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Constraining the spin parameter of near-extremal black holes
using LISA
Ollie Burke,1, 2, ∗ Jonathan R. Gair,1, 2 Joan Simón,2 and
Matthew C. Edwards2, 31Max Planck Institute for Gravitational
Physics (Albert Einstein Institute),
Am Mühlenberg 1, Potsdam-Golm 14476, Germany2School of
Mathematics, University of Edinburgh, James Clerk Maxwell
Building,
Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK3Department of
Statistics, University of Auckland,38 Princes Street, Auckland
1010, New Zealand
We describe a model that generates first order adiabatic EMRI
waveforms for quasi-circularequatorial inspirals of compact objects
into rapidly rotating (near-extremal) black holes. Usingour model,
we show that LISA could measure the spin parameter of near-extremal
black holes (fora & 0.9999) with extraordinary precision, ∼ 3-4
orders of magnitude better than for moderate spins,a ∼ 0.9. Such
spin measurements would be one of the tightest measurements of an
astrophysicalparameter within a gravitational wave context. Our
results are primarily based off a Fisher matrixanalysis, but are
verified using both frequentist and Bayesian techniques. We present
analyticalarguments that explain these high spin precision
measurements. The high precision arises from thespin dependence of
the radial inspiral evolution, which is dominated by geodesic
properties of thesecondary orbit, rather than radiation reaction.
High precision measurements are only possible ifwe observe the
exponential damping of the signal that is characteristic of the
near-horizon regimeof near-extremal inspirals. Our results
demonstrate that, if such black holes exist, LISA would beable to
successfully identify rapidly rotating black holes up to a = 1 −
10−9 , far past the Thornelimit of a = 0.998.
I. INTRODUCTION
Extreme mass ratio inspirals (EMRIs) are one of themost exciting
possible sources of gravitational radiationfor the space-based
detector LISA [1], but also one ofthe most challenging to model and
extract from the datastream. An EMRI involves the slow inspiral of
a stellar-origin compact object (CO) of mass µ ∼ 10M� into amassive
black hole in the centre of a galaxy. For a cen-tral black hole
with massM ∼ 10(5−7)M�, EMRIs emitgravitational waves (GWs) in the
mHz frequency bandand so are prime sources for the LISA detector.
EM-RIs begin when, as a result of scattering processes inthe
stellar cluster surrounding the massive black hole,the CO becomes
gravitationally bound to the primary.The subsequent inspiral of the
CO towards the horizonof the primary is driven by radiation
reaction throughthe emission of gravitational waves. EMRI
waveformsare very complicated and EMRIs can be present in theLISA
frequency band for several years prior to plunge, somodelling the
full observable signal is a complex task [2].EMRI orbits are
expected to be both eccentric and in-clined even up to the last few
cycles before plunging intothe primary black hole. For these
reasons, EMRIs posea challenging problem for both waveform
modellers [2–4]and data analysts.
This same complexity also makes EMRIs one of therichest sources
of gravitational waves. Typically an
∗ [email protected]
EMRI will be observable for 1/(mass ratio) ∼ 105−7 cy-cles
before plunge and the emitted gravitational wavesthus provide a
very precise map of the spacetime geom-etry of the primary hole
[5–8]. Through accurate detec-tion and parameter inference, one can
conduct tests ofgeneral relativity to very high precision [6,
9].
It is well known that the information about the sourceis carried
through the time evolution of the phase in agravitational wave [10,
11]. The slow evolution of EM-RIs means that a large number of
cycles can be observedduring the inspiral, which will provide
constraints onthe parameters of the source with remarkable
precision[12, 13]. Previous work has indicated that LISA will
beable to place constraints on the dimensionless Kerr
spinparameter, a, of the primary black hole in an EMRI,at the level
of 1 part in 104 for moderately spinning,a ∼ 0.9, primaries [3, 4,
14, 15]. In this paper, weexplore how well LISA will be able to
measure the spinparameter for very rapidly rotating black holes,
i.e., sys-tems in which the spin parameter is close to the maxi-mum
value allowed by general relativity.
Super massive black holes with a large spin parame-ter are
abundant throughout our universe. Observationsindicate that massive
BHs reside in the centres of mostgalaxies, where these black holes
are known to accretematter and hence are predicted to have very
high spins[11, 16–20]. The dimensionless Kerr spin parameter ofa
Kerr black hole, cannot exceed 1, since the result-ing spacetime
contains a naked singularity no longer en-cased within a well
defined horizon. Thorne [21] showedthat a moderately spinning black
hole cannot be spun upby thin-disc accretion above a spin of a ≈
0.998. How-
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ever, in principle primordial black holes could be formedwith
spins exceeding that value [22]. “Near-extremal”black holes with
spins close to the limit of a = 1 have in-teresting properties and
we focus our attention on thesehere.
This past decade, researchers [23–32] have exploredthe rich
properties of near-extremal EMRIs. The gravi-tational radiation
emitted from these systems is unique,and would prove a smoking gun
for the existence ofthese near-extremal systems (see [24]). In this
paper,we show qualitatively that the inspiraling dynamics ofthe
compact object into an near-extremal massive blackhole is very
different from that into a moderately spin-ning black hole, and
these differences are reflected inthe emitted gravitational waves.
As such, in order todetect and correctly perform parameter
estimation onthese near-extremal sources, it is essential to
updateour family of waveform models to include them. Wewill argue
throughout this work that, if observed, near-extreme black holes
offer significantly greater precisionmeasurements on the Kerr spin
parameter than moder-ately spinning systems. In particular, LISA
will havethe capability to successfully conclude whether the
cen-tral object in an EMRI system is truly a near-extremalblack
hole. Thus, if near-extremal black holes exist,LISA observations of
EMRIs may be one of the bestways to find them.
In this paper we will consider only EMRIs on circu-lar and
equatorial orbits around near-extremal primaryblack holes. This
choice is made primarily for com-putational convenience, but there
are also astrophysi-cal scenarios that produce such systems. As
discussedin [33], compact objects can form within accretion
disksaround massive black holes. When these objects fallinto the
central black hole, the resultant EMRI will becircular and
equatorial. Super-Eddington accretion canprovide a means to spin up
a black hole past the Thornelimit [34], and so it is not
unreasonable to expect thatthis EMRI formation channel would be
more importantfor near-extremal systems. The standard EMRI
for-mation channel, involving capture of a compact objectvia
scattering interactions, tends to form EMRIs withmoderate initial
eccentricities. However, this eccentric-ity decreases during the
inspiral due to the emission ofgravitational radiation [35]. This
decrease in eccentric-ity continues until the orbit reaches a
critical radius atwhich is starts to increase again [36, 37]. The
criticalradius moves closer to the last stable orbit as the
spinparameter increases and for near-extremal systems islocated
within the regime where transition from inspi-ral to plunge occurs
[38, 39]. Additionally, the increasein eccentricity is a
subdominant effect throughout thetransition regime [40]. As the
spin increases, we there-fore expect that for an object captured at
a fixed ra-dius, the amount of eccentricity dissipated before
thecritical radius increases, and the eccentricity gained af-
ter the critical radius decreases. Therefore, even in
thestandard capture picture it is reasonable to assume
theeccentricity is small at the end of the inspiral. We willshow in
this paper that very precise measurements ofspin for near-extremal
systems are possible, but this pre-cision comes from observation of
features [24] in the finalphase of the inspiral, which is where the
near-circularassumption is most likely to be valid.
The main results of the paper are given in figures 9and 10 in
section VI. Readers who wish to understandwhy near-extremal systems
offer greater precision spinmeasurements than moderate spin systems
should directtheir attention to section III.
This paper is organised as follows. In section II,we set
notation and discuss the trajectory of a com-pact object on a
circular and equatorial orbit arounda near-extremal Kerr BH. In
section III, we show thatthe spin dependence of kinematical
quantities appear-ing in the radial evolution rather than
radiation-reactiveeffects dominate the spin precision measurements
fornear-extremal EMRI systems. Our Teukolsky basedwaveform
generation schemes are outlined in section IV.We discuss prospects
for detection in section V, argu-ing that LISA is more sensitive to
heavier mass sys-tems M ∼ 107M� than lighter systems M ∼ 106M�.Our
Fisher Matrix results are presented in section VI.Here we show that
we can constrain the spin parameter∆a ∼ 10−10, even when
correlations amongst other pa-rameters are taken into account.
Finally, in section VII,we perform a Bayesian analysis to verify
our Fisher ma-trix results, before finishing with conclusions and
out-looks in section VIII.
II. BACKGROUND
We consider the inspiral of a secondary test particleof mass µ
on a circular, equatorial orbit around a pri-mary super massive
Kerr black hole with mass M andKerr spin parameter a . 1 where the
mass ratio is as-sumed small η = µ/M � 1. The secondary is on
aprograde orbit aligned with the rotation of the primaryblack hole
with a > 0 and dimensionful angular mo-mentum L > 0. Unless
stated otherwise, throughoutthis paper any quantity with an
over-tilde is dimension-less, e.g., r̃ = r/M and t̃ = t/M etc. The
one exceptionis the dimensionless spin parameter, which we denoteby
a without a tilde. Quantities with an over-dot willdenote
coordinate time derivatives, e.g., ṙ = dr/dt. Weuse geometrised
units such that G = c = 1.
In Boyer-Lindquist [41] coordinates, the metric of a
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Kerr black hole for θ = π/2 is given by
g = −(
1− 2r̃
)dt̃2 +
r̃2
∆̃dr̃2+(
r̃2 + a2 +2a2
r̃
)dφ2 − 4a
r̃dt̃dφ, (1)
where ∆̃ = r̃2 − 2r̃ + a2 and a is the dimensionlessspin
parameter introduced earlier. This is related tothe mass, M , and
angular momentum, J , of the Kerrblack hole via a = J/M and lies in
the range a ∈ [0, 1].The event horizon is located on the surface
defined by∆̃ = 0, when
r̃+ = 1 +√
1− a2. (2)
Introducing an extremality parameter �� 1
� =√
1− a2, (3)
the event horizon is at
r̃+ = 1 + �. (4)
The trajectory of the secondary confined to the equato-rial
plane of a central Kerr hole is governed by the Kerrgeodesic
equations [42](r̃2dr̃
dτ̃
)2= [Ẽ(r̃2 + a2)− aL̃]2 −∆[(L̃− aẼ)2 − r̃2]
r̃2dφ
dτ̃= −(aẼ − L̃) + a
r̃(Ẽ[r̃2 + a2]− aL̃)
r̃2dt̃
dτ̃= −a(aẼ − L̃) + r̃
2 + a2
∆(Ẽ[r̃2 + a2]− aL̃),
in which τ̃ denotes the proper-time coordinate for
theinspiraling object. The dimensionless conserved quan-tities Ẽ =
E/µ and L̃ = L/(Mµ) are related to theenergy, E, and angular
momentum, L, measured at in-finity. For the circular and equatorial
orbits consideredhere, the energy Ẽ and angular frequency Ω̃ can
be ex-pressed analytically
Ẽ =1− 2/r̃ + ã/r̃3/2√1− 3/r̃ + 2a/r̃3/2
(5)
Ω̃ =1
r̃3/2 + a, (6)
in which the dimensionless angular frequency Ω̃ is de-fined
through Ω = Ω̃/M = dφ/dt.
