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Given the compact binary evolution problem of numerical relativity, in the finite-difference, block-based, adaptive mesh refinement context, choices must be made on how evolved fields are to be discretized. In GR-Athena++, the space-time solver was previously fixed to be vertex-centered. Here, our recent extensions to a cell-centered treatment, are described. Simplifications in the handling of variables during the treatment of general relativistic magneto-hydrodynamical (GRMHD) evolution are found. A novelty is that performance comparison for the two choices of grid sampling is made within a single code-base. In the case of a binary black hole inspiral-merger problem, by evolving geometric fields on vertex-centers, an average $\sim 20\%$ speed increase is observed, when compared against cell-centered sampling. The opposite occurs in the GRMHD setting. A binary neutron star inspiral-merger-collapse problem, representative of typical production simulations is considered. We find that cell-centered sampling for the space-time solver improves performance, by a similar factor.

We present a self-contained overview of GR-Athena++, a general-relativistic magnetohydrodynamics (GRMHD) code, that incorporates treatment of dynamical space-time, based on the recent work of (Daszuta+, 2021)[49] and (Cook+, 2023)[45]. General aspects of the Athena++ framework we build upon, such as oct-tree based, adaptive mesh refinement (AMR) and constrained transport, together with our modifications, incorporating the Z4c formulation of numerical relativity, judiciously coupled, enables GRMHD with dynamical space-times. Initial verification testing of GR-Athena++ is performed through benchmark problems that involve isolated and binary neutron star space-times. This leads to stable and convergent results. Gravitational collapse of a rapidly rotating star through black hole formation is shown to be correctly handled. In the case of non-rotating stars, magnetic field instabilities are demonstrated to be correctly captured with total relative violation of the divergence-free constraint remaining near machine precision. The use of AMR is show-cased through investigation of the Kelvin-Helmholtz instability which is resolved at the collisional interface in a merger of magnetised binary neutron stars. The underlying task-based computational model enables GR-Athena++ to achieve strong scaling efficiencies above $80\%$ in excess of $10^5$ CPU cores and excellent weak scaling up to $\sim 5 \times 10^5$ CPU cores in a realistic production setup. GR-Athena++ thus provides a viable path towards robust simulation of GRMHD flows in strong and dynamical gravity with exascale high performance computational infrastructure., Comment: Invited chapter for the edited book {\it New Frontiers in GRMHD Simulations} (Eds. C. Bambi, Y. Mizuno, S. Shashank and F. Yuan, Springer Singapore, expected in 2024)

We study initially unbound systems of two black holes using numerical relativity (NR) simulations performed with GR-Athena++. We focus on regions of the parameter space close to the transition from scatterings to dynamical captures, considering equal mass and spin-aligned configurations, as well as unequal mass and nonspinning ones. The numerical results are then used to validate the effective-one-body (EOB) model TEOBResumS-Dal\'i for dynamical captures and scatterings. We find good agreement for the waveform phenomenologies, scattering angles, mismatches, and energetics in the low energy regime ($E_0\lesssim 1.02\,M$). In particular, mismatches are typically below or around the $1\%$ level, with only a few cases -- corresponding to spinning binaries -- slightly above the $3\%$ threshold. We also discuss dynamical captures in the test-mass limit by solving numerically the Zerilli equation with the time domain code RWZHyp. The latter analysis provides valuable insights into both the analytical noncircular corrections of the EOB waveform and the integration of NR Weyl scalars., Comment: 31 pages, 18 figures

In a systematic study, we use an equivalent pair of improved numerical relativity codes based on a tetrad-formulation of the classical Einstein-scalar field equations to examine whether slow contraction or inflation (or both) can resolve the homogeneity, isotropy and flatness problems. Our finding, based on a set of gauge/frame invariant diagnostics, is that slow contraction robustly and rapidly smooths and flattens spacetime beginning from initial conditions that are outside the perturbative regime of the flat Friedmann-Robertson-Walker metric, whereas inflation fails these tests. We present new numerical evidence supporting the conjecture that the combination of ultralocal evolution and an effective equation-of-state with pressure much greater than energy density is the key to having robust and rapid smoothing. The opposite of ultralocality occurs in expanding spacetimes, which is the leading obstruction to smoothing following a big bang., Comment: 18 pages, 6 figures, 1 table

