A high-order, conservative integrator with local time-stepping

We present a family of multistep integrators based on the Adams-Bashforth methods. These schemes can be constructed for arbitrary convergence order with arbitrary step size variation. The step size can differ between different subdomains of the system. It can also change with time within a given subdomain. The methods are linearly conservative, preserving a wide class of analytically constant quantities to numerical roundoff, even when numerical truncation error is significantly higher. These methods are intended for use in solving conservative PDEs in discontinuous Galerkin formulations or in finite-difference methods with compact stencils. A numerical test demonstrates these properties and shows that significant speed improvements over the standard Adams-Bashforth schemes can be obtained.Comment: 29 page

High precision calculation of generic extreme mass ratio inspirals

Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Physics, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 63-64).Orbits around black holes evolve due to gravitational-wave emission, losing energy and angular momentum, and driving the orbiting body to slowly spiral into the black hole. Recent theoretical advances now make it possible to model the impact of this wave emission on generic (eccentric and inclined) black hole orbits, allowing us to push beyond the handful of constrained (circular or equatorial) cases that previous work considered. This thesis presents the first systematic study of how generic black hole orbits evolve due to gravitational-wave emission. In addition to extending the class of orbits which can be analyzed, we also introduce a new formalism for solving for the wave equation which describes radiative backreaction. This approach is based on a spectral decomposition of the radiation field originally introduced by Mano, Suzuki, and Takasugi (MST), and was then adapted for numerical analysis by Fujita and Tagoshi (FT). We find that the MST-FT formalism allows us to compute various quantities significantly more accurately than previous work, even in strong field regimes. We use this code to explore the location in orbital parameter space of the surface at which the evolution of orbital eccentricity changes sign from negative (orbits circularize) to positive (orbits become more eccentric).by William Throwe.S.B

A High-Order, Conservative Integrator with Local Time-Stepping

We present a family of multistep integrators based on the Adams--Bashforth methods. These schemes can be constructed for arbitrary convergence order with arbitrary step size variation. The step size can differ between different subdomains of the system. It can also change with time within a given subdomain. The methods are linearly conservative, preserving a wide class of analytically constant quantities to numerical roundoff, even when numerical truncation error is significantly higher. These methods are intended for use in solving conservative PDEs in discontinuous Galerkin formulations or in finite-difference methods with compact stencils. A numerical test demonstrates these properties and shows that significant speed improvements over the standard Adams--Bashforth schemes can be obtained

What does a binary black hole merger look like?

We present a method of calculating the strong-field gravitational lensing caused by many analytic and numerical spacetimes. We use this procedure to calculate the distortion caused by isolated black holes and by numerically evolved black hole binaries. We produce both demonstrative images illustrating details of the spatial distortion and realistic images of collections of stars taking both lensing amplification and redshift into account. On large scales the lensing from inspiraling binaries resembles that of single black holes, but on small scales the resulting images show complex and in some cases self-similar structure across different angular scales.Comment: 10 pages, 12 figures. Supplementary images and movies can be found at http://www.black-holes.org/the-science-numerical-relativity/numerical-relativity/gravitational-lensin

Extreme Mass-Ratio Inspirals in the Effective-One-Body Approach: Quasi-Circular, Equatorial Orbits around a Spinning Black Hole

We construct effective-one-body waveform models suitable for data analysis with LISA for extreme-mass ratio inspirals in quasi-circular, equatorial orbits about a spinning supermassive black hole. The accuracy of our model is established through comparisons against frequency-domain, Teukolsky-based waveforms in the radiative approximation. The calibration of eight high-order post-Newtonian parameters in the energy flux suffices to obtain a phase and fractional amplitude agreement of better than 1 radian and 1 % respectively over a period between 2 and 6 months depending on the system considered. This agreement translates into matches higher than 97 % over a period between 4 and 9 months, depending on the system. Better agreements can be obtained if a larger number of calibration parameters are included. Higher-order mass ratio terms in the effective-one-body Hamiltonian and radiation-reaction introduce phase corrections of at most 30 radians in a one year evolution. These corrections are usually one order of magnitude larger than those introduced by the spin of the small object in a one year evolution. These results suggest that the effective-one-body approach for extreme mass ratio inspirals is a good compromise between accuracy and computational price for LISA data analysis purposes.Comment: 21 pages, 8 figures, submitted to Phys. Rev.

