Systematic errors in low latency gravitational wave parameter estimation impact electromagnetic follow-up observations

4.7 MB·

by  Tyson B. Littenberg

Draftversion March2,2016
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
SYSTEMATIC ERRORS IN LOW-LATENCY GRAVITATIONAL WAVE PARAMETER ESTIMATION IMPACT
ELECTROMAGNETIC FOLLOW-UP OBSERVATIONS
Tyson B. Littenberg
CenterforInterdisciplinaryExplorationandResearchinAstrophysics(CIERA)andDepartmentofPhysicsandAstronomy,
NorthwesternUniversity,2145SheridanRoad,Evanston, IL60208,USA
6 Ben Farr
1 EnricoFermiInstitute, DepartmentofPhysics,andKavliInstituteforCosmologicalPhysics,UniversityofChicago,Chicago,IL60637,
0 USA
2

r Scott Coughlin
a
CardiffUniversity,CardiffCF243AA,UK
M

1 Vicky Kalogera
CenterforInterdisciplinaryExplorationandResearchinAstrophysics(CIERA)andDepartmentofPhysicsandAstronomy,
] NorthwesternUniversity,2145SheridanRoad,Evanston, IL60208
E (Dated: March 2, 2016)
Draft version March 2, 2016
H
. ABSTRACT
h
p Among the most eagerly anticipated opportunities made possible by Advanced LIGO/Virgo are
- multimessenger observations of compact mergers. Optical counterparts may be short-lived so rapid
o characterizationof gravitationalwave (GW) events is paramount for discoveringelectromagnetic sig-
r
natures. One way to meet the demand for rapid GW parameter estimation is to trade off accuracy
t
s for speed, using waveform models with simplified treatment of the compact objects’ spin. We report
a
on the systematic errors in GW parameter estimation suffered when using different spin approxi-
[
mations to recover generic signals. Component mass measurements can be biased by > 5σ using

2 simple-precession waveforms and in excess of 20σ when non-spinning templates are employed. This
v suggeststhatelectromagneticobservingcampaignsshouldnottakeastrictapproachtoselectingwhich
1 LIGO/Virgocandidateswarrantfollow-upobservationsbasedonlow-latencymassestimates. Forsky
6 localization, we find searched areas are up to a factor of ∼2 larger for non-spinning analyses,and are
6 systematically larger for any of the simplified waveforms considered in our analysis. Distance biases
2 for the non-precessing waveforms can be in excess of 100% and are largest when the spin angular
0 momenta are in the orbital plane of the binary. We confirm that spin-aligned waveforms should be
.
1 usedforlow-latencyparameterestimationattheminimum. Includingsimpleprecession,thoughmore
0 computationally costly, mitigates biases except for signals with extreme precession effects. Our re-
6 sults shine a spotlight onthe criticalneed for developmentof computationally inexpensive precessing
1 waveforms and/or massively parallel algorithms for parameter estimation.
:
v
i
X 1. INTRODUCTION ral and merger signals will probe regimes of strong-
field, dynamical gravity inaccessible to any other obser-
r Ground-based gravitational wave (GW) interferome-
a ters aremost sensitiveat frequencies between10 Hz and vational or experimental techniques, testing the general
relativistic field equations (Cornish et al. 2011; Li et al.
2 kHz. Binary systems comprising of compact stellar
2012; Sampson et al. 2013; Berti et al. 2015) and exam-
remnants such as black holes (BHs) and neutron stars
ining the equation of state of matter at supranuclear
(NSs) will experience the late stages of orbit inspiral
density (Flanagan and Hinderer 2008; Read et al. 2009;
and will merge at these frequencies, making them the
primetargetfortheAdvancedLIGO(Abadie et al.2014) Lackey et al. 2014; Del Pozzo et al. 2013; Wade et al.
and Advanced Virgo (Acernese et al. 2015) network of 2014). The mergerrates andphysicalproperties of com-
pact binaries are sensitive to physical processes of com-
detectors. The scientific payoff from observing a pop-
pact object formation and binary evolution; their mea-
ulation of compact binaries in GWs is hard to over-
surements will provide strong constraints on theoretical
state. Mergers of binary neutron star (BNS) and/or
models for population synthesis (Belczynski et al. 2010,
neutron star-black hole (NSBH) systems are the lead-
2012;Mandel et al. 2015). Lastbut notleast, welllocal-
ing candidates for the progenitorsof short-hardgamma-
izedbinarieswillserveascosmological“standardsirens,”
ray bursts (sGRBs). Comparing rates of GW detec-
further constraining the scale and expansion rate of
tionsandsGRBswillconstrainthebeamingangleforthe
the local universe (Schutz 1986; Holz and Hughes 2005;
gamma-ray jet while GW signatures from such systems
Nissanke et al. 2010).
