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9.3 Detectors and templates 331

technique – pulsar timing – that has been used already to establish upper limits on the

gravitational radiation in this band. Radio pulsars are extremely precise clocks, but the

pulses’ arrival times on Earth may be modulated by the presence of gravitational waves.

Making these measurements sufﬁciently accurate requires integrating the pulsar data for

at least a few months, which sets the higher limit of the frequency band; on the other hand,

we have observed pulsars only for a few decades, which sets the lower limit. Pulsar timing

analysis has been used to establish upper limits both on stochastic background radiation

and on supermassive black hole binaries that have such large masses or binary separations

that they radiate in the very low frequency band.

The ultra low frequency band spans frequencies of approximately 10

−18

∼

−13

Hz, or wavelengths that are comparable to the Universe’s Hubble length. Waves in this

frequency band may be generated by quantum ﬂuctuations in the early Universe and are

ampliﬁed during inﬂation. Measuring the amplitude of this gravitational radiation would

therefore probe the inﬂation epoch and help discriminate between competing models.

Gravitational waves in this frequency range leave an imprint in the cosmic microwave

background, and may therefore be detected indirectly by analyzing the cosmic microwave

background and its polarization.

9.3 Detectors and templates

In many ways, gravitational wave detection is more like hearing than seeing.

In most

other astronomical observations we detect photons, which behave very differently from

their gravitational analogs. Photons typically have wavelengths that are much shorter than

the emitting object, so that we can create images. Gravitational waves, on the other hand,

have wavelengths that are larger than or at least comparable to the size of the emitting

object. That means that we cannot use gravitational waves to create an image of the

emitting object. In analogy to hearing we cannot even locate a gravitational wave source in

the sky with just one detector. This makes it so important to operate a number of different

gravitational wave detectors, spread far apart over the Earth or in space.

Photons are also emitted incoherently from very small regions within the emitting object,

usually from atoms or electrons, and we therefore observe the radiation’s intensity – which

measures the time-average of the square of the wave amplitude – rather than the individual

waveform itself. By contrast, gravitational waves are created coherently by the bulk motion

of the emitting object, and we observe the gravitational waveform directly. This difference

has two very important consequences.

The ﬁrst consequence is related to the fact that the wave amplitude falls off with one

over the distance from the emitting object, while the intensity falls of with one over the

square of the distance. That means that an increase of a factor of 2, say, in the sensitivity of

See, e.g., Smith et al. (2006).

Flanagan and Hughes (2005) explore this analogy in detail.

332 Chapter 9 Gravitational waves

our gravitational wave detector doubles the distance out to which we can observe certain

objects. The total observable volume of the Universe then increases by a factor of eight –

meaning that doubling the sensitivity increases the expected event rate by almost a fac-

tor of 10! For instruments that measure the intensity of electromagnetic radiation, the

corresponding increase in the event rate is smaller by the square root of eight.

The second consequence is that gravitational wave astronomy beneﬁts from theoreti-

cally predicted waveforms. Gravitational wave detectors measure gravitational amplitudes

directly, as a function of time, and this measurement can be compared with theoretical

models. Clearly such a comparison will be necessary for the physical interpretation of any

observed signal. In addition, “matched-ﬁltering” techniques – in which the noisy output of

the detector is compared with a catalog of theoretical gravitational waveform templates –

dramatically increase the likelihood of identifying a particular signal. Using this tech-

nique, the distance out to which an object can be observed increases approximately with

the square root of the number of wave cycles – which again may increase the effective

event rate signiﬁcantly.

The ﬁrst gravitational wave detector was a bar detector constructed by Joseph Weber,

one of the pioneers of gravitational wave astronomy. Several bar detectors are still oper-

ational. These detectors have a high sensitivity only in a very narrow frequency range.

While these detectors are probably not as promising as the new generation of gravita-

tional wave interferometers, they may provide some useful and important complementary

information.

Gravitational wave interferometers currently come in two different types: ground-

based and space-based. As we discussed in Section 9.2, the ground-based detectors are

designed to observe gravitational wave sources in the high frequency band (Section 9.2.1),

while space-based detectors will be able to detect sources in the low frequency band

(Section 9.2.2).

