Gravitational waves from inspiraling compact binaries: Validity of the stationary-phase approximation to the Fourier transform

8 Pages·0.3 MB·English

by  Serge Droz

Gravitational waves from inspiraling compact binaries:
Validity of the stationary-phase approximation to the Fourier transform
Serge Droz∗, Daniel J. Knapp†, Eric Poisson
Department of Physics, University of Guelph, Guelph, Ontario, Canada N1G 2W1
Benjamin J. Owen
Theoretical Astrophysics 130-33, California Institute of Technology, Pasadena, California 91125, and
Max Planck Institut fu¨r Gravitationsphysik, Schlaatzweg 1, D-14473 Potsdam, Germany
(Draft of January 26, 1999)
Weprovethattheoft-usedstationary-phasemethodgivesaveryaccurateexpressionfortheFourier
transform of the gravitational-wave signal produced by an inspiraling compact binary. We give
three arguments. First, we analytically calculate the next-order correction to the stationary-phase
approximation, and show that it is small. This calculation is essentially an application of the
9
steepest-descent method to evaluate integrals. Second, we numerically compare the stationary-
9
phase expression to the results obtained by Fast Fourier Transform. We show that the differences
9
can be fully attributed to the windowing of the time series, and that they have nothing to do with
1
an intrinsic failure of the stationary-phase method. And third, we show that these differences are

n negligible for thepractical application of matched filtering.
a
J
Pacs numbers: 04.25.Nx; 04.30.Db; 04.80.Nn

6
2

1 I. INTRODUCTION AND SUMMARY alytic expression rather than as a table of values; efforts
v
to obtain such an analytical expression,even if it is only
6
7 The ongoing construction of kilometer-size interfero- approximate, are therefore well justified. It can also be
0 metric gravitational-wave detectors, such as the Ameri- arguedthattoexpressthewaveforminthefrequencydo-
1 can LIGO (Laser Interferometer Gravitational-waveOb- main is in a sense more natural than to express it in the
0 servatory), the French-Italian VIRGO, and the British- time domain. One of the reasons is that the orbital en-
9 German GEO600,has motivated a lot of recent work on ergy,whoseexpressionisrequiredinthederivationofthe
9
strategies to analyze the data to search for and measure waveform,is primarily a function of F, the gravitational
/
c the gravitational-wavesignals [1–16]. Much of this work wave’s instantaneous frequency (which will defined pre-
q has been devoted to inspiraling compact binaries, com- ciselybelow);itisthereforenaturaltoexpresshalsoasa
-
r posedofneutronstars and/orblack holes,whichare one functionofF,whichunliketisacoordinate-independent
g of the most promising sources of gravitational waves for quantity. Another reason resides with the fact that the
:
v these detectors. The key idea behind this work is that post-Newtonianexpressionforthe relationF(t) suggests
i the gravitational-wavesignals will be detected and mea- that the instantaneous frequency is not always a mono-
X
sured by matched filtering [17], a well-known technique tonically increasing function of time [18], a nonphysical
ar by which the detector’s data stream is cross-correlated behaviorthatsignalstheeventualbreakdownofthepost-
with a set of model waveforms (called templates), and Newtonian expansion; in contrast, the post-Newtonian
the signal-to-noise ratio maximized as a function of the expression for the inverted relation t(F) is monotonic, a
template parameters. property that suggests that F is indeed a more natural
While the detector output is representedby a discrete time variable.
timeseries,theoperationsassociatedwithmatchedfilter- In the past, and in the specific context of inspiral-
ing are usually carried out in (discrete) frequency space, ing binaries, analytic expressions for h˜(f) have been ob-
which requirestaking the (discrete) Fourier transformof tained within the stationary-phase approximation [19],
thetimeseries;themethodofchoicehereisthestandard essentially the leading-order term in an expansion in
Fast Fourier Transform (FFT). At the same time, it is powers of the small quantity (radiation-reaction time
necessary to compute the Fourier transform of h(t), the scale)/(orbital period). While it has generally been felt
model waveform. The waveform h(t) can easily be dis- that this approximation is adequate, this belief has not
cretized to mimic the discrete sampling of the detector yet been backed up (in the published literature) with a
output, andthe discretetime seriescaneasilybeFourier detailed quantitative analysis. Claims in the literature
transformedbyFFTtoyieldthediscreteanalogueofthe [14]tothe effectthatthis approximationis notadequate
(continuous) Fourier transform h˜(f). have prompted us to examine this question.