coincident with GRBs will confirm the binary merger
Clearlythe adventofGWastronomyandastrophysics
scenario (Chen and Holz 2013). Details of the inspi-
2
is highly anticipated in its own right. Furthermore, GW tive amplitude, polarization, and time of arrival of the
discoveries with identified electromagnetic (EM) coun- signal at each detector. To meet the low-latency de-
terparts would allow us to reach the full potential of mands placed on GW parameter estimation for prompt
multimessengerastronomy. GWobservatoriesareall-sky EM follow-up, a hierarchical strategy has been devised
monitorsandarewellapproximatedbyaquadrupolean- that takes advantage of the separability between intrin-
tenna with maximum sensitivity directly above and be- sic and extrinsic parameters to rapidly report parame-
low the plane of the detector. Because the antenna pat- ters for a candidate GW event(see Farr et al. (2014) for
tern varies slowly with angular distance away from the discussion on separate intrinsic and extrinsic analyses).
detector plane zenith, single detectors have poor sky- In this paradigm computation time is saved by reduc-
localization capabilities. Differences in the arrival time, ing the number of intrinsic degrees of freedom in the
phase,andpolarizationofthe GW signalindifferentde- GWmodelbymakingsimplifyingassumptionsaboutthe
tectors separated by large baselines enable more precise spin of the compact objects, either assuming particular
sky-locationmeasurementsbecausearelativelysmallre- alignments for the spin angular momenta or fixing these
gion of the sky could produce a particular coherent sig- terms to be zero. The scheme first relies on the rela-
nal. tive signal-to-noiseratio (S/N) and phase in each detec-
Because the GW measurement will not be able to tor with latency of minutes (Singer and Price 2015), a
pinpoint the location of a source alone, great effort parameter estimation analysis where the templates are
is needed to conduct electromagnetic observing cam- restricted by assuming either non-spinning or aligned
paigns. Because any optical counterpart may be short- spin binaries (substantially decreasing run time to a few
lived, and a typical EM telescope will have to use sev- hours) (Singer et al. 2014; Berry et al. 2015); finally the
eral pointings to cover the LIGO/Virgo error region full parameter estimation analysis that uses generically
on the sky, time is of the essence. Unfortunately, ob- spinning waveforms (Farr et al. 2015).
taining the most precise and accurate estimates for a The separability of intrinsic and extrinsic parameters
GW event’s physical parameters is a time-consuming is a fair approximation when dealing with binary NS
computational challenge. LIGO/Virgo parameter es- systems where spins are expected to be small, possibly
timation is performed using stochastic Bayesian sam- less than 10% of the maximum value allowed by general
pling algorithms such as Markov Chain Monte Carlo relativity. This separability breaks down when the bi-
(MCMC) (Gamerman 1997), Nested Sampling (Skilling narycontainsoneor twoBHs,as their spins canbe near
2004), or MultiNest (Feroz and Hobson 2007). Circu- maximal (see Fragos and McClintock (2015) and refer-
larized binary inspiral signals are characterized by 15 ences therein). Through relativistic couplings between
parameters. The 15-dimensional posterior distribution the spin-spin and spin-orbit angular momenta, rapidly
function is highly structured, often exhibiting multiple spinning misaligned binaries undergo dramatic preces-
modes and strongcorrelationsbetween parameters. The sion of the orbital plane, thereby coupling the intrinsic
samplingalgorithmsperformafundamentallyserialran- and extrinsic parameters. As a result, restricting the
domexplorationofparameterspace,ateachpointevalu- spin degreesof freedomfor template waveformscanbias
atinghowwellatrialsignalmatchesthedata. Eachstep the parameters we wish to measure rapidly, namely sky
of the analysis requires the computation of a model GW location of the GW source and the distance to a binary
waveform,or“template,”whichmustberepeatedO(107) signal, as well as to whether the source involves a NS
times to sufficiently characterize the 15-dimensional pa- or not (the latter has implications for the potential exis-
rameter space. The standard sampling algorithms take tence of an EM counterpart).
