PHYSICAL REVIEW D 102, 124054 (2020)
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. Edwards2,3 1Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam-Golm 14476, Germany 2School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom 3Department of Statistics, University of Auckland, 38 Princes Street, Auckland 1010, New Zealand
(Received 12 October 2020; accepted 24 November 2020; published 22 December 2020)
We describe a model that generates first order adiabatic extreme mass ratio inspiral waveforms for quasi- circular equatorial inspirals of compact objects into rapidly rotating (near-extremal) black holes. Using our model, we show that LISA could measure the spin parameter of near-extremal black holes (for a ≳ 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 astrophysical parameter within a gravitational wave context. Our results are primarily based off a Fisher matrix analysis, but are verified using both frequentist and Bayesian techniques. We present analytical arguments that explain these high spin precision measurements. The high precision arises from the spin dependence of the radial inspiral evolution, which is dominated by geodesic properties of the secondary orbit, rather than radiation reaction. High precision measurements are only possible if we observe the exponential damping of the signal that is characteristic of the near-horizon regime of near-extremal inspirals. Our results demonstrate that, if such black holes exist, LISA would be able to successfully identify rapidly rotating black holes up to a ¼ 1 − 10−9, far past the Thorne limit of a ¼ 0.998.
DOI: 10.1103/PhysRevD.102.124054
I. INTRODUCTION the LISA frequency band for several years prior to plunge, so modeling the full observable signal is a complex task [2]. Extreme mass ratio inspirals (EMRIs) are one of the most EMRI orbits are expected to be both eccentric and inclined exciting possible sources of gravitational radiation for the even up to the last few cycles before plunging into the space-based detector LISA [1], but also one of the most primary black hole. For these reasons, EMRIs pose a challenging to model and extract from the data stream. An – EMRI involves the slow inspiral of a stellar-origin compact challenging problem for both waveform modelers [2 4] and data analysts. object (CO) of mass μ ∼ 10 M⊙ into a massive black hole in the center of a galaxy. For a central black hole with mass This same complexity also makes EMRIs one of the ð5−7Þ richest sources of gravitational waves. Typically an EMRI M ∼ 10 M⊙, EMRIs emit gravitational waves (GWs) in 5−7 will be observable for 1=ðmass ratioÞ ∼ 10 cycles before the mHz frequency band and so are prime sources for the plunge and the emitted gravitational waves thus provide a LISA detector. EMRIs begin when, as a result of scattering very precise map of the spacetime geometry of the primary processes in the stellar cluster surrounding the massive hole [5–8]. Through accurate detection and parameter black hole, the CO becomes gravitationally bound to the inference, one can conduct tests of general relativity to primary. The subsequent inspiral of the CO toward the very high precision [6,9]. horizon of the primary is driven by radiation reaction It is well known that the information about the source is through the emission of gravitational waves. EMRI wave- carried through the time evolution of the phase in a forms are very complicated and EMRIs can be present in gravitational wave [10,11]. The slow evolution of EMRIs means that a large number of cycles can be observed during *[email protected] the inspiral, which will provide constraints on the parameters of the source with remarkable precision [12,13]. Previous Published by the American Physical Society under the terms of work has indicated that LISAwill be able to place constraints the Creative Commons Attribution 4.0 International license. on the dimensionless Kerr spin parameter, a, of the primary Further distribution of this work must maintain attribution to 104 the author(s) and the published article’s title, journal citation, black hole in an EMRI, at the level of 1 part in for and DOI. Open access publication funded by the Max Planck moderately spinning, a ∼ 0.9, primaries [3,4,14,15]. In this Society. paper, we explore how well LISAwill be able to measure the
2470-0010=2020=102(12)=124054(29) 124054-1 Published by the American Physical Society BURKE, GAIR, SIMÓN, and EDWARDS PHYS. REV. D 102, 124054 (2020) spin parameter for very rapidly rotating black holes, i.e., emission of gravitational radiation [35]. This decrease in systems in which the spin parameter is close to the maximum eccentricity continues until the orbit reaches a critical value allowed by general relativity. radius at which is starts to increase again [36,37]. The Super massive black holes with a large spin parameter critical radius moves closer to the last stable orbit as the are abundant throughout our universe. Observations indi- spin parameter increases and for near-extremal systems is cate that massive BHs reside in the centers of most galaxies, located within the regime where transition from inspiral to where these black holes are known to accrete matter and plunge occurs [38,39]. Additionally, the increase in eccen- hence are predicted to have very high spins [11,16–20]. The tricity is a subdominant effect throughout the transition dimensionless Kerr spin parameter of a Kerr black hole, regime [40]. As the spin increases, we therefore expect that cannot exceed 1, since the resulting spacetime contains a for an object captured at a fixed radius, the amount of naked singularity no longer encased within a well-defined eccentricity dissipated before the critical radius increases, horizon. Thorne [21] showed that a moderately spinning and the eccentricity gained after the critical radius black hole cannot be spun up by thin-disc accretion above a decreases. Therefore, even in the standard capture picture spin of a ≈ 0.998. However, in principle primordial black it is reasonable to assume the eccentricity is small at the end holes could be formed with spins exceeding that value [22]. of the inspiral. We will show in this paper that very precise “Near-extremal” black holes with spins close to the limit of measurements of spin for near-extremal systems are pos- a ¼ 1 have interesting properties and we focus our atten- sible, but this precision comes from observation of features tion on these here. [24] in the final phase of the inspiral, which is where the This past decade, researchers [23–32] have explored the near-circular assumption is most likely to be valid. rich properties of near-extremal EMRIs. The gravitational The main results of the paper are given in Figs. 9 and 10 radiation emitted from these systems is unique, and would in Sec. VI. Readers who wish to understand why near- prove a smoking gun for the existence of these near- extremal systems offer greater precision spin measurements extremal systems (see [24]). In this paper, we show than moderate spin systems should direct their attention to qualitatively that the inspiralling dynamics of the compact Sec. III. object into an near-extremal massive black hole is very This paper is organized as follows. In Sec. II, we set different from that into a moderately spinning black hole, notation and discuss the trajectory of a compact object on a and these differences are reflected in the emitted gravita- circular and equatorial orbit around a near-extremal Kerr tional waves. As such, in order to detect and correctly BH. In Sec. III, we show that the spin dependence of perform parameter estimation on these near-extremal kinematical quantities appearing in the radial evolution sources, it is essential to update our family of waveform rather than radiation-reactive effects dominate the spin models to include them. We will argue throughout this precision measurements for near-extremal EMRI systems. work that, if observed, near-extreme black holes offer Our Teukolsky based waveform generation schemes are significantly greater precision measurements on the Kerr outlined in Sec. IV. We discuss prospects for detection in spin parameter than moderately spinning systems. In Sec. V, arguing that LISA is more sensitive to heavier mass 7 6 particular, LISA will have the capability to successfully systems M ∼ 10 M⊙ than lighter systems M ∼ 10 M⊙. conclude whether the central object in an EMRI system is Our Fisher Matrix results are presented in Sec. VI. Here we truly a near-extremal black hole. Thus, if near-extremal show that we can constrain the spin parameter Δa ∼ 10−10, black holes exist, LISA observations of EMRIs may be one even when correlations among other parameters are taken of the best ways to find them. into account. Finally, in Sec. VII, we perform a Bayesian In this paper we will consider only EMRIs on circular analysis to verify our Fisher matrix results, before finishing and equatorial orbits around near-extremal primary black with conclusions and outlooks in Sec. VIII. holes. This choice is made primarily for computational convenience, but there are also astrophysical scenarios that II. BACKGROUND produce such systems. As discussed in [33], compact objects can form within accretion disks around massive We consider the inspiral of a secondary test particle of black holes. When these objects fall into the central black mass μ on a circular, equatorial orbit around a primary hole, the resultant EMRI will be circular and equatorial. super massive Kerr black hole with mass M and Kerr spin Super-Eddington accretion can provide a means to spin up parameter a ≲ 1 where the mass ratio is assumed small a black hole past the Thorne limit [34], and so it is not η ¼ μ=M ≪ 1. The secondary is on a prograde orbit unreasonable to expect that this EMRI formation channel aligned with the rotation of the primary black hole with would be more important for near-extremal systems. The a>0 and dimensionful angular momentum L>0. Unless standard EMRI formation channel, involving capture of a stated otherwise, throughout this paper any quantity with an compact object via scattering interactions, tends to form overtilde is dimensionless, e.g., r˜ ¼ r=M and t˜ ¼ t=M etc. EMRIs with moderate initial eccentricities. However, The one exception is the dimensionless spin parameter, this eccentricity decreases during the inspiral due to the which we denote by a without a tilde. Quantities with an
124054-2 CONSTRAINING THE SPIN PARAMETER OF NEAR-EXTREMAL … PHYS. REV. D 102, 124054 (2020) overdot will denote coordinate time derivatives, e.g., Circular orbits exist only outside the innermost stable r_ ¼ dr=dt. We use geometrized units such that G ¼ c ¼ 1. circular orbit (ISCO). For radii smaller than the ISCO, the In Boyer-Lindquist [41] coordinates, the metric of a Kerr secondary will start to plunge toward the horizon of the black hole for θ ¼ π=2 is given by primary. The ISCO for equatorial orbits is at [43] 2 2 ˜ 3 − 3 − 3 2 1=2 2 r˜ 2a risco ¼ þ Z2 ½ð Z1Þð þ Z1 þ Z2Þ ð7aÞ g ¼ − 1 − dt˜2 þ dr˜2 þ r˜2 þ a2 þ dϕ2 r˜ Δ˜ r˜ 2 1=3 1=3 1=3 Z1 ¼ 1 þð1 − a Þ ½ð1 þ aÞ þð1 − aÞ ð7bÞ 4a − dtd˜ ϕ; ð1Þ r˜ 2 2 1=2 Z2 ¼ð3a þ Z1Þ : ð7cÞ where Δ˜ ¼ r˜2 − 2r˜ þ a2 and a is the dimensionless spin For near extremal orbits, using (3) and (7), and expand- parameter introduced earlier. This is related to the mass, M, ing for ϵ ≪ 1, we obtain and angular momentum, J, of the Kerr black hole via a ¼ ∈ 0 1 J=M and lies in the range a ½ ; . The event horizon is ˜ 1 21=3ϵ2=3 O ϵ4=3 risco ¼ þ þ ð Þ; ð8Þ located on the surface defined by Δ˜ ¼ 0, when pffiffiffiffiffiffiffiffiffiffiffiffiffi and deduce ˜ 1 1 − 2 rþ ¼ þ a : ð2Þ ˜ − ˜ O ϵ2=3 ϵ ≪ 1 jrisco rþj¼ ð Þ; for : ð9Þ Introducing an extremality parameter ϵ ≪ 1 The radial coordinate separation between the ISCO and pffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ 1 − 2 horizon is determined by the spin parameter. In the limit, ¼ a ; ð3Þ ϵ → 0 ˜ → ˜ → 1 , then risco rþ in Boyer-Lindquist coordinates. the event horizon is at A. Radiation reaction r˜ ¼ 1 þ ϵ: ð4Þ To compute circular and equatorial adiabatic inspirals, a þ detailed knowledge of the radial self-force is required (see, for example, [44] for a detailed review). In this paper, we The trajectory of the secondary confined to the equatorial will work at leading order, including the radiative (dis- plane of a central Kerr hole is governed by the Kerr sipative) part of the radial self-force at first order, but geodesic equations [42] neglecting first order conservative effects and all second order in mass-ratio effects. The first order dissipative force dr˜ 2 ˜2 ˜ ˜2 2 − ˜ 2 − Δ ˜ − ˜ 2 − ˜2 can be computed by solving the Teukolsky equation [45]. r τ˜ ¼½Eðr þ a Þ aL ½ðL aEÞ r d The rate of emission of energy is given by ϕ ˜2 d − ˜ − ˜ a ˜ ˜2 2 − ˜ r τ˜ ¼ ðaE LÞþ˜ ðE½r þ a aLÞ ˜_ ˜_ ˜_ ∞ ˜_ H d r h−Ei¼hEGWi¼hE iþhE i ˜ ˜2 2 2 dt ˜ ˜ r þ a ˜ 2 2 ˜ X∞ Xl r˜ ¼ −aðaE − LÞþ ðE½r˜ þ a − aLÞ; ˜_ ∞ ˜_ H dτ˜ Δ ¼ 2 ðhElmiþhElmiÞ: ð10Þ l¼2 m¼1 in which τ˜ denotes the proper-time coordinate for the ˜_ t −1 inspiraling object. The dimensionless conserved quantities where hEGWi¼h−ðu Þ Fti, h·i denotes coordinate time t E˜ ¼ E=μ and L˜ ¼ L=ðMμÞ are related to the energy, E, and averaging over several periods of the orbit. The quantity u angular momentum, L, measured at infinity. For the is the t component of the four velocity and Ft the t circular and equatorial orbits considered here, the energy component of the gravitational self-force at first order in the η η ≪ 1 E˜ and angular frequency Ω˜ can be expressed analytically mass ratio . This expression is valid only if , that is, when orbits evolve adiabatically such that the timescale on 3 2 which the orbital parameters evolve is much longer than the 1 − 2=r˜ þ a=˜ r˜ = E˜ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ orbital period. 3=2 _ ∞ _ H 1 − 3=r˜ þ 2a=r˜ The fluxes hE˜ i and hE˜ i denote the (dimensionless and orbit averaged) dissipative fluxes of gravitational radiation 1 Ω˜ emitted toward infinity and toward the horizon respectively. ¼ 3 2 ; ð6Þ r˜ = þ a From here on, we shall drop the angular brackets hE_ i → E_ , to avoid cumbersome notation. The quantities jmj ≤ l are ˜ in which the dimensionless angular frequency Ω is defined angular multiples which appear in the decomposition of through Ω ¼ Ω˜ =M ¼ dϕ=dt. the emitted radiation into a sum of spheroidal harmonics.
