# The needle in the 100 deg haystack: The hunt for binary neutron star mergers with \textlsLigo and \textlsPalomar Transient Factory

## Abstract

The Advanced \textlsLIGO and Virgo experiments are poised to detect gravitational waves directly for the first time this decade. The ultimate prize will be joint observation of a compact binary merger in both gravitational and electromagnetic channels. However, \textlsGW sky locations that are uncertain by hundreds of square degrees will pose a challenge. I describe a real-time detection pipeline and a rapid Bayesian parameter estimation code that will make it possible to search promptly for optical counterparts in Advanced \textlsLIGO. Having analyzed a comprehensive population of simulated \textlsGW sources, we describe the sky localization accuracy that the \textlsGW detector network will achieve as each detector comes online and progresses toward design sensitivity. Next, in preparation for the optical search with the \textls\textls[0]iPTF, we have developed a unique capability to detect optical afterglows of gamma-ray bursts detected by the *Fermi* Gamma-ray Burst Monitor (\textlsGBM). Its comparable error regions offer a close parallel to the Advanced \textlsLIGO problem, but *Fermi*’s unique access to MeV—GeV photons and its near all–sky coverage may allow us to look at optical afterglows in a relatively unexplored part of the \textlsGRB parameter space. We present the discovery and broadband follow-up observations (X–ray, UV, optical, millimeter, and radio) of eight \textlsGBM—\textls\textls[0]iPTF afterglows. Two of the bursts (\textlsGRB 130702A / iPTF13bxl and \textlsGRB 140606B / iPTF14bfu) are at low redshift ( and , respectively), are sub–luminous with respect to “standard” cosmological bursts, and have spectroscopically confirmed broad–line type Ic supernovae. These two bursts are possibly consistent with mildly relativistic shocks breaking out from the progenitor envelopes rather than the standard mechanism of internal shocks within an ultra-relativistic jet. On a technical level, the \textlsGBM–\textls\textls[0]iPTF effort is a prototype for locating and observing optical counterparts of \textlsGW events in Advanced \textlsLIGO with the \textlsZwicky Transient Facility.

2015

To the love of my life, my wife Kristin, and our precious son Isaac.

Bacon in his instruction tells us that the scientific student ought not to be as the ant, who gathers merely, nor as the spider who spins from her own bowels, but rather as the bee who both gathers and produces. All this is true of the teaching afforded by any part of physical science. Electricity is often called wonderful, beautiful; but it is so only in common with the other forces of nature. The beauty of electricity or of any other force is not that the power is mysterious, and unexpected, touching every sense at unawares in turn, but that it is under law, and that the taught intellect can even now govern it largely. The human mind is placed above, and not beneath it, and it is in such a point of view that the mental education afforded by science is rendered super-eminent in dignity, in practical application and utility; for by enabling the mind to apply the natural power through law, it conveys the gifts of God to man.

Michael Faraday, Lecture notes of 1858, quoted in

The Life and Letters of Faraday(1870) by Bence Jones, Vol. 2, p. 404

###### Acknowledgements.

This is \textlsLIGO Document Number \textlsLIGO–P1400223–v10. I carried out the work presented in this thesis within the \textls Scientific Collaboration (\textlsLSC) and the Intermediate \textls (\textls\textls[0]iPTF) collaboration. The methods and results I present are under review and are potentially subject to change. The opinions expressed here are my own and not necessarily those of the \textls Scientific Collaboration (\textlsLSC) or Intermediate \textls (\textls\textls[0]iPTF). I gratefully acknowledge funding from the United States National Science Foundation (\textlsNSF) for the construction and operation of the \textlsLIGO Laboratory, which provided support for this work. \textlsLIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation (\textlsNSF) and operates under cooperative agreement PHY–0107417. I thank the \textlsNSF for supporting my research directly through a Graduate Research Fellowship. This work is based on observations obtained with the \textlsPalomar 48–inch Oschin telescope and the \textlsrobotic Palomar 60–inch telescope at the Palomar Observatory as part of the Intermediate Palomar Transient Factory project, a scientific collaboration among the California Institute of Technology, Los Alamos National Laboratory, the University of Wisconsin, Milwaukee, the Oskar Klein Center, the Weizmann Institute of Science, the TANGO Program of the University System of Taiwan, and the Kavli Institute for the Physics and Mathematics of the Universe. The work in this thesis is partly funded by*Swift*Guest Investigator Program Cycle 9 award 10522 (NASA grant NNX14AC24G) and Cycle 10 award 10553 (NASA grant NNX14AI99G). Thank you, Mom, thank you Dad, for an upbringing full of love, learning, and love of learning. Thank you, my wife Kristin, thank you, my son Isaac, for your love and for your patience with me. Thank you, Susan Bates, for your tutoring in problem solving that has resonated with me from elementary school through every day of my scientific career. Thank you, John Jacobson, Amanda Vehslage, Tambra Walker, and Dr. Philip Terry–Smith, for the most inspiring courses in my high school education, and for molding me into a responsible and well–rounded individual. Thank you, Profs. Luis Orozco and Betsy Beise, for your mentoring and friendship as well as the University of Maryland undergraduate physics courses that I enjoyed so much. Thank you for initiating me into physics research, and for sending me to graduate school so well prepared. Thank you, Prof. Alan Weinstein, for being an outstanding (and, when necessary, forbearing) thesis advisor, for engineering the many wonderful collaborations that I have been a part of at Caltech, and for showing me how to thrive within a Big Science experiment. Thank you, Prof. Shri Kulkarni, for recruiting me into \textlsPTF, for engineering a totally original cross-disciplinary research opportunity in physics and astronomy, and for placing trust in me. I am continually in awe of how that trust has paid off. I thank my colleagues in \textlsPTF for welcoming me into their highly capable and exciting team. Thank you, Prof. Christian Ott, for teaching me two formative courses. I was able to write \textlsBAYESTAR, my greatest contribution so far to Advanced \textlsLIGO, only because the latter of these courses (Ay 190: Computational Astrophysics) was fresh in my head. Thank you, Prof. David Reitze, for making me feel like the success of Advanced \textlsLIGO depends upon me. (I think that you inspire that same feeling in everyone at \textlsLIGO Laboratory.) Thank you, Rory Smith, my officemate, for ducking good-naturedly whenever I wanted to chuck a chair out the window of 257 West Bridge. (Despite many strong oaths, no chairs were actually chucked during the writing of this thesis.) Thank you, Nick Fotopoulos, Larry Price, Brad Cenko, and Mansi Kasliwal, for your collaboration and friendship throughout my studies, friendships that I hope to keep and to nurture.

