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).
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most of their lives at low frequencies. The Nyquist theorem states that a signal which is band limited to 0 ≤ f ≤ fhigh can be fully reconstructed, provided that the signal is sampled at the critical frequency fs = 2fhigh or higher. On the other hand, the merger frequency for low mass compact binaries, which traditionally sets the analysis sampling rate, is usually in the kHz range.
While treating the signal as a whole, we therefore require sampling rates of the order a few kHz, but we are actually over-sampling the signal throughout most of its duration. Thus, one approach to mitigating the computational cost for filtering is to break the signal into contiguous band-limited time intervals and process each interval separately at possibly lower sample rates, as depicted in Fig. 5.5. By the Nyquist theorem, one can fully reconstruct the whole SNR time series that one would have obtained by filtering the whole signal at a single, higher sampling rate.
To make the above statements more quantitative, we return to Eqn. 5.1, which gives the time- frequency relation for a compact binary inspiral at Newtonian order. Higher order post-Newtonian terms, including those from spin and merger effects, will significantly alter the time-frequency rela- tion, which should be accounted for in an actual analysis, but here we are simply trying to set the scale. We see from Eqn. 5.1 that an inspiralling binary spends a time
∆t t1
= 1− f1
f2
8/3
, (5.2)
emitting gravitational waves in the frequency band [f1, f2]. Heret1 is the time to coalescence from initial frequencyf1 and ∆tis the time it takes for the binary to evolve fromf1 tof2. Of common practical interest is the case f2 = 2f1; since Fourier transforms are most efficiently executed on arrays having power-of-two lengths, we typically only ever change the sampling rate by factors of two. Putting this numerical value into Eqn. 5.2, we find
∆t
t1 ≈0.84. (5.3)
This result says that the signal spends about 84% of its remaining lifetime in the band [f1,2f1].
In particular, a signal which lasts for 30 minutes starting from 10 Hz will spend only five minutes above 20 Hz, and only 45 seconds above 40 Hz. To filter such a signal, we can safely sample all but the last minute at fs ∼100 Hz, typically at least a factor of ten smaller than the usual sampling rate.
Our strategy is to reduce the filtering cost of a large fraction of the waveform by computing
Figure 5.5: Computational benefits of multibanding and the singular value decomposition. We depict (left) the decomposition of an inspiralling compact binary signal into separate frequency bands in which the signal can be sampled at lower rates than would be required for the signal as a whole. This particular signal lasts
∼1100 s, reaching at coalescence gravitational wave frequencies off∼1 kHz. However, all but the last 12.5 seconds of the signal can be sampled atfs≤128 Hz without aliasing. Decomposing a template bank of 1314 templates sorted by chirp mass, including the example signal, we indicate (right) the number of basis vectors required to achieve a specified level of SVD tolerance. Even for a SVD tolerance of 10−6, corresponding to the right most column, the SVD decomposition reduces the number of filter templates by more than an order of magnitude.
part of the convolution at a lower sample rate. Similar techniques were successfully implemented in iLIGO for the purpose of achieving low latency trigger generation and exercising the procedures for electromagnetic followup [94]. One example is MBTA [59, 95], which was deployed in S6/VSR3.
MBTA consists of multiple, usually two, template banks for different frequency bands, one which is matched to the early inspiral and the other which is matched to the late inspiral. An excursion in the output of any filter bank triggers coherent reconstruction of the full matched filtered output.
Final triggers are built from the reconstructed matched filter output.
In thegstlalimplementation, we divide the templates into time slices in a time-domain analog to the frequency-domain decomposition employed by MBTA. We consider a bank of filtersB ={hi} expressed in the time domain and sampled initially at sufficiently high frequencyf0such that there is no aliasing in any template in the bank. The templates in the bank are all zero-padded as necessary so that they have the same number of samplesN. We trivially decompose each templatehi into a sum ofS non-overlapping templateshsi such that
hi[k] =
S−1X
s=0
hsi[k] ifts6k/f0< ts+1
0 otherwise
(5.4)
forS integers{f0ts}such that 0 =f0t0< f0t1<· · ·< f0tS =N. We filter the data separately with
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each time-sliced filter, and the outputs of these new time-sliced filters form an ensemble of partial SNR streams. By linearity of the filtering process, these partial SNR streams can be summed to reproduce the SNR of the full template.
Since waveforms with neighboring intrinsic source parameters have similar time-frequency evolu- tion, it is possible to design computationally efficient time slices for an extended region of parameter space rather than having to design different time slices for each template. We construct from our time-sliced filters S sub-banks Bs ={hsi}i, and within each sub-bank we choose time slice bound- aries with the smallest power-of-two sample rates that sub-critically sample all time-sliced templates in that bank. The time slices consist of theS intervals [t0, t1),[t1, t2), . . . ,[tS−1, tS), sampled at frequenciesf0, f1, . . . , fS−1, wherefs is at least twice the highest nonzero frequency component of any filter in the bankBsfor thesth time slice. The time-sliced templates can then be downsampled in each interval without aliasing, so we define them as
hsi[k]≡
hi
hkffsi
ifts6k/fs< ts+1
0 otherwise.
(5.5)
We note that the time slice decomposition in Eqn. 5.4 is manifestly orthogonal since the time slices are disjoint in time. In the next section, we examine how to reduce the number of filters within each time slice via the singular value decomposition (SVD) of the time-sliced templates.
In the case where the time-frequency relationship is not known precisely, as for example during merger, we can still apply this multibanding prescription. We treat these cases as we have done here, except that we err on the side of over sampling.