Blanchet L., Spallicci A., Whiting B. (Eds.) Mass and Motion in General Relativity

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High-Accuracy Comparison Between the PN and SF Dynamics of Black-Hole Binaries 441

Acknowledgements The authors acknowledge the 2008 Summer School on Mass and Motion,

organized by A. Spallicci and supported by the University of Orl´eans and the CNRS, through

which we experienced an extensive opportunity to understand each other’s perspective and make

rapid progress on this work. SD and BFW acknowledge support through grants PHY-0555484 and

PHY-0855503 from the National Science Foundation. LB and ALT acknowledge support from the

Programme International de Coop´eration Scientiﬁque (CNRS–PICS).

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LISA and Capture Sources

Oliver Jennrich

Abstract LISA is a joint ESA/NASA mission to detect and observe gravitational

waves. It is designed to register the change in distance between free-falling reference

points to picometer accuracy, allowing to measure the effect of gravitational waves

created by the coalescence of massive black holes almost anywhere in the universe,

stellar mass black holes and neutron stars spiraling into massive black holes in other

galaxies at intermediate distances, and tightly orbiting binary stars in our galaxy.

LISA will be able to detect gravitational waves from coalescing massive black holes

to redshifts of z  10 and higher, allowing an unprecedented view into the early

stages of galaxy formation. The signals from the many million binary stars in our

galaxy yield information about the evolution and the morphology of our galaxy,

giving a view of the population of binary stars unobstructed by dust. Among the

most challenging, yet scientiﬁcally interesting sources are the captures of a small

massive object by massive black holes where the mass ratio exceeds 1,000. Those

events, named extreme mass ratio inspirals (EMRI), create very complex waveforms

and allow to test general relativity to very high precision. LISA has been recently

conﬁrmed as a candidate for the L1 mission in ESA’s Cosmic Vision program and

is foreseen to be launched in the 2018 time frame.

1 LISA – A Mission to Detect and Observe Gravitational Waves

LISA is a spaceborne interferometric gravitational wave detector, jointly planned by

ESA and NASA [16,24,48,75] based on earlier ideas to build a gravitational wave

detector in space [20, 30]. In contrast to ground-based gravitational wave detectors

[1, 4, 33, 45, 66,70, 71, 82] that have a typical sensitivity in the range from 1 Hz to

1 kHz, the sensitivity for LISA stretches between 0:1mHzand0:1 Hz, accessing

a frequency window that is inaccessible to ground-based detectors due to seismic

noise and gravity gradient noise.

O. Jennrich (



)

ESA/ESTEC, Noordwijk, The Netherlands

e-mail: oliver.jennrich@esa.int

L. Blanchet, A. Spallicci, and B. Whiting (eds.), Mass and Motion in General Relativity,

Fundamental Theories of Physics 162, DOI 10.1007/978-90-481-3015-3

16,

 Springer Science+Business Media B.V. 2011

443

444 O. Jennrich

The sensitivity for low frequencies allows LISA to assess gravitational waves

that are emitted by some of the most violent events in the universe, such as the

coalescence of massive black holes (m

 10

10

) that occur during

the formation and the growth of galaxies, signals from galactic binary systems, and

the capture of stellar-size compact objects by massive black holes. It is even possible

that LISA will be able to detect gravitational waves from times shortly after the Big

Bang. The general science of LISA is discussed in detail in [3,19,25,42,43,53,54];

the focus of this paper is on the capture of compact objects by massive black holes.

1.1 Mission Concept

LISA comprises three spacecraft in a heliocentric orbit, forming an equilateral tri-

angle with 5 10

km a side (the “armlength”). The plane of the triangle is inclined

by 60

with respect to the ecliptic, causing the constellation to counter-rotate while

orbiting around the Sun. This peculiar arrangement of the satellites results in a sta-

ble constellation, as orbital mechanics cause the constellation to rigid: it maintains

its size and shape closely during the nominal mission time of 5 years [32] allowing

to operate LISA without further station keeping maneuvers.

The rigidity of the constellation is only approximate, so that the distances be-

tween the satellites change by up to 1%, causing a differential velocity along the

line of sight of up to 15 m/s, and resulting in the constellation to slightly change its

form.

The distance to the Earth of about 50  10

km (Fig. 1), corresponding to a trail-

ing (or leading) angle of 20

, has been chosen as a compromise between long-term

stability of the constellation and communications requirements.

