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Model of gravitational waves from precessing black-hole binaries through merger and ringdown

Eleanor Hamilton, Lionel London, Jonathan E. Thompson, Edward Fauchon-Jones, Mark Hannam, Chinmay Kalaghatgi, Sebastian Khan, Francesco Pannarale, and Alex Vano-Vinuales
Phys. Rev. D 104, 124027 – Published 9 December 2021
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  • INTRODUCTION
  • NUMERICAL RELATIVITY WAVEFORMS
  • WAVEFORM FRAMES, CONVENTIONS, AND…
  • SPIN PARAMETRIZATION
  • COPRECESSING-FRAME MODEL
  • PRECESSION ANGLE MODEL: INSPIRAL
  • PRECESSION ANGLE MODEL: MERGER RINGDOWN
  • FULL INSPIRAL-MERGER-RINGDOWN ANGLE…
  • PHYSICAL FEATURES OF THE WAVEFORMS
  • TIME DOMAIN VALIDATION
  • MODEL VALIDATION: MATCHES
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We present phenompnr, a frequency-domain phenomenological model of the gravitational-wave signal from binary-black-hole mergers that is tuned to numerical relativity (NR) simulations of precessing binaries. In many current waveform models, e.g., the “phenom” and “eobnr” families that have been used extensively to analyse LIGO-Virgo GW observations, analytic approximations are used to add precession effects to models of nonprecessing (aligned-spin) binaries, and it is only the aligned-spin models that are fully tuned to NR results. In phenompnr we incorporate precessing-binary numerical relativity results in two ways: (i) we produce the first numerical relativity-tuned model of the signal-based precession dynamics through merger and ringdown, and (ii) we extend a previous aligned-spin model, phenomd, to include the effects of misaligned spins on the signal in the coprecessing frame. The numerical relativity calibration has been performed on 40 simulations of binaries with mass ratios between 1∶1 and 1∶8, where the larger black hole has a dimensionless spin magnitude of 0.4 or 0.8, and we choose five angles of spin misalignment with the orbital angular momentum. phenompnr has a typical mismatch accuracy within 0.1% up to mass ratio 1∶4 and within 1% up to mass ratio 1∶8.

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    • Received 21 July 2021
    • Accepted 19 October 2021

    DOI:https://doi.org/10.1103/PhysRevD.104.124027

    © 2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Classical black holesGeneral relativity equations & solutionsGravitational wave sources
    Gravitation, Cosmology & Astrophysics

    Authors & Affiliations

    Eleanor Hamilton1,2, Lionel London3, Jonathan E. Thompson1, Edward Fauchon-Jones1, Mark Hannam1, Chinmay Kalaghatgi4,5,6, Sebastian Khan1, Francesco Pannarale7,8, and Alex Vano-Vinuales9

    • 1School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA, United Kingdom
    • 2Physik-Institut, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
    • 3MIT-Kavli Institute for Astrophysics and Space Research and LIGO Laboratory, 77 Massachusetts Avenue, 37-664H, Cambridge, Massachusetts 02139, USA
    • 4Nikhef-National Institute for Subatomic Physics, Science Park, 1098 XG Amsterdam, The Netherlands
    • 5Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University, Princetonplein 1, 3584 CC Utrecht, The Netherlands
    • 6Institute for High-Energy Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
    • 7Dipartimento di Fisica, Università di Roma “Sapienza”, Piazzale A. Moro 5, I-00185 Roma, Italy
    • 8INFN, Sezione di Roma, Piazzale A. Moro 5, I-00185 Roma, Italy
    • 9Centro de Astrofísica e Gravitação - CENTRA, Departamento de Física, Instituto Superior Técnico IST, Universidade de Lisboa UL, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal

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    Issue

    Vol. 104, Iss. 12 — 15 December 2021

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    Images

    • Figure 1
      Figure 1

      Frequency domain comparison of numerical relativity and model waveforms in the coprecessing frame. Top: phase derivative for the (q,χ1,θLS)=(4,0.8,60°) configuration, which illustrates the variation in the inspiral phase. Middle and bottom: phase derivative and amplitude for the (q,χ1,θLS)=(4,0.8,150°) configuration, which demonstrates the shift in effective ringdown frequency.

