The Emergence of Gravitational Wave Science: 100 Years of Development of Mathematical Theory, Detectors, Numerical Algorithms, and Data Analysis Tools

On September 14, 2015, the newly upgraded Laser Interferometer Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW) signal, emitted a billion light-years away by a coalescing binary of two stellar-mass black holes. The detection was announced in February 2016, in time for the hundredth anniversary of Einstein's prediction of GWs within the theory of general relativity (GR). The signal represents the first direct detection of GWs, the first observation of a black-hole binary, and the first test of GR in its strong-field, high-velocity, nonlinear regime. In the remainder of its first observing run, LIGO observed two more signals from black-hole binaries, one moderately loud, another at the boundary of statistical significance. The detections mark the end of a decades-long quest, and the beginning of GW astronomy: finally, we are able to probe the unseen, electromagnetically dark Universe by listening to it. In this article, we present a short historical overview of GW science: this young discipline combines GR, arguably the crowning achievement of classical physics, with record-setting, ultra-low-noise laser interferometry, and with some of the most powerful developments in the theory of differential geometry, partial differential equations, high-performance computation, numerical analysis, signal processing, statistical inference, and data science. Our emphasis is on the synergy between these disciplines, and how mathematics, broadly understood, has historically played, and continues to play, a crucial role in the development of GW science. We focus on black holes, which are very pure mathematical solutions of Einstein's gravitational-field equations that are nevertheless realized in Nature, and that provided the first observed signals.Comment: 41 pages, 5 figures. To appear in Bulletin of the American Mathematical Societ

A multi-block infrastructure for three-dimensional time-dependent numerical relativity

We describe a generic infrastructure for time evolution simulations in numerical relativity using multiple grid patches. After a motivation of this approach, we discuss the relative advantages of global and patch-local tensor bases. We describe both our multi-patch infrastructure and our time evolution scheme, and comment on adaptive time integrators and parallelisation. We also describe various patch system topologies that provide spherical outer and/or multiple inner boundaries. We employ penalty inter-patch boundary conditions, and we demonstrate the stability and accuracy of our three-dimensional implementation. We solve both a scalar wave equation on a stationary rotating black hole background and the full Einstein equations. For the scalar wave equation, we compare the effects of global and patch-local tensor bases, different finite differencing operators, and the effect of artificial dissipation onto stability and accuracy. We show that multi-patch systems can directly compete with the so-called fixed mesh refinement approach; however, one can also combine both. For the Einstein equations, we show that using multiple grid patches with penalty boundary conditions leads to a robustly stable system. We also show long-term stable and accurate evolutions of a one-dimensional non-linear gauge wave. Finally, we evolve weak gravitational waves in three dimensions and extract accurate waveforms, taking advantage of the spherical shape of our grid lines.Comment: 18 pages. Some clarifications added, figure layout improve

A sparse representation of gravitational waves from precessing compact binaries

Many relevant applications in gravitational wave physics share a significant common problem: the seven-dimensional parameter space of gravitational waveforms from precessing compact binary inspirals and coalescences is large enough to prohibit covering the space of waveforms with sufficient density. We find that by using the reduced basis method together with a parametrization of waveforms based on their phase and precession, we can construct ultra-compact yet high-accuracy representations of this large space. As a demonstration, we show that less than $100$ judiciously chosen precessing inspiral waveforms are needed for $200$ cycles, mass ratios from $1$ to $10$ and spin magnitudes $\le 0.9$. In fact, using only the first $10$ reduced basis waveforms yields a maximum mismatch of $0.016$ over the whole range of considered parameters. We test whether the parameters selected from the inspiral regime result in an accurate reduced basis when including merger and ringdown; we find that this is indeed the case in the context of a non-precessing effective-one-body model. This evidence suggests that as few as $\sim 100$ numerical simulations of binary black hole coalescences may accurately represent the seven-dimensional parameter space of precession waveforms for the considered ranges.Comment: 5 pages, 3 figures. The parameters selected for the basis of precessing waveforms can be found in the source file

Turduckening black holes: an analytical and computational study

We provide a detailed analysis of several aspects of the turduckening technique for evolving black holes. At the analytical level we study the constraint propagation for a general family of BSSN-type formulation of Einstein's field equations and identify under what conditions the turducken procedure is rigorously justified and under what conditions constraint violations will propagate to the outside of the black holes. We present high-resolution spherically symmetric studies which verify our analytical predictions. Then we present three dimensional simulations of single distorted black holes using different variations of the turduckening method and also the puncture method. We study the effect that these different methods have on the coordinate conditions, constraint violations, and extracted gravitational waves. We find that the waves agree up to small but non-vanishing differences, caused by escaping superluminal gauge modes. These differences become smaller with increasing detector location.Comment: Minor changes to match the final version to appear in PR

A complete gauge-invariant formalism for arbitrary second-order perturbations of a Schwarzschild black hole

Using recently developed efficient symbolic manipulations tools, we present a general gauge-invariant formalism to study arbitrary radiative $(l\geq 2)$ second-order perturbations of a Schwarzschild black hole. In particular, we construct the second order Zerilli and Regge-Wheeler equations under the presence of any two first-order modes, reconstruct the perturbed metric in terms of the master scalars, and compute the radiated energy at null infinity. The results of this paper enable systematic studies of generic second order perturbations of the Schwarzschild spacetime. In particular, studies of mode-mode coupling and non-linear effects in gravitational radiation, the second-order stability of the Schwarzschild spacetime, or the geometry of the black hole horizon.Comment: 14 page