We introduce a bilinear form for Weyl scalar perturbations of Kerr. The form is symmetric and conserved, and we show that, when combined with a suitable renormalization prescription involving complex r integration contours, quasinormal modes are orthogonal in the bilinear form for different $(l, m, n)$. These properties are not in any straightforward way consequences of standard properties for the radial and angular solutions to the decoupled Teukolsky relations and rely on the Petrov type D character of Kerr and its $t$–$\phi$ reflection isometry. Finally, we show that quasinormal mode excitation coefficients are given precisely by the projection with respect to our bilinear form. We believe that these properties can make our bilinear form useful to set up a framework for nonlinear quasinormal mode coupling in Kerr. We include a general discussion on conserved local currents and their associated local symmetry operators for metric and Weyl perturbations of Kerr. In particular, we obtain an infinite set of conserved, local, gauge invariant currents associated with Carter’s constant for metric perturbations, containing $2n + 9$ derivatives.

We combine amortized neural posterior estimation with importance sampling for fast and accurate gravitational-wave inference. We first generate a rapid proposal for the Bayesian posterior using neural networks, and then attach importance weights based on the underlying likelihood and prior. This provides (1) a corrected posterior free from network inaccuracies, (2) a performance diagnostic (the sample efficiency) for assessing the proposal and identifying failure cases, and (3) an unbiased estimate of the Bayesian evidence. By establishing this independent verification and correction mechanism we address some of the most frequent criticisms against deep learning for scientific inference. We carry out a large study analyzing 42 binary black hole mergers observed by LIGO and Virgo with the SEOBNRv4PHM and IMRPhenomXPHM waveform models. This shows a median sample efficiency of ≈10% (two orders-of-magnitude better than standard samplers) as well as a ten-fold reduction in the statistical uncertainty in the log evidence. Given these advantages, we expect a significant impact on gravitational-wave inference, and for this approach to serve as a paradigm for harnessing deep learning methods in scientific applications.

Gravitational-wave observations of black hole ringdowns are commonly used to characterize binary merger remnants and to test general relativity. These analyses assume linear black hole perturbation theory, in particular that the ringdown can be described in terms of quasinormal modes even for times approaching the merger. Here we investigate a nonlinear effect during the ringdown, namely how a mode excited at early times can excite additional modes as it is absorbed by the black hole. This is a third-order secular effect: the change in the black-hole mass causes a shift in the mode spectrum, so that the original mode is projected onto the new ones. Using nonlinear simulations, we study the ringdown of a spherically-symmetric scalar field around an asymptotically anti-de Sitter black hole, and we find that this “absorption-induced mode excitation” (AIME) is the dominant nonlinear effect. We show that this effect takes place well within the nonadiabatic regime, so we can analytically estimate it using a sudden mass-change approximation. Adapting our estimation technique to asymptotically-flat Schwarzschild black holes, we expect AIME to play a role in the analysis and interpretation of current and future gravitational wave observations.