###### Contents:

- \thechapter The road to Advanced \textls
- \thechapter Range and sky resolution of \textls detector networks
- \thechapter Early warning \textls detection
- \thechapter \textls: Rapid Bayesian sky localization of \textls mergers
- \thechapter The first two years of \textls follow-up with Advanced \textls and Virgo
- \thechapter Discovery and redshift of an optical afterglow in 71 square degrees: iPTF13bxl and \textls 130702A
- \thechapter
*Fermi*, \textls, and the \textls–supernova connection - \thechapter Conclusion
- \thechapter Computer codes
- A Astropy
- B GSTLAL
- C GStreamer
- D HEALPix
- E LALSuite
- F Code listing: sky resolution from Fisher matrices
- \thechapter Low frequency cutoff for inspiral searches
- \thechapter Sky map file format
- G Mandatory
- H Optional
- I Example code in Python
- J Specimen
- \thechapter “First Two Years” data release
- \thechapter Observations of iPTF/GBM afterglows

###### List of Figures:

- 1 Effect of a \textls on a…
- 2 A basic \textls inspiral waveform…
- 3 Aerial photographs of ground-based \textls detectors
- 4 \textls and \textls cameras
- 5 Example \textls localization, comparing…
- 6 Discovery of GRB 130702A / iPTF13bxl…
- 7 Antenna patterns
- 8 Position resolution as a function of…
- 9 Log ratios of \textls in pairs…
- 10 Improvement in sky resolution due to…
- 11 Fisher matrix error ellipses of a…
- 12 Expected number of \textls sources detectable…
- 13 Localization area as a function of…
- 14 Sky resolution versus time before merger…
- 15 Schematic of \textls pipeline
- 16 Mass parameters of sub-bank
- 17 Mismatch between \textls filter response…
- 18 Timeline of \textls \textls…
- 19 \textls on time and…
- 20 Ratio of Fisher matrix elements between…
- 21 Initial subdivisions for radial integral
- 22 \textls run time
- 23 Amplitude spectral densities for early Advanced…
- 24 \textls —…
- 25 Cumulative histograms of sky areas
- 26 Cumulative histograms of angle offsets
- 27 The HL degeneracy
- 28 A typical bimodal localization, circa 2015…
- 29 A typical unimodal localization, circa 2015…
- 30 Frequency of localizations with one, two,…
- 31 Breakdown of 2016 scenario by detector…
- 32 Typical HLV localization, circa 2016
- 33 Typical HV localization, circa 2016
- 34 Discovery of \textls 130702A /…
- 35 Calibration of
*Fermi*systematic errors - 36 Light curve of GRB 130702A /…
- 37 Broadband \textls of GRB…
- 38 Optical spectra
- 39 Light curve and \textls of \textls…
- 40 Light curve of \textls 131011A /…
- 41 Light curve and \textls of \textls…
- 42 Light curve and \textls of \textls…
- 43 Light curve and \textls of \textls…
- 44 Light curve and \textls of \textls…
- 45 Light curve of \textls 140623A /…
- 46 Light curve and \textls of \textls…
- 47 Sensitivity curves for various squeezing configurations…
- 48 The \textls camera
- 49 Kilonova phase space accessible with various…
- 50 Accumulated \textls as a…

###### List of Tables:

- 1 Comparison of the survey speeds of…
- 2 Revised sky resolution predictions for Advanced…
- 3 Number of sources that are localizable…
- 4 Notation used to describe filters
- 5 Filter design for a sub-bank of…
- 6 Computational cost of \textls versus conventional…
- 7 Detection rate and sky localization accuracy…
- 8 Fractional accumulated \textls for selected…
- 9 Simulated BNS signals for 2015 scenario…
- 10 Detections and sky localization areas for…
- 11 Simulated BNS signals for 2016 scenario…
- 12 Detections and sky localization areas for…
- 13 Optical observations of \textls–\textls…
- 14 Radio observations of \textls–\textls…