However,intheoreticalinvestigations[1–13,15,16]it is Ourobjectiveinthispaperistoprove,beyondanyrea-
often much more convenient to deal with h˜(f) as an an- sonable doubt, that the stationary-phase approximation
to the Fourier transform of an inspiraling-binary signal
1
is in fact very accurate. First, we calculate the next- where v (π f)1/3 1, and
≡ M ≪
order correction to the stationary-phase approximation,
π 3
and show that it is indeed small. This calculation is es-
ψ(v)=2πft Φ + . (1.5)
sentially an application of the steepest-descent method c− c− 4 128v5
to evaluate integrals [19]. We then numerically compare
Two types of corrections to this result are calculated in
the stationary-phase expression for h˜(f) to the results
Sec. II.
obtained by FFT. We show that the differences can be
Thefirsttypeofcorrectionconstitutesanintrinsicim-
fully attributed to the windowing of the time series, and
provement on the stationary-phase approximation. We
that they are irrelevant for matched filtering.
show that the steepest-descent evaluation of the Fourier
We now present and explain our main results. As we
transformreturnsthesameamplitudeasbefore,butthat
shalljustifybelow,itissufficientforourpurposestocon-
the phase is altered by a term δψ: ψ ψ+δψ, where
sider the leading-order expression for the gravitational- →
wave signal, obtained by assuming that the binary’s or- 92
δψ(v)= v5+O(v10). (1.6)
bital motion (taken to be circular) is governed by New- 45
tonian gravity, with an inspiral — caused by the loss
of energy and angular momentum to gravitational radi- Notice that this is a correction of order v10 relative to
ation — governed by the Einstein quadrupole formula. ψ(v). This is much smaller than post-Newtonian correc-
This so-called Newtonian signal is given by [1] tions to the phase, which appear at relative order v2 [5].
Incorporating these post-Newtonian corrections into our
h(t)=Q(angles)M(π F)2/3e−iΦ, (1.1) calculationwouldonlychangeEq.(1.6)byaddingaterm
r M of order v7 to the right-hand side. This justifies the fact
that it was not necessary, for the purposes of this inves-
where Q is a function of all the relevant angles (posi-
tigation, to use more accurate versions of Eqs. (1.2) and
tion of the source in the sky, inclination of the orbital
(1.3).
plane, orientation of the gravitational-wave detector),
The second type of correction addresses an implicit
= (m m )3/5/(m + m )1/5 (with m and m de-
M 1 2 1 2 1 2 assumption of the stationary-phase method, that the
noting the individual masses) is the chirp mass, r is the
function h(t) has support in the complete time interval
distance to the source, F(t) the instantaneous frequency
< t < t . Physically, this assumption means that
(twice the orbitalfrequency), andΦ(t)= 2πF(t′)dt′ is −∞ c
the binary system must have formed in the infinite past
thephase. Thefunctionh(t)representsthRegravitational
(a reasonable assumption given the long lives of com-
wavemeasuredatthedetectorsite. Itisalinearsuperpo-
pact binaries), and that the inspiral must continue until
sition of the wave’s two fundamental polarizations, and
F =F(t )= (anunrealisticassumption). Ifwechoose
we choose to express it in this complex form for conve- c ∞
instead to restrict the time interval to t < t < t ,
nience. The relation between F and t is given by [1] min max
suchthatF F(t )>0andF F(t )< ,
min min max max
≡ ≡ ∞
3/8 then the Fourier transform will be affected, and it will
5
π F(t)= M , (1.2) differfromh˜spa(f). ThevalueofFmin istypicallychosen
M (cid:20)256(t t)(cid:21)
c to reflect the lower bound of the instrument’s frequency
−
band. The value of F could be chosen to reflect the
wheret ,the“timeatcoalescence”,issuchthatformally, max
c
upper bound of the instrument’s frequency band, or for
F(t )= . The relation between Φ and t is
c
∞ some binaries it can be chosen to correspond to the ap-
5/8 proximatefrequencyofthelaststableorbit,atwhichthe
1 256(t t)
c
Φ(t)=Φc − , (1.3) inspiral signal changes over to a poorly-known merger
− 16(cid:20) 5 (cid:21)
M signal.
We shall refer to this truncation of the time interval
where Φ , the “phase at coalescence”, is equal to Φ(t ).
