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Compact binary coalescences: constraints on waveforms


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


  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.


  1. 1.

    Boyle, M., et al.: The SXS Collaboration catalog of binary black hole simulations. Class. Quantm Gravity 36, 195006 (2019).

    ADS  Article  Google Scholar 

  2. 2.

    Ajith, P., Hannam, M., Husa, S., Chen, Y., Brügmann, B., Dorband, N., Müller, D., Ohme, F., Pollney, D., Reisswig, C., Santamaría, L., Seiler, J.: Inspiral-merger-ringdown waveforms for black-hole binaries with nonprecessing spins. Phys. Rev. Lett. 106, 241101 (2011).

    ADS  Article  Google Scholar 

  3. 3.

    Santamaría, L., Ohme, F., Ajith, P., Brügmann, B., Dorband, N., Hannam, M., Husa, S., Mösta, P., Pollney, D., Reisswig, C., Robinson, E.L., Seiler, J., Krishnan, B.: Matching post-Newtonian and numerical relativity waveforms: systematic errors and a new phenomenological model for nonprecessing black hole binaries. Phys. Rev. D 82, 064016 (2010).

    ADS  Article  Google Scholar 

  4. 4.

    Husa, S., Khan, S., Hannam, M., Pürrer, M., Ohme, F., Jiménez Forteza, X., Bohé, A.: Frequency-domain gravitational waves from nonprecessing black-hole binaries. I. New numerical waveforms and anatomy of the signal. Phys. Rev. D 93, 044006 (2016).

    ADS  Article  Google Scholar 

  5. 5.

    Khan, S., Husa, S., Hannam, M., Ohme, F., Pürrer, M., Jiménez Forteza, X., Bohé, A.: Frequency-domain gravitational waves from nonprecessing black-hole binaries. II. A phenomenological model for the advanced detector era. Phys. Rev. D 93, 044007 (2016).

    ADS  Article  Google Scholar 

  6. 6.

    Hannam, M., Schmidt, P., Bohé, A., Haegel, L., Husa, S., Ohme, F., Pratten, G., Pürrer, M.: Simple model of complete precessing black-hole-binary gravitational waveforms. Phys. Rev. Lett. 113, 151101 (2014).

    ADS  Article  Google Scholar 

  7. 7.

    Damour, T., Nagar, A.: In: Haardt, F., Gorini, V., Moschella, U., Treves, A., Colpi, M. (eds.)Astrophysical Black Holes, Lecture Notes in Physics, vol. 905, pp. 273–312. Springer, Berlin (2016).

  8. 8.

    Buonanno, A., Pan, Y., Pfeiffer, H.P., Scheel, M.A., Buchman, L.T., Kidder, L.E.: Effective-one-body waveforms calibrated to numerical relativity simulations: coalescence of non-spinning, equal-mass black holes. Phys. Rev. D 79, 124028 (2009).

    ADS  Article  Google Scholar 

  9. 9.

    Taracchini, A., Buonanno, A., Pan, Y., Hinderer, T., Boyle, M., Hemberger, D.A., Kidder, L.E., Lovelace, G., Mroué, A.H., Pfeiffer, H.P., Scheel, M.A., Szilágyi, B., Taylor, N.W., Zenginoglu, A.: Effective-one-body model for black-hole binaries with generic mass ratios and spins. Phys. Rev. D 89, 061502 (2014).

    ADS  Article  Google Scholar 

  10. 10.

    Bohé, A., et al.: Improved effective-one-body model of spinning, nonprecessing binary black holes for the era of gravitational-wave astrophysics with advanced detectors. Phys. Rev. D 95, 044028 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

    Pan, Y., Buonanno, A., Taracchini, A., Kidder, L.E., Mroué, A.H., Pfeiffer, H.P., Scheel, M.A., Szilágyi, B.: Inspiral-merger-ringdown waveforms of spinning, precessing black-hole binaries in the effective-one-body formalism. Phys. Rev. D 89, 084006 (2014).

