Beyond the Stationary Phase Approximation: an alternative approach to model gravitational waves from precessing compact binaries in the Fourier Domain

[eng] The first detection of gravitational waves of a black hole binary [1] opened the current observational era of gravitational wave astronomy. Several gravitational waves from merging compact binaries have already been observed during the three observational “runs” [2, 3], with the expectation of increasing the detection rate with upgraded and upcoming detectors. Sophisticated data analysis methods are indispensable for the detection of gravitational waves and it requires theoretical models to estimate the source parameters. Through the “matched filtering” method, the theoretical templates are cross correlated against the observed signal at the detector, so one can infer the source parameters using Bayesian inference. In order to sample the posterior probability distribution of the parameters, Bayesian inference requires at least millions of evaluations of the likelihood function. The better the sensitivity of the detectors, the more accurate and computationally efficient the signal templates need to be. This is why one of the main efforts of the gravitational wave group at the UIB is to improve the current Inspiral-Merger-Ringdown (IMR) waveforms in the Fourier domain, which describe the amplitude and phase needing a low computational cost to evaluate them and hence, making them a reasonable template for applications in Bayesian inference. In this work I focus on the challenging effect that a complete representation of the spins implies on waveform models, known as spin precession. In case of having a black hole binary with misaligned spins, i.e. when the angular momenta of the individual black holes are not orthogonal to the orbital plane, the spin-orbit and the spin-spin couplings induce a precession of the orbital plane and of the spins themselves. This precession leads to a modulation of the signal as seen by the observer, and increases the dimensionality of the problem, which makes it difficult to cover the large parameter space with numerical relativity simulations. However, the fact that the acceleration due to the orbital motion dominates and the power radiated due to precession can be neglected in the inspiral gives rise to a fruitful approach to modelling this effect [4–7]: One can use a (non-inertial) co-precessing frame in which the decomposed waveform is similar to a non-precessing one, performing a time-dependent rotation that follows the precession of the orbital plane. In order to create efficient Fourier domain models, one needs to understand how to translate the time rotation from an inertial frame to the co-precessing one into a Fourier domain transfer function. Our purpose is thus to implement the formalism developed in [8] in order to process the time domain modulation necessary to treat precession in the Fourier domain, while retaining the compactness of a Fourier domain amplitude and phase representation of the signal. This new formalism, based on the separation of time-scales between precession and orbital motion directly in the Fourier domain, seeks to overcome the limitations of the Stationary Phase Approximation (SPA) [9]. This method can only be applied to compute the Fourier transform of non-precessing systems, i.e. with aligned spins, during the inspiral, and hence, it is not applicable to IMR precessing waveforms. A better approach than SPA is crucial to deal with the challenging events we may detect with the upgraded and upcoming detectors