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LISA and Capture Sources 451
can be combined with red-shift measurements of the host galaxy, an independent
redshift–distance relationship can be given.
In the case of EMRIs, a direct detection of the host galaxy is unlikely, as EMRI
are not expected to produce an electromagnetic signal. It is, however, possible to
make a statistical argument: Provided LISA detects a suitable number of EMRIs
(20 to a redshift of z D 0:5), it is possible to establish such an independent
redshift–distance relationship, thus determining Hubble’s constant to a precision
of better than 1% [53].
A more detailed and very extensive review of the sources, the astrophysics, sci-
ence, and detection of EMRIs using LISA can be found in [3].
It is worthwhile noting that the sensitivity of LISA is “just right” for the detection
of EMRI in the sense that a significant degradation of the sensitivity in the frequency
range of a few millihertz would diminish the science return for the EMRIs drasti-
cally. However, the cost benefit of giving up sensitivity in this frequency range as
part of a descoping scenario is marginal and would at present not be justified [74].
4 Detection
Before exploiting the scientific potential of EMRI and IMRI observations, it is
necessary to be able to detect those signals in the LISA data stream. Three main
issues arise: the complexity of the signal, the algorithms used for detection, and the
expected rate for EMRI and IMRI events.
Furthermore, the capture signals fall at least partly in the frequency range where
the sensitivity is limited by the signal confusion of the galactic binary sources (see
Section 4.3).
4.1 Capture Rates
Robust estimates for the rates of EMRI and IMRI require a model of the stellar
distribution and dynamics in the galactic nucleus, taking into account the interaction
between the stars, the relativistic effects close to the central black hole, and the
relative abundance of the various stellar types.
Taking the relative abundance from stellar evolution simulations or observations,
the two major approaches to the problem of stellar dynamics are based on direct
N -body integration and on statistical methods.
The direct integration poses a significant numerical challenge, as a large number
of objects has to be integrated and the equation of motion has to include post-
Newtonian terms to accurately predict the number of captured objects [44].
Dedicated hardware and the general progress in computational speeds will allow to
perform simulations yielding more accurate results.
452 O. Jennrich
Table 1 Number of EMRI detections for LISA, SNR, and distance thresholds. The
predicted number of detectable EMRIs by LISA for a downgraded mission scenario
(3-year mission, poor removal of confusion noise, see Section 4.3) and a nominal
mission scenario (5-year mission, good removal of confusion noise). It should be
noted that the estimates for the event rates are optimistic; true rates could be a factor
of up to 100 lower, causing all detection numbers to scale accordingly. Numbers are
taken from original data in [35]. The first two columns are the black hole mass M
and the compact body mass m
2
. The third column gives the SNR at a luminosity
distance of 1 Gpc and the fourth column gives the maximum redshift at which such
a source could be detected, given a detection threshold of SNR D 30. Values taken
from [3]
M
(10
6
M
ˇ
) m
2
(M
ˇ
)SNRat1Gpcz
max
Nominal # Downgraded #
0.3 0.6 18 0.13 8 0.7
0.3 10 73 0.44 700 89
0.3 100 620 2.5 1 1
1 0.6 30 0.21 94 9
1 10 210 1.0 1,100 660
1 100 920 3.5 1 1
3 0.6 25 0.17 67 2
3 10 270 1.3 1,700 134
3 100 1,500 5.2 2 1
Statistical methods such as approaches based on the Fokker–Planck equation [26]
or Monte Carlo methods are generally numerically cheaper. However, they have a
harder time than direct integration methods in predicting EMRI rates, as the EMRI
events are not only intrinsically rare, but also very sensitive to the details of the stel-
lar dynamics around the central black hole as the capture “event” is rather gradual,
as it requires a large number of orbits around the central black hole to allow the
object to dissipate enough energy and bring it close enough to the black hole to be
detectable in the LISA band[2].
The current implementations of statistical approaches have various shortcom-
ings, so that a full understanding of the processes that lead to EMRIs and hence a
robust prediction for their rate and their parameters, such as eccentricities, masses,
stellar types, etc. are not yet available. Consequently, the rates for EMRIs assumed
for LISA have quite a large spread, from a few to a few hundreds during the lifetime
of the mission (see Table 1).
4.2 Signal Characteristics
The signals emitted from EMRI can be very complex due to the fact that the inspi-
raling object explores the metric close to the central black hole and experiences the
full wealth of relativistic effects, which makes it difficult and numerically costly to
create precise waveforms. It is, on the other hand, precisely this complexity that
LISA and Capture Sources 453
makes the EMRI signals distinctive with respect to the precise geometry of the
space-time around a black hole.
The standard approach, exploiting the post-Newtonian approximation of the fully
relativistic equation of motions cannot work in the case of EMRI. Typically, the
post-Newtonian expansion converges poorly when v=c & 0:3 [21, 73]. As EMRI
typicallyspenduptoafew10
6
orbits in this regime, using v=c as an expansion
parameter is not useful.
