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Compact binary coalescences: constraints on waveforms

Abstract

Gravitational waveforms for compact binary coalescences (CBCs) have been invaluable for detections by the LIGO-Virgo collaboration. They are obtained by a combination of semi-analytical models and numerical simulations. So far systematic errors arising from these procedures appear to be less than statistical ones. However, the significantly enhanced sensitivity of the new detectors that will become operational in the near future will require waveforms to be much more accurate. This task would be facilitated if one has a variety of cross-checks to evaluate accuracy, particularly in the regions of parameter space where numerical simulations are sparse. Currently errors are estimated by comparing the candidate waveforms with the numerical relativity (NR) ones, which are taken to be exact. The goal of this paper is to propose a qualitatively different tool. We show that full non-linear general relativity (GR) imposes an infinite number of sharp constraints on the CBC waveforms. These can provide clear-cut measures to evaluate the accuracy of candidate waveforms against exact GR, help find systematic errors, and also provide external checks on NR simulations themselves.

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Fig. 1

Notes

  1. 1.

    This notion is weaker than Penrose’s original definition of asymptotic simplicity which requires that every null geodesic in M should have endpoints on \(\mathfrak {I}^{\pm }\); our conditions refer only to properties of spacetime geometry near infinity.

  2. 2.

    It has a clear-cut geometric meaning in the conformally completed spacetime: \(N_{ab}\) is the conformally invariant part of the curvature of the intrinsic connection on \(\mathfrak {I}^{+}\) [55].

  3. 3.

    If we change the Bondi-frame, the vector field \(\mathring{n}^{a}\) is rescaled as in Eq. (1). Since the supertranslation is given by \(\xi ^{a} = f \mathring{n}^{a} = \tilde{f} \mathring{\tilde{n}}^{a}\), the labels f and \(\tilde{f}\) in the two frames are related by \(\tilde{f} = \omega f\). In the Penrose conformal picture, f is not a scalar but carries a conformal weight 1. It turns out that the notion of a BMS translation is invariant with respect to this change of the Bondi-frame, but the notion of a ‘pure supertranslation’ is not.

  4. 4.

    Throughout, we assume that if a field \(F(u,\theta ,\phi ) = O(1/|u|^{\alpha })\)—i.e., if \(|u|^{\alpha }F(u,\theta ,\varphi )\) admits smooth limits \(F_{\pm }(\theta ,\varphi )\) as \(u\rightarrow \pm \infty \)—then its mth  u-derivative, \(\partial _{u}^{m} F(u,\theta ,\varphi )\) is \(O(1/|u|^{m+\alpha })\).

  5. 5.

    This transformation property follows from the conformal rescaling of \(\mathring{n}^{a}\) given in Eq. (1), the normalization condition \(g_{ab}\ell ^a n^b=-1\) in physical spacetime, the definition of \(\varPsi _{2}^{\circ }(u,\theta ,\phi )\) given in Eq. (6) and the fact that the radial coordinate changes via \(r'=\gamma \,(1-\tfrac{\mathbf {v}}{c}\cdot \hat{x})\, r +O(1)\) under a boost.

  6. 6.

    More precisely, we need to work with a finite time analog of (30), obtained by integrating (28) against \(Y_{2,\,0}(\theta ,\varphi )\) and setting \(u_{1}=-\infty \). In our setup \(\sigma ^{o}\) vanishes at \(u=-\infty \). Therefore, this procedure expresses \(\oint \mathrm {d}^{2} \mathring{V}\, Y_{2,0} (\theta ,\varphi )\,\sigma ^{\circ }(u_{2},\theta ,\varphi )\) as a sum of the (finite-time) non-linear memory that is monotonic, and a (finite-time) linear memory term which turns out to be oscillatory.

  7. 7.

    We chose a time-translation just for definiteness: the argument continues to hold if the ‘angle dependent translation’ is generic.

