Measuring the non-Gaussian stochastic gravitational-wave background: a method for realistic interferometer data

0.38 MB·

by  Eric Thrane

Measuring the non-Gaussian stochastic gravitational-wave background: a method for
realistic interferometer data
Eric Thrane1,a
1LIGO Laboratory, California Institute of Technology, MS 100-36, Pasadena, CA, 91125, USA
A stochastic gravitational-wave background (SGWB) can arise from the superposition of many
independentevents. Iftherateofeventsperunittimeissufficientlyhigh,theresultingbackgroundis
Gaussian,whichistosaythatitischaracterizedonlybyagravitational-wavestrainpowerspectrum.
Alternatively, if the event rate is low, we expect a non-Gaussian background, characterized by
intermittent sub-threshold bursts. Many experimentally accessible models of the SGWB, such as
the SGWB arising from compact binary coalescences, are expected to be of this non-Gaussian
3 variety. Primordial backgrounds from the early universe, on the other hand, are more likely to be
1 Gaussian. Measuring the Gaussianity of the SGWB can therefore provide additional information
0 about its origin. In this paper we introduce a novel maximum likelihood estimator that can be
2 used to estimate the non-Gaussian component of an SGWB signature measured in a network of
interferometers. This method can be robustly applied to spatially separated interferometers with
n
a colored, non-Gaussian noise. Furthermore, it can be cast as a generalization of the widely used
J stochastic radiometer algorithm.

2
PACSnumbers: 95.55.Ym

]
M
I. INTRODUCTION of isotropy provide yet another means of characterizing
I different models [32, 37, 45].
.
h A stochastic gravitational-wave background (SGWB) InthispaperweexploretheGaussianityoftheSGWB.
p
- is expected to arise from the superposition of many sys- A Gaussian SGWB is described only by its strain power
o tems, which are individually too weak to detect, but spectrum, while a non-Gaussian SGWB (sometimes re-
r which combine to produce a GW signature character- ferred to as an SGWB in the “popcorn” or “shot noise”
t
s ized by its ensemble statistical properties. In astrophys- regime) consists of a series of discrete sub-threshold
a
[ ical models [1], an SGWB can arise from objects such bursts (see Figure 1). Non-Gaussian signals can be de-
as compact binary coalescences [2–6], neutron stars (in- scribed with a probability distribution of burst wave-
1 cluding highly magnetized neutron stars) [7–12], young formsanda duty cycleξ, whichwe define asthe fraction
v or spun-up neutron stars [9, 13–16], core collapse super- of data segments during which a GW source somewhere
3
novae [17–20], and white dwarf binaries [21]. Cosmolog- inthe universeemitsGWs insomeanalysisband. Inthe
6
2 ical/primordial sources, meanwhile, can arise from infla- analysis that follows, it will be useful to divide the full
0 tionary physics [22–26], cosmic strings [27–32], and pre- GW observing band into smaller analysis bands.
. Big-Bang models [33, 34]. In this work we assume the duty cycle is less than
1
0 The initial LIGO and Virgo experiments have yielded unity, i.e., it is rare for two or more events to simulta-
3 a number of constraints on the SGWB [35–37] includ- neously emit in the same band. For many astrophysical
1 ing limits on the energy density of GWs, which sur- models, this is a good approximation. In the case of the
v: pass indirect bounds from Big Bang nucleosynthesis and SGWB from binary neutron star coalescence, for exam-
i measurements of the cosmic microwave background. A ple,weexpectthatξ ≈0.5%fora4Hz-widebincentered
X worldwidenetworkofsecond-generationGWinterferom- at 100Hz. We refer to both Gaussian and non-Gaussian
r eters are expected to begin taking data in 2015 [38–42], signals as “stochastic” since both can be described in
a
and recent work [5] indicates that realistic astrophysical terms of the ensemble behaviorof many individually un-
models can be probed with second-generation advanced detectable bursts. For a more nuanced discussion of rel-
interferometers—most notably, the SGWB arising from evant terminology, see Section A in the appendix.
binary neutron star and binary black hole coalescences. Different models of the SGWB predict different levels
In the eventof a detection, itmay notbe immediately of Gaussianity (see, e.g., [46]). Early-universe scenar-
clearwhichsystemsgiverisetotheobservedSGWB.The ios, as a rule of thumb, produce nearly Gaussian signa-
strainpowerspectrumprovidesonetoolfordisentangling tures whereas astrophysical scenarios tend to be more
differentpossiblesources[43,44]. Also,measurementsof non-Gaussian [47]. Moreover, a single model can pro-
the SGWB can be compared with measurements of GW ducearangeofdifferentdutycyclesandburstamplitudes
transients in order to indirectly infer information about depending on its parameters. Measurements of SGWB
theSGWB[43]. Finally,skymapsofGWpowerandtests Gaussianity can provide an important probe to distin-
guish between models and also to estimate cosmological
parameters such as the GW burst rate.