## Chapter \thechapter The road to Advanced \textlsLigo

Einstein’s general theory of relativity holds that the laws of motion play out in a curved space–time, with curvature caused by the presence of matter and energy. This strange statement has some even stranger consequences. One of the earliest solutions of Einstein’s equation predicted black holes, stars made of pure space-time curvature, whose gravitational wells are so deep that nothing, not even light, can escape. We now know that when a massive star exhausts the last of its fuel, it can collapse to form a neutron star (\textlsNS)—the densest possible stable arrangement of matter, something akin to a gigantic nucleus with atomic number —or a stellar-mass \textlsBH. This gravitational collapse can be messy and loud. It may produce a relativistic shock wave that powers a long \textlsGRB, and it may drive a supernova explosion that outshines the late star’s host galaxy in visible light for several weeks. Long afterward, the strong gravitational field of a compact object can have other interesting consequences. If the star has a binary companion from which it can accrete matter, it can power a wide range of high-energy transient phenomena. However, all of these processes occur in basically static (but strongly curved) space-time.

In the dynamical regime, Einstein’s theory predicts GWs that transmit energy via propagating disturbances in space–time, much as the dynamical solutions of Maxwell’s equations carry energy as light. Operationally, the effect of a passing \textlsGW is to slightly change the separation between free–falling objects (see Figure 1). The brightest source of gravitational waves that we think nature can make is a binary system of two compact objects (NSs and/or BHs). If a compact binary is in a tight enough orbit, gravitational radiation can efficiently carry away energy and angular momentum. This orbital decay was famously observed in the binary pulsar PSR 1913+16 (Hulse & Taylor, 1975; Taylor & Weisberg, 1982), for which Hulse and Taylor received the Nobel Prize in Physics in 1993. The energy loss eventually will become a runaway process, as the orbital separation decreases and the system radiates even more gravitational waves. See Figure 2 for an illustration of the basic \textlsGW “inspiral” waveform due to a compact binary coalescence (\textlsCBC). Ultimately, the two compact objects will coalesce: they will become a single perturbed \textlsBH, which will ring down as it settles into rotationally symmetric stationary state.

If one or both of the binary companions is a \textlsNS, the the merger process itself can also be messy and loud. The immense tidal forces can tear apart the \textlsNS before it takes the final plunge. The resultant hot, highly magnetized accretion flow may create the conditions necessary for a highly relativistic jet (Rezzolla et al., 2011). This process is thought to power short GRBs (Paczynski, 1986; Eichler et al., 1989; Narayan et al., 1992; Rezzolla et al., 2011).

The Laser Interferometer \textls Observatory (\textlsLIGO) and Virgo have been constructed with the aim of directly detecting GWs from CBCs of binary neutron stars, among other potential sources. This will provide a singularly dramatic confirmation of Einstein’s relativity in the otherwise largely untested strong–field dynamical regime. \textlsGW observations could even test alternative theories of gravity (Section 6, Will, 2006; Del Pozzo et al., 2013) or constrain the NS equation of state (Read et al., 2009). A temporal coincidence between a \textlsCBC event and a short \textlsGRB would also settle the question of the progenitors of at least some of these elusive explosions. No \textlsGW events were detected in Laser Interferometer \textls Observatory (\textlsLIGO)–Virgo observing runs at an initial sensitivity (Abadie et al., 2012c). However, \textlsLIGO is just now finishing its transformation into Advanced \textlsLIGO, with Advanced Virgo soon to follow suit. Both designed to be ultimately ten times more sensitive than their predecessors, they will be able to monitor a thousand times more volume within the local Universe. The first detections are expected over the next few years (Abadie et al., 2010b).

A perhaps even greater prize would be detecting both the \textlsGW signal and an optical transient resulting from the same \textlsBNS merger event. An optical afterglow (van Eerten & MacFadyen, 2011) would aid in the understanding of the physics of the relativistic jet (for the “classic” model, see Sari et al. 1998), and a bright on-axis afterglow would be the most obvious signpost by which to locate the host galaxy. These signatures, however, are expected to be rare because, like the short \textlsGRB itself, we have to be inside the collimated cone of the jet to see them. Perhaps a more promising optical signature (Metzger & Berger, 2012) would be that of a roughly omnidirectional “kilonova” powered by the radioactive decay of the hot -process ejecta (Li & Paczyński, 1998; Barnes & Kasen, 2013a) or a “kilonova precursor” powered by free neutrons in the fast-moving outer layers of the ejecta (Metzger et al., 2015). A kilonova could inform us about the nature and distribution of the ejecta, and tell us whether the merged compact object collapsed directly to a black hole or went through a brief phase as a hyper–massive neutron star (Metzger & Fernández, 2014). See Figure LABEL:fig:sgrb-kcorrected for typical -band light curves of these optical signatures. If we could detect \textlsGW and electromagnetic (\textlsEM) emission from a sufficiently large number of CBCs, then we could simultaneously measure their luminosity distances and redshifts, thereby adding an almost calibration-free “standard siren” to the cosmological distance ladder (Schutz, 1986; Holz & Hughes, 2005; Dalal et al., 2006; Nissanke et al., 2010).^{1}

### 1 Challenges

Some first steps toward multimessenger observations were taken in the last \textlsLIGO–Virgo science run. The first low–latency \textlsCBC search was deployed, including an online matched filter analysis for detection (\textlsMulti-Band Template Analysis; \textlsMBTA), a fast but ad hoc algorithm for sky localization (Abadie et al., 2012a), and a system for sending alerts to optical facilities (Kanner et al., 2008). The whole process from data acquisition to alerts took about half an hour (dominated by a final human–in–the–loop check; see Chapter \thechapter for a full timing budget). This period also saw the development of the first practical Bayesian parameter estimation codes, which at the time took a few weeks to thoroughly map the parameter space of any detection candidate (Aasi et al., 2013b). A consortium of X–ray, optical, and radio telescopes participated in searching for \textlsEM counterparts (Abadie et al., 2012b; Aasi et al., 2014).