c c
as “windowing”, and for concreteness, we will assume
The Newtonian signal is the leading-order term in the
that the signalis started abruptly at t=t and ended
expansionofthegravitationalwavesinpowersofV 1, min
where V = (π F)1/3 is (up to a numerical factor)≪the abruptly at t = tmax. Thus, h(t) is assumed to be given
M by Eq. (1.1) in the interval t < t < t , and is as-
orbital velocity. Post-Newtonian corrections to this re- min max
sumed to be zero outside this interval. In Sec. II we
sult come with a relative factor of order V in the ampli-
tude, and a relative factor of order V2 in the phase [20]. showthatwindowing affectsboth the amplitude andthe
phase of the Fourier transform. The amplitude acquires
Throughoutthis paper we usegeometrizedunits, setting
an extra factor 1+δA , where
G=c=1. w
In the stationary-phase approximation, the Fourier
12 x 2
transform of the function h(t) appearing in Eq. (1.1) is δA (v)= v7/2 min cos(φ + π)
given by [5] w −√30π (cid:20)v3 xmin3 min 4
−
x 2
h˜spa(f)= √2340π QMr 2 v−7/2eiψ, (1.4) + xmaxm3ax−v3 cos(φmax+ π4)(cid:21), (1.7)
2
where x (π F )1/3 and II. CALCULATION OF THE FOURIER
min min
≡ M TRANSFORM
5v3 8x 3 3
min
φ = − + , (1.8)
min 128xmin8 128v5 Webeginwiththetime-domainsignalofEq.(1.1),and
we assumethatthe signalbegins abruptly atatime t
with similar equations holding for x and φ . On min
max max andendsabruptlyatatimet . WeletF F(t )
the other hand, the phase acquires an extra term δψ max min ≡ min
w andF F(t ) be the correspondinginstantaneous
given by max ≡ max
frequencies. TherelationbetweenF andtisobtainedby
integrating
12 x 2
δψ (v)= v7/2 min sin(φ + π)
w √30π (cid:20)v3−xmin3 min 4 dF = 96 (π F)11/3, (2.1)
x 2 dt 5π 2 M
+ max sin(φ + π) . (1.9) M
x 3 v3 max 4 (cid:21)
max which leads to
−
The fact that δA and δψ both diverge at the bound- 5
w w t(F)=t M(π F)−8/3, (2.2)
ariesv =xminandv =xmax signalsthebreakdownofour c− 256 M
approximations there. Away from the boundaries, δA
w
andδψw arebounded, andthey oscillate asa functionof wheretc (“time atcoalescence”)is aconstantofintegra-
tion. The phase function is then given by
frequency. Thus, these corrections represent amplitude
and phase modulations induced by the abrupt cutoffs of
1
the function h(t) at the boundary points. This is to be Φ(F)=Φc (π F)−5/3, (2.3)
− 16 M
contrastedwithδψ,whichrepresentsasteadyphasedrift.
Our complete expression for the steepest-descent ap- whereΦ (“phaseatcoalescence”)is anotherconstantof
c
proximation to the Fourier transform is therefore integration.
The Fourier transform,
h˜ (f)=˜h (f)(1+δA )ei(δψ+δψw). (1.10)
sda spa w
˜h(f)= h(t)e2πiftdt, (2.4)
In Sec. III we show that the only noticeable corrections Z
toh˜ (f)aretheamplitudeandphasemodulationsthat
spa
come as a consequence of windowing; in particular, the is evaluated by introducing a new integration variable,
intrinsic correction δψ is too small to be noticeable in
x (π F)1/3, (2.5)
therelevantfrequencyinterval. Wedothisbycomparing
≡ M
h˜ (f) to h˜ (f), the discrete Fourier transform of the
spa fft which can be related to t via Eqs. (2.1) and (2.2). After
windowed time series h(t); this comparison reveals that
some re-arrangement,we obtain
any discrepancy between the two versions of the Fourier
transformcanbe fully accountedfor by the modulations 5Q 2 3
δAw and δψw. This allows us to conclude that window- h˜(f)= 32Mr exp(cid:20)i(cid:18)2πftc−Φc+ 128v5(cid:19)(cid:21)
ing,andwindowingonly,mustbeheldresponsibleforany
discrepancy between h˜ (f) and h˜ (f). While limited I(v,xmin,xmax), (2.6)
spa fft ×
to the Newtonian signal of Eq. (1.1), there is no reason
where x =(π F )1/3, x =(π F )1/3, and
to believe that this conclusionwould be invalidated by a min M min max M max
full post-Newtonian analysis [21]. v (π f)1/3 1. (2.7)
Finally, in Sec. IV we calculate the matched-filtering ≡ M ≪
overlap between h˜spa(f) and ˜hfft(f), and show that the We have introduced the Fourier integral
modulations do not significantly affect the overlap. This
result,togetherwithourpreviousfindings,leadustocon- xmax
clude that for the purposes of matched filtering, h˜spa(f) I(v,xmin,xmax)=Zxmin x−7e−iφdx, (2.8)
and h˜ (f) are essentially equivalent representations of
fft
the gravitational-wavesignal. where
Themainconclusionofthisworkisthatthestationary- 5v3 8x3 3
phase method returns a sufficiently accurate expression φ(x;v)= − + . (2.9)
for h˜(f). What’s more, from the fact that δψ/ψ = 128x8 128v5
O(v10), we can be sure that the method will stay ac- It is easy to check that φ(v;v) = φ′(v;v) = 0, where a
curate for as long as the post-Newtonian expansion of primedenotesdifferentiationwithrespecttox. Thefunc-
h(t) in powers of V is itself an accurate approximation tion φ(x;v) initially decreases from φ (v) φ(x ;v)
min min
≡
to the gravitational-wavesignal. to zero as x increases from x to v, and then increases
min
from zero to φ (v) φ(x ;v) as x increases from v
max max
≡
to x .