We introduce a maximum likelihood statistic to es-
aElectronicaddress: [email protected] timate the duty cycle and other parameters associated
2
2 6000 II. FORMALISM
1
4000
A. Gaussian searches
h 0
2000
−1 In order to motivate our non-Gaussian statistic, we
begin by describing how traditional (Gaussian) SGWB
−2 0
0 500 1000 −2 −1 0 1 2 analyses are carried out. The basic idea behind a Gaus-
t h
sianSGWBmeasurementistocross-correlatestraintime
100 1500
series from a pair of detectors I and J to create a cross
50 correlationstatistic,whichisanestimatorfortheenergy
1000
density spectrum of GWs (see, e.g., [35, 53]):
h 0
500
−50 f dρGW
Ω (f)= . (2.1)
GW
ρ df
−100 0 c
0 500 1000 −100 −50 0 50 100
t h
Here f is the GW frequency, dρ is the energy density
GW
of GWs in a frequency band (f,f +df), and ρ is the
FIG. 1: An illustration of Gaussian and non-Gaussian sig- c
criticalenergydensityofthe universe. Bysummingdata
nals. Top-left: a time series of sine-Gaussian bursts with a
from many time segments and frequency bins, it is pos-
low duty cycle produces a non-Gaussian signal. Top-right:
histogram of this non-Gaussian signal. Bottom-left: a time sible to observesignalsordersofmagnitude smallerthan
seriesofsine-Gaussianburstswithahighdutycycleproduces theinstantaneousnoisecurve. Foradditionaldetails,the
an approximately Gaussian signal. Bottom-right: histogram interested reader is referred to [53].
of this Gaussian signal. TheSGWBisassumedtobeisotropic,stationary,and
Gaussian [53, 54]. The measured strain time series in
detector I can be written as
s (t)=h (t)+n (t). (2.2)
I I I
Here h (t) and n (t) are respectively the astrophysical
I I
withnon-GaussianSGWB signaturesinGW interferom-
strain signal and the strain noise. For spatially sepa-
eters. In IIC, we show that this statistic can be cast
ratedinterferometers,n (t)andn (t)areexpectedtobe
I J
as a generalization of the stochastic radiometer [48, 49].
uncorrelated. Signal and noise are, of course, expected
Previousworkhasyieldedanearlyoptimaltechnique(in
to be uncorrelated. The signals in I and J, however,are
the statistical sense) for the special case of colocated,
expected to be highly correlated:
co-aligned interferometers characterized by stationary,
Gaussian white noise [50]. More recently, Seto has pro- hh˜⋆(t;f)h˜ (t;f)i6=0. (2.3)
I J
posedtheuseofhigherordermomentsformeasuringthe
Gaussianity of the SGWB [51, 52]. Here,tildesdenotediscreteFouriertransforms,and(t;f)
are spectrogram indices for time and frequency bins re-
spectively.
RealGWinterferometerdata,however,isfarfromide-
From Eq. 2.2-2.3, we can construct an estimator from
alized noise. It is colored, non-Gaussian,non-stationary,
the cross-power spectrum of the I and J strain chan-
and colocated interferometers suffer from correlated
nels [53]:
noise. These factors have make it challenging to imple-
ment the method from [50] in practice. The maximum 2
Yˆ(t;f)= Q(f)Re[s˜⋆(t;f)s˜ (t;f)]. (2.4)
likelihoodmethodproposedhereisapplicabletorealistic T I J
GW interferometer noise.
T is the segment duration and Q(f) is a filter function,
In Section II, we review the canonical framework for f3
Q(f)≡ (2.5)
analyzing the Gaussian SGWB and introduce a new γ(f)Ω (f)
M
formalism for handling non-Gaussian signals; in Sec-
tion III, we demonstrate the non-Gaussian formalism which emphasizes certain frequency bins depending on
with a Monte Carlo simulation; in Section IV we discuss thespectralshapeΩ (f)ofaparticularmodel,andage-
M
thesensitivityofSGWBsearchesusingthenon-Gaussian ometricfactorcalledthe overlapreductionfunctionγ(f)
formalism. Finally,inSectionV,wesummarizeourfind- (see Figure 2), which takes into account the fact that
ings and describe the next steps necessary to implement uncorrelated GW signals from different parts of the sky
thenon-Gaussianformalismwithactualdataandforspe- interfereto reducethe observedcorrelationbetweenspa-
cific models. tially separated interferometers.
3
LHO−LLO overlap reduction function that go into the calculation of Yˆ and σˆ [55–57].
tot tot
1 Recent work utilizes spectrograms of cross- and auto-
powertosearchforlong-livedGWtransientsandtoiden-
tify sources of environmental noise contaminating strain
0.5 data channels. Building on this work, we cast our non-
Gaussian statistic in terms of these intermediate data.
To begin, we define a complex estimator
γ(f) 0 Yˆ′(t;f)=Q(f)s˜⋆(t;f)s˜ (t;f), (2.9)
I J
which is a simple generalization of Eq. 2.4. Unlike a
−0.5
Gaussianbackground,whichis isotropicateveryinstant
in time, non-Gaussian bursts are associated with indi-
vidualskylocations,(evenif,onaverage,theyaredrawn
−1 from an isotropic distribution). This information is en-
50 100 150 200 250 300 350 400
f (Hz) coded in the phase of Yˆ′(t;f), and so it is necessary to
work with both real and imaginary components.
