Two-harmonic approximation for gravitational waveforms from precessing binaries

Abstract

Binary-black-hole orbits precess when the black-hole spins are misaligned with the binary's orbital angular momentum. The apparently complicated dynamics can in most cases be described as simple precession of the orbital angular momentum about an approximately fixed total angular momentum. However, the imprint of the precession on the observed gravitational-wave signal is yet more complicated, with a nontrivial time-varying dependence on the black-hole dynamics, the binary's orientation and the detector polarization. As a result, it is difficult to predict under which conditions precession effects are measurable in gravitational-wave observations, and their impact on both signal detection and source characterization. We show that the observed waveform can be simplified by decomposing it as a power series in a new precession parameter b ¼ tanðβ=2Þ, where β is the opening angle between the orbital and total angular momenta. The power series is made up of five harmonics, with frequencies that differ by the binary's precession frequency, and individually do not exhibit amplitude and phase modulations. In many cases, the waveform can be well approximated by the two leading harmonics. In this approximation we are able to obtain a simple picture of precession as caused by the beating of two waveforms of similar frequency. This enables us to identify regions of the parameter space where precession is likely to have an observable effect on the waveform, and to propose a new approach to searching for signals from precessing binaries, based upon the two-harmonic approximation.

GW190412: Observation of a binary-black-hole coalescence with asymmetric masses

“…This suggests that by changing those parameters, our models can effectively treat the amplitude and phase of the higher multipoles as tunable parameters that make their contributions more or less pronounced. If the data only contained the dominant quadrupole mode and Gaussian noise, the squared orthogonal SNR in the subdominant multipole will be χ 2 distributed with 2 degrees of freedom [140,141,150]. This was verified by analyzing an injection with parameters close to GW190412.…”

“…Gravitational waveforms from precessing binaries can be decomposed into an expansion in terms of the opening angle, β JL , between the total and orbital angular momenta (see Sec. III in [140], and [141]). Considering only l ¼ 2 modes, this expansion contains five terms proportional to tan k ðβ JL =2Þ (k ¼ 0; …; 4), and each term alone does not show the characteristic phase and amplitude modulations of a precessing signal.…”

Measuring precession in asymmetric compact binaries

“…Recently, Ref. [120] introduced a frequentist framework to estimate the contribution to the SNR ρ that stems from precession, referred to as the precessing SNR ρ p . The formalism decomposes the GW signal into two harmonics, each of which is equivalent to the emission of a nonprecessing binary.…”

“…In the absence of precession, ρ p is χ 2 distributed with two degrees of freedom. A simple criterion for precession to be considered observable is the requirement that ρ p 2.1 [120,121]. Here we will assess the significance of ρ p via the single-sided p value associated with the mean of the distribution.…”

“…Here we will assess the significance of ρ p via the single-sided p value associated with the mean of the distribution. The two-harmonic formalism relies on several assumptions [120], which are valid for the signals considered in this paper. We will thus use it as a complementary quantifier to assess the measurability of spin precession.…”