Circular orbits exist only outside the innermost stablecircular
orbit (ISCO). For radii smaller than the ISCO,the secondary will
start to plunge towards the horizonof the primary. The ISCO for
equatorial orbits is at [43]
r̃isco = 3 + Z2 − [(3− Z1)(3 + Z1 + 2Z2)]1/2 (7a)Z1 = 1 + (1−
a2)1/3[(1 + a)1/3 + (1− a)1/3] (7b)Z2 = (3a
2 + Z21 )1/2. (7c)
For near extremal orbits, using (3) and (7), and expand-ing for
�� 1, we obtain
r̃isco = 1 + 21/3�2/3 +O(�4/3), (8)
and deduce
|r̃isco − r̃+| = O(�2/3), for �� 1. (9)
The radial coordinate separation between the ISCO andhorizon is
determined by the spin parameter. In thelimit, � → 0, then r̃isco →
r̃+ → 1 in Boyer-Lindquistcoordinates.
A. Radiation Reaction
To compute circular and equatorial adiabatic inspi-rals, a
detailed knowledge of the radial self force is re-quired (see, for
example, [44] for a detailed review). Inthis paper, we will work at
leading order, including theradiative (dissipative) part of the
radial self force at firstorder, but neglecting first order
conservative effects andall second order in mass-ratio effects. The
first order dis-sipative force can be computed by solving the
Teukolskyequation [45]. The rate of emission of energy is given
by
〈− ˙̃E〉 = 〈 ˙̃EGW 〉 = 〈 ˙̃E∞〉+ 〈 ˙̃EH〉
= 2
∞∑l=2
l∑m=1
(〈 ˙̃E∞lm〉+ 〈˙̃EHlm〉). (10)
where 〈 ˙̃EGW 〉 = 〈−(ut)−1Ft〉, 〈·〉 denotes coordinatetime
averaging over several periods of the orbit. Thequantity ut is the
t component of the four velocity andFt the t component of the
gravitational self force atfirst order in the mass ratio η. This
expression is validonly if η � 1, that is, when orbits evolve
adiabaticallysuch that the timescale on which the orbital
parametersevolve is much longer than the orbital period.
The fluxes 〈 ˙̃E∞〉 and 〈 ˙̃EH〉 denote the (dimensionlessand
orbit averaged) dissipative fluxes of gravitational ra-diation
emitted towards infinity and towards the horizonrespectively. From
here on, we shall drop the angularbrackets 〈Ė〉 → Ė, to avoid
cumbersome notation. Thequantities |m| ≤ l are angular multiples
which appear inthe decomposition of the emitted radiation into a
sumof spheroidal harmonics. The components of the fluxes˙̃E are
obtained by numerical solution of the Teukolsky
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equation sourced by a point particle (the secondary).There
exists an open source code in the Black Hole Per-turbation Toolkit
(BHPT) [46] to do this for circularand equatorial orbits -
specifically the Teukolsky pack-age.
The enhancement of symmetry in the near horizon ge-ometry of
extreme Kerr [47] provides an additional toolto compute the fluxes
˙̃E analytically from first princi-ples. See [23–32] for a
description of this work. Forcircular equatorial orbits near the
horizon of a near-extremal black hole, there is a remarkably simple
ap-proximation for the total flux [25], which takes the form
˙̃ENHEKGW = η(C̃H+C̃∞)(r̃−r̃+)/r̃+,r̃ − r̃+r̃+
� 1. (11)
The quantities C̃H and C̃∞ are constants representingthe
emission towards the horizon and infinity respec-tively. These
constants are given analytically in equa-tions (76) and (77) of
[25] and codes in the BHPT canbe used to evaluate them. Numerically
evaluating themand summing the contribution of the first |m| ≤ l =
30modes gives C̃H ≈ 0.987 and C̃∞ ≈ −0.133. Eq.(11)is useful when
working within the near-horizon geome-try of the rapidly rotating
hole, but it breaks down farfrom the horizon and extra terms would
be required tocompute reliable fluxes.
All the numerical work presented in this paper, whichis found in
section V onwards, will use the exact fluxesobtained from BHPT.
However, to understand our nu-merical results, we develop a set of
new analytic tools insections IIIA and III C. These will partially
make useof the leading contribution to (11)
˙̃ENHEKGW ≈ η(C̃H + C̃∞)x , x = r̃ − 1� 1. (12)
This differs from (11) by O(�) contributions since r̃ −
1measures the BL radial distance to the extremal horizonand not the
radial distance to the near-extremal hori-zon r̃+. The
approximation (12) can be derived fromfirst principles by solving
the Teukolsky equation in theNHEK region1. Our numerical analysis
based on theBHPT, suggests the spin dependence of certain
observ-ables, to be discussed in section III B, is better
capturedby (12). Table I compares the flux at r̃isco computedusing
BHPT to that obtained from the near-extremalapproximations of
Eq.(11) and Eq.(12). This table cor-roborates that (12) is a good
approximation to the totalenergy flux, particularly in the limit as
a→ 1, where itoutperforms the full expression, (11).
1 This follows by measuring the radial distance to the
extremalhorizon by λ, defined through r̃ = r̃++r̃−
2+λr̃, and then taking
the decoupling limit λ→ 0.
B. Inspiral and Waveform
The radial evolution of the secondary can be foundby taking a
coordinate time derivative of the circularenergy relation (5)
dr̃
dt̃= −PGW
∂r̃Ẽ(13)
where we defined PGW :=˙̃EGW. As the ISCO is ap-
proached, the denominator ∂r̃Ẽ tends to zero, markinga break
down of the quasi-circular approximation. TheODE (13) is easily
numerically integrated given an ex-pression for the flux PGW.
The outgoing gravitational wave energy flux measuredat infinity
has a harmonic decomposition [11]
˙̃E∞m = AmηΩ̃2+2m/3Ė∞m , (14)
where
Am =2(m+ 1)(m+ 2)(2m)!m2m−1
(m− 1)[2mm!(2m+ 1)!!]2, (15)
and (2m+ 1)!! = (2m+ 1)(2m− 1) . . . 3 · 1. Here Ė∞m isthe
relativistic correction to the Newtonian expressionfor the flux in
harmonic m.
In this work, we shall consider two different waveformmodels.
For the analytic discussion in section III, wewill use the waveform
model in [11], whereas for thenumerical analysis in later sections,
we will use the fullTeukolsky based waveform.
Let us first review the main features of the model dis-cussed in
[11] for the waveform observed by the detectorin the source frame.
This model is written
h(t̃;θ) ≈∞∑m=2
ho,m sin(2πf̃mt̃+ φ0) . (16)
Some remarks are in order. First, we ignore the m =
1contribution since, as argued in [11], this is subleadingto the m
≥ 2 contributions. Second, the amplitudeho,m =
√〈h2+m + h2×m〉 corresponds to the root mean
square (RMS) amplitude of gravitational waves emittedtowards
infinity in harmonic m. These are averaged 〈·〉over the viewing
angle2 and over the period of the waves.Third, the oscillatory
phase depends on the initial phaseφ0 and the frequency f̃m of each
waveform harmonic isgiven by
f̃m =m
2πΩ̃ . (17)
2 The (normalised) spheroidal harmonics −2SamΩ̃ml are
integratedout over the 2-sphere.
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a ˙̃EExact/η˙̃E+NHEK/η
˙̃ENHEK/η | ˙̃E+NHEK −˙̃EExact|/η | ˙̃ENHEK − ˙̃EExact|/η
1− 10−5 0.0264197 0.0261523 0.0300885 0.0002674 0.00366881− 10−6
0.0129344 0.0125200 0.0137455 0.0004143 0.00081111− 10−7 0.0061516
0.0059484 0.006333 0.0002031 0.00018141− 10−8 0.0028875 0.0028082
0.0029294 0.0000793 0.00004191− 10−9 0.0013472 0.0013193 0.0013575
0.0000280 0.00001031− 10−10 0.0006273 0.0006176 0.0006296 0.0000097
0.00000231− 10−11 0.0002915 0.0002883 0.0002922 0.0000031
0.00000071− 10−12 0.0001354 0.0001344 0.0001356 0.0000009
0.0000002
Table I: NHEK fluxes at the ISCO computed using the
approximations Eq. (11) (denoted ˙̃E+NHEK) and Eq. (12)(denoted
˙̃ENHEK), and computed exactly using BHPT (denoted
˙̃EExact and based on the first thirty m and lmodes).
The relation between the RMS amplitude and the out-going
radiation flux in harmonic m is
ho,m =2
√η ˙̃E∞m
mΩ̃D̃(18)
where D̃ = D/M is the distance to the source fromearth. Using
Eq.(14), we can rewrite ho,m as
ho,m =
√8(m+ 1)(m+ 2)(2m)!m2m−1
(m− 1)[2mm!(2m+ 1)!!]2Ė∞m√η
D̃Ω̃m/3
(19)for m ≥ 2. We note that the effect of the averaging isthat
this waveform model does not represent the wave-form measured by
any physical observer. However, itcaptures the main physical
features of the waveformwhich encode information about the source
parameters.
Given the nature of our orbits, our parameter spacewill only be
six dimensional θ = {r̃0, a, µ,M, φ0, D̃},where r̃0 stands for the
initial size of the circular or-bit. We stress this waveform model
does not includethe LISA response functions, which affect the
ampli-tude evolution of the signal and induce modulations,due to
Doppler shifting, through the motion of the LISAspacecraft [48,
49]. Since these response functions donot depend on the intrinsic
parameters of the systemthat we are most interested in, we omit
these here and,consequently, they will also be omitted in our
analyticdiscussion based on this waveform model.
Let us now review the full Teukolsky based waveformmodel that we
will use in our numerical study. This isgiven by
h+ − ih× =µ
D̃
∑ml
1
m2Ω̃2Gml exp(−i[φ0 +mΩ̃t̃]) (20)
where
Gml =−2 SamΩ̃ml (θ) exp(iφ)Z
∞ml(r̃, a) (21)
depends on the radial Teukolsky amplitude at infinity,Z∞ml(r̃,
a), and the viewing angle (θ, φ). The latter de-pendence is through
the spin-weight minus 2 spheroidalharmonics −2SamΩ̃ml (θ, φ) =−2
S
amΩ̃ml (θ) exp(iφ). This
work will consider two viewing angles: face on (θ, φ) =(0, 0)
and edge on (θ, φ) = (π/2, 0). Using the identities
−2Sa(−m)Ω̃(−m)l (π/2, 0) = (−1)
l−2S̄
amΩ̃ml (π/2, 0) (22)
Z∞(−m)l = (−1)lZ̄∞ml (23)
where barred quantities are complex conjugates, we canwrite
equation (20) as
h+ =2µ
D
( ∞∑m=1
1
m2Ω̃2exp(−i[φ0 +mΩ̃t̃])
∞∑l=m
Gml
),
(24)for the edge-on case, and as
h+ − ih× ≈µ
4Ω̃2DG22 exp(−i[φ0 + 2Ω̃t̃]), (25)
for the face-on case. Note we have neglected higherorder l modes
withm = 2 fixed in the last equation sincethe Teukolsky amplitudes
Z∞l2 for l > 2 are negligiblein comparison to the dominant
quadrupolar l = m =2 mode. Figure 1. in [25] further justifies our
claimthat higher order m modes when l = 2 can be ignoredfor face-on
sources. Furthermore, the only spheroidalharmonics that are
non-vanishing at θ = 0 are thosewith m = −s, or m = 2 [50, 51].
