LISA data analysis: Source identification and subtraction

15 Pages·0.51 MB·English

by  Neil J. Cornish

LISA data analysis: Source identification and subtraction
Neil J. Cornish
Department of Physics, Montana State University, Bozeman, MT 59717
Shane L. Larson
Space Radiation Laboratory, California Institute of Technology, Pasadena, CA 91125
TheLaser Interferometer SpaceAntenna(LISA)will operateas an AM/FMreceiverfor gravita-
tional waves. For binary systems, thesource location, orientation and orbital phase are encoded in
the amplitude and frequency modulation. The same modulations spread a monochromatic signal
overa range of frequencies, making it difficult to identify individual sources. We present a method
for detecting and subtracting individual binary signals from a data stream with many overlapping
signals.
3
0
0 I. INTRODUCTION detectandstudy gravitationalwavesfromother sources,
2 such as the extreme mass ratio inspiral of compact ob-
jects into supermassive black holes [6, 7].
n Estimates of the low frequency gravitational wave
a background below 3 mHz [1, 2] have suggested that As will be seen, the problem of subtracting a binary
J ∼ outofthedatastreamisintimatelytiedtotheproblemof
the profusion of binary stars in the galaxy will be a sig-
8 sourceidentification,whichiscomplicatedbythemotion
nificantsourceofnoiseforspacebasedgravitationalwave
2 of the LISA detector. Several authors [8, 9] have previ-
observatorieslike LISA [3]. Most of these binary sources
are expected to be monochromatic, evolving very little ouslyexaminedtheangularresolutionoftheobservatory
2
as a function of the time dependent orientation. The bi-
v over the lifetime of the LISA mission; they will thus be
nary subtractionproblemhas receivedsome attention in
8 ever present in the data stream, and data analysis tech-
the past[10], but the work was not published.
4 niques will need to be developed to deal with them.
5 This paper examines the problem of binary subtrac-
Below 3 mHz, it is expected that there will be more
1 ∼ tion using a variant of the CLEAN algorithm [11] from
than one binary contributing to the gravitational wave
0 electromagnetic astronomy as a model for the subtrac-
background in a given frequency resolution bin. Predic-
3 tionprocedure. TheCLEANalgorithmmaybeconcisely
0 tions suggest that the population of binaries will be so
described in a few steps:
/ largeastoproduceaconfusionlimitedbackgroundwhich
h
will effectively limit the performance of the instrument.
p Identify the brightest source in the data.
- In this regime, it is likely that time delay interferome- •
o trytechniquescanbeemployedtocharacterizethe back- Using a model of the instrument’s response func-
r •
t ground [4]. At higher frequencies, open bins appear and tion, subtract a small portion of signal out of the
s individual galactic binaries (in principle) become resolv- data, centered on the bright source.
a
: able as single monochromatic lines in the Fourier record
v (a “binary forest”). Complications arise, however, from Remember how much was subtracted and where.
i •
X theorbitalmotionoftheLISAdetector,whichwillmod-
Iterate the first three steps until some prescribed
r ulate the signal froman individual source,spreading the •
level in the data is reached.
a signal over many frequency bins. A quick method for
demodulating the effect of the orbital motion on contin-
From the stored record of subtractions, rebuild in-
uousgravitationalwavesourceshasrecentlybeendemon- •
dividual sources[20].
strated [5].
Unlike sources for ground based observatories, the TheimplementationoftheCLEANalgorithminthispa-
gravitational waves from low frequency galactic bina- per is built around a search through a multidimensional
ries are expected to be well understood. In principle, template space which covers a binary source’s frequency
it should be possible to use knowledge about the ex- andamplitude,skyposition,inclination,polarizationand
pected gravitational wave signals to “subtract” individ- orbital phase.
ual sources out of the LISA data stream, both at high The format of this paper will be as follows: Section
frequencieswhereindividualsourcesareresolvableandat II describes the modulation of gravitational wave sig-
lower frequencies where single bright sources will stand nals by the motion of the LISA detector with respect
outabovethermsleveloftheconfusionbackground. The to the sky. Sections III and IV outline the description
ability to performbinary subtractionin LISA data anal- of the binaries used in this work, and the effect of the
ysis is particularly important in the regime of the LISA detector motion on their signals. Section V describes
floor (from 3 mHz to the LISA transfer frequency, the template space used to implement the gravitational
∼
f∗ = c/(2πL) 10 mHz), where an overlapping popula- wave CLEAN (“gCLEAN”) algorithm. Section VI re-
∼
tion of galactic binaries will severely limit our ability to views the expected contributions of instrumental noise
2
and the effects on the data analysis procedure. Section The strain produced in the detector is given by
VII describes and demonstrates the gCLEAN procedure
in detail. Lastly, a discussion of outstanding problems s(t)=A+F+(t)cosΦ(t)+A×F×(t)sinΦ(t), (7)
and future work is given in Section VIII.