    ADS  Article  Google Scholar 

  12. 12.

    Nagar, A., Messina, F., Rettegno, P., Bini, D., Damour, T., Geralico, A., Akcay, S., Bernuzzi, S.: Nonlinear-in-spin effects in effective-one-body waveform models of spin-aligned, inspiralling, neutron star binaries. Phys. Rev. D 99, 044007 (2019).

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    Nagar, A., Bernuzzi, S., Del Pozzo, W., Riemenschneider, G., Akcay, S., Carullo, G., Fleig, P., Babak, S., Tsang, K.W., Colleoni, M., et al.: Time-domain effective-one-body gravitational waveforms for coalescing compact binaries with nonprecessing spins, tides, and self-spin effects. Phys. Rev. D 98, 104052 (2018)

    ADS  Article  Google Scholar 

  14. 14.

    Field, S.E., Galley, C.R., Hesthaven, J.S., Kaye, J., Tiglio, M.: Fast prediction and evaluation of gravitational waveforms using surrogate models. Phys. Rev. X 4, 031006 (2014).

    Article  Google Scholar 

  15. 15.

    Varma, V., Field, S.E., Scheel, M.A., Blackman, J., Kidder, L.E., Pfeiffer, H.P.: Surrogate model of hybridized numerical relativity binary black hole waveforms. Phys. Rev. D 99, 064045 (2019).

    ADS  MathSciNet  Article  Google Scholar 

  16. 16.

    Rifat, N.E.M., Field, S.E., Khanna, G., Varma, V.: Surrogate model for gravitational wave signals from comparable and large-mass-ratio black hole binaries. Phys. Rev. D 101, 081502 (2020).

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    Lackey, B.D., Pürrer, M., Taracchini, A., Marsat, S.: Surrogate model for an aligned-spin effective one body waveform model of binary neutron star inspirals using Gaussian process regression. Phys. Rev. D 100, 024002 (2019).

    ADS  Article  Google Scholar 

  18. 18.

    Kumar, P., Chu, T., Fong, H., Pfeiffer, H.P., Boyle, M., Hemberger, D.A., Kidder, L.E., Scheel, M.A., Szilagyi, B.: Accuracy of binary black hole waveform models for aligned-spin binaries. Phys. Rev. D 93, 104050 (2016).

    ADS  Article  Google Scholar 

  19. 19.

    Pürrer, M., Haster, C.J.: Gravitational waveform accuracy requirements for future ground-based detectors. Phys. Rev. Res. 2, 023151 (2020).

    Article  Google Scholar 

  20. 20.

    Aasi, J., et al.: Advanced LIGO. Class. Quantum Gravity 32, 074001 (2015).

    ADS  Article  Google Scholar 

  21. 21.

    Acernese, F., et al.: Advanced Virgo: a second-generation interferometric gravitational wave detector. Class. Quantum Gravity 32, 024001 (2015).

    ADS  Article  Google Scholar 

  22. 22.

    Aso, Y., Michimura, Y., Somiya, K., Ando, M., Miyakawa, O., Sekiguchi, T., Tatsumi, D., Yamamoto, H.: Interferometer design of the KAGRA gravitational wave detector. Phys. Rev. D 88, 043007 (2013).

    ADS  Article  Google Scholar 

  23. 23.

    Unnikrishnan, C.S.: IndIGO and LIGO-India: Scope and plans for gravitational wave research and precision metrology in India. Int. J. Mod. Phys. D 22, 1341010 (2013).

    ADS  Article  Google Scholar 

  24. 24.

    Punturo, M., et al.: The Einstein Telescope: a third-generation gravitational wave observatory. Class. Quantum Gravity 27, 194002 (2010).

    ADS  Article  Google Scholar 

  25. 25.

    Reitze, D., et al.: Cosmic explorer: the U.S. contribution to gravitational-wave astronomy beyond LIGO. Bull. Am. Astron. Soc. 51, 035 (2019)

    Google Scholar 

  26. 26.