To perform our numerics, the spheroidal harmonicsare calculated
using the SpinWeightedSpheroidalHar-monics mathematica package in
the BHPT, whereasthe Teukolsky amplitudes Z∞ml are calculated using
theTeukolsky package from the same toolkit. For rea-sons discussed
later, we generate both amplitudes andspheroidal harmonics for a
fixed spin parameter a =1− 10−9. For the remainder of this study,
we will onlyconsider the plus polarised signal h(t;θ) ≡ h+(t;θ)
forthe face-on and edge-on observations.
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We finish this waveform discussion with a commentregarding the
relation between the two models consid-ered in this work. The
(dimensionful) Teukolsky ampli-tudes are related to the energy flux
for each (l,m) modeby
Ė∞lm =|Z∞lm|2
4πm2Ω2. (26)
Hence |Z̃∞ml| ∼M Ω̃√η ˙̃Elm. Averaging over the sky and
ignoring the phase of the radial amplitude Z∞ml, theTeukolsky
waveform (20) reduces to (16). Our numeri-cal results indicate that
the spin precision measurementsare driven by the radial trajectory
given by (13), whichis common to both (16) and (20), while not
being largelyinfluenced by the spin dependence on the waveform
am-plitude. Given this fact and since it is analytically mucheasier
to analyse the waveform model (16), this is theone being discussed
in the analytics section III to ex-plain the increase in the spin
precision measurement fornear-extremal primaries.
C. Gravitational Wave Data Analysis
The data stream of a gravitational wave detector,d(t) = h(t;θ) +
n(t), is typically assumed to consist ofprobabilistic noise n(t)
and (one or more) deterministicsignals, h(t;θ), with parameters θ.
Assuming that thenoise is a weakly stationary Gaussian random
processwith zero mean, the likelihood is [52]
p(d|θ) ∝ exp[−1
2(d− h|d− h)
](27)
with inner product
(b|c) = 4Re∫ ∞
0
b̂(f)ĉ∗(f)
Sn(f)df. (28)
Here b̂(f) is the continuous time fourier transform(CTFT) of the
signal b(t) and Sn(f) the power spec-tral density (PSD) of the
noise. Here we use the ana-lytical PSD given by Eq.(1) in [53]. We
do not includethe galactic foreground noise in the PSD to ensure
allnoise realisations generated through Sn(f) are station-ary. This
is not a serious restriction as for the sourceswe consider here,
the majority of the GW emission isat higher frequencies where the
galactic foreground liesbelow the level of instrumental noise in
the detector.
The optimal signal to noise ratio (SNR) of a source isgiven
by
ρ2 = (h|h). (29)
This is the SNR that would be realised in a matchedfiltering
search and is a measure of the brightness,
or ease of detectability, of a gravitational wave sig-nal.
Measures of the similarity of two template wave-forms h1 := h(t;θ1)
and h2 := h(t;θ2) are the overlapO(h1, h2) ∈ [−1, 1] and mismatch
M(h1, h2) functions
O(h1, h2) =(h1|h2)√
(h1|h1)(h2|h2)(30)
M(h1, h2) = 1−O(h1, h2). (31)
If O(h1, h2) = 1 then the shape of the two waveformsmatches
perfectly. Waveforms with O(h1, h2) = 0 areorthogonal, being as
much in phase as out of phase overthe observation.
Consider θ = θ0+∆θ for ∆θ a small deviation aroundthe true
parameters θ0. Assuming that the waveformh(t;θ) has a valid first
order expansion3 in ∆θ, we sub-stitute into (27) and expand up to
second order in ∆θ
p(d|θ) ∝ exp
−12
∑i,j
Γij(∆θi −∆θibf)(∆θj −∆θ
jbf)
,(32)
where ∆θibf = (Γ−1)ij(∂jh|n) and Γij is the Fisher Ma-
trix given by
Γij =
(∂h
∂θi
∣∣∣∣ ∂h∂θj). (33)
The Fisher Matrix Γ ∼ ρ2 and therefore ∆θ scales
like(Γ−1)ij(∂jh|n) ∼ ρ−1. The linear signal approximationis
therefore valid for high SNR, ρ� 1.
The Fisher Matrix Γ, evaluated at the true parame-ters θ0,
provides an estimate of the width of the like-lihood function (27).
Hence, it can be used as a guideto how precisely you can measure
parameters. The in-verse of the Fisher matrix is an approximation
to thevariance-covariance matrix Σ on parameter precisions∆θi
Cov(∆θi,∆θj
)≈ (Γ−1)ij . (34)
The square route of the diagonal elements of the in-verse fisher
matrix provide estimates on the precisionof parameter measurements,
accounting for correlationsbetween the parameters.
III. ANALYTIC ESTIMATES OF SPINPRECISION
Before discussing numerical results on the measure-ment
precisions for the parameters θ of near-extremal
3 In the literature, this is called the linear signal
approximation.It is a good approximation for sufficiently small ∆θ,
such that∆θ ∂2θh� ∂θ∆h.
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7
EMRIs, we would like to develop some analytic toolsthat will
allow us to understand the precisions we findnumerically. In
particular the fact that spin measure-ments for near-extremal
primaries are noticeably tighterthan those obtained for more
moderately rotating pri-maries. Throughout this section, we will
use the wave-form model (16) for analytical convenience. We
willfocus on the spin-spin component of the Fisher matrix
Γaa = 4
∫df|∂ĥ(f, r(a), t;θ)/∂a|2
Sn(f)(35)
in the following analytic discussion. Our numerical
andstatistical analysis will be more general and employ
theTeukolsky based model (20). In future work, we will ex-tend this
analytic considerations to multiple parameterstudy.
If all other parameters were known perfectly, the es-timated
precision on the spin parameter would be
∆a ≈ 1/√
Γaa. (36)
Thus, to compare precisions between near-extremal (de-noted ext)
and moderately rotating (denoted mod) pri-maries one is led to
study the ratio
ΓextaaΓmodaa
. (37)
Consider the (semi-analytic) gravitational wave am-plitude
(16)
h(t) =∑m
hm(t) ≈∑m
2√Ė∞m
mΩ̃D̃sin(mΩ̃t̃) , (38)
where we have chosen the initial phase φ0 = 0 for sim-plicity.
The Fisher Matrix depends on the PSD ofthe detector. In the
numerical calculations presentedlater we will use the full
frequency dependent PSD,but to derive our analytic results we will
approximateSn(f) ≈ Sn(f◦), a constant. The rationale for this
isthat EMRIs evolve quite slowly and so the total changein the PSD
over the range of frequencies present inthe signal is small.
Between 1 mHz and 100 mHz, the(square root of the) LISA PSD changes
by just one or-der of magnitude, which is much smaller than the
threeorders of magnitude improvement in spin measurementprecision
that we find numerically. Additionally, the dif-ference in the ISCO
frequencies across all combinationsof mass and spin considered in
our numerical analysis isless than a factor of 2.5. PSD variations
can not there-fore explain the numerical results, and so we can
ignorethese in deriving the analytic results which do explainthe
numerics. Under this approximation
Γaa ≈4
Sn(f◦)
∫dt (∂ah(t))
2 . (39)
We additionally assume that the choice of f◦ does notdepend on
the spin, and therefore the ratio (37) is inde-pendent of Sn(f◦).
Again, this approximation could in-troduce at most an order of
magnitude uncertainty, andmost likely much less than that. Once the
Fisher matrixis written in the form (39), we can use the
semi-analyticwaveform model (38) to evaluate it. In appendix A,
weargue the dominant contribution can be approximatedby
Γaa ≈8M
D̃2 Sn(fo)
∑m
Γaa,m
Γaa,m ≈∫ t̃cutt̃0
dt̃ ˙̃E∞m (Ω̃t̃)2
(1 +
3
2
√r̃ ∂ar̃
)2.
(40)
Here t̃0 is the coordinate time at which the observationstarts
and t̃cut is the coordinate time at the end of theobservation. For
the results in this paper, we analyse∼ 1 year long signals and fix
t̃cut independently of spin,such that all inspirals terminate
before r̃isco is reached.
As seen in (40), a proper understanding of the preci-sion in the
spin measurement requires quantifying thespin dependence of the
inspiral trajectory of the sec-ondary, i.e. ∂ar̃.
A. Spin dependence on the radial evolution
Our primary goal here is to understand the spin de-pendence on
the radial trajectory of the secondary (∂ar̃)for any spin parameter
a of the primary.
The trajectory of the secondary is the integral of theinspiral
equation
∂r̃Ẽ(r̃, a)dr̃
dt̃= −PGW(r̃, a) . (41)
This follows from energy conservation, where Ẽ(r̃, a) isthe
energy of a circular orbit (5) and PGW :=
˙̃EGW(r̃, a)is the energy rate carried away by gravitational
waves(10). While Ẽ(r̃, a) is kinematic, that is, derivedthrough
geodesic properties, PGW is dynamic, that is,it is a radiation
reactive term determined by solvingTeukolsky’s equation for a point
particle source. Theformer is under analytic control, whereas the
latter typ-ically requires numerical treatment.
The quantity ∂ar̃ captures the change in the sec-ondary’s
trajectory when the spin parameter a of theprimary varies, keeping
the remaining primary and sec-ondary parameters fixed, including
t̃. More explicitly,the integral r̃(r̃0, a) of (41) depends on the
initial con-dition r̃(t̃0) = r̃0 and it depends on the spin
parametera both through (∂r̃Ẽ) and (PGW) information, but
notthrough t̃, which is simply labelling the points in
thetrajectory. We will comment on the possible spin de-pendence on
the initial condition r̃0 below.
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8
One possibility to compute ∂ar̃ is to integrate (41)and to take
the spin derivative explicitly afterwards. Asecond, equivalent, way
is to observe r̃ is a monotonicfunction of t̃ at fixed spin and
initial radius r̃0. Hence,it can be used as the integration
coordinate to study∂ar̃(r̃). To do this, notice that the total spin
derivativeof the kinematic and dynamic functions in (41), at
fixedr̃0 and t̃, is
∂
∂a∂r̃Ẽ(r̃(a), a)
∣∣∣∣r̃0,t̃
= (∂2r̃ Ẽ) ∂ar̃ + ∂2ar̃Ẽ ,
∂PGW∂a
∣∣∣∣r̃0,t̃
= (∂r̃PGW) ∂ar̃ + ∂aPGW .