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˜_ þ TABLE I. NHEK fluxes at the ISCO computed using the approximations Eq. (11) (denoted ENHEK) and Eq. (12) ˜_ ˜_ (denoted ENHEK), and computed exactly using BHPT (denoted EExact and based on the first thirty m and l modes).
a ˜_ η ˜_ þ η ˜_ η ˜_ þ − ˜_ η ˜_ − ˜_ η EExact= ENHEK= ENHEK= jENHEK EExactj= jENHEK EExactj= 1 − 10−5 0.0264197 0.0261523 0.0300885 0.0002674 0.0036688 1 − 10−6 0.0129344 0.0125200 0.0137455 0.0004143 0.0008111 1 − 10−7 0.0061516 0.0059484 0.006333 0.0002031 0.0001814 1 − 10−8 0.0028875 0.0028082 0.0029294 0.0000793 0.0000419 1 − 10−9 0.0013472 0.0013193 0.0013575 0.0000280 0.0000103 1 − 10−10 0.0006273 0.0006176 0.0006296 0.0000097 0.0000023 1 − 10−11 0.0002915 0.0002883 0.0002922 0.0000031 0.0000007 1 − 10−12 0.0001354 0.0001344 0.0001356 0.0000009 0.0000002
_ 1 The components of the fluxes E˜ are obtained by numerical solving the Teukolsky equation in the NHEK region. Our solution of the Teukolsky equation sourced by a point numerical analysis based on the BHPT, suggests the spin particle (the secondary). There exists an open source code dependence of certain observables, to be discussed in in the black hole perturbation toolkit (BHPT) [46] to do Sec. III B, is better captured by (12). Table I compares ˜ this for circular and equatorial orbits—specifically the the flux at risco computed using BHPT to that obtained from Teukolsky package. the near-extremal approximations of Eq. (11) and Eq. (12). The enhancement of symmetry in the near horizon This table corroborates that (12) is a good approximation to → 1 geometry of extreme Kerr [47] provides an additional tool the total energy flux, particularly in the limit as a , _ to compute the fluxes E˜ analytically from first principles. where it outperforms the full expression, (11). See [23–32] for a description of this work. For circular equatorial orbits near the horizon of a near-extremal black B. Inspiral and waveform hole, there is a remarkably simple approximation for the The radial evolution of the secondary can be found by total flux [25], which takes the form taking a coordinate time derivative of the circular energy ˜ − ˜ relation (5) ˜_ NHEK η ˜ ˜ ˜ − ˜ ˜ r rþ ≪ 1 EGW ¼ ðCH þ C∞Þðr rþÞ=rþ; ˜ : ð11Þ rþ dr˜ P − GW 13 ˜ ¼ ∂ ˜ ð Þ ˜ ˜ dt r˜E The quantities CH and C∞ are constants representing the emission toward the horizon and infinity respectively. ≔ ˜_ where we defined PGW EGW. As the ISCO is These constants are given analytically in Eqs. (76) and ˜ approached, the denominator ∂r˜E tends to zero, marking (77) of [25] and codes in the BHPT can be used to evaluate a break down of the quasicircular approximation. The ODE them. Numerically evaluating them and summing the (13) is easily numerically integrated given an expression for ≤ 30 ˜ ≈ contribution of the first jmj l ¼ modes gives CH the flux P . ˜ GW 0.987 and C∞ ≈−0.133. Equation (11) is useful when The outgoing gravitational wave energy flux measured at working within the near-horizon geometry of the rapidly infinity has a harmonic decomposition [11] rotating hole, but it breaks down far from the horizon and _ ∞ 2 2 3 _ ∞ extra terms would be required to compute reliable fluxes. E˜ ¼ A ηΩ˜ þ m= E ; ð14Þ All the numerical work presented in this paper, which is m m m found in Sec. Vonwards, will use the exact fluxes obtained where from BHPT. However, to understand our numerical results, we develop a set of new analytic tools in 2ðm þ 1Þðm þ 2Þð2mÞ!m2m−1 Secs. III A and III C. These will partially make use of Am ¼ ; ð15Þ ðm − 1Þ½2mm!ð2m þ 1Þ!! 2 the leading contribution to (11) _ ∞ and ð2m þ 1Þ!! ¼ð2m þ 1Þð2m − 1Þ…3 · 1. Here E is ˜_ NHEK ˜ ˜ m EGW ≈ ηðCH þ C∞Þx; x ¼ r˜ − 1 ≪ 1: ð12Þ the relativistic correction to the Newtonian expression for the flux in harmonic m. This differs from (11) by OðϵÞ contributions since r˜ − 1 measures the BL radial distance to the extremal horizon and 1 This follows by measuring the˜ radial˜ distance to the extremal ˜ rþþr− not the radial distance to the near-extremal horizon rþ. The horizon by λ, defined through r˜ ¼ 2 þ λr˜, and then taking the approximation (12) can be derived from first principles by decoupling limit λ → 0.
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In this work, we shall consider two different waveform omit these here and, consequently, they will also be omitted models. For the analytic discussion in Sec. III, we will use in our analytic discussion based on this waveform model. the waveform model in [11], whereas for the numerical Let us now review the full Teukolsky based waveform analysis in later sections, we will use the full Teukolsky model that we will use in our numerical study. This is based waveform. given by Let us first review the main features of the model X discussed in [11] for the waveform observed by the detector μ 1 ˜ h − ih ¼ G expð−i½ϕ0 þ mΩ t˜ Þ ð20Þ in the source frame. This model is written þ × ˜ 2Ω˜ 2 ml D ml m X∞ ˜; θ ≈ 2π ˜ ˜ ϕ hðt Þ ho;m sinð fmt þ 0Þ: ð16Þ where m¼2 amΩ˜ ∞ Gml ¼ −2S ðθÞ expðiϕÞZ ðr;˜ aÞð21Þ Some remarks are in order. First, we ignore the m ¼ 1 ml ml contribution since, as argued in [11], this is subleading to depends on the radial Teukolsky amplitude at infinity, ≥ 2 ∞ ptheffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim contributions. Second, the amplitude ho;m ¼ Z ðr;˜ aÞ, and the viewing angle ðθ; ϕÞ. The latter depend- 2 2 ml hhþm þ h×mi corresponds to the root mean square ence is through the spin-weight minus 2 spheroidal (RMS) amplitude of gravitational waves emitted toward amΩ˜ θ ϕ amΩ˜ θ ϕ harmonics −2Sml ð ; Þ¼−2Sml ð Þ expði Þ. This work infinity in harmonic m. These are averaged h·i over the θ ϕ 0 0 2 will consider two viewing angles: face on ð ; Þ¼ð ; Þ viewing angle and over the period of the waves. Third, the and edge on ðθ; ϕÞ¼ðπ=2; 0Þ. Using the identities oscillatory phase depends on the initial phase ϕ0 and ˜ ˜ the frequency fm of each waveform harmonic is given by að−mÞΩ π 2 0 −1 l ¯ amΩ˜ π 2 0 −2Sð−mÞl ð = ; Þ¼ð Þ −2Sml ð = ; Þð22Þ m f˜ ¼ Ω˜ : ð17Þ m 2π ∞ −1 l ¯ ∞ Zð−mÞl ¼ð Þ Zml ð23Þ The relation between the RMS amplitude and the outgoing radiation flux in harmonic m is where barred quantities are complex conjugates, we can qffiffiffiffiffiffiffiffiffi write Eq. (20) as _ ∞ 2 ηE˜ X∞ X∞ m 2μ 1 ˜ ho;m ¼ ˜ ˜ ð18Þ h ¼ expð−i½ϕ0 þ mΩ t˜ Þ G ; ð24Þ mΩ D þ D 2Ω˜ 2 ml m¼1 m l¼m where D˜ ¼ D=M is the distance to the source from earth. for the edge-on case, and as Using Eq. (14), we can rewrite ho;m as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ pffiffiffi − ≈ − ϕ 2Ω˜ ˜ 8 1 2 2 ! 2m−1 η h ih× G22 expð i½ 0 þ t Þ; ð25Þ ðm þ Þðm þ Þð mÞ m _ ∞ 3 þ 4Ω˜ 2 h ¼ E Ω˜ m= ð19Þ D o;m ðm − 1Þ½2mm!ð2m þ 1Þ!! 2 m D˜ for the face-on case. Note we have neglected higher order l for m ≥ 2. We note that the effect of the averaging is that modes with m ¼ 2 fixed in the last equation since the ∞ 2 this waveform model does not represent the waveform Teukolsky amplitudes Zl2 for l> are negligible in com- measured by any physical observer. However, it captures parison to the dominant quadrupolar l ¼ m ¼ 2 mode. the main physical features of the waveform which encode Figure 1 in [25] further justifies our claim that higher order information about the source parameters. m modes when l ¼ 2 can be ignored for face-on sources. Given the nature of our orbits, our parameter space will Furthermore, the only spheroidal harmonics that are non- ˜ only be six dimensional θ ¼fr˜0;a;μ;M;ϕ0; Dg, where r˜0 vanishing at θ ¼ 0 are those with m ¼ −s,orm ¼ 2 [50,51]. stands for the initial size of the circular orbit. We stress this To perform our numerics, the spheroidal harmonics waveform model does not include the LISA response are calculated using the SpinWeightedSpheroidal functions, which affect the amplitude evolution of the Harmonics Mathematica package in the BHPT, whereas ∞ signal and induce modulations, due to Doppler shifting, the Teukolsky amplitudes Zml are calculated using the through the motion of the LISA spacecraft [48,49]. Since Teukolsky package from the same toolkit. For reasons these response functions do not depend on the intrinsic discussed later, we generate both amplitudes and spheroidal parameters of the system that we are most interested in, we harmonics for a fixed spin parameter a ¼ 1 − 10−9.Forthe remainder of this study, we will only consider the plus 2 amΩ˜ polarized signal hðt; θÞ ≡ h ðt; θÞ for the face-on and edge- The (normalized) spheroidal harmonics −2Sml are integrated þ out over the 2-sphere. on observations.