    Amaro-Seoane, P., et al.: Laser Interferometer Space Antenna, arXiv e-prints (2017)

  27. 27.

    Luo, J., et al.: TianQin: a space-borne gravitational wave detector. Class. Quantum Gravity 33, 035010 (2016).

    ADS  Article  Google Scholar 

  28. 28.

    Kawamura, S., et al.: The Japanese space gravitational wave antenna DECIGO. Class. Quantum Gravity 23, S125 (2006).

    Article  Google Scholar 

  29. 29.

    Buonanno, A.: New approaches to GW source modeling: Overview (2019). Talk at the discussion meeting on The Future of Gravitational Wave Astronomy, held at ICTS, Bangalore (August 19–22 2019)

  30. 30.

    Yunes, N., Pretorius, F.: Fundamental theoretical bias in gravitational wave astrophysics and the parameterized post-einsteinian framework. Phys. Rev. D 80, 122003 (2009).

    ADS  Article  Google Scholar 

  31. 31.

    London, L., Khan, S., Fauchon-Jones, E., García, C., Hannam, M., Husa, S., Jiménez-Forteza, X., Kalaghatgi, C., Ohme, F., Pannarale, F.: First higher-multipole model of gravitational waves from spinning and coalescing black-hole binaries. Phys. Rev. Lett. 120, 161102 (2018).

    ADS  Article  Google Scholar 

  32. 32.

    Cotesta, R., Buonanno, A., Bohé, A., Taracchini, A., Hinder, I., Ossokine, S.: Enriching the symphony of gravitational waves from binary black holes by tuning higher harmonics. Phys. Rev. D 98, 084028 (2018).

    ADS  Article  Google Scholar 

  33. 33.

    Khan, S., Ohme, F., Chatziioannou, K., Hannam, M.: Including higher order multipoles in gravitational-wave models for precessing binary black holes. Phys. Rev. D 101, 024056 (2020).

    ADS  MathSciNet  Article  Google Scholar 

  34. 34.

    Khan, S., Chatziioannou, K., Hannam, M., Ohme, F.: Phenomenological model for the gravitational-wave signal from precessing binary black holes with two-spin effects. Phys. Rev. D 100, 024059 (2019).

    ADS  MathSciNet  Article  Google Scholar 

  35. 35.

    Blackman, J., Field, S.E., Scheel, M.A., Galley, C.R., Hemberger, D.A., Schmidt, P., Smith, R.: A surrogate model of gravitational waveforms from numerical relativity simulations of precessing binary black hole mergers. Phys. Rev. D 95, 104023 (2017).

    ADS  MathSciNet  Article  Google Scholar 

  36. 36.

    Apostolatos, T.A., Cutler, C., Sussman, G.J., Thorne, K.S.: Spin induced orbital precession and its modulation of the gravitational wave forms from merging binaries. Phys. Rev. D 49, 6274 (1994).

    ADS  Article  Google Scholar 

  37. 37.

    Varma, V., Field, S.E., Scheel, M.A., Blackman, J., Gerosa, D., Stein, L.C., Kidder, L.E., Pfeiffer, H.P.: Surrogate models for precessing binary black hole simulations with unequal masses. Phys. Rev. Res. 1, 033015 (2019).

    Article  Google Scholar 

  38. 38.

    Kidder, L.E., et al.: SpECTRE: a task-based discontinuous Galerkin code for relativistic astrophysics. J. Comput. Phys. 335, 84 (2017).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Pan, Y., Buonanno, A., Baker, J.G., Centrella, J., Kelly, B.J., McWilliams, S.T., Pretorius, F., van Meter, J.R.: A Data-analysis driven comparison of analytic and numerical coalescing binary waveforms: nonspinning case. Phys. Rev. D 77, 024014 (2008).

    ADS  Article  Google Scholar 

  40. 40.