(42)
To ease our notation, all spin partial derivatives in therhs,
and in the forthcoming discussion, should be under-stood as
computed at fixed r̃0 and t̃. Defining u = ∂ar̃(to ease notation)
and computing the total spin deriva-tive of equation (41), we
obtain[
u∂2r̃ Ẽ + ∂2ãrẼ + ∂r̃Ẽ
du
dr̃
]dr̃
dt̃= −dPGW
da. (43)
Plugging in the radial velocity using (41) one obtains
du
dr̃+
(∂2r̃ Ẽ
∂r̃Ẽ− ∂r̃PGW
PGW
)u = −∂
2ar̃Ẽ
∂r̃Ẽ+∂aPGWPGW
. (44)
This is a first order linear ODE, valid for any spin andfor any
location of the secondary, whose solution de-scribes the desired
spin dependence in the radial trajec-tory ∂ar̃(r̃).
Its general solution is a sum of the homogeneous solu-tion uh
and a particular solution up. It will depend onan initial condition
u(r̃0). The initial condition of theradial trajectory is
r̃(r̃0, a, t = 0) = r̃0 ⇒∂r̃
∂a
∣∣∣∣r0,t=0
= 0, (45)
from which we deduce u(t = 0) = 04.The homogeneous version of
equation (44) is equiva-
lent to
duhuh
+ d log
(∂r̃Ẽ
PGW
)= 0⇒ uh = k0
PGW
∂r̃Ẽ(46)
4 The initial condition u(r̃0) can play an important role
whengluing a numerical calculation for ∂ar̃ with an analytic one
insome specific piece of the trajectory where the information
de-termining the solution to (44) is under analytic control. We
willbe more explicit about this when we discuss ∂ar̃ in the
regionclose to ISCO.
where k0 is an arbitrary integration constant. We followa
standard approach and look for a particular solutionof the form up
= k(r̃, a)uh. Plugging this into (44) gives
k(r̃, a) =
∫∂r̃Ẽ
PGW
(−∂
2ar̃Ẽ
∂r̃Ẽ+∂aPGWPGW
)dr̃ (47)
= −∫
∂r̃Ẽ
PGW∂a log
(∂r̃Ẽ
PGW
)dr̃. (48)
Combining our results, we obtain
∂ar̃ =PGW
∂r̃Ẽ
(k0 −
∫∂r̃Ẽ
PGW∂a log
(∂r̃Ẽ
PGW
)dr̃
). (49)
This is valid for any spin, for any location of the sec-ondary
and for any flux PGW. This analytic result willallow us to
determine what the dominant source of thespin dependence is in
different regions of the trajectory.
In figure 1 we show the near perfect agreement be-tween the
solution to (49) and our numerical calculationof ∂ar̃ using finite
difference method
∂ar̃ ≈r̃(a+ δ, t̃, Ė(a+ δ))− r̃(a− δ, t̃, Ė(a− δ))
2δ. (50)
the method used to calculate year-long trajectories usedfor our
Fisher matrix results in later sections, for bothmoderately and
rapidly rotating primaries.
Following [11], we express the energy flux as a rel-ativistic
correction factor, Ė , times the leading orderNewtonian flux
PGW =32
5η Ω̃10/3Ė . (51)
Plugging this into Eq. (49) gives
∂ar̃ =1
Q
(k0 −
∫Q ∂a logQ dr̃
), Q = ∂r̃Ẽ
Ω̃10/3Ė.
(52)Decomposing the source term
Q ∂a logQ =∂r̃Ẽ
Ω̃10/3Ė
(∂2ar̃Ẽ
∂r̃Ẽ− ∂aĖĖ
+10
3Ω̃
), (53)
we see that the first and third terms are kinematic, i.e.,driven
by geodesic physics, whereas the second is dy-namical, i.e., driven
by the energy flux. Comparisonbetween these terms at different
stages of the inspiral,as a function of the spin, can help us to
determine whatthe driving source of spin dependence is in each
case. Inthe next subsection, we investigate the contribution ofboth
the geodesic and radiation reactive terms to ∂ar̃.
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9
1.05 1.10 1.15 1.20r
3
2
1
0
1
2
3
4
log 1
0(ar
)(r0, , a) = (1.225, 2 × 10 6, 1 10 6)
Finite differencing - equation (50)Solution to equation (49)
2.5 3.0 3.5 4.0 4.5 5.0r
6
4
2
0
2
4
log 1
0(ar
)
(r0, , a) = (5.185, 2 × 10 6, 0.9)Finite differencing - equation
(50)Solution to equation (49)
Figure 1: The dashed curves (black dashed and yellow dashed) on
each figure is the solution to (49) with k0 = 0corresponding to
∂ar̃(r̃0) = 0. In both plots, the solid colours (blue and violet)
are ∂ar̃ calculated using a fifth
order stencil method. In each plot, the intrinsic parameters
given in the titles.
B. Comparison of radial evolution for moderateand near-extremal
black holes
Despite the universality of (49) or (52), the depen-dence on the
energy flux makes it not feasible to ana-lytically integrate ∂ar̃
along the entire secondary trajec-tory. However, we can integrate
(49) in specific regionsof the secondary trajectory.
It is possible to prove that d∂ar̃/dr̃ < 0 and hencethat ∂ar̃
grows monotonically over the inspiral. It istherefore natural to
study the behaviour of ∂ar̃ close toISCO, where its contribution to
the Fisher matrix (40)will be maximal. We first compare the
kinematic anddynamical contributions to (53). Using results from
theBHPT, we have numerically calculated the spin deriva-tive of Ė
for two primaries with spin parameters a = 0.9and a = 1 − 10−6.
These are compared with the kine-matic sources in (53) in figure 2.
These figures showthat ∣∣∣∣∂2ar̃Ẽ∂r̃Ẽ + 103 Ω̃
∣∣∣∣� ∣∣∣∣∂aĖĖ∣∣∣∣, (54)
for both spin parameters. This suggests it is the kine-matic
sources in (53) that drive the spin dependenceof the secondary
trajectory, particularly close to ISCO.Although we have only
verified it for two choices of spinparameter, we will assume this
approximation holds forany spin parameter a ≥ 0.9.
We first consider moderately spinning black holesclose to ISCO.
Dropping the dynamical contribution to
(53), we can compare the two remaining terms. Theangular
velocity piece is bounded and order one, but∂r̃Ẽ tends to zero at
ISCO. This means that ∂2ar̃Ẽ/∂r̃Ẽdominates close to ISCO,
allowing us to use the approx-imation
∂r̃Ẽ
Ω̃10/3Ė
(∂2ar̃Ẽ
∂r̃Ẽ− ∂aĖĖ
+10
3Ω̃
)≈ ∂
2ar̃Ẽ
Ω̃10/3Ė. (55)
Since, for moderate spins, the variation of Ω̃ and Ė withradius
close to ISCO is negligible compared to the varia-tion in ∂2ar̃Ẽ,
we will approximate them by their valuesat r̃isco. This allows us
to integrate (49) to give thespin dependence of the radial
trajectory for moderatelyspinning black holes
∂ar̃ ≈1
∂r̃Ẽ
(kmodΩ̃
10/3isco Ė0(a, r̃isco)− ∂aẼ
), (56)
where kmod is an arbitrary constant. Since
∂r̃Ẽ =r̃2 − 3a2 + 8a
√r̃ − 6r̃
2r̃7/4(r̃3/2 − 3
√r̃ + 2a
)3/2 , (57)it follows from eq. (A5) in [40] that ∂r̃Ẽ(r̃isco) =
0. Formoderately rotating primaries and near ISCO, we canexpand Ẽ
≈ Ẽ(r̃isco) + 12 ∂
2r̃ Ẽ(r̃isco) (r̃− r̃isco)2 leading to
∂aẼ ≈ ∂aẼ(r̃isco) + ∂2r̃ Ẽ(r̃isco) (r̃ −
r̃isco)(−∂ar̃isco)∂r̃Ẽ ≈ ∂2r̃ Ẽ(r̃isco) (r̃ − r̃isco)
(58)
http://bhptoolkit.org
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10
2 3 4 5 6 7 8 9 10r
1.0
0.5
0.0
0.5
1.0
1.5
2.0
2.5M
agni
tude
Non-extremal a = 0.9
log10| 2arE/ rE + 10 /3|log10| a / |
2 4 6 8 10r
2
0
2
4
6
8
Mag
nitu
de
Near-extremal a = 0.999 999
log10| 2arE/ rE + 10 /3|log10| a / |
Figure 2: The top plot compares the kinematic andradiation
reaction quantities given in (53) for a spin ofa = 0.999999. The
bottom plot is the same but for aspin parameter of a = 0.9. Notice
that in these twocases the kinematical quantities dominate over
the
relativistic correction terms.
Using these expansions in Eq. (56) we deduce∂ar̃ = k̃mod/(r̃ −
r̃isco) + ∂ar̃isco, where k̃mod =kmodΩ̃
10/3isco Ė0(a, r̃isco)/∂2r̃ Ẽ(r̃isco). Assuming that r̃0
is
sufficiently close to r̃isco that this approximation
holdsthroughout the range [r̃isco, r̃0], we can use the
boundarycondition (45) to determine k̃mod = ∂ar̃isco(r̃isco− r̃0)
andhence
∂ar̃ ≈ (−∂ar̃isco)r̃0 − r̃r̃ − r̃isco
. (59)
We now repeat this analysis for near-extremal pri-maries. Near
ISCO, the energy flux can be approxi-mated by the NHEK flux (x ≡ r̃
− 1� 1)
PGW ≈ η(C̃∞ + C̃H)x . (60)
Using this approximation, there is no explicit spin de-pendence
and so the ∂aPGW term in Eq. (47) vanishes.Expanding (57) for x =
r̃ − 1 � 1 and � � 1, thedenominator involves
r̃3/2−3√r̃+2a =
3
4x2−�2− 1
4x3+
9
64x4+O(x5, x�2, �4),
while the numerator has the expansion
r̃2−3a2 +8a√r̃−6r̃ = 1
2x3−�2− 5
32x4 +O(x5, x�2, �4) .
We conclude
∂r̃Ẽ ≈2
3√
3
(1− 11
8x− x
3isco
x3
)+O(x2, �2/x2). (61)
Using this approximation in Eq. (47) we find
k(r̃, a) ≈ −∫∂2ar̃Ẽ
PGWdr̃ (62)
≈ 2x2isco
η(C̃∞ + C̃H)√
3
∂xisco∂a
∫x−4dx (63)
≈ 89√
3η(C̃∞ + C̃H)
1
x3(64)
where we have used xisco ≈ 21/3�2/3 and PGW defined in(60). This
is valid for x� 1 and includes the correctionsdue to x ∼ xisco.