Computation of Displacement and Spin Gravitational Memory in Numerical Relativity

We present the first numerical relativity waveforms for binary black hole mergers produced using spectral methods that show both the displacement and the spin memory effects. Explicitly, we use the SXS Collaboration's $\texttt{SpEC}$ code to run a Cauchy evolution of a binary black hole merger and then extract the gravitational wave strain using $\texttt{SpECTRE}$'s version of a Cauchy-characteristic extraction. We find that we can accurately resolve the strain's traditional $m=0$ memory modes and some of the $m\not=0$ oscillatory memory modes that have previously only been theorized. We also perform a separate calculation of the memory using equations for the Bondi-Metzner-Sachs charges as well as the energy and angular momentum fluxes at asymptotic infinity. Our new calculation uses only the gravitational wave strain and two of the Weyl scalars at infinity. Also, this computation shows that the memory modes can be understood as a combination of a memory signal throughout the binary's inspiral and merger phases, and a quasinormal mode signal near the ringdown phase. Additionally, we find that the magnetic memory, up to numerical error, is indeed zero as previously conjectured. Lastly, we find that signal-to-noise ratios of memory for LIGO, the Einstein Telescope (ET), and the Laser Interferometer Space Antenna (LISA) with these new waveforms and new memory calculation are larger than previous expectations based on post-Newtonian or Minimal Waveform models.Comment: 20 pages, 11 figures; 10.1103/PhysRevD.102.104007. Corrected a minor sign error in Eqs. 27, 40, 42, 43, and 5

Computation of displacement and spin gravitational memory in numerical relativity

We present the first numerical relativity waveforms for binary black hole mergers produced using spectral methods that show both the displacement and the spin memory effects. Explicitly, we use the SXS (Simulating eXtreme Spacetimes) Collaboration’s SpEC code to run a Cauchy evolution of a binary black hole merger and then extract the gravitational wave strain using SpECTRE’s version of a Cauchy-characteristic extraction. We find that we can accurately resolve the strain’s traditional m=0 memory modes and some of the m≠0 oscillatory memory modes that have previously only been theorized. We also perform a separate calculation of the memory using equations for the Bondi-Metzner-Sachs charges as well as the energy and angular momentum fluxes at asymptotic infinity. Our new calculation uses only the gravitational wave strain and two of the Weyl scalars at infinity. Also, this computation shows that the memory modes can be understood as a combination of a memory signal throughout the binary’s inspiral and merger phases, and a quasinormal mode signal near the ringdown phase. Additionally, we find that the magnetic memory, up to numerical error, is indeed zero as previously conjectured. Last, we find that signal-to-noise ratios of memory for LIGO, the Einstein Telescope, and the Laser Interferometer Space Antenna with these new waveforms and new memory calculation are larger than previous expectations based on post-Newtonian or minimal waveform models

Collective filters: a new approach to analyze the gravitational-wave ringdown of binary black-hole mergers

We propose two frequency-domain filters to analyze ringdown signals of binary black hole mergers. The first rational filter is constructed based on a set of (arbitrary) quasi-normal modes (QNMs) of the remnant black holes, whereas the second full filter comes from the transmissivity of the remnant black holes. The two filters can remove corresponding QNMs from original time-domain ringdowns, while changing early inspiral signals in a trivial way - merely a time and phase shift. After filtering out dominant QNMs, we can visualize the existence of various subdominant effects. For example, by applying our filters to a GW150914-like numerical relativity (NR) waveform, we find second-order effects in the (l = 4, m = 4), (l = 5, m = 4) and (l = 5, m = 5) harmonics; the spherical-spheroidal mixing mode in the (l = 2,m = 2) harmonic; and a mixing mode in the (l = 2,m = 1) harmonic due to a gravitational recoil. In another NR simulation where two component spins are anti-aligned with the orbital angular momentum, we also find retrograde modes. Additionally, we propose to use the rational filter to estimate the start time of a QNM. The filters are sensitive to the remnant properties (i.e., mass and spin) and thus have a potential application to future data analyses and parameter estimations. We also investigate the stability of the full filter. Its connection to the instability of QNM spectra is discussed