FIG. 2: The overlap reduction function γ(f) for the LIGO We take as our starting point Yˆ′(t;f) and σˆ(t;f) and
Hanford -LIGO Livingston detectorpair. Thered diamonds their ratio
markfrequencybinsoff =14Hzand74Hz,whicharesingled
out below to illustrate how the distribution of non-Gaussian ρ(t;f)=Yˆ′(t;f)/σˆ(t;f). (2.10)
signal varies with frequency.
The quantity ρ(t;f) has useful properties for our pur-
poses. First, it is well-studied and already in use in
An estimator for the variance of Yˆ(t;f) is given by stochastic analyses. Second, its statistical behavior can
be probed robustly through time slides in which one
|Q(f)|2
σˆ2(t;f)= P (t;f)P (t;f), (2.6) strain time series is offset by an amount greater than
I J
2Tδf the GW traveltime betweendetectorsinorderto obtain
many independent realizations of noise.
where δf is the frequency resolution and P (t;f) is
I Inordertoderiveournon-Gaussianitystatistic,ween-
the auto-power in detector I in time-frequency bins
deavor to answer a simple question: how does the signal
(t;f) [65]. Armed with Yˆ(t;f) and σˆ(t;f), the opti-
distribution of ρ(t;f), denoted S, differ from the back-
mal estimator for dfΩ (f) at time t is given by a
GW ground distribution of of ρ(t;f), denoted B. We take
weighted average:
eachvalueof(t;f)—pixelsinspectrogramsofρ(t;f)—to
R
be a separate measurement.
Yˆ(t)= Yˆ(t;f)σˆ−2(t;f)/ σ−2(t;f) For our purposes, any measurement in which there is
f f a non-Gaussian burst signal in (t;f) is drawn from S.
X X (2.7)
σˆ−2(t)= σˆ−2(t;f). Allothermeasurementsareconsideredtobebackground.
This definition of signal and backgroundis useful, but it
f
X can be counterintuitive. The signal distribution, as we
The optimal estimator for the entire data-taking period have defined it, is determined not only by properties of
is a weighted averageover time: the non-Gaussian SGWB; it is also determined in part
by properties of the detector noise. Defining signal and
Yˆ(t)σˆ−2(t) backgroundlikethiswillbeusefultoderiveanestimator
Yˆ = t
tot σˆ−2(t) for duty cycle.
P t Having pointed out these subtleties, we can now write
σˆt−o2tP= σˆ−2(t) (2.8) the signal distribution as
t
X
SNRtot =Yˆtot/σˆtot. S(ρ(t;f)|~θsignal,~θnoise), (2.11)
SNR is expectedto be well-approximatedby anormal where θ~ is a vector of parameters describing the
tot signal
distribution by the central limit theorem, and indeed, SGWB and ~θ describes the detector noise. In the
noise
this is born out empirically [35, 37]. same vein, we can write the background distribution as
B(ρ(t;f)|~θ ). (2.12)
noise
B. A non-Gaussian statistic
As we proceed we shall refer to simply S(Θ~) and B(Θ~),
Anumber ofapplicationshaveemergedthatmake use using capital Θ~ as shorthand for the appropriate vector
of Yˆ(t;f) and σˆ(t;f)—the intermediate data products ofparameters. Further,wedrop(t;f)argumentsinfavor
4
of i, which denotes separate measurements. We return tion, however, can be a useful starting point. S(ρ |Θ~)
i
later to asymmetries in the t and f variables. and B(ρ |Θ~) are used to weight data as more or less like
i
Examples of S(ρ |Θ~) and B(ρ |Θ~) obtained using signal. To the extent that they differ from the true dis-
i i
Monte Carlo are shown in Figure 3. A detailed descrip- tributions, the likelihood statistic will be less effective
tionoftheMonteCarlosimulationisprovidedinthesub- distinguishingbetweensignalandbackground. However,
sequent section. For now, we simply describe the panels if the likelihood statistic is ultimately tested empirically
of Figure 3 in general terms and point out some of the on time-shifted data, then we can avoid bias in detec-
interesting features. tion or parameter estimation—even if we construct the
The top-left panel shows a scatter plot of ρ for Gaus- estimatorwithanonlyapproximateMonte Carlomodel.
siannoise. (Inthiscase,weconsiderafrequencybincen- Armedwithdistributions ofofsignalandbackground,
tered on f = 74Hz, but the distribution looks the same the likelihood of observing ρ can be written as
i
at all frequencies.) The distribution has a mean of zero
and and is narrow compared to both non-Gaussian sig- P (ξ|ρ ,Θ~)=ξS(ρ |Θ~)+(1−ξ)B(ρ |Θ~). (2.13)
i i i i
nal and non-Gaussiannoise. The top-right panel, mean-
while, shows the case of a non-Gaussian signal in the Here ξ is the probability that the measurement is drawn
presenceofGaussiannoiseinafrequencybincenteredon from the signal distribution. We can interpret ξ as the
f =14Hz. At14Hz,theGWwavelengthλ≈2×104km duty cycle for our non-Gaussian signal model. This
is large compared to the separation of the interferome- formulation is similar to a maximum likelihood ap-
ters,≈3000kmforthecaseoftheLIGOHanfordObser- proach used in neutrino and gamma-ray astronomy, see,
vatory (LHO) and LIGO Livingston Observatory (LLO) e.g., [58–60].