    Lindblom, L., Owen, B.J., Brown, D.A.: Model waveform accuracy standards for gravitational wave data analysis. Phys. Rev. D 78, 124020 (2008).

    ADS  Article  Google Scholar 

  41. 41.

    MacDonald, I., Mroue, A.H., Pfeiffer, H.P., Boyle, M., Kidder, L.E., Scheel, M.A., Szilagyi, B., Taylor, N.W.: Suitability of hybrid gravitational waveforms for unequal-mass binaries. Phys. Rev. D 87, 024009 (2013).

    ADS  Article  Google Scholar 

  42. 42.

    Kumar, P., Barkett, K., Bhagwat, S., Afshari, N., Brown, D.A., Lovelace, G., Scheel, M.A., Szilágyi, B.: Accuracy and precision of gravitational-wave models of inspiraling neutron star-black hole binaries with spin: comparison with matter-free numerical relativity in the low-frequency regime. Phys. Rev. D 92, 102001 (2015).

    ADS  Article  Google Scholar 

  43. 43.

    Bondi, H., van der Burg, M.G.J., Metzner, A.W.K.: Gravitational waves in general relativity. VII. Waves from axisymmetric isolated systems. Proc. R. Soc. Lond. Ser. A 269, 21 (1962).

    ADS  Article  MATH  Google Scholar 

  44. 44.

    Sachs, R.K., Bondi, H.: Gravitational waves in general relativity. VIII. Waves in asymptotically flat space-time. Proc. R. Soc. Lond. Ser. A 270, 103 (1962).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  45. 45.

    Infeld, I. (ed.): Relativistic Theories of Gravitation. Pergamon Press, Oxford (1964)

    MATH  Google Scholar 

  46. 46.

    Khera, N., Krishnan, B., Ashtekar, A., De Lorenzo, T.: Inferring the gravitational wave memory for binary coalescence events, arXiv e-prints (2020)

  47. 47.

    Mitman, K., Iozzo, D., Khera, N., Boyle, M., De Lorenzo, T., Kidder, L., Moxon, J., Pfeiffer, H., Scheel, M.A., Teukolsky, S.A.: Adding Gravitational Memory to the SXS Catalog using BMS Balance Laws. (2020) arXiv:2011.01309v1

  48. 48.

    Ashtekar, A., De Lorenzo, T., Khera, N.: Compact binary coalescences: the subtle issue of angular momentum. Phys. Rev. D 101, 044005 (2020).

    ADS  MathSciNet  Article  Google Scholar 

  49. 49.

    Ashtekar, A., Khera, N., Krishnan, B.: Using balance laws to infer the spin of the final black hole for binary coalescence events (2020) (in preparation)

  50. 50.

    Geroch, R.: In: Esposito, F.P., Witten, L. (eds.) Asymptotic Structure of Space-Time, pp. 1–105. Springer US, Boston (1977).

  51. 51.

    Ashtekar, A., Streubel, M.: Symplectic geometry of radiative modes and conserved quantities at null infinity. Proc. R. Soc. Lond. Ser. A 376, 585 (1981).

    ADS  MathSciNet  Article  Google Scholar 

  52. 52.

    Ashtekar, A.: In: Bieri, L., Yau, S.T. (eds.) Surveys in Differential Geometry 2015: 100 Years of General Relativity. A Jubilee Volume on General Relativity and Mathematics, pp. 99–122. International Press, Boston (2015).

  53. 53.

    Penrose, R.: Zero rest mass fields including gravitation: asymptotic behavior. Proc. R. Soc. Lond. Ser. A 284, 159 (1965).

    ADS  Article  MATH  Google Scholar 

  54. 54.

    Bohé, A., Shao, L., Taracchini, A., Buonanno, A., Babak, S., Harry, I.W., Hinder, I., Ossokine, S., Pürrer, M., Raymond, V., et al.: Improved effective-one-body model of spinning, nonprecessing binary black holes for the era of gravitational-wave astrophysics with advanced detectors. Phys. Rev. D 95, 044028 (2017)

    ADS  Article  Google Scholar 

  55. 55.

