and as expected should be a product of slowly varying amplitude and a rapidly varying cosine function with argument . Due to the virtue of Riemann-Lebesgue lemma, as noted in Ref. Bender and Orszag (1999), the Fourier transform of becomes
[TABLE]
It is not difficult to gather that the argument of the exponential function vanishes at the stationary point such that . This allows us to invoke the approach of SPA to obtain the asymptotic behaviour of by the following expression:
[TABLE]
where the Fourier phase is defined as
[TABLE]
Note that and therefore its value at the stationary point should be . Interestingly, a rather identical computation can be done to obtain the Fourier transform of a similar sinusoidal time series to be .
To make operational the above expression for , we require an explicit expression for the above defined Fourier phase at the stationary point , namely
[TABLE]
This is done by defining such that and become
[TABLE]
where and are the orbital phase and time at coalescence. In the present context, is defined using our 3PN-accurate expression for given by Eq. (25). Additionally, we require 3PN-accurate expression, namely 3PN extension of Eq. (16), for computing these integrals analytically. The expression for obtained using Eq. (38) and (39) in (37) may be written as
[TABLE]
where . In the present context, we need to evaluate the above integral at a point of time where the orbital frequency is related to the Fourier frequency by . A close inspection of Eq. (III.1) reveals that our expression for the quadrupolar order time domain response function is structurally similar to the above displayed cosine time series and therefore we can easily adapt these results to obtain the Fourier transform of our quadrupolar order . However, the SPA based will have contributions from a number of distinct stationary points. This is primarily due to the fact that Eq. (III.1) consists of cosine functions of three different arguments, namely and . Note that there are only three distinct types of cosine arguments as we restricted our attention to the quadrupolar order GW response function for eccentric inspirals. However, we infer from our 1PN-accurate , given by Eq. (II.2), that there are nine distinct types of cosine functions with arguments where . The associated nine stationary points are computed by demanding that , where .
For computing Fourier transform of Eq.(III.1), we solve to get the relevant stationary points and this leads to
[TABLE]
where and this by definition is . The treatment of requires PN approximation as equals ( this is because ). We need to express in terms of and this leads to as . For computing Fourier phase analytically, we express as , where stands for the 3PN-accurate expression for that incorporates contributions accurate to . The resulting expression reads
[TABLE]
With the help of these inputs, the stationary points , where vanish, are given by
[TABLE]
In other words, the stationary phase condition is given by
[TABLE]
Rewriting using relation between and gives . We are now in a position to obtain analytic PN-accurate expressions for Fourier phases, associated with these stationary points. With Eq. (38) and (39), our Eq. (40) becomes
[TABLE]
Note that takes values [math] and as we are dealing with quadrupolar order GW response function given by Eq.(III.1). However, varies from [math] to if the underlying GW response function contains 1PN-accurate amplitude corrections that include at each PN order eccentricity corrections accurate to . Further, we do not display here 3PN-accurate expression for that includes the leading order corrections, listed as Eqs. (6.7a) and (6.7b) in Ref. Moore et al. (2016). However, we do list below the explicit 3PN-accurate that incorporates leading order contributions at each PN order:
[TABLE]
A few comments are in order. To obtain the circular limit, we require to impose in and let . This is indeed due to the fact that does not go to zero in the circular limit. Additionally, we have verified that the resulting expression in the limit is identical to 3PN accurate version of Eq. (6.26) in Ref. Moore et al. (2016) while neglecting the spin contributions. It is natural to expect that the version of our above equation should be identical to Eq. (6.26) of Ref. Moore et al. (2016). This is because this equation indeed provided quadrupolar with 3PN-accurate Fourier phase while incorporating leading order corrections at each PN order by extending the post-circular approach of Ref. Yunes et al. (2009). However, our expression for is not identical to Eq. (6.26) of Ref. Moore et al. (2016). This is because that effort did not incorporate the effect of periastron advance while obtaining analytic expression for their Fourier phase. A close inspection of the version of our Eq. (44) reveals that it will still be influenced by our PN-accurate expression for . This clearly shows that it is rather impossible to remove the effect of periastron advance from our Eq. (44). Therefore, our Eq. (III.1) will be different from Eq. (6.26) of Ref. Moore et al. (2016) which, as noted earlier, neglected the effect of periastron advance. The differences may be attributed to the physical fact that we are providing an analytic expression for associated with compact binaries inspiraling along PN-accurate eccentric orbits. In contrast, Ref. Moore et al. (2016) models inspiral GWs from compact binaries spiraling in along Newtonian orbits though frequency evolution in both cases are fully 3PN accurate. Additionally, we are unable to match with the 2PN order results of Ref. Tanay et al. (2016) due to similar reasons. We note in passing that the explicit 3PN-accurate contributions to and the associated 3PN accurate expression are provided in the Appendix C .
