Self-consistent adiabatic inspiral and transition motion
aa r X i v : . [ g r- q c ] F e b Self-consistent adiabatic inspiral and transition motion
Geoffrey Comp`ere ♣∗ and Lorenzo K¨uchler ♦† ♣♦ Universit´e Libre de Bruxelles and International Solvay Institutes, C.P. 231, B-1050 Bruxelles, Belgium ♦ Institute for Theoretical Physics, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium
The transition motion of a point particle around the last stable orbit of Kerr is described at leadingorder in the transition timescale expansion. Taking systematically into account all self-force effects,we prove that the transition motion is still described by the Painlev´e transcendent equation of thefirst kind. Using an asymptotically matched expansions scheme, we consistently match the quasi-circular adiabatic inspiral with the transition motion. The matching requires to take into accountthe secular change of angular velocity due to radiation-reaction during the adiabatic inspiral, whichconsistently leads to a leading-order radial self-force in the slow timescale expansion.
PACS numbers: 04.30.-w, 04.25.-g, 04.25.Nx, 11.10.Jj
Binary coalescences are the loudest signals of all cur-rent and prospective gravitational wave observatories [1–4]. Current waveform models of such events are so-phisticated interpolations between results from numer-ical relativity, EOB methods, the post-Newtonian/post-Minkowskian formalism and black hole perturbation the-ory, see Table III of [3] for a list of references. The currentand future high precision tests of General Relativity [5]strongly motivate the understanding of possible system-atic errors in current waveform models.In particular, the modeling of the transition from in-spiral to merger of binaries is notoriously difficult since itoccurs in the strong field regime. In the quasi-circular ap-proximation, non-perturbative resummation techniqueshave been used to obtain explicit models for comparablemass binaries [6–10] and small mass ratio binaries [11–14]. The deviation from quasi-circularity was estimatedto be numerically small, d log r/dφ . .
05, even aroundthe innermost stable circular orbit [6]. In the small massratio expansion, the transition timescale regime was de-fined in the quasi-circular approximation and neglectingself-force effects [15] but it was later shown that non-quasi-circular corrections occur at the same order in thistransition regime [16]. For extensions, see [17–21].The main aim of this paper is to provide an accurateand complete treatment of the matching between theinspiral and transition motion in the small mass ratioregime, taking into account all self-force effects, with themotivation to extend current self-force models and pro-vide more faithful EOB models. The inspiral motion canbe studied via the slow timescale expansion [22], whichbreaks at the separatrix between bound and plunging or-bits [23] or during resonances [24, 25]. Radiation-reactionrequires an inspiral with dynamical angular velocity. Wewill first derive such an inspiral in the adiabatic regimearound a Kerr black hole, thereby extending the cur- ∗ [email protected] † [email protected] rent quasi-circular parametrization with geodesic circu-lar angular velocity [26, 27] to dynamical angular