asusedinthissimulation. Thus,ρisdistributedapprox-
Given N measurements, we can construct a likelihood
imately as it would be for a colocated pair of interfer-
function
ometers. The presence of a signal is evidenced by the
shift of the distribution away from zero. The fact that N
the shift is negative is due to relative orientations of the L(ξ|{ρ },Θ~)= P (ρ |ρ ,Θ~). (2.14)
i i i i
LHO and LLO detectors,which is encoded inthe signof i
Y
the overlapreduction function (see Figure 2).
The lower-left panel shows the distribution of ρ for Here,wehaveimplicitlyassumedthateachmeasurement
non-Gaussian signal in the presence of Gaussian noise isstatisticallyindependent. Thisassumptionisexpected
in a frequency bin centered on f = 74Hz. At 74Hz, tobeapproximatelyvalidforreasonablystationarynoise
the interferometers no longer behave as though they are andforsignalsthatdonotrepeatedlywanderinandout
colocated. The presenceofa signalshifts the meanaway of the same frequency bin. Subtle effects from overlap-
from zero, but it also changes the width and shape of ping data segments may require a more careful treat-
the distribution. Since 74Hz occurs in between the first ment. An estimator for ξ (denoted ξˆ) is given simply by
and second zeros of the overlap reduction function, the maximizing L(ξ). Confidence intervals for ξ are calcu-
meanispositive(seeFigure2). Forillustrativepurposes, lated straightforwardly from L(ξ|{ρ },Θ~). To illustrate,
i
thenon-Gaussianburstsusedtomakethisplotaremade we perform a simple Monte Carlo calculation, described
loud compared to the detector noise. in detail in Section III and summarized in Figure 4.
The lower-right panel shows non-Gaussian detector Finally,weconsiderthe questionofhowbestto detect
noise (also called “glitchy” noise) simulated by taking non-Gaussianity in an SGWB signal. We can frame this
the same bursts used to simulate a non-Gaussian sig- questionmorepreciselyas: whatis the appropriatemet-
nal (lower-left), but generating a signal in only one de- ric for determining whether ξ is non-zero. A convenient
tector at a time; i.e., we assume the noise at each site measure of non-Gaussian signal strength is the ratio of
is uncorrelated. The glitchiness of the signal widens thelikelihoodevaluatedatitsmaximumtothelikelihood
the distribution compared to Gaussian noise, but not evaluated at ξ =0 (see [58–60]):
as much as coincident non-Gaussian signals, which sug-
geststhatwecandifferentiatepopulationofglitchesfrom Λ=2log L(ξˆ)/L(0) . (2.15)
non-Gaussiansignals. Additionally,themeanofthenon-
Gaussian noise distribution is still zero, like Gaussian h i
noise. By comparing these distributions of different sig- If S(ρ |Θ~) and B(ρ |Θ~) are accurate descriptions of the
i i
nal and noise distributions, we can build a statistical signalandbackground,thenthe probabilisticinterpreta-
framework to differentiate them. This is the goal of the tionofΛas(twicethelogof)alikelihoodratioisstraight-
remainder of this section. forward. However, even if S(ρ |Θ~) and B(ρ |Θ~) are only
i i
In practice, S(ρ |Θ~) and B(ρ |Θ~) can be calculated approximatelyknown, we can calculate Λ for many real-
i i
fromMonteCarloorfrompseudoexperimentsperformed izations of time-shifted data. In this way, we can obtain
with time-shifted data. The former method is computa- a robust and empirical means of converting Λ to a false
tionally cheaper,but yields a lessaccuratedescriptionof alarm probability—even if our noise and signal models
the noise. Even an approximate Monte Carlo descrip- are only roughly approximate.
5
8 8
6 6
4 4
2 2
ρ) ρ)
m( 0 m( 0
I I
−2 −2
−4 −4
−6 −6
−8 −8
−5 0 5 −5 0 5
Re(ρ) Re(ρ)
8 8
6 6
4 4
2 2
ρm() 0 ρm() 0
I I
−2 −2
−4 −4
−6 −6
−8 −8
−5 0 5 −5 0 5
Re(ρ) Re(ρ)
FIG. 3: Scatter plots of ρ for different signal and background models. Top-left: Gaussian noise with no signal (arbitrary
frequencyband). Top-right: Gaussiannoiseinthepresenceofanon-Gaussiansignal;(f =14Hz). Bottom-left: Gaussiannoise
inthepresenceofanon-Gaussiansignal(f =74Hz). Bottom-right: Gaussiannoisewithnon-Gaussianglitchesidenticaltothe
non-Gaussian signals, butonly in onedetectorduringeach measurement (f =74Hz). Ineach plot, thefrequencybin widthis
4Hz.