We discuss refinement criteria for the Berger-Rigoutsos (block-based) refinement algorithm in our numerical relativity code GR-Athena++ in the context of binary black hole merger simulations. We compare three different strategies: the "box-in-box" approach, the "sphere-in-sphere" approach and a local criterion for refinement based on the estimation of truncation error of the finite difference scheme. We extract and compare gravitational waveforms using the three different mesh refinement methods and compare their accuracy against a calibration waveform and demonstrate that the sphere-in-sphere approach provides the best strategy overall when considering computational cost and the waveform accuracy. Ultimately, we demonstrate the capability of each mesh refinement method in accurately simulating gravitational waves from binary black hole systems -- a crucial aspect for their application in next-generation detectors. We quantify the mismatch achievable with the different strategies by extrapolating the gravitational wave mismatch to higher resolution., Comment: 23 pages, 7 figures. Matches the version accepted on Classical and Quantum Gravity

We present the extension of GR-Athena++ to general-relativistic magnetohydrodynamics (GRMHD) for applications to neutron star spacetimes. The new solver couples the constrained transport implementation of Athena++ to the Z4c formulation of the Einstein equations to simulate dynamical spacetimes with GRMHD using oct-tree adaptive mesh refinement. We consider benchmark problems for isolated and binary neutron star spacetimes demonstrating stable and convergent results at relatively low resolutions and without grid symmetries imposed. The code correctly captures magnetic field instabilities in non-rotating stars with total relative violation of the divergence-free constraint of $10^{-16}$. It handles evolutions with a microphysical equation of state and black hole formation in the gravitational collapse of a rapidly rotating star. For binaries, we demonstrate correctness of the evolution under the gravitational radiation reaction and show convergence of gravitational waveforms. We showcase the use of adaptive mesh refinement to resolve the Kelvin-Helmholtz instability at the collisional interface in a merger of magnetised binary neutron stars. GR-Athena++ shows strong scaling efficiencies above $80\%$ in excess of $10^5$ CPU cores and excellent weak scaling is shown up to $\sim 5 \times 10^5$ CPU cores in a realistic production setup. GR-Athena++ allows for the robust simulation of GRMHD flows in strong and dynamical gravity with exascale computers., Comment: 27 pages, 23 figures

Gravitational wave astronomy pipelines rely on template waveform models for searches and parameter estimation purposes. For coalescing binary neutron stars (BNS), such models need to accurately reproduce numerical relativity (NR) up to merger, in order to provide robust estimate of the stars' equation of state - dependent parameters. In this work we present an improved version of the Effective One Body (EOB) model $\tt TEOBResumS$ for gravitational waves from BNS systems. Building upon recent post-Newtonian calculations, we include subleading order tidal terms in the waveform multipoles and EOB metric potentials, as well as add up to 5.5PN terms in the gyro-gravitomagnetic functions entering the spin-orbit sector of the model. In order to further improve the EOB-NR agreement in the last few orbital cycles before merger, we introduce next-to-quasicircular corrections in the waveform -- informed by a large number of BNS NR simulations -- and introduce a new NR-informed parameter entering the tidal sector of our conservative dynamics. The performance of our model is then validated against 14 new eccentricity reduced simulations of unequal mass, spinning binaries with varying equation of state. A time-domain phasing analysis and mismatch computations demonstrate that the new model overall improves over the previous version of $\tt TEOBResumS$. Finally, we present a closed-form frequency domain representation of the (tidal) amplitude and phase of the new model. This representation accounts for mass-ratio, aligned spin and (resummed) spin-quadrupole effects in the tidal phase and -- within the calibration region -- it is faithful to the original model., Comment: 17 pages, 6 figures

Dynamical captures of black holes may take place in dense stellar media due to the emission of gravitational radiation during a close passage. Detection of such events requires detailed modelling, since their phenomenology qualitatively differs from that of quasi-circular binaries. Very few models can deliver such waveforms, and none includes information from Numerical Relativity (NR) simulations of non quasi-circular coalescences. In this study we present a first step towards a fully NR-informed Effective One Body (EOB) model of dynamical captures. We perform 14 new simulations of single and double encounter mergers, and use this data to inform the merger-ringdown model of the TEOBResumS-Dali approximant. We keep the initial energy approximately fixed to the binary mass, and vary the mass-rescaled, dimensionless angular momentum in the range $(0.6, 1.1)$, the mass ratio in $(1, 2.15)$ and aligned dimensionless spins in $(-0.5, 0.5)$. We find that the model is able to match NR to $97%$, improving previous performances, without the need of modifying the base-line template. Upon NR informing the model, this improves to $99%$ with the exception of one outlier corresponding to a direct plunge. The maximum EOBNR phase difference at merger for the uninformed model is of $0.15$ radians, which is reduced to $0.1$ radians after the NR information is introduced. We outline the steps towards a fully informed EOB model of dynamical captures, and discuss future improvements., Comment: 21 pages, 15 figures