veloc-ity. As we will demonstrate, consistency of the match-ing between the forced inspiral motion and the transi-tion motion will lead to a leading-order radial self-forcein the slow timescale expansion that is determined by thesecular change of angular velocity required by radiation-reaction.In the following, we will review the equations of equato-rial forced geodesics, solve them in the adiabatic approxi-mation for non-geodesic angular velocity and expand thesolution close to the last stable orbit. We will then solvethe equations again but in the transition timescale ex-pansion. We will finally match the two expansions usingthe method of asymptotically matched expansions andconclude. Conventions
We use geometrical units G = c = 1. Allquantities are made dimensionless using the mass M ofthe Kerr background, including the Kerr angular mo-mentum a , the binary mass ratio η = m/M , the properparticle energy e = − p t /m and the proper particle az-imuthal angular momentum ℓ = p φ / ( mM ). Spacetimeindices are lowered and raised with the Kerr metric inBoyer-Linquist coordinates g µν . The outer horizon is thelargest root of ∆ = r − r + a . EQUATORIAL FORCED GEODESICS
We consider equatorial orbits around the Kerr blackhole with position z µ = ( t, r, π , R Ω dt ) where Ω = dφ/dt is the orbital frequency. We denote as s = sign(Ω), i.e. s = +1 for prograde orbits and s = − v µ = dz µ /dτ and, in particular, theredshift is denoted as U = dt/dτ where τ is the dimen-sionless proper time. In terms of the angular momentum ℓ = v φ and energy e = − v t one has U = − g tt e + g tφ ℓ, Ω = − g tφ e + g φφ ℓ − g tt e + g tφ ℓ . (1)It will be convenient to introduce δ as the deviation fromthe geodesic angular velocity Ω geo = s/ ( r / + sa ) asΩ = sr / + sa + δ , δ = s (Ω − − Ω − ) . (2)The forced geodesic equations v α ∇ α v µ = f µ are equiva-lent to (i) the radial equations (cid:18) drdτ (cid:19) = e − V geo , d rdτ + 12 ∂V geo ∂r = f r , (3a) V geo ≡ − r + ℓ + a (1 − e ) r + 2( ℓ − ae ) r ; (3b)(ii) the energy and angular momentum flux-balance equa-tions dℓdτ = f φ , dedτ = − f t ; (4)(iii) the normalisation of the velocity v µ v µ = − U − = − g µν dz µ dt dz ν dt ; (5)and, (iv) the orthogonality of the force with the velocity, f µ v µ = 0, which can be written asΩ − dedτ − dℓdτ = f r drdτ (Ω U ) − . (6) SLOW TIMESCALE EXPANSION
We now restrict our analysis to the small mass ratiolimit η ≪ X = X (0) (˜ τ ) + O ˜ τ ( η ) (7)where all variables are collectively denoted as X =( a, δ, r, Ω , U, e, ℓ ). Here, the slow proper time is defined as˜ τ ≡ η τ . Since all quantities have been made dimension-less using a rescaling with the mass M , we will disregardthe slow time evolution of M . The symbol O ˜ τ ( η ) refersto the limit η → τ . Neglectingsuch corrections defines the adiabatic approximation.Eqs. (4) and (6) then lead to the expansion f a = η f a (1) (˜ τ ) + O ˜ τ ( η ) , a = t, φ, (8) f r = f r (0) (˜ τ ) + η f µ (1) (˜ τ ) + O ˜ τ ( η ) . (9)As we will discuss below, the consistent matching of theadiabatic inspiral with the transition motion will requirea leading-order radial self-force f r (0) (˜ τ ), see (46) below. Quasi-circular adiabatic inspiral