Assuming r̃0 is close to ISCO, sothat the initial condition (45)
holds, we conclude thatthe spin dependence in the near-ISCO region
of a near-extremal black hole is
∂ar̃ ≈8
9√
3x2∂r̃Ẽ
(1− x
3
x30
). (65)
Figure 3 compares (59) and (65) to the full ∂ar̃ com-puted
numerically without using the near-ISCO approx-imations. We see
that the approximations are very ac-curate in the region close to
the ISCO where they arevalid.
Before continuing, we will comment further on thechoice of flux
(12) instead of (11). The latter has anexplicit dependence on r̃+ =
1 + �. Consequently, itcarries an additional spin dependence. In
particular,∂ar̃+ = −a/�. Thus, for near-extremal primaries thisspin
dependence can induce extra diverging sources for∂ar̃ with a very
specific sign. We can easily computetheir effects by integrating
the ODE with such an en-ergy flux source. The result one finds does
not agreewith the numerical evaluation of ∂ar̃ generated fromthe
BHPT, which computes the exact flux5. We con-clude that (12)
appears to capture the spin dependenceof our observable (the
amplitude of the gravitationalwave) more accurately than (11). This
is in fact thereason we chose to work with (12).
5 In particular, it is no longer the case that ∂ar̃ is
monotonicallyincreasing all along the inspiral trajectory, whereas
the BHPTdata is monotonically increasing, a feature our ODE with
flux(12) reproduces.
http://bhptoolkit.orghttp://bhptoolkit.org
-
11
2.325 2.350 2.375 2.400 2.425 2.450 2.475 2.500r
3
2
1
0
1
2
log 1
0(ar
)(r0, , a) = (2.5, 2 × 10 6, 0.9)
Finite differencing - equation (50)Solution to (59)
1.05 1.10 1.15 1.20r
3
2
1
0
1
2
3
4
log 1
0(ar
)
(r0, , a) = (1.225, 2 × 10 6, 1 10 6)Finite differencing -
equation (50)Solution to (65)
Figure 3: The yellow dashed and black dashed curves are
solutions to (59) and (65). The purple and blue curvesare the true
solutions to ∂ar̃ obtained numerically without near-ISCO
simplifications. We see both
approximations capture the leading order behaviour of the spin
derivative of the radial trajectory very well.
Let us close this discussion with a brief comparisonbetween the
analytic results for moderate and near-extremal spins. We write r̃−
r̃isco ∼ δ > η2/5, the latterinequality ensuring that we avoid
entering the transi-tion region [40, 54]. Expanding Eq. (61) we
find fornear-extremal black holes
∂r̃Ẽ ≈2
3√
3
(3δ
xisco− 11
8δ − 11
8xisco + · · ·
).
The first term is dominant unless |δ . x2isco ∼ �4/3.The
constraint δ > η2/5 ensures this is only violatedif � >
η3/10. This will be satisfied for all the casesthat we consider in
this paper, but we emphasise thisis not a physical constraint. When
this constraint isviolated, additional terms become important in
the ex-pansion which we have ignored, and these ensure that∂r̃Ẽ →
0 at r̃isco. We conclude the scaling of ∂r̃Ẽ isδ/�2/3 for
near-extremal black holes, compared to δ formoderate spins.
It follows using (59) and (65) that
∂ar̃ ∝
{1δ , moderate spins
�2/3
δ(δ+�2/3)2, near-extremal spins
. (66)
The spin dependence on the radial trajectory for near-extremal
primaries is larger than for moderately rotat-ing ones.
C. Precision of spin measurement
In the previous section we showed the effect that thespin
parameter has on the radial trajectory. This wasachieved by
studying the general linear ODE for ∂ar̃,Eq. (49). By arguing that
the kinematic terms dom-inate the behaviour of ∂ar̃, for both the
near-extremeand moderately spinning black holes, analytic
solutionswere found near the ISCO. We were able to concludethat
∂ar̃ grows much more rapidly close to the ISCO fornear-extreme
black holes than for moderately spinningblack holes. We also
emphasise that Eq. (54) showsthat corrections to ∂ar̃ of the form
∂aĖ are subdomi-nant. We now explore the consequences of these
resultsfor the precision of spin measurements, computed usingthe
Fisher Matrix formalism.
Due to the large number of observable gravitationalwave cycles
that are generated while the secondary iswithin the strong field
gravity region outside the pri-mary Kerr black hole, extreme
mass-ratio inspirals willprovide measurements of the system
parameters withunparalleled precision [1]. In particular, it has
beenshown that our ability to constrain the spin parameter ais
expected to be O(10−6) [3, 4, 15, 55]. It has also beenshown that
the measurements are more precise for pro-grade inspirals into more
rapidly spinning black holes,when the secondary spends more orbits
closer to theevent horizon of the primary (see Fig.(11) in [55]).
Insubsequent sections we show through numerical calcu-lation that
spin measurements are even more precise for
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12
EMRIs into near-extremal black holes. We now try tounderstand
this result using Eq. (40).
Inspection of (40) suggests there are two main effects:the
dependence on t̃2 and the dependence on (∂ar̃)2.First, the fact
that t̃ ∼ O(η−1) follows from integrat-ing (41), and therefore the
contribution to the Fishermatrix due to t̃2 is large and scales
like η−2. Second,∂ar̃ is monotonically increasing as the secondary
spiralsinwards. Thus, its maximal contribution comes fromthe region
close to ISCO, which supports results in [55].Eq. (66) shows this
contribution is largest in the laststages before entering into the
transition regime. Aschanges in Ω̃ close to ISCO are negligible,
the factor(Ω̃t̃)2 is, approximately, the square of the number of
cy-cles, a proxy widely used in the literature in discussionsof the
precision of measurements. Our estimate (40)confirms this intuition
and shows the spin precision willbe further increased by large
values of the radial spinderivative, ∂ar̃.
D. Comparison of spin measurement precision formoderate and
near-extremal black holes
The Fisher matrix estimate (40) depends on the spinderivative of
the radial evolution, on the duration of theinspiral and on the
energy flux. Eq. (66) shows that,at a fixed distance to the
corresponding ISCO, ∂ar̃ islarger for a near-extremal primary than
for a moder-ately rotating primary. As a consequence of time
dila-tion near the black hole horizon, Ė → 0 near the ISCOfor
near-extremal primaries, but remains finite for mod-erately
rotating ones. This means that the energy fluxfor near-extremal
inspirals is much smaller than thatfor moderate spins, but the
duration of the inspiral islonger. However, we can write (40) as an
integral over
the BL radial coordinate r̃. In that case, the integrandis
proportional to
˙̃E∞m˙̃EGW
(t̃ Ω̃)2 (∂r̃Ẽ) (∂ar̃)2
close to the relevant ISCO. While the energy fluxes aremuch
smaller for near-extremal inspirals, the ratio of en-ergy fluxes
appearing above is an order one quantity forall spin parameters.
The expression above is therefore aproduct of factors that have
been argued to be either ofcomparable magnitude or much smaller for
moderatelyrotating primaries. We therefore expect the precision
ofspin measurements to be much higher for near-extremalEMRIs.
A quantitative comparison between the near-extremaland the
moderately spinning sources requires a precisecalculation of the
ratio (37) computed along the entirerespective trajectories. In
general, this is a hard ana-lytic task since both energy fluxes
˙̃EGW and
˙̃E∞m mustbe handled through numerical means and long
obser-vations of inspirals (starting in the weak field)
requirecalculations performed in the frequency domain whereSn(f)
shows non-trivial (non-constant) behaviour. Thiswould be no more
straightforward than direct numeri-cal computation of the Fisher
Matrix and so we do notpursue it here.
For any sources whose trajectory lies entirely lie inthe
near-ISCO region, these analytic approximations al-low us to
compute the ratio (37) reliably. This can beexploited to obtain an
analytic approximation to theFisher Matrix for such sources and
this calculation willbe pursued elsewhere. Additionally, earlier
argumentstell us that it is the near-ISCO regime that dominatesthe
spin precision and so these expressions are sufficientto understand
the increase in spin precision seen fornear-extremal inspirals.
a. Near-extremal source. From (65), it follows
∂ar̃ ≈4
3x30
x
x3 − x3isco(x30 − x3) . (67)
Since√r̃∂ar̃ grows fast and the rate of change of r̃ and Ω̃ is
small, near-ISCO, we can approximate (40) by
Γaa ≈ 18µ
(ηD̃)2 Sn(f◦)r̃ext Ω̃
2ext
∑m
∫ t̃cut0
d(ηt̃)dẼ∞mηdt̃
(ηt̃)2 (∂ar̃)2. (68)
Here, t̃cut is the time at the end of the integration, where x =
xcut. Our approximations break down when thetransition regime
breaks down, so we can assume xcut ∼ η2/5 + �2/3, which is a small
quantity. Using dẼ∞m /dt̃ =ηC̃∞m x and assuming x0 ≥ x� xisco, so
that the trajectory can be approximated by x(t̃) ≈ x0 e−y with y =
αηt̃ ≡3√
32 (C̃H + C̃∞) ηt̃, the Fisher matrix reduces to
Γextaa ≈64µ
(ηD̃)2 Sn(f◦)
r̃ext Ω̃2ext
(3x0α)3·G(ycut)
(∑m
C̃∞m
), (69)
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13
with eycut = x0/xcut and
G(ycut) = −9y3cut + (9y2cut + 2) sinh 3ycut − 6ycut cosh 3ycut
≈x30
2x3cut
[(3 log
x0xcut− 1)2 + 1
], (70)
where in the last step we used x0 � xcut.b. Moderately spinning
source. Using the same kind of approximations as above, but taking
into account the
different energy flux and different trajectory
(r̃ − r̃isco)2 − (r̃0 − r̃isco)2 ≈64
5η Ω̃
10/3isco
Ė0(a)∂2r̃ Ẽ(r̃isco)
t̃ , (71)
one can approximate the Fisher matrix for moderate spins by
Γmodaa ≈ 18
(5∂2r̃ Ẽ(r̃isco)
64Ė0
)3µ
(ηD̃)2 Sn(f◦)
r̃isco (∂ar̃isco)2
Ω̃8isco(r̃0 − r̃isco)6 F (δ)
(∑m
dẼ∞mη dt̃
∣∣∣∣∣isco
), (72)
where δ ≡ r̃cut−r̃iscor̃0−r̃isco < 1 and
F (δ) = −2 log δ − 4(1− δ)− (1− δ2) + 83
(1− δ3)− 12
(1− δ4)− 45
(1− δ5) + 13
(1− δ6) . (73)
c. Ratio of Fisher matrices. Within these approximations, the
ratio (37) now reduces to
ΓextaaΓmodaa
≈ 2569
(64
45√
3 ∂2r̃ Ẽ(r̃isco)
)3r̃extΩ̃
2ext
r̃iscoΩ̃2isco (∂ar̃isco)2
G(ycut)
x30 (r̃0 − r̃isco)6 F (δ)T ,
T =∑m C̃∞m
(C̃H + C̃∞)3(Ω̃10isco Ė30 )∑m
dẼ∞mη dt̃
∣∣∣isco
(74)
The most relevant feature for our current discussion isthe
quotient dependence
G(ycut)
x30 (r̃0 − r̃isco)6 F (δ)≈ 1x3cut (r̃0 − r̃isco)6F (δ)
(75)
The first two denominator factors increase the ratio,since xcut
� 1 and r̃0− r̃isco < 1. The last could in prin-ciple be large,
due to the logarithmic term. However δand xcut have similar scaling
and therefore xcutF (δ)� 1.We deduce that the spin component of the
Fisher Ma-trix is much larger for near-extremal inspirals than
formoderate spins. This is confirmed by the numerical re-sults that
will be reported in subsequent sections.