    Ashtekar, A.: Radiative degrees of freedom of the gravitational field in exact general relativity. J. Math. Phys. 22, 2885 (1981).

    ADS  MathSciNet  Article  Google Scholar 

  56. 56.

    Newman, E.T., Penrose, R.: New conservation laws for zero rest-mass fields in asymptotically flat space-time. Proc. R. Soc. Lond. Ser. A 305, 175 (1968).

    ADS  Article  Google Scholar 

  57. 57.

    Dray, T.: Momentum flux at null infinity. Class. Quantum Gravity 2, L7 (1985).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  58. 58.

    Kesavan, A.: Asymptotic structure of space-time with a positive cosmological constant. Ph.D. thesis, Penn State U. (2016-06-02).

  59. 59.

    Friedrich, H.: Peeling or not peeling–is that the question? Class. Quantum Gravity 35, 083001 (2018).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  60. 60.

    Blanchet, L., Damour, T.: Radiative gravitational fields in general relativity. I. General structure of the field outside the source. Philos. Trans. R. Soc. Lond. Ser. A 320, 379 (1986).

  61. 61.

    Blanchet, L.: Gravitational radiation from post-Newtonian sources and inspiralling compact binaries. Living Rev. Relativ. 17, 2 (2014).

    ADS  Article  MATH  Google Scholar 

  62. 62.

    Ashtekar, A.: Asymptotic Quantization: Based on 1984 Naples Lectures, Monographs and Textbooks in Physical Science, vol. 2. Bibliopolis, Naples (1987)

    MATH  Google Scholar 

  63. 63.

    Sachs, R.K.: Asymptotic symmetries in gravitational theory. Phys. Rev. 128, 2851 (1962).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  64. 64.

    Newman, E.T., Penrose, R.: Note on the Bondi–Metzner–Sachs group. J. Math. Phys. 7, 863 (1966).

    ADS  MathSciNet  Article  Google Scholar 

  65. 65.

    Ashtekar, A., Magnon-Ashtekar, A.: On the symplectic structure of general relativity. Commun. Math. Phys. 86, 55 (1982).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  66. 66.

    Winicour, J.: Some total invariants of asymptotically flat space-times. J. Math. Phys. 9, 861 (1968).

    ADS  Article  MATH  Google Scholar 

  67. 67.

    Bramson, B.D., Penrose, R.: Relativistic angular momentum for asymptotically flat Einstein–Maxwell manifolds. Proc. R. Soc. Lond. Ser. A 341, 463 (1975).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  68. 68.

    Prior, C.R., Hawking, S.W.: Angular momentum in general relativity. I. Definition and asymptotic behaviour. Proc. R. Soc. Lond. Ser. A 354, 379 (1977).

    ADS  MathSciNet  Article  Google Scholar 

  69. 69.

    Streubel, M.: “Conserved” quantities for isolated gravitational systems. Gen. Relativ. Gravit. 9, 551 (1978).

    ADS  MathSciNet  Article  Google Scholar 

  70. 70.

    Geroch, R., Winicour, J.: Linkages in general relativity. J. Math. Phys. 22, 803 (1981).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  71. 71.

    Ashtekar, A., Winicour, J.: Linkages and hamiltonians at null infinity. J. Math. Phys. 23, 2410 (1982).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  72. 72.

    Dray, T., Streubel, M.: Angular momentum at null infinity. Class. Quantum Gravity 1, 15 (1984).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  73. 73.

    Streubel, M.: “Conserved” quantities related to asymptotic symmetries for isolated systems in general relativity. Ph.D. thesis, Max Planck Institut für Astrophysik, München MPI PAE/ Astro 165 (1978)

  74. 74.

    Baker, J.G., Boggs, W.D., Centrella, J., Kelly, B.J., McWilliams, S.T., Miller, M.C., van Meter, J.R.: Modeling kicks from the merger of nonprecessing black hole binaries. Astrophys. J. 668, 1140 (2007).