Although these were important proofs of concept, the increased sensitivity of the Advanced \textlsLIGO detectors will force us toward more sophisticated approaches at each stage of the process (detection, sky localization, and \textlsEM follow–up). The first challenge is longer \textlsGW signals. A significant part of Advanced \textlsLIGO’s expanded detection volume comes from better sensitivity at low frequencies, moving the seismic noise cutoff from 40 Hz to 10 Hz (see Appendix \thechapter for a detailed discussion of the sensitivity as a function of low frequency cutoff). \textlsCBC signals are chirps, ramping from low to high frequency as . Although a typical \textlsBNS merger signal would remain in band for Initial \textlsLIGO for about 25 s, it would be detectable by Advanced \textlsLIGO for as long as 1000 s (see Equation (67) in Chapter \thechapter). A second consequence of longer signals is that the signal can accumulate more power and a larger total phase shift while in band, improving the ability to measure the mass of the binary but dramatically increasing the number of \textlsGW templates required to adequately tile the parameter space. A third problem is that we cannot assume that the detector and the instrument noise are in a stationary state for the durations of these long signals; we must adaptively condition or *whiten* the data as the noise level rises or falls, and we must be able to carry on integrating the signal over gaps or glitches. These are all formidable problems for traditional fast Fourier transform (\textlsFFT)-based matched filter pipelines, which have inflexible data handling, whose latency grows with the length of the signal, and whose computational requirements increase with both the length and number of template signals. To effectively search for these signals in real time we need a detection pipeline whose latency and computational demands do not scale much with the duration of the \textlsGW signal.

The second challenge is that the sky localization most be both fast *and* accurate. The original rapid sky localization and full Bayesian parameter estimation codes entailed an undesirable tradeoff of response time and accuracy: the former took only minutes, but produced sky areas that were 20 times larges than the latter, which could take days (Sidery et al., 2014). This compromise was somewhat acceptable in Initial \textlsLIGO because, given the small number of galaxies within the detectable volume, one could significantly reduce the area to be searched by selecting fields that contained nearby galaxies (Kopparapu et al., 2008; White et al., 2011). With the expanded range of Advanced \textlsLIGO enclosing many more galaxies, this will still be a valuable strategy, but will be somewhat less effective (Nissanke et al., 2013). Given that the predicted optical signatures of \textlsBNS mergers are faint (with kilonovae predicted to be fainter than mag) and may peak in under a day, it is essential that the rapid localization be as accurate as possible. Ideally, it should be just as accurate as the localization from the full Bayesian parameter estimation.

Third and most importantly, we have to build the instruments, software, collaborations, and observational discipline to search through areas of hundreds of deg for the faint, rapidly fading optical counterparts. We need deep, wide–field optical survey telescopes to scan the \textlsGW localizations and detect new transient or variable sources, robotic follow-up telescopes to track photometric evolution and obtain color information, a network of 5–m class and larger telescopes to secure spectroscopic classifications, as well as X–ray and radio telescopes that can act on targets of opportunity. To identify which among tens or hundreds of thousands of optical transients to follow up we need real-time image subtraction, machine learning, and integration with archival survey data, not to mention a team of human observers in the loop and executing deep spectroscopic observations on large–aperture telescopes.

There is a fortunate convergence between the construction of Advanced \textlsLIGO and Virgo and the deployment of deep, high–cadence, synoptic, optical transient surveys. Experiments like the \textlsPalomar Transient Factory (\textlsPTF; Rau et al. 2009; Law et al. 2009) have focused on discovering rare or rapidly rising optical transients, but should also be well suited to searching for optical counterparts of \textlsGW sources. The key instrument in Palomar Transient Factory (\textlsPTF) is the Canada–France–Hawaii pixel CCD mosaic (\textlsCFH12k) camera (Rahmer et al., 2008) on the Palomar 48–inch Oschin telescope (\textlsP48), capable of reaching limits of mag in 60 s over a wide, 7.1 deg field of view (\textlsFOV). In its planned successor, the Zwicky Transient Facility (\textlsZTF), this will be replaced by a new 47 deg camera. With a larger \textlsFOV and faster readout electronics, Zwicky Transient Facility (\textlsZTF) will achieve an order of magnitude faster volumetric survey rate (see Figure 4 for an illustration of the \textlsPTF and \textlsZTF cameras and Table 1 for a comparison of survey speeds). A real–time image subtraction and machine learning pipeline supplies a stream of new optical transient candidates from which a team of human observers selects the most interesting targets for multicolor photometry on the robotic Palomar 60–inch telescope (\textlsP60) and spectroscopic classification on the Palomar 200–inch Hale telescope (\textlsP200) and other large telescopes. Lessons learned by \textlsPTF will inform the planning and operation of future optical transient surveys such as BlackGEM^{2}

\textlsPTF | \textlsZTF | |
---|---|---|

Active area | 7.26 deg | 47 deg |

Overhead time | 46 s | s |

Optimal exposure time | 60 s | 30 s |

Relative areal survey rate | 1x | 15.0x |

Relative volumetric survey rate | 1x | 12.3x |

### 2 Aims of this thesis

The aim of my thesis is to deliver the major, fully and realistically characterized and tested, pieces of the search for optical counterparts of \textlsBNS mergers, including detection and parameter estimation as well as the optical transient search itself. Here is a chapter–by–chapter summary of the content of this thesis.