We now employ fully the final result of SPA, namely Eq. (36), to compute the Fourier transform of Eq. (III.1). This gives us
[TABLE]
where we have used the quadrupolar (Newtonian) order differential equation for the orbital frequency, available in Refs. Peters and Mathews (1963); Damour et al. (2004), to compute the amplitudes of . Note that we require to employ the earlier defined stationary points to replace . In practice, we employ the unperturbed stationary points, namely , while evaluating the amplitudes of .
In what follows, we collect the above pieces together to display the quadrupolar order that incorporates fourth order orbital eccentricity contributions while including the effects due to 3PN-accurate frequency, eccentricity evolution and periastron advance as
[TABLE]
where the Fourier amplitudes are now given by
[TABLE]
and takes values 0 and 2. A crucial expression that will be required to operationalize the above , namely 3PN-accurate expression for in terms of and , is listed as Eq. (63) in the Appendix. Note that the approach to obtain such an expression for is detailed in Ref. Tanay et al. (2016) and briefly summarized in Sec. II.1. Finally, the fully 3PN-accurate expression for that incorporates fourth order orbital eccentricity contributions at each PN order is displayed as Eq. (67) in the Appendix. It should be noted that the approach of SPA demands the evaluation of Fourier amplitudes, and Fourier phases, at .
We have extended these calculations by including 1PN-accurate amplitude corrections to and with the help of Eqs. (27),(28a),(28b) and (28c). Additionally, we have included initial eccentricity corrections, accurate to , in our 3PN-accurate and expressions. We note in passing that these expressions are available in the accompanying Mathematica file. The resulting expression for may be symbolically written as
[TABLE]
In the above expression, the Fourier amplitudes are given by
[TABLE]
where the superscript takes values and in our amplitude corrected . Further, we have used the 1PN-accurate differential equation for the orbital frequency while obtaining the Fourier amplitude expressions. This expression, adaptable from Eqs. (B8a) and (B9a) of Ref. Tanay et al. (2016), reads
[TABLE]
The explicit expressions for and that incorporate the next to leading order corrections at each PN order, as noted earlier, are listed in the Appendix C.
We move on to contrast our approach with other attempts in the literature. The Sec. VI of Ref. Yunes et al. (2009) indeed sketched a road map to include PN corrections to their Newtonian waveform family. This road map included a suggestion to incorporate the effect of periastron advance into their quadrupolar order GW polarization states, influenced by Ref. Damour et al. (2004). Their suggestion involves splitting the orbital phase evolution into two parts where one part remains linear in the mean anomaly while the other part is periodic in . These considerations influenced them to re-write our Eq. (3) essentially to be
[TABLE]
where stands for the 1PN accurate expression for , given by , expanded to the sixth order in (see our Eq.(III.1)). It is not difficult to see that the associated SPA based Fourier phase takes the following form:
[TABLE]
where
[TABLE]
It turned out that by construction. The use of in the above Fourier phase expression essentially ensures that the suggestion of Ref. Yunes et al. (2009) leads to what is detailed in Ref. Tanay et al. (2016). Note that Ref. Tanay et al. (2016) provided in terms of infinite set of harmonics with quadrupolar order amplitudes and 2PN-accurate Fourier phase. We observe that Ref. Yunes et al. (2009) indeed commented on the absence of side bands in their prescription in comparison with what was reported in Refs. Moreno-Garrido et al. (1994); Mediavilla et al. (1995) and suggested future investigations to clarify the issue. In contrast, the present investigation employs Eqs. (II.2) that explicitly incorporates the effect of periastron advance both in the amplitude and phase of GW polarization states, as detailed in Ref. Boetzel et al. (2017). The use of such an expression ensures that our analytic Fourier domain expression does indeed contain periastron advance induced frequency side bands. Additionally, Refs. Gondán et al. (2018); Mikóczi et al. (2012) employed the dominant order periastron advance induced decomposition of Fourier phases, associated with quadrupolar order gravitational waveform, while exploring LISA and aLIGO relevant parameter estimation studies. A close comparison of Eqs. (B10) and (B11) of Ref. Mikóczi et al. (2012) and Eqs. (35) and (36) of Ref. Gondán et al. (2018) with our Eq. (3.10) reveals fairly identical expressions for the Fourier phases.
