We have demonstrated a computationally feasible filtering algorithm for the rapid and even early-warning detection of G W s emitted during the coalescence of N S s and stellar-mass B H s. It is one part of a complicated analysis and observation strategy that will unfortunately have other sources of latency. However, we hope that it will motivate further work to reduce such technical sources of G W observation latency and encourage the possibility of even more rapid E M follow-up observations to catch prompt emission in the advanced detector era.
C B C events may be the progenitors of some short hard G R B s and are expected to be accompanied by a broad spectrum of E M signals. Rapid alerts to the wider astronomical community will improve the chances of detecting an E M counterpart in bands from gamma-rays down to radio. In the Advanced L I G O era, it appears possible to usefully localize a few rare
events prior to the G R B, allowing multi-wavelength observations of prompt emission. More frequently, low-latency alerts will be released after merger but may still yield extended X-ray tails and early on-axis afterglows.
The L L O I D method is as fast as conventional F F T-based, F D convolution but allows for latency free, real-time operation. We anticipate requiring&40 modern multi-core computers to search for binary N S s using coincident G W data from a four-detector network. In the future, additional computational savings could be achieved by conditionally reconstructing the S N R time series only during times when a composite detection statistic crosses a threshold (Cannon et al., 2011). However, the anticipated required number of computers is well within the current computing capabilities of the L I G O Data Grid.
We have shown a prototype implementation of the L L O I D algorithm using GStreamer, an open-source signal processing platform. Although our prototype already achieves latencies of less than one second, further fine tuning may reduce the latency even further. Ultimately the best possible latency would be achieved by tighter integration between data acquisition and analysis with dedicated hardware and software. This could be considered for third-generation detector design. Also possible for third-generation instruments, the L L O I D method could provide the input for a dynamic tuning of detector response via the signal recycling mirror to match the frequency of maximum sensitivity to the instantaneous frequency of the G W waveform. This is a challenging technique, but it has the potential for substantial gains in S N R and timing accuracy (Meers et al., 1993).
Although we have demonstrated a computationally feasible statistic for advance detection, we have not yet explored data calibration and whitening, triggering, coincidence, and ranking of G W candidates in a framework that supports early E M follow-up. One might explore these, and also use the time slice decomposition and the S V D to form low-latency signal-based vetoes (e.g., χ2statistics) that have been essential for glitch rejection used in previous G W C B C searches. These additional stages may incur some extra overhead, so computing requirements will likely be somewhat higher than our estimates.
Future work must more deeply address sky localization accuracy in a realistic setting as well as observing strategies. Here, we have followed Fairhurst (2009) in estimating the area of 90% localization confidence in terms of timing uncertainties alone, but it would be advantageous to use a galaxy catalog to inform the telescope tiling (Nuttall & Sutton, 2010). Because early detections
will arise from nearby sources, the galaxy catalog technique might be an important ingredient in reducing the fraction of sky that must be imaged. Extensive simulation campaigns incorporating realistic binary merger rates and detector networks will be necessary in order to fully understand the prospects for early-warning detection, localization, and E M follow-up using the techniques we have described.
Acknowledgements
L I G O was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the N S F and operates under cooperative agreement PHY-0107417. C.H. thanks Ilya Mandel for many discussions about rate estimates and the prospects of early detection, Patrick Brady for countless fruitful conversations about low-latency analysis methods, and John Zweizig for discussions about L I G O data acquisition. N.F. thanks Alessandra Corsi and Larry Price for illuminating discussions on astronomical motivations. L.S. thanks Shaun Hooper for productive conversations on signal processing. This research is supported by the N S F through a Graduate Research Fellowship to L.S. and by the Perimeter Institute for Theoretical Physics through a fellowship to C.H. D.K. is supported from the Max Planck Gesellschaft. M.A.F. is supported by N S F Grant PHY-0855494.