Another technique that has seen dramatic and spectacular progress in the last
few years, numerical relativity, is despite all this progress not very useful either.
In contrast to (almost) equal mass mergers, where only a few orbits have to be
calculated, the large number of orbits required to get a waveform for EMRI makes
numerical relativity too slow for that purpose.
One of the methods that can be successfully employed to calculate at least ap-
proximate waveforms is based on perturbation theory. As the mass ratio in EMRIs
is very small, the inspiralling object can be regarded as a small perturbation to the
very well known background space-time of the central black hole. The computa-
tional complexity in this case stems from the accurate calculation of the gravitational
self-force of the particle. While the mathematical theory of the self-force has been
developed in the last decade ([64,65] and references therein) a full, generic inspiral
trajectory and a waveform from a particle on that trajectory is out of reach for the
near future, as the calculations are computationally very expensive.
It is, however, much easier to compute adiabatic waveforms by calculating a
sequence of geodesic trajectories and stitching together the waveforms resulting
from these trajectories. These waveforms are called adiabatic, as this procedure
only works as long as the timescale for the changes in the orbit are long compared
to the orbital period itself, allowing to calculate average rates of change for the
orbital constants. For a more detailed discussion see for example [3].
While the adiabatic waveforms are much easier to generate than the full wave-
form, their computation is still too expensive for the needs of data analysis in the
current stage. In order to test algorithms and find the most suitable one for detect-
ing EMRI in the LISA data stream, a very large number of signal templates has
to be generated so that the whole parameter space is covered. Two families of ap-
proximate “kludge” waveforms have been developed, the analytic kludge and the
numeric kludge. The purpose of these kludge waveforms is to allow their easy (and
fast) calculation while at the same time being reasonably accurate, that is, yielding
a good enough overlap (typically 95%) with the real waveform, so as not to lose too
much signal to noise.
The analytic kludge is based on the gravitational wave emission from a parti-
cle on a Keplerian orbit and is then augmented by relativistic precession of the
pericentre and the orbital plane and the inspiral, both taken from post-Newtonian
calculations [12]. The fact that those waveforms can be expressed in an analytic way
makes them very fast to calculate. However, their accuracy is limited in the later
stages of the inspiral, as the approximation of the true orbits by Keplerian orbits
becomes less and less accurate as the compact object comes close to the central
black hole.
454 O. Jennrich
The numeric kludge waveforms are designed to overcome that problem by using
the true geodesic orbit of the infalling particle, integrating the geodesic equation
numerically. The building of the numerical kludge waveforms involves construct-
ing a inspiral trajectory in phase space, for which a description that is accurate until
the very end of the inspiral has been obtained [36] allowing to integrate the geodesic
equation to get the trajectory of the inspiraling object in the Kerr background metric.
The trajectory is then used to calculate the waveforms using standard radiation emis-
sion formulas. The numerical kludge waveforms show a very high overlap (95%)
with the adiabatic waveforms for objects with a pericenter of larger than about 5M
(where M is the mass of the black hole), making them a very useful tool in the
LISA Data Analysis for detection and preliminary source identification, limiting the
parameter range for a follow-up search with more accurate templates significantly.
4.3 Data Analysis
The LISA data analysis in general and for EMRI in particular makes significant use
of template matching, where the noise-weighted inner product between the signal
h.t/ and the template r.tIq/ is evaluated as
.hjr/ D
1
Z
0
Q
h.!/Qr
.!Iq/ C
Q
h
.!/Qr.!Iq/
S.!/
d!; (2)
where
Q
h denotes the Fourier transform of h (with respect to t), S.!/ is the power
spectral density of the noise entering the data stream and q stands for the parameters
of the template.
The relevant noise for the detection of EMRI is not only the instrumental noise
as given in Fig. 5 but includes also the so called galactic binary foreground. While
reaching the instrumental sensitivity is a challenging problem, particularly in the
face of laser frequency noise and relative spacecraft motion, it is well understood
and techniques for dealing with the laser frequency noise have been developed
[5, 23, 27, 28, 59, 68, 69, 76–80]. The galactic binary background is caused by mil-
lions of galactic binaries emitting gravitational waves [15, 17, 31, 41, 61], in the
frequency band below a few millihertz, overlapping in time and frequency and cre-
ating a strongly coloured confusion noise.
It should be noted that depending on the event rates, signals from capture sources
might contribute to the confusion noise [13].
Identifying the most likely parameters q
0
of the template and determining the
probability density function (pdf) p.q/ of the parameters q, usually referred to as
parameter estimation, requires one way or the other comparing the signal against
a large number of templates. Currently, the data analysis for LISA is still in the
exploratory phase, although quite a lot of progress has been made over the last few
years, partly driven by the Mock LISA Data Challenge [6–9,58].