We finish by noting that the Fisher matrices increasein
magnitude as the trajectory is cut off closer to r̃isco.In the case
of moderate spin, we already noted thelogarithmic dependence of F
(δ) as δ → 0. This haspreviously been observed in the literature,
see for ex-ample Fig.(11) in [55]. For near-extremal EMRIs, ifxcut
∼ xisco ∼ �2/3, then for fixed x0 and as � → 0the spin Fisher
matrix scales as Γaa ∼ (log(�)/�)2. Wededuce that observing the
latter stages of inspiral is im-
portant for precise parameter measurement, for any pri-mary
spin.
In summary, we have derived an analytic approxima-tion, valid
close to ISCO, for the spin component of theFisher Matrix. This
indicates that this component ismuch larger for near-extremal spins
and therefore weexpect much more precise measurements of the
spinparameter in that case. The approximation dependssensitively on
certain quantities, such as the cut-off ra-dius, xcut, that are
somewhat arbitrary. However, forany choice the near-extremal
precision is a few orders ofmagnitude better. This provides support
for the numer-ical results that we will obtain in Sec.(VI), which
showa similar trend.
IV. WAVEFORM GENERATION
In this section we provide more details on how we con-struct the
waveform model used to compute the FisherMatrix in the next
section. The waveform model waspreviously given in Eq. (16) and Eq.
(20). Here, we de-
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14
scribe how the various terms entering these equationsare
evaluated.
A. Energy Flux
Both waveform models (16) and (20) depend on theradial
trajectory r̃(t̃, a, η, Ė). The amplitude evolu-tion using the
Teukolsky formalism depends on thespheroidal harmonics −2SamΩ̃ml
(θ, φ) and Teukolsky am-plitudes at infinity Z∞ml(r̃, a). The
energy flux at infinity˙̃E∞ml(r̃, a) is related to the Teukolsky
amplitudes Z
∞ml
through equation (26). Thus, to accurately generatethe waveforms
(16) and (20) far from the horizon wherenear-extremal
simplifications can not be made, the var-ious radiation reactive
terms Z∞ml,
˙̃E(Ė), ˙̃E∞m (Ė∞m ) haveto be handled numerically. This
section outlines ournumerical routines to do so.
We use the BHPT to calculate the first order dissipa-tive radial
fluxes ˙̃EGW for a = 1−{10−i}i=9i=3 from whichĖ in Eq. (51) can be
computed. We used the Teukolskymathematica script in the toolkit
and tuned the numer-ical precision to ∼ 240 decimal digits to avoid
numericalinstabilities when computing ˙̃EGW in the
near-horizonregime for rapidly rotating holes. For moderately
spin-ning holes a . 0.999, we used the tabulated data inTable II of
[11].
Each coefficient appearing in Eq. (19) is itself a sumover l
modes, ˙̃E∞m =
∑∞|l|=m
˙̃E∞ml. Both the sum over land the sum over m in Eq. (19) can be
truncated with-out appreciable loss of accuracy. As discussed in
[32],near-extremal EMRIs require a significant number ofharmonics
to be included to obtain an accurate repre-sentation of the
gravitational wave signal. To illustratefor a high spin of a = 1 −
10−9, we used the BHPTto compute ˙̃E∞ml for harmonics |m| ≤ l ∈ {2,
. . . , 15}.Figure 4 illustrates the convergence as the number
ofharmonics is increased.
Based on these results, we go further by includingharmonics with
l ≤ lmax = 20 to calculate the totalenergy flux ˙̃EGW
˙̃EGW =
lmax∑|l|=2
∑|m|≤l
( ˙̃E∞ml +˙̃EHml), (76)
using the Teukolsky package in the BHPT. In the samenumerical
routine, we compute ˙̃E∞m =
∑lmax|l|=m
˙̃E∞ml usinglmax = 20 for m ≤ 20. These formulas are rearranged
toobtain Ė and Ė∞m using (51) and (14).
Finally for our Teukolsky based waveforms used innumerics
section VI, we use the BHPT to extract theTeukolsky amplitudes
Z∞ml(r̃, a) and build an interpolant
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0r
0.00
0.02
0.04
0.06
0.08
0.10
Ener
gy fl
ux a
t inf
inity
High spin: a = 1 10 9
lmax = 15lmax = 11lmax = 6lmax = 2
Figure 4: Comparison of the total energy flux atinfinity (black
curve) including different harmonic ˙̃E∞lm
contributions. Note that at r̃ ≈ 1.3, the l = 2harmonic energy
flux ˙̃E∞2 contributes ∼ 32% of the
total energy flux, whereas including the first lmax =
11harmonics (violet curve) contributes more than ∼ 98%.
over r for each harmonic m = {1, . . . , lmax = 20}
Gm(r̃, a) =
∞∑l=m
−2SamΩ̃ml (θ) exp(iφ)Z
∞ml(r̃, a) (77)
for each viewing angle (θ, φ) = (π/2, 0) and (θ, φ) =(0, 0). To
summarise, we use (77) in (20) to computeFisher matrices
numerically in section VI. To aid ouranalytic study, we use the
computed Ė∞,m in the wave-form model (16) when evaluating the
ratio (74).
B. Radial trajectory & Waveform
The radial trajectory can be constructed by numeri-cally
integrating the ODE (13) using an interpolant forĖ(r̃) and
suitable initial conditions. As before, we usethe spin independent
initial condition r̃(t̃0 = 0) = r̃0.Fig.(5) shows some example
radial trajectories for var-ious spin parameters, computed using
flux data fromthe BHPT. In the high spin regime, the exponential
de-cay of the radial coordinate is prominent as discussedin [24,
56]. Throughout our simulations, the observa-tion ends after a
fixed amount of time, chosen such thatthis is before the transition
to plunge for all parame-ter values used to compute the Fisher
Matrix. This isimportant to avoid introducing artifacts from the
ter-mination of the waveform, given that the transition toplunge is
not properly included in this waveform model.It is clear from
Figure (5) that larger the spin parame-ter, the longer the
secondary spends in the dampeningregime. See equation (22) of [24]
for further details.
The spin dependence of the radial evolution can becalculated by
integrating (13) and then taking numericalderivatives. We consider
two reference cases, both withcomponent masses µ = 10M� andM =
2×106M�, but
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15
1.0 1.5 2.0 2.5 3.0 3.5r
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1 dr/d
tRate of change of r(t)
a = 0.99a = 0.999a = 0.999 9a = 0.999 999a = 0.999 999 999 9
0 1 2 3 4 5 6 7t × /M
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
r(t)
Radial Trajectorya = 0.999 999 999a = 0.999 999a = 0.999
Figure 5: The top panel shows how dr̃/dt̃ varies withr̃. The
higher the spin parameter, the more time thesecondary spends in the
throat before plunge. The
lower panel shows the corresponding inspiraltrajectory. The
dampening is clearly shown when the
primary is near maximal spin, as seen in [24].
with different spin parameters a = 0.9 and a = 1−10−6.We compute
one year long trajectories, with r̃(0) = 5.08in the first case and
r̃(0) = 4.315 in the second. The spinderivative of the radial
evolution can be calculated byperturbing the spin and using the
symmetric differenceformula for δ � 1
∂r̃
∂a≈ r̃(a+ δ, t̃, Ė(a+ δ))− r̃(a− δ, t̃, Ė(a− δ))
2δ. (78)
Figure 6 plots the quantity |∂ar̃|2 appearing in theFisher
matrix estimation (40). By inspection, it is clearthat |∂ar̃|2 is
largest when the spin parameter is closeto unity and when the
radius is close to r̃isco, match-ing our analytical conclusions
using approximations (65)and (56).
Using the semi-analytic model (16) we now evaluatethe estimate
(74), for the same two systems, but differ-ent r0 to ensure that
the assumptions made in derivingEq. (74) still hold (r̃0 = 2.85 for
a = 0.9 and r̃0 = 1.2for a = 1− 10−6). We choose termination points
r̃cut =r̃isco + λ with λ ∼ {λext = 10−4, λmod = 10−2}, just
0 50 100 150 200 250 300 350time [days]
10
5
0
5
2log
10|
ar|
Spin dependence on radial evolutiona = 0.999 999a = 0.9
Figure 6: The blue curve is ∂ar̃ for a = 0.999999. Theorange
curve is ∂ar̃ for a = 0.9. Notice that the spindependence on r
grows rapidly in the near-ISCO
region of the rapidly rotating hole.
outside the transition region. Finally, the expression∑C∞,m was
calculated using the high_spin_fluxes.nb
mathematica notebook in the BHPT, including harmon-ics up to m =
10. We find the ratio to be
ΓextaaΓmodaa
∼ 500. (79)
giving a rough estimate that the spin precision in-creases by at
least two orders of magnitude for thesetwo sources.
This verifies claims made in section III C. When cor-relations
with other parameters and the shape of thePSD are ignored, we
predict a precision on the spin pa-rameter roughly two orders of
magnitude higher thanfor moderately spinning black holes.
To generate gravitational waveforms for the nu-merical study we
use the Teukolsky waveform model(20). The waveform depends on
parameters θ ={a, r̃0, µ,M, φ0, D̃}. We will consider two classes
ofnear-extremal source, differentiated by the magnitudeof their
component masses and mass ratio. The first“heavier" source has
parameters
θheavy = {r̃(t0 = 0) = 1.225, a = 1−10−6, µ = 20M�,M = 107M�, φ0
= π,
D = {Dedge = 1.8, Dface = 3}Gpc} (80)
and the second “lighter” source has
θlight = {r̃(t0 = 0) = 4.3, a = 1− 10−6, µ = 10M�,M = 2× 106M�,
φ0 = π,D = {Dedge = 1, Dface = 4}Gpc}. (81)
whereDedge andDface refer to the distance if each sourceis
viewed edge-on/face-on respectively. The distances
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16
are fine tuned6 so that we achieve a signal to noise ra-tio of ρ
∼ 20. This is discussed later in section V. Thelighter source is
sampled with sampling interval ∆ts ≈ 4seconds and the heavier one
with ∆ts ≈ 25 seconds. Wenote here that ∆ts = M∆t̃ where ∆t̃ is the
dimension-less sampling interval used to integrate (13). The
sam-pling interval is chosen from Shannon’s sampling theo-rem such
that ∆ts < 1/(2fmax), where
f edgemax =20
2π
Ω̃iscoM
, f facemax =2
2π
Ω̃iscoM
(82)
are the highest frequencies present in the waveform forthe
edge-on and face-on cases respectively. To illus-trate,
near-extremal waveforms with parameters θlightfor both edge-on and
face-on viewing angles are plottedin Fig.(7).