O(days) of modern high-performance computing time Miller et al. (2015) systematically compared the re-
to converge–latencies which can be longer in duration sults of parameter estimation from using different wave-
thanthelifetimeofproposedEMsignaturessuchaskilo- forms, varying different spin parameterizations as well
nova (Metzger et al. 2010). as post-Newtonian families, finding that biases in in-
Thetwophilosophicalapproachestoreducingthereal- trinsic parameter estimation (i.e. masses and spins) are
time latency for parameter estimation are to use com- comparable to the systematic differences between post-
putationally inexpensive approximations to the tem- Newtonianfamilies;theydidnotaddressissuesrelatedto
plate models or a simplification of the likelihood func- extrinsic parametersof importance, such as sky location
tion. Several groups have explored novel, massively and distance.
parallel, parameter estimation strategies to further re- Inthispaper,usingaconsistentpost-Newtoniantreat-
ducethetimerequiredforlow-latencyparameterestima- ment for all waveforms, we put the biases due to reduc-
tion (Pankow et al. 2015; Haster et al. 2015; Farr 2015). ing the spin degrees of freedom into astrophysical con-
Whilethemassivelyparallelparameterestimationmeth- text. Weaddressthefollowingkeyquestion: dothespin-
ods have shown promise in providing prompt, robust, constrained, fast, frequency-domain waveforms planned
signal characterization none have reached sufficient ma- forlow-latencyanalysisintroduceimportantbiasesinpa-
turity to be deployed in the real-time analysis for the rameter estimation, specifically relevant to EM follow-
early Advanced LIGO observing runs. up work (which is the reason for low-latency analysis)?
GWparametersareoftendividedintoextrinsicparam- We find that systematic errors incurred by using non-
eters (orientation, sky location, distance), which depend spinning or spin-aligned templates will be significant,
on the observer, and intrinsic parameters (masses and with respect to both the statistical errors and the as-
spins of the compact objects), which depend on the bi- trophysical interpretation of the results. For component
nary system. Generally speaking, intrinsic parameters masses that may be used to determine which candidate
influence the phase evolution of the binary waveform events should be followed up with telescopes, the biases
while extrinsic parameters are responsible for the rela- can be in excess of several M⊙, opening the possibility
3
of misidentifying a NS as a BH or vice versa. Electro- for parameter estimation, LALInference. LALInference
magnetic searches will have to cover an area on the sky is a code library built to provide robust Bayesian anal-
larger by a factor of . 2 to have the same probability ysis of GW signals and is part of the LIGO Analysis
of pointing at the true source location compared to the Library (LAL). For this study we use the lalinfer-
sky localization of the fully spinning/precessing analy- ence mcmcapplicationinLALInference,aparalleltem-
sis. While the biases systematically decrease as we use pered MCMC algorithm. A full description of the LAL-
increasingly complete waveforms, we find no substitute, Inference package, and the custom features found in lal-
fast, frequency-domain template waveform that is im- inference mcmc for analysis of compact binaries, can be
mune to introducing substantial biases in some parts of found in Veitch et al. (2015).
parameter space. TheLALInferenceanalysisis repeatedusingfourtem-
In the near term our results should be used to guide plate waveforms: SpinTaylorT2, which provides bias-
follow-up observing strategies. For the future we hope free parameter estimation of our simulated signals (“full
that our finding helps stimulate further development of precession”), and three other faster, frequency-domain
fast,accuratewaveformsthatcontainallofthenecessary waveforms, all inspiral-only, for consistency with the
degrees of freedom for accurate parameter estimation. source signals: SpinTaylorF2, which uses a simplified
treatment of the precession that conserves the open-
2. SETUP ing angle β = cos−1 Jˆ·Lˆ (“simple precession”); Tay-
(cid:16) (cid:17)
We simulate 1000 LIGO/Virgo signals randomly se-
lorF2Aligned, which fits for spin magnitudes assuming
lected from a uniform distribution in component mass
(m1,m2) drawn from m∈[1,30] M⊙ with an additional βfor=ms0(“(“naolnig-snpeidnnsipnign””)).; aFnigduzreero1-ssphionwTsatyhleorcFo2mwpuavtae--
constraintonthe totalmass ofthe binarym +m ≤30
1 2 tionalcostperlikelihoodevaluationbetweenthedifferent
M⊙; six parameters describing the two spin vectors
S ,S withmagnitudesdrawnfromU[0,1]andisotropic waveformsemployedinthiswork. Theapproximatecost
1 2 for each waveformmodel is determined by using the lal-
orientation with respect to the orbital angular momen-
tum L; sky location (α,δ) distributed uniformly on inference mcmcsoftwaretodraw1000independentsam-
ples for the prior distribution. The results are normal-
the celestial sphere; and three angles drawn uniformly
ized by the computation time for the non-spinning Tay-
over a sphere that describe the orientation of the bi-
lorF2 waveforms because they are the fastest available
nary (θ ,ψ,φ ). The θ parameter is the angle be-
J,N 0 J,N and the baseline model for rapid parameter estimation.