9.3.1 Ground-based gravitational wave interferometers

Ground-based gravitational wave detectors are giant Michelson–Morley interferometers.

As illustrated in Figure 9.2, the basic idea is to send a laser beam on a beam splitter that

splits the beam into two. The two beams are then sent down two orthogonal vacuum tubes,

or “arms”, reﬂected by mirrors at their end, and reassembled upon returning to the beam

splitter, possibly after several return trips through the arms. A passing gravitational wave

will distort the relative length of the two arms according to equation (9.9),

and will

therefore modify the interference pattern of the two returning light beams.

The challenge, of course, lies in the fact that gravitational radiation is so weak. As we have

seen from equation (9.66), even a reasonably optimistic estimate predicts a gravitational

wave strain in the order of h ∼ 10

−20

.From(9.9) we then have δξ ∼ 10

−20

ξ; inserting

a few kilometers for ξ (which is the arm length for the largest ground-based detectors),

Recall the effect of the two gravitational wave polarizations illustrated in Figure 9.1.

9.3 Detectors and templates 333

photodiode

beam

splitter

laser

mirrors

mirror

Figure 9.2 Schematic diagram of a Michelson–Morley interferometer as used in laser interferometer

gravitational wave detectors. [From Abramovici et al. (1992).]

we still end up with a dauntingly small distortion, similar in size to a small fraction of

the nucleus of the hydrogen atom. To measure such a small effect is an extraordinary

proposition indeed! Undeterred by these challenges a generation of gravitational wave

pioneers have not only designed the necessary instruments, but also demonstrated that

these instruments are capable of detecting such exceedingly small distortions.

Several ground-based detectors are currently either operational or in the planning phase.

The Laser Interferometer Gravitational-wave Observatory – LIGO – in the US oper-

ates three interferometers: one interferometer with an arm length of four kilometers in

Livingston, Lousiana, and a pair of interferometers with arm lengths of two and four kilo-

meters in Hanford, Washington. Figure 9.3 shows an areal view of the Hanford facility.

VIRGO is a joint French–Italian instrument, located in Pisa, Italy, with an arm length of

3 km. GEO600 is a detector with an arm length of 600 m, operated by a English–German

collaboration near Hannover, Germany. While this instrument has a slightly smaller arm

length, it uses a very advanced interferometer technology that makes its sensitivity com-

parable to those with longer arm lengths. It is also being used as a test-bed for advanced

technologies to be used in the next generation of larger detectors. TAMA300, a 300-m

interferometer located close to Tokyo, Japan, was the ﬁrst operational, large-scale gravita-

tional wave interferometer, and a successor with a longer arm length is being planned. The

only gravitational wave interferometer in the southern hemisphere, ACIGA, is being con-

structed near Perth in Australia. The current instrument is an 80-meter interferometer, but

the hope is that it can be extended to a multi-kilometer instrument some time in the future.

Figure 9.4 shows noise curves

h( f ) for the LIGO instruments, together with the sig-

nal strengths

( f ) for several different gravitational wave sources that we discussed in

Section 9.2. Ignoring for simplicity the exact deﬁnition of these quantities

we merely

note that a point on the signal curve that lies above a noise curve in the graph can be

See Cutler and Thorne (2002).

334 Chapter 9 Gravitational waves

Figure 9.3 The LIGO observatory in Hanford, Washington. [From LIGO website,

http://www.ligo-wa.caltech.edu.]

detected with the corresponding instrument with a false-alarm probability of less than one

percent.

The upper curve in Figure 9.4 is the design noise curve for LIGO-I, which already

has been achieved by the LIGO collaboration. The graph then indicates that with this

instrument we would be able to detect a black hole binary inspiral of two 10 M



black

holes at a distance of 100 Mpc. Whether or not we will indeed observe such an inspiral

with this generation of gravitational wave interferometers depends primarily on nature’s

generosity, i.e., on whether or not such an inspiral and merger happens to occur within this

distance while the detectors are operating.

Figure 9.4 also includes two noise curves for the next-generation of detector, LIGO-

II or Advanced LIGO. One of these curves, labeled “WB LIGO-II”, is basically the

improved “wide-band” version of LIGO-I with reduced noise sources. However, LIGO-II

will also have the capability of ﬁne-tuning the noise-curve to a narrow band of promising

sources. The ﬁgure shows an example, labeled “NB LIGO-II”, that is ﬁne-tuned to typical

frequencies in low-mass X-ray binaries. Clearly, LIGO-II should be able to detect many

more potential gravitational wave sources than its predecessor LIGO-I.