The lighter source is interesting because it exhibitsboth an
“inspiral" regime and a exponentially decayingregime that we will
refer to as “dampening". The heaviersource is interesting because
the dampening regime lastsmore than one year and so the signal is
in the dampeningregion for the entire duration of the observation.
In thenext section, we discuss detectability of these two typesof
sources by LISA.
V. DETECTABILITY
The LISA PSD reaches a minimum around 3mHz, andis fairly flat
within the band from 1 to 100mHz. For
an edge-on near-extremal inspiral with primary massof ∼ 107M�,
the dominant harmonic has a frequencyof ∼ 3.2mHz at plunge, while
the m = 20 harmonichas frequency of 64 mHz. Such heavy sources are
thusideal systems for observing the near-ISCO dynamics.For the
lighter mass considered, 2 × 106M�, the near-ISCO dynamics are at
frequencies a factor of 5 higher,where the LISA PSD starts to rise.
While the near-ISCO radiation will still be observable for these
systems,its relative contribution to the signal will be
relativelyreduced. We therefore expect to obtain more precisespin
measurements for the heavier of the two referencesystems.
The discrete analogue of the optimal matched filteringSNR
defined in Eq. (29)
ρ2 ≈ 4∆tsN
bN/2+1c∑i=0
|h̃(fi)|2
Sn(fi). (83)
Here N is the length of the time series, ∆ts thesampling
interval (in seconds) and fi = i/N∆ts arethe Fourier frequencies.
In Eq.(83), the discrete timeFourier transform (DTFT) h̃(fj) is
related to the CTFTthrough ĥ(f) = ∆ts ·h̃(f). To avoid problems
with spec-tral leakage, prior to computing the Fourier transform,we
smoothly taper the end points of our signals usingthe Tukey
window
w[n] =
12 [1 + cos(π(
2nα(N−1) − 1))] 0 ≤ n ≤
α(N−1)2
1 α(N−1)2 ≤ n ≤ (N − 1)(1− α/2)12 [1 + cos(π(
2nα(N−1) −
2α + 1))] (N − 1)(1− α/2) ≤ n ≤ (N − 1).
(84)
here n is defined through t̃n = n∆t̃. The tunable pa-rameter α
defines the width of the cosine lobes on eitherside of the Tukey
window. If α = 0 then our window isa rectangular window offering
excellent frequency reso-lution but is subject to high leakage
(high resolution).If α = 1 then this defines a Hann window, which
haspoor frequency resolution but has significantly reducedleakage
(high dynamic range). For the heavier source,we use α = 0.25 to
reduce leakage effects significantlyand frequency resolution is not
a problem since the fre-
6 Strictly speaking, distance here is not a physical
parametersince our waveform model does not include the LISA
responseto the strains h+ and h×.
quencies of the signal are contained within the LISA fre-quency
band (for all harmonics). For the lighter source,we use α = 0.05 to
reduce edge effects while retainingthe ability to resolve the
frequencies where the signal isdampened. We found that calculated
SNRs and param-eter measurement precisions are insensitive to the
choiceof α in the heavier system. The lighter system is
moresensitive: for larger α, more of the dampening regime islost,
with a corresponding impact on the measurementprecisions. We
believe that α = 0.05 is large enough toreduce leakage but small
enough to resolve as much ofthe dampening regime as possible.
After tapering, we zero pad our waveforms to an in-teger power
of two in length, in order to facilitate rapidevaluation of the
DTFT using the fast Fourier trans-
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17
0 50 100 150 200 250 300 350time [days]
1.0
0.5
0.0
0.5
1.0h +
1e 22 Face on: ( , ) = (0, 0)
0 50 100 150 200 250 300 350time [days]
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
h +
1e 22 Edge on: ( , ) = ( /2, 0)
Figure 7: A near-extremal waveform with parameters θlight viewed
face-on (left) and edge-on (right). Thedampening region lasts ∼ 55
days. The edge-on case is asymmetric due to the large number of l =
20 modes and
shows prominent relativistic beaming near the ISCO as observed
in figure 3.b) of [24].
form. Computing the SNR in this way gives ρ ∼ 20 forthe light
and heavy sources respectively when viewedboth edge-on and face-on
under the configuration of pa-rameters θlight and θheavy.
In all cases we marginally exceed the threshold of ρ ≈20 which
is typically assumed to be required for EMRIdetection in the
literature [4, 55].
As mentioned above, the lighter source exhibits tworegimes of
interest - the initial gradually chirping phase,where the waveform
resembles those for moderatelyspinning primaries, and then the
exponentially dampedphase while the secondary is in the
near-horizon regime.It is natural to ask what proportion of the
SNR, andlater what proportion of the spin measurement preci-sion,
is contributed by each regime. For both edge-onand face-on systems,
we separate the two parts of thewaveform using Tukey windows and
compute the SNRcontributed by each part to find
ρ2face-on ∼
{83% Outside Dampening region17% Dampening region.
(85)
ρ2edge-on ∼
{96% Outside Dampening region4% Dampening region.
(86)
For the face-on source, there is just a single dominantharmonic,
and the frequency of this harmonic is suchthat it lies in the most
sensitive part of the LISA fre-quency range. This helps to enhance
the relative SNRcontributed by the dampening region. The
edge-onsource, by contrast, has multiple contributing harmon-ics,
which are spread over a range of frequencies, andthe proportional
contribution of the dampening regionto the overall SNR is therefore
diminished.
For a non-evolving signal the SNR accumulates like√Tobs, where
Tobs is the total observation time. The pre-
dampening regime lasts 308 days, and so from durationalone we
would expect a fraction
√308/365 ≈ 93% of
SNR to be accumulated there. The difference to whatwe find above
is explained by differences in amplitudesof the individual
harmonic(s). The heavier system iswithin the dampening regime
throughout the last yearof inspiral and so all of the SNR of ρ ∼ 20
is accumu-lated there. This may seem counter-intuitive given
theexponential decay of the signal during the dampeningregime.
However, the exponential decay rate is rela-tively slow, a large
number of harmonics contribute tothe SNR and the emission is all
within the most sensitiverange of the LISA detector. This is clear
from lookingat the time-frequency spectrogram of the heavier
signalshown in Fig.(8). What we learn from this figure is thatthere
are a significant number of harmonics that havecomparable power to
the dominant m = 2 harmonic.We see also that the angular velocity
at each harmonic,and thus fm, shows little rate of change forM ∼
107 andη ∼ 10−6. This is consistent with [24, 32], where it
wasshown that a large number ofm harmonics is required toproduce an
accurate representation of the gravitationalwave signal for a
near-extremal EMRI, particularly fornear edge-on viewing angles.
For moderately spinningblack holes a ∼ 0.9 there are not as many
dominantharmonics, so those waveforms are cheaper to evaluate.
We are now ready to move on to compute FisherMatrix estimates of
parameter measurement precisions.This will be the focus of the next
section.
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18
Figure 8: Here we plot the spectrogram of h(θheavy; t) viewed
edge on. We see 20 tracks in the time-frequencyplane corresponding
to the m ∈ {1, . . . , 20} harmonics. The colorbar shows that the m
= 2 harmonic (second
lowest track in frequency) is dominant, but that there are
several other harmonics which contribute significantlyto the
radiated power
.
VI. NUMERICS: FISHER MATRIX
We now compute (33) numerically without mak-ing the simplifying
assumptions used in Sections III B
and III C. We will use one simplification, which isto ignore the
spin dependence in Ė , Z∞lm(r̃, a) and−2S
amΩ̃lm (θ, φ) and fix these at the values computed for
a = 1 − 10−9 using the BHPT. We argued in Eq. (54)
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19
that the spin dependence of the flux correction is a
sub-dominant contribution in the near-ISCO regime, andthis is
further justified in Appendix B (see Fig. 16 inparticular). While
∂aĖ does grow as the ISCO is ap-proached, it remains sub-dominant
to the spin depen-dence of the kinematic terms. This approximation
isprobably conservative in the sense that we are
removinginformation about the spin from the waveform modeland so
the true measurement precision is most likelyhigher. Nonetheless we
expect this to be a small effect,and have verified that relaxing
this assumption does notsignificantly change the result for the
heavier referencesource (see Figure 9). We note that we make this
as-sumption only for computational convenience. Wave-form models
used for parameter estimation on actualLISA data should use the
most complete results avail-able to ensure maximum sensitivity and
minimal pa-rameter biases.
To compute the waveform derivatives required to eval-uate (33),
we use the fifth order stencil method
∂f
∂x≈ −f2 + 8f1 − 8f−1 + f−2
12δx, (87)
for δx � 1 and fi = f(x + iδx). To avoid numericalinstability of
∂ah for the near-extremal spin values ofa ≤ 1 − 10−9, we ensure
that δx < 1 − a so the per-turbed waveform does not have spin
exceeding a = 1.We further assume that ∂at̃end is zero so there is
no spindependence on the total observation time.
In addition to the sources with parameters θheavy andθlight, we
now consider a third source with parameters
θmod = {r̃(t0 = 0) = 5.01, a = 0.9, µ = 10M�,M = 2 · 106M�, φ0 =
π,Dedge = 1Gpc}, (88)
with SNR ∼ 20.Fisher matrix estimates of parameter
measurement
precisions for all three sources viewed edge-on are shownin
Figure 9. We do not present the results for a face-onobservation as
they are near-equivalent to the measure-ments presented in figure 9
for equivalent SNR.
We see from this figure that we should be able toconstrain the
spin parameter of near-extremal EMRIsources to a precision as high
as ∆a ∼ 10−10, evenwhen accounting for correlations amongst the
waveformparameters. This is true for both the lighter and
theheavier sources viewed edge-on and face-on, with a con-straint a
factor of a few better for the heavier source.The right panel of
the figure compares the contribu-tion to the measurement precision
for the lighter sourcefrom the two different phases of the signal.
We see thatthe high spin precision comes almost entirely from
theobservation of the dampening regime and this phaseof the signal
contributes much more information than
we would expect based on its contribution to the totalSNR7.
The spin measurement precision for the near-extremalsystems is
three orders of magnitude better than for thesystem with moderate
spin, while all other parametermeasurements are comparable.
Comparing to the exact Fisher matrix result with spindependence
included in all the various terms, we see thatthe two precisions
are almost identical : the exact resultoffers precisions that are
marginally better in compar-ison to our approximate result
(removing spin depen-dence from the corrections). This Figure thus
justifiesignoring the spin dependence of Ė , since relaxing
thatassumption makes almost no difference to the results.This
numerically confirms our belief that the spin de-pendence in the
corrections to the fluxes are subdomi-nant in the analysis leading
to (40). In the same plot 9,we also compare results of
near-extremal black holes tomoderately spinning holes. A direct
comparison showsan increase in the spin precision by ∼ 3 orders of
magni-tude, which agrees with the intuition given by the
earlieranalytic analysis, Eq. (79).