tween the wave propagation direction Nˆ (opposite the
Includingthetwospinmagnitudesinaconstrained,fully
observer’sline of sight k) and the total angular momen-
aligned configuration (TaylorF2aligned) has a negligible
tumJ. Inthelimitofnon-spinningbinariesthisisdegen-
effect on the computational cost of the model. Allowing
erate with the inclination angle ι=cos−1(kˆ·Lˆ) which is for simple precession (SpinTaylorF2) comes with a fac-
more traditionally used in the GW literature. We prefer tor of ∼2 increase in run time, while the time-domain,
θJ,N becauseithaslessvariationduringtheorbitalevolu- fully precessing waveform (SpinTaylorT2) is over an or-
tion,whilethedirectionofListime-varyingforspincon- der of magnitude more computationally taxing. This
figurations that induce significant precession (Farr et al. simplebenchmarkingtestdoesnottakeintoaccountthe
2014). The luminosity distance, DL, of each source is increased difficulty of sampling higher dimension poste-
drawnuniformlyinvolumewithanadditionaldetectabil- riors, which can increase the convergence time for the
ityrequirementthatthebinaryhasasignal-to-noiseratio MCMC analysis, but serves as a useful reference for the
S/N>5inatleasttwodetectors. ThisS/Nrequirement tradeoff between waveform accuracy and speed.
producesastrongselectioneffectduetothequadrupolar The output of lalinference mcmc is a series of inde-
emission of gravitational radiation. While GW emission pendent samples drawn from the posterior distribution
froma compactbinary is omnidirectional, it is strongest function for the template parameters that are used to
alongthedirectionoftheorbitalangularmomentumvec- draw inferences about the astrophysical nature of the
tor L of the binary. A system that is “face on,” i.e. the GW source.
line of sight to the observer k and L, is nearly parallel
(kˆ ·Lˆ ∼ 1), is detectable out to a larger distance than 3. RESULTS:INTRINSIC PARAMETERS
the same system in an “edge on” ( kˆ·Lˆ ∼ 0 ) configu- We begin by focusing on intrinsic parameter biases.
ration. Assuming that compact binaries are distributed Because the spin and mass parameters governthe phase
uniformly in volume, far more face-on binaries will be evolution of the binary, artificially constraining spin de-
detected than edge-on binaries (see Fig. 2 and related greesoffreedomforgenericbinarieswillhaveadramatic
discussion in Littenberg et al. (2015). effect on mass measurement. Figure 2 summarizes the
In our simulations, we generate the source GW sig- consequences of restricting spin degrees of freedom for
nals using the SpinTaylorT2waveforms,which are time- mass determination. Results shown in this figure come
domain inspiral-only waveforms that allow for generic frommarginalizedposteriordistributions for the smaller
spin magnitude and orientation, and hence for full pre- massinthebinary,m . Weusem asaproxyforwhether
2 2
cession. The waveforms do not include any treatment of a GW signalis a goodcandidate for electromagnetic ob-
themergerorringdownsignal,andassumequasi-circular servations. While BNS and NSBH systems are consid-
orbitsandnegligiblefinite-sizeeffectsofthecomponents. ered possible progenitors of short-duration GRBs, there
For each binary we compute the response of Advanced are no strong theoretical candidates from electromag-
LIGO/Virgoatdesignsensitivity Aasi et al. (2013) with neticsignalsfrommergingstellar-massBHs. Aresource-
a low-frequency cutoff at 20 Hz. Each simulated signal limited electromagnetic follow-up campaign may con-
is processed by the LIGO/Virgo data analysis pipeline siderusingtheinferredm todeterminewhichcandidate
2
4
value (open circle) is typically well outside of the region
in m –m space coveredby the posterior and often con-
1 2
finedinthewrongregionofparameterspaceentirely(e.g.
