A generalized precession parameter ${\ensuremath{\chi}}_{\mathrm{p}}$ to interpret gravitational-wave data

A generalized precession parameter $χ_{p}$ to interpret gravitational-wave data

Davide Gerosa, Matthew Mould, Daria Gangardt, Patricia Schmidt, Geraint Pratten, and Lucy M. Thomas

Phys. Rev. D 103, 064067 – Published 25 March 2021

Abstract

Originally designed for waveform approximants, the effective precession parameter $χ_{p}$ is the most commonly used quantity to characterize spin-precession effects in gravitational-wave observations of black-hole binary coalescences. We point out that the current definition of $χ_{p}$ retains some, but not all, variations taking place on the precession timescale. We rectify this inconsistency and propose more general definitions that either fully consider or fully average those oscillations. Our generalized parameter $χ_{p} \in [0, 2]$ presents an exclusive region $χ_{p} > 1$ that can only be populated by binaries with two precessing spins. We apply our prescriptions to current LIGO/Virgo events and find that posterior distributions of $χ_{p}$ tend to show longer tails at larger values. This appears to be a generic feature, implying that (i) current $χ_{p}$ measurement errors might be underestimated, but also that (ii) evidence for spin precession in current data might be stronger than previously inferred. Among the gravitational-wave events released to date, that which shows the most striking behavior is GW190521.

Received 24 November 2020
Accepted 1 March 2021

DOI:https://doi.org/10.1103/PhysRevD.103.064067

Physics Subject Headings (PhySH)

Astrophysical studies of gravity Classical black holes Gravitation Gravitational wave detection Gravitational wave sources Gravitational waves

Astronomical black holes

Gravitational wave detectors

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Davide Gerosa ^*, Matthew Mould , Daria Gangardt, Patricia Schmidt , Geraint Pratten, and Lucy M. Thomas

School of Physics and Astronomy & Institute for Gravitational Wave Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom

^*d.gerosa@bham.ac.uk

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Vol. 103, Iss. 6 — 15 March 2021

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Figure 1
Evolution of $χ_{p}$ during the inspiral of four representative BH binaries. The heuristic $χ_{p}$ from Eq. (14) is shown in blue, the asymptotic $χ_{p}$ from Eq. (19) (here used inappropriately at finite separations) is shown in orange, the generalized $χ_{p}$ from Eq. (15) is shown in green, and the averaged $χ_{p}$ from Eq. (16) is shown red. Binaries are evolved from $r = 100 M$ to $r = 10 M$ with initial conditions set by the values indicated to the right of each panel. The top axis indicates the corresponding GW frequency $f = \sqrt{M / π^{2} r^{3}}$ for nominal sources with $M = 60 M_{⊙}$ .
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Figure 2
Distribution of $χ_{p}$ using the heuristic [Eq. (14), blue], asymptotic [Eq. (19), orange], generalized [Eq. (15), green], and averaged [Eq. (16), red] definitions. Each panel contains a population of $10^{4}$ sources with fixed values of $q$ , $χ_{1}$ , $χ_{2}$ (as indicated in the figure), and isotropic spin directions. Sources are taken at $r ≃ 14 M$ or $f = 20 Hz$ for a total mass $M = 60 M_{⊙}$ .
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Figure 3
Difference $Δ χ_{p}$ between the averaged [Eq. (16)] and heuristic [Eq. (14)] estimates for the same population of sources shown in Fig. 2. Each histogram is produced assuming isotropic spin directions and fixed values of $q$ , $χ_{1}$ , and $χ_{2}$ as indicated in the legend. The parameter $Δ χ_{p}$ is evaluated at $r ≃ 14 M$ , equivalent to $f = 20 Hz$ for $M = 60 M_{⊙}$ .
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Figure 4
Posterior distributions for the heuristic [Eq. (14), blue] and averaged [Eq. (16), red] estimate of $χ_{p}$ for 45 GW events from the first, second, and third LIGO/Virgo observing runs. Posteriors for the asymptotic (generalized) $χ_{p}$ are qualitatively similar to the heuristic (averaged) results and are omitted for clarity. We report the generic tendency of tails extending toward the region where $χ_{p} ≳ 1$ .
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Figure 5
Statistical properties of the precession parameter $χ_{p}$ for current GW events. We contrast the averaged [Eq. (16), $x$ -axis] and heuristic [Eq. (14), $y$ -axis] definitions of $χ_{p}$ . Scatter points show the medians of the posterior distributions (orange circles), the width of their 90% confidence interval (purple triangles), and the KL divergence between prior and posterior measured in bits (teal squares). The KL divergence of GW190814 is $\sim 4.3$ bits, which is off the scale of this figure in the direction of the arrow.
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Figure 6
Precession parameter $χ_{p}$ for GW190521. Colors indicate the four definitions of $χ_{p}$ described in this paper: heuristic [Eq. (14), blue], asymptotic [Eq. (19), orange], generalized [Eq. (15), green], and averaged [Eq. (16), red]. The left panel shows prior (dashed) and posterior (solid) distributions obtained by combining samples from different waveform approximants. The right panel shows posterior distributions obtained with three different waveform models: NRSur7dq4 (“NRsur,” solid), SEOBNRv4PHM (“EOB,” dashed), and IMRPhenomPv3HM (“Phenom,” dotted).
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Physical Review D

covering particles, fields, gravitation, and cosmology

A generalized precession parameter $χ_{p}$ to interpret gravitational-wave data

Davide Gerosa, Matthew Mould, Daria Gangardt, Patricia Schmidt, Geraint Pratten, and Lucy M. Thomas

Phys. Rev. D 103, 064067 – Published 25 March 2021

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Physical Review D

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A generalized precession parameter χp to interpret gravitational-wave data

Davide Gerosa, Matthew Mould, Daria Gangardt, Patricia Schmidt, Geraint Pratten, and Lucy M. Thomas

Phys. Rev. D 103, 064067 – Published 25 March 2021

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A generalized precession parameter $χ_{p}$ to interpret gravitational-wave data