Chapter \thechapter introduces the basic principles of a matched filter bank \textlsGW search. We describe the range of a \textlsGW detector in terms of its directional sensitivity or antenna pattern, its noise power spectral density (\textlsPSD), and the signal–to–noise ratio (\textlsSNR). We then apply the Fisher information matrix formalism to compute the approximate sky resolution of a network of \textlsGW detectors. There is a great deal of prior literature on this topic that considers \textlsGW sky localization in terms of timing triangulation (see, for instance, Fairhurst 2009). Our calculation captures the additional contributions of the phases and amplitudes on arrival at the detectors, which we show to be significant, especially near the plane of the detectors where timing triangulation is formally degenerate. Our derivation is extremely compact, and evaluating it is only marginally more complicated than the timing triangulation approach. We discuss the sky localization accuracy as a function of direction in the sky, and build some intuition that we will rely upon in future chapters. This chapter is in preparation as a separate paper and as a proposed update to a living document describing the \textlsGW detector commissioning and observing schedule (Aasi et al., 2013c).

Chapter \thechapter describes a novel matched filtering algorithm that is capable of detecting a \textlsGW signal within seconds after the merger, or even seconds *before*. This algorithm, called \textlsLLOID, uses orthogonal decomposition and multirate signal processing to bring the computational demands of an online \textlsBNS search within the scope of current resources. My contributions to \textlsLLOID include: working on the pipeline to drive the latency to 10 s, improving data handling to be able to skip over glitches in the data efficiently without unduly sacrificing \textlsSNR, studying the signal processing and computational aspects of the algorithm, improving the time and phase accuracy of triggers, and preparing the first complete description of it for the literature (Cannon et al., 2012). \textlsLLOID has been extensively tested both offline and in real time with simulated and commissioning data in a series of “engineering runs,” and will serve as the flagship low–latency \textlsBNS detection pipeline in Advanced \textlsLIGO. This chapter is in preparation as a standalone technical paper.

Chapter \thechapter develops a new rapid sky localization algorithm, \textlsBAYESTAR, that takes just tens of seconds, but achieves about the same accuracy (see Figure 5) as the full parameter estimation. It owes its speed to three innovations. First, like the ad hoc Initial \textlsLIGO code, it takes as input the matched filter parameter estimates from the detection pipeline rather than the full \textlsGW time series. Second, using a result from Chapter \thechapter, it concerns itself with sky location only and not the masses of the signal, exploiting the fact that the errors in the intrinsic and extrinsic parameters of a \textlsBNS signal are approximately uncorrelated. Third, though fully Bayesian, unlike the full parameter estimation it does not use Markov chain Monte Carlo (\textlsMCMC) sampling; instead it uses an adaptive sampling grid and low-order Gaussian quadrature. The result is both inherently fast and also highly parallelizable. Like \textlsLLOID, it has been tested with both extensive offline simulations and in online engineering runs.

Having assembled the Advanced \textlsLIGO real–time \textlsBNS pipeline in Chapters \thechapter and \thechapter, in Chapter \thechapter we provide a detailed description of what the first Advanced \textlsLIGO detections and sky localizations may look like. Because our new rapid sky localization algorithm is orders of magnitude faster than the full parameter estimation, for the first time we can perform end-to-end analyses of thousands of events, thereby providing a statistically meaningful and comprehensive description of the areas and morphologies that will arise in the early Advanced \textlsLIGO configurations. The first 2015 observing run is expected to involve only the two \textlsLIGO detectors in Hanford, Washington, and Livingston, Louisiana, and not the Virgo detector in Cascina, Italy. Aasi et al. (2013c) assumed, based on timing triangulation considerations, that two detector networks would always produce localizations that consist of degenerate annuli spanning many thousands of deg. We find that the interplay between the phase and amplitude on arrival (i.e., the \textlsGW polarization) and prior distribution powerfully break this degeneracy (see also Raymond et al. 2009; Kasliwal & Nissanke 2014), limiting *almost all* areas to below 1000 deg, with a median of about 600 deg. We elucidate one curious degeneracy that survives, that causes most source localizations to equally favor the true position of the source as well as a position at the polar opposite. We then model the 2016 observing run, which has the \textlsLIGO detectors operating with somewhat deeper sensitivity and has Advanced Virgo online. Even with Virgo’s sensitivity delayed with the staggered commissioning, adding the third detector shrinks areas to a median of 200 deg. As the detectors continue approaching final design sensitivity and as more detectors come online, areas will continue to shrink to 10 deg and below. This chapter is published as Singer et al. (2014). The supplementary data described in Appendix \thechapter contains a browsable catalog of simulated \textlsGW sky maps, in the format that will be used for sending \textlsGW alerts (which is described in Appendix \thechapter).