where
Φ(t)=2πft+ϕ +φ (t). (8)
II. SIGNAL MODULATION 0 D
Hereφ (t)describestheDopplermodulationandF+(t),
D
LISA’s orbital motion around the Sun introduces am- F×(t) are the detector beam patterns
plitude, frequency and phase modulation into the ob-
served gravitational wave signal. The amplitude mod- F+(t) = 1 cos2ψD+(t) sin2ψD×(t)
ulation results from the detector’s antenna pattern be- 2 −
ing swept across the sky, the frequency modulation is F×(t) = 1(cid:2)sin2ψD+(t)+cos2ψD×(t)(cid:3), (9)
due to the Doppler shift from the relative motion of the 2
detector and source, and the phase modulation results (cid:2) (cid:3)
where
from the detector’s varying response to the two gravi-
tational wave polarizations. The general expression de- √3
scribingthestrainmeasuredbytheLISAdetectorisquite D+(t)= 36sin2θsin(2α(t) 2λ)
64 − −
complicated[12],butweneedonlyconsiderlowfrequency,
h
monochromatic plane waves. Here low frequency is de- +(3+cos2θ) cos2φ(9sin2λ sin(4α(t) 2λ))
− −
fined relative to the transfer frequency[13] of the LISA (cid:16)
detector, f∗ ≈ 10 mHz. The low frequency LISA re- +sin2φ(cos(4α(t)−2λ)−9cos2λ)
sponse function was first derived by Cutler[8], but we (cid:17)
4√3sin2θ sin(3α(t) 2λ φ)
shall use the simpler description given in Ref. [12].
− − −
AmonochromaticplanewavepropagatingintheΩdi- (cid:16)
3sin(α(t) 2λ+φ) , (10)
rection can be decomposed: − −
(cid:17)i
b
h(t,f)=A+cos(2πft+ϕ0)ǫ++A×sin(2πft+ϕ0)ǫ×, and
(1) 1
where A+ and A× are the amplitudes of the two polar- D×(t) = 16 √3cosθ 9cos(2λ−2φ)
ization states and h (cid:16)
cos(4α(t) 2λ 2φ)
ǫ+ = pˆ pˆ qˆ qˆ, − − −
(cid:17)
ǫ× = pˆ⊗qˆ+−qˆ⊗pˆ, (2) − 6sinθ cos(3α(t)−2λ−φ)
⊗ ⊗
(cid:16)
+3cos(α(t) 2λ+φ) . (11)
are polarization tensors. Here pˆ and qˆ are vectors that
−
point along the principal axes of the gravitational wave. (cid:17)i
Thequantityα(t)=2πf t+κdescribestheorbitalphase
For a source located in the nˆ = Ω direction described m
− of the LISA constellation, whichorbits the Sun with fre-
by the ecliptic coordinates (θ,φ) we can construct the
quency f =year−1. The constants κ and λ specify the
orthogonaltriad b m
initial orbital phase and orientation of the detector[12].
uˆ=cosθcosφxˆ+cosθsinφyˆ sinθzˆ We set κ = 0 and λ = 3π/4 in order to reproduce the
−
initialconditionschosenbyCutler[8]. TheDopplermod-
vˆ=sinφxˆ cosφyˆ
− ulation depends on the source location and frequency,
nˆ =sinθcosφxˆ+sinθsinφyˆ+cosθzˆ. (3)
andon the velocityof the guiding center of the detector:
This allows us to write R
φ (t)=2πf sinθcos(2πf t φ) (12)
D m
ǫ+ = cos2ψe+ sin2ψe×, c −
−
ǫ× = sin2ψe++cos2ψe×, (4) HereRistheseparationofthedetectorfromthebarycen-
ter,soR/cisthelighttraveltimefromtheguidingcenter
where of the detector to the barycenter.
The expression for the strain in the detector can be
e+ = uˆ uˆ vˆ vˆ,
re-arrangedusing double angle identities to read:
⊗ − ⊗
e× = uˆ vˆ+vˆ uˆ, (5)
⊗ ⊗ s(t)=A(t)cosΨ(t) (13)
and the polarization angle ψ is defined by
where
vˆ pˆ
tanψ = · . (6)
−uˆ pˆ Ψ(t)=2πft+ϕ0+φD(t)+φP(t). (14)
·
3
The amplitude modulation A(t) and phase modulation source,anddependsontheanglebetweenthewaveprop-
φP(t) are given by agationdirectionΩandthevelocityvectoroftheguiding
center. Pure Doppler modulation takes the form
A(t) = (A+F+(t))2+(A×F×(t))2 1/2 (15) b
s(t)=Acos[2πft+βcos(2πf t+δ)+ϕ ] , (22)
m 0
(cid:2) (cid:3)
A×F×(t) where β and δ are constants. Using the Jacobi-Anger
φ (t) = arctan . (16)
P − A F+(t) identity to write the Fourier expansion as
(cid:18) + (cid:19)
Eachofthemodulationfunctionsareperiodicinharmon- ∞
ics of fm. To get a feel for how each modulation affects eiβsin(2πfmt) = Jn(β)e2πifmnt (23)
thesignal,webeginbyturningoffallbutonemodulation n=−∞
X
at a time and look at how each individual term affects
allows the signal in Eq. (22) to be written
the signal.