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We finish this waveform discussion with a comment of detectability, of a gravitational wave signal. Measures of regarding the relation between the two models considered the similarity of two template waveforms h1 ≔ hðt; θ1Þ and in this work. The (dimensionful) Teukolsky amplitudes are h2 ≔ hðt; θ2Þ are the overlap Oðh1;h2Þ ∈ ½−1; 1 and related to the energy flux for each ðl; mÞ mode by mismatch Mðh1;h2Þ functions
∞ 2 h1 h2 _ ∞ jZlmj O pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið j Þ Elm ¼ 2 2 : ð26Þ ðh1;h2Þ¼ ð30Þ 4πm Ω ðh1jh1Þðh2jh2Þ qffiffiffiffiffiffiffiffiffiffi _ ˜ ∞ ∼ Ω˜ η ˜ Mðh1;h2Þ¼1 − Oðh1;h2Þ: ð31Þ Hence jZmlj M Elm. Averaging over the sky and ∞ ignoring the phase of the radial amplitude Zml, the O h1;h2 1 Teukolsky waveform (20) reduces to (16). Our numerical If ð Þ¼ then the shape of the two waveforms matches perfectly. Waveforms with O h1;h2 0 are results indicate that the spin precision measurements are ð Þ¼ orthogonal, being as much in phase as out of phase over driven by the radial trajectory given by (13), which is the observation. common to both (16) and (20), while not being largely Consider θ θ0 Δθ for Δθ a small deviation around influenced by the spin dependence on the waveform ¼ þ the true parameters θ0. Assuming that the waveform amplitude. Given this fact and since it is analytically much hðt; θÞ has a valid first order expansion3 in Δθ,we easier to analyze the waveform model (16), this is the one substitute into (27) and expand up to second order in Δθ being discussed in the analytics Sec. III to explain the increase in the spin precision measurement for near- 1X extremal primaries. p d θ ∝ − Γ Δθi −Δθi Δθj −Δθj ; ð j Þ exp 2 ijð bfÞð bfÞ ð32Þ i;j C. Gravitational wave data analysis Δθi Γ−1 ij ∂ Γ where bf ¼ð Þ ð jhjnÞ and ij is the Fisher matrix The data stream of a gravitational wave detector, given by dðtÞ¼hðt; θÞþnðtÞ, is typically assumed to consist of probabilistic noise nðtÞ and (one or more) deterministic ∂h ∂h ; θ θ Γ : 33 signals, hðt Þ, with parameters . Assuming that the noise ij ¼ ∂θi ∂θj ð Þ is a weakly stationary Gaussian random process with zero mean, the likelihood is [52] The Fisher matrix Γ ∼ ρ2 and therefore Δθ scales like Γ−1 ij ∂ ∼ ρ−1 1 ð Þ ð jhjnÞ . The linear signal approximation is θ ∝ − − − ρ ≫ 1 pðdj Þ exp 2 ðd hjd hÞ ð27Þ therefore valid for high SNR, . The Fisher matrix Γ, evaluated at the true parameters θ0, provides an estimate of the width of the likelihood function with inner product (27). Hence, it can be used as a guide to how precisely you Z can measure parameters. The inverse of the Fisher matrix is ∞ bˆðfÞcˆ ðfÞ ðbjcÞ¼4Re df: ð28Þ an approximation to the variance-covariance matrix Σ on 0 SnðfÞ parameter precisions Δθi ˆ Here bðfÞ is the continuous time fourier transform (CTFT) CovðΔθi; ΔθjÞ ≈ ðΓ−1Þij: ð34Þ of the signal bðtÞ and SnðfÞ the power spectral density (PSD) of the noise. Here we use the analytical PSD given The square route of the diagonal elements of the inverse by Eq. (1) in [53]. We do not include the galactic fore- fisher matrix provide estimates on the precision of param- ground noise in the PSD to ensure all noise realizations eter measurements, accounting for correlations between generated through SnðfÞ are stationary. This is not a serious the parameters. restriction as for the sources we consider here, the majority of the GW emission is at higher frequencies where the III. ANALYTIC ESTIMATES OF SPIN PRECISION galactic foreground lies below the level of instrumental noise in the detector. Before discussing numerical results on the measurement The optimal signal to noise ratio (SNR) of a source is precisions for the parameters θ of near-extremal EMRIs, we given by would like to develop some analytic tools that will allow us to understand the precisions we find numerically. In ρ2 ¼ðhjhÞ: ð29Þ 3In the literature, this is called the linear signal approximation. This is the SNR that would be realized in a matched It is a good approximation for sufficiently small Δθ, such that 2 filtering search and is a measure of the brightness, or ease Δθ∂θh ≪ ∂θΔh.
124054-6 CONSTRAINING THE SPIN PARAMETER OF NEAR-EXTREMAL … PHYS. REV. D 102, 124054 (2020) particular the fact that spin measurements for near-extremal independent of Snðf∘Þ. Again, this approximation could primaries are noticeably tighter than those obtained for introduce at most an order of magnitude uncertainty, and more moderately rotating primaries. Throughout this sec- most likely much less than that. Once the Fisher matrix is tion, we will use the waveform model (16) for analytical written in the form (39), we can use the semianalytic convenience. We will focus on the spin-spin component of waveform model (38) to evaluate it. In the Appendix A,we the Fisher matrix argue the dominant contribution can be approximated by Z ˆ 2 8 X j∂hðf; rðaÞ;t; θÞ=∂aj Γ ≈ M Γ Γaa ¼ 4 df ð35Þ aa ˜ 2 aa;m S f D SnðfoÞ nð Þ Z m t˜ 3 pffiffiffi 2 cut ˜ ˜_ ∞ ˜ ˜ 2 in the following analytic discussion. Our numerical and Γaa;m ≈ dtEm ðΩ tÞ 1 þ r˜∂ar˜ : ð40Þ ˜ 2 statistical analysis will be more general and employ the t0 Teukolsky based model (20). In future work, we will extend Here t˜0 is the coordinate time at which the observation this analytic considerations to multiple parameter study. starts and t˜ is the coordinate time at the end of the If all other parameters were known perfectly, the cut observation. For the results in this paper, we analyze estimated precision on the spin parameter would be ∼1 ˜ year long signals and fix tcut independently of spin, pffiffiffiffiffiffiffi such that all inspirals terminate before r˜ is reached. Δ ≈ 1 Γ isco a = aa: ð36Þ As seen in (40), a proper understanding of the precision in the spin measurement requires quantifying the spin Thus, to compare precisions between near-extremal dependence of the inspiral trajectory of the secondary, (denoted ext) and moderately rotating (denoted mod) i.e., ∂ar˜. primaries one is led to study the ratio ext A. Spin dependence on the radial evolution Γaa mod : ð37Þ Γaa Our primary goal here is to understand the spin depend- ence on the radial trajectory of the secondary (∂ar˜) for any Consider the (semianalytic) gravitational wave ampli- spin parameter a of the primary. tude (16) The trajectory of the secondary is the integral of the pffiffiffiffiffiffiffi inspiral equation X X 2 E_ ∞ hðtÞ¼ h ðtÞ ≈ m sinðmΩ˜ t˜Þ; ð38Þ m mΩ˜ D˜ ˜ dr˜ m m ∂ ˜Eðr;˜ aÞ ¼ −P ðr;˜ aÞ: ð41Þ r dt˜ GW where we have chosen the initial phase ϕ0 ¼ 0 for ˜ ˜ simplicity. The Fisher matrix depends on the PSD of the This follows from energy conservation, where Eðr; aÞ is the ≔ ˜_ ˜ detector. In the numerical calculations presented later we energy of a circular orbit (5) and PGW EGWðr; aÞ is the will use the full frequency dependent PSD, but to derive our energy rate carried away by gravitational waves (10). While ≈ ˜ analytic results we will approximate SnðfÞ Snðf∘Þ,a Eðr;˜ aÞ is kinematic, that is, derived through geodesic constant. The rationale for this is that EMRIs evolve quite properties, PGW is dynamic, that is, it is a radiation reactive slowly and so the total change in the PSD over the range of term determined by solving Teukolsky’s equation for a frequencies present in the signal is small. Between 1 mHz point particle source. The former is under analytic control, and 100 mHz, the (square root of the) LISA PSD changes whereas the latter typically requires numerical treatment. by just one order of magnitude, which is much smaller than The quantity ∂ r˜ captures the change in the secondary’s the three orders of magnitude improvement in spin meas- a trajectory when the spin parameter a of the primary varies, urement precision that we find numerically. Additionally, keeping the remaining primary and secondary parameters the difference in the ISCO frequencies across all combi- fixed, including t˜. More explicitly, the integral r˜ðr˜0;aÞ of nations of mass and spin considered in our numerical ˜ ˜ ˜ analysis is less than a factor of 2.5. PSD variations can not (41) depends on the initial condition rðt0Þ¼r0 and it ∂ ˜ therefore explain the numerical results, and so we can depends on the spin parameter a both through ð r˜EÞ and ˜ ignore these in deriving the analytic results which do ðPGWÞ information, but not through t, which is simply explain the numerics. Under this approximation labeling the points in the trajectory. We will comment on Z the possible spin dependence on the initial condition 4 ˜ Γ ≈ ∂ 2 r0 below. aa dtð ahðtÞÞ : ð39Þ One possibility to compute ∂ r˜ is to integrate (41) and to Snðf∘Þ a take the spin derivative explicitly afterwards. A second, ˜ We additionally assume that the choice of f∘ does not equivalent, way is to observe r˜ is a monotonic function of t depend on the spin, and therefore the ratio (37) is at fixed spin and initial radius r˜0. Hence, it can be used as
124054-7 BURKE, GAIR, SIMÓN, and EDWARDS PHYS. REV. D 102, 124054 (2020) Z the integration coordinate to study ∂ r˜ðr˜Þ. To do this, notice ∂ ˜ ∂ ˜ a − r˜E ∂ r˜E ˜ that the total spin derivative of the kinematic and dynamic ¼ a log dr: ð48Þ PGW PGW functions in (41), at fixed r˜0 and t˜,is Combining our results, we obtain ∂ ˜ 2 ˜ 2 ˜ ∂r˜Eðr˜ðaÞ;aÞ ¼ð∂ ˜ EÞ∂ar˜ þ ∂ ˜E; Z ∂ ˜ ˜ r ar ∂ ˜ ∂ ˜ a r0;t PGW r˜E r˜E ∂ r˜ k0 − ∂ log dr˜ : 49 ∂ a ¼ ∂ ˜ a ð Þ PGW ∂ ∂ ˜ ∂ r˜E PGW PGW ¼ð r˜PGWÞ ar þ aPGW: ð42Þ ∂a ˜ ˜ r0;t This is valid for any spin, for any location of the secondary To ease our notation, all spin partial derivatives in the rhs, and for any flux PGW. This analytic result will allow us to and in the forthcoming discussion, should be understood as determine what the dominant source of the spin dependence is in different regions of the trajectory. computed at fixed r˜0 and t˜. Defining u ¼ ∂ r˜ (to ease a In Fig. 1 we show the near perfect agreement between notation) and computing the total spin derivative of ∂ ˜ Eq. (41), we obtain the solution to (49) and our numerical calculation of ar using finite difference method ˜ ∂2 ˜ ∂2 ˜ ∂ ˜ du dr − dPGW _ _ u ˜ E þ E þ r˜E ¼ : ð43Þ r˜ða þ δ; t;˜ Eða þ δÞÞ − r˜ða − δ; t;˜ Eða − δÞÞ r are dr˜ dt˜ da ∂ ˜ ≈ ar 2δ : ð50Þ Plugging in the radial velocity using (41) one obtains the method used to calculate year-long trajectories used for 2 ˜ 2 ˜ our Fisher matrix results in later sections, for both mod- du ∂ E ∂ ˜P ∂ E ∂ P r˜ − r GW u − ar˜ a GW : 44 erately and rapidly rotating primaries. ˜ þ ∂ ˜ ¼ ∂ ˜ þ ð Þ dr r˜E PGW r˜E PGW Following [11], we express the energy flux as a relativ- istic correction factor, E_, times the leading order Newtonian This is a first order linear ODE, valid for any spin and for flux any location of the secondary, whose solution describes the desired spin dependence in the radial trajectory ∂ r˜ðr˜Þ. 32 a ηΩ˜ 10=3E_ Its general solution is a sum of the homogeneous PGW ¼ 5 : ð51Þ solution uh and a particular solution up. It will depend on an initial condition uðr˜0Þ. The initial condition of the Plugging this into Eq. (49) gives radial trajectory is Z 1 ∂ ˜ ∂ ˜ − Q∂ Q ˜ Q r˜E ∂˜ ar ¼ k0 a log dr ; ¼ 10 3 _ : ð52Þ ˜ ˜ 0 ˜ ⇒ r 0 Q Ω˜ = E rðr0;a;t¼ Þ¼r0 ∂ ¼ ; ð45Þ a r0;t¼0 Decomposing the source term 4 from which we deduce uðt ¼ 0Þ¼0. ∂ ˜ ∂2 ˜ ∂ E_ 10 The homogeneous version of Eq. (44) is equivalent to Q∂ Q r˜E ar˜E − a Ω˜ a log ¼ 10 3 _ ˜ _ þ ; ð53Þ Ω˜ = E ∂ ˜E E 3 ∂ ˜ r duh r˜E 0 ⇒ PGW þ d log ¼ uh ¼ k0 ˜ ð46Þ we see that the first and third terms are kinematic, i.e., uh PGW ∂r˜E driven by geodesic physics, whereas the second is dynami- where k0 is an arbitrary integration constant. We follow a cal, i.e., driven by the energy flux. Comparison between standard approach and look for a particular solution of the these terms at different stages of the inspiral, as a function form up ¼ kðr;˜ aÞuh. Plugging this into (44) gives of the spin, can help us to determine what the driving source Z of spin dependence is in each case. In the next subsection, ˜ 2 ˜ ∂ ˜E ∂ E ∂ P we investigate the contribution of both the geodesic and k r;˜ a r − ar˜ a GW dr˜ ð Þ¼ ˜ þ ð47Þ radiation reactive terms to ∂ar˜. PGW ∂r˜E PGW B. Comparison of radial evolution for moderate and near-extremal black holes 4 The initial condition uðr˜0Þ can play an important role when Despite the universality of (49) or (52), the dependence ∂ ˜ gluing a numerical calculation for ar with an analytic one in on the energy flux makes it not feasible to analytically some specific piece of the trajectory where the information ∂ ˜ determining the solution to (44) is under analytic control. We integrate ar along the entire secondary trajectory. will be more explicit about this when we discuss ∂ar˜ in the region However, we can integrate (49) in specific regions of the close to ISCO. secondary trajectory.