max
3
The Fourier integral is evaluated by using φ as the the integral. The advantage of this choice of contour is
integration variable. Because φ is not monotonic in the thattheintegrandisexponentiallysuppressedawayfrom
interval [x ,x ], the integral must be broken down β =0, ensuring a rapid convergence of the integral.
min max
intotwoparts. Thefirstpartcoverstheinterval[x ,v), We evaluate the contributionfromC to J (v,x )by
min 1 min
while the second part covers the interval (v,x ]. It is substituting Eq. (2.14) into (2.11), replacing φ by iβ
max
−
easy to check that the integral can be expressed as and using β = 0 and β = as limits. The integrations
∞
give rise to Γ-functions, and we obtain
16
I(v,x ,x )= J (v,x )+J (v,x ) ,
min max 5 h 1 min 2 max i JC = √−30iπv−7/2 5iv−1+ 23√30iπv3/2
(2.10) 1 24 − 9 270
19232
v4+O(v13/2). (2.16)
where − 6075
φmin x2 A similar calculation also reveals
J (v,x )= e−iφdφ (2.11)
1 min Z v3 x3
0 − √ 30iπ 5i 23√30iπ
JC = − v−7/2+ v−1+ v3/2
and 2 24 9 270
19232
φmax x2 + v4+O(v13/2). (2.17)
J (v,x )= e−iφdφ. (2.12) 6075
2 max Z x3 v3
0 − The contribution from C′ to J (v,x ) is calculated
1 min
We recall that φ = φ(x ;v) and φ = φ(x ;v) by letting φ = φ iβ, and expressing the function
are functions of vm.in min max max f(φ) x2/(v3 xm3)ina−saTaylorseriesaboutφ . Thus,
min
The coordinate transformation x φ is defined im- f(φ ≡ iβ)=−f(φ ) if′(φ )β+ is substituted
min min min
→ − − ···
plicitly by Eq.(2.9). Itwillprovesufficienttoinvertthis into Eq. (2.11), whose limits are replaced by β = and
∞
relation in a neighborhood of x = v, or φ = 0. The β = 0. The resulting integrations are again elementary,
following relations are established by Taylor expansion: and we obtain
x=v 4√30v7/2φ1/2+ 256v6φ 10016√30v17/2φ3/2 J1C′ =ie−iφming(v,xmin), (2.18)
∓ 15 45 ∓ 2025
where
4281344
+ v11φ2+O(v27/2φ3/2), (2.13)
30375 x2 16i x10(x3+2v3)
g(v,x)= + + . (2.19)
wheretheuppersignreferstotheinterval[x ,v),while v3 x3 5 v3 x3 3 ···
min − −
the lower sign refers to (v,x ]. We also have (cid:12) (cid:12) (cid:12) (cid:12)
max A similar ca(cid:12)lculation(cid:12) also reve(cid:12)als (cid:12)
x2 √30 5 23√30
= v−7/2φ−1/2+ v−1 v3/2φ1/2 JC′ =ie−iφmaxg(v,x ). (2.20)
v3 x3 24 9 − 135 2 max
−
19232
+ v4φ+O(v13/2φ3/2) (2.14) We note that when x 1, the function g(v,x) can be
6075 well approximated by i≪ts first term, x2/v3 x3 , except
| − |
when x v. In this situation, the expansion for g(v,x)
in the first interval, and
≃
does not converge,and our method of calculationbreaks
x2 √30 5 23√30 down. Thus, our expression for the Fourier transform
= v−7/2φ−1/2 v−1 v3/2φ1/2 will be accurate only when v3 π f is not too close
x3−v3 24 − 9 − 135 to either x 3 π F or≡x M3 π F . In
19232v4φ+O(v13/2φ3/2) (2.15) other wordsm,inour≡expMressimoinn will bmeaxina≡ccurMate nmeaaxr the
− 6075
boundaries f =F and f =F .