5× 10

Fig. 1 Schematics of the LISA constellation orbiting the Sun. The constellation is inclined by

with respect to the ecliptic and trails the Earth by about 20

, resulting in a distance of about

50 10

km to Earth. The distance between the satellites (the armlength) is nominally 5 10

LISA and Capture Sources 445

As gravitational waves cause a strain in the space-time [57], measuring the

change in the distance of spatially separated objects is a common concept for mea-

suring the effect of gravitational waves. Each spacecraft houses two test masses,

kept in free fall, that form the reference points for an interferometric measurement

of the inter-spacecraft distance.

1.2 Sensitivity

It is customary in the ﬁeld of gravitational wave detection to state the sensitivity and

noise ﬁgures in terms of a linear spectral density S.f /, that is, quantities are given

in units

of 1/

Hz; RMS values of the quantities can be retrieved by integrating

the spectral density S

.f / over the frequency band f of interest and taking the

square root:

x

rms

f

.f /

2

(1)

The interferometric measurement allows to assess the distance between the

(almost) free-falling test masses to a level of 10 pm/

Hz, resulting in a strain sensi-

tivity of about e-20 /

Hz, enough to detect, for example, the coalescence of massive

black holes even at redshifts of z D 20 with a signal-to-noise ratio of several hun-

dreds [43, 49, 50]. The science requirements for LISA specify the sensitivity in a

frequency window of 0:03 mHz to 0:1Hz(Fig.2)asthisiswheremostofthe

sources for LISA are expected to emit.

The limitation to the sensitivity at frequencies below approximately 3 mHz is

given by residual acceleration noise of the order of 3e-15 m/s acting on the test

masses. Any acceleration along the line connecting the test masses in two differ-

ent satellites mimics the action of a gravitational wave and therefore limits LISA’s

sensitivity.

At higher frequencies, the noise associated with the position measurement limits

the sensitivity to about 10 pm/

Hz. This noise is partly due to the unavoidable shot

noise of the photons and partly due to imperfections and digitization noise in the

phase meter.

The sensitivity is further reduced for frequencies above about 30 mHz by the

transfer function of the detector: the effect of gravitational waves cancels out as the

wavelength of the gravitational wave equals an integer multiple of the optical path

in the detector [51].

The sensitivity of LISA to changes in the distance between the spacecraft is

small, compared with ground-based detectors. However, as gravitational waves pro-

duce a strain, or a fractional change in distance, the large distance between the

satellites provides a sensitivity to gravitational waves comparable to those of the

much shorter ground-based detectors.

Some authors use the spectral density,thatis,S

.f / for the same purpose.

446 O. Jennrich

-15

-16

-6

-7

-8

-9

-10

-11

-17

-18

-19

-20

-21

0.0001 0.001 0.01 0.1 1

Frequency (Hz)

RequirementPerformance

Strain sensitivity (1/ Hz)

Displacement sensitivity (m/ Hz)

Fig. 2 Sensitivity requirements (green solid) and current best estimate of the performance (red

dots). The strain sensitivity (left axis) is given for the full instrument, the displacement sensitivity

(right axis) has been converted to a single arm link, that is represents the equivalent accuracy for

the armlength measurement, combining all noise sources. The design of the instrument is such that

a 35% performance margin is included, resulting in the values for the current best estimate to be

consistently lower than the requirement

1.3 Measurement Principle

For practical reasons, the interferometric measurement in LISA is broken up into

three distinct parts: the measurement between the spacecraft, that is, between the

optical benches that are ﬁxed to the spacecraft, and the measurement between each

of the test masses and its respective optical bench (see Fig. 4). While such a partition

of the measurement is usually avoided as it increases the noise introduced by the

detectors, it does little harm in the case of LISA, as the noise budget is dominated

by the contribution of the shot noise in the measurement between the spacecraft.

The distance between the test mass and the optical bench is measured by re-

ﬂecting light off the test mass and combining this measurement beam with a local

oscillator on the optical bench (“test mass interferometer”).