These considerations allowed us to state that our expression for , given by Eqs. (III.1),(III.1),(63),(67), provides analytic PN-accurate Fourier domain templates for compact binaries inspiraling along PN-accurate precessing eccentric orbits. We are now in a position to explore basic GW data analysis implications of our inspiral templates.
III.2 Preliminary GW data analysis implications
We employ the familiar match computations to probe basic GW data analysis implications of our PN-accurate inspiral templates. Following Ref. Damour et al. (1998), the match between members of two waveform classes, namely signal and template , is computed by maximizing a certain overlap integral with respect to the kinematic variables of the template waveform. In other words,
[TABLE]
where and are the detector arrival time and the associated arrival phase of our template. The overlap integral involves the interferometer-specific normalized inner product between members of and families; it reads
[TABLE]
where and are the Fourier transforms of the and inspiral waveforms. Further, denotes the one-sided power spectral density of the detector noise. In the following, we employ the zero-detuned, high power (ZDHP) noise configuration of Advanced LIGO at design sensitivity Harry and LIGO Scientific Collaboration (2010). In our estimates, we let be Hz, corresponding to the lower cut-off frequency of Advanced LIGO. The upper frequency limit is chosen to be the usual of the last stable circular orbit. We have verified that orbital eccentricities of compact binaries reduce to well below at , thereby justifying the use of the last stable circular orbit frequency for the upper frequency limit.
We require additional steps to operationalize our inspiral templates while performing the computations. Clearly, these waveform families should only be implemented within the physically allowed frequency intervals. This is to ensure that the many higher harmonics present in these waveform families do not cross the above listed upper frequency limit. Influenced by Ref. (Yunes et al., 2009), we invoke the Unit Step function () to operationalize our inspiral templates. This step function allows us to appropriately terminate the waveform as for and zero otherwise. The structure of our quadrupolar amplitude inspiral family, given by Eqs. (III.1), compels us to invoke functions such that
[TABLE]
Note that we have appropriately shifted the upper frequency limits to ensure that higher harmonics are suitably terminated. While implementing our we have encountered the violation of the stationary phase condition, namely Eq. (43), at a few Fourier frequencies corresponding to lower harmonic indices (). We infer that the periastron advance induced shift of these harmonics can lead to negative GW frequencies. Therefore, we have discarded such Fourier components. Interestingly, Ref. Tessmer and Gopakumar (2007) showed that these harmonics provide negligible contributions to the GW power spectrum, which may be used to justify our neglect of such Fourier components in the implementation of our waveform families. The above steps ensure smoothly varying templates which we will use in the following to pursue match computations. We provide three frequency series of the same length (corresponding to and inspiral families and the ZDHP noise power spectral density) and employ a routine from the free and open software package PyCBC Nitz et al. (2019) to compute various estimates.