min max
in the second interval. It is easy to see from Eqs. (2.18) and (2.19) that the
To evaluate the integrals J1(v,xmin) and J2(v,xmax), contribution from C′ to J1(v,xmin) vanishes in the limit
we let φ=α iβ and deform the contour of integration xmin 0. Similarly, it can be shown that the contri-
into the com−plex plane. While the original contour is bution→from C′ to J2(v,xmax) vanishes (as xmax−6) in
along the α axis, from 0 to φmin or φmax, we take the the limit xmax → ∞. [This behavior is not revealed by
new contour to be the union of C and C′, where C is Eq. (2.19), which gives g(v,x) as a series expansion for
the curve α = 0 with β running from 0 to , while C′ small values of x. An alternative expression for g(v,x),
is the curve α = φ or φ with β runn∞ing from appropriateforlargevaluesofx,caneasilybeobtained.]
min max
back to 0. The contour is completed by joining C an∞d This allows us to conclude that if the boundaries are
Cor′φwmitahx;ththeiscupravretβof=th∞e,cwonitthouαrrduonensinngotfrcoomnt0ribtoutφemtion cpounsthreidbuttoesFtmoinJ1=(v0,xamnind)FamndaxJ=2(v∞,x,mtahxe)n. C′ no longer
4
Gathering the results, Eqs. (2.10) and (2.16)–(2.20), III. COMPARISON WITH DISCRETE FOURIER
we find that the Fourier integral can be expressed as TRANSFORM
I(v,x ,x )= 4√30π e−iπ/4v−7/2 1+ 92iv5 The preceding analysis reveals that apart from win-
min max
15 (cid:20) 45 dowingissues,thestationary-phaseapproximationtothe
Fourier transformis extremely accurate: Apartfrom the
+O(v10)+R(v,xmin,xmax)(cid:21), (2.21) modulationsδAw andδψw,h˜spa(f)differsfromh˜(f)only
by a small phase drift δψ of relative order v10. In this
where section,wefirmupthis conclusionby comparingh˜spa(f)
to h˜ (f), the discrete Fourier transform of the function
12 fft
R(v,x ,x )= e−iπ/4v7/2 e−iφming(v,x ) h(t).
min max min
−√30π h The discrete Fourier transform is evaluated by Fast
+e−iφmaxg(v,x ) . (2.22) Fourier Transform(FFT), using the routines of Numeri-
max
i cal Recipes [22]. The time series is prepared as follows.
We begin with h(t) as displayed in Eq. (1.1), with the
SubstitutingthisintoEq.(2.6),treatingRandtheO(v5)
irrelevant factor Q /r set to unity. Thus,
term in Eq. (2.21) as small quantities, we arrive at the M
following expression for the Fourier transform: h(t)=(π F)2/3e−iΦ, (3.1)
M
h˜(f)=h˜ (f)(1+δA )ei(δψ+δψw). (2.23)
spa w where F(t) and Φ(t) are given by Eqs. (1.2) and (1.3).
This function is assumed to be nonzero only in the in-
Here,
terval F < F(t) < F . The duration of the signal
min max
is
√30π Q 2
h˜ (f)= M v−7/2eiψ, (2.24)
spa
24 r T =t(F ) t(F ), (3.2)
max min
−
with
while the total number of wave cycles is
π 3
ψ(v)=2πft Φ + , (2.25) 1
c− c− 4 128v5 = Φ(Fmax) Φ(Fmin) . (3.3)
N 2π −
(cid:2) (cid:3)
is the stationary-phase approximation to the Fourier
The values of h(t) at the endpoints t(F ) and t(F )
min max
transform. Our calculation reveals the existence of two
do not agree. This is a potential difficulty for the FFT,
types of correction terms. The first is
which considers the signal to be periodic with period T.
To remedy this, we prepare our time series by padding
92
δψ(v)= v5+O(v10), (2.26) h(t)with zerosonboth sides. Moreprecisely,welet h(t)
45
be zero inthe interval0<t<T,be equalto the expres-
which represents a small, but steadily growing phase sion (3.1) in the interval T < t < 2T, and be zero again
drift. Notice that δψ is of order O(v10) relative to ψ. inthe interval2T <t<4T. Thus,the effective duration
The othercorrectiontermscomeasa consequenceofthe ofthe time seriesis four times the durationofthe actual
abrupt cutoffs imposed at F = F and F = F . signal. We choose the value of the parameters t and Φ
min max c c
They are such that t(F ) T and Φ(F ) 0. This particular
min min
≡ ≡
paddingofthetime seriesissomewhatad hoc(ithasnot
12
δA (v)= v7/2 g(v,x )cos(φ + π) been carefully chosen to be the smallest needed to make
w −√30π h min min 4 negligiblethecircularcorrelationsoftheFFT),butaswe
shallsee the finalresults areverygoodand optimization
+g(v,x )cos(φ + π) (2.27)
max max 4 would be redundant.