To measure the distance between the spacecraft, about 2 W of infrared light

(1:064 nm) is sent through a 40 cm telescope, used for receiving and transmit-

ting, to the respective far spacecraft, that is, spacecraft A transmits to spacecraft

B and C (nomenclature as in Fig. 3) etc., forming six single laser links. In an

ordinary interferometer, the received light would be directly reﬂected back to the

transmitting spacecraft A where the light would then be combined with a local os-

cillator, completing the measurement. However, due to the large distance between

the two spacecrafts, a direct reﬂection of the light is not feasible. Diffraction widens

LISA and Capture Sources 447

Fig. 3 Schematics of the LISA constellation. Each satellite transmits laser light to the two other

satellites and receives laser light from the two other satellites. The laser used for transmitting is

phase-locked to the received laser light, establishing a transponder scheme. As transmitted and

received light share the same telescope, a polarization multiplexing scheme is used to separate the

two beams. The use of linear polarisations requires slightly different optical benches on the sending

and the receiving spacecraft

Measurement S/C to test mass Measurement S/C to test mass

S/C to S/C measurement

Fig. 4 The measurement of the distance between the test masses is broken up into three parts: The

measurement between the two spacecraft, that is, the optical benches and one measurement each

between the test mass and its optical bench

448 O. Jennrich

the transmitted laser beam to many kilometers at the receiving spacecrafts and the

received power is of the order of 100 pW (or about 5  10

photons per second),

resulting in a dilution factor of 10

10

, resulting in a photon rate of about 1 photon

per second if direct reﬂection was attempted.

Therefore, a transponder scheme with offset phase-locking is implemented.

The received light on spacecraft B is combined with a local oscillator derived

from the transmitting laser (B) and the phase difference is measured. The frequency

of the laser (B) can then be controlled so that the beat note corresponds precisely to

an electronic offset frequency, rendering the phase of the transmitting laser a true

copy of the phase of the received light. Recombining the light on spacecraft A then

yields a beat note that contains the electronic offset frequency on spacecraft B, the

Doppler shift (that can be as large as 15 m/s), and the signal due to the gravitational

wave (“science interferometer”). To be able to distinguish a gravitational wave

signal from a noise in the electronic offset frequency, the latter needs to be known

precisely enough. This is achieved by multiplying the clock frequency on each

spacecraft up to a few GHz, impose it as sidebands to the laser. On the other space-

craft, these sidebands can be compared to the local clock, allowing to assess and

correct the noise of the local clock.

Extending the phase-locking scheme so that the two lasers on one spacecraft are

phase-locked to each other as well, it is obvious that all six lasers can be phase-

locked to one (arbitrarily chosen) master laser. Such a scheme requires then nine in-

dividual phase measurements, two for each arm and one each between the two lasers

on a single spacecraft. In addition, each measurement of the test masses with respect

to the optical bench requires another six phase measurements, so that a total of

15 phase measurements carry the complete information on the gravitational waves.

These signals will be sent to ground and combined on ground to form the LISA

data streams that will be made available to the scientiﬁc community.

A more detailed overview of the instrumentation of LISA and the technology

employed by LISA can be found in [47].

2 Capture Sources

The capture of compact objects by massive black holes is believed to occur regularly

at the centers of galaxies where the central black hole is surrounded by a dense stel-

lar environment. A “captured” object does not necessarily fall into the central black

hole in the observation period, although eventually, most of them will. The term

refers to an object that is in a bound orbit around the central black hole and comes

close enough to it to emit a detectable amount of gravitational radiation. Such a

scenario is possible only for compact objects, as non-compact objects (i.e., normal

stars, molecular clouds) will be tidally disrupted long before they reach those orbits

and will not emit gravitational waves in the LISA band.

Out of the stellar cloud surrounding the central black hole, compact stars, such

as neutron stars and black holes, can be deﬂected toward the central black hole

through interaction with other objects. If these “infalling” compact objects come

LISA and Capture Sources 449

close enough to the central black hole, the energy loss through the emission of

gravitational radiation is large enough to put them into a bound orbit around the

central black hole [40, 44, 46, 63, 72]. Even low-mass main sequence stars can be

captured [34], despite the fact that main sequence stars usually suffer from early

tidal disruption.

Another mechanism that can bring compact objects close to the central black

hole is based on tidal effects: Binary systems can be disrupted by tidal forces, leav-

ing one constituent close to the central black hole [56]; compact cores of giant stars

can be tidally stripped of their gas envelope by the central black hole and end up

close to it [29].

Yet another class of compact objects might have formed in situ: black holes that

have formed near the central black hole by the evolution of giant stars such as those

seen near the central black hole in the Milky Way [52, 60, 62]orinthecenterof

M31 [18].