LISA and Capture Sources 455
4 × 10
−22
3 × 10
−22
2 × 10
−22
1 × 10
−22
0
−1 × 10
−22
−2 × 10
−22
−3 × 10
−22
−4 × 10
−22
0 1000 2000 3000 4000 5000
Time (M/10
6
M
)
Strain (m/10M
× 1 Gpc/D)
A
B
Fig. 5 Typical EMRI signals showing the complexity of the signals. Signals A (red, solid)andB
(green, dashed) originate from identical systems, a 10 M
ˇ
black hole orbiting a 10
6
M
ˇ
central
black hole, but differ in sky position and spin of the central black hole, given in the table below.
The different initial conditions lead to a different time to plunge t
c
, which for both systems lies
far enough in the future to justify the use of analytic kludge waveforms. In the plot, time is given
in units of the mass of the central object (10
6
M
ˇ
equals about 5 s) and the strain in the ratio of
the mass of the compact object to the luminosity distance D, normalized to 1 Gpc=10 M
ˇ
.The
signals are obtained through the lisatools software which is freely available at http://code.
google.com/p/lisatools/
Position Spin t
c
Lat. (
ı
) Long. (
ı
)Mag.(S=M
2
)Polar(
ı
)Azimuth(
ı
) (days)
A 15.4 77.6 0.52 95.1 8.1 384
B 34.4 113.3 0.56 125.53 2.2 698
As the size of the template space will ultimately dominate the computational
cost of the data analysis, a brief analysis of the expected size is helpful. Without
any further assumptions, N cycles of observed gravitational wave radiation and d
parameters to be fit would lead to about N
d
templates for an optimal matched-filter
search for EMRI waveforms. As noted before, EMRIs should emit about 10
5
cycles
of radiation in the LISA band, resulting in 10
5d
templates. Even for a small number
of parameters as d D 7, this is an enormous number, exceeding the capabilities of
any present computing architecture by many orders of magnitude.
Fortunately, the situation becomes a lot better, when the search is restricted to
signals with a “high enough” SNR. For strong signals, hierarchical methods, such as
ones that use crude templates [10,36,37], matching only short stretches of the signal,
456 O. Jennrich
or that sample the template space very coarsely, have been shown to work. The false
alarm rate will be quite high for the initially estimated candidates, but drops off very
quickly when follow-up searches around the candidates are performed.
For LISA, several detailed implementations of the first step of a hierarchical
method have been investigated, such as semi-coherent analysis [35]andtime-
frequency analysis [37–39,81].
Another approach is based on a stochastic search, usually conducted with some
variant of the Metropolis–Hastings algorithm. Recent progress [11, 22]showsthat
stochastic searches are a promising method for tackling the problem of parameter
estimation for EMRIs as well.
The actual performance of LISA with regard to the parameter estimation of
EMRI depends to some degree on how well the signals from the galactic binaries
can be identified, limiting the contamination of the EMRI signal by the signals of
galactic binaries. The higher in frequency the EMRI are, the higher the “effective”
SNR. Typical parameter accuracies for EMRI are very high [12]. Mass and spin of
the central black hole, the mass of the inspiralling object, and the eccentricity of the
orbit (at some time; eccentricity is a function of time) get estimated with a fractional
accuracy of about 10
4
. The inclination cos.i/ of the orbit as measured with respect
to the spin of the central black hole can be determined to about 10
3
to 10
2
.The
location of the EMRI in the sky can be constrained to a few square degrees and
the luminosity distance to about 5%. The parameters for IMRI can be determined
with even greater accuracy due to the higher mass of the IMRI that causes stronger
gravitational waves.
5 Summary and Conclusions
Following [3], the science connected to EMRI and IMRI observations with LISA
can be summarized in five categories: the characterization of the internal dynamics,
the nature of the inspiralling objects, the characteristics of the central black hole,
cosmology and early structure formation, and testing general relativity.
The characterization of the internal dynamics will yield information about the
capture process itself, as different mechanisms lead to different eccentricities.
Furthermore, different types of capture objects have different rates, lead to different
orbits and different signal strengths, giving a handle on population studies. The char-
acteristics of the central black hole can be assessed through the dependence of the
signals on its spin and mass. EMRI observations can make a contribution to cos-
mology when used to establish an independent distance scale through either direct
observation of the host galaxy or statistical methods whereas some information on
structure formation and the growth of black holes can be obtained through the ob-
servation of EMRI rates. EMRI enable high-precision tests of general relativity, as
they map out in detail the space-time around the central black hole.
EMRI pose the most difficult data analysis problem that LISA faces, because of
the large number of potentially distinguishable signals and because of the challenges
LISA and Capture Sources 457
in general relativity of computing reliable orbits over long durations. The numeri-
cal and conceptual problems of the data analysis are still a field of active research,
however, the recent years have brought significant progress in algorithms and nu-
merics so that the current situation allows to look forward in optimism regarding the
remaining challenges.
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