i
The zero-padded function h(t) is discretely sampled
and
at times t = k∆t, where k = 0,1,...,4N 1 and
k
−
12 ∆t = 4T/(4N 1), with 4N denoting the number of
δψ (v)= v7/2 g(v,x )sin(φ + π) −
w √30π h min min 4 sampled points. The Nyquist frequency [22] is given
by f = 1/(2∆t), and N must be adjusted so that
Ny
+g(v,xmax)sin(φmax+ π4) . (2.28) fNy > Fmax. The FFT returns the Fourier transform
i
of the time series, discretely sampled in the frequency
Notice that δA represents an amplitude modulation, domain. The frequency resolution is ∆f =(4N∆t)−1.
w
whileδφ isaphasemodulation;bothoscillateasafunc- We denote the discrete Fourier transform of the zero-
w
tion of frequency. The suffix “w” indicates that these padded time series h(t) by h˜ (f), and we wish to com-
fft
corrections are associated with “windowing”. parethis to h˜ (f), the stationary-phaseapproximation
spa
given by Eqs. (2.24) and (2.25). (It is understood that
5
1.25
SPA analytical
0.012 FFT 1.05 numerical
1.2
0.01 1.15 1
Fourier transform (modulus) 000...000000468 0.009 amplitude modulation 011..90.1515 0.9549 50 51
0.008 0.9
0.002
0.85
48 49 50 51 52
0 0.8
40 45 50 55 60 40 45 50 55 60
f (Hz) f (Hz)
FIG. 1. The solid curve labeled “SPA” is a plot of FIG. 2. The solid curve labeled “analytical” is a plot of
mod[h˜spa(f)], and the dashed curve labeled “FFT” is a plot 1+δAw(f), and the dashed curve labeled “numerical” is a
ofmod[h˜ (f)]. Theinsetshowsthesamecurvesinasmaller plot ofA (f). Theinset shows thesamecurvesin asmaller
fft rel
frequency interval. frequency interval.
the correct values for t and Φ are substituted in these the discrete Fourier transform and the stationary-phase
c c
equations.) To do this we define the relative amplitude approximation must be attributed to windowing. This
A and relative phase ψ by conclusion is confirmed by Fig. 3, which shows plots of
rel rel
ψ (f) — defined in Eq. (3.5) — and δψ (f) — defined
rel w
A (f)=mod ˜h (f)/h˜ (f) (3.4) inEq.(1.9). Againweseeanear-perfectagreement,indi-
rel fft spa
h i catingthatwindowingaccountsfullyforanydiscrepancy
and between h˜spa(f) and h˜fft(f).
Equations(1.7)and(1.9)giveapproximateexpressions
ψ (f)=arg h˜ (f)/h˜ (f) , (3.5) for δA (f) and δψ (f), and we should expect that in
rel fft spa w w
h i somesituations,therecouldbe noticeabledifferencesbe-
wheremod(z)=risthemodulusofthecomplexnumber tween these quantities and the numerically-determined
z =reiθ, while arg(z)=θ is its argument. If h˜fft(f) and Arel(f)andψrel(f). Figure4indicatesthatsuchisindeed
h˜ (f) were in perfect agreement, then A = 1 and the case when the frequency interval is expanded. Here,
spa rel
thesignalispreparedwiththesamechirpmassasbefore,
ψ =0.
rel
Figure 1 shows plots of mod(h˜ ) and mod(h˜ ) as but the frequency interval is now between Fmin =40 Hz
fft spa
and F =1300 Hz; other relevant quantities are listed
functions of f for a signal prepared with = 1.25 M max
⊙
M in Table 1 below. Although the agreement is no longer
inthe intervalbetweenF =40 Hz andF =60Hz.
min max
near-perfect, it is still remarkably good, and this re-
(Such a narrow band is not at all typical for inspiralsig-
enforcesourclaimthatanydiscrepancybetweenthe dis-
nals sought by any interferometer; we use it merely to
creteFouriertransformandthestationary-phaseapprox-
exaggeratethe errorscausedby the stationaryphase ap-
imationisentirelyanartifactofwindowing. Wehavever-
proximation to the level where they are visible.) The
ified that the intrinsic correctionto the stationary-phase
duration of such a signal is T = 15.8 s, and the to-
approximation, δψ(f) given by Eq. (1.6), is irrelevant in
tal number of wave cycles is = 749. The FFT was
N this frequencyinterval: This phase shift is just too small
taken with N = 2048, giving a Nyquist frequency of
to be noticeable at these frequencies.
f = 65 Hz. The figure shows that the agreement is
Ny
not perfect: While the stationary-phase approximation
seemsto givethe meancurve,the discrete Fouriertrans-
IV. OVERLAP INTEGRAL
formdisplaysoscillationsaboutthemean,andthesegrow
near the boundary points, f = F and f = F . We
min max
shall argue that the discrepancy is entirely caused by We have shown that any discrepancy between h˜fft(f)
windowing. and h˜spa(f) can be fully attributed to windowing, which
InFig.2weshowplotsofA (f)—definedinEq.(3.4) induces amplitude and phase modulations in the Fourier
rel
—and1+δA (f)—definedinEq.(1.7). Werecallthat transform. In this section we show that these modula-
w
δA (f)representsthe amplitude modulationinducedby tions have no significant effect on operations associated
w
windowing. The near-perfectagreementbetween A (f) with matched filtering.