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Acknowledgements

We would like to thank K. G. Arun, A. Gupta, and B. Sathyaprakash for comments on the manuscript; Bala Iyer and Luc Blanchet for extensive correspondence on the PN methods; Alessandra Buonanno for discussions on EOB, and Larry Kidder, on NR; and Badri Krishnan for several detailed discussions on waveforms in general and Phenom models in particular. We are also indebted to participants of the APS meeting in Denver, IGC@25 conference at Penn State, GR22/Amaldi13 conference in Valencia, and the Discussion meeting on Future of Gravitational Waves at ICTS, Bangalore for questions and suggestions. This work was supported in part by the NSF Grant PHY-1806356, Grant UN2017-92945 from the Urania Stott Fund of Pittsburgh Foundation and the Eberly research funds of Penn State.

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Null infinity and emergence of supertranslations

Null infinity and emergence of supertranslations

In Sect. 2 we used several known facts about the structure of null infinity and properties of fields thereon. Since some in the PN and NR communities may not be familiar with them, in this Appendix we present a brief summary, focusing only on those features that we need.

Eventhough Einstein, Eddington and others explored the properties of gravitational waves in the weak field approximation around Minkowski space soon after the discovery of general relativity, there was considerable confusion about the reality of gravitational waves in full, nonlinear GR for several subsequent decades largely because of the coordinate freedom: What appeared to be a wave-like behavior in one coordinate system could appear stationary in another. This confusion was resolved only in the 1960s when Bondi, Sachs and others showed that one can unambiguously disentangle gravitational waves by moving away from isolated sources in retarded null directions, i.e., in the usual terminology, by taking the limit \(r \rightarrow \infty \) keeping the retarded time constant \(u=\mathrm{const}\). The asymptotic boundary conditions introduced by Bondi and Sachs [43, 44] were geometrized by Penrose [53] through the notion of a conformal completion of spacetime, i.e., by attaching to spacetime a 3-dimensional boundary \(\mathfrak {I}^{+}\), representing ‘future null infinity’.

These frameworks provided a definitive, coordinate invariant characterization of gravitational radiation in asymptotically flat spacetimes and introduced techniques to analyze its properties in exact, non-linear general relativity. However, initially there were concerns as to whether the underlying assumptions are too strong to be satisfied by realistic isolated systems such as compact binaries (see, e.g., [60]). The current consensus is that they are not too strong. In particular, the asymptotic form of the PN metric is completely consistent with the Bondi–Sachs–Penrose framework, as shown for instance by Theorem 4 in [61]. Similarly, the key notions of this framework—such as the radiation field \(\varPsi _{4}^{\circ }\), the Bondi news N, and the asymptotic shear \(\sigma ^{\circ }\)—and their properties are heavily used in numerical simulations of waveforms and calculations of energy and momentum flux in NR. These notions and properties are summarized in Sect. 2.

The detailed analysis of gravitational radiation at null infinity brought to forefront an unforeseen result that plays a key role in Sect. 2: Even though spacetimes representing isolated gravitating systems are asymptotically Minkowskian, the asymptotic symmetry group is not the Poincaré group \(\mathcal {P}\), but rather an infinite dimensional generalization thereof, the BMS group \(\mathfrak {B}\). This is a consequence of the fact that gravitational radiation is on a genuinely different footing—from, say, the electromagnetic one—in one important respect. It introduces ripples in spacetime curvature that extend all the way to infinity, i.e., to \(\mathfrak {I}^{+}\), making it impossible to single out a preferred Poincaré group \(\mathcal {P}\) using asymptotic Killing vectors. This difficulty can be seen in concrete terms as follows. Suppose we have a metric \(g_{ab}\) that is asymptotically flat in the sense of Bondi and Sachs; so it approaches a Minkowski metric \(\eta _{ab}\), with \(g_{ab} = \eta _{ab} +{\mathcal {O}}(1/r)\), as \(r \rightarrow \infty \) keeping \(u=t-r\) constant. Therefore, Poincaré transformations of \(\eta _{ab}\) provide us with asymptotic Killing fields for \(g_{ab}\). Now consider a diffeomorphism \( t \rightarrow t^{\prime } = t +f(\theta ,\varphi ), \,\, \mathbf {x} \rightarrow \mathbf {x}^{\prime } = \mathbf {x}\) where \(t,\mathbf {x}\) are Cartesian coordinates of \(\eta _{ab}\). This is an angle dependent translation, whence the metric \(\eta _{ab}\) is sent to a distinct flat metric \(\eta ^{\prime }_{ab}\).Footnote 7

