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

∗ ollie.burke@aei.mpg.de

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

3

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.

http://bhptoolkit.orghttp://bhptoolkit.orghttp://bhptoolkit.orghttp://bhptoolkit.orghttp://bhptoolkit.org

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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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6

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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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.

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.

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

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

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)

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-

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-

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.

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.

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