2.2 The physics of gravitational waves from compact binary coalescences
2.2.1 The mathematical formulation of gravitational waves
2.2.1.1 Effective-one-body + numerical relativity (EOBNR) waveforms
The EOBNR waveforms combine the effective-one-body (EOBNR) formalism with NR results. effective- one-body (EOB) methods map the dynamics onto a test particle in an external effective metric. Though the EOB equations can be expressed analytically, what is done in practice is a non-perturbative resummation of the PN expansion of the equations of motion [48]. For a single system, EOB waveforms are calculated for each leading l, mmode (using spherical harmonic notation), but still have a few tunable parameters.
Each waveform is calculated separately in two parts: theinspiral-plunge and themerger-ringdown. For the systems whose waveforms have been calculated with NR, Buonanno et al. calibrate the inspiral-plunge EOB waveforms against the NR waveforms and set the tunable parameters to achieve the greatest amplitude and phase consistency between the two [49]. The inspiral-plunge waveform is then stitched to the merger- ringdown waveform, which is a sum of 8 quasinormal modes. The tuned EOBNR waveforms used in the
Figure 2.6: A screenshot, at merger, from a NR simulation of two black holes with a mass ratio of 6:1 and non-aligned spins of .91 and .3, respectively. Note the amplitude and phase modulation, which is due to the precession of the orbital plane resulting from the spin-orbit coupling of the non-aligned spins. Visit http://www.black-holes.org/explore2.html for the full movie and more animated simulations.
search for high-mass CBCs have been tuned using NR for mass ratiosm1/m2= 1, 2, 3, 4 ,6 and total masses M = 20−200M[49].
There are two different versions of EOBNR waveforms used in this thesis. EOBNRv1 is used, for his- torical reasons, to create the template banks as discussed in Section 7.3.2. EOBNRv2 is used to create the simulated signals we use to test the sensitivity of our pipeline and create upper limits. Though the EOBNR approach works for waveforms from systems where the compact objects are spinning, the code was not re- viewed in time for it to be included in the search described in this thesis. Figure 2.7 and Figure 2.8 show the EOBNRv2 waveforms for the equal-mass case, as compared to the waveforms discussed in the following section. Although EOBNRv2 waveforms were only tested for mass ratios up to 6, they should be valid in the limit of large mass ratios, as they are created on the model of a test particle orbiting an effective potential;
Figure 2.9 and Figure 2.10 show these EOBNRv2 waveforms for the asymmetric mass ratios on the template bank for the highmass search (25 - 100M).
The EOBNR waveforms are created in the time domain and are fast-Fourier transformed (FFTed) before they are used in the analysis. The FFT waveform multiplied by the square root of the frequency can be laid atop the strain amplitude sensitivity of the detectors, allowing us to easily visualize our ability to detect a particular signal. The strain amplitude sensitivity of the detectors is a result of design choices and known and unknown noise sources, which will be described in Section 3.4.
Figure 2.7: Time-domain waveforms for a 12.5M+ 12.5Msystem.
Figure 2.8: Time-domain waveforms for a 45M+ 45Msystem.
Figure 2.9: An EOBNRv2 time-domain waveform for a 1M+ 24Msystem. Note that the merger and ringdown are present even though not visible due to the scale of the plot. The IMRPhenomB waveform is not plotted, as it is not valid for this mass ratio.
Figure 2.10: An EOBNRv2 time-domain waveform for a 1M+ 99Msystem. The IMRPhenomB wave- form is not plotted, as it is not valid for this mass ratio.
Figure 2.11: Waveforms for a 12.5M+ 12.5M system in the frequency domain, compared to the mode of H1’s noise amplitude spectral density during S6 [4]. The EOBNR waveform was originally in the time domain, and was fast Fourier transformed into the frequency domain, resulting in non-physical wiggles. The green dashed curve indicates the frequency journey of an inspiral-only waveform, whose amplitude has been set by the IMRPhenomB waveform. Merger is short and has an undefined duration. The Fourier transform of a ringdown is the imaginary part of a Lorentzian, and can be seen in this plot beginning when the blue or red curve deviates (has a less steep slope) from the green dashed curve and continuing through the steeper negative slope towards the right of the plot, remembering that the wiggles on the blue curve are non-physical.
Figure 2.12: Waveforms for a 50M+ 50Msystem in the frequency domain, compared to the mode of H1’s noise amplitude spectral density during S6 [4]. The EOBNR waveform was originally in the time domain, and was fast Fourier transformed into the frequency domain, resulting in non-physical wiggles throughout the waveform, since the waveform has a finite duration. The green dashed curve indicates the frequency journey of an inspiral-only waveform, whose amplitude has been set by the IMRPhenomB waveform. Merger is short and has an undefined duration. The Fourier transform of a ringdown is the imaginary part of a Lorentzian, and can be seen in this plot beginning when the blue or red curve deviates (has a less steep slope) from the green dashed curve and continuing through the steeper negative slope towards the right of the plot, remembering that the wiggles on the blue curve are non-physical.