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FIG. 1. The dashed curves (black dashed and yellow dashed) on each figure is the solution to (49) with k0 ¼ 0 corresponding to ˜ ˜ ˜ ∂arðr0Þ¼0. In both plots, the solid colors (blue and violet) are ∂ar calculated using a fifth order stencil method. In each plot, the intrinsic parameters given in the titles.
It is possible to prove that d∂ar=d˜ r<˜ 0 and hence that This allows us to integrate (49) to give the spin dependence ∂ar˜ grows monotonically over the inspiral. It is therefore of the radial trajectory for moderately spinning black natural to study the behavior of ∂ar˜ close to ISCO, where holes its contribution to the Fisher matrix (40) will be maximal. We first compare the kinematic and dynamical contribu- tions to (53). Using results from the BHPT, we have numerically calculated the spin derivative of E_ for two primaries with spin parameters a ¼ 0.9 and a ¼ 1 − 10−6. These are compared with the kinematic sources in (53) in Fig. 2. These figures show that
∂2 ˜ 10 ∂ E_ ar˜E Ω˜ ≫ a ˜ þ 3 _ ; ð54Þ ∂r˜E E for both spin parameters. This suggests it is the kinematic sources in (53) that drive the spin dependence of the secondary trajectory, particularly close to ISCO. Although we have only verified it for two choices of spin parameter, we will assume this approximation holds for any spin parameter a ≥ 0.9. We first consider moderately spinning black holes close to ISCO. Dropping the dynamical contribution to (53),we can compare the two remaining terms. The angular velocity ˜ piece is bounded and order one, but ∂r˜E tends to zero at ∂2 ˜ ∂ ˜ ISCO. This means that ar˜E= r˜E dominates close to ISCO, allowing us to use the approximation ∂ ˜ ∂2 ˜ ∂ E_ 10 ∂2 ˜ r˜E ar˜E − a Ω˜ ≈ ar˜E 10 3 _ ˜ _ þ 10 3 _ : ð55Þ Ω˜ = E ∂ ˜E E 3 Ω˜ = E r FIG. 2. The top plot compares the kinematic and radiation a 0 999999 Ω˜ E_ reaction quantities given in (53) for a spin of ¼ . . The Since, for moderate spins, the variation of and with bottom plot is the same but for a spin parameter of a ¼ 0.9. radius close to ISCO is negligible compared to the variation Notice that in these two cases the kinematical quantities dominate ∂2 ˜ ˜ in ar˜E, we will approximate them by their values at risco. over the relativistic correction terms.
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2 Z 1 10 3 2 ∂ ∂ ˜ ≈ Ω˜ = E_ ˜ − ∂ ˜ xisco ffiffiffi xisco −4 ar ˜ ðkmod isco 0ða; riscoÞ aEÞ; ð56Þ ≈ p x dx ð63Þ ∂ ˜E ˜ ˜ ∂ r ηðC∞ þ CHÞ 3 a where kmod is an arbitrary constant. Since 8 1 ≈ pffiffiffi ð64Þ ffiffiffi 9 3η ˜ ˜ x3 2 2 p ðC∞ þ CHÞ ˜ r˜ − 3a þ 8a r˜ − 6r˜ ∂ ˜E ¼ pffiffiffi ; ð57Þ r 7=4 3=2 3=2 ≈ 21=3ϵ2=3 2r˜ ðr˜ − 3 r˜ þ 2aÞ where we have used xisco and PGW defined in (60). This is valid for x ≪ 1 and includes the corrections ∂ ˜ ˜ 0 ∼ ˜ it follows from Eq. (A5) in [40] that r˜EðriscoÞ¼ .For due to x xisco. Assuming r0 is close to ISCO, so that the moderately rotating primaries and near ISCO, we can initial condition (45) holds, we conclude that the spin ˜ ≈ ˜ ˜ 1 ∂2 ˜ ˜ ˜ − ˜ 2 dependence in the near-ISCO region of a near-extremal expand E EðriscoÞþ2 r˜ EðriscoÞðr riscoÞ leading to black hole is ˜ ˜ 2 ˜ ∂ E ≈ ∂ Eðr˜ Þþ∂ ˜ Eðr˜ Þðr˜ − r˜ Þð−∂ r˜ Þ a a isco r isco isco a isco 8 x3 ∂ ˜ ≈ ∂2 ˜ ˜ ˜ − ˜ ∂ r˜ ≈ pffiffiffi 1 − : ð65Þ r˜E r˜ EðriscoÞðr riscoÞð58Þ a 2 ˜ 3 9 3x ∂r˜E x0 ∂ ˜ ˜ Using these expansions in Eq. (56) we deduce ar¼kmod= Figure 3 compares (59) and (65) to the full ∂ar˜ computed ˜−˜ ∂ ˜ ˜ Ω˜ 10=3E_ ˜ ðr riscoÞþ arisco, where kmod ¼kmod isco 0ða;riscoÞ= numerically without using the near-ISCO approximations. ∂2 ˜ ˜ ˜ ˜ We see that the approximations are very accurate in the r˜ EðriscoÞ. Assuming that r0 is sufficiently close to risco that this approximation holds throughout the range region close to the ISCO where they are valid. ˜ ˜ Before continuing, we will comment further on the ½risco; r0 , we can use the boundary condition (45) to ˜ choice of flux (12) instead of (11). The latter has an determine k ¼ ∂ r˜ ðr˜ − r˜0Þ and hence mod a isco isco ˜ 1 ϵ explicit dependence on rþ ¼ þ . Consequently, it carries ˜ − ˜ an additional spin dependence. In particular, ∂ r˜ ¼ −a=ϵ. ∂ ˜ ≈ −∂ ˜ r0 r a þ ar ð ariscoÞ ˜ − ˜ : ð59Þ Thus, for near-extremal primaries this spin dependence can r risco induce extra diverging sources for ∂ar˜ with a very specific We now repeat this analysis for near-extremal primaries. sign. We can easily compute their effects by integrating the Near ISCO, the energy flux can be approximated by the ODE with such an energy flux source. The result one finds ∂ ˜ NHEK flux ðx ≡ r˜ − 1 ≪ 1Þ does not agree with the numerical evaluation of ar generated from the BHPT, which computes the exact flux.5 ˜ ˜ We conclude that (12) appears to capture the spin depend- P ≈ ηðC∞ þ C Þx: ð60Þ GW H ence of our observable (the amplitude of the gravitational Using this approximation, there is no explicit spin depend- wave) more accurately than (11). This is in fact the reason ence and so the ∂ P term in Eq. (47) vanishes. we chose to work with (12). a GW Let us close this discussion with a brief comparison Expanding (57) for x ¼ r˜ − 1 ≪ 1 and ϵ ≪ 1, the denom- between the analytic results for moderate and near-extremal inator involves ˜ − ˜ ∼ δ η2=5 spins. We write r risco > , the latter inequality pffiffiffi 3 1 9 ensuring that we avoid entering the transition region ˜3=2 − 3 ˜ 2 2 −ϵ2 − 3 4 O 5 ϵ2 ϵ4 r rþ a ¼ 4x 4x þ 64x þ ðx ;x ; Þ; [40,54]. Expanding Eq. (61) we find for near-extremal black holes while the numerator has the expansion ˜ 2 δ 11 11 ∂ ˜E ≈ pffiffiffi 3 − δ − x þ : pffiffiffi 1 5 r 3 3 x 8 8 isco ˜2 − 3 2 8 ˜ − 6˜ 3 − ϵ2 − 4 O 5 ϵ2 ϵ4 isco r a þ a r r ¼ 2x 32x þ ðx ;x ; Þ: δ ≲ 2 ∼ ϵ4=3 The first term is dominant unless | xisco . The We conclude constraint δ > η2=5 ensures this is only violated if ϵ > η3=10. This will be satisfied for all the cases that we consider in 2 11 3 ˜ xisco 2 2 2 this paper, but we emphasize this is not a physical ∂ ˜E ≈ pffiffiffi 1 − x − þ Oðx ; ϵ =x Þ: ð61Þ r 3 3 8 x3 constraint. When this constraint is violated, additional terms become important in the expansion which we have Using this approximation in Eq. (47) we find 5 ∂ ˜ Z In particular, it is no longer the case that ar is monotonically 2 ˜ increasing all along the inspiral trajectory, whereas the BHPT ∂ ˜E kðr;˜ aÞ ≈− ar dr˜ ð62Þ data is monotonically increasing, a feature our ODE with flux PGW (12) reproduces.
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FIG. 3. The yellow dashed and black dashed curves are solutions to (59) and (65). The purple and blue curves are the true solutions to ˜ ∂ar obtained numerically without near-ISCO simplifications. We see both approximations capture the leading order behavior of the spin derivative of the radial trajectory very well.