∞
s(t)= A J (β)e2πi(f+fmn)teiϕ0ein(δ+π/2) .
n
A. Amplitude modulation ℜ n=−∞ !
X
(24)
Amplitude modulation derives from the sweep of the Onceagain,the Fourierpowerspectrumofs(t) willhave
detector’s antenna pattern across the sky due to the ob- sidebands about the carrier frequency f, spaced by the
servatory’s orbital motion, which for LISA gives a mod- modulating frequency fm. The bandwidth of the signal
ulation frequency, f = 1/year. Pure amplitude modu- is given by
m
lation takes the form
B =2(1+β)f (25)
m
s(t)=A(t)cos(2πft+ϕ ). (17)
0 ForLISA,theparameterβ (calledthemodulation index),
whichencodesthedescriptionofthedetectormotionrel-
Theamplitude, A(t)ismodulatedbytheorbitalmotion,
ative to the source, is given by
and may be expanded in a Fourier series:
∞ R
β =2πf sinθ. (26)
A(t)= a e2πifmnt (18) c
n
n=−∞
X Sourcesinthe equatorialplanehavebandwidths ranging
which allows the signal in Eq. (17) to be written fromB =2.6 10−4mHzatf =1mHztoB =2.1 10−3
× ×
mHz at f =10 mHz.
∞
s(t)= a e2πi(f+fmn)teiϕ0 . (19)
n
ℜ n=−∞ ! C. Phase modulation
X
Thus, the Fourier power spectrum of s(t) will have side-
Phase modulation is a consequence of the fact that
bandsaboutthecarrierfrequencyf ofthesignal,spaced
thedetectorhasdifferentsensitivitiestothetwogravita-
by the modulating frequency f . The bandwidth, B, of
m
tional wave polarization states, + and , characterized
the signal is defined to be the frequency interval which by the two detector beam patterns, F×+(t) and F×(t).
contains 98% of the total power:
The variationofthese beampatterns is a function ofthe
B =2Nf , (20) detector motion (see Eq. (9)), and modulates the phase.
m
Phase modulation takes a similar form to the frequency
where N is determined empirically by modulation. Expanding φ (t) in a Fourier sine series
P
yields a signal
N ∞
a 2 0.98 a 2. (21)
n n s(t)=Acos(2πft+ϕ + β sin(2πf nt+δ )). (27)
| | ≥ | | 0 n m n
nX=−N n=X−∞ n
X
Typical LISA sources give rise to an amplitude modu- Again, the the Fourier power spectrum of s(t) has side-
lation with N = 2 and using Eq. (20) a bandwidth of bands about the carrier frequency f, spaced by the
B =4fm =1.3×10−4 mHz. modulating frequency fm. The main difference is that
the Fourier amplitude of the kth sideband (located at
f +kf ) is given by
m
B. Frequency modulation
c =A J (β )eilnδneiϕ0 where k = l .
k ln n n
Doppler (frequency) modulation of signals occurs be- Yn Xln Xn
cause of relative motion between the detector and the (28)
4
Since the β ’sfor LISAareindependent offrequency(at It follows that the Fourier expansion of s(t) is described
n
least in the low frequency approximationused here), the by the triple sum
bandpassofthephasemodulatedsignalisindependentof
1
tBheca1r0r−ie4rmfrHeqzu.ency. Empiricallywefindthebandwidth sq = 2eiϕ0 (A+pl+ei3π/2A×cl) an Jk(β),
≈ Xl Xn Xk
(34)
whereq =k+l+n. Thelimitedbandwidthofthevarious
D. Total modulation modulations allows us to restrict the sums: (1+β)
− ≤
k (1+β), 4 l 4 and 10<n int(f/f )<10.
m
≤ − ≤ ≤ − −
Itis possible tocombine the amplitude, frequency and Using(34)wecancomputethediscreteFouriertransform
phase modulations together to arrive at an analytic ex- of s(t) very efficiently.
pression for the full signal modulation. The carrier fre- The source identificationandsubtractionscheme used
quency f develops sidebands spaced by the modulation in this work depends on the development and use of a
frequency f . The total modulation is most easily com- template bank covering a large parameter space. As
m
puted beginning from (7). One can write such, issues related to efficient computing are of interest
in order to make the problem tractable in a reasonable
4
amount of time. A number of simplifying factors allow
F+ = p e2πifmnt
n the problem to be compactified significantly, with great
n=−4
X savings in computational efficiency.