C. Relationship to radiometer statistic The signal distribution is given approximately by [55]
1
The stochastic radiometerstatistic [48, 49] applies the S(ρ |Ω)∝exp − (ρ −Ω/σ )2 . (2.17)
i i i
2
Gaussianisotropicformalismof[53]tothecaseofaGaus- (cid:18) (cid:19)
sian point source. In this subsection we show how the
Sincethesourceis,byassumption,alwaysemittingGWs,
radiometer statistic can be cast as a special case of our
we can set ξ =1, which implies that there is GW signal
non-Gaussian statistic. We begin by defining the signal
present in every data segment and that we can ignore
model. We consider a point source associated with a
B(ρ |Θ~) entirely. It follows from Eqs. 2.13-2.14 that the
fixed sky location nˆ. We assume that the source is char- i
likelihood can be written as
acterized by a stationary GW energy density spectrum
Ω(f)=Ω. (For the sake of simplicity, we assume that it
N
1
is constant with respect to f.) L(ξ =1|{ρ },Ω)∝exp − (ρ −Ω/σ )2
i i i
2
For a single point source, there is a known phase rela- ! (2.18)
i
X
tionship between I and J and so Eq. 2.4 becomes [48]
∝exp −(Ω−Yˆ )2/2σ2 ,
tot tot
Yˆ(t;f)=Q(f)Re e−2πifτ(nˆ,t)s˜⋆I(t;f)s˜J(t;f) . (2.16) where Yˆ is simply the o(cid:16)ptimal estimator from(cid:17) Eq. 2.8.
tot
h i Thus, the radiometer statistic is a special case of the
Hereτ(nˆ,t)isthedelaytimebetweentheinterferometers. non-Gaussianstatisticinwhichthesignalmodelfixesthe
6
skylocationofthesourceandthedutycycleissettoξ = In order to construct the likelihood function used
1. The likelihood function can then be used to estimate in Figure 4, we construct distributions of S(ρ |Θ~) and
i
the energy density spectrum Ω. In principle, a similar B(ρ |Θ~) using 107 trials of Monte Carlo data. We then
i
analogyispossiblebetweentheisotropicstatistic[53]and useanindependentdatasetconsistingof500MonteCarlo
the non-Gaussian statistic. However, the fact that both measurements, which—following Eq. 2.14—we compare
the radiometer and the non-Gaussian statistic assume to S(ρ |Θ~) and B(ρ |Θ~) in order to measure the duty
i i
GW point sources makes the analogy shown here more cycle ξ [66].
straightforward.
The results of our simulation are illustrated in Fig-
ure 4. We test three datasets: one consisting of pure
background(ξ =0,dash-dotblue),oneconsistingofpure
D. Comparison to other methods signal (ξ =1, solid red), and a third consisting of a 50%
mixtureofeach(ξ =0.5,dashedgreen). Inallthreecases
wefindthattheobservedposteriordistributionsarecon-
One of the first papers to address the topic of detect-
sistentwith the true value ofξ. This demonstrates,with
ing a non-Gaussian SGWB is [50]. Our method differs
a very simple toy model, how our formalismcan be used
in several important ways. In this work we relax the as-
to measure ξ in the presence of colorednoise in spatially
sumptionsfrom [50]thatthenoiseisGaussianandwhite.
separated interferometers.
Instead of relying on two colocated detectors as in [50],
we assume two spatially separate detectors. Unlike [50],
ourlikelihoodstatisticisframedintermsofcross-power, x 10−3

with auto-power terms used only for normalization. pure S
8
Since the statistic in [50] is nearly optimal, it is very half−and−half
likely to provide a more sensitive measurement within 7 pure B
its domain of utility, compared to the method described
6
here. However,ourstatistic(builtfromcross-power)can
be extended straightforwardly to interferometers with 5
ξ)
colored, non-Gaussian noise, and they need not be colo- p(4
cated. Spatiallyseparatedinterferometers,inturn,allow
fortheuseofrobustbackgroundestimationthroughtime 3
slides. 2
Statistics utilizing forth-order (kurtosis) strain mo-
1
mentshavebeenproposedasprobesfornon-Gaussianity
in the SGWB [51, 52]. It is presently difficult to evalu- 0
0 0.2 0.4 0.6 0.8 1
ate the relative merits of different non-Gaussianity tech- ξ
niques,sincenonehasbeenutilizedwithrealinterferom-
eter data. Clearly, the next step is to carry out analyses
FIG. 4: Example posteriors for duty cycle ξ using Monte
with real data. Carlo data for pure background ξ = 0 (dash-dot blue), pure
signal ξ = 1 (solid green), and an even mixture of the two
ξ=0.5 (dashed green).