The compact objects (neutron stars, black holes, and white dwarfs) resulting from

those mechanisms would then survive as quasi point-like objects orbiting the central

black hole until eventually their orbital energy is carried away through the emission

of gravitational radiation and they vanish at the horizon. During the period of or-

biting the central black hole, they would radiate waves in the LISA band for a long

time, many of them emitting approximately 10

cycles or more of radiation before

falling into the central black hole.

Two principal classes of systems can be distinguished according to their astro-

physical characteristics and their mass ratio: extreme mass ratio inspirals (EMRI)

and intermediate mass ratio inspirals (IMRI). For EMRIs, the ratio of the mass of

the inspiraling object to that of the central black hole is 10

4

or smaller. This class

comprises white dwarfs, neutron stars, or stellar-mass black holes. As black holes

are the most massive objects in this class, they sink towards the denser regions of

the galactic nucleus and will be therefore over-represented among the EMRI popu-

lation. In addition, their higher mass makes for a stronger gravitational wave signal,

allowing detection of them from larger distances, thus increasing their event rate

further. The typical expected EMRI system is therefore a stellar-mass black hole

with a mass of about 10 M

falling into a 10

massive black hole.

For IMRIs, the ratio between compact object and massive black hole is some-

what larger, between 10

4

and 10

2

. The inspiraling object here is expected to be

an intermediate mass black hole, such as may have been formed in the ﬁrst gener-

ation of star formation (Population III stars) [55]. The canonical expected source

is an intermediate-mass black hole with a mass of 10

falling into a 10

massive black hole.

3 Science Return

Capture sources ﬁgure among the principal sources that allow to conduct funda-

mental tests of general relativity with LISA. Their signals contain rich information

about the geometry around the central black holes and allow to test certain aspects

of general relativity to unprecedented precision.

450 O. Jennrich

To emit gravitational radiation that can be detected with LISA, these objects are

necessarily so close to their central black holes that it completely determines the or-

bital dynamics – surrounding stars or even accretion disks will have at most a small

effect on the orbits. The compact object then acts as the textbook “test particle,” ex-

ploring the background space-time around the central black hole. The evolution of

the signals, lasting for thousands or even hundreds of thousands of cycles, reﬂects

in detail the near-geodesic orbits they follow around the central black hole; the or-

bits in turn map the space-time [67]. Comparing this phase evolution to the phase

evolution of a signal from a compact object in a pure Kerr geometry allows to look

for deviations from the geometry predicted by general relativity, thereby testing one

of the most fundamental theorems of Einstein’s theory, that black holes are uniquely

determined by their mass and their angular momentum [14].

An even more fundamental test of GR will be the direct observation of the event

horizon. Direct evidence will come from seeing the signals changing characteristics

as the compact object approaches the horizon and transfers from the inspiral phase

to the coalescence or merger phase. If no such change will be observed, this would

be strong evidence for the lack of an event horizon, and therefore ruling out a black

hole as the central object.

Beyond fundamental physics, there are astrophysical payoffs for detecting these

signals as well, as both EMRIs and IMRIs sample the stellar population near the

central black holes. The different formation channels for EMRI lead to distinguish-

able properties of their orbits and consequently to different characteristics of the

gravitational wave signals: direct captures of objects into close orbits are typically

characterized by high eccentricities that can be clearly identiﬁed by the harmonic

content of the gravitational wave signal; tidally captured objects will have nearly

circular orbits with more or less random inclination, depending on their origin; and

objects that evolved in the vicinity of the central black hole will occur in circular

orbits that are most likely aligned with the accretion disk, and therefore with the

spin of the black hole.

The observed rate of IMRI events will give a valuable handle on the demograph-

ics of the intermediate-mass black holes and their progenitors in the nucleus of

galaxies. With additional information from simulations of mass segregation, stellar

interactions and population evolution, the rates, and mass spectrum of the different

classes of detected objects will provide unique insight into the coevolution of black

holes and their host galaxies.

Capture sources can even play an important role in cosmology. The signals from

EMRI and IMRI, depend only on a few parameters such as the masses of the cen-

tral black hole and the compact object, the initial orbital angular momentum, the

spin of the black hole and the initial separation, in particular, the frequency of the

signals is determined by the mass of the central black hole, whereas the amplitude

is determined by the mass of the infalling object.

As the signal shape and amplitude can be very precisely determined and as

gravitational waves do not suffer from dispersion or absorption by the interstellar

medium, the ratio of the observed signal strength to the calculated signal strength is

a direct measure for the luminosity distance of the sources. If these measurements