4 The quantities pn and cn only depend on θ, φ, and ψ,
F× = c e2πifmnt, (29) so they can be pre-computed and stored. The complete
n
n=−4 template bank can then be built using (34) by stepping
X
where the small number of non-zero Fourier coefficients through a grid in f, ϕ0 and the ratio A×/A+. The com-
putationalsavingascomparedtodirectlygeneratings(t)
can be attributed to the quadrupole approximation for
for each of the six searchparameters is a factor of 105
the beam pattern. The coefficients pn and cn can be in computer time. ∼
read off from (9) in terms of (θ, φ) and ψ.
Another big saving in computer time is based on the
Our next task is to Fourier expand cosΦ(t) and
followingobservation: The Fourierexpansionsof sources
sinΦ(t), being careful to take into account the fact that
a and b with all parameters equal save their frequencies,
we aredoing finite time Fouriertransforms,sof willnot
whichdifferbyanintegermultiple,m,ofthemodulation
be an integer multiple of f . In other words, writing
m frequency f , are related:
m
N
e2πift =n=−Nane2πifmnt, (30) saq −sbq+m ≃πmfmRc sinθ(saq+1−saq−1). (35)
X
we find in the limit N 1 that Thus, so long as m < 104, we have sa sb . This
≫ allows us to use a se∼t of templates genqe≈rateqd+mat a fre-
f
a sinc(πx )eiπxn where x = n. (31) quency f to coverfrequenciesbetweenf 104f . These
n ≃ n n f − ± m
m savings mean that our Fourier space approach to calcu-
The coefficients are highly peaked about n=int(f/f ), lating the template bank are a factor of 109 times faster
m
where the function “int” returns the nearest integer to than a direct computation in the time domain!
its argument. The maximum bandwidth occurs when
the remainder f/f n equals 1/2; the max bandwidth
m
is equal to 20f for−98% power (36f for 99% power). III. BINARY SOURCES
m m
Putting everything together we find
With the exception of systems that involve super-
F+cosΦ(t) = J (β)e2πifmkteik(π/2−φ) massive black holes, all of the binary systems that can
k
ℜ(cid:18)" k # be detected by LISA are well described by the post-
X Newtonian approximation to general relativity. Most
eiϕ0 p e2πifmlt a e2πifmnt . (32) of these sources can be adequately described as circu-
× " l l #" n n #(cid:19) larNewtonianbinaries,andthe gravitationalwavesthey
X X produce can be calculated using the quadrupole approx-
and imation. In terms of these approximations, a circular
Newtonian binary produces waves propagating in the Ω
F×sinΦ(t) = J (β)e2πifmkteik(π/2−φ) direction with amplitudes
k
ℑ(cid:18)" k # b
X
A = 1+(L Ω)2
+
eiϕ0 c e2πifmlt a e2πifmnt . (33) A ·
× "Xl l #"Xn n #(cid:19) A× = 2A(cid:16)L·Ωb b (cid:17) (36)
b b
5
where IV. BINARY SIGNAL MODULATION
2M M
= 1 2. (37) Theeffectsofamplitude,frequencyandphasemodula-
A rd tionontwobinarysourceswithbarycenterfrequenciesof
10 and 1 mHz are shownin Figures 1 and2 respectively.
Herer isthedistancebetweenmassM andM ,disthe
1 2 Thesourceshaveallparametersequalsavetheirfrequen-
distance between the source and the observer, and L is
cies, and are located close to the galactic center. We see
a unit vector parallelto the binary’sangularmomentum
that frequency modulation dominates at 10 mHz, while
vector. The gravitationalwaves have frequency b frequency and phase modulation become comparable at
1 mHz.
1 M +M
1 2
f =2f = . (38)
orb π r3 0.12 FM PM
r
0.1
0.6
The generalization to elliptical Newtonian binaries is 0.08
given in Peters and Mathews[14]. They found that ellip- 0.06 0.4
ticalbinariesproducegravitationalwavesatharmonicsof
0.04
the orbital frequency f . For small eccentricities, most 0.2
orb
0.02
of the power is radiated into the second harmonic, with
the portion of the power radiated into higher harmon- 00.009999 0.01 0.010001 00.009999 0.01 0.010001
ics increasing with increasing eccentricity. From a data
P(f) AM P(f) TM
analysis perspective, an eccentric binary looks like a col- 1.5 0.06
lection of circular binaries located at the same position
on the sky, with frequencies separated by multiples of 1.0 0.04
f . One strategy to search for eccentric binaries would
orb
be to conduct a search for individual circular binaries, 0.5 0.02
then check to see if binaries at a certain location form
part of a harmonic series. If they do, the relative am- 0 0
0.009999 0.01 0.010001 0.009999 0.01 0.010001
plitude of the harmonics can be used to determine the f f
FIG.1: Powerspectrashowingtheeffectsoffrequency(FM),
eccentricity.
phase(PM) and amplitude (AM) modulation separately and
Thepolarizationangleofacircularbinaryisrelatedto
all together (TM). The gravitational wave has frequency 10
its angular momentum vector orientation, L (θL,φL), mHz.