∂ ˜ → 0 ˜ ignored, and these ensure that r˜E at risco.We [1]. In particular, it has been shown that our ability to ˜ 2=3 O 10−6 conclude the scaling of ∂r˜E is δ=ϵ for near-extremal constrain the spin parameter a is expected to be ð Þ black holes, compared to δ for moderate spins. [3,4,15,55]. It has also been shown that the measurements It follows using (59) and (65) that are more precise for prograde inspirals into more rapidly 8 spinning black holes, when the secondary spends more < 1 ; moderate spins orbits closer to the event horizon of the primary (see δ Fig. (11) in [55]). In subsequent sections we show through ∂ r˜ ∝ 2 3 : a : ϵ = ð66Þ 2 3 2 ; near-extremal spins δðδþϵ = Þ numerical calculation that spin measurements are even more precise for EMRIs into near-extremal black holes. We The spin dependence on the radial trajectory for near- now try to understand this result using Eq. (40). Inspection of (40) suggests there are two main effects: extremal primaries is larger than for moderately rotat- ˜2 ∂ ˜ 2 ing ones. the dependence on t and the dependence on ð arÞ . First, the fact that t˜ ∼ Oðη−1Þ follows from integrating (41), and therefore the contribution to the Fisher matrix due to t˜2 is C. Precision of spin measurement −2 large and scales like η . Second, ∂ar˜ is monotonically In the previous section we showed the effect that the spin increasing as the secondary spirals inwards. Thus, its parameter has on the radial trajectory. This was achieved by maximal contribution comes from the region close to ∂ ˜ studying the general linear ODE for ar, Eq. (49).By ISCO, which supports results in [55]. Equation (66) shows arguing that the kinematic terms dominate the behavior of this contribution is largest in the last stages before entering ∂ ˜ ar, for both the near-extreme and moderately spinning into the transition regime. As changes in Ω˜ close to ISCO black holes, analytic solutions were found near the ISCO. Ω˜ ˜ 2 ∂ ˜ are negligible, the factor ð tÞ is, approximately, the We were able to conclude that ar grows much more square of the number of cycles, a proxy widely used in rapidly close to the ISCO for near-extreme black holes than the literature in discussions of the precision of measure- for moderately spinning black holes. We also emphasize ments. Our estimate (40) confirms this intuition and shows ∂ ˜ ∂ E_ that Eq. (54) shows that corrections to ar of the form a the spin precision will be further increased by large values are subdominant. We now explore the consequences of of the radial spin derivative, ∂ar˜. these results for the precision of spin measurements, computed using the Fisher matrix formalism. Due to the large number of observable gravitational wave D. Comparison of spin measurement precision for cycles that are generated while the secondary is within the moderate and near-extremal black holes strong field gravity region outside the primary Kerr black The Fisher matrix estimate (40) depends on the spin hole, extreme mass-ratio inspirals will provide measure- derivative of the radial evolution, on the duration of the ments of the system parameters with unparalleled precision inspiral and on the energy flux. Equation (66) shows that, at
124054-11 BURKE, GAIR, SIMÓN, and EDWARDS PHYS. REV. D 102, 124054 (2020) Z X ˜ a fixed distance to the corresponding ISCO, ∂ r˜ is larger for μ tcut a Γ ≈ 18 r˜ Ω˜ 2 dðηt˜Þ a near-extremal primary than for a moderately rotating aa η ˜ 2 ext ext ð DÞ Snðf∘Þ m 0 primary. As a consequence of time dilation near the black dE˜ ∞ hole horizon, E_ → 0 near the ISCO for near-extremal m η˜ 2 ∂ ˜ 2 × η ˜ ð tÞ ð arÞ : ð68Þ primaries, but remains finite for moderately rotating ones. dt This means that the energy flux for near-extremal inspirals ˜ is much smaller than that for moderate spins, but the Here, tcut is the time at the end of the integration, where duration of the inspiral is longer. However, we can write x ¼ xcut. Our approximations break down when the tran- (40) as an integral over the BL radial coordinate r˜. In that sition regime breaks down, so we can assume ∼ η2=5 ϵ2=3 case, the integrand is proportional to xcut þ , which is a small quantity. Using ˜ ∞ ˜ η ˜ ≥ ≫ dEm =dt ¼ C∞mx and assuming x0 x xisco, so that −y the trajectory can be approximated by xðt˜Þ ≈ x0e with ∞ pffiffi ˜_ 3 3 ˜ ˜ Em ˜ 2 ˜ 2 y ¼ αηt˜ ≡ ðC þ C∞Þηt˜, the Fisher matrix reduces to ðt˜ ΩÞ ð∂ ˜EÞð∂ r˜Þ 2 H ˜_ r a EGW 64μ r˜ Ω˜ 2 X Γext ≈ ext ext ˜ aa ˜ 2 3 · GðycutÞ C∞m ; ð69Þ η 3x0α close to the relevant ISCO. While the energy fluxes are much ð DÞ Snðf∘Þ ð Þ m smaller for near-extremal inspirals, the ratio of energy fluxes ycut appearing above is an order one quantity for all spin with e ¼ x0=xcut and parameters. The expression above is therefore a product of −9 3 9 2 2 3 − 6 3 factors that have been argued to be either of comparable GðycutÞ¼ ycut þð ycut þ Þ sinh ycut ycut cosh ycut magnitude or much smaller for moderately rotating primar- 3 2 ≈ x0 3 x0 − 1 1 ies. We therefore expect the precision of spin measurements 3 log þ ; ð70Þ 2x x to be much higher for near-extremal EMRIs. cut cut A quantitative comparison between the near-extremal where in the last step we used x0 ≫ x . and the moderately spinning sources requires a precise cut calculation of the ratio (37) computed along the entire respective trajectories. In general, this is a hard analytic task 2. Moderately spinning source ˜_ ˜_ ∞ since both energy fluxes EGW and Em must be handled Using the same kind of approximations as above, but through numerical means and long observations of inspirals taking into account the different energy flux and different (starting in the weak field) require calculations performed trajectory in the frequency domain where SnðfÞ shows nontrivial (nonconstant) behavior. This would be no more straightfor- 64 E_ 2 2 ˜ 10=3 0ðaÞ r˜ − r˜ − r˜0 − r˜ ≈ ηΩ t;˜ 71 ward than direct numerical computation of the Fisher ð iscoÞ ð iscoÞ 5 isco ∂2 ˜ ˜ ð Þ Matrix and so we do not pursue it here. r˜ EðriscoÞ For any sources whose trajectory lies entirely lie in the near-ISCO region, these analytic approximations allow us one can approximate the Fisher matrix for moderate to compute the ratio (37) reliably. This can be exploited to spins by obtain an analytic approximation to the Fisher Matrix for 5∂2E˜ r˜ 3 μ r˜ ∂ r˜ 2 such sources and this calculation will be pursued else- Γmod ≈ 18 r˜ ð iscoÞ iscoð a iscoÞ aa _ ˜ 2 ˜ 8 where. Additionally, earlier arguments tell us that it is the 64E0 ðηDÞ S ðf∘Þ Ω n isco near-ISCO regime that dominates the spin precision and so X ˜ ∞ 6 dEm these expressions are sufficient to understand the increase × ðr˜0 − r˜ Þ FðδÞ ; ð72Þ isco ηdt˜ in spin precision seen for near-extremal inspirals. m isco ˜ −˜ where δ ≡ rcut risco < 1 and r˜0−r˜ 1. Near-extremal source isco 8 From (65), it follows δ −2 δ − 4 1 − δ − 1 − δ2 1 − δ3 Fð Þ¼ log ð Þ ð Þþ3 ð Þ 1 4 1 4 x − ð1 − δ4Þ − ð1 − δ5Þþ ð1 − δ6Þ: ð73Þ ∂ r˜ ≈ x3 − x3 : 2 5 3 a 3 3 3 − 3 ð 0 Þ ð67Þ x0 x xisco ffiffiffi 3. Ratio of Fisher matrices p ˜ Since r˜∂ar˜ grows fast and the rate of change of r˜ and Ω is Within these approximations, the ratio (37) now small, near-ISCO, we can approximate (40) by reduces to
124054-12 CONSTRAINING THE SPIN PARAMETER OF NEAR-EXTREMAL … PHYS. REV. D 102, 124054 (2020) Γext 256 64 3 ˜ Ω˜ 2 A. Energy flux aa ≈ ffiffiffi rext ext mod p 2 ˜ 2 2 Γ 9 45 3∂ E˜ ðr˜ Þ r˜ Ω ð∂ r˜ Þ Both waveform models (16) and (20) depend on the aa r˜ isco isco isco a isco _ radial trajectory r˜ðt;˜ a; η; EÞ. The amplitude evolution using GðycutÞ T × 3 6 ; the Teukolsky formalism depends on the spheroidal har- x ðr˜0 − r˜ Þ FðδÞ ˜ 0 isco amΩ θ ϕ P 3 monics −2Sml ð ; Þ and Teukolsky amplitudes at infinity ˜ Ω˜ 10 E_ _ ∞ T mC∞m ð isco 0Þ ∞ ˜ ˜ ˜ ¼ 3 P ˜ ∞ ð74Þ Zmlðr; aÞ. The energy flux at infinity Emlðr; aÞ is related to ˜ ˜ dEm ∞ ðCH þ C∞Þ ˜ j m ηdt isco the Teukolsky amplitudes Zml through Eq. (26). Thus, to accurately generate the waveforms (16) and (20) far from the horizon where near-extremal simplifications can The most relevant feature for our current discussion is ∞ not be made, the various radiation reactive terms Zml; the quotient dependence ˜_ _ ˜_ ∞ _ ∞ EðEÞ; Em ðEm Þ have to be handled numerically. This section outlines our numerical routines to do so. Gðy Þ 1 We use the BHPT to calculate the first order dissipative cut ≈ _ 3 ˜ − ˜ 6 δ 3 ˜ − ˜ 6 δ ð75Þ ˜ 1 − 10−i i¼9 E_ x0ðr0 riscoÞ Fð Þ xcutðr0 riscoÞ Fð Þ radial fluxes EGW for a ¼ f gi¼3 from which in Eq. (51) can be computed. We used the TEUKOLSKY Mathematica script in the toolkit and tuned the numerical The first two denominator factors increase the ratio, since precision to ∼240 decimal digits to avoid numerical x ≪ 1 and r˜0 − r˜ < 1. The last could in principle be _ cut isco instabilities when computing E˜ in the near-horizon large, due to the logarithmic term. However δ and x have GW cut regime for rapidly rotating holes. For moderately spinning similar scaling and therefore x FðδÞ ≪ 1. We deduce that cut holes a ≲ 0.999, we used the tabulated data in Table II the spin component of the Fisher matrix is much larger for of [11]. near-extremal inspirals than for moderate spins. This is Each coefficient appearing in Eq. (19) is itself a sum over confirmed by the numerical results that will be reported in P ˜_ ∞ ∞ ˜_ ∞ subsequent sections. l modes, Em ¼ jlj¼m Eml. Both the sum over l and the We finish by noting that the Fisher matrices increase in sum over m in Eq. (19) can be truncated without appreci- ˜ magnitude as the trajectory is cut off closer to risco. In the able loss of accuracy. As discussed in [32], near-extremal case of moderate spin, we already noted the logarithmic EMRIs require a significant number of harmonics to be dependence of FðδÞ as δ → 0. This has previously been included to obtain an accurate representation of the observed in the literature, see for example Fig. (11) in [55]. gravitational wave signal. To illustrate for a high spin of ∼ ∼ ϵ2=3 −9 ˜_ ∞ For near-extremal EMRIs, if xcut xisco , then for a ¼ 1 − 10 , we used the BHPT to compute E for ϵ → 0 ml fixed x0 and as the spin Fisher matrix scales as harmonics jmj ≤ l ∈ f2; …; 15g. Figure 4 illustrates the Γ ∼ ϵ ϵ 2 aa ðlogð Þ= Þ . We deduce that observing the latter convergence as the number of harmonics is increased. stages of inspiral is important for precise parameter Based on these results, we go further by including measurement, for any primary spin. ≤ 20 harmonics with l lmax ¼ to calculate the total energy In summary, we have derived an analytic approximation, _ flux E˜ valid close to ISCO, for the spin component of the Fisher GW matrix. This indicates that this component is much larger for near-extremal spins and therefore we expect much more precise measurements of the spin parameter in that case. The approximation depends sensitively on certain quan- tities, such as the cutoff radius, xcut, that are somewhat arbitrary. However, for any choice the near-extremal precision is a few orders of magnitude better. This provides support for the numerical results that we will obtain in Sec. VI, which show a similar trend.
IV. WAVEFORM GENERATION In this section we provide more details on how we FIG. 4. Comparison of the total energy flux at infinity (black construct the waveform model used to compute the Fisher ˜_ ∞ matrix in the next section. The waveform model was curve) including different harmonic Elm contributions. Note that ˜_ ∞ previously given in Eq. (16) and Eq. (20). Here, we at r˜ ≈ 1.3, the l ¼ 2 harmonic energy flux E2 contributes ∼32% 11 describe how the various terms entering these equations of the total energy flux, whereas including the first lmax ¼ are evaluated. harmonics (violet curve) contributes more than ∼98%.
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Xlmax X B. Radial trajectory and waveform _ _ ∞ _ H E˜ E˜ E˜ ; 76 GW ¼ ð ml þ mlÞ ð Þ The radial trajectory can be constructed by numerically jlj¼2 jmj≤l _ integrating the ODE (13) using an interpolant for Eðr˜Þ and suitable initial conditions. As before, we use the spin using the Teukolsky package in the BHPT. In the same ˜ ˜ 0 ˜ ∞ P ∞ independent initial condition rðt0 ¼ Þ¼r0. Figure 5 ˜_ lmax ˜_ numerical routine, we compute Em ¼ jlj¼m Eml using shows some example radial trajectories for various spin 20 ≤ 20 lmax ¼ for m . These formulas are rearranged to parameters, computed using flux data from the BHPT. In _ _ ∞ the high spin regime, the exponential decay of the radial obtain E and Em using (51) and (14). Finally for our Teukolsky based waveforms used in coordinate is prominent as discussed in [24,54]. numerics Sec. VI, we use the BHPT to extract the Throughout our simulations, the observation ends after a ∞ ˜ fixed amount of time, chosen such that this is before the Teukolsky amplitudes Zmlðr; aÞ and build an interpolant 1 … 20 transition to plunge for all parameter values used to over r for each harmonic m ¼f ; ;lmax ¼ g compute the Fisher matrix. This is important to avoid X∞ introducing artifacts from the termination of the waveform, ˜ amΩ˜ θ ϕ ∞ ˜ Gmðr; aÞ¼ −2Sml ð Þ expði ÞZmlðr; aÞð77Þ given that the transition to plunge is not properly included l¼m in this waveform model. It is clear from Fig. 5 that larger the spin parameter, the longer the secondary spends in the for each viewing angle ðθ; ϕÞ¼ðπ=2; 0Þ and dampening regime. See Eq. (22) of [24] for further details. ðθ; ϕÞ¼ð0; 0Þ. To summarize, we use (77) in (20) to The spin dependence of the radial evolution can be compute Fisher matrices numerically in Sec. VI. To aid our calculated by integrating (13) and then taking numerical _ analytic study, we use the computed E∞;m in the waveform derivatives. We consider two reference cases, both with 6 model (16) when evaluating the ratio (74). component masses μ ¼ 10 M⊙ and M ¼ 2 × 10 M⊙,but with different spin parameters a ¼ 0.9 and a ¼ 1 − 10−6. We compute one year long trajectories, with r˜ð0Þ¼5.08 in the first case and r˜ð0Þ¼4.315 in the second. The spin derivative of the radial evolution can be calculated by perturbing the spin and using the symmetric difference formula for δ ≪ 1
_ _ ∂r˜ r˜ða þ δ; t;˜ Eða þ δÞÞ − r˜ða − δ; t;˜ Eða − δÞÞ ≈ : ð78Þ ∂a 2δ
2 Figure 6 plots the quantity j∂ar˜j appearing in the Fisher 2 matrix estimation (40). By inspection, it is clear that j∂ar˜j is largest when the spin parameter is close to unity and
FIG. 5. The top panel shows how dr=d˜ t˜ varies with r˜. The higher the spin parameter, the more time the secondary spends in FIG. 6. The blue curve is ∂ar˜ for a ¼ 0.999999. The orange the throat before plunge. The lower panel shows the correspond- curve is ∂ar˜ for a ¼ 0.9. Notice that the spin dependence on r ing inspiral trajectory. The dampening is clearly shown when the grows rapidly in the near-ISCO region of the rapidly rotating primary is near maximal spin, as seen in [24]. hole.