One can verify that since \(g_{ab}\) approaches \(\eta _{ab}\) as 1/r à la Bondi–Sachs, it also approaches \(\eta ^{\prime }_{ab}\) as \(1/r^{\prime }\) à la Bondi–Sachs! Therefore, the Poincaré transformations of \(\eta ^{\prime }_{ab}\) are also asymptotic Killing fields of our physical \(g_{ab}\). But since the two Minkowski metrics are distinct, their isometry groups \(\mathcal {P}\) and \(\mathcal {P}'\) are also distinct. The BMS group can be interpreted as a ‘consistent union’ of Poincaré groups associated with all these Minkowski metrics, related to one another by ‘angle dependent translations.’ These are known as supertranslations. Detailed examination brought out another subtlety: All these Poincaré groups define the same translation subgroup asymptotically, whence the BMS group does admit a canonical, 4-dimensional translation subgroup \(\mathcal {T}\) [63]. However, the Lorentz subgroups \(\mathfrak {L}\) of various Poincaré groups are different even asymptotically. Recall that the Poincaré group \(\mathcal {P}\) admits a 4-parameter family of Lorentz subgroups—each of which defines rotations and boosts about one specific origin in Minkowski space—related to one another by a translation. By contrast, the BMS group \(\mathfrak {B}\) admits an infinite parameter family of Lorenz subgroups that are related to one another by supertranslations. This gives rise to the well-known ‘supertranslation ambiguity’ in the notion of angular momentum at null infinity. We discuss this issue in detail in [48], again in the context of CBC.

In this paper we focused on supertranslations. Just as the translational symmetries of the Minkowski metric lead to the notion of energy-momentum for fields in Minkowskian physics, supertranslation symmetries on \(\mathfrak {I}^{+}\) lead to the notion of supermomenta. In the case of energy-momentum, we have two different quantities available at \(\mathfrak {I}^{+}\). The first is the Bondi 4-momentum—a 2-sphere integral on a cross-section C at \(\mathfrak {I}^{+}\), representing the energy momentum of the system, left over at the retarded instant of time \(u=u_{0}\) defined by C. The second is the notion of flux of energy-momentum carried away by gravitational waves through a ‘patch’ \(\varDelta {\mathfrak {I}^{+}}\) of \(\mathfrak {I}^{+}\). As a consequence we have a balance law: The difference between the Bondi 4-momentum evaluated on two different cross-sections \(C_{1}\) and \(C_{2}\) of \(\mathfrak {I}^{+}\) is the flux of energy-momentum across the patch \(\varDelta \mathfrak {I}^{+}\) bounded by them. It turns out that the same is true for supermomentum. Thus we have an infinite number of balance laws—Eqs. (25) (or  (28)) in the main text—each characterized by a function on a 2-sphere defining the supertranslation. As we discussed in Sect. 2, these lead to an infinite set of constraints—imposed by full, non-linear GR—that any waveform must satisfy in a CBC.

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Ashtekar, A., De Lorenzo, T. & Khera, N. Compact binary coalescences: constraints on waveforms. Gen Relativ Gravit 52, 107 (2020). https://doi.org/10.1007/s10714-020-02764-1

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Keywords

  • Gravitational waves
  • Asymptotic flatness
  • Waveform modeling