Chapter \thechapter confronts the search for optical transients in large error regions with the \textlsIntermediate \textls; (\textls\textls[0]iPTF; Kulkarni 2013) and its planned successor, the \textlsZwicky Transient Facility (\textlsZTF; Kulkarni 2012; Bellm 2014; Smith et al. 2014b). The surest way to convince ourselves that the search for optical counterparts of \textlsGW transients will be effective is to try it out and discover something. As a model problem, for the past year we have searched for optical counterparts of GRBs detected by the \textlsGamma-ray Burst Monitor (\textlsGBM; Meegan et al. 2009a) instrument onboard the *Fermi* satellite. Like \textlsLIGO, *Fermi* \textlsGBM produces coarse localizations that are uncertain by 100 deg, and though afterglows of long GRBs are much brighter than anticipated \textlsLIGO optical counterparts, the important timescales for follow-up observations are similar. *Fermi* \textlsGBM bursts are also interesting in their own right. *Fermi* \textlsGBM and the *Swift* \textlsBurst Alert Telescope (\textlsBAT; Barthelmy et al. 2005) have highly complementary strengths: fields of view of 70% and 10% of the sky respectively, and energy bandpasses of few keV to 300 GeV (when including the *Fermi* \textlsLarge Area Telescope or \textlsLAT; Atwood et al. 2009) and 15—150 keV respectively. However, with the \textlsGBM’s coarse localization, very few *Fermi* bursts have been studied outside the gamma–ray band (the exception being bursts that are coincidentally also detected by *Fermi* \textlsLAT or *Swift* Burst Alert Telescope (\textlsBAT)). In this chapter, we relate the discovery, redshift, and broadband observations of \textlsGRB 130702A and its optical afterglow, iPTF13bxl Singer et al. (2013). This is the first discovery of an optical afterglow based solely on a *Fermi* \textlsGBM localization (see discovery image in Figure 6). This is a notable event in and of itself for several other reasons. First, its redshift places it among the nearest GRBs ever recorded. Second, its prompt energy release in gamma rays is intermediate between bright, cosmologically distant, “standard” bursts, and nearby low–luminosity \textlss which comprise many of the well–studied \textlsGRB– supernova (\textlsSN). Finally, because of its low redshift we were able to spectroscopically detect its associated \textlsbroad–line type Ic \textls, establishing it as a test for the \textlsGRB–\textlsSN connection.

Chapter \thechapter reports on the total of eight \textlsGBM–\textls\textls[0]iPTF afterglows that we have discovered in one year of this experiment. In this chapter, we present our broadband follow–up including spectroscopy as well as X–ray, UV, optical, sub–millimeter, millimeter, and radio observations. We study possible selection effects in the context of the total *Fermi* and *Swift* \textlsGRB samples. We identify one new outlier on the Amati relation, challenging its application to standardize \textlsGRB luminosities. We find that two bursts are consistent with a mildly mildly relativistic shock breaking out from the progenitor star, rather than the ultra–relativistic internal shock mechanism that powers standard cosmological bursts. Finally, in the context of the Zwicky Transient Facility (\textlsZTF), we discuss how we will continue to expand this effort to find optical counterparts of \textlsbinary neutron star mergers that should soon be detected Advanced \textlsLIGO and Virgo.

## Chapter \thechapter Range and sky resolution of \textlsGw detector networks

This chapter is reproduced from a work in preparation, of which I will be the sole author. Section 4 is reproduced from Singer et al. (2014), copyright © 2014 The American Astronomical Society.

In this chapter, we will use a basic description of the signal and noise received by a \textlsGW detector network to derive a matched filter bank, the prevailing technique used to search for well–modeled \textlsCBC signals in \textlsLIGO data. This model will allow us to calculate the range and angular resolution of a network of detectors.

### 3 Basic matched filter search

With interferometric detectors like LIGO and Virgo, the astrophysical signal is embedded in a time series measurement, the strain or the differential change in the lengths of the detectors’ two arms. Many noise sources enter the detector in different subsystems, get filtered by the detector’s response, and add to the measured strain. There are “fundamental” noise sources, such as quantum fluctuations in the laser field that result in shot noise at low frequency and radiation pressure noise at high frequency. Other noise sources are “technical,” meaning that they arise from the implementation of the detector as a realizable non-ideal system; examples include glitches due to scattered light, laser frequency fluctuations, cross–coupling between length degrees of freedom, coupling between angular and length degrees of freedom, and time–varying alignment drifts. Other noise sources are “environmental,” such seismic or anthropogenic ground motion noise.

For the purpose of \textlsGW data analysis, the most important division is between quasi-stationary Gaussian-like noise and transient noise sources (“glitches”). Extracting astrophysical signals from the data requires frequency domain (\textlsFD) techniques (whitening, matched filtering) to suppress the former and time domain (\textlsTD) approaches (coincidence, candidate ranking, time slides) to deal with the latter.

CBC searches are greatly aided by the fact that their \textlsGW signals can (at least in principle) be predicted with exquisite precision throughout LIGO’s sensitive band. Therefore, a standard approach to \textlsCBC detection is matched filtering; a representative set of model waveforms is assembled into a template bank with which the data is convolved.

In the \textlsTD, the strain observed by a single \textlsGW interferometer is

(1) |

In the \textlsFD,

(2) |

where is the \textlsGW signal given a parameter vector that describes the \textlsGW source, and is that detector’s Gaussian noise with one–sided \textlsPSD . We shall denote the combined observation from a network of detectors as .

