Measuring coalescing massive binary black holes with gravitational waves: The impact of spin-induced precession

TL;DR

This paper demonstrates that including spin-induced precession in gravitational-wave waveform models for merging massive black-hole binaries measured by LISA significantly improves parameter estimation. By numerically integrating full precession (including spin-spin effects) and applying a Fisher-information framework across a 17-parameter space, the study shows dramatic gains in mass accuracy (by 1–3 orders of magnitude) and enables spin magnitude measurements (0.1%–10% for favorable, low-redshift cases). Extrinsic parameters such as sky position and luminosity distance also improve, though more modestly, yielding sky-localization on the order of a few tens of arcminutes at z ~ 1, which enhances prospects for electromagnetic follow-up. The work underscores the scientific payoff of LISA in constraining black-hole mass and spin evolution over cosmic time and highlights the need for more robust statistical methods and waveform models to fully quantify uncertainties.

Abstract

The coalescence of massive black holes generates gravitational waves (GWs) that will be measurable by space-based detectors such as LISA to large redshifts. The spins of a binary's black holes have an important impact on its waveform. Specifically, geodetic and gravitomagnetic effects cause the spins to precess; this precession then modulates the waveform, adding periodic structure which encodes useful information about the binary's members. Following pioneering work by Vecchio, we examine the impact upon GW measurements of including these precession-induced modulations in the waveform model. We find that the additional periodicity due to spin precession breaks degeneracies among certain parameters, greatly improving the accuracy with which they may be measured. In particular, mass measurements are improved tremendously, by one to several orders of magnitude. Localization of the source on the sky is also improved, though not as much -- low redshift systems can be localized to an ellipse which is roughly ${a few} \times 10$ arcminutes in the long direction and a factor of 2-4 smaller in the short direction. Though not a drastic improvement relative to analyses which neglect spin precession, even modest gains in source localization will greatly facilitate searches for electromagnetic counterparts to GW events. Determination of distance to the source is likewise improved: We find that relative error in measured luminosity distance is commonly $\sim 0.2%-0.7%$ at $z \sim 1$. Finally, with the inclusion of precession, we find that the magnitude of the spins themselves can typically be determined for low redshift systems with an accuracy of about $0.1%-10 %$, depending on the spin value, allowing accurate surveys of mass and spin evolution over cosmic time.

Figures (9)

• Figure 1: These figures depict the "polarization amplitude" $A_{\mathrm{pol}}(t)$ of the signal measured in detector I as a function of time. The curves are as follows: solid line, $\chi_1 = \chi_2 = 0$; dashed line (nearly overlapping the solid line), $\chi_1 = \chi_2 = 0.1$; dotted line, $\chi_1 = \chi_2 = 0.5$; and dot-dashed line, $\chi_1 = \chi_2 = 0.9$. ($\chi = S/m^2$ is the dimensionless spin parameter.) The top figure covers the last two years of inspiral. The spinless curve has periodicity of one year, corresponding to the motion of LISA around the sun. Notice that as spin is introduced, the curves become more strongly modulated, with the number of additional oscillations growing as the spin is increased. By tracking these spin-precession-induced modulations, it becomes possible to better measure parameters like mass and sky position and measure spin for the first time. The bottom figure shows a close-up of the final months of inspiral.
• Figure 2: Distribution of errors in chirp mass $\mathcal{M}$ for $10^4$ binaries with $m_1 = 10^6 M_\odot$ and $m_2 = 3 \times 10^5 M_\odot$ at $z = 1$. The dashed line is the precession-free calculation; the solid line includes precession. Precession reduces the measurement error by about an order of magnitude.
• Figure 3: Distribution of errors in reduced mass $\mu$ for $10^4$ binaries with $m_1 = 10^6 M_\odot$ and $m_2 = 3 \times 10^5 M_\odot$ at $z = 1$. The dashed line is the precession-free calculation; the solid line includes precession. Precession has an enormous effect on the reduced mass, which was previously highly correlated with the parameters $\beta$ and $\sigma$.
• Figure 4: Distribution of errors in individual hole masses for $10^4$ binaries at $z = 1$. The solid line is $m_1 = 10^6 M_\odot$, while the dashed line is $m_2 = 3 \times 10^5 M_\odot$. The individual masses are not determined as well as $\mathcal{M}$ and $\mu$, but they are better behaved parameters when precession is introduced.
• Figure 5: Distribution of errors in dimensionless spin parameters $\chi_1$ (solid line) and $\chi_2$ (dashed line) for $10^4$ binaries with $m_1 = 10^6 M_\odot$ and $m_2 = 3 \times 10^5 M_\odot$ at $z = 1$. In each binary, the spin values are randomly selected between 0 and 1. The higher mass then has, on average, higher total spin and more effect on the precession.