III. SIMULATION AND RESULTS
In order to demonstrate our likelihood formalism, we
perform a Monte Carlo simulation. We generate three IV. SENSITIVITY
types of data: Gaussian noise, Gaussian noise + non-
Gaussian GW bursts (signal), and non-Gaussian noise. A natural question is: if the non-Gaussian search
The Gaussian noise is colored according to the design presented here incorporates information about the non-
sensitivity of Advanced LIGO [38] and we assume a net- Gaussiancharacterofthepopcornsignalweseektomea-
work consisting of 4km detectors at LHO and LLO. We sure,canit,insomecases,provideamoresensitivedetec-
employatoysignalmodelconsistingofrandomlyarriving tionstatisticthantheGaussianstatisticusedinprevious
200mswhite-noiseburstswithastrainamplitudedensity stochastic searches [35–37]? A detailed analysis, beyond
of ≈ 3×10−24Hz−1/2. These bursts are marginal com- our present scope, is required to answer this question
paredtothenoise—theaverage|ρˆ|is0.43forburstsplus thoroughly. However,thereareseveralpointsworthnot-
noise and only slightly less (0.41) for noise alone—and ing.
can therefore be characterizedas sub-threshold. First,inthelimitof(highlyidealized)stationaryGaus-
We calculate spectrograms of ρ(t;f) (Eq. 2.10) using siannoise,weexpectthe non-Gaussianstatisticwillout-
4Hz×1s pixels. Since we are presently concerned only perform the Gaussian statistic. To illustrate, we note
with demonstration, this choice of pixel size is arbitrary. thatthe greendata in Figure4 havea Gaussianstatistic
The issue of pixel size is revisited in the appendix. signature of SNR < 1 (typical of pure noise) whereas
tot
7
the non-Gaussian statistic Λ = 15 represents a strong dation and operates under cooperative agreement PHY-
detection. We also expect, however, that as the data 0757058. This paper has been assigned LIGO document
becomes glitchier, the advantage of the non-Gaussian number ligo-p1200141.
approach will diminish, since both glitches and non-
Gaussian bursts will have a tendency to perturb higher-
order moments of the distribution of ρ(t;f) (albeit in
Appendix A: Terminology
different ways).
Second, the Gaussian statistic is almost completely
Rosado [47] has reviewed the SGWB literature and
insensitive to stochastic signals in frequency bins corre-
attempted to provide a comprehensive and standardized
sponding to the zeros of the overlap reduction function
glossaryofterminology. Wherepossible,wetrytofollow
γ(f). These zeros represent frequencies at which the de-
the terminology of [47], referring, for example, to the
tectorpairareaslikelytobeoutofphaseasinphase,and
objects,which combine to create a SGWB as “systems.”
so the integrated signal is zero. Since the non-Gaussian
However,whileweareloathtoaddtotheSGWBlexicon,
technique presented here incorporates higher-order mo-
some definitions and distinctions are necessary for our
ments in the distribution of ρ(t;f) (beyond the mean),
present purpose.
it will have at least some sensitivity to the SGWB even
Rosado makes a distinction between (in-principle) re-
when γ(f)≈0.
solvable and unresolvable SGWBs. Unresolvable SGWB
Third, while there are potential advantagesassociated
signals, according to [47], are present when, on average,
with the non-Gaussianstatistic, it is worthwhileto men-
twoormore systemssimultaneously createstrainsignals
tionseveraladvantagespossessedbytheGaussianstatis-
inthesamefrequencybin. Thisdistinctionismostuseful
tic. It is very well studied and has been shown to yield
inthecontextofafar-futuredetectorwithsufficientsen-
reliable results [35–37, 53, 61–63], it is simple to under-
sitivitytosubtractoutresolvablesignalsinordertomea-
stand and implement, and since it utilizes the sum of a
sure an underlying primordial SGWB [64]. Near-future
great many numbers, it is very robust to non-stationary
detectors will lack the sensitivity to separately measure
noise artifacts. Thus, the Gaussian statistic is likely to
the systems contributing to the “resolvable” SGWB. It
provide an important benchmark and cross-check to re-
is therefore useful to define sub-threshold bursts as the
sults obtained with the non-Gaussian statistic.
components of an in-principle resolvable SGWB, which
cannot be resolved in practice. An SGWB consisting
of sub-threshold bursts is always resolvable according to
V. CONCLUSIONS
thedefinitionin[47]. WhetherornotaresolvableSGWB
consistsofsub-thresholdburstswilldependonthedetec-
Some of the most promising sources of the stochastic
tor used to measure it. The non-Gaussian SGWB con-
gravitational-wave background (such as compact binary
sidered in this paper consist of sub-threshold bursts.
coalescences) are likely to be non-Gaussian. By measur-
ing the non-Gaussianity of the stochastic background,
we can learnmore aboutits origin. To this end, we have
Appendix B: Details
presented a maximum likelihood estimator that can be
used to measure the non-Gaussianity of the stochastic
gravitational-wavebackgroundutilizingrealisticinterfer- Herewepointoutdetailsthatwillrequiremorecareful
ometerdata. UsingMonteCarlodata,weillustratedhow attentioninordertoimplementthismethodforaspecific
thecalculationcanbecarriedout,anddemonstratedthat SGWB model. Our aim is not to provide a systematic
wecanestimatethe dutycycleoftheburststhatcharac- treatment,butrathertohighlightsomeofthefinerpoints
terizeanon-Gaussiansignal. Weoutlinedthenextsteps, worthy of attention.
which must be undertaken in order to tune the analysis Probability density functions.—The distributions for
for specific astrophysical models such as the stochastic S(ρ |Θ~) and B(ρ |Θ~) must be sampled with sufficient
i i
background arising from compact binary coalescences. resolutiontodistinguishbetweensignalandbackground.
