bank. While the signals we are looking for may be as long as an hour and extend up to very high frequencies, the signal spends most of its time within a very small bandwidth, allowing for the down- sampling of the filters without significant loss of SNR. Additionally, template banks are by design highly redundant. As a result, an orthogonal decomposition of the template bank can greatly reduce the required number of templates. We filter with the non-physical orthogonal waveforms instead of the redundant physical templates and only reconstruct the SNR when an excess power is found within the orthogonal projection.
We have implemented these techniques in a new pipeline, known asgstlal [89,90], which is built upon a stream-based infrastructure known asGStreamer [91] and the sameLALlibrary [92] upon whichIHOPEis built. The stream-based infrastructure provides an elegant solution to a number of other computational problems, such as the ability to filter over gaps in the data, a crucial requirement for signals that exceed the length of a typical stable lock stretch. Furthermore, the use of stream- based technology allows for a tunable latency of trigger generation, ideal for electromagnetic followup of mergers involving neutron stars, for which there may be bright electromagnetic counterparts [93].
In the following chapters, we use thegstlal pipeline described here to address the problem of extending the search parameter space to include spin.
5.1 Scaling CBC Searches from Initial LIGO to Advanced
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Figure 5.1: Comparison of initial and advanced LIGO spectra. The bandwidth of aLIGO will extend beyond that of iLIGO in the low frequency regime. With this added bandwidth comes an increased computational scale for CBC searches in aLIGO. Even the early-stage detector design (operating at low power without a signal recycling mirror) indicates a low frequency sensitivity well beyond that of iLIGO and is in fact quite comparable to the late-stage detector design.
Early and late stage aLIGO detectors may have non-negligible sensitivity down to frequencies as low asf ∼10 Hz. Extending the templates down to this frequency requires at the lowest masses filters nearly half an hour long.
Recall that to account for the non-stationary behavior of the detectors, we break the data into small segments which can be analyzed independently and during which the background can be considered approximately stationary. Estimating the background PSD requires using segments which are factors of∼15 longer than the signal itself; otherwise, the signal corrupts the PSD estimation.
In our iLIGO analyses, we used a segment length of 2048 seconds and measured the PSD in sixteen partially overlapping sub-segments over the course of the larger 2048 second segment [63]. Given that the longest template for flow = 40 Hz is shorter than 60 seconds, any signal existing in the data would span no more than two of these measurements. We take the median of the measured PSDs to minimize the impact of the signal on measuring the background. If we apply this method of PSD estimation directly to an aLIGO search, we will need to have contiguous coincident segments that are nearly half a day long to accommodate waveforms as long as half an hour. Restricting an analysis only to segments of such a long duration would be disastrous for the search.
A natural question to ask while examining Fig. 5.2 is whether there are small sacrifices we can
(a)m1/m2= 1 (b)m1/m2= 4
Figure 5.2: Dependence of coalescence signal duration on the masses of the system. The duration of an inspiral template is a strong function the low frequency cutoff chosen for filtering. In the plots above, we see that templates exceeding several minutes and as long as half an hour will be required to filter LIGO data down to 10 Hz. We show the dependence of duration on mass for mass ratiosq= 1 (left) andq= 4 (right).
For fixed total mass, high mass ratio systems tend to be longer in duration compared to nearly equal mass systems. Spin effects can further increase the duration of the signal.
Figure 5.3: Signal-to-noise available in the low frequency band of aLIGO.For detectors with aLIGO sensi- tivities, a non-negligible amount of the SNR is contained in the band 10–40 Hz, a frequency range which was inaccessible to iLIGO observations. Above we show the fraction of the recovered SNR as a function of the low frequency cutoff of integration for three different noise curves. Each curve uses the same inspiral-only approximation for a binary neutron star signal withm1=m2 = 1.4M. We see that for such a signal, as much as 50% of the available SNR may be contained in the low frequency band 10–40 Hz.
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Figure 5.4: Number of templates required to cover a space as a function of low frequency cutoff. We show the number of templates required to fully cover the parameter space 1M≤mi≤24Mand 2M≤Mtotal≤ 25M at a minimal match ofM = 0.97 for a late aLIGO sensitivity as we vary the low frequency cutoff.
We see clearly that the required number of templates increases precipitously as the low frequency cutoff approachesflow = 10 Hz. We have furthermore illustrated the concentration of templates as a function of total mass as we varyflow, which shows that the plurality of templates is in the low mass range.
make in order to avoid using the long filters that go all the way down toflow= 10 Hz. The figure shows that the long duration templates span only a small subset of the much larger parameter space.
One should consider whether the loss of SNR from neglecting the low frequency content of the signal is sufficiently small to not warrant having to develop new data analysis solutions. To quantitatively understand the impact of the low frequency sensitivity for signal recovery, we must also fold in information about the detectors. In Fig. 5.3, we show the fractional accumulated SNR as a function of the low frequency cutoff and detector noise model for a BNS system with mi = 1.4M. We see that for an aLIGO sensitivity as much as 20% of the available SNR is accumulated between 10 Hz and 20 Hz for both the early and late aLIGO models. In terms of detection rates, including low frequency content for the filter could yield up to a factor (1/0.8)3≈2 increase. We simply cannot afford to neglect the low frequency content of the signal. We must be able to handle filters that span up to half an hour in duration.
Long templates increase the computational cost of the searchper filter. To compound on this situ- ation, longer filters require many more templates to cover a given parameter space. As the waveforms get longer and longer, there are more cycles in band and a longer time over which two similar signals can dephase. In Fig. 5.4, we show the number of templates generated by lalapps tmpltbank [92]
for the same mass range covered by the low mass search presented in the last chapter as a function
of the low frequency cutoff (namely components inmi∈[1,24]M andMtotal∈[2,25]M). We use for this figure a minimal match ofM = 0.97. We see that moving fromflow= 20 Hz toflow= 10 Hz increases the number of required templates by more than a factor of two. Relative toflow= 40 Hz, the required number of templates increases by a factor of ten in going toflow= 10 Hz.
It is worth pointing out the historical difference between detector design and actual performance.
Note the curious feature in Fig. 5.3, in which there appears to be roughly 20% of the available SNR for an analysis with an iLIGO sensitivity in the region between 30 Hz and 40 Hz. However, our most recent searches only used filters down to 40 Hz. The reason for this difference is that the initial detectors never reached design sensitivity below 40 Hz, as one can see in Fig. 5.1. Thus, in the actual pipeline analyses discussed in the previous chapter, there was very little sensitivity lost by choosing a 40 Hz low frequency cutoff instead of a 30 Hz cutoff. It is possible that it will take some time for aLIGO detectors to reach their design sensitivity at the lowest frequencies, and that some of the problems we consider here will be less relevant.
All these considerations pertain only to the problem of developing an analysis strategy that does essentially the same kind of analysis as done for iLIGO. We have already incurred two factors of ten;
one from the increased cost of filtering longer templates, and one from having to use more filters to cover the parameter space. However, we also want to take the pipeline beyond what we have done in the past. In particular, we want to see if we can improve the sensitivity of the pipeline to spinning signals with the use of spinning templates. As we have discussed, spin in compact binary systems is expected to be significant, but our iLIGO analyses used non-spinning templates, as no known methods were available to profitably extend the parameter space to include spin. Including more physical parameters in our search templates will also increase the computational scale of the analysis. Even if we consider only aligned spins, the inclusion of spin in the template banks can further increase the required number of templates by another factor of ten (see Chap. 6).