6.5 A community CCE tool

The importance of accurate waveforms has prompted development of a newly-designed CCE tool [16Jump To The Next Citation Point], which meets the advanced LIGO accuracy standards. Preliminary progress was reported in [17]. The CCE tool is available for use by the numerical relativity community under a general public license as part of the Einstein Toolkit [104]. It can be applied to a generic Cauchy code with extraction radius as small as r = 20 M, which provides flexibility for many applications besides binary black holes, such as waveform extraction from stellar collapse.

The matching interface was streamlined by introducing a pseudospectral decomposition of the Cauchy metric in the neighborhood of the extraction worldtube. This provides economical storage of the boundary data for the characteristic code so that the waveform at ℐ+ can be obtained in post-processing with a small computational burden compared to the Cauchy evolution. The new version incorporates stereographic grids with circular patch boundaries [13], which eliminates the large error from the corners of the square patches used previously. The finite-difference accuracy of the angular derivatives was increased to fourth order. Bugs were eliminated that had been introduced in the process of parallelizing the code using the Cactus framework [292]. In addition, the worldtube module, which supplies the inner boundary data for the characteristic evolution, was revamped so that it provides a consistent, second-order–accurate start-up algorithm for numerically-generated Cauchy data. The prior module required differentiable Cauchy data, as provided by analytic testbeds, to be consistent with convergence.

These changes led to clean second-order convergence of all evolved quantities at finite locations. Because some of the hypersurface equations become degenerate at ℐ+, certain asymptotic quantities, in particular the Bondi news function, are only first-order accurate. However, the clean first-order convergence allows the application of Richardson extrapolation, based upon three characteristic grid sizes, to extract waveforms with third-order accuracy.

The error norm for the extracted news function, ℰ1(N ) as defined in Equation (85View Equation), has been measured for the simulation of the inspiral of equal mass, non-spinning black holes obtained via a BSSN simulation [16Jump To The Next Citation Point]. The advanced LIGO criterion for detection (87View Equation), was satisfied for 𝜖max = .005 (which corresponds to less than a 10% signal loss) and for values of C1 throughout the entire binary mass range. The criterion (88View Equation) for measurement is more stringent. For the expected lower bound of the calibration factor ηmin = 0.4, for C1 = .24 (corresponding to the most demanding small mass limit) and for the most optimistic advanced LIGO signal-to-noise ratio ρ = 100, the requirement for measurement is −4 ℰ1(N ) ≤ 9.6 × 10. This measurement criterion was satisfied throughout the entire binary mass range by the numerical truncation error ℰ1(N ) in the CCE waveform.

These detection and measurement criteria were satisfied for a range of extraction worldtubes extending from R = 20 M to R = 100M. The ℰ1(N ) error norm decreased with larger extraction radius, as expected since the error introduced by characteristic evolution depends upon the size of the integration region between the extraction worldtube and ℐ+. However, the modeling error corresponding to the difference in waveforms obtained with extraction at R = 50 M as compared to RE = 100 M only satisfied the measurement criterion for signal-to-noise ratios ρ < 25 (which would still cover the most likely advanced LIGO events). This modeling error results from the different initial data, which correspond to different extraction radii. This error would be smaller for longer simulations with a higher number of orbits. The results suggest that the choice of extraction radius should be balanced between a sufficiently large radius to reduce initialization effects and a sufficiently small radius where the Cauchy grid is more highly refined and outer boundary effects are better isolated.

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