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0(a) Mismatch versus S V D tolerance
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Figure 3.6 Box-and-whisker plot of mismatch between nominal template bank and L L O I D measured impulse responses. The upper and lower boundaries of the boxes show the upper and lower quartiles; the lines in the center denote the medians. The whiskers represent the minimum and maximum mismatches over all templates. In (a) the interpolation filter length is held fixed at N =192, while the S V D tolerance is varied from 1−10−1to 1−10−6. In (b), the S V D
Chapter 4
B AY E S TA R: Rapid Bayesian sky
localization of B N S mergers
This chapter is reproduced from a work in preparation for Physical Review D. The authors will be Leo P. Singer and Larry R. Price. The introduction of this chapter is reproduced in part from Singer et al. (2014), copyright © 2014 The
American Astronomical Society.
We expect this decade to bring the first direct detection of G W s from compact objects. The L I G O and Virgo detectors are being rebuilt with redesigned mirror suspensions, bigger optics, novel optical coatings, and higher laser power (Harry, 2010; Acernese et al., 2013). In their final configuration, Advanced L I G O and Virgo are expected to reach∼10 times further into the local universe than their initial configurations did. The best-understood sources for L I G O and Virgo are B N S mergers. They also offer a multitude of plausible E M counterparts (Metzger & Berger, 2012) including collimated short-hard gamma-ray bursts (short GRBs; see for example Paczynski 1986; Eichler et al. 1989; Narayan et al. 1992; Rezzolla et al. 2011) and X-ray/optical afterglows, near-infrared kilonovae (viewable from all angles; Li & Paczy ´nski, 1998; Barnes & Kasen, 2013b, etc.), and late-time radio emission (Nakar & Piran, 2011; Piran et al., 2013). Yet, typically poor G W localizations of&100 deg2will present formidable challenges to observers hunting for their E M counterparts.
The final Initial L I G O–Virgo observing run pioneered the first accurate, practical parameter estimation and position reconstruction methods for B N S signals. This included a prompt,
semi-coherent, ad hoc analysis (Timing++, designed to work with M B TA; Abadie et al. 2012a), and the first version of a rigorous Bayesian M C M C analysis (L A L I N F E R E N C E; Aasi et al. 2013b). Though important milestones, there was an undesirable compromise made between accuracy and speed: though the former analysis took only minutes, it produced areas that were typically 20 times larges than the latter, which could take days (Sidery et al., 2014). To increase the odds of finding a relatively bright but rapidly fading afterglow, one wants localizations that are both prompt and accurate, to begin optical searches within minutes to hours of a G W detection. To increase the odds of finding a kilonova, one wants localizations that are reliably available in under one day, to allow as much time as possible for multiple deep exposures. See Figure 4.1 for a timeline of the most promising E M counterparts as compared to the response times of the various steps in the G W analysis.
To that end, in this chapter we develop a rapid and accurate Bayesian sky localization method that takes mere seconds but achieves approximately the same accuracy as the full M C M C analysis. We call this algorithm BAYESian TriAngulation and Rapid localization (B AY E S TA R)1. It differs from existing techniques in several important ways. In the first place, we treat the matched-filter detection pipeline as a measurement system in and of itself, treating the point-parameter estimates that it provides as the experimental input data rather than the full G W time series that is used by the M C M C analysis. This drastically reduces the dimensionality of both the data and the signal model. It also permits us to avoid directly computing the expensive post-Newtonian model waveforms, making the likelihood itself much faster to evaluate. Finally, instead of of using M C M C or some similar method for statistical sampling, we make use of a deterministic quadrature scheme. This algorithm is unique in that it bridges the detection and parameter estimation of G W signals, two tasks that have until now involved very different numerical methods and time scales. We expect that B AY E S TA R will take on a key role in observing C B C events in both G W and optical channels during the Advanced L I G O era.
1A pun on the Cylon battleships in the American television series Battlestar Galactica. The defining characteristic of the Cylons is that they repeatedly defeat humanity by using their superhuman information-gathering ability to coordinate overwhelming forces. The name also suggests that, like the Cylons, G W detectors may some day rise against us humans. We do not like to mention the final ‘L’ in the acronym, because then it would be pronounced B AY E S TA R L, which sounds stupid.