rel
and 1 + δA (f) shows that any discrepancy between
w
6
Thestandardtheoryofmatchedfilteringpredictsthat
the loss of signal-to-noise ratio incurred when filtering a
signal h˜ (f) with a filter h˜ (f) is equal to [8]
fft spa
h h
fft spa
= | . (4.1)
analytical O (cid:0) (cid:1)
0.3 0.05 numerical q hfft|hfft hspa|hspa
(cid:0) (cid:1)(cid:0) (cid:1)
We will refer to this quantity as the overlap between the
0.2 0 two expressions for the signal’s Fourier transform. An
overlap close to unity indicates that the filter is an ac-
modulation 0.1 -0.0549 50 51 cfiultrearteinraenparelyszeinntgattihoendoaftathweilslirgentaulr,natnhdeltahragtesutspinogsstibhlies
phase 0 signal-to-noise ratio. We use the notation
Fmax a˜∗(f)˜b(f)+a˜(f)˜b∗(f)
(ab)=2 df, (4.2)
-0.1 | Z S (f)
Fmin n
whereS (f)is the spectraldensityofthe detector noise.
-0.2 n
40 45 50 55 60 Notice that the point of view expressed here is that
f (Hz)
h˜ (f) is an exact representation of the signal’s Fourier
FIG. 3. The solid curve labeled “analytical” is a plot of fft
transform, while h˜ (f) is an approximate filter. How-
δψw(f), and the dashed curve labeled “numerical” is a plot spa
ever,because of its symmetry in these quantities, also
of ψrel(f). The inset shows the same curves in a smaller fre- O
represents the loss in signal-to-noise ratio incurred when
quency interval. Notice that the error in the phase does not
accumulate, and that it is always much smaller than 2π. filtering a signal h˜spa(f) with a filter ˜hfft(f). This is the
opposite point view, in which the stationary-phase ap-
proximation is viewed as an exact representation of the
Fourier transform.
We evaluate the integrals in Eq. (4.1) by turning
them into discrete sums, using the sampled frequencies
f = k∆f returned by the FFT. Thus, α(f)df
k
≈
kα(fk)∆f. It is sufficient for our purpoRses to use a
sPimpleanalyticmodelforthenoise’sspectraldensity. We
choose a noise curve that roughly mimics the expected
noise spectrum of the initial LIGO detector, and set [5]
0.4
analytical
numerical
S (f)=S (f /f)4+2+2(f/f )2 (4.3)
n 0 0 0
0.3 h i
for f > 40 Hz, with f = 200 Hz. The value of S
0 0
0.2 is irrelevant for our purposes, and S (f) is taken to be
n
phase modulation 0.1 imsnioafiTnsnhsiietessec.olbTeveaehlrroe:lwarFpe4os0ruOlaHtlslizsa.carcesaedlsci,suplalat>yeed0d.f9ion9r7T,aainbndleuicm1a.tbiTenrhgeotchfoacnthctilhrupe-
0 O
amplitude and phase modulations have very little effect
onmatchedfiltering. Inviewofthefactthatthemodula-
-0.1
tions oscillate and never get large, especially away from
F and F where the instrument is most sensitive,
min max
-0.2 200 400 600 800 1000 1200 this is the expected conclusion.
f (Hz) It should be noted that for most of the binaries listed
FIG.4. The thin,solid curvelabeled “analytical” is aplot in Table I, the adopted value for F exceeds the fre-
max
ofδψw(f),andthethick,dashedcurvelabeled“numerical”is quency at which the last stable orbit is expected to be
aplotofψrel(f). Noticethatherealso,theerrorinthephase found. (For equal-mass systems, this frequency is given
does not accumulate, and is always muchsmaller than 2π. approximatelyby 1910(M / ) Hz.) Our gravitational-
⊙
M
wave signals are therefore not realistic at high frequen-
cies, something which is already made clear by the fact
that we do not incorporate post-Newtonian corrections
into our waveforms. Since our purpose here is simply to
7
TABLE I. The last column gives the overlap O between
stationary phase and FFT waveforms prepared with (identi-
cal) chirp mass M given in solar masses in the first column,
initial frequency Fmin given in Hzin thesecond column, and
final frequency Fmax given in Hz in the third column. The
waveforms have a duration T given in seconds in the fourth ∗ Present address: Institute for Theoretical Physics, Uni-
columnandanumberofwavecyclesN giveninthefifthcol- versity of Zurich, CH-8057 Zurich, Switzerland.