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FIG. 7. A near-extremal waveform with parameters θlight viewed face-on (left) and edge-on (right). The dampening region lasts ∼55 days. The edge-on case is asymmetric due to the large number of l ¼ 20 modes and shows prominent relativistic beaming near the ISCO as observed in figure 3(b) of [24].
˜ θ ˜ 0 4 3 1 − 10−6 μ 10 when the radius is close to risco, matching our analytical light ¼frðt0 ¼ Þ¼ . ;a¼ ; ¼ M⊙; conclusions using approximations (65) and (56). 6 M 2 × 10 M⊙; ϕ0 π; Using the semianalytic model (16) we now evaluate the ¼ ¼ 1 4 estimate (74), for the same two systems, but different r0 to D ¼fDedge ¼ ;Dface ¼ g Gpcg; ð81Þ ensure that the assumptions made in deriving Eq. (74) still hold (r˜0 ¼ 2.85 for a ¼ 0.9 and r˜0 ¼ 1.2 for 1 − 10−6 ˜ ˜ where Dedge and Dface refer to the distance if each source is a ¼ ). We choose termination points rcut ¼ risco þ λ λ ∼ λ 10−4 λ 10−2 viewed edge-on/face-on respectively. The distances are fine with f ext ¼ ; mod ¼ g, justP outside the 6 tuned so that we achieve a signal to noise ratio of ρ ∼ 20. transition region. Finally, the expression C∞ was ;m This is discussed later in Sec. V. The lighter source is calculated using the high_spin_fluxes.nb Mathematica sampled with sampling interval Δts ≈ 4 seconds and the notebook in the BHPT, including harmonics up to Δ ≈ 25 10 heavier one with ts seconds. We note here that m ¼ . We find the ratio to be ˜ ˜ Δts ¼ MΔt where Δt is the dimensionless sampling Γext interval used to integrate (13). The sampling interval is aa ∼ 500: ð79Þ ’ Γmod chosen from Shannon s sampling theorem such that aa Δ 1 2 ts < =ð fmaxÞ, where giving a rough estimate that the spin precision increases by at least two orders of magnitude for these two sources. 20 Ω˜ 2 Ω˜ edge isco face isco This verifies claims made in Sec. III C. When correla- fmax ¼ ;f¼ ð82Þ 2π M max 2π M tions with other parameters and the shape of the PSD are ignored, we predict a precision on the spin parameter roughly two orders of magnitude higher than for moder- are the highest frequencies present in the waveform for the ately spinning black holes. edge-on and face-on cases respectively. To illustrate, near- θ To generate gravitational waveforms for the numerical extremal waveforms with parameters light for both edge-on study we use the Teukolsky waveform model (20). The and face-on viewing angles are plotted in Fig. 7. ˜ waveform depends on parameters θ ¼fa; r˜0; μ;M;ϕ0; Dg. The lighter source is interesting because it exhibits both We will consider two classes of near-extremal source, an “inspiral” regime and a exponentially decaying regime differentiated by the magnitude of their component masses that we will refer to as “dampening.” The heavier source is and mass ratio. The first “heavier” source has parameters interesting because the dampening regime lasts more than one year and so the signal is in the dampening region for the θ ˜ 0 1 225 1 − 10−6 μ 20 heavy ¼frðt0 ¼ Þ¼ . ;a¼ ; ¼ M⊙; entire duration of the observation. In the next section, we 7 discuss detectability of these two types of sources by LISA. M ¼ 10 M⊙; ϕ0 ¼ π; 1 8 3 D ¼fDedge ¼ . ;Dface ¼ g Gpcgð80Þ 6Strictly speaking, distance here is not a physical parameter since our waveform model does not include the LISA response to “ ” and the second lighter source has the strains hþ and h×.
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V. DETECTABILITY The discrete analogue of the optimal matched filtering SNR defined in Eq. (29) The LISA PSD reaches a minimum around 3 mHz, and is fairly flat within the band from 1 to 100 mHz. For an edge- 7 on near-extremal inspiral with primary mass of ∼10 M⊙, bN=X2þ1c 2 4Δt jh˜ðf Þj the dominant harmonic has a frequency of ∼3.2 mHz at ρ2 ≈ s i : ð83Þ N S f plunge, while the m ¼ 20 harmonic has frequency of i¼0 nð iÞ 64 mHz. Such heavy sources are thus ideal systems for observing the near-ISCO dynamics. For the lighter mass Δ 2 106 Here N is the length of the time series, ts the sampling considered, × M⊙, the near-ISCO dynamics are at Δ frequencies a factor of 5 higher, where the LISA PSD starts interval (in seconds) and fi ¼ i=N ts are the Fourier frequencies. In Eq. (83), the discrete time Fourier transform to rise. While the near-ISCO radiation will still be observ- ˜ ˆ able for these systems, its relative contribution to the signal (DTFT) hðfjÞ is related to the CTFT through hðfÞ¼ ˜ will be relatively reduced. We therefore expect to obtain Δts · hðfÞ. To avoid problems with spectral leakage, prior more precise spin measurements for the heavier of the two to computing the Fourier transform, we smoothly taper the reference systems. endpoints of our signals using the Tukey window
8 > α −1 > 1 1 cos π 2n − 1 0 ≤ n ≤ ðN Þ > 2 þ αðN−1Þ 2 <> 1 αðN−1Þ ≤ ≤ − 1 1 − α 2 w½n ¼> 2 n ðN Þð = Þ ð84Þ > > :> 1 1 π 2n − 2 1 − 1 1 − α 2 ≤ ≤ − 1 2 þ cos αðN−1Þ α þ ðN Þð = Þ n ðN Þ:
˜ ˜ here n is defined through tn ¼ nΔt. The tunable parameter As mentioned above, the lighter source exhibits two α defines the width of the cosine lobes on either side of the regimes of interest—the initial gradually chirping phase, Tukey window. If α ¼ 0 then our window is a rectangular where the waveform resembles those for moderately spin- window offering excellent frequency resolution but is ning primaries, and then the exponentially damped phase subject to high leakage (high resolution). If α ¼ 1 then while the secondary is in the near-horizon regime. It is natural this defines a Hann window, which has poor frequency to ask what proportion of the SNR, and later what proportion resolution but has significantly reduced leakage (high of the spin measurement precision, is contributed by each α dynamic range). For the heavier source, we use ¼ regime. For both edge-on and face-on systems, we separate 0 25 . to reduce leakage effects significantly and frequency the two parts of the waveform using Tukey windows and resolution is not a problem since the frequencies of the compute the SNR contributed by each part to find signal are contained within the LISA frequency band (for α 0 05 all harmonics). For the lighter source, we use ¼ . to 83% 2 Outside Dampening region reduce edge effects while retaining the ability to resolve the ρ − ∼ ð85Þ frequencies where the signal is dampened. We found that face on 17% Dampening region: calculated SNRs and parameter measurement precisions are α insensitive to the choice of in the heavier system. The 96% Outside Dampening region lighter system is more sensitive: for larger α, more of the ρ2 ∼ ð86Þ edge−on 4% dampening regime is lost, with a corresponding impact on Dampening region: the measurement precisions. We believe that α ¼ 0.05 is large enough to reduce leakage but small enough to resolve For the face-on source, there is just a single dominant as much of the dampening regime as possible. harmonic, and the frequency of this harmonic is such that After tapering, we zero pad our waveforms to an integer it lies in the most sensitive part of the LISA frequency range. power of two in length, in order to facilitate rapid evaluation This helps to enhance the relative SNR contributed by the of the DTFT using the fast Fourier transform. Computing the dampening region. The edge-on source, by contrast, has SNR in this way gives ρ ∼ 20 for the light and heavy sources multiple contributing harmonics, which are spread over a respectively when viewed both edge-on and face-on under range of frequencies, and the proportional contribution of the θ θ the configuration of parameters light and heavy. dampening region to the overall SNR is therefore diminished. ρ ≈ 20 For a nonevolving signal the SNR accumulates like In all cases we marginally exceed the threshold of pffiffiffiffiffiffiffiffi which is typically assumed to be required for EMRI Tobs, where Tobs is the total observation time. The pre- detection in the literature [4,55]. dampening regime lasts 308 days, and so from duration
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within the most sensitive range of the LISA detector. This is clear from looking at the time-frequency spectrogram of the heavier signal shown in Fig. 8. What we learn from this figure is that there are a significant number of harmonics that have comparable power to the dominant m ¼ 2 harmonic. We see also that the angular velocity at each harmonic, and thus fm, shows little rate of change for M ∼ 107 and η ∼ 10−6. This is consistent with [24,32], where it was shown that a large number of m harmonics is required to produce an accurate representation of the gravitational wave signal for a near-extremal EMRI, particularly for near edge-on viewing angles. For moderately spinning black holes a ∼ 0.9 there are not as many dominant har- monics, so those waveforms are cheaper to evaluate. We are now ready to move on to compute Fisher matrix estimates of parameter measurement precisions. This will be the focus of the next section.
VI. NUMERICS: FISHER MATRIX We now compute (33) numerically without making the θ ; simplifying assumptions used in Secs. III B and III C.We FIG. 8. Here we plot the spectrogram of hð heavy tÞ viewed edge on. We see 20 tracks in the time-frequency plane corre- will use one simplification, which is to ignore the spin E_ ∞ ˜ amΩ˜ θ ϕ sponding to the m ∈ f1; …; 20g harmonics. The color bar shows dependence in , Zlmðr; aÞ and −2Slm ð ; Þ and fix these that the m ¼ 2 harmonic (second lowest track in frequency) is at the values computed for a ¼ 1 − 10−9 using the BHPT. dominant, but that there are several other harmonics which We argued in Eq. (54) that the spin dependence of the flux contribute significantly to the radiated power. correction is a subdominant contribution in the near-ISCO regime, and this is further justified in Appendix B (see pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂ E_ 308 365 ≈ 93% Fig. 16 in particular). While a does grow as the ISCO is alone we would expect a fraction = of approached, it remains subdominant to the spin dependence SNR to be accumulated there. The difference to what we of the kinematic terms. This approximation is probably find above is explained by differences in amplitudes of the conservative in the sense that we are removing information individual harmonic(s). The heavier system is within the about the spin from the waveform model and so the true dampening regime throughout the last year of inspiral and measurement precision is most likely higher. Nonetheless we ρ ∼ 20 so all of the SNR of is accumulated there. This may expect this to be a small effect, and have verified that relaxing seem counterintuitive given the exponential decay of the this assumption does not significantly change the result for signal during the dampening regime. However, the expo- the heavier reference source (see Fig. 9). We note that we nential decay rate is relatively slow, a large number of make this assumption only for computational convenience. harmonics contribute to the SNR and the emission is all Waveform models used for parameter estimation on actual
FIG. 9. Left plot: parameter measurement precision, as estimated using the Fisher matrix formalism, for the three 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 relativistic corrections, E_ in the waveform model for the heavy source. Right plot: parameter measurement precisions for the source with parameters θlight, computed using the full waveform (blue asterix), only the inspiral phase (blue dot) and only the dampening phase (green diamond).