2
F in the NSBH region when the true signal was a binary
or
yl 101 BH of comparable mass). When including some free-
a
T dominthe spinparameters,whetherthatbe the aligned
o
e t (green, bottom left) or simple precession case (blue, top
v right), we find more consistent overlap with the bias-
elati free posteriors. However, neither the aligned spin nor
e r the simple-precession waveforms can generically be sub-
pl stituted for the full precession waveforms without risk-
m
a ing some substantial biases in the determination of m
s 1
∆t/ 100 andm2. For example, both the alignedspin and simple-
precession waveforms misidentified a source with a low-
mass secondary (m1 ∼19M⊙ and m2 ∼4.5M⊙) as a bi-
TaylorF2 TaylorF2Aligned SpinTaylorF2 SpinTaylorT2 nary BH system of comparable mass. We conclude that
low-latency mass estimates are not generically reliable,
Fig.1.—Relativespeedfordifferentwaveformstestedinthis
and that electromagnetic observing campaigns should
work estimated by drawing 1000 independent samples from
nottakeastrictapproachtoselectingwhichLIGO/Virgo
the lalinference mcmc prior. All times are measured rela-
tivetotheTaylorF2 (non-spinning)waveforms. Spin-aligned candidateswarrantfollow-upobservationsbasedonearly
(TaylorF2Aligned) waveforms add negligible cost. Simple mass estimates.
precession(SpinTaylorF2)isafactorof∼2morecostly,while
the full precession waveform (SpinTaylorT2) is slower by an 4. RESULTS:EXTRINSICPARAMETERS
order of magnitude. These results do not take into account The most pertinent measurements needed for low-
theadded inefficiency of sampling thehigher dimension pos- latencyGWsearchesaretheskylocationanddistanceto
teriors. thesource. Theskylocationisofobviousimportancefor
follow-upelectromagneticobservations,andthe distance
LIGO/Virgodetections are worth telescope time, select- canbeusedtoeitherselectnearbycandidateGWsources
ing only those systems that have an m2 measurement forobservationortoconvolvetheGWmeasurementwith
consistent with NS masses. galaxycatalogs(Nissanke et al. 2013;Hanna et al.2014;
The left panel is a histogram of the bias on m2. The Singer et al. 2014; Fan et al. 2014).
biasismeasuredasthedifference∆m2betweenthemean Forskylocalization,ourfigureofmeritisthe“searched
of the m2 posterior and the true value, normalized by area” for different waveforms. The searched area is
the posterior’s standard deviation σ. Using this defini- a measure of the number of square degrees contained
tion, the bias is a measure of how many standard de- within the credible interval of the posterior that inter-
viations (σ) away from the true value the posterior dis- sects the true signal parameters. If an electromagnetic
tribution is peaked. In this figure and throughout this follow-up strategy is to sequentially point a telescope at
paper, the gray, blue, green, and red curves correspond fields on the sky ranked by their posterior weight, the
to the precessing, simple precession, aligned spin, and searched area gives a sense of how large a region needs
non-spinning waveforms, respectively. In Fig. 2 we find to be searched to have a certain probability of the true
thatthepossiblebiasescanbeinexcessof5σ evenwhen source location being included among the observations.
usingthesimple-precessionwaveforms,andreachbeyond Figure 4 shows the cumulative distribution function
20σ for the non-spinning waveforms often used for low- of searched area for the precessing, simple precession,
latency parameter estimation. The right panel is the aligned spin, and non-spinning waveforms. We see that
distribution of standard deviations for the same simu- searched areas are systematically shifted to larger val-
lated signals, which are strongly peaked around 1 M⊙ ues as the spin degrees of freedomare further restricted,
withalongtaildownto ∼10−2 M⊙ forthe non-spinning meaning that sky-locationbiases incurredby the simpli-
waveforms. Not only is a 10σ or 20σ bias statistically fied waveforms translate to more telescope pointings to
alarming,but because ofthe typicalstandarddeviations ensure the same probability of imaging the true source.
for this parameter, these biases can have astrophysical For most signals in our simulated population, the differ-
repercussions as well (Littenberg et al. 2015). encesinsearchedareabetweenthefullyprecessingwave-
Figure 3 shows 11 randomly selected binaries (open forms andthe non-spinning waveformsare arounda fac-
circles) from our simulated population with a scatter toroftwo. Usingspin-alignedwaveformsyieldssearched
plot of the m1–m2 posterior samples for each source. areas that are typically ∼30% to ∼50% larger than the
The heavy vertical and horizontal lines denote the pu- full-precession results. Simple-precession searched areas
tative “mass gap” between the theorized highest mass are,atmost,∼30%largerbuttypicallyarewithin∼20%
NSs (∼3 M⊙) and the observed lowest mass BHs (∼5 of the full-precession areas.