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    • Figure 2
      Figure 2

      Comparison of the post-Newtonian expressions for each of the precession angles (blue dotted line) with the numerical relativity data (black solid line) for the case with (q,χ,θLS)=(8,0.8,60°). The gray vertical lines indicate the ISCO frequency (Mf=0.0287) of the final black hole, which has final spin magnitude χf=0.799 and final mass Mf=0.981M.

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    • Figure 3
      Figure 3

      The Euler angles (α,β,γ) that make up the precession angles that describe the transformation from the fixed-J^ frame into a coprecessing frame. There are different choices for the definition of V: the QA direction, the Newtonian orbital angular momentum, and varying orders of the post-Newtonian orbital-angular momentum. These directions are all approximately the same, as discussed in the text.

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    • Figure 4
      Figure 4

      Frequency domain comparison of numerical relativity and model waveforms in the coprecessing frame. Top: phase derivative for the (q,χ,θLS)=(4,0.8,60°) configuration, which illustrates the variation in the inspiral phase. Middle and bottom: phase derivative and amplitude for the (q,χ,θLS)=(4,0.8,150°) configuration, which demonstrates the shift in effective ringdown frequency.

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    • Figure 5
      Figure 5

      Opening angles for the (q,χ,θLS)=(8,0.8,60°) configuration. Solid black: the numerical relativity opening angle of the QA frame, β. Dotted blue: the PN opening angle of the orbital plane, ι. Dashed magenta: Approximate QA angle β as a function of ι; see text for details.

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    • Figure 6
      Figure 6

      Various options for the PN expression for the opening angle. The left-hand panel shows the PN value of ι for a two-spin system (blue) and for the equivalent single-spin system (green) calculated using the expressions used in phenompv3. In light blue is shown the effect of tapering the two-spin oscillations to the single-spin value at the connection frequency fc, shown as a gray vertical line. In the right-hand panel, the value for β used in the model (pink) is compared with the numerical relativity value of β found for this case. We only show ι and β up to fc, since the merger-ringdown model is used at higher frequencies. The configuration shown is SXS1397 in Table 2.

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    • Figure 7
      Figure 7

      Comparison of the phenomenological ansatz presented in Eq. (49) (solid lines) with the numerical relativity data (translucent lines) over the frequency range to which the coefficients in the ansatz were tuned for a selection cases in the numerical relativity catalogue with θLS=90° at varying mass ratios. We have made use of the freedom to choose a constant offset in α in order to offset the curves shown here to make them easier to distinguish.

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    • Figure 8
      Figure 8

      Comparison of the phenomenological ansatz presented in Eq. (50) (solid lines) with the numerical relativity data (translucent lines) over the frequency range to which the coefficients in the ansatz were tuned for a selection of cases in the numerical relativity catalogue with θLS=90° at varying mass ratios.

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    • Figure 9
      Figure 9

      Comparison of the complete model for each of the precession angles (thick red line) with the numerical relativity data (thin black line). The MSA angles (blue dotted line) are shown for reference. The left hand column shows the case with (q,χ,θ)=(1,0.4,30°). The right hand column shows the case with (q,χ,θ)=(8,0.8,60°). The vertical black lines show the connection frequencies for α and β.

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    • Figure 10
      Figure 10

      Possible morphologies of the ansatz given by Eq. (50) depending on the values taken by the coefficients in different regions of the parameter space. From left to right, the panels show systems with (q,χ,θLS)=(8,0.2,155°), (2.5, 0.4, 90°), and (5, 0.8, 160°). The red dots mark the extrema. The green crosses show the inflection points, and the blue dot indicates the inflection point chosen as described in Sec. 8d. The points of maximum curvature around this inflection point are shown by the black lines, which give a measure of the width of the turnover. The solid black line in the shaded region indicates the frequency region that will be used as the merger-ringdown portion of the full angle model. All cases within our calibration region will have the morphology shown in the middle panel; the outer panels show that a reasonable choice is made outside the calibration region.