We qualify the implications of our match estimates on GW data analysis by considering the threshold , denoted in the presentation of results in Figs. 1, 2 and 3 by solid black lines. This limit corresponds to a loss of less than of all signals in the matched filter searches. In regions of parameter space where the computed matches are high, i.e., , waveform models are generally considered both effectual templates for the detection of fiducial GW signals and reasonably faithful in the estimation of GW source parameters Damour et al. (1998). However, even if larger than , certain errors in the model waveform (due to unmodeled effects of, e.g., eccentricity) may become distinguishable from noise at high signal-to-noise ratio (SNR) and can affect the accuracy of source parameter estimation. Negligible systematic errors in parameter estimation – despite differences between the true signal waveform and the template model – can be guaranteed only if , the so-called indistinguishability criterion Creighton and Anderson (2011). In other words, such systematic errors in the estimated source parameters may become significant when the mismatch and clearly depend on the amplitude of the signal. In the following analysis, we let the signal-to-noise ratio of our fiducial GW signals be SNR (corresponding to the SNR of the binary neutron star inspiral GW170817) and probe the distinguishability of certain effects in our model waveforms for inspiraling eccentric binaries. In the inset plots of Figs. 1 and 2, we zoom into those regions of parameter space where we can expect waveform uncertainties to become indistinguishable from noise for SNR; the corresponding distinguishable limit is represented by the dashed black lines.
We first probe the importance of higher-order eccentricity corrections in the GW phasing. For this purpose, we let the signal family to be our quadrupolar-order , with a 3PN-accurate Fourier phase that includes next-to-next-to leading order, eccentricity corrections at each PN order. The template family is given by a quadrupolar-order in the low-eccentricity limit, incorporating only the leading-order, eccentricity contributions in the 3PN-accurate Fourier phase. We consider the traditional non-spinning compact binary sources relevant for Advanced LIGO: namely, binary neutron stars (NS-NS), NS-BH systems and binary black holes (BH-BH), with NS and BH components of and , respectively. For each of these three configurations, we compute the match between signal and template waveforms for different values of the initial orbital eccentricity between [math] and (defined at the cut-off frequency Hz). Fig. 1 suggests that the importance of higher-order eccentricity corrections for GW data analysis is strongly dependent on the total mass of an eccentric compact binary source. Given the same but for configurations with increasing total mass, we find that templates restricted to leading-order eccentricity corrections become increasingly faithful representations of those inspiral waveforms that include higher-order eccentricity effects at each PN order. This is expected, as compact binaries with higher total mass provide a smaller number of inspiral GW cycles in the frequency window of Advanced LIGO. Therefore, these systems require larger initial eccentricities to bring on a substantial de-phasing and subsequent mismatch between our inspiral signal and template families. Fig. 1 indicates that a waveform model restricted to only leading-order eccentricity corrections would be an effectual template family for the detection of GWs from even moderately eccentric inspirals (with and for our traditional NS-NS and BH-BH binaries, respectively). However, the inset of Fig. 1 suggests that waveform effects of higher-order eccentricity corrections become distinguishable from detector noise at significantly lower initial eccentricities ( and for GWs from NS-NS and BH-BH systems with SNR). In this region of parameter space, we should expect systematic errors in source parameter estimation with inspiral templates that are accurate only to leading order in eccentricity . The inclusion of higher-order eccentricity corrections in waveform modeling is therefore desirable for an accurate follow-up of eccentric GW signals.
