10.2 shows one example of IMR and ringdown-only waveforms produced for a system with spins of 0.88 and 0.84 and masses of 8.9M and 6.3 M respectively. The end time of the inspiral and the start time of the ringdown are marked, and it is only after the latter point that the amplitude of the ringdown-only injection becomes non-zero.
This is best seen in figure 10.3 which zooms in on the transition between the inspiral and ringdown. After this point the amplitudes match exactly. Figure 10.4 shows the frequency evolution of the same IMR waveform with the inset showing just the final 20 ms of the waveform. Here we see how the inspiral frequency increases rapidly until the constant ringdown frequency is reached. This marks the ringdown start time. In figure 10.5 we plot the same quantity in the dimensionless quantities. The final spin of the black hole is 0.9 and the ringdown M ωR = 0.68. This is in agreement with figure 3.3 for the l =m = 2 mode. We can also compare this to figure 3.7.
If we want to evaluate how well we are recovering the injections it is necessary to calculate and record the ringdown parameters to compare with the output of the ringdown filter. Solvinga+and a× forAandιin equations (7.1) and (7.2) enables us to calculate the effective distance and the percentage of mass radiated as gravitational waves . For every injection these parameters are written out to a sim ringdown table.
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Figure 10.6: Number of IMR and ringdown-only injections recovered in H1 as a function of the frequency of the final ringdown. The vertical lines denote the template bank boundaries.
Figure 10.7: Detected versus injected ringdown frequency for IMR and ringdown-only waveforms in H1.
us to count how many injections were found and how well the parameters were re- covered. As shown in figure 10.6, we found that approximately 18% more injections were found in the IMR run than in the ringdown-only run, with the excess appearing above an injected ringdown frequency finj of ∼ 600 Hz. Figure 10.7 shows that of the injections that were found in common by the two runs, the detected frequency fdet of injections withfinj below about 200 Hz were fairly consistent between the two runs. Those injections with finj >200 Hz were not so consistent; the ringdown-only injections were found with templates close to the injected ringdown parameters, but the IMR injections were mostly found by templates in the 100–200 Hz band.
This can be explained as follows: for IMR injections with low ringdown frequency the inspiral part of the waveform is outside the LIGO band. As we increase the ringdown frequency, an increasing proportion of the inspiral and merger enters the band, matching an increasing number of templates. This is demonstrated in figures 10.8 and 10.9, which show the templates that rang off during the 120 ms around a ringdown-only and an IMR injection, respectively, where the ringdown frequency was finj ∼ 1500 Hz. In the ringdown-only case the only templates that ring are close in frequency to the frequency of the injection and do so right at the time of peak amplitude of the waveform, as indicated by the dashed lines. For the IMR case, however, most of the templates in the bank ring off. The templates ring off just as expected for the characteristic chirp frequency evolution of an inspiral; the inspiral enters the LIGO band at low frequency and its frequency increases until it reaches the ringdown. The template that rings up the loudest (and hence is the template associated with the injection) in this case is at ∼110 Hz, as indicated in the plot by the red horizontal line. This is far from the ringdown frequency, denoted by the black horizontal line, but it is where the LIGO strain sensitivity is best (see figure 6.2).
In figure 10.10 we plot the initial component masses of all the IMR injections with finj > 50 Hz that were found at the correct ringdown frequency in red, and those that were found incorrectly in green. The plot shows that the majority of the injections that were found incorrectly by the ringdown search fall within the scope of the S4 BBH search, and thus that search should be able to find the signal and correctly
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Figure 10.8: Frequency versus time for the templates that rang up around the time of a ringdown-only injection. The colour of the data points represents the quality factor of the template. The black lines represent the frequency and time of the injection, and the red lines represent the frequency and time of the template with the largest signal-to-noise ratio.
Figure 10.9: Frequency versus time for the templates that rang up around the time of an IMR injection. The colour of the data points represents the quality factor of the template. The black lines represent the frequency and time of the injection, and the red lines represent the frequency and time of the template with the largest signal-to-noise ratio.
identify the component masses. Most of those injections that were correctly identified lie outside that region, and so we can conclude that between the two searches the mass space is covered quite well.
Our results concur with a study by Baumgarte et al. [92] in which two numerical waveforms — one with ringdown frequency of∼80 Hz and the other withf ∼280 Hz
— were filtered with ringdown templates. For the former waveform the best-matched template triggered at the time of the ringdown, whereas in the latter case, when the ringdown frequency was higher than LIGO’s most sensitive band, the best match occurred earlier, during the inspiral phase.
Figure 10.10: Initial masses of the binary components for IMR injections found by correct (red) and incorrect (green) ringdown templates. The black line represents the upper limit to the mass range of the S4 binary black hole inspiral search.