Futureworkwillfocusoncarryingoutthis optimization. Models with very low-level bursts may require very high
resolutions, and so significant computational resources
may be necessary to compute S(ρ |Θ~) and B(ρ |Θ~).
i i
Acknowledgments Pixel size.—Pixel size can be chosen to optimize the
sensitivity of a search. The pixel dimensions should be
We thank Joseph Romano and Tania Regimbau for chosen so as to be comparable to the time and band-
thoughtful comments on a draft of this paper. This widthofthe non-Gaussianburstthatis the targetofthe
workisbyamemberoftheLIGOLaboratory,supported search. Pixels thatareverylong/shortintime areunde-
by funding from United States National Science Foun- sirablebecause,inthefirstcase,thesignalwillbediluted
dation. LIGO was constructed by the California Insti- with more noise than necessary, and in the second case,
tute of Technology and Massachusetts Institute of Tech- the signal will be spread thinly over many pixels. An
nology with funding from the National Science Foun- analogous argument can be made for the frequency bin
8
width. Numerical studies determine a suitable pixel size ξ from different bands, it may therefore be necessary to
appropriate for a given model. apply normalization factors to take into account the ex-
Broadband analysis.—The behavior of S(ρ |Θ~) and pected frequency evolution of the signal.
i
B(ρ |Θ~) varies significantly depending on the frequency
i Time-varying detector performance.—For a variety of
band of interest. For example, the mean of S(ρ |Θ~) can
i reasons, real GW detectors vary in performance on
be positive, negative, or zero depending on the value of
timescales ranging from minutes to months. As exam-
the overlap reduction function at the frequency in ques-
ples, anthropogenic noise can cause elevated noise levels
tion (see Figure 2). (For bursts drawn from an isotropic
duringthelocalrushhour,andnoiseperformancecanim-
distribution, hρ i is always real since ρ’s drawn from
i prove month to month following commissioning breaks.
some direction nˆ have, on average, the opposite imagi-
Inthe exampleplotsshowedinFigure3,weassumethat
nary component of ρ’s drawn from the antipodal direc-
the detector noise is stationary.
tion −nˆ.) Therefore, it may be desirable to calculate
S(ρ |Θ~) and B(ρ |Θ~) for many different bands. In order take into account the variability of the noise
i i
Additional complications may arise from the fact that asa function oftime, itmay proveuseful to addanother
the signal may not spend the same duration emitting variable to Θ~ describing the variability of the noise. An-
in every band. Compact binary coalescences, for exam- otheroptioncouldbetosimplyuseasubsetofthehighest
ple, emit at a frequency that accelerates as a function qualitydatainwhichthestrainsensitivityandglitchiness
of time. In order to combine posterior distributions of are relatively uniform.
[1] T. Regimbau, Res. Astr. Astrop.11, 369 (2011). [24] R. Easther and E. A.Lim, JCAP 0604, 010 (2006).
[2] E. S.Phinney,Astrophys.J. Lett. 380, L17 (1991). [25] N. Barnaby, E. Pajer, and M. Peloso, Phys. Rev. D 85,
[3] D.I.KosenkoandK.A.Postnov,Astron.&Astrop.336, 023535 (2012).
786 (1998). [26] J. Cook and L. Sorbo, Phys.Rev. D 85, 023534 (2012).
[4] X.-J. Zhu, E. Howell, T. Regimbau, D. Blair, and Z.-H. [27] R. R. Caldwell and B. Allen, Phys. Rev. D 45, 3447
Zhu,Astrophys.J. 739, 86 (2011). (1992).
[5] C. Wu, V. Mandic, and T. Regimbau, Phys. Rev. D 85, [28] T. Damour and A. Vilenkin, Phys. Rev. Lett. 85, 3761
104024 (2012). (2000).
[6] T.RegimbauandJ.A.deFreitasPacheco,Astrophys.J. [29] T. Damour and A. Vilenkin, Phys. Rev. D 71, 063510
642, 455 (2006). (2005).
[7] C. Cutler, Phys.Rev.D 66, 084025 (2002). [30] X. Siemens, V. Mandic, and J. Creighton, Phys. Rev.
[8] T.RegimbauandV.Mandic,Class. QuantumGrav.25, Lett. 98, 111101 (2007).
184018 (2008). [31] S.Olmez,V.Mandic,andX.Siemens,Phys.Rev.D81,
[9] T. Regimbau and J. A. de Freitas Pacheco, Astron. and 104028 (2010).
Astrophys.376, 381 (2001). [32] S. Olmez, V. Mandic, and X. Siemens, arXiv:1106.5555
[10] P.A. Rosado, Phys.Rev.D 86, 104007 (2012). (2011).
[11] E. Howell, T. Regimbau, A. Corsi, D. Coward, and [33] M. Gasperini andG.Veneziano,Astropart.Phys.1,317
R.Burman, Mon. Not.R. Ast.Soc. 410, 2213 (2010). (1993).