We move on to probe data analysis implications of including the effect of periastron advance in our eccentric inspiral waveforms . In our match calculation , the signal waveforms employ our quadrupolar-order given by Eq. (III.1), including both and effects to the sixth order in at each PN order. We build a template family that neglects effects of periastron advance, by extending to 3PN order previously developed eccentric inspiral waveforms (provided with 2PN-accurate Fourier phase in Ref. Tanay et al. (2016)). In other words, we construct quadrupolar templates with the help of Eq. (5) and the 3PN extension of our Newtonian Eq. (11) for while incorporating all corrections at each PN order. Additionally, we evaluate the Fourier phase at the unperturbed stationary point Yunes et al. (2009). It is important to note that such a template waveform family ignores the effect of periastron advance in its Fourier phase evolution. We consider the same NS-NS, NS-BH and BH-BH systems as before and compute the match between signal and template waveforms for discrete values of initial orbital eccentricity at Hz, . From our results, presented in Fig. 2, we learn that the significance of periastron advance effects for GW data analysis is rather independent of the total mass of the source, with similar match estimates for all three traditional compact binaries under consideration. Periastron advance starts to influence the effectualness of GW templates for detection only for systems that have eccentricities at Hz. This agrees with our observation that -induced modulations in the inspiral waveforms presented in Fig. 5 and 6 of Ref. Damour et al. (2004) become clearly visible only for moderate values of initial orbital eccentricity. However, we can expect systematic biases in the source parameter estimation for much smaller values of orbital eccentricity. The inset of Fig. 2 suggests that periastron advance effects in an eccentric GW signal with SNR would already become distinguishable from noise for eccentricities at Hz, leading to systematic errors in the recovered source parameters when waveform models neglect periastron advance.
Lastly, we explore the relevance of PN-accurate amplitude corrections while constructing realistic analytic Fourier-domain waveforms for eccentric inspirals. For these estimates, we invoke as the expected GW signal our 1PN-accurate amplitude corrected , given by Eq. (III.1), including the effects of 3PN-accurate periastron advance, frequency and eccentricity evolution accurate to sixth order in orbital eccentricity. For the template family, we are utilizing a quadrupolar-order , given by Eq. (III.1), that includes the same order effects of 3PN-accurate periastron advance and 3PN-accurate frequency and eccentricity evolution as above. We consider five compact binary configurations with a fixed total mass and varying mass ratios . For each of these configurations, we pursue match computations for different choices of initial orbital eccentricity at Hz, resulting in Fig. 3. We observe that amplitude corrections are rather unimportant while constructing template waveforms for equal-mass binaries in eccentric orbits. This is expected, as the dominant amplitude corrections – appearing at 0.5PN order in Eq. (III.1) – are proportional to and therefore vanish for equal-mass binaries. Our plots suggest that the effect of amplitude corrections on the faithfulness of eccentric inspiral waveforms crucially depends on the mass ratio of a binary system, with rapidly dropping below the critical value of 0.97 as , even for systems with negligible initial eccentricities. This is a familiar result from the modeling of compact binary inspiral along circular orbits and points to the relevance of higher modes for GWs from binaries with asymmetric masses Van Den Broeck and Sengupta (2007). In other words, our plots in Fig. 3 essentially confirm previous literature that compared restricted and amplitude-corrected for quasi-circular inspiral. Interestingly, we find that the -dependent effect of amplitude corrections on the faithfulness of eccentric inspiral waveforms is largely unaffected by the value of initial eccentricity .
IV Conclusions
We have provided fully analytic PN-accurate Fourier domain gravitational waveforms for compact binaries inspiraling along precessing moderately eccentric orbits. Our inspiral approximant contains 1PN-accurate amplitude corrections and its Fourier phase incorporates the effects of 3PN-accurate periastron advance and GW emission. Additionally, the eccentricity effects are accurate to sixth order in at each PN order. We infer from our analytic waveform expression that the orbital eccentricity induced higher harmonics are no longer integer multiples of orbital frequency due to the influence of periastron advance. This substantiates and extends what is detailed in Ref. Boetzel et al. (2017) for compact binaries inspiraling along PN-accurate precessing eccentric orbits. Preliminary GW data analysis implications of our waveforms are probed with the help of the usual match computations.
In what follows, we provide a step-by-step summary of our effort.
We start from our Eqs. (18) and (19) that provide quadrupolar order GW polarization states from compact binaries in PN-accurate eccentric orbits as a sum over various harmonics. 2. 2.
