Spectral characteristic evolution: a new algorithm for gravitational wave propagation

Abstract

Abstract. We present a spectral algorithm for solving the full nonlinear vacuum Einstein field equations in the Bondi framework. Developed within the Spectral Einstein Code (SpEC), we demonstrate spectral characteristic evolution as a technical precursor to Cauchy Characteristic Extraction (CCE), a rigorous method for obtaining gauge-invariant gravitational waveforms from existing and future astrophysical simulations. We demonstrate the new algorithm's stability, convergence, and agreement with existing evolution methods. We explain how an innovative spectral approach enables a two orders of magnitude improvement in computational efficiency. What is Characteristic Evolution, and why?As an international network of gravitational wave observatories come online, the race to the first direct detection of gravitational waves is expected to herald the beginning of gravitational wave astronomy. Detectors such as Advanced LIGO, VIRGO, GEO, and KAGRA aim for strain sensitivities approaching 10 −24 [1,2,3,4]. Nevertheless, signal candidates from compact binaries or supernovae will be on the cusp of detectability, with very poor signal to noise ratios requiring matched filtering for detection [5]. Matched filtering requires a comprehensive template bank, the generation of which has been a primary goal of the field of numerical relativity. These templates cover a range of expected astronomical phenomena, and are generated by a variety of numerical codes, including the Spectral Einstein Code (SpEC) [6]. Filling out the template bank requires a balance of numerical relativity and analytic waveforms and effective-one-body[8]), with waveform models often calibrated using numerical results [9,10,11].One technical challenge facing the construction of a large template bank is extraction of gauge invariant waveforms from simulations. In general, large computationally intensive simulations are needed to describe the physics of events such as supernovae and compact object binary inspirals. While a waveform-like signal can readily be extracted from arXiv:1406.7029v2 [gr-qc] 24 Sep 2014 anywhere in the computational domain, waveforms are only rigorously defined at future null infinity. Finite radii waveform approximations are universally contaminated by coordinate system dynamics, or gauge effects, which are poorly understood and nearly impossible to remove or even quantify. Comparison with Cauchy Characteristic Extraction, or CCE, an alternative which applies Characteristic Evolution to enable waveform computation at future null infinity, suggests that extrapolation gauge errors could dominate the global error [12]. Characteristic Evolution has been previously implemented at up to 4 th order radial accuracy [13], while complete extraction has been achieved with finite difference/volume methods up to 2 nd order [14,15]. Here we implement inner boundary extraction and evolution. This must be combined with an appropriate outer boundary algorithm to generate the gauge-invariant news at future null infinity. In a hypothetical complete ...

Targeted numerical simulations of binary black holes for GW170104

“…Gravitational wave extraction is done by three independent methods: direct extraction of the Newman-Penrose quantity ψ 4 at finite radius [58,70,79], extraction of the strain h by matching to solutions of the Regge-WheelerZerilli-Moncrief equations at finite radius [80,81], and Cauchy-characteristic extraction [82][83][84][85][86]. The latter method directly provides gravitational waveforms at future null infinity, while for the former two methods the waveforms are computed at a series of finite radii and then extrapolated to infinity [87].…”

Gauge invariant spectral Cauchy characteristic extraction

“…The spectral algorithm used to solve these equations and the treatment of the nonlinear terms N β , N Q , N U , N W , N J are detailed in Handmer and Szilágyi [15]. Here, we extend the characteristic spectral algorithm to calculating the gauge invariant Bondi news at I + .…”

“…It has more recently been implemented as a spectral code within the Spectral Einstein Code (SpEC) by Handmer and Szilágyi [15], upon which the present work is based.…”

Spectral Cauchy characteristic extraction of strain, news and gravitational radiation flux

“…Our results are based upon the same generic precessing binary black hole run taken from Taylor et al [45], which was also used in [21] to calibrate the SpEC characteristic code and in [22] to compare waveform extraction using the SpEC and Pitt null codes using the gauge invariant version of the of the news function computed in computational coordinates. The physical parameters are mass ratio q = 3, black hole dimensionless spins χ 1 = (0.7, 0, 0.7)/ √ 2 and χ 2 = (−0.3, 0, 0.3)/ √ 2, number of orbits 26, total time T = 7509M , initial eccentricity 10 −3 , initial frequency ω ini = 0.032/M and extraction radius R = 100M , where M is the total mass of the black holes.…”

“…For a review, see [15]. A version of this initial-boundary value problem based upon a timelike worldtube [16] has been implemented as a characteristic evolution code, the PITT null code [17,18,19,20], and more recently as the SpEC characteristic code [21,22], both of which incorporate a Penrose compactification of the exterior spacetime extending to I + . In this way, characteristic evolution coupled to Cauchy evolution has been implemented to give an accurate numerical computation of the Bondi news function, which determines both the waveform and the radiated energy-momentum.…”