9.3.2 Space-based detectors

As we have seen in Section 9.2, gravitational waves in the low frequency band cannot be

observed with ground-based detectors. Observing gravitational radiation in this frequency

9.3 Detectors and templates 335

−24

−23

−22

20 50 100 200

frequency, Hz

Vela Spindown

Upper Limit

Crab Spindown

Upper Limit

Known Pulsar

=10

−6

r=10kpc

ε=10

−7

Ω=10

−7

Ω=10

−9

Ω=10

−11

LIGO-I

10M

/10M

BH/BH inspiral 100 Mpc

BH/BH Merger, z=2

BH/BH Inspiral 400 Mpc

WB LIGO-II

NS/NS Inspiral 300Mpc; NS/BH Inspiral 650Mpc

BH/BH Inspiral, z=0.4

Merger

50M

/50M

Merger

NB LIGO-II

Sco X-1

20M

/20M

BH/BH Merger, z=1

LMXBs

500 1000

h( f ) = S

1/2

, Hz

−1/2

Figure 9.4 The noise curve

h( f ) for several planned LIGO Interferometers. Also included is the signal strength

( f ) for a number of promising sources in the high frequency band (see Section 9.2.1). Whenever a point on the

signal curve lies above the interferometer’s noise curve, the signal is detectable with a false alarm probability of

less than one percent. [From Cutler and Thorne (2002).]

band therefore requires space-based instruments. One such instrument, the Laser Interfer-

ometer Space Antenna – LISA – is currently being planned as a joint venture between the

US space agency NASA and its European counterpart ESA.

As currently envisioned, LISA will be made up of three spacecraft located at the corners

of a triangle with sides of length of about 5 ×10

km. Each spacecraft houses two lasers

that send laser beams along the sides of the triangle. In essence, each side of the triangle

functions as the arm of a large interferometer. As illustrated in Figure 9.5, the instrument

as a whole will be placed in a solar orbit, trailing the Earth in its orbit about the Sun

by about 20

◦

. To maintain an approximately equal distance between the spacecraft over

the course of a year, each spacecraft is placed into a slightly eccentric orbit, with a

phase difference of 120

◦

between each one of them. The resulting motion of an individual

spacecraft is illustrated in the bottom panel of Figure 9.5. The whole conﬁguration therefore

exhibits a “rolling” motion, which turns out to be very useful for locating sources in the

sky.

336 Chapter 9 Gravitational waves

Figure 9.5 LISA’s orbit about the Sun. The top image shows how the triangle of three LISA spacecrafts trails

Earth’s orbit about the Sun by 20

◦

. The bottom image follows one of the three satellites’ orbit about the Sun,

demonstrating how the triangle’s orientation changes during the course of a year. [From LISA web site,

http://lisa.nasa.gov.]

In Figure 9.6 we show the projected noise curve for LISA, together with some of

the potential low frequency band sources that we discussed in Section 9.2.2.Atlow

frequencies, the noise arising from unresolved white dwarf binaries is expected to exceed

LISA’s intrinsic instrument noise. Evidently LISA has great potential to detect and discover

extremely interesting objects.

Another space-based detector is the gravitational wave antenna DECIGO (Deci-hertz

Interferometer Gravitational Wave Observatory ), proposed by scientists in Japan. This

instrument will bridge the gap between LISA and terrestrial detectors like LIGO, as it is

most sensitive to gravitational waves in the frequency range f ∼ 0.1–10 Hz.