M⊙) (Farr et al. 2011; O¨zel et al. 2010; Kreidberg et al. The distance to the source is a more difficult param-
2012). Each panel shows the recovered masses for the eter to measure. As noted throughout the GW litera-
different waveforms tested superimposed with the bias- ture, first in Cutler and Flanagan(1994) and further in-
free full precession posteriors, which are shown in gray vestigatedin Nissanke et al. (2010) and Rodriguez et al.
in each panel, and alone in the top left. Here we find (2014), there is a strong degeneracy between the orien-
thatthe non-spinningwaveforms(red,bottomright)are tation of the binary and the distance which causes the
highly prone to misclassifying the GW source. The true waveforms of near by, edge-on systems to be nearly in-
5
100 6
Precessing
Simple precession 5
Aligned spin
10-1 Non-spinning
4
10-2 3
2
10-3
1
10-4 0
-20 -15 -10 -5 0 5 10 15 20 10-3 10-2 10-1 100 101
∆m /σ σ (M )
2 solar
Fig. 2.— Left: bias in component mass recovery for full precession (gray, solid), simple precession (blue, dashed), aligned
spin (green, dotted), and non-spinning (red, dotted-dashed) templates measured in standard deviations of the full-precession
posterior. We find large biases when using simplified treatments of the spin parameters. Right: same color scheme as the left
panel, now showing the distribution of the standard deviation for the component mass posteriors, which are typically ∼1 M⊙.
The biases we findare astrophysically significant.
30 30
full precession full precession
25 injection 25 simple precession
injection
20 20
m1 15 m1 15
10 10
5 5
0 0
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
m m
2 2
30 30
full precession full precession
25 aligned spin 25 non-spinning
injection injection
20 20
m1 15 m1 15
10 10
5 5
0 0
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
m m
2 2
Fig.3.—A selection of m1–m2 posteriors compared to the true source values (open circles) for the four different spin config-
urations: full precession (gray points in each panel, plotted by themselves in the top left; simple precession (blue, top right);
alignedspin(green,bottomleft);andnon-spinning(red,bottomright). Heavyverticalandhorizontallinesmarkthe“massgap”
boundarybetweenhigh-massNSsandlow-massBHs. Theinferredcomponentmassesfromnon-spinningwaveformsgenerically
miss the true source value and often incorrectly classify the binary. Including aligned spins or simple precession improves the
agreement with the full-precession results but misclassification is still possible. Low-latency mass estimates should be taken
with caution.
6
detectable signal.
1
Precessing The right column in Figure 5 presents the fractional
Simple precession errorin distance ∆DL as a functionof the opening angle
0.8 NAolnig-nspeidn nsipnign DL
/beta between total and orbital angular momenta. Here
we see that the biases become larger than the statistical
0.6 errorforthe aligned(thirdrow,green)andnon-spinning
d.f. (bottom row, red) waveforms for β > 10◦ and can be
c. 0.4 larger than 100% for β > 30◦. At β = 90◦ the spin an-
gular momentum vector lies in the orbital plane of the
binary, maximizing the orbital precession. The aligned-
0.2
spin and non-spinning waveforms assume a constant bi-
nary orientation, making these waveforms a poor match
0 to the true signal. Except for rare instances the simple-
10-2 10-1 100 101 102 103 precession waveforms (second row, blue) return qualita-
searched area (deg2) tively similar distance posteriors to the full-precession
waveforms.
Fig.4.—Cumulativedistributionfunctionsforthesearched None of the results shown have assumed anything
sky area of each injection when using the non-spinning tem- aboutthe GW sourcebeyondthe implicit validity of the
plates(red,dotted-dashed),alignedspin(green,dotted),sim- waveformmodelsusedforsignalrecovery. Aspecialcase
ple precession (blue, dashed), and the precessing templates for ground-based GW astrophysics is the simultaneous
(gray, solid). The non-spinning analysis systematically re- observation of the GW and EM signature from a com-
quires more searched area to find the true signal. Includ- pactmerger. The mostplausibleEMcounterpartsto bi-
ing more accuracy in the waveform systematically improves
nary mergers involving at least one NS are short GRBs.
searchedareas. Noneofthesimplifiedwaveformsmatchesthe
Itisgenerallyassumedthatthe gamma-raysareemitted
performance of thefully precessing templates.
along a narrow, relativistic jet with an opening angle of
order 10◦. In a joint GRB/GW analysis, assuming the
distinguishable from distant, face-on systems at plausi-
jet is parallel to the angular momentum vector, gamma
ble S/Ns for Advanced LIGO and Virgo observations.
ray detections place a strong constraint on the allowed
Convolving this with the assumption that binaries are
values of θ , and consequently D for the binary.
uniformly distributed within the LIGO/Virgo horizon J,N L
Forasenseofhowtheinferreddistancedependsondif-
results in distance posteriors heavily influenced by the
ferentspin approximations,we selectNSBH events from
prior, which strongly favors distant, face-on systems.