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θ 1 − 10−i ∈ 4 … 9 FIG. 10. We keep lightnfag fixed and vary a ¼ for i f ; ; g while computing estimates on the precision of the measured parameters using the Fisher Matrix. Results are shown for sources viewed face-on (left) and edge-on (right).
LISA data should use the most complete results available more information than we would expect based on its to ensure maximum sensitivity and minimal parameter contribution to the total SNR.7 biases. The spin measurement precision for the near-extremal To compute the waveform derivatives required to evalu- systems is three orders of magnitude better than for the ate (33), we use the fifth order stencil method system with moderate spin, while all other parameter measurements are comparable. ∂f −f2 þ 8f1 − 8f−1 þ f−2 Comparing to the exact Fisher matrix result with spin ≈ ; ð87Þ ∂x 12δx dependence included in all the various terms, we see that the two precisions are almost identical: the exact result for δx ≪ 1 and fi ¼ fðx þ iδxÞ. To avoid numerical offers precisions that are marginally better in comparison to instability of ∂ah for the near-extremal spin values of our approximate result (removing spin dependence from a ≤ 1 − 10−9, we ensure that δx<1 − a so the perturbed the corrections). This figure thus justifies ignoring the spin _ waveform does not have spin exceeding a ¼ 1. We further dependence of E, since relaxing that assumption makes ∂ ˜ assume that atend is zero so there is no spin dependence on almost no difference to the results. This numerically the total observation time. confirms our belief that the spin dependence in the θ In addition to the sources with parameters heavy and corrections to the fluxes are subdominant in the analysis θ light, we now consider a third source with parameters leading to (40). In the same plot 9, we also compare results of near-extremal black holes to moderately spinning holes. A direct comparison shows an increase in the spin precision θ ¼fr˜ðt0 ¼ 0Þ¼5.01;a¼ 0.9; μ ¼ 10 M⊙; mod by ∼3 orders of magnitude, which agrees with the intuition 2 106 ϕ π 1 M ¼ × M⊙; 0 ¼ ;Dedge ¼ Gpcg; ð88Þ given by the earlier analytic analysis, Eq. (79). To our knowledge, these are the first circular and with SNR ∼ 20. equatorial parameter precision studies for EMRIs that have Fisher matrix estimates of parameter measurement pre- employed Teukolsky-based adiabatic waveforms, rather cisions for all three sources viewed edge-on are shown in than approximate waveform models (or “kludges”), which Fig. 9. We do not present the results for a face-on have been used for many studies [3,4,15]. Comparing our observation as they are near-equivalent to the measure- results for the moderately spinning system to these previous ments presented in Fig. 9 for equivalent SNR. studies, we find that our results are very comparable, but a We see from this figure that we should be able to factor of a few tighter. This could be because we are constrain the spin parameter of near-extremal EMRI including only a subset of parameters and ignoring the sources to a precision as high as Δa ∼ 10−10, even when details of the LISA response, or because we have a more accounting for correlations among the waveform parame- complete treatment of relativistic effects. A more in depth ters. This is true for both the lighter and the heavier sources study addressing both of these limitations would be needed viewed edge-on and face-on, with a constraint a factor of a to understand the origin of the differences. However, the few better for the heavier source. The right panel of the agreement between our results and previous studies is figure compares the contribution to the measurement sufficiently close, and considerably less than the difference precision for the lighter source from the two different phases of the signal. We see that the high spin precision 7 ˜ In (40), the growth of ∂ar exceeds the growth of SnðfÞ ∼ comes almost entirely from the observation of the damp- const in the dampening regime. This sources the high precision ening regime and this phase of the signal contributes much measurement.
124054-18 CONSTRAINING THE SPIN PARAMETER OF NEAR-EXTREMAL … PHYS. REV. D 102, 124054 (2020) we find between the moderate and near-extremal spin log pðθjdÞ ∝ log pðdjθÞþlog pðθÞð89Þ cases, to give us confidence that our results are not being unduly influenced by these simplifications. where pðdjθÞ is the likelihood function, and pðθÞ is the In Fig. 10 we show how the parameter estimation prior probability distribution on the parameters. In our θ precision for the source with parameters light changes as context the likelihood is given by Eq. (27) and we will we vary the spin parameter, while keeping all other assume independent priors such that parameters unchanged. We present results for both face- on and edge-on viewing angles. This shows that while the 1 X θ ∝− − ; θ − ; θ θi measurement precision for most of the parameters is largely log pð jdÞ 2 ðd hðt Þjd hðt ÞÞ þ log pð Þ: independent of spin in the near-extremal regime, the spin θi∈θ precision steadily increases as a → 1. We note that even at a ð90Þ spin of 1 − 10−9, the measurement precision satisfies the Δ 1 − ; θ constraint a 124054-19 BURKE, GAIR, SIMÓN, and EDWARDS PHYS. REV. D 102, 124054 (2020) θ FIG. 11. The diagonal plots represent the marginalized posterior distributions on the parameters heavy. The plots below the diagonal are the joint two-dimensional posterior distributions. The red lines indicate the true values of the injected signal. to the inverse Fisher matrix, evaluated at the known dependence induced through the kinematic terms present injection parameters. in (20), as discussed in Sec. VI, and not its dynamical terms, justifying our approximation. B. Results The priors on a; ϕ0, and D for both sources were We compute MCMC posteriors for the two signals θ θ ; −4 −8 hðf heavy; lightg tÞ for the waveform model (20) for the a ∼ 1 − U½10 ; 10 face-on case only. As before, we construct waveforms ϕ0 ∼ U½0; 2π ignoring the spin dependence in E_, the Teukolsky ampli- ∞ aΩ˜ θ ϕ D ∼ U½1; 8 Gpc: tudes Zlm and the spheroidal harmonics −2Slm ð ; Þ.We evaluate these for a fixed spin parameter of a ¼ 1 − 10−9. We remind the reader the main drive for the tight The priors on μ, M, and r˜0 were chosen differently for the constraints on the spin parameter is due to the spin heavy and light source as 124054-20 CONSTRAINING THE SPIN PARAMETER OF NEAR-EXTREMAL … PHYS. REV. D 102, 124054 (2020) FIG. 12. As Fig. 11, but now for the source with parameters θlight. μ ∼ 18 22 heavy U½ ; M⊙ are confident these restrictions are not influencing the results. μ ∼ U 8; 12 M⊙ light ½ Evaluating the likelihood for EMRI waveforms is an ∼ 0 9 1 1 107 Mheavy U½ . ; . × M⊙ expensive procedure. In order to obtain a sufficient number 6 of samples from the posterior, we used high performance M ∼ U½1.9; 2.1 × 10 M⊙: light computing facilities and ran 20 unique chains for N ¼ 40, heavy r˜0 ∼ U½1.2; 1.3 000 iterations. All chains analyzed the same input dataset, but with different initial random seeds. This ensures that the ˜light ∼ 4 2 4 4 r0 U½ . ; . : dynamics of the chains are different but the noise realiza- tions are the same for each MCMC procedure. The prior on a ensures that we do not move outside the The marginal posterior distributions and two-dimen- range in which our approximations are valid, a ≳ 0.9999. sional contour plots for the two sources are shown in The tight priors on the individual component masses helped Figs. 11 and 12. These plots confirm the high precisions of to improve the computational efficiency of our algorithm. parameter measurements that were seen with the Fisher However, there was no evidence of the MCMC chains matrix. The relative uncertainties Δθ=θ are similar for the reaching the edges of the priors in our simulations, so we two sources, although we can measure the spin parameter 124054-21 BURKE, GAIR, SIMÓN, and EDWARDS PHYS. REV. D 102, 124054 (2020) FIG. 13. For the source with parameters θheavy,we compare the one-dimensional marginalized posterior distributions (orange histograms) to a Gaussian distribution (blue solid line), centered at the posterior mean, and with standard deviation set to the prediction of the Fisher matrix. more precisely for the heavier source. For the most part the waveform phase shifting by one cycle within the late posteriors are unimodal, apart from the spin posterior of the dampening regime. We also note that shifts in the peak lighter source. We have verified that the secondary modes of the posterior away form the true value are larger for the are real features of the likelihood, and correspond to the heavier source than for the lighter source. This appears to θ FIG. 14. As Fig. 13, but now for the lighter source with parameters light. 124054-22 CONSTRAINING THE SPIN PARAMETER OF NEAR-EXTREMAL … PHYS. REV. D 102, 124054 (2020) be due to the particular noise realization. For other noise removing the spin dependence in corrections to the radial realizations the noise-induced biases for the heavier source fluxes. This meant we could build an interpolant, valid for are smaller. For noise-free datasets, we find posterior quasicircular and equatorial inspirals, into rapidly spinning distributions peaked at the true parameters, as expected. black holes with a ≳ 0.999. Using this approximation, we The primary reason for doing the MCMC simulations were able to build a waveform model valid for near- was to verify the Fisher matrix results found earlier. In extremal spins that could be used for parameter precision Figs. 13 and 14, we plot the marginalized posteriors on the and estimation studies. θ θ parameters f light; heavyg alongside a Gaussian distribution Our analysis showed that the LISA optimum masses for with variance given by the Fisher matrix and centered at the parameter estimation and precision studies are heavier mass 7 mean value of the posterior distributions pðθjdÞ. systems M ≈ 10 M⊙. For heavier sources, the emitted These results nicely confirm the accuracy of the Fisher frequencies will lie within the minimum of the LISA PSD matrix results for these sources. In each case, the 1σ during the dampening regime; when the signal provides precision predicted by the Fisher matrix is slightly smaller most accurate information about the source. Analyzing than the width of the numerically computed posterior. This different parts of the signal, we were able to conclude that is to be expected as the Fisher matrix also provides the the end part of the signal will be “loud” in the data stream Cramer-Rao lower bound on parameter uncertainties. for spins approaching near-extremality a → 1. For lighter 6 However, the difference is very small. We are thus con- mass sources M ∼ 10 M⊙, the accumulated SNR of the fident that all of our Fisher matrix predictions are accurate, dampened part of the signal is weak. Fully numerical Fisher including the exploration of parameter space shown in matrix analysis revealed that we can constrain the spin Fig. 10. We conclude that even at the near-threshold SNR parameter of a near-extremal EMRI ∼ 3 orders of magni- of this source, ρ ∼ 20, the Fisher matrix can be used to tude higher than moderately spinning black holes a ∼ 0.9. confidently estimate the precision of parameters for near- For very near-extremal primaries a ∼ 1 − 10−9, one is able extremal sources. to constrain the spin parameter with precision Δa ∼ 10−10. This is, as far as we know, the tightest constraint on VIII. CONCLUSION astrophysical orbital parameter found in the literature. This work has shown that we expect tighter spin In the final section, with the view of verifying our Fisher constraints on the Kerr spin parameter for EMRIs when matrix calculations, we performed a parameter estimation the primary is rapidly rotating rather than moderately study on these near-extremal sources. We showed that the rotating. We argued that the spin precision is sourced signal parameters are able to be extracted with precision through the rapid growth of the spin dependence on the comparable to our Fisher matrix estimates. radial trajectory throughout the near-ISCO regime. It was To conclude, we can safely say that the spin parameter of shown that radiation-reaction effects are subdominant to near-extremal EMRIs can be measured with excellent geodesic motion, which is what ultimately drives the precision. We are also confident that the increase in excellent constraint on the spin parameter. precision is governed by the spin dependence on the radial This work has tried to answer the question of how well trajectory. If the dampened part of the signal is not observed ∂ ˜ ≫ 1 LISA can constrain the spin parameter of an EMRI source, (where ar ), then we cannot make such precise conditioned on the primary rapidly rotating. In the earlier statements on the spin of the primary black hole. sections of this