    ADS  Article  Google Scholar 

  75. 75.

    Campanelli, M., Lousto, C.O., Zlochower, Y., Merritt, D.: Large merger recoils and spin flips from generic black-hole binaries. Astrophys. J. Lett. 659, L5 (2007).

    ADS  Article  Google Scholar 

  76. 76.

    De Lorenzo, T.: Constraints on GW Waveforms (2020). Talk at Virtual April APS Meeting, Session T16: Approximate Methods in Gravitational Astrophysics

  77. 77.

    Kidder, B.: Personal communication to AA (2019)

  78. 78.

    Iozzo, D.: Personal communication to TDL and NK (2019)

  79. 79.

    Iyer, B.: Personal communication to AA (2019)

  80. 80.

    Poisson, E.: Personal communication to AA (2019)

  81. 81.

    Beig, R., Simon, W.: The stationary gravitational field near spatial infinity. Gen. Relativ. Gravit. 12, 1003 (1980).

    ADS  MathSciNet  Article  Google Scholar 

  82. 82.

    Satishchandran, G., Wald, R.M.: Asymptotic behavior of massless fields and the memory effect. Phys. Rev. D 99, 084007 (2019).

    ADS  MathSciNet  Article  Google Scholar 

  83. 83.

    Abbott, B.P., et al.: GWTC-1: a gravitational-wave transient catalog of compact binary mergers observed by LIGO and virgo during the first and second observing runs. Phys. Rev. X 9, 031040 (2019).

    Article  Google Scholar 

  84. 84.

    Littenberg, T.B., Baker, J.G., Buonanno, A., Kelly, B.J.: Systematic biases in parameter estimation of binary black-hole mergers. Phys. Rev. D 87, 104003 (2013).

    ADS  Article  Google Scholar 

  85. 85.

    Brown, D.A., Kumar, P., Nitz, A.H.: Template banks to search for low-mass binary black holes in advanced gravitational-wave detectors. Phys. Rev. D 87, 082004 (2013).

    ADS  Article  Google Scholar 

  86. 86.

    Capano, C., Pan, Y., Buonanno, A.: Impact of higher harmonics in searching for gravitational waves from nonspinning binary black holes. Phys. Rev. D 89, 102003 (2014).

    ADS  Article  Google Scholar 

  87. 87.

    Varma, V., Ajith, P., Husa, S., Bustillo, J.C., Hannam, M., Pürrer, M.: Gravitational-wave observations of binary black holes: effect of nonquadrupole modes. Phys. Rev. D 90, 124004 (2014).

    ADS  Article  Google Scholar 

  88. 88.

    Babak, S., Taracchini, A., Buonanno, A.: Validating the effective-one-body model of spinning, precessing binary black holes against numerical relativity. Phys. Rev. D 95, 024010 (2017)

    ADS  Article  Google Scholar 

  89. 89.

    Boyle, M.: Transformations of asymptotic gravitational-wave data. Phys. Rev. D 93, 084031 (2016).

    ADS  MathSciNet  Article  Google Scholar 

  90. 90.

    Garfinkle, D.: A simple estimate of gravitational wave memory in binary black hole systems. Class. Quantum Gravity 33, 177001 (2016).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  91. 91.

    Pollney, D., Reisswig, C.: Gravitational memory in binary black hole mergers. Astrophys. J. 732, L13 (2011).

    ADS  Article  Google Scholar 

  92. 92.

    Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space (PMS-41). Princeton University Press, Princeton (1994).

  93. 93.

    Chrusciel, P.T., Delay, E.: Existence of non-trivial, vacuum, asymptotically simple spacetimes. Class. Quantum Gravity 19, L71 (2002).

    ADS  MathSciNet  Article  MATH  Google Scholar 

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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).

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  • Gravitational waves
  • Asymptotic flatness
  • Waveform modeling