With above inputs, we compute the time domain GW detector response function and express it as a summation of several cosine functions whose arguments are sum of integer multiples of and associated with the orbital and periastron motions. Amplitudes of these functions are expressed in terms of , and the angles that specify the antenna patterns and the direction of the orbital angular momentum vector. The quadrupolar version of that explicitly incorporates the next to leading order corrections is given by Eq. (II.2) and associated expressions like Eqs. (59) and (60). Its 1PN extension is symbolically provided by Eq. (II.2) and the accompanying Mathematica file provide the explicit expressions for various PN coefficients while incorporating corrections. 3. 3.
We also provide a prescription to obtain temporally evolving for compact binaries inspiraling due to 3PN accurate GW emission along precessing 3PN accurate orbits of moderate eccentricities. This involves imposing temporal evolution for and with the help of PN accurate differential equations. The relative 3PN accurate equations for and are due to the emission of GWs, as evident from our Eqs. (25) and (26). The conservative 3PN accurate differential equation for arises essentially due to periastron advance as evident from Eq. (24). The differential equation for is kinematical in nature as . 4. 4.
The structure of the time domain response function allows us to involve the method of stationary phase approximation to compute its Fourier transform. The crucial Fourier phases and the associated ’nine’; stationary points may be concisely written as , where takes values . The nine stationary points, associated with 1PN accurate amplitude corrected , essentially provide relations between the orbital and Fourier frequencies , where is related to the rate of periastron advance per orbit. The explicit expression for the resulting 3PN -accurate Fourier phases with leading order initial eccentricity corrections are provided by Eq. (III.1). Gathering various results, we obtain Eqs. (III.1) and (48) that provide the quadrupolar order while incorporating fourth order orbital eccentricity contributions along with the effects due to 3PN-accurate frequency, eccentricity evolution and periastron advance. Additionally, we have extended these results by including 1PN accurate amplitude corrections and six order eccentricity contributions. 5. 5.
A crucial ingredient to obtain a fully analytic involves a derivation, detailed in Sec. II, that provides PN accurate analytic expression for in terms of . We have obtained 3PN accurate expression for by extending the post-circular scheme of Refs. Yunes et al. (2009); Tanay et al. (2016).
A number of extensions are possible. Influenced by Refs. Königsdörffer and Gopakumar (2005); Kidder (1995), we are incorporating the effects of leading order aligned spin-orbit and spin-spin interactions into these waveforms. It will be interesting to explore data analysis implications of our present waveforms. A possible avenue is to explore the astrophysical implications of using PN-accurate periastron advance contributions that depend both on and , influenced by Refs. Mikóczi et al. (2012); Seto (2001). There are on-going efforts to construct analytic IMR templates to model eccentric compact binary coalescence Huerta et al. (2017, 2018). The present waveform family will be relevant to construct IMR templates for moderately eccentric compact binary mergers which can be used to extract orbital eccentricity and periastron advance as done in Ref Mroué et al. (2010). Efforts are on-going to obtain various constructs, using elements of our post-circular Fourier domain approximant, that should allow us to make comparisons with brand new PN-accurate frequency domain waveform family, developed in Refs. Moore et al. (2018); Moore and Yunes (2019) for moderate eccentricities.
V Acknowledgements
We thank Yannick Boetzel for helpful discussions, suggestions and providing us the lengthy eccentricity enhancement functions. We are grateful to Marc Favata and Blake Moore for their helpful comments. M. H. acknowledges support from Swiss National Science Foundation (SNSF) grant IZCOZ0_177057. We have used software packages from PyCBC Nitz et al. (2019) and Matplotlib Hunter (2007) to compute and plot match estimates.
