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Many codes have been developed to study highly relativistic, magnetized flows around and inside compact objects. Depending on the adopted formalism, some of these codes evolve the vector potential $\mathbf{A}$, and others evolve the magnetic field $\mathbf{B}=\nabla\times\mathbf{A}$ directly. Given that these codes possess unique strengths, sometimes it is desirable to start a simulation using a code that evolves $\mathbf{B}$ and complete it using a code that evolves $\mathbf{A}$. Thus transferring the data from one code to another would require an inverse curl algorithm. This paper describes two new inverse curl techniques in the context of Cartesian numerical grids: a cell-by-cell method, which scales approximately linearly with the numerical grid, and a global linear algebra approach, which has worse scaling properties but is generally more robust (e.g., in the context of a magnetic field possessing some nonzero divergence). We demonstrate these algorithms successfully generate smooth vector potential configurations in challenging special and general relativistic contexts., Comment: 20 pages, 9 figures

Numerical relativity codes that do not make assumptions on spatial symmetries most commonly adopt Cartesian coordinates. While these coordinates have many attractive features, spherical coordinates are much better suited to take advantage of approximate symmetries in a number of astrophysical objects, including single stars, black holes and accretion disks. While the appearance of coordinate singularities often spoils numerical relativity simulations in spherical coordinates, especially in the absence of any symmetry assumptions, it has recently been demonstrated that these problems can be avoided if the coordinate singularities are handled analytically. This is possible with the help of a reference-metric version of the Baumgarte-Shapiro-Shibata-Nakamura formulation together with a proper rescaling of tensorial quantities. In this paper we report on an implementation of this formalism in the Einstein Toolkit. We adapt the Einstein Toolkit infrastructure, originally designed for Cartesian coordinates, to handle spherical coordinates, by providing appropriate boundary conditions at both inner and outer boundaries. We perform numerical simulations for a disturbed Kerr black hole, extract the gravitational wave signal, and demonstrate that the noise in these signals is orders of magnitude smaller when computed on spherical grids rather than Cartesian grids. With the public release of our new Einstein Toolkit thorns, our methods for numerical relativity in spherical coordinates will become available to the entire numerical relativity community., Comment: 14 pages, 8 figures, published in Phys. Rev. D

We report on a new open-source, user-friendly numerical relativity code package called SENR/NRPy+. Our code extends previous implementations of the BSSN reference-metric formulation to a much broader class of curvilinear coordinate systems, making it ideally-suited to modeling physical configurations with approximate or exact symmetries. In the context of modeling black hole dynamics, it is orders of magnitude more efficient than other widely used open-source numerical relativity codes. NRPy+ provides a Python-based interface in which equations are written in natural tensorial form and output at arbitrary finite difference order as highly efficient C code, putting complex tensorial equations at the scientist's fingertips without the need for an expensive software license. SENR provides the algorithmic framework that combines the C codes generated by NRPy+ into a functioning numerical relativity code. We validate against two other established, state-of-the-art codes, and achieve excellent agreement. For the first time we demonstrate--in the context of puncture, trumpet, and dual black hole evolutions--nearly exponential convergence of constraint violation and gravitational waveform errors to zero as the order of spatial finite difference derivatives is increased, while holding the spherical-like coordinate grids fixed at moderate resolution. Such behavior outside the horizons is remarkable, as numerical errors do not converge to zero inside horizons, and all points along the polar axis are coordinate singularities. The formulation addresses such coordinate singularities via cell-centered grids and a simple change of basis that analytically regularizes tensor components with respect to the coordinates. Future plans include extending this formulation to allow dynamical coordinate grids and bispherical-like distribution of points to efficiently capture orbiting compact binary dynamics., Comment: 23 pages, 7 figures, matches published version

We evolve a binary black hole system bearing a mass ratio of $q=m_1/m_2=2/3$ and individual spins of $S^z_1/m_1^2=0.95$ and $S^z_2/m_2^2=-0.95$ in a configuration where the large black hole has its spin antialigned with the orbital angular momentum, $L^z$, and the small black hole has its spin aligned with $L^z$. This configuration was chosen to measure the maximum recoil of the remnant black hole for nonprecessing binaries. We find that the remnant black hole recoils at 500km/s, the largest recorded value from numerical simulations for aligned spin configurations. The remnant mass, spin, and gravitational waveform peak luminosity and frequency also provide a valuable point in parameter space for source modeling., Comment: 8 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1706.01980