Hougardy and Schroeder (WG 2014) proposed a combinatorial technique for pruning the search space in the traveling salesman problem, establishing that, for a given instance, certain edges cannot be present in any optimal tour. We describe an implementation of their technique, employing an exact TSP solver to locate k-opt moves in the elimination process. In our computational study, we combine LP reduced-cost elimination together with the new combinatorial algorithm. We report results on a set of geometric instances, with the number of points n ranging from 3,038 up to 115,475. The test set includes all TSPLIB instances having at least 3,000 points, together with 250 randomly generated instances, each with 10,000 points, and three currently unsolved instances having 100,000 or more points. In all but two of the test instances, the complete-graph edge sets were reduced to under 3n edges. For the three large unsolved instances, repeated runs of the elimination process reduced the graphs to under 2.5n edges.

We present the second data release of gravitational waveforms from binary neutron star merger simulations performed by the Computational Relativity (CoRe) collaboration. The current database consists of 254 different binary neutron star configurations and a total of 590 individual numerical-relativity simulations using various grid resolutions. The released waveform data contain the strain and the Weyl curvature multipoles up to $\ell=m=4$. They span a significant portion of the mass, mass-ratio,spin and eccentricity parameter space and include targeted configurations to the events GW170817 and GW190425. CoRe simulations are performed with 18 different equations of state, seven of which are finite temperature models, and three of which account for non-hadronic degrees of freedom. About half of the released data are computed with high-order hydrodynamics schemes for tens of orbits to merger; the other half is computed with advanced microphysics. We showcase a standard waveform error analysis and discuss the accuracy of the database in terms of faithfulness. We present ready-to-use fitting formulas for equation of state-insensitive relations at merger (e.g. merger frequency), luminosity peak, and post-merger spectrum., Comment: 33 pages, 14 figures

Last-mile routing refers to the final step in a supply chain, delivering packages from a depot station to the homes of customers. At the level of a single van driver, the task is a traveling salesman problem. But the choice of route may be constrained by warehouse sorting operations, van-loading processes, driver preferences, and other considerations, rather than a straightforward minimization of tour length. We propose a simple and efficient penalty-based local-search algorithm for route optimization in the presence of such constraints, adopting a technique developed by Helsgaun to extend the LKH traveling salesman problem code to general vehicle-routing models. We apply his technique to handle combinations of constraints obtained from an analysis of historical routing data, enforcing properties that are desired in high-quality solutions. Our code is available under the open-source MIT license. An earlier version of the code received the $100,000 top prize in the Amazon Last Mile Routing Research Challenge organized in 2021.

The numerical solution of relativistic hydrodynamics equations in conservative form requires root-finding algorithms that invert the conservative-to-primitive variables map. These algorithms employ the equation of state of the fluid and can be computationally demanding for applications involving sophisticated microphysics models, such as those required to calculate accurate gravitational wave signals in numerical relativity simulations of binary neutron stars. This work explores the use of machine learning methods to speed up the recovery of primitives in relativistic hydrodynamics. Artificial neural networks are trained to replace either the interpolations of a tabulated equation of state or directly the conservative-to-primitive map. The application of these neural networks to simple benchmark problems shows that both approaches improve over traditional root finders with tabular equation-of-state and multi-dimensional interpolations. In particular, the neural networks for the conservative-to-primitive map accelerate the variable recovery by more than an order of magnitude over standard methods while maintaining accuracy. Neural networks are thus an interesting option to improve the speed and robustness of relativistic hydrodynamics algorithms., Comment: 17 pages, 12 figures, 2 tables

Strong magnetic fields play an important role in powering the emission of neutron stars. Nevertheless a full understanding of the interior configuration of the field remains elusive. In this work, we present General Relativistic MagnetoHydroDynamics simulations of the magnetic field evolution in neutron stars lasting 500 ms (5 Alfven crossing times) and up to resolutions of 0.231 km using Athena++. We explore two different initial conditions, one with purely poloidal magnetic field and the other with a dominant toroidal component, and study the poloidal and toroidal field energies, the growth times of the various instability-driven oscillation modes and turbulence. We find that the purely poloidal setup generates a toroidal field which later decays exponentially reaching 1% of the total magnetic energy, showing no evidence of reaching equilibrium. The initially stronger toroidal field setup, on the other hand, loses up to 20% of toroidal energy and maintains this state till the end of our simulation. We also explore the hypothesis, drawn from previous MHD simulations, that turbulence plays an important role in the quasi equilibrium state. An analysis of the spectra in our higher resolution setups reveal, however, that in most cases we are not observing turbulence at small scales, but rather a noisy velocity field inside the star. We also observe that the majority of the magnetic energy gets dissipated as heat increasing the internal energy of the star, while a small fraction gets radiated away as electromagnetic radiation., Comment: 11 pages, 8 figures; Accepted version in MNRAS