Under the assumptions that the detector noise is Gaussian and that the noise from different detectors are uncorrelated, the likelihood of the observation, , conditioned on the value of , is a product of Gaussian distributions:

(3) |

A \textlsCBC source is specified by a vector of extrinsic parameters describing its position and orientation, and intrinsic parameters describing the physical properties of the binary components:

(4) |

This list of parameters involves some simplifying assumptions. Eccentricity is omitted: although it does play a major role in the evolution and waveforms of \textls—\textls (\textlsNSBH) and binary black hole (\textlsBBH) sources formed by dynamical capture (East et al., 2013), \textlsBNS systems formed by binary stellar evolution should almost always circularize due to tidal interaction (Belczynski et al., 2002) and later \textlsGW emission (Peters, 1964) long before the inspiral enters LIGO’s frequency range of 10—1000 kHz. Tidal deformabilities of the NSs are omitted because the signal imprinted by the companions’ material properties is so small that it will only be detectable by an Einstein Telescope–class \textlsGW observatory (Hinderer et al., 2010). Furthermore, in \textlsGW detection efforts, especially those focused on \textlsBNS systems, the component spins and are often assumed to be nonprecessing and aligned with the system’s total angular momentum and condensed to a single scalar parameter , or even neglected entirely: .

Assuming circular orbits and no spin precession, we can write the \textlsGW signal in each detector as a linear combination of two basis waveforms, and . For nonprecessing systems, and are approximately “in quadrature” in the same sense as the sine and cosine functions, being nearly orthogonal and out of phase by at all frequencies. If and are Fourier transforms of real functions, then and , and we can write (assuming an arbitrary phase convention)

(5) |

For brevity, we define and write all subsequent equations in terms of the basis vector alone. Then, we can write the signal model in a way that isolates all dependence on the extrinsic parameters, , into the coefficients and all dependence on the intrinsic parameters, , into the basis waveform, by taking the Fourier transform of Equation (2.8) of Harry & Fairhurst (2011b):

(6) |

for , where

(7) |

The quantities and are the dimensionless detector antenna factors, defined such that . They depend on the orientation of detector as well as the sky location (as depicted in Figure 7) and sidereal time of the event and are presented in Anderson et al. (2001). In a coordinate system with the and axes aligned with the arms of a detector, its antenna pattern is given in spherical polar coordinates as

(8) | ||||

(9) |

The unit vector represents the position of detector in units of light travel time.^{3}

(10) |

More succinctly, we can write the signal received by detector in terms of observable extrinsic parameters , the amplitude , phase , and time delay on arrival at detector :

(11) |

The prevailing technique for detection of GWs from CBCs is to realize a \textls estimator (\textlsMLE) from the likelihood in Equation (3) and the signal model in Equation (11). Concretely, this results in a bank of matched filters, or the cross-correlation between a template waveform and the incoming data stream,

(12) |

The maximum likelihood (\textlsML) point estimates of the signal parameters, , are given by

(13) | |||||

(14) | |||||

(15) |

A detection candidate consists of . There are various ways to characterize the significance of a detection candidate. In Gaussian noise, the maximum likelihood for the network is obtained by maximizing the network \textlsSNR, ,

(16) |

this, therefore, is the simplest useful candidate ranking statistic.

### 4 Measures of detector sensitivity

The sensitivity of a single GW detector is customarily described by the horizon distance, or the maximum distance at which a particular source would create a signal with a maximum fiducial single–detector \textlsSNR, . It is given by

(17) |

where is Newton’s gravitational constant, is the speed of light, the sum of the component masses, the reduced mass, the approximate power spectral density (PSD) of the inspiral signal, and the PSD of the detector’s noise. The lower integration limit is the low–frequency extent of the detector’s sensitive band. For the Advanced LIGO and Virgo detectors, ultimately limited at low frequency by ground motion (Adhikari, 2014), we take Hz. Using a typical value of the detector sensitivity , we can write Equation (17) as a scaling law:

(18) |

For BNS masses, the inspiral ends with a merger and black hole ring down well outside LIGO’s most sensitive band. A reasonable approximation is to simply truncate the \textlsSNR integration at the last stable orbit of a Schwarzschild black hole with the same total mass (Maggiore, 2008),

(19) |

Usually, is assumed because signals in two detectors (for a root–sum–squared network \textlsSNR of ) is nearly adequate for a confident detection (see discussion of detection thresholds in Section 24). Another measure of sensitivity is the BNS range , the volume-, direction-, and orientation–averaged distance of a source with , drawn from a homogeneous population. Due to the directional sensitivity or antenna pattern of interferometric detectors, the range is a factor of 2.26 smaller than the horizon distance for the same \textlsSNR threshold. See also Allen et al. (2012); Abadie et al. (2012d).^{4}

### 5 Fisher information matrix: single detector

We can predict the uncertainty in the \textlsML estimates without working out its full distribution. The Cramér—Rao lower bound (\textlsCRLB) gives its covariance in the asymptotic limit of high \textlsSNR. The \textlsCRLB has been widely applied in \textlsGW data analysis to estimate parameter estimation uncertainty (for example, Balasubramanian et al. 1996; Fairhurst 2009; Ajith & Bose 2009; Wen & Chen 2010; Aasi et al. 2013c; Fairhurst 2014)^{5}

(20) |

with given by Equation (11).