→
by [12]
b
cosθcos(φ φ )sinθ cosθ sinθ
tanψ = − L L− L , (39) 0.6 FM PM
sinθ sin(φ φ ) 0.8
L L
−
0.6
0.4
The inclination of a circular binary, ı is given by
0.4
0.2
cosı = L Ω 0.2
− ·
= cosθ cosθ+sinθ sinθcos(φ φ) (40)
b Lb L L− 0.0009999 0.001 0.0010001 00.0009999 0.001 0.0010001
It follows that the amplitude and phase modulation de- 2.0 P(f) AM P(f) TM
0.4
pendonfourparameters. Twoaretheskyposition(θ,φ)
1.5
and the other two are either the the angular momen- 0.3
tum direction (θ ,φ ), or the polarization angle ψ and 1.0
L L
the inclination ı. We found (ı,ψ) to be easier to work 0.2
withasthe quadrupoledegeneracybetweensourceswith 0.5
0.1
parameters (ψ,ϕ ) and (ψ +π/2,ϕ +π) is explicit in
0 0 0
these coordinates. The total gravitational wave signal 0.0009999 0.001 0.0010001 0.0009999 0.001 0.0010001
f f
from a Newtonian binary depends on seven parameters: FIG.2: Powerspectrashowingtheeffectsoffrequency(FM),
~λ (θ, φ, ı, ψ, ϕ , f, ). The parameter space has phase(PM) and amplitude (AM) modulation separately and
0
top→ology S2 T3 R2.AThe parameters θ and φ range all together (TM). The gravitational wave has frequency 1
over their usu×al in×tervals: θ [0,π] and φ [0,2π]. The mHz.
∈ ∈
inclination and polarization have ranges: ı [0,π] and
∈
ψ [0,π]. Because of the quadrupole degeneracy dis- Oneofthemaineffectsofthemodulationsistospread
cus∈sed above, we restrict the range of the orbital phase: the power across a bandwidth B 2(1+2πfRsinθ)f .
≃ c m
ϕ [0,π). This,combinedwithLISA’santennapattern,meansthat
0
∈
6
The average strain spectral density in the detector, hD,
TABLEI:Propertiesofthesixnearestinteractingwhitedwarf f
isbetween5andtentimesbelowthestrainspectralden-
binaries. Physical data from Hellier [15], periods taken from
sity at the barycenter hB. The effect is more significant
NSSDC catalog 5509[16]. Spectral amplitudes are computed f
at higher frequencies since the bandwidth increases with
using themethods of this paper for one year of observations.
frequency.
ThemassesquotedinunitsofthesolarmassM⊙,theorbital
periods are in seconds, the distances are in parsecs and the
strain spectral densities are in unitsof 10−19 Hz−1/2. The signalsfromthreeofthese binaries,averagedover
their bandwidths,areplotted againstthe standardLISA
Name m1 m2 Porb d hBf hDf hDf /hBf noise curve in Figure 3. The complete signals for all
six binaries are shown in Figure 4. The strain spectra
AM CVn 0.5 0.033 1028.76 100 21.2 2.34 0.111 appear as nearly vertical lines of dots due to the highly
compressed frequency scale.
CP Eri 0.6 0.02 1723.68 200 5.19 1.06 0.205
CR Boo 0.6 0.02 1471.31 100 11.5 1.63 0.141
GP Com 0.5 0.02 2791.58 200 3.32 0.44 0.133
HP Lib 0.6 0.03 1118.88 100 20.7 4.53 0.219
V803 Cen 0.6 0.02 1611.36 100 10.9 1.89 0.174
the strain in the detector is often considerably less than
the strain of the wave. The effect can be quantified in
terms of the amplitude of the detector response, A, and
the intrinsic amplitude of the source . Suppose that
A
a source is responsible for a strain in the detector s(t).
Defining A as the orbit-averagedresponse:
1 T
A2 = s2(t)dt, (41)
T
Z0
we find from (7) that
FIG.3: Thestrainspectraldensitiesofthreenearbyinteract-
1 ing white dwarf binaries plotted against the standard LISA
A2 = 2 (1+cos2ı)2 F2 +4cos2ı F2 , (42)
2A h +i h ×i noise curve.