We demonstrate that numerical relativity codes based on the moving punctures formalism are capable of evolving nearly maximally spinning black hole binaries. We compare a new evolution of an equal-mass, aligned-spin binary with dimensionless spin chi=0.99 using puncture-based data with recent simulations of the SXS Collaboration. We find that the overlap of our new waveform with the published results of the SXS Collaboration is larger than 0.999. To generate our new waveform, we use the recently introduced HiSpID puncture data, the CCZ4 evolution system, and a modified lapse condition that helps keep the horizon radii reasonably large., Comment: Version accepted to PRD. 7 pages, 8 figures

We use fully nonlinear numerical relativity techniques to study high energy head-on collision of nonspinning, equal-mass black holes to estimate the maximum gravitational radiation emitted by these systems. Our simulations include improvements in the construction of initial data, subsequent full numerical evolutions, and the computation of waveforms at infinity. The new initial data significantly reduces the spurious radiation content, allowing for initial speeds much closer to the speed of light, i.e. $v\sim0.99c$. Using these new techniques, We estimate the maximum radiated energy from head-on collisions to be $E_{\text{max}}/M_{\text{ADM}}=0.13\pm0.01$. This value differs from the second-order perturbative $(0.164)$ and zero-frequency-limit $(0.17)$ analytic computations, but is close to those obtained by thermodynamic arguments $(0.134)$ and by previous numerical estimates $(0.14\pm0.03)$., Comment: 11 pages, 10 figures

We solve the Hamiltonian and momentum constraints of general relativity for two black holes with nearly extremal spins and relativistic boosts in the puncture formalism. We use a non-conformally-flat ansatz with an attenuated superposition of two Lorentz-boosted, conformally Kerr or conformally Schwarzschild 3-metrics and their corresponding extrinsic curvatures. We compare evolutions of these data with the standard Bowen-York conformally flat ansatz (technically limited to intrinsic spins $\chi=S/M^2_{\text{ADM}}=0.928$ and boosts $P/M_{\text{ADM}}=0.897$), finding, typically, an order of magnitude smaller burst of spurious radiation and agreement with inspiral and merger. As a first case study, we evolve two equal-mass black holes from rest with an initial separation of $d=12M$ and spins $\chi_i=S_i/m_i^2=0.99$, compute the waveforms produced by the collision, the energy and angular momentum radiated, and the recoil of the final remnant black hole. We find that the black-hole trajectories curve at close separations, leading to the radiation of angular momentum. We also study orbiting nonspinning and moderate-spin black-hole binaries and compare these with standard Bowen-York data. We find a substantial reduction in the nonphysical initial burst of radiation which leads to cleaner waveforms. Finally, we study the case of orbiting binary black-hole systems with spin magnitude $\chi_i=0.95$ in an aligned configuration and compare waveform and final remnant results with those of the SXS Collaboration, finding excellent agreement. This represents the first moving punctures evolution of orbiting and spinning black holes exceeding the Bowen-York limit. Finally, we study different choices of the initial lapse and lapse evolution equation in the moving punctures approach to improve the accuracy and efficiency of the simulations., Comment: 23 pages, 32 figures. Version accepted to PRD

Calculating the expected number of misclassified outcomes is a standard problem of particular interest for rare-event searches. The Clopper-Pearson method allows calculation of classical confidence intervals on the amount of misclassification if data are all drawn from the same binomial probability distribution. However, data is often better described by breaking it up into several bins, each represented by a different binomial distribution. We describe and provide an algorithm for calculating a classical confidence interval on the expected total number of misclassified events from several bins, based on calibration data with the same probability of misclassification on a bin-by-bin basis. Our method avoids a computationally intensive multidimensional search by introducing a Lagrange multiplier and performing standard root finding. This method has only quadratic time complexity as the number of bins, and produces confidence intervals that are only slightly conservative., Comment: 12 pages, 4 figures, to be published in NIMA