We study the detailed process by which slow contraction smooths and flattens the universe using an improved numerical relativity code that accepts initial conditions with non-perturbative deviations from homogeneity and isotropy along two independent spatial directions. Contrary to common descriptions of the early universe, we find that the geometry first rapidly converges to an inhomogeneous, spatially-curved and anisotropic ultralocal state in which all spatial gradient contributions to the equations of motion decrease as an exponential in time to negligible values. This is followed by a second stage in which the geometry converges to a homogeneous, spatially flat and isotropic spacetime. In particular, the decay appears to follow the same history whether the entire spacetime or only parts of it are smoothed by the end of slow contraction., Comment: 27 pages, 10 figures

Numerical relativity is central to the investigation of astrophysical sources in the dynamical and strong-field gravity regime, such as binary black hole and neutron star coalescences. Current challenges set by gravitational-wave and multi-messenger astronomy call for highly performant and scalable codes on modern massively-parallel architectures. We present GR-Athena++, a general-relativistic, high-order, vertex-centered solver that extends the oct-tree, adaptive mesh refinement capabilities of the astrophysical (radiation) magnetohydrodynamics code Athena++. To simulate dynamical space-times GR-Athena++ uses the Z4c evolution scheme of numerical relativity coupled to the moving puncture gauge. We demonstrate stable and accurate binary black hole merger evolutions via extensive convergence testing, cross-code validation, and verification against state-of-the-art effective-one-body waveforms. GR-Athena++ leverages the task-based parallelism paradigm of Athena++ to achieve excellent scalability. We measure strong scaling efficiencies above $95\%$ for up to $\sim 1.2\times10^4$ CPUs and excellent weak scaling is shown up to $\sim 10^5$ CPUs in a production binary black hole setup with adaptive mesh refinement. GR-Athena++ thus allows for the robust simulation of compact binary coalescences and offers a viable path towards numerical relativity at exascale.

We present numerical relativity simulations of cosmological scenarios in which the universe is smoothed and flattened by undergoing a phase of slow contraction and test their sensitivity to a wide range of initial conditions. Our numerical scheme enables the variation of all freely specifiable physical quantities that characterize the initial spatial hypersurface, such as the initial shear and spatial curvature contributions as well as the initial field and velocity distributions of the scalar that drives the cosmological evolution. In particular, we include initial conditions that are far outside the perturbative regime of the well-known attractor scaling solution. We complement our numerical results by analytically performing a complete dynamical systems analysis and show that the two approaches yield consistent results., Comment: 41 pages, 18 figures; accepted for publication in JCAP

Performing a fully non-perturbative analysis using the tools of numerical general relativity, we demonstrate that a period of slow contraction is a `supersmoothing' cosmological phase that homogenizes, isotropizes and flattens the universe both classically and quantum mechanically and can do so far more robustly and rapidly than had been realized in earlier studies., Comment: 6 pages, 3 figures

We compute the gravitational wave energy $E_{\rm rad}$ radiated in head-on collisions of equal-mass, nonspinning black holes in up to $D=8$ dimensional asymptotically flat spacetimes for boost velocities $v$ up to about $90\,\%$ of the speed of light. We identify two main regimes: Weak radiation at velocities up to about $40\,\%$ of the speed of light, and exponential growth of $E_{\rm rad}$ with $v$ at larger velocities. Extrapolation to the speed of light predicts a limit of $12.9\,\%$ $(10.1,~7.7,~5.5,~4.5)\,\%$. of the total mass that is lost in gravitational waves in $D=4$ $(5,\,6,\,7,\,8)$ spacetime dimensions. In agreement with perturbative calculations, we observe that the radiation is minimal for small but finite velocities, rather than for collisions starting from rest. Our computations support the identification of regimes with super Planckian curvature outside the black-hole horizons reported by Okawa, Nakao, and Shibata [Phys.~Rev.~D {\bf 83} 121501(R) (2011)]., Comment: minor corrections to references; 10 pages, 5 figures, in print in Physical Review D