our population of simulated signals that had θ <20◦
To quantify the difference in distance measurement J,N
(or > 160◦) as a set of potential short GRBs. If co-
we use the bias-free fully precessing distance estimates
incident GRB detections were made we could adopt a
as the baseline against which the other waveforms’ per-
strictpriorontheorientationofthebinary,whichwould
formances are measured. Figure 5 shows the fractional
helpconstrainthedistancetothesource. Figure6shows
difference ∆D /D in distance D , defined as the dif-
L L L 90%credibleintervalsforthedistancewhenweadoptthe
ference between the peaks of the distance posteriors for
“GRB prior” for θ as a function of the true signals’
the waveformunder investigationandthe full-precession J,N
S/N. The black circles denote the true distance. Once
waveform divided by the peak of the full-precession dis-
more we find significant bias in the distance estimate
tance posterior. The errorbarsin Figure5 are one stan-
when non-spinning (red, dotted-dashed) waveforms are
dard deviation of the posterior normalized by the peak
used for the analysis. The aligned spin (green, dotted)
of the full-precession distance distribution. The result
analysis excludes the true distance for three of the 10
is a measure of the bias and fractional error in distance
simulation. For this small sample the simple precession
measurement from using waveforms that make simpli-
(blue,dashed)andfullprecession(gray,solid)waveforms
fying approximations to the spin. From top to bottom,
return qualitatively similar distance posteriors, and in
therowsofplotscorrespondtothefullprecession(gray),
only one event do they differ significantly.
simple precession (blue), aligned spin (green), and non-
spinning (red) templates used for recovery. Our use of 5. DISCUSSION
the mode of the distance posterior as the point estimate Among the most eagerly anticipated discoveries made
againstwhichwemakecomparisonswasforconvenience. possiblebyAdvancedLIGO/VirgoarejointGWandEM
Macroscopicdifferencesbetweenthedistributionsarein- observations of compact mergers. Optical counterparts
sensitive to the particular choice of definition for bias in of compact mergers may be short-lived so a rapid re-
distance. sponse of electromagnetic follow-up observations to GW
The left column in Figure 5 shows ∆DDLL as a function triggers is paramount for discovering the EM signature
of the true signal’s network S/N. When using the same of GW detections. One way to meet the demand for
waveform for recovery as we do for simulating the data rapid GW parameter estimation is to trade off accu-
(top row) we find statistical errors of 10-30%. We find racy for speed, using computationally inexpensive wave-
that the distance biases grow and exceed the statistical form models that make simplifying assumptions about
uncertainty as the waveforms employed for recovery use the spin of the constituent compact objects.
simpler treatment of the spin. The biases in distance In this paper we studied the biases in parameter es-
determination are not found to be strongly dependent timation for generic GW signals when non-spinning,
on S/N. As a consequence, for accurate distance estima- aligned-spin, and simple-precession templates are used
tion spin precession will have to be considered for any for the analysis. We find non-negligible systematic
7
2 2
full precession
1 1
L
D
/L
D
∆
0 0
-1 -1
10 20 30 40 50 0.1 1 10 100
2 2
simple precession
1 1
L
D
/L
D
∆
0 0
-1 -1
10 20 30 40 50 0.1 1 10 100
2 2
aligned spin
1 1
L
D
/L
D
∆
0 0
-1 -1
10 20 30 40 50 0.1 1 10 100
2 2
non-spinnning
1 1
L
D
/L
D
∆
0 0
-1 -1
10 20 30 40 50 0.1 1 10 100
SNR acos|S•L| (deg)
Fig.5.— Left column: fractional differences in distance as a function of S/N. From top to bottom the results correspond
to the full precession, simple precession, aligned spin, and non-spinning analyses. Systematic errors in distance are largest
for waveforms without precession and the errors do not strongly scale with S/N. Right column: same ordering but now the
independent variable is the opening angle β between the total and orbital angular momenta. Here we see that the bias is
strongest for opening angles close to 90◦, which are cases that exhibit themost dramatic precession effects.