Appendix A and coefficients
We list the and coefficients appearing in Eq.(21). The relevant expressions read
[TABLE]
The counterparts of above expressions read
[TABLE]
Appendix B 3PN accurate and
We give here the 3PN accurate expressions for temporal evolution of and for obtaining associated with compact binaries inspiraling along precessing eccentric orbits. 1PN accurate and with eccentricity corrections are given by Eq.(25) and Eq.(26) respectively. The 1.5PN - 3PN contributions to appearing in Eq. 25 with corrections are,
[TABLE]
where stands for the Euler-Mascheroni constant. The 1.5PN - 3PN contributions to appearing in Eq. 26 with corrections are,
[TABLE]
Appendix C 3PN accurate analytic expressions for and
We display explicit expressions for 3PN-accurate and Fourier phases that incorporate next to leading order corrections at each PN order. These expressions along with Eqs. (III.1), (48), (31), (59) and (60) are required to make operational the fully analytic frequency domain quadrupolar order GW response function for eccentric inspirals that includes corrections at every PN order. We begin by listing explicit expression for the 3PN accurate in terms of and . The underlying computation is detailed in Ref. Tanay et al. (2016) and requires 3PN-accurate expressions for and , given by Eqs. (25) and (26). The fully 3PN accurate expression that accounts for all the contributions read
[TABLE]
The coefficients with next to leading order eccentricity corrections at each PN order can be listed as,
[TABLE]
Due to the lengthy nature of 3PN order terms in , we split it in two parts as
[TABLE]
The explicit form of these two contributions are
[TABLE]
We have pursued careful checking of our results with what is available in Ref. Tanay et al. (2016) and observed a slight typo in the contributions for the expression (Eq. (A6e)) of Ref. Tanay et al. (2016). The independent term present in the coefficient of should be instead of . Note that the above expression is required while computing the Fourier amplitudes . Additionally, it is a crucial ingredient while computing analytic expression for our Fourier phases . It should be obvious that its frequency dependence is encapsulated in and the PN expansion parameter .
We now display our 3PN accurate closed form expression for the Fourier phases . Recall that nine different Fourier phases appear in our 1PN accurate amplitude corrected expression, given by Eq. (III.1). To circumvent the task of displaying all the 9 different Fourier phases separately, we provide a general expression for these phases as where . It is not very difficult to obtain from by replacing with appropriate sign in the expression. The general expression for 3PN accurate Fourier phase reads
[TABLE]
Various PN coefficients with next to leading order eccentricity contributions are given by
[TABLE]
For the ease of presentation, we split the 3PN contributions to in to three parts
[TABLE]
Various contributions to are given by,
[TABLE]
Let us emphasize that the above expression indeed provides all the required Fourier phases, ’s that appear in Eq. (III.1) for . For instance, Fourier phases present in the quadrupolar order , namely, , and are obtained by putting in Eq. (67) , respectively. Further, one should evaluate these Fourier phases at the correct stationary points and this requires us to use x=\bigg{\{}\frac{G\,m\,2\,\pi\,f}{c^{3}\,\left(j-(j\pm n)k^{(6)}_{(3)}\right)}\bigg{\}}^{2/3}. We note in passing that the 3PN accurate and expressions along with 1PN accurate Fourier amplitudes while incorporating eccentricity corrections to at each PN order can be found in the attached Mathematica notebook.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1Abbott et al. (2016 a) B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, et al., Physical Review Letters 116 , 131103 (2016 a), eprint 1602.03838.
- 2Acernese et al. (2015) F. Acernese, M. Agathos, K. Agatsuma, D. Aisa, N. Allemandou, A. Allocca, J. Amarni, P. Astone, G. Balestri, G. Ballardin, et al., Classical and Quantum Gravity 32 , 024001 (2015), eprint 1408.3978.
- 3Abbott et al. (2016 b) B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, et al., Physical Review Letters 116 , 061102 (2016 b), eprint 1602.03837.
- 4Abbott et al. (2016 c) B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, et al., Physical Review Letters 116 , 241103 (2016 c), eprint 1606.04855.
- 5Abbott et al. (2017 a) B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, et al., Physical Review Letters 118 , 221101 (2017 a), eprint 1706.01812.
- 6Abbott et al. (2017 b) B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, et al., Ap J Letters 851 , L 35 (2017 b), eprint 1711.05578.
- 7Abbott et al. (2017 c) B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, et al., Physical Review Letters 119 , 141101 (2017 c), eprint 1709.09660.
- 8Abbott et al. (2017 d) B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, V. B. Adya, et al., Physical Review Letters 119 , 161101 (2017 d), eprint 1710.05832.