9.4 Extracting gravitational waveforms 337

−17

−18

−19

−22

−21

0.0001

= (f s

)

1/2

, Hz

−1/2

0.001

/10

BH Inspiral at 3Gpc

/10

BH Inspiral at 3Gpc

/10

BH Inspiral at 3Gpc

BH@1Gpc

10 M

BH into

maximal spin

brightest NS/NS binaries

white dwarf binary

LISA noise

no spin

noise

0.01

Frequency, Hz

0.1

∗

4U1820-30

WD 0957-666

RXJ1914+245

merger

waves

merger

waves

Figure 9.6 Projected noise curve h

( f ) for LISA, together with a number of promising souces of gravitational

radiation in the low frequency band (see Section 9.2.2). The thick line is LISA’s instrument noise; the

thick-dashed curve is the noise curve incorporating the expected stochastic background of gravitational radiation

from unresolved white-dwarf binaries. Wave strengths above these curves have sufﬁciently high signal-to-noise

ratio that they should be detectable using optimal signal processing. [From Cutler and Thorne (2002).]

9.4 Extracting gravitational waveforms

We now turn to the topic that is most important from a numerical relativity perspective:

the extraction of gravitational waveforms in a numerical simulation. One of the major

motivations for performing numerical relativity simulations is the accurate calculation of

gravitational waveforms from promising sources in order that these theoretically computed

signals can be compared with observational data from gravitational wave detectors. As we

discussed in Section 9.3, such a comparison will be needed not only for the physical inter-

pretation of any observed data, but also to increase signiﬁcantly the liklihood of a detection

in the ﬁrst place. The latter can be accomplished by employing a “matched-ﬁltering” tech-

nique that compares observed signals with templates of theoretical waveforms.

Far from the sources, gravitational radiation is weak and can be described very con-

veniently in terms of the formalism introduced in Section 9.1. In particular, the wave

information can be expressed in terms of two polarization amplitudes in the TT gauge.

Numerical relativity simulations, however, focus on the strong-ﬁeld regime of the sources,

and compute a 3 + 1 spacetime metric decomposed in terms of a lapse function, shift vector

and spatial 3-metric. The functional form of this spacetime metric is strongly dependent

on the chosen coordinate system. Under some special gauge conditions, we can directly

read off the wave polarization amplitudes h

and h

from the evolved metric.

In general,

See Shibata (1999a,b) for an example.

338 Chapter 9 Gravitational waves

however, it is not trivial to extract, in a gauge-invariant way, the linearized wave quantities

that we introduced in Section 9.1.

In this section we focus on two different strategies for this gravitational wave extrac-

tion.

The ﬁrst approach, described in Section 9.4.1, employs the gauge-invariant Moncrief

formalism, and involves decomposing the metric into spherical harmonics and then com-

bining the latter in a gauge-invariant way. The second method, summarized in Section 9.4.2,

is based on the Newman–Penrose formalism and uses Weyl scalars, which are computed

from a projection of the Weyl tensor onto a null tetrad.

9.4.1 The gauge-invariant Moncrief formalism

The Moncrief formalism is based on a perturbative decomposition of the metric.

particular, we assume that at a large distance from any source we can decompose the

spacetime metric g

into a background metric g

, which we take to be the Schwarzschild

metric, and a perturbation h

= g

+ h

. (9.69)

The basic idea is then the following: As we did for the linearized wave solutions (9.48)

and (9.60), we split the perturbative metric h

into even- and odd-parity parts h

and h

and then decompose both in terms of spherical harmonics,

∞



l=2



m=−l

elm

+ h

olm

). (9.70)

For each mode we can form a gauge-invariant Moncrief function that satisﬁes a certain

wave equation. From these Moncrief functions we can then determine the gravitational

wave forms h

and h

The decomposition into spherical harmonics involves scalar, vector and tensor spherical

harmonics, which we review in Appendix D. As a brief reminder, any scalar on the unit

sphere can be decomposed into scalar spherical harmonics Y

. To represent a vector on

the unit sphere, we need two basis vectors E

and S

, which we can express in terms of

derivatives of the scalar spherical harmonics on the unit sphere. To represent a symmetric,

traceless, rank-2 tensor on the unit sphere we require two basis tensors, Z

and S

which we can express in terms of second derivatives of the scalar spherical harmonics.

As we might expect, the decomposition of the even-parity perturbations will involve those

spherical harmonics with even parity (namely the scalar spherical harmonics Y

, the vector

spherical harmonics E

, and the tensor spherical harmonics Z

), while the decomposition

We are greatly indepted to Y. T. Liu, who provided useful notes for this section.

In this section we will present most results without proof; we refer to the original paper by Moncrief (1974), as well

as the review by Nagar and Rezzolla (2005), for more details.