umn. The numberof sampled times is 4N, where N is given † Presentaddress: DepartmentofPhysicsandAstronomy,
in the sixth column, corresponding to a Nyquist frequency McMaster University, Hamilton, Ontario, Canada L8S
fNy given in Hzin theseventh column. Inall cases thevalue 4M1.
ofFmax issufficientlylarge thatthecontributiontotheover- [1] K.S.Thorne,in300YearsofGravitation,editedbyS.W.
lap from higher frequencies can be neglected (assuming the Hawking and W. Israel (Cambridge University Press,
initial LIGO noise spectrum given in the text). Cambridge, England, 1987), p. 330.
[2] B.F. Schutz, in The Detection of Gravitational waves,
M F F T N N f O
min max Ny
editedbyD.G.Blair(CambridgeUniversityPress,Cam-
M⊙ Hz Hz s Hz
bridge, England, 1991), p. 406.
[3] B.S. Sathyaprakashand S.V.Dhurandhar,Phys.Rev.D
1.00 40 900 34.6 2200 216 948 0.9999
44, 3819 (1991).
1.25 40 1300 23.8 1521 216 1375 0.9998
[4] S.V.DhurandharandB.S. Sathyaprakash,Phys.Rev.D
1.50 40 1300 17.6 1122 216 1863 0.9997
49, 1707 (1994).
1.75 40 1200 13.6 868 215 1204 0.9997
[5] C. Cutler and E´.E´. Flanagan, Phys. Rev. D 49, 2658
2.00 40 1500 10.9 695 215 1505 0.9996
(1994).
2.25 40 900 8.9 570 214 916 0.9994
[6] R. Balasubramanian and S.V. Dhurandhar, Phys. Rev.
2.50 40 1000 7.5 478 214 1091 0.9994
D 50, 6080 (1994).
2.75 40 1200 6.4 409 214 1279 0.9994
[7] B.S. Sathyaprakash,Phys.Rev.D 50, 7111 (1994).
3.00 40 1400 5.5 354 214 1479 0.9994
[8] T.A. Apostolatos, Phys. Rev.D 52, 605 (1996).
10.00 40 1300 0.7 48 211 1374 0.9972
[9] E. Poisson and C.M. Will, Phys.Rev.D 52, 848 (1995).
[10] R.Balasubramanian,B.S.Sathyaprakash,andS.V.Dhu-
randhar, Phys. Rev.D 53, 3033 (1996).
establishthe validity ofthe stationary-phaseapproxima- [11] B.J. Owen,Phys. Rev.D 53, 6749 (1996).
[12] T.A. Apostolatos, Phys. Rev.D 54, 2421 (1996).
tion, this lack of realism is not too important. To pro-
[13] S. Droz and E. Poisson, Phys.Rev.D 56, 4449 (1997).
ducearealisticwaveformathighfrequencies,inaregime
[14] T. Damour, B.R. Iyer, and B.S. Sathyaprakash, Phys.
where the slow inspiral gives way to a rapid merger of
Rev. D 57, 885 (1998).
the twostars,is stillanopenandchallengingproblemin
[15] R. Balasubramanian and S.V. Dhurandhar, Phys. Rev.
gravitational-waveresearch.
D 57, 3408 (1998).
[16] B.J. Owen and B.S. Sathyaprakash,gr-qc/9808076.
[17] L.A.WainsteinandV.D.Zubakov,Extraction of Signals
ACKNOWLEDGMENTS
from Noise (Prentice-Hall, Englewood Cliffs, 1962).
[18] B.
The authors express gratitude toward P.R. Brady, Allen et al., GRASP software package, http://www.lsc-
A. Gopakumar,B.S. Sathyaprakash,and A.G. Wiseman group.phys.uwm.edu/˜ballen/grasp-distribution/.
for discussions. The work in Guelph was supported by [19] C.M. Bender and S.A. Orszag, Advanced Mathematical
the Natural Sciences and Engineering Research Council MethodsforScientistsandEngineers(McGrawHill,New
of Canada. The work in Pasadena was supported by York, 1978) Chap. 6.
the National Science Foundation under the grant PHY- [20] L. Blanchet, B.R. Iyer, C.M. Will, and A.G. Wiseman,
9424337, and under the NSF graduate program. E.P. Class. Quantum Grav. 13, 575 (1996).
wishes to thank Kip Thorne for his kind hospitality at [21] A. Gopakumar, in preparation.
the CaliforniaInstitute ofTechnology,where partofthis [22] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T.
Vetterling, Numerical Recipes (Cambridge University
work was carried out.
Press, Cambridge, England, 1986) Chap. 12.
8