(cid:0) (cid:1)
wherethe orbit-averageddetectorresponsesaregivenby
1
hF+2i= 4 cos22ψhD+2i−sin4ψhD+D×i+sin22ψhD×2i
1(cid:0) (cid:1)
hF×2i= 4 cos22ψhD×2i+sin4ψhD+D×i+sin22ψhD+2i
(cid:0)1 (cid:1)
hF+F×i= 8 sin4ψ(hD+2i−hD×2i)+2cos4ψhD+D×i
(cid:0) (4(cid:1)3)
and
243
D+D× = cosθsin2φ(2cos2φ 1)(1+cos2θ)
h i 512 −
3
D2 = 120sin2θ+cos2θ+162sin22φcos2θ
×
h i 512
3 (cid:0) (cid:1)
D2 = 487+158cos2θ+7cos4θ
h +i 2048
16(cid:0)2sin22φ(1+cos2θ)2 . (44)
− FIG. 4: The modulated strain spectral densities of the six
TherelativeamplitudeA/ dependsont(cid:1)hesourcedecli- nearest interacting white dwarf binaries plotted against the
nationθ,rightascensionφA,inclinationıandpolarization standardLISAnoisecurve. Crosses markstandardestimates
angle ψ. for the known binaries, while alternate symbols mark modu-
lated Fourier components.
Table 1 illustrates the power spreading effect for the
six nearest interacting white dwarf binaries. Random
numberswereusedforthe unknownparametersı andψ.
7
V. TEMPLATE OVERLAP Here T = 1 year is the observation time. We have to go
beyond the Doppler approximation to find metric com-
A. Template metric ponentsthatinvolveıandψ. Thecomputationsarecon-
siderably more involved, as are the resulting expression.
For example, including all modulations we find
The templates areconstructed by choosingthe six pa-
rameters~λ (f,θ,φ,ı,ψ,ϕ )andformingthe noise-free
detector res→ponse correspon0ding to a source with those g = 2hF+2ihF×2isin2ı sin2ı+2cos4ı (52)
parameters: ıı (1+cos2ı)2hF+2i(cid:0)+4cos2ıhF×2i(cid:1)2
s(t,~λ)=A(t,~λ)cosΨ(t,~λ) (45) and (cid:0) (cid:1)
We need to determine how closely the templates need to g = 2(1+cos2ı)2hF×2i+4cos2ıhF+2i
be spaced to give a desired level of overlap. The overlap ψψ (1+cos2ı)2 F2 +4cos2ı F2
of templates with parameters~λ1 and~λ2 is defined: 2hsin+4iı F+F× h ×i
h i (53)
R(~λ ,~λ )= hs(t,~λ1)|s(t,~λ2)i , − (1+cos2ı)2hF+2i+4cos2ıhF×2i 2
1 2 hs(t,~λ1)|s(t,~λ1)i1/2hs(t,~λ2)|s(t,~λ2)i1/2 Wehavebeenab(cid:0)letoderivedexactexpressionsfo(cid:1)rallthe
(46)
metric components, but they are cumbersome and not
with the inner product
very informative. For most purposes the simple Doppler
metric is sufficient.
T
a(t)b(t) = a(t)b(t)dt. (47)
h | i
Z0
Suppose we have two templates, one with parameters ~λ B. Overlap of parameters
and the other with parameters~λ+δ~λ. To leading order
in δ~λ the overlapis given by [17] AnimportantapplicationoftheDopplermetricinEq.
(51) is the determination of parameter overlap, which
R(~λ,~λ+δ~λ)=1 g (~λ)∆λi∆λj (48) has great bearing on the placement of templates. In re-
ij
− gionswherelargevariationsoftheoverlapcanbeseenfor
where g is the template space metric small changes in parameters, templates must be spaced
ij
closely to distinguish between different realizable physi-
∂ s(t,~λ)∂ s(t,~λ) cal situations. In regions where the change in overlap is
g (~λ) = h i | j i smallforsmallchangesinparameters,the templates can
ij 2 s(t,~λ)s(t,~λ)
be spaced more widely. The Doppler metric depends on
h | i
s(t,~λ)∂ s(t,~λ) s(t,~λ)∂ s(t,~λ) onlyfourparameters. Takingconstantslicesthroughthe
i j
h | ih | i.(49)
− 2 s(t,~λ)s(t,~λ) 2 parameter space for any two of the four will produce a
h | i metricwhichcanbe usedtoplotlevelcurvesoftheover-
UsingthefactthatΨ(t,~λ)variesmuchfasterthanA(t,~λ) lap function R(~λ1,~λ2) as a function of two parameters.
we find
0.6
∂ A∂ A + A∂ ΨA∂ Ψ
g (~λ) = h i | j i h i | j i
ij
2 AA
0.4
h | i
A∂ A A∂ A
i j
h | ih | i. (50)
− 2 AA 2
0.2
h | i
Ignoring the sub-dominantamplitude and phase mod-
ulation allows us to analytically compute the “Doppler (cid:1)'0 0
(cid:25)
metric”
–0.2
ds2 = g (~λ)∆λi∆λj
ij
2π2 1
= T2df2+πTdfdϕ + dϕ2 –0.4
3 0 2 0
R
2πf Tdf(cosθsinφdθ+sinθcosφdφ)
− c –0.6 –0.4 –0.2 0 0.2 0.4 0.6
+π2f2 Rc 2 cos2θdθ2+sin2θdφ2 . (51) FIG.5: Thetemplateoverlapc(cid:1)onft=ofmursonthe(f,ϕ0)cylinder.