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    • Figure 11
      Figure 11

      Effective frequency-domain ringdown frequencies for (q,χ)=(8,0.8), as modeled by phenompv3 and phenomdcp. Additional lines show QNM frequencies predicted from standard perturbation theory methods using the remnant BH’s mass and spin [92]. The solid thick gray line traces prograde QNM frequencies, and the dashed thick gray line traces the retrograde QNM frequencies. All curves are bound between prograde and retrograde QNM frequencies. phenompv3 displays a discontinuity near θLS=120°, while numerical relativity data and phenompnr do not.

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    • Figure 12
      Figure 12

      Comparison of analytic ringdown estimate [Eq. (57b) of Ref. [19] ] and numerical relativity for (top) (q,χ,θLS)=(4,0.4,60°) and (bottom) (q,χ,θLS)=(8,0.4,30°).

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    • Figure 13
      Figure 13

      Amplitudes of the ℓ=2 multipoles for the (q,χ,θLS)=(4,0.4,90°) configuration at 100  M⊙. The numerical relativity data are shown in black on all panels, with phenompnr (top panel in blue), phenompv3 and phenomxp (central panel in purple), and seobnrv4p (bottom panel in red). The vertical line indicates the frequency of the peak (2,2) amplitude. The small vertical line segments indicate the turnover frequency of each of the subdominant modes.

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    • Figure 14
      Figure 14

      A comparison of the time domain obtained from phenompnr with the numerical relativity data. The top panel shows the case (q,χ,θLS)=(4,0.8,60°), while the bottom panel shows the case (q,χ,θLS)=(8,0.8,60°). Both are for a face on (θLN=0°) binary with a total mass of 100  M⊙. For comparison, we also show the waveform produced using seobnrv4p. The match values for the specific configuration for each of the waveforms plotted are given in the legend.

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    • Figure 15
      Figure 15

      Comparison of the time domain precession angles for the phenompv3, phenomxp, seobnrv4p, and phenompnr models with the numerical relativity data. These angles are for the case with (q,χ,θLS)=(4,0.4,60°).

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    • Figure 16
      Figure 16

      Comparison of the time domain precession angles for the phenompv3, phenomxp, seobnrv4p, and phenompnr models with the numerical relativity data. These angles are for the case with (q,χ,θLS)=(8,0.8,60°).

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    • Figure 17
      Figure 17

      Mismatches for each of the BAM calibration and verification waveforms, at a total mass of 100  M⊙. Mismatches are between the symmetrized coprecessing numerical relativity waveforms and phenomdcp (purple diamonds), modified phenomd (blue circles), and modified phenomxas (red squares). The configuration mass ratio increases from left to right (with q∈{1,2,4,8}). Solid black lines separate cases mass ratios, and dotted lines separate spin magnitudes.

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    • Figure 18
      Figure 18

      SNR-weighted mismatches for the same configurations as in Fig. 17, averaged over inclination. These mismatches are between the symmetrized numerical relativity waveforms in the J-aligned frame and the coprecessing numerical relativity waveform twisted up with the angle model presented here (purple diamonds) and twisted up with the angle model used by phenompv3 (steel blue triangles).

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    • Figure 19
      Figure 19

      Comparison MSA α (blue dashed line) with the value calculated from the numerical relativity waveform (black solid line) for the (q,χ,θLS)=(8,0.8,120°) configuration. In order to see a region over which the two values agree well, we would need a longer numerical relativity waveform; see text for more details.

