Spectral Methods for Numerical Relativity

Version published online by Living Reviews in Relativity.International audienceEquations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses on a class called spectral methods where, typically, the various functions are expanded onto sets of orthogonal polynomials or functions. A theoretical introduction on spectral expansion is first given and a particular emphasis is put on the fast convergence of the spectral approximation. We present then different approaches to solve partial differential equations, first limiting ourselves to the one-dimensional case, with one or several domains. Generalization to more dimensions is then discussed. In particular, the case of time evolutions is carefully studied and the stability of such evolutions investigated. One then turns to results obtained by various groups in the field of General Relativity by means of spectral methods. First, works which do not involve explicit time-evolutions are discussed, going from rapidly rotating strange stars to the computation of binary black holes initial data. Finally, the evolutions of various systems of astrophysical interest are presented, from supernovae core collapse to binary black hole mergers

Black holes in cosmological spacetimes and alternative theories of gravity

This thesis is dedicated to the study of spacetimes surrounding black holes within the context of cosmology, high energy physics and modified theories of gravity. We do this by applying and adapting modern numerical relativity techniques to probe the inhomogeneous and strong field regime in a number of different scenarios. The first application we consider is the nonlinear evolution of unstable flux compactifi- cations in a low-energy limit of string theory. Going beyond stationary solutions and their perturbations, we find rich dynamics, in some cases finding that the evolution from an unstable homogeneous state to a stable warped compactification can serve as a toy-model for slow-roll inflation, while in other cases finding solutions that eventually evolve to a singular state. We then apply the methods for numerically evolving scalar fields coupled to the Ein- stein field equations to address several problems in early universe cosmological scenarios. We study the conditions under which inflation can arise from very inhomogeneous initial conditions. To do so, we introduce and compare several different ways of constucting ini- tial data with large inhomogeneities in both the scalar field and time derivative profiles, by solving for the coupled Einstein constraint equations. We then study the evolution of various classes of initial conditions in both single- and two-field inflationary models. In some of the cases studied, the initial gradient and kinetic energy are much larger than the inflationary energy scale such that black holes can form. Taken together, our results suggest inflation can arise from highly inhomogeneous conditions. Using the same numerical techniques, we study the nonlinear classical dynamics and evo- lutions of black holes in a particular nonsingular bouncing cosmology. We find that for sufficiently large black holes the black hole apparent horizon can disappear during the contraction phase. Despite this, we show that most of the local cosmological evolution remains largely unaffected by the presence of the black hole. For all the cases explored, the black hole’s event horizon persists throughout the bounce, suggesting the nonsingular bouncing model under study is fairly robust to large perturbations. Finally, we use and further develop a novel formulation of the Einstein field equations for evolving a large class of modified theories of gravity. We use this formulation to study the nonlinear dynamics of binary black hole mergers in a specific class of theories, where the black holes acquire a scalar charge. We consider quasi-circular inspirals with different mass-ratios, varying the coupling parameter introducing deviations from General Relativity and quantifying the impact on the emitted scalar and gravitational waveforms. We also compare our numerical results to analytic post-Newtonian calculations of the radiation emitted during the inspiral

Investigating the build-up of precedence effect using reflection masking

The auditory processing level involved in the build‐up of precedence [Freyman et al., J. Acoust. Soc. Am. 90, 874–884 (1991)] has been investigated here by employing reflection masked threshold (RMT) techniques. Given that RMT techniques are generally assumed to address lower levels of the auditory signal processing, such an approach represents a bottom‐up approach to the buildup of precedence. Three conditioner configurations measuring a possible buildup of reflection suppression were compared to the baseline RMT for four reflection delays ranging from 2.5–15 ms. No buildup of reflection suppression was observed for any of the conditioner configurations. Buildup of template (decrease in RMT for two of the conditioners), on the other hand, was found to be delay dependent. For five of six listeners, with reflection delay=2.5 and 15 ms, RMT decreased relative to the baseline. For 5‐ and 10‐ms delay, no change in threshold was observed. It is concluded that the low‐level auditory processing involved in RMT is not sufficient to realize a buildup of reflection suppression. This confirms suggestions that higher level processing is involved in PE buildup. The observed enhancement of reflection detection (RMT) may contribute to active suppression at higher processing levels

Aeronautical engineering: A continuing bibliography with indexes (supplement 279)

This bibliography lists 759 reports, articles, and other documents introduced into the NASA scientific and technical information system in May 1992. Subject coverage includes: design, construction, and testing of aircraft and aircraft engines; aircraft components, equipment, and systems; ground support systems; and theoretical and applied aspects of aerodynamics and general fluid dynamics