Binary black hole mergers

SpECTRE can simulate black holes that orbit each other and eventually merge, emitting gravitational waves. SpECTRE uses the Generalized Harmonic formulation of Einstein's equations of general relativity to solve this problem. Since we expect the solution of Einstein's equations to be smooth for the BBH problem, we represent our solution using the Discontinuous Galerkin (DG) method because of its ability to represent smooth functions to high accuracy. Also, DG allows SpECTRE to parallelize the BBH problem.

Publications:

Lovelace, Nelli, et al. (2024). Simulating binary black hole mergers using discontinuous Galerkin methods. [129]

Apparent horizons of an equal mass non-spinning BBH. The colorful surface represents the lapse value in the equatorial plane and the arrows depict the shift vector. Image credit: Alex Carpenter, CSUF

Overview of a binary black hole simulation in SpECTRE. Upper left panel: computational grid and shape of black hole horizons during merger. A common horizon has formed (blue) that envelops the two original horizons (black). Upper right panel: trajectories of the black holes until merger. This inspiral is 18 orbits long and approximately circular. Bottom panel: gravitational waveform extracted with CCE (see below). Image credit: Geoffrey Lovelace, CSUF

Binary black hole initial data

SpECTRE can generate initial data to start simulations of merging black holes. This problem involves solving the elliptic constraint sector of the Einstein equations for a slice of spacetime that contains two black holes with the requested parameters. SpECTRE uses the XCTS formulation with a non-conformally-flat background defined by the superposed Kerr-Schild formalism to reach high spins. Black holes are represented by excisions and boundary conditions.

Publications:

Vu et al. (2022). A scalable elliptic solver with task-based parallelism for the SpECTRE numerical relativity code. [197]
Vu (2024). A discontinuous Galerkin scheme for elliptic equations on extremely stretched grids. [199]

An initial slice of spacetime containing two black holes in orbit around each other. Shown is the lapse variable. The two black holes are represented as boundary conditions on excised regions of the computational domain. Image credit: Nils Vu, Caltech

Cauchy-characteristic evolution (CCE)

SpECTRE implements a novel Cauchy-characteristic evolution (CCE) system for extracting gravitational waveforms from our simulations. It evolves the Einstein equations on null slices to infinity, which is more accurate than extrapolation and allows us to extract the gravitational memory effect. The CCE waveform extraction is publicly available as a standalone module.

Tutorial:

Running CCE

Publications:

Moxon et al. (2023). SpECTRE Cauchy-characteristic evolution system for rapid, precise waveform extraction. [142]
Moxon et al. (2020). Improved Cauchy-characteristic evolution system for high-precision numerical relativity waveforms. [141]

Binary neutron star mergers

SpECTRE can simulate merging neutron stars and other general-relativistic magneto-hydrodynamic (GRMHD) problems with dynamic gravity. Our DG-FD hybrid scheme accelerates smooth regions of the grid with high-order spectral methods (see DG-FD hybrid method).

Publications:

Deppe et al. (2024). Binary neutron star mergers using a discontinuous Galerkin-finite difference hybrid method. [51]

Simulation of two merging neutron stars. The colors show density contours. Video credit: Nils Vu, Caltech

Curved and moving meshes with control systems

Our computational domains in SpECTRE are designed to adapt to the geometry of the problems we want to solve. They can be curved, e.g. to wrap around excision regions in binary black hole problems (see Binary black hole mergers) or to resolve the wavezone in binary neutron star merger. They can also rotate and deform in time using control systems, which reactively adjust coordinate maps to track the position and shape of the black hole excisions or neutron stars.

Adaptive mesh refinement

Our discontinuous Galerkin methods allow two types of mesh refinement: splitting elements in half along any dimension (h-refinement) or increasing their polynomial expansion order (p-refinement). The former allows us to distribute computational cost to supercomputers, while the latter allows us to use these resources efficiently by decreasing the numerical error exponentially with the number of grid points where the solution is smooth. Our adaptive mesh refinement technology decides which type of refinement to apply in each region of the domain.

DG-FD hybrid method

Our hydrodynamical simulations use a discontinuous Galerkin-finite difference (DG-FD) hybrid method: smooth regions of the simulation are evolved with an efficient DG scheme and non-smooth regions fall back to a robust FD method. Shocks and discontinuities on the grid are tracked with a troubled-cell indicator (TCI) to switch between DG and FD. This approach accelerates our simulations by reducing the computational resources spent on smooth regions of the grid, e.g. when evolving inspiral binary neutron stars and their gravitational radiation.

Publications:

Deppe et al. (2022). A high-order shock capturing discontinuous Galerkin-finite difference hybrid method for GRMHD. [53]
Deppe et al. (2022). Simulating magnetized neutron stars with discontinuous Galerkin methods. [50]

Simulation of the Kelvin-Helmholtz instability (KHI). Squares indicate cells that have switched to a finite-difference method. They track shocks and discontinuities in the solution. The rest of the domain uses an efficient DG method. Video credit: Nils Deppe, Cornell University

Binary black holes in scalar Gauss-Bonnet gravity

SpECTRE can generate initial data for binary black holes in scalar Gauss-Bonnet gravity, evolve the modified Einstein equations, and extract the gravitational and scalar radiation.

Publications:

Nee et al. (2024). Quasistationary hair for binary black hole initial data in scalar Gauss-Bonnet gravity [145]
Lara et al. (2024). Scalarization of isolated black holes in scalar Gauss-Bonnet theory in the fixing-the-equations approach. [120]

Binary black hole initial data in scalar Gauss-Bonnet gravity, in a configuration where the two black holes have opposite charge. The scalar field is solved such that it is in equilibrium with the gravity background, minimizing initial transients in the evolution and giving control over the simulation parameters. Image credit: Peter James Nee, MPI for Gravitational Physics Potsdam, Germany

Thermal noise in gravitational wave detectors

We have applied the SpECTRE technology to an interdisciplinary problem, simulating the Brownian thermal noise in the mirrors of interferometric gravitational-wave detectors at unprecedented accuracy. It uses the SpECTRE elliptic solver [197] to solve an elasticity problem, which connects to the thermal noise problem through the fluctuation dissipation theorem.

Publications:

Vu et al. (2024). High-accuracy numerical models of Brownian thermal noise in thin mirror coatings. [198]