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    • Figure 20
      Figure 20

      Mismatch as a function of the inclination of the binary, quantified by the angle between the line of sight and the total angular momentum θJN, for four cases, at 100  M⊙. These mismatches consider the coprecessing NR waveform twisted up with the angle model used by phenompv3 (steel blue) and phenompnr (purple). The solid markers show the SNR-weighted average mismatch, while the shaded regions show the variation with respect to signal polarization and phase.

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    • Figure 21
      Figure 21

      Histograms of the SNR-weighted mismatches between the numerical relativity waveforms listed in Tables 1 and 2 and the waveform models phenompnr and phenompv3. Each subplot contains the SNR-weighted mismatches for all total masses separated by inclination descending as θLN∈[0°,30°,60°,90°]. The mismatches for all total mass values listed in Sec. 11e are included at each inclination. The results for phenompnr are present in solid black, while the results for phenompv3 are given in dashed red.

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    • Figure 22
      Figure 22

      Histograms of the SNR-weighted mismatches between various models in comparison and the numerical relativity waveforms listed in Tables 1 and 2. The mismatches for all inclination and total mass values listed in Sec. 11e are included. In all three subplots, the results for phenompnr (“PNR”) are presented with a solid black outline, with the other model results given with dashed outlines from left to right as phenomxp (“XP”) in red, seobnrv4p (“EOB”) in blue, and nrsur7dq4 (“SUR”) in green. For the comparison plot between phenompnr and nrsur7dq4 we only include results of numerical relativity waveforms for which both models are run.

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    • Figure 23
      Figure 23

      SNR-weighted mismatches averaged over total mass and inclination between the precessing waveform models phenompnr (“PNR”), phenomxp (“XP”), and seobnrv4p (“EOB”), and the numerical relativity waveforms listed in Tables 1 and 2, shown in order of the table listings. For the BAM cases, the solid vertical lines separate cases by mass ratio, and the dashed vertical lines separate spin magnitude. For the SXS and Maya cases, the solid vertical line splits by numerical relativity catalogue, and the dashed vertical line indicates a transition from single-spin to two-spin cases.

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    • Figure 24
      Figure 24

      SNR-weighted mismatches computed at 100  M⊙ and averaged over inclination between the precessing waveform model phenompnr (“PNR”) and the numerical relativity waveforms listed in Table 1, shown in order of the table listings. Alongside these results are plotted the SNR-weighted mismatches computed between the numerical relativity waveforms in the initially J^-aligned frame and the fixed-J^ frame (“NR”). The solid vertical lines separate cases by mass ratio, and the dashed vertical lines separate spin magnitude.

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    • Figure 25
      Figure 25

      Amplitude parameters for tuned coprecessing waveform model, phenomdcp. The fits are shown as two-dimensional surfaces covering the parameter space described by η and cosθLS. On the left in blue are the fits for the simulations with χ=0.4, and on the right in red are the fits for χ=0.8. Above each of these surfaces are shown the residuals.

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    • Figure 26
      Figure 26

      Phase parameters for tuned coprecessing waveform model, phenomdcp. The fits are shown as two-dimensional surfaces covering the parameter space described by η and cosθLS. On the left in blue are the fits for the simulations with χ=0.4, and on the right in red are the fits for χ=0.8. Above each of these surfaces are shown the residuals.

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    • Figure 27
      Figure 27

      Comparison of the fits for each of the coefficients for the ansatz for α given in Eq. (49) with the coefficients found from the data as described in Sec. 7. The fits are shown as two-dimensional surfaces covering the parameter space described by η and cosθLS. On the left in blue are the fits for the simulations with χ=0.4, and on the right in red are the fits for χ=0.8. Above each of these surfaces are shown the residuals.

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    • Figure 28
      Figure 28

      Comparison of the fits for each of the coefficients for the ansatz for β given in Eq. (50) with the coefficients found from the data as described in Sec. 7. The fits are shown as two-dimensional surfaces covering the parameter space described by η and cosθLS. On the left in blue are the fits for the simulations with χ=0.4, and on the right in red are the fits for χ=0.8. Above each of these surfaces are shown the residuals.

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