The Missing Multipole Problem: Investigating biases from model starting frequency in gravitational-wave analyses

TL;DR

The paper investigates biases introduced when time-domain gravitational-wave templates start at frequencies that omit low-frequency content from higher-order multipoles, focusing on heavy ($M\gtrsim 200\,M_\odot$) binary black holes. Using zero-noise injections with NRSur7dq4 across $200\le M\le 450\,M_\odot$, $0.25\le q\le1$, and varying spins and inclinations, it demonstrates that starting at $f_{22}=20$ Hz biases recovery for $M\gtrsim 250$–$300\,M_\odot$, while including $(3,3)$ via $f_{22}=13$ Hz or lower mitigates biases up to $\rho\lesssim70$, and including $(4,4)$ with $f_{22}=10$ Hz becomes essential at higher masses or SNRs. The study quantifies biases using the Mahalanobis recovery score and Bayesian evidences, and confirms findings on a real event GW231123_135430, offering practical guidelines for selecting template starting frequencies in current and future gravitational-wave analyses. Overall, it highlights the importance of including higher-order multipoles for accurate parameter estimation in short-duration, high-mass systems and provides mass-, spin-, inclination-, and SNR-dependent recommendations to minimize systematic biases.

Abstract

Our ability to infer the true source properties of colliding black holes from gravitational wave observations requires not only accurate waveform models but also their correct use. A key property when evaluating time-domain models is when to start the waveform: choosing a time that is too late can omit low-frequency power from higher order multipoles. By focusing on binary systems with total mass $\ge 200 \, M_{\odot}$, we show that current detectors are sensitive to this missing power and biased source properties can be obtained. We show that for systems with total mass $\lesssim 300 \, M_{\odot}$, mass ratio $\gtrsim 0.33$, and signal-to-noise ratio $ρ\gtrsim 20$, templates starting at $20 \, \mathrm{Hz}$ recover biased source properties. As the total mass increases, and the component masses become more asymmetric, templates starting from $13 \, \mathrm{Hz}$ recover biased properties. If the gravitational-wave signal is observed at signal-to-noise ratio $ρ< 20$, time-domain models can start from $20\, \mathrm{Hz}$ as statistical uncertainties dominate.

Figures (11)

• Figure 1: Plot showing the amplitude of the plus polarization $h_{+}$ of a Fourier transformed signal for a simulated light system produced with the NRSur7dq4 waveform model Varma:2019csw. The left panel shows a signal when the starting frequency of the $(\ell, m) = (2, 2)$ multipole is $f_{22} = 20\, \mathrm{Hz}$. We also show a selection of higher order multipoles for the $20 \, \mathrm{Hz}$ case. The right panel compares the same signal produced with starting frequencies $f_{22} = 10 \, \mathrm{Hz}$ (blue), $f_{22} = 13 \, \mathrm{Hz}$ (green), and $f_{22} = 20 \, \mathrm{Hz}$ (orange) over a reduced frequency range. In both panels, the black dotted lines indicate the starting frequency of the $(\ell, m) = (2, 2), (3, 3)$ and $(4, 4)$ multipoles, see Eq. \ref{['equ:Harmonic_multipoles']}. In all cases, the gravitational-wave is produced from the same simulated system with total mass $M = 300\, M_{\odot}$, mass ratio $q=0.25$, spin magnitudes $\chi_{1} = \chi_{2} = 0.7$ and observed an inclination angle angle of $\theta_{\mathrm{JN}} = \pi / 3\, \mathrm{rad}$.
• Figure 2: The two-dimensional marginalised posterior distribution for the inferred total mass $M$ and mass ratio $q$ for our golden injection, see Sec. \ref{['sec:golden']}. In blue, green and orange we show the posterior distribution obtained when the template starting frequency is $f_{22} = 10, 13$ and $20\, \mathrm{Hz}$ respectively. The contours represent the inferred 90% credible interval and the black horizontal and vertical lines show the true value.
• Figure 3: The one-dimensional marginalised posterior distribution for the inferred total mass $M$, mass ratio $q$, primary and secondary spin magnitudes $\chi_{1}$, $\chi_{2}$ respectively and inclination angle of the binary $\theta_{\mathrm{JN}}$ when varying the total mass of the binary. In blue, green and orange we show the posterior distribution obtained when the template starting frequency is $f_{22} = 10, 13$ and $20\, \mathrm{Hz}$ respectively. In all panels, the black solid vertical line shows the true source properties of the binary. For the left column we shift the posterior distribution by the injected total mass to centre the true value around $0\, M_{\odot}$. The different rows shows the results for different total mass injections.
• Figure 4: log$_{10}$ Bayes factors comparing analyses with starting frequencies $f_{22} = 13 \, \mathrm{Hz}$ and $f_{22} = 20 \, \mathrm{Hz}$ against $f_{22} = 10 \, \mathrm{Hz}$ across the total mass series. Higher Bayes factors indicate stronger preference for the lower starting frequency analysis. In solid we show the Bayes factors for binaries with spin magnitudes $\chi_{i} = 0.7$ and in dashed we show the Bayes factors for binaries with spin magnitudes $\chi_{i} = 0.9$. Given that the $f_{22} = 10 \, \mathrm{Hz}$ generally well recovers the injected parameters, larger Bayes factors highlight increasing bias in parameter recovery as the total mass increases. The red, orange, green, and blue shaded regions indicate no substantial, substantial, strong, and decisive evidence in favour of the $f_{22} = 10 \, \mathrm{Hz}$ analysis Kass:1995loi.
• Figure 5: Two-dimensional marginalised posterior distribution for the inferred total mass $M$ and mass ratio $q$ when varying the total mass of injection. In blue, orange and green we show the posterior distribution obtained when the template starting frequency is $f_{22} = 10, 13$ and $20\, \mathrm{Hz}$ respectively. In the top row we show results for binaries with spin magnitude $\chi_{i} = 0.7$, and in the bottom row we show results for binaries with $\chi_{i} = 0.9$. The left, middle and right columns show the results for injection with injected total mass $M = 250 \, M_{\odot}, 350\, M_{\odot}$, and $450\, M_{\odot}$ respectively. In all panels, the black solid vertical and horizontal lines show the true source properties of the binary and contours represent the 90% credible interval.