9.4 Extracting gravitational waveforms 339

of the odd-parity perturbations only involves odd-parity spherical harmonics (the vector

spherical harmonics S

, and the tensor spherical harmonics S

We now alert the reader to a departure from our general notation convention. Instead

of separating spatial indices from spacetime indices, it is more convenient, in the context

of the spherical coordinates used in this section, to separate the angular coordinates θ

and φ from the remaining spacetime coordinates r and t. Following this convention, we

will label the former (θ and φ) with lower-case letters a, b,..., and the latter (t and r )

with upper-case letters A, B,... In fact, we have already adopted this convention for the

vector and tensor spherical harmonics in the previous paragraph, which only have θ and φ

components. Similarly, the 2-dimensional metric σ

on the unit sphere S

,givenby

= dθ

+ sin

θdφ

, (9.71)

has only angular components.

Before discussing the speciﬁc decompositions of the even- and odd-parity perturbations

we can already anticipate their general character. The completely angular parts of the pertur-

bative metric, h

, will involve tensor spherical harmonics. For the odd-parity perturbations

this can only be the S

, but for the even-parity perturbations this could be a combination

of the tensor spherical harmonics Z

and a scalar spherical harmonic multiplying the

2-dimensional metric, Y

. Similarly, we will express the mixed angular-nonangular

parts of the metric, h

, as vector spherical harmonics. For the even-parity perturbations

this will involve the E

; for the odd-parity parturbations the S

. Finally, the nonangular

parts of the metric, h

, decompose like scalars. Since we can express scalars only in

terms of the even-parity scalar spherical harmonics Y

, these terms can be nonzero only

for the even-parity perturbations.

Even-parity (polar) modes

As we discussed above, we can express the nonangular parts of the perturbative metric,

elm

, in terms of scalar spherical harmonics. We denote these components as

elm



1 −



0lm

(t, r)Y

(θ,φ), (9.72)

elm

= H

1lm

(t, r)Y

(θ,φ), (9.73)

elm

2lm

(t, r)

1 − 2M/r

(θ,φ), (9.74)

where H

0lm

, H

1lm

and H

2lm

are expansion coefﬁcients. The mixed angular-nonangular

parts of the metric, h

elm

, are decomposed into electric-type vector spherical

Here we follow the notation of Shibata (1999a).

340 Chapter 9 Gravitational waves

harmonics E

elm

= h

Alm

(t, r)E

(θ,φ), (9.75)

where we will use labels 0 for A = t and1forA = r in the expansion coefﬁcients h

Alm

Finally, we decompose the angular part of the metric h

elm

according to

elm

= r



(t, r)σ

(θ)Y

(θ,φ) + 2G

(t, r)Z

(θ,φ)



, (9.76)

where the expansion coefﬁcient K

captures the trace and G

the traceless part of h

elm

In matrix form, the perturbative metric then appears as

elm

µν







H

0lm

1lm

0lm

sym H

2lm

/ h

1lm

sym sym r

+ G

) r

sym sym sym r

sin

θ(K

− G

)







(9.77)

wherewehaveabbreviated = 1 − 2M/r, and used equation (D.25) to express Z

terms of the functions X

and W

(see equations D.10 and D.11). We have also denoted

components that can be inferred from symmetry with a sym.

A numerical simulation will result in a spacetime metric. In most cases, this space-

time metric is decomposed into the lapse α, the shift β

, and the spatial metric γ

.We

now would like to express the asymptotic metric in the form of equation (9.77). To do

so, we compute the expansion coefﬁcients appearing in equation (9.77) using the orthog-

onality relations for the spherical harmonics (see Appendix D), evaluating the surface

integrals at a large radius. In particular, we ﬁnd the following expressions for the spatial

components:

2lm





1 −



∗

d (9.78)

1lm

l(l + 1)



)

∗

d

l(l + 1)





(∂

)

∗

rθ

+ (∂

)

∗

rφ

sin



d (9.79)



∗

d =



∗

d (9.80)

(l − 1)l(l + 1)(l + 2)r



)

∗

d

2(l − 1)l(l + 1)(l + 2)r





−

∗

2γ

θφ

sin

∗



d, (9.81)