8
Distancedeterminationcanbeusedforselectingwhich
2500
Precessing candidates to observe, optimizing exposure times, and
Simple precession
employing galaxy catalogs to help guide EM observa-
Aligned spin
2000 Non-spinning tions. Distance biases for the non-precessing waveforms
injections canbeinexcessof100%andarelargestwhenthespinan-
c) 1500 gularmomentaareintheorbitalplaneofthebinary. The
p simple-precessionwaveformsproduceconsistentlysimilar
M
(L distance posteriors to the full-precession waveforms and
D 1000 are the only reliable low-latency option if distance esti-
∆
mates are deemed important for electromagnetic follow-
up observations. For potential GRB sources in our sim-
500
ulated population–NSBH injections with orbital angu-
larmomentumorientedwithin20◦–thesimple-precession
0 waveforms produced distance estimates consistent with
5 10 15 20 25 30 the full-precession waveforms while the aligned-spin re-
SNR covery excluded the true distance at the 90% level for
three of the 10 injections. The non-spinning waveforms
Fig.6.— Examples of marginalized distance posteriors for
showed a substantial bias, often overestimating the dis-
fullprecession (gray,solid),simpleprecession (blue,dashed),
tance to the source by hundreds of Mpc.
aligned spin (green, dotted), and non-spinning (red, dotted-
dashed)distanceposteriors. ResultscorrespondtotheNSBH In the near term, we confirm that spin-aligned wave-
systems in our injection set with face-on (θJ,N <20◦) orien- forms should be used for low-latency parameter estima-
tations. Black circles mark the true value. Points are the tion at the minimum. The computational cost is nearly
median values and error bars span the 90% credible inter- identical to that of the non-spinning waveforms, and
val of the posterior. For all cases the simple precession and the inclusion of spin dramatically improves mass esti-
precessing recovery include the injection. The aligned-spin mates and distance estimates for face-on systems. The
analysis excludes the true distance for three of the 10 simu- aligned-spin models are still subject to significant bi-
lations. Thenon-spinninganalysisexcludesthetruedistance
ases in the distance when the spin angular momenta
in all but four of thesimulations.
are in the orbital plane of the binary, and yield larger
errors on both intrinsic (masses) and extrinsic (loca- searched areas than the fully precessing waveforms. At
tion and orientation) parameters. Our study builds on the added cost of a factor of ∼2 in computational ef-
Miller et al. (2015), which investigated systematic dif- fort, the simple-precession waveforms correct biases in
ferences between post-Newtonianwaveforms,by putting the distance and make further improvements to the sky
the biases resulting from restricting the spin degrees of localization. However,atthis time there isno substitute
freedomintoastrophysicalcontext. Weconcludethatthe for the full spinning precessing analysis for accurate pa-
biasesinparameterestimationofspinningbinariesusing rameter estimation in any of the parameterswe studied.
non-spinning templates will be large enough to impact Ourresultsshineaspotlightontheimportanceofefforts
astrophysicalinterpretation of the results. todevelopcomputationallyinexpensiveprecessingwave-
Component mass measurements, which may be used forms (Field et al. 2014; Lundgren and O’Shaughnessy
to determine which candidate events should be followed 2014; Chatziioannou et al. 2014; Klein et al. 2014), and
up with telescopes, can be biased by >5σ using simple- for novel, embarrassingly parallel, parameter estimation
precession waveforms and in excess of 20σ when non- methods (Pankow et al. 2015; Haster et al. 2015; Farr
spinning templates are employed. These systematic er- 2015). There is a critical need for low-latency param-
rors translate to biases several M⊙ large, potentially eter estimation methods which are able to account for
leading to misclassification of NSs and BHs. This sug- all known features in the signals.
gests that electromagnetic observing campaigns should
nottakeastrictapproachtoselectingwhichLIGO/Virgo
candidateswarrantfollow-upobservationsbasedonlow- 6. ACKNOWLEDGMENTS
latency mass estimates. We thank Atul Adhikari, Claudeson Azurin, Brian
Without question the most pertinent parameter esti- Klein,BrandonMiller,LeahPerri,BenSandeen,Jeremy
mationresultofGWsearchesistheinferredskylocation Vollen, MichaelZevinfor helprunning the MCMC anal-
of the source. We find that searched areas are up to a ysis. T.B.L. and V.K. acknowledge NSF award PHY-
factor of ∼2 larger for non-spinning analyses, and are 1307020. B.F. was supported by the Enrico Fermi Insti-
systematically largerfor any of the simplified waveforms tute at the University of Chicago as a McCormick Fel-
considered in our analysis. While the searched areas low. S.C. thanks the US-UK Fulbright Commission for
systematically decrease as we use increasingly complete personal financial support during this research period.
waveforms,wefindnosubstitutetemplatewaveformthat Computational resources were provided by the North-
is able to reproduce the sky-localization performance of westernUniversityGrailclusterthroughNSFMRIaward
the fully spinning/precessing waveforms. PHY-1126812.
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