100 101 102 103 104 105 106 107
Detection Rapid localization Full parameter estimation
← GRB X-ray/optical afterglow Kilonova Radio afterglow → t −tmerger(s)
Figure 4.1 Rough timeline of compact binary merger electromagnetic emissions in relation to the timescale of the Advanced LIGO/Virgo analysis described in this thesis. The time axis measures seconds after the merger.
4.1
Bayesian probability and parameter estimation
In the Bayesian framework, parameters are inferred from the G W data by forming the posterior distribution, p(θ|Y), which describes the probability of the parameters given the observations.
Bayes’ rule relates the likelihood p(Y|θ)to the posterior p(θ|Y),
p(θ|Y) = p(Y|θ)p(θ)
p(Y) , (4.1)
introducing the prior distribution p(θ) which encapsulates previous information about the
paramters (for example, arising from earlier observations or from known physical bounds on the parameters), and the evidence p(Y). In parameter estimation problems such as those we are concerned with in this chapter, the evidence serves as nothing more than a normalization factor. However, when comparing two models with different numbers of parameters, the ratio of the evidences quantifies the relative parsimony of the two models, serving as a precise form of Occam’s razor.
The parameters that describe a B N S merger signal are listed in Equation (2.4). The choice of prior is determined by one’s astrophysical assumptions, but during L I G O’s sixth science run (S 6) when LIGO’s Bayesian C B C parameter estimation pipelines were pioneered, the prior was taken to be isotropic in sky location and binary orientation, and uniform in volume, arrival time, and the component masses (Aasi et al., 2013b).
In Bayesian inference, although it is often easy to write down the likelihood or even the full posterior in closed form, usually one is interested in only a subset β of all of the model’s parameters, the others, λ, being nuisance parameters. In this case, we integrate away the nuisance
parameters, forming the marginal posterior
p(β|Y) =
Z p(Y|
β, λ)p(β, λ)
p(Y) dλ, (4.2)
with θ= (β, λ). For instance, for the purpose of locating a G W source in the sky, all parameters
(distance, time, orientation, masses, and spins) except for(α, δ)are nuisance parameters.
Bayesian parameter estimation has many advantages, including broad generality and the ability to make probabilistically meaningful statements even with very low S N R measurements. However, in problems of even modest complexity, the marginalization step involves many-dimensional, ill-behaved integrals. The powerful M C M C integration technique has become almost synonymous with Bayesian inference. Though powerful, M C M C is inherently non-deterministic and resistant to parallelization, as well as (at least historically) slow. With the most sophisticated C B C parameter estimation codes, it still takes days to process a single event. This delay is undesirable for planning targeted E M follow-up searches of L I G O events.
In what follows, we describe a complementary rapid parameter estimation scheme that can produce reliable positions estimates within minutes of a detection. We can even use our scheme to speed up the full M C M C analysis and make the refined parameter estimates available more quickly. The key difference is that we start not from the G W signal itself, but from the point parameter estimates from the detection. By harnessing the detection pipeline in this manner, we arrive at a simpler Bayesian problem that is amenable to straightforward, deterministic, numerical quadrature.
There are many practical advantages of doing so. For one, there are difficulties in synchronously gathering together the calibrated G W strain data, auxiliary instrument channels, and data quality vetoes from all of the sites. The data consumed by the real-time detection pipeline are not necessarily final. Longer-running follow-up analyses can benefit from offline calibration, whereas the rapid sky localization need not re-analyze the online data. Moreover, the dimensionality of the problem is greatly reduced, and the problem becomes computationally easier. Finally, by breaking free of the M C M C framework, the results are much easier to use for pointing telescopes. With M C M C algorithms, it is often desirable to bin or interpolate the cloud of sample points to provide a smooth, high-resolution representation of the probability distribution. Reliable post-processing of M C M C chains often relies on clustering and kernel density estimation, both of which prohibit
very large numbers of samples due to rapid growth of computational cost. With B AY E S TA R, the natural form of the result is an adaptively sampled mesh with high resolution only where it is needed. The output from B AY E S TA R is therefore extremely convenient for packaging into a Flexible Image Transport System (F I T S) file for transmission with G W alerts (see Appendix C for details).