(cid:18) (cid:19)
(cid:0) (cid:1)
8
Setting dθ =dφ=0leavesthetwodimensionalmetric
on the (f,ϕ ) cylinder:
0 0.04
2
2 π 3 1
ds2 = df + dϕ + dϕ2. (54)
2 3(cid:18)fm 4 0(cid:19) 8 0 0.02
Using this metric to plot the level sets of the overlap
function, the contours for 90%, 80% and 70% overlap 0
(cid:1)(cid:18)
are shown in Figure 5. One rather surprising result is
that the template overlap drops very quickly with ∆f.
AccordingtoNyquist’stheorem,thefrequencyresolution –0.02
observations of time T should equal f = 1/T. But we
N
seefromFigure5thattheoverlapdropsto90%for∆f
f /10. (Since we are using T = 1 year, it so happen∼s –0.04
N
that the Nyquist frequency f equals the modulation
N –0.04 –0.02 0 0.02 0.04
frequency f .)
m
(cid:1)(cid:30)
FIG.7: Thetemplateoverlapcontoursonthesky2-spherein
theneighborhood of θ=π/4.
FIG. 8: The overlap of templates with all parameters equal
save sky position. The reference template has a frequency of
f =10 mHzand a sky location of (θ,φ)=(π/4,0).
It is clear from this form of the metric that the angular
resolution improves at higher frequencies, and that the
FIG. 6: The overlap of templates with all parameters equal
savefrequencyandorbitalphase. Thereferencetemplatehas metric is not that of a round 2-sphere. The cos2θ factor
(f,ϕ0)=(0.01,π). Thefrequencyresolution wasfoundtobe in front of dθ tells us that the θ resolution drops as we
independentof thereference frequency. near the equator. This is might seem counter-intuitive
since the Doppler modulation is maximal at the equator
Since the metric (51)wasderivedby neglectingampli- (it depends on sinθ). But, the θ resolution depends on
tudeandphasemodulation,itonlygivesanapproximate the rate of change of the Doppler modulation with θ,
determination of the template overlap. Moreover, the which goes as cosθ. Setting f = 10 mHz, we plot the
approximate metric neglects the (ı,ψ) dependence com- template overlap contours in the neighborhood of θ =
pletely. In order to have a more reliable determination π/4 and θ =π/2 in Figures 7 and 9 respectively.
of the template overlap we generated a large template The template overlaps shown in Figures 8 and 10 (θ
bank and studied the template overlap directly. We see vs. φ) agree with those in found in Figures 7 and 9.
thatthe template overlapshowninFigure 6for f vs. ϕ The overlaps shown in Figures 11 and 12 (also θ vs. φ)
0
agrees with what was found in Figure 5. confirm our expectation that the angular resolution de-
Setting df = dϕ = 0 gives us the metric on the sky creases with decreasing frequency.
0
2-sphere: The template overlap as a function of inclination and
polarization angle turns out to be a very sensitive func-
R 2 tion of location in parameter space. While independent
ds2 =π2f2 cos2θdθ2+sin2θdφ2 . (55)
2 c of frequency and orbital phase, the metric functions gıı
(cid:18) (cid:19)
(cid:0) (cid:1)
9
0.2
0.1
0
(cid:1)(cid:18)
–0.1
FIG. 11: The overlap of templates with all parameters equal
save sky position. The reference template has a frequency
–0.2 –0.1 0 0.1 0.2
of f = 1 mHz and is close to the galactic center, (θ,φ) =
(1.66742,4.65723).
(cid:1)(cid:30)
FIG.9: Thetemplateoverlapcontoursonthesky2-spherein
theneighborhood of θ=π/2.
FIG. 12: The overlap of templates with all parameters equal
FIG. 10: The overlap of templates with all parameters equal save sky position. The reference template has a frequency of
save sky position. The reference template has a frequency f =1 mHzand a sky location of (θ,φ)=(π/4,0).
of f = 10 mHz and is close to the galactic center, (θ,φ) =
(1.66742,4.65723).
able to distinguish between sources above and below the
equator. This strong degeneracy is ameliorated by the
andg rangebetween0and326as(θ,φ,ı,ψ)arevaried.
ψψ amplitude and phase modulations, which are sensitive
Takingauniformsampleof1.6 109pointsin(θ,φ,ı,ψ),
× to which hemisphere the source is located in. In the
we found that g had a mean value of 0.4664, a median
ıı courseofapplyingthegCLEANprocedurewediscovered
valueof0.043,andthat90%ofallpointshadg <1.273.