The Fisher information matrix for a measurement described by the unknown parameter vector is the conditional expectation value

(21) |

Note that if is twice differentiable in terms of , then the Fisher matrix can also be written in terms of second derivatives as

(22) |

In this form, we can recognize the Fisher matrix as the expectation value, conditioned on the true parameter values, of the Hessian matrix of the log likelihood. It describes how strongly the likelihood depends, on average, on the parameters. If is an unbiased estimator of , is the measurement error, and is the covariance of the measurement error, then the \textlsCRLB says that , in the sense that is positive semi–definite.

When (as in our assumptions) the likelihood is Gaussian, Equation (21) simplifies to

(23) |

This form is useful because it involves manipulating the signal rather than the entire observation . In terms of the th \textlsSNR-weighted moment of angular frequency,

(24) |

the Fisher matrix for the signal in the th detector is

(25) |

where

(26) |

(This is equivalent to an expression given in Grover et al. 2014.) The information matrix elements that relate to the intrinsic parameters can also be expressed as linear combinations of the angular frequency moments. However, as we will see in the next section, we need not compute these matrix elements if we are only interested in sky localization accuracy.

### 6 Independence of intrinsic and extrinsic errors

If all of the detectors have the same noise PSDs up to multiplicative factors, , then we can show that the errors in the intrinsic parameters (masses) are not correlated with sky position errors. This is because we can change variables from amplitudes, phases, and times to amplitude ratios, phase differences, and time differences. With detectors, we can form a single average amplitude, time, and phase, plus linearly independent differences. The averages are correlated with the intrinsic parameters, but neither are correlated with the differences. Since only the differences inform sky location, this gives us license to neglect uncertainty in masses when we are computing sky resolution.

This is easiest to see if we make the temporary change of variables . This allows us to factor out the \textlsSNR dependence from the single-detector Fisher matrix. The extrinsic part becomes

(27) |

Due to our assumption that the detectors’ PSDs are proportional to each other, the noise moments are the same for all detectors, . Then we can write the single-detector Fisher matrix as

(28) |

with the top-left block comprising the extrinsic parameters and the bottom-right block the intrinsic parameters.

Information is additive, so the Fisher matrix for the whole detector network is

(29) |

Now we introduce the change of variables that sacrifices the th detector’s extrinsic parameters for the network averages,

(30) | ||||

and replaces the first detectors’ extrinsic parameters with differences, | ||||

(31) |

The Jacobian matrix that describes this change of variables is

(32) |

The transformed network Fisher matrix is block diagonal,

(33) |

The top–left block contains relative amplitudes, phases, and times on arrival, all potentially correlated with each other. The bottom–right block contains the average amplitudes, phases, and times, as well as the masses. The averages and the masses are correlated with each other, but are not correlated with the differences. Because only the differences are informative for sky localization, we drop the intrinsic parameters from the rest of the Fisher matrix calculations in this chapter.

### 7 Interpretation of phase and time errors

We take a brief digression to discuss the physical interpretation of the time and amplitude errors. For our likelihood, the \textlsCRLB implies that

(34) |

where . Reading off the element of the covariance matrix reproduces the timing accuracy in Equation (24) of Fairhurst (2009),

(35) |

The Fisher matrix in Equation (27) is block diagonal, which implies that estimation errors in the signal amplitude are uncorrelated with the phase and time . A sequence of two changes of variables lends some physical interpretation to the nature of the coupled estimation errors in and .

First, we put the phase and time on the same footing by measuring the time in units of with a change of variables from to :

(36) |

The second change of variables, from and to , diagonalizes the Fisher matrix:

(37) |

Thus, in the appopriate time units, the sum and difference of the phase and time of the signal are measured independently.

### 8 Position resolution

Finally, we will calculate the position resolution of a network of \textlsGW detectors. We could launch directly into computing derivatives of the full signal model from Equation (6) with respect to all of the parameters, but this would result in a very complicated expression. Fortunately, we can take two shortcuts. First, since we showed in Section 6 that the intrinsic parameters are correlated only with an overall nuisance average arrival time, amplitude, and phase, we need not consider the derivatives with respect to mass at all. Second, we can reuse the extrinsic part of the single detector Fisher matrix from Equation (27) by computing the much simpler Jacobian matrix to transform from the time, amplitude, and phase on arrival, to the parameters of interest.

We begin by transforming the single–detector Fisher matrix from a polar to a rectangular representation of the complex amplitude given in Equations (14, 13), :

(38) |

Consider a source in a “standard” orientation with the direction of propagation along the axis, such that the \textlsGW polarization tensor may be written in Cartesian coordinates as

(39) |

Now introduce a rotation matrix that actively transforms this source to the Earth-relative polar coordinates , and gives the source a polarization angle (adopting temporarily the notation ):

(40) | ||||

(41) |

(The rightmost rotation reverses the propagation direction so that the wave is traveling *from* the sky position .) With the (symmetric) detector response tensor , we can write the received amplitude and arrival time as

(42) | ||||

(43) |

Equivalently, we can absorb the rotation and the horizon distance into the polarization tensor, detector response tensors, and positions,

(44) | ||||

(45) | ||||

(46) | ||||

(47) |

Now the model becomes

(48) | ||||

(49) | ||||

(50) | ||||

where | ||||

(51) | ||||

(52) |

We insert an infinitesimal rotation to perturb the source’s orientation from the true value:

(53) | ||||

(54) |

We only need a first order expression