[12] S.Marassi,R.Ciolfi,R.Schneider,L.Stella,andV.Fer- [34] V. Mandic and A. Buonanno, Phys. Rev. D 73, 063008
rari, 411, 2549 (2011). (2006).
[13] B.J.Owen,L.Lindblom,C.Cutler,B.F.Schutz,A.Vec- [35] B. Abbott et al., Nature460, 990 (2009).
chio,andN.Andersson,Phys.Rev.D58,084020(1998). [36] J. Abadieet al., Phys.Rev. D 85, 122001 (2012).
[14] S. Chandrasekhar, ”Ellipsoidal Figures of Equilibrium” [37] J. Abadie et al. (LIGO Scientific Collaboration, Virgo
(NewHaven,Yale Univ.Press, 1969). Collaboration), Phys. Rev.Lett. 107, 271102 (2011).
[15] J. L. Houser, J. M. Centrella, and S. C. Smith, Phys. [38] G. M. Harry (for the LIGO Scientific Collaboration),
Rev.Lett. 72, 1314 (1994). Class. Quantum Grav. 27, 084006 (2010).
[16] D.LaiandS.L.Shapiro,Astrophys.J.442,259 (1995). [39] https://wwwcascina.virgo.infn.it/advirgo/docs.html
[17] P. Sandick, K. A. Olive, F. Daigne, and E. Vangioni, (2012).
Phys.Rev.D 73, 104024 (2006). [40] B. Willke et al., Class. QuantumGrav.23, S207 (2006).
[18] E. Howell, D. Coward, R. Burman, D. Blair, and [41] T. Uchiyama et al., Class. Quantum Grav. 23, S1161
J. Gilmore, Mon. Not. R.Ast. Soc. 351, 1237 (2004). (2004).
[19] ABuonannoandGSiglandGGRaffeltandH-TJanka [42] KurodaKfortheLCGTCollaboration, Class.Quantum
and E Mu¨ller, Phys.Rev.D 72, 084001 (2005). Grav. 27, 084004 (2010).
[20] S. Marassi, R. Schneider, and V. Ferrari, Mon. Not. R. [43] V.Mandic, E.Thrane, S.Giampanis, andT. Regimbau,
Ast.Soc. 398, 293 (2009). Phys. Rev.Lett. (2012), forthcoming.
[21] A. J. Farmer and E. Phinney, Mon. Not. R. Ast. Soc. [44] N. Seto and K. Kyutoku,prd 86, 042002 (2012).
346, 1197 (2003). [45] E. Thrane et al., Phys. Rev.D 80, 122002 (2009).
[22] L. P.Grishchuk, Sov.Phys. JETP 40, 409 (1975). [46] T.Regimbau,S.Gimapanis,X.Siemens,andV.Mandic,
[23] A.A. Starobinskii, JETP Lett. 30, 682 (1979). Phys. Rev.D 85, 066001 (2012).
9
[47] P.A. Rosado, Phys.Rev.D 84, 084004 (2011). [58] E.Thraneetal.(TheSuper-KamiokandeCollaboration),
[48] S.W. Ballmer, Class. Quantum Grav. 23, S179 (2006). Astrophys. J. 704, 503 (2009).
[49] S.Ballmer,Ph.D.thesis,MassachusettsInstituteofTech- [59] E.Thraneetal.(TheSuper-KamiokandeCollaboration),
nology (2006). Astrophys. J. 697, 730 (2009).
[50] S. Drasco and E. Flanagan, Phys. Rev. D 67, 082003 [60] J. Braun et al., Astroparticle Physics 29, 299 (2008).
(2003). [61] B. Abbott et al., Phys. Rev.D 69, 122004 (2004).
[51] N.Seto, Phys. Rev.D 80, 043003 (2009). [62] B. Abbott et al., Phys. Rev.Lett. 95, 221101 (2005).
[52] N.Seto, Astrophys.J. Lett. 683, L95 (2012). [63] B. Abbott et al., Astrophys.J. 659, 918 (2007).
[53] B. Allen and J. D. Romano, Phys. Rev. D 59, 102001 [64] C.CutlerandJ.Harms,Phys.Rev.D73,042001(2006).
(1999). [65] Typically, auto-power is calculated using adjacent time
[54] N.Christensen, Phys.Rev.D 46, 4250 (1992). bins in order to avoid a bias that results from using the
[55] E.Thrane,S.Kandhasamy,C.D.Ott,etal.,Phys.Rev. same data to calculate both Yˆ(t;f) and σˆ(t;f)
D 83, 083004 (2011). [66] Using this pixel resolution, one year’s worth of data
[56] T. Prestegard, E. Thrane, et al., Class. Quantum Grav. would correspond to a significantly larger dataset—≈
28, 095018 (2012). 3×107 measurements for every frequency bin—so this
[57] MichaelW Coughlin (fortheLIGOScientificCollabora- demonstrationisnotmeanttoberepresentativeofatyp-
tionandtheVirgoCollaboration),Class.QuantumGrav. ical year-long stochastic search.
28, 235008 (2011).