ıı several other much stronger non-local parameter degen-
Similarly,g hadameanvalueof2.101,amedianvalue
ψψ eracies. By far the worst were those that involved fre-
of 2.0, and that 90% of all points had g <2.251. The
ψψ quency and sky location. The secondary maxima some-
analytic expressions for g , g and g were found to
ψψ ıı ψı timeshadoverlapsashighas90%. Anexampleofanon-
agree with direct numerical calculations of the template
local parameter degeneracy in the f - θ plane is shown
overlap.
in Figure 13. The reference template has f = 4.999873
mHz, θ = 0.5690, φ = 0.643, ı = 1.57, ψ = 0.314, and
ϕ = 0.50. The strongest of the secondary maxima is
0
C. Degeneracies
located at f = 4.999910 mHz, θ = 0.4615, and has an
overlapof90%with the referencetemplate. Inprinciple,
What the metric can not tell us about are the non- a sufficiently fine template grid should always find the
localdegeneraciesthatoccurinparameterspace. Amild global maxima, but in practice, detector noise and in-
example of a non-local degeneracy can be seen in Figure terference from other sources can cause gCLEAN to use
8, where there are secondary maxima in the template templates from secondary maxima. We will return to
overlap in the southern hemisphere. Physically this oc- this issue when discussing the source identification and
curs because the dominant Doppler modulation is un- reconstruction procedure.
10
VI. INSTRUMENT NOISE
Inorderto constructa demonstrationofthe gCLEAN
method, it is necessary not only to characterize the bi-
nary signals themselves, but also the noise in the detec-
tor. Instrumentalnoisecanhaveimportantconsequences
for the gCLEAN process, particularly in low signal to
noiseratiosituations,whererandomfeaturesinthenoise
spectrum of the instrument could conspire to approxi-
mate the modulated signal from a binary.
The total output of the interferometer is given by the
sum of the signal and the noise:
h(t)=s(t)+n(t). (56)
Assuming the noise is Gaussian, it can be fully charac-
terized by the expectation values
1
∗ ′ ′
n˜(f) =0, and n˜ (f)n˜(f ) = δ(f f )S (f),
n
h i h i 2 −
(57)
whereS (f)istheone-sidednoisepowerspectraldensity.
n
It is defined by
FIG. 13: An example of non-local parameter degeneracies in
thef, θ plane. ∞
n2(t) = dfS (f) , (58)
n
h i
Z0
D. Counting templates where the angle brackets denote an ensemble average.
The one-sided power spectral density is related to the
strain spectral density by S (f)= h˜ (f)2.
n n
Deciding what level of template spacing is acceptable | |
Expressing the noise as a discrete Fourier transform:
dependsontwofactors: thesignal-to-noiselevelandcom-
puting resources. Given a signal-to-noise level of SNR,
n(t)= n e2πifmjt, (59)
there is no point having the template overlap exceeding j
(1 1/SNR) 100%. Forthesearchesdescribedinthe Xj
∼ − ×
next section we made a trade-off between coverage and
arealizationofthenoisecanbemadebydrawingthereal
speed,andchosetemplatespacingsthatgaveaminimum
and imaginary parts of n from a Gaussian distribution
j
template overlap of 75% in each parameter direction
∼ with zero mean and standard deviation
(i.e. with all parameters equal save the one that is var-
ied). We chose to study sources with frequencies near h˜ (f j)
n m
5 mHz, and used a uniform template grid with spacings σj = . (60)
√2
∆f = f /5, ∆ϕ = π/4, ∆θ = ∆φ = 3.7o, ∆ı = π/7
m 0
and ∆ψ = π/9. A better approach would be to vary
Thesignal-to-noiseratioinagravitationalwavedetec-
the template spacing according to where the templates
tor is traditionally defined as
lie in parameter space. As explained earlier, each set of
5 4 3072 7 9=3,870,720templatescanbeusedto
co×ver×afrequ×enc×yrangeof104fm. Atworst,asourcemay SNR(f)= Ss(f) , (61)
liehalfwaybetweentwotemplates,soa 75%template sSn(f)
∼
overlaptranslatesintoa 92%sourceoverlap. Afterthe
coarsetemplate bankhas∼beenusedto findabestmatch whereSs(f)istheone-sidedpowerspectraldensityofthe
with the data, we refine the search in the neighborhood instrumentalsignal[21]. Givenaparticularsetofsources,
of the best match using templates that are spaced twice eachwiththeirownmodulationpattern,andaparticular
as finely in each parameter direction. realization of the noise, the quantity SNR(f) will vary
wildlyfrombintobin. Amoreusefulquantityisobtained
To implement the gCLEAN procedure, a template
by comparing the signal to noise over some frequency
bank was constructed by gridding the sky using the
interval of width ∆f centered at f:
HEALPIX hierarchical, equal area pixelization scheme
[18]. The HEALPIX centers provide sky locations (θ,φ)
to build up families of templates distributed across the Ss(f)
SNR(f,∆f)= { } , (62)
parameters (f,ı,ψ,ϕ0). s{Sn(f)}