We study gravitational wave emission and the structure and formation of apparent horizons in orbiting black-hole binary systems in higher-dimensional general relativity. For this purpose we present an apparent horizon finder for use in higher dimensional numerical simulations and test the finder's accuracy and consistency in single and binary black-hole spacetimes. The black-hole binaries we model in $D=6$ dimensions complete up to about one orbit before merging or scatter off each other without formation of a common horizon. In agreement with the absence of stable circular geodesic orbits around higher-dimensional black holes, we do not find binaries completing multiple orbits without finetuning of the initial data. All binaries radiate about $0.13\,\%$ to $0.2\,\%$ of the total mass-energy in gravitational waves, over an order of magnitude below the radiated energy measured for four-dimensional binaries. The low radiative efficiency is accompanied by relatively slow dynamics of the binaries as expected from the more rapid falloff of the binding gravitational force in higher dimensions., Comment: 24 pages, 3 Figures, 3 Tables; to match version published in CQG

The grand challenges of contemporary fundamental physics---dark matter, dark energy, vacuum energy, inflation and early universe cosmology, singularities and the hierarchy problem---all involve gravity as a key component. And of all gravitational phenomena, black holes stand out in their elegant simplicity, while harbouring some of the most remarkable predictions of General Relativity: event horizons, singularities and ergoregions. The hitherto invisible landscape of the gravitational Universe is being unveiled before our eyes: the historical direct detection of gravitational waves by the LIGO-Virgo collaboration marks the dawn of a new era of scientific exploration. Gravitational-wave astronomy will allow us to test models of black hole formation, growth and evolution, as well as models of gravitational-wave generation and propagation. It will provide evidence for event horizons and ergoregions, test the theory of General Relativity itself, and may reveal the existence of new fundamental fields. The synthesis of these results has the potential to radically reshape our understanding of the cosmos and of the laws of Nature. The purpose of this work is to present a concise, yet comprehensive overview of the state of the art in the relevant fields of research, summarize important open problems, and lay out a roadmap for future progress., Comment: White Paper for the COST action "Gravitational Waves, Black Holes, and Fundamental Physics", 272 pages, 12 figures; v4: updated references and author list. Overall improvements and corrections. To appear in Classical and Quantum Gravity

This paper addresses the problem of verifying equivalence between a pair of programs that operate over databases with different schemas. This problem is particularly important in the context of web applications, which typically undergo database refactoring either for performance or maintainability reasons. While web applications should have the same externally observable behavior before and after schema migration, there are no existing tools for proving equivalence of such programs. This paper takes a first step towards solving this problem by formalizing the equivalence and refinement checking problems for database-driven applications. We also propose a proof methodology based on the notion of bisimulation invariants over relational algebra with updates and describe a technique for synthesizing such bisimulation invariants. We have implemented the proposed technique in a tool called Mediator for verifying equivalence between database-driven applications written in our intermediate language and evaluate our tool on 21 benchmarks extracted from textbooks and real-world web applications. Our results show that the proposed methodology can successfully verify 20 of these benchmarks.

The collision of black holes and the emission of gravitational radiation in higher-dimensional spacetimes are of interest in various research areas, including the gauge-gravity duality, the TeV gravity scenarios evoked for the explanation of the hierarchy problem, and the large-dimensionality limit of general relativity. We present numerical simulations of head-on collisions of nonspinning, unequal-mass black holes starting from rest in general relativity with $4 \leq D\leq 10$ spacetime dimensions. We compare the energy and linear momentum radiated in gravitational waves with perturbative predictions in the extreme mass ratio limit, demonstrating the strength and limitations of black-hole perturbation theory in this context., Comment: 10 pages, 7 Figures; to match published version

Gravitational waves are one of the most important diagnostic tools in the analysis of strong-gravity dynamics and have been turned into an observational channel with LIGO's detection of GW150914. Aside from their importance in astrophysics, black holes and compact matter distributions have also assumed a central role in many other branches of physics. These applications often involve spacetimes with $D>4$ dimensions where the calculation of gravitational waves is more involved than in the four dimensional case, but has now become possible thanks to substantial progress in the theoretical study of general relativity in $D>4$. Here, we develop a numerical implementation of the formalism by Godazgar and Reall (Ref.[1]) -- based on projections of the Weyl tensor analogous to the Newman-Penrose scalars -- that allows for the calculation of gravitational waves in higher dimensional spacetimes with rotational symmetry. We apply and test this method in black-hole head-on collisions from rest in $D=6$ spacetime dimensions and find that a fraction $(8.19\pm 0.05)\times 10^{-4}$ of the Arnowitt-Deser-Misner mass is radiated away from the system, in excellent agreement with literature results based on the Kodama-Ishibashi perturbation technique. The method presented here complements the perturbative approach by automatically including contributions from all multipoles rather than computing the energy content of individual multipoles., Comment: 30 pages, 2 figures