To our knowledge, these are the first circularand equatorial
parameter precision studies for EMRIsthat have employed
Teukolsky-based adiabatic wave-forms, rather than approximate
waveform models (or“kludges”), which have been used for many
studies [3, 4,15]. Comparing our results for the moderately
spinningsystem to these previous studies, we find that our re-sults
are very comparable, but a factor of a few tighter.This could be
because we are including only a subsetof parameters and ignoring
the details of the LISA re-sponse, or because we have a more
complete treatmentof relativistic effects. A more in depth study
addressingboth of these limitations would be needed to
understandthe origin of the differences. However, the
agreementbetween our results and previous studies is
sufficientlyclose, and considerably less than the difference we
findbetween the moderate and near-extremal spin cases, togive us
confidence that our results are not being undulyinfluenced by these
simplifications.
In Figure (10) we show how the parameter estimationprecision for
the source with parameters θlight changesas we vary the spin
parameter, while keeping all otherparameters unchanged. We present
results for bothface-on and edge-on viewing angles. This shows
thatwhile the measurement precision for most of the param-eters is
largely independent of spin in the near-extremalregime, the spin
precision steadily increases as a → 1.We note that even at a spin
of 1 − 10−9, the measure-ment precision satisfies the constraint ∆a
< |1− a| and
7 In (40), the growth of ∂ar̃ exceeds the growth of Sn(f) ∼
constin the dampening regime. This sources the high precision
mea-surement.
-
20
a M 0 D r0Parameter uncertainties
10 9
10 7
10 5
10 3
10 1
101M
agni
tude
(log
scal
e)Comparison of parameter precisions [edge-on]
M 106, (Exact) Moderate spinM 106, (Approx) High spinM 107,
(Approx) High spinM 107, (Exact) high spin
a mu M 0 D r0Parameter uncertainties
10 8
10 6
10 4
10 2
100
Mag
nitu
de (l
og sc
ale)
Comparison: Inspiral and Dampening [edge-on]Full InspiralUp to
DampeningThroughout Dampening
Figure 9: (Left plot) Parameter measurement precision, as
estimated using the Fisher Matrix formalism, for thethree reference
sources, with parameters θlight (green diamonds), θheavy (purple
crosses) and θmod (blue asterisks).
The black diamonds show the precisions obtained when including
the spin-dependence of the relativisticcorrections, Ė in the
waveform model for the heavy source. (Right plot) Parameter
measurement precisions for thesource with parameters θlight,
computed using the full waveform (blue asterix), only the inspiral
phase (blue dot)
and only the dampening phase (green diamond).
therefore a LISA EMRI observation would be able to re-solve that
the system was not maximally extremal, i.e.,that a < 1. We stop
at 1 − 10−9 since the derivativeusing Eq.(87) begins to
misbehave.
Due to large condition numbers, inverting Fisher ma-trices for
EMRI sources is a highly non-trivial task. Inappendix C, we provide
multiple diagnostic tests of ourFisher matrix algorithm and verify
that, in the single pa-rameter case, the spin parameter precision
is a suitablerepresentation of the 1σ width of the Gaussian
likeli-hood as shown in figure 18. These single parameter testsof
the Fisher matrix are useful tests to verify that a sin-gle
parameter algorithm yields sensible results. How-ever, real
instabilities of the numerical procedure areprominent the moment
the inverse of the Fisher matrixis performed when correlations are
present. Hence, it isboth necessary and sufficient to verify our
Fisher matrixcalculations using an independent procedure. The
nextsection is dedicated to performing a parameter estima-tion
study on both near-extremal EMRIs with parame-ters θlight and
θheavy.
VII. NUMERICS: MARKOV CHAIN MONTECARLO
The Fisher Matrix is a local approximation to thelikelihood,
valid in the limit of sufficiently high signal-to-noise ratio. We
can verify that this local approxima-tion is correctly representing
the parameter measure-ment uncertainties by numerically evaluating
the likeli-hood using Markov Chain Monte Carlo. To reduce
thecomputational cost of these simulations we use a face-on viewing
profile and thus only consider the m = 2harmonic. We have shown in
figure (10) that param-eter precision measurements are not largely
dependent
on the choice of viewing angle for the lighter source. Wehave
further verified this claim for the heavier source.
A. Markov Chain Monte Carlo
Markov Chain Monte Carlo (MCMC) methods weredeveloped for
Bayesian inference to sample from the pos-terior probability
distribution, p(θ|d), which is given byBayes’ theorem as
log p(θ|d) ∝ log p(d|θ) + log p(θ) (89)
where p(d|θ) is the likelihood function, and p(θ) is theprior
probability distribution on the parameters. In ourcontext the
likelihood is given by Eq. (27) and we willassume independent
priors such that
log p(θ|d) ∝ −12
(d− h(t;θ)|d− h(t;θ)) +∑θi∈θ
log p(θi).
(90)We generate a data set d(t) = h(t;θtr)+n(t) by specify-ing
the waveform parameters, θtr, of the injected signaland generating
noise in the frequency domain
ñ(fi) ∼ N(0, σ2(fi)), σ2(fi) ≈NSn(fi)
4∆t. (91)
We use MCMC to sample from the posterior distribu-tion (90),
employing a standard Metropolis algorithmwith proposal distribution
q(θ?|θi−1) equal to a multi-variate normal distribution, centred at
the current pointand with a fixed covariance. We take flat priors
on allof the waveform parameters, since the goal is to checkthe
validity of the Fisher matrix approximation to thelikelihood. The
algorithm proceeds as follows
-
21
4 5 6 7 8 9log10(1 a)
10
8
6
4
2
log 1
0(/
)Parameter Precisions - face on: Varying Spin
a/a/
M/M0/ 0
D/Dr0/r0
4 5 6 7 8 9log10(1 a)
12
10
8
6
4
2
log 1
0(/
)
Parameter Precisions - edge on: Varying Spin
a/a/
M/M0/ 0
D/Dr0/r0
Figure 10: We keep θlight\{a} fixed and vary a = 1− 10−i for i ∈
{4, . . . , 9} while computing estimates on theprecision of the
measured parameters using the Fisher Matrix. Results are shown for
sources viewed face-on (left)
and edge-on (right).
1. We start the algorithm close to the true valuesθ0 = θtr + δ
for ||δ|| � 1. For iteration i =1, 2, . . . , N
2. Draw new candidate parameters θ? ∼ q and gen-erate the
corresponding signal template h(t;θ?).
3. Using (90), compute the log acceptance probabil-ity
log(α) = min[0, logP (θ?|d,θi−1)− logP (θi−1|d,θ?)].
We note that we are using a symmetric proposaldistribution and
so the usual proposal ratio is notrequired.
4. Draw u ∼ U [0, 1].
(a) If log u < logα we accept the proposed pointand set θi =
θ?.
(b) Else we reject the proposed point and setθi = θi−1.
5. Increment i→ i+ 1 and go back to step 2 until Niterations
have been completed.
Since we know the true parameters we can start thealgorithm in
the vicinity of the true parameters anddo not need to discard the
initial samples as burn-in,allowing us to generate useful samples
more quickly.
In principle, the MCMC algorithm should convergefor any choice
of proposal distribution, but propos-als that more closely match
the shape of the posteriorshould lead to more rapid convergence. As
we expectthat the proposal should be approximated by the
FisherMatrix, we set the covariance matrix of the normal pro-posal
distribution to be equal to the inverse Fisher ma-trix, evaluated
at the known injection parameters.
B. Results
We compute MCMC posteriors for the two signalsh({θheavy,θlight};
t) for the waveform model (20) for theface-on case only. As before,
we construct waveformsignoring the spin dependence in Ė , the
Teukolsky am-plitudes Z∞lm and the spheroidal harmonics −2S
aΩ̃lm (θ, φ).
We evaluate these for a fixed spin parameter of a =1− 10−9.
We remind the reader the main drive for the tightconstraints on
the spin parameter is due to the spin de-pendence induced through
the kinematic terms presentin (20), as discussed in section VI, and
not its dynamicalterms, justifying our approximation.
The priors on a, φ0 and D for both sources were
a ∼ 1− U [10−4, 10−8]φ0 ∼ U [0, 2π]D ∼ U [1, 8]Gpc.
The priors on µ,M and r̃0 were chosen differently forthe heavy
and light source as
µheavy ∼ U [18, 22]M�µlight ∼ U [8, 12]M�
Mheavy ∼ U [0.9, 1.1]× 107M�Mlight ∼ U [1.9, 2.1]× 106M�.
r̃heavy0 ∼ U [1.2, 1.3]
r̃light0 ∼ U [4.2, 4.4]
The prior on a ensures that we do not move outsidethe range in
which our approximations are valid, a &0.9999. The tight priors
on the individual componentmasses helped to improve the
computational efficiencyof our algorithm. However, there was no
evidence ofthe MCMC chains reaching the edges of the priors inour
simulations, so we are confident these restrictionsare not
influencing the results.
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22
Evaluating the likelihood for EMRI waveforms is anexpensive
procedure. In order to obtain a sufficientnumber of samples from
the posterior, we used high per-formance computing facilities and
ran 20 unique chainsfor N = 40, 000 iterations. All chains analysed
the sameinput data set, but with different initial random
seeds.This ensures that the dynamics of the chains are dif-ferent
but the noise realisations are the same for eachMCMC procedure.
The marginal posterior distributions and two-dimensional contour
plots for the two sources are shownin figures (11) and (12). These
plots confirm the highprecisions of parameter measurements that
were seenwith the Fisher Matrix. The relative uncertainties ∆θ/θare
similar for the two sources, although we can mea-sure the spin
parameter more precisely for the heaviersource. For the most part
the posteriors are unimodal,apart from the spin posterior of the
lighter source. Wehave verified that the secondary modes are real
featuresof the likelihood, and correspond to the waveform
phaseshifting by one cycle within the late dampening regime.We also
note that shifts in the peak of the posterior awayform the true
value are larger for the heavier source thanfor the lighter source.
This appears to be due to the par-ticular noise realisation. For
other noise realisations thenoise-induced biases for the heavier
source are smaller.For noise-free data sets, we find posterior
distributionspeaked at the true parameters, as expected.
The primary reason for doing the MCMC simulationswas to verify
the Fisher Matrix results found earlier. Infigures 13 and 14, we
plot the marginalised posteriors onthe parameters {θlight,θheavy}
alongside a Gaussian dis-tribution with variance given by the
Fisher matrix andcentred at the mean value of the posterior
distributionsp(θ|d).
These results nicely confirm the accuracy of the Fishermatrix
results for these sources. In each case, the1σ precision predicted
by the Fisher matrix is slightlysmaller than the width of the
numerically computed pos-terior. This is to be expected as the
Fisher matrix alsoprovides the Cramer-Rao lower bound on parameter
un-certainties. However, the difference is very small. Weare thus
confident that all of our Fisher matrix predic-tions are accurate,
including the exploration of param-eter space shown in Figure 10.
We conclude that evenat the near-thr