paper, we were able to derive a general There are a few extensions to this work. The first would linear ODE whose solution describes the spin dependence be to include the results of the transition from inspiral to on the radial trajectory. Using this ODE, we were able to plunge in [40] to our analysis here. We suspect that, argue that this spin dependence is much more significant although the SNR accumulation would be weak, the extra ∂ ˜ for near-extremal primaries rather than moderately spin- contribution from ar could only improve the precision ning primaries a ∼ 0.9. It was also shown that the spin measurement. Another interesting direction to explore dependence on the radial evolution is dominated more by would be the effect of systematic errors (through inaccurate geodesics, rather than the radiation reactive terms. waveform modeling) in our EMRI PE studies. Through the Using reasonable assumptions, analytic Fisher scalars Cutler and Vallisneri formalism [56], one could investigate were derived for both rapidly rotating and moderately whether parameter estimation/precision studies on near- spinning black holes. It was then shown, explicitly, that the extremal waveforms could be dominated by systematic precision on spin parameters for near-extreme black holes errors rather than statistical ones induced purely by noise should exceed that of moderately spinning ones by a few realizations from the detector. Given that we have shown orders of magnitude. The exceptional precision is governed precisions on the spin parameter on the order Δa ∼ 10−10,it by the growth of the spin dependence of the trajectory near would be of utmost importance to have immensely accurate the ISCO. waveform models in order to keep the systematic error to a Given that the kinematical terms dominate the precision, minimum. This would be to ensure that parameter estimates we were able to build a near-extremal waveform model by are not affected by (deterministic) biases due to waveform 124054-23 BURKE, GAIR, SIMÓN, and EDWARDS PHYS. REV. D 102, 124054 (2020) ˜ ˜ ˜ ˜ ˜ ˜ modelling errors rather than (probabilistic) biases due to ∂ahmðtÞ¼jhmðtÞjfsinðmΩ tÞBm þðmt∂aΩÞ cosðmΩ tÞg noise fluctuations of the LISA detector. Another obvious extension would be to introduce more ðA2Þ parameters in the parameter space, such as eccentricity and inclination. In this case, we are unsure whether the where jhmðtÞj stands for the amplitude without oscillatory kinematic terms would dominate over the flux corrections. factor and we defined As such, we would require a more complex and general ∂ _ ∞ ∂ Ω˜ waveform model. As seen in [24], eccentric trajectories of aEm a B ≡ − : ðA3Þ m _ ∞ Ω˜ secondaries in the vicinity of near-extremal black holes 2Em exhibit inverse zoom-whirl behavior. That is, the “zoom” part of the inspiral (near aperiastron) outputs larger ampli- Using Parseval’s identity, the integrand in the Fisher matrix tude radiation than the “whirl” phase. Perhaps this extra can be decomposed information, unique to near-extreme EMRIs, could provide X X even tighter constraints on the spin parameter a. It would be 2 2 j∂ahðtÞj ¼ ð∂ahmÞ þ 2 ∂ahn∂ahm; ðA4Þ interesting to see whether the larger number of parameters m n 124054-24 CONSTRAINING THE SPIN PARAMETER OF NEAR-EXTREMAL … PHYS. REV. D 102, 124054 (2020) which is satisfied for all secondary locations x before behavior of the relativistic corrections becomes somewhat entering the transition regime. universal as a → 1. In other words, as the near extremal The discussion of the off-diagonal terms is very similar. parameter approaches unity, the corrections approach a Any such term can be written as constant value. This means that the correction values at some r˜ will not differ much as the spin parameter is jhmjjhnj ˜ ˜ ˜ ˜ 2 changed providing the spin parameter is close to one. That ∂ahm∂ahn ¼ fcosððm−nÞΩtÞ½BmBn þmnðt∂aΩÞ 2 is, spin dependence on E_ is weak “far” from ISCO. ˜ ˜ ˜ ˜ 2 ∂ E_ ≈ 0 þcosððmþnÞΩtÞ½−BmBn þmnðt∂aΩÞ We emphasize that this does not mean that a throughout the entire inspiral. Here we have shown three þsinððmþnÞΩ˜ t˜Þ½nt˜∂ Ω˜ B þmt˜∂ Ω˜ B a m a n radial coordinates which are considerably far away from ˜ ˜ ˜ ˜ ˜ ˜ þsinððm−nÞΩtÞ½nt∂aΩBm −mt∂aΩBn g: the ISCO in the near-extreme spin parameter case. The closer in radial coordinate to the ISCO, the larger the spin ðA8Þ dependence in the relativistic corrections. This can be observed in Fig. 16. For a weak gravitational field, the The key point is that all such terms involve oscillatory 2 spin dependence on the flux corrections for near-extreme functions, even the ones having Oðt˜ Þ dependence. Hence, ˜_ when compared to the contribution from the diagonal ones, inspirals is small, approaching E → 0 for a → 1 as the they are subleading. This justifies our claim in (40). ISCO is approached. Much smaller than when compared to moderately spinning holes, where the total energy flux is approximately constant. Only when very close to the ISCO APPENDIX B: GENERAL RELATIVISTIC does the spin dependence on E_ grow significantly for near- CORRECTIONS extremal holes. _ _ −9 We begin by focusing on the total energy flux E˜ . The left A comparison of our interpolant, E for a ¼ 1 − 10 _ most panel of Fig. 15 shows Eðr˜Þ at fixed a and the right with high spinning fluxes computed from the BHPT are _ given in Fig. 17. A useful approximant for the relativistic most panel EðaÞ at fixed r˜. The left plot of Fig. 15, shows E_ that for moderately rotating holes with a ≲ 0.99, the corrections ðrÞ is given by corrections E_ are not strongly dependent on the radial coordinate r˜. However, approaching an extremal spin _ parameter a indicates that E rapidly goes to zero as the _ EðxÞ¼ax arctanðbxcÞ;x¼ r˜ − 1 ðB1Þ ISCO is approached. This matches the description given by both NHEK fluxes (12) and (11). Using a Teukolsky solver from the BHPT, we generated a ≥ 0 999 our own energy flux values for spin parameters . . with constants ða; b; cÞ¼ð0.6897912; 1.084803; The corresponding corrections are shown in the right panel −0.936685Þ. This approximates the high spin corrections of Fig. 15 for fixed coordinate radii. This shows that at a E_ 1 − 10−9 ˜ ≲ 6 ∼1% fixed coordinate radius r˜, the corrections approach a ðr; Þ for r with largest fractional error constant as a → 1. What we learn from this is that the at the ISCO. _ FIG. 15. The figure on the left shows the values of the GRCs Eðr˜Þ evaluated at a fixed spin parameter. The purple dashed line (bottom left of leftmost plot) uses the near-extremal approximation to the flux (11). The other colors are the tabulated values GRCs presented in Table IV in Ref. [11]. The figure on the right shows the GRCs evaluated at a fixed radial coordinate r˜ while varying the spin parameter a. The GRCs for a>0.999 were computed using the BHPT [46]. 124054-25 BURKE, GAIR, SIMÓN, and EDWARDS PHYS. REV. D 102, 124054 (2020) precision measurements. One first has to be sure that the regime of SNR is high enough so that the linear signal approximation applies when truncating the perturbed wave- form at first order. Numerical derivatives also exhibit convergence problems and are very problematic if your waveforms are not smooth. One must also make sure to taper each derivative so that “hard cutoffs” do not feature in the waveform [64]. Finally, one has to invert the Fisher matrix and, if the Fisher matrix is ill conditioned (as normally the case for EMRIs), it could lead to catastrophic errors in the elements of Γ−1. The condition number of a matrix is defined through FIG. 16. Here we compute ∂ E_ for the two spin parameters a a 0 999999 a 0 9 ∂ E_ ¼ . (orange) and ¼ . (blue). The growth in a for max fλ g rapidly rotating holes is larger closer to ISCO than for moderately Γ i i Condð Þ¼ λ ; ðC1Þ spinning holes. minjf jg where λi is an ith eigenvalue of the matrix Γ. In the case of EMRIs, the computation of the fisher matrix is affected by numerical instabilities. A small perturbation to the systems (intrinsic) parameters lead to a large overall change in the waves phase evolution. It is for this reason EMRI obser- vations permit parameter constraints with such high pre- cision. Since waveforms are sensitive to small perturbations of the source parameters, the numerical derivatives ∂θhðθ; tÞ are large and so the elements of the Fisher matrix can be enormous in magnitude. In other words, there is a significant amount of “information” about the parameters θ encoded in the waveforms, which are then reflected by the large elements of the Fisher matrix. However, not all derivatives are large and the differences between the size of the elements for different parameters leads to signifi- cantly varying eigenvalues. In our case, the measurements of spin and distance are ∼8 orders of magnitude apart. A consequence of this is that the condition number of the Fisher matrix is CondðΓÞ ∼ 1021. In light of these known instabilities, we spend the next few subsections providing suitable tests to verify our Fisher matrix in the single parameter case, assuming all other parameters are known perfectly. In order to verify Fisher matrices over multiple parameters, Bayesian techniques like MCMC are required (see Sec. VII). FIG. 17. On the left panel, we have used the BHPT to compute We used three different methods to verify the Fisher the total energy flux for near-extremal spin parameters with our matrix calculations without performing a Bayesian interpolant (black dashed line) overlaid. The right panel is a zoom in of the left plot giving visual aid as to why our interpolant can be analysis: used to approximate a larger regime of spin parameters. (1) Verification of the linear signal approximation ∂h hða þ ΔaÞ ≈ hðaÞþΔa : APPENDIX C: VERIFICATION OF FISHER ∂a MATRIX CALCULATIONS (2) Overlaps defined through (30)—perturbing the spin 1. Ill-conditioned Fisher matrix parameter by Δa given by the Fisher matrix should It is well known throughout the literature [6,58–63] that return overlaps close to one. a Fisher matrix analysis, when not handled carefully, has (3) Likelihood—The log-likelihood is maximized at the the potential to underestimate (or, worse, overestimate) true parameters θ0. If a parameter is perturbed by the 124054-26 CONSTRAINING THE SPIN PARAMETER OF NEAR-EXTREMAL … PHYS. REV. D 102, 124054 (2020) Fisher matrix estimate then it should be a measure of the 1σ width of this log-likelihood. As a proof of principle of these methods, we will compute θ the precision on the spin parameter for a heavy source heavy [see (80)], viewed face-on with ρ ∼ 20. For a source with Δ ∼ 2 this configuration of parameters, we found aNHEK × 10−10 numerically. a. Linear-signal approximation In the derivation of the Fisher matrix, we used the linear- signal approximation so a first test would be to test whether it is valid in our analysis. To test whether the expansion is FIG. 18. The blue curve is the likelihood (C5) evaluated on a valid in the regime of SNR we are considering, we compute grid of points. The red the true value a ¼ 1 − 10−6 and orange the the overlap precision measurement predicted by the 1 parameter Fisher matrix Δa . ∂ NHEK h −5 O hða þ ΔaÞjhðaÞþΔa ≈ 1 − 10 ðC2Þ ∂a a¼atrue Since the noise realization in the data stream dðtÞ¼ hðt; aÞþnðtÞ induces a bias to the maximum likelihood Hence we conclude that our waveform model at ρ ∼ 20 estimate, and does not affect the likelihood width, we shall does not violate the linear-signal approximation. ignore the noise in this case. As such, we will consider a zero noise approximation and use d ¼ hðaÞ with signal b. Overlaps templates h ≔ hða þ Δa; tÞ. Substituting this into the In the limit of high SNR, the inner product (28) can be likelihood above we find that expanded in ρ. Observe for small Δa ∂ Δ 2 ∂2 1 2 h a h 2 ðhða þ ΔaÞjhðaÞÞ ≈ ρ þ Δa h þ h log pðdjaÞ ≈ ðρ − 1Þ: ðC4Þ ∂a 2 ∂a2 2 2 where ðhjhÞ¼ρ as in (29). Since the SNR is fixed, it’s −1 2 Here we have assumed that Γ ¼ Δa . Calculating easy to show that ð∂ hjhÞ¼0 and Γ ¼ −ð∂2hjhÞ. It can aa a aa a log pðdjaÞ for d ¼ hða þ ΔaÞ with Δa our Fisher matrix then be shown for ρ ≫ 1 estimate we found that log pðdjaÞ ≈ 199.46 which agrees 1 with the above formula, to a precision of 0.05%. O Δ ≈ 1 − ρ−4 ðhða þ aÞ;hðaÞÞ ρ2 þ Oð Þ: ðC3Þ Since we are interested in only one parameter, it is evaluate the likelihood function Δ Δ Substituting a ¼ aNHEK into the left-hand side of (C3), we numerically find an agreement of ∼0.01%. pðdjaÞ ∝ exp½−ðd − hjd − hÞ=2 ðC5Þ c. Likelihood In the high SNR limit, the precision predicted by the on a grid of spin parameter values. In doing so, we find Fisher matrix should approximate the 1σ width of the Fig. 18. The area between the yellow line and red line is likelihood function. Using Eq. (27), we can write the log- approximately 31.51%, which is a reasonable approxima- likelihood as tion to the true 1 − σ ≈ 34% width of likelihood. 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