The adiabatic solution without eccentricity to Eqs.(1)-(3)-(4)-(5)-(6) can be found straightforwardly. In or-der to write compact expressions, it is convenient to de-fine the coefficients A = r − r + 2 sa (0) r / ,B = r − sa (0) r / + a , (10) C = r / − r / + sa (0) , D = B (4 Ar − − , and their δ (0) -corrected version, A δ = A + 2 Cδ (0) + (1 − r − ) δ , (11a) B δ = B − sa (0) r − δ (0) , (11b) C δ = C + (1 − r − ) δ (0) , (11c) D δ = 4 A δ ( Br − + δ (0) r − / (1 + a r − )) − ∆(3 B δ + 2 sδ (0) (2 + r − / δ (0) )Ω − r − ) . (11d)Importantly, the function D admits a single root outsidethe horizon at the location of the geodesic ISCO r ∗ , DB | ∗ = r ∗ − r (0) ∗ + 8 sa √ r (0) ∗ − a = 0 . (12)The unique solution to (1)-(3)-(5) can be written asΩ (0) (˜ τ ) = s ( r / + sa (0) + δ (0) ) − , (13a) U (0) (˜ τ ) = sA − / δ Ω − = A − / δ | Ω (0) | − , (13b) ℓ (0) (˜ τ ) = sB δ A − / δ , e (0) (˜ τ ) = C δ A − / δ . (13c)The radial self-force is algebraically determined as [32] f r (0) ( r (0) , δ (0) , a (0) ) = ∆ U Ω δ (0) r ( δ (0) + 2 r / ) . (14)Eq. (6) is then equivalent to da (0) d ˜ τ = 0 , (15)which implies that a (0) is a constant. The flux-balanceequations (4) are equivalent to dr (0) d ˜ τ = r A δ δ (0) ( δ (0) + 2 r / ) (cid:16) e (0) f t (1) − ℓ (0) f φ (1) (cid:17) , (16) dδ (0) d ˜ τ = − A / δ B δ f t (1) + √ r (0) D δ B δ dr (0) d ˜ τ . (17)These equations are linear in the first order self-force andnon-linear in the kinematic parameters r (0) (˜ τ ), δ (0) (˜ τ )and a (0) . Since the self-force is an integral over the pastmotion of the source, these evolution equations are re-tarded integro-differential equations. Inspiral towards the last stable orbit (LSO)
The adiabatic expansion of the inspiral breaks downat the LSO where a singularity in the evolution equation(16) must appear. None of the quantities f a (1) , A δ , B δ , C δ , ∆ can diverge along the trajectory. We deduce that δ (0) vanishes at the LSO, δ (0) ∗ ≡ δ (0) (˜ τ ∗ ) = 0 where ˜ τ ∗ isthe slow proper time at the LSO. Even in the presenceof self-force, the radial potential is given by the geodesicpotential (3b). The LSO is therefore the innermost stablecircular orbit of Kerr (ISCO) that is reached at the radiallocation r (0) = r (0) ∗ where ∂ V geo ( e, ℓ, r, a ) ∂r (cid:12)(cid:12)(cid:12)(cid:12) ∗ = 0 . (18)Since δ (0) ∗ = 0 this condition amounts to the radialgeodesic location (12) for which D δ | ∗ = D | ∗ = 0. Wedenote as e (0) ∗ = C/ √ A | ∗ , ℓ (0) ∗ = sB/ √ A | ∗ , Ω (0) ∗ = s/ ( r / ∗ + sa (0) ) the energy, angular momentum and an-gular velocity at the ISCO in the adiabatic limit.The asymptotic behavior towards the ISCO can be ob-tained by first expanding ℓ (0) − ℓ (0) ∗ = κ ∗ (0) , (˜ τ ∗ − ˜ τ ) + κ ∗ (0) , (˜ τ ∗ − ˜ τ ) / + O (˜ τ ∗ − ˜ τ ) , (19) e (0) − e (0) ∗ = Ω (0) ∗ κ ∗ (0) , (˜ τ ∗ − ˜ τ ) + e ∗ (0) , (˜ τ ∗ − ˜ τ ) / + O (˜ τ ∗ − ˜ τ ) . (20)The inspiral motion then implies the following expansion r (0) − r (0) ∗ = r ∗ (0) , (˜ τ ∗ − ˜ τ ) / + O (˜ τ ∗ − ˜ τ ) , (21) δ (0) = δ ∗ (0) , (˜ τ ∗ − ˜ τ ) + O (˜ τ ∗ − ˜ τ ) / , (22)Ω (0) − Ω (0) ∗ = Ω ∗ (0) , (˜ τ ∗ − ˜ τ ) / + O (˜ τ ∗ − ˜ τ ) , (23) f r (0) = 83 r − / ∗ δ (0) + O (˜ τ ∗ − ˜ τ ) / , (24)after using 4 A ∗ = 3 r (0) ∗ ∆ ∗ . Solving Eqs. (4)-(13)-(16)-(17), we obtain Ω ∗ (0) , = − sr / ∗ Ω ∗ r ∗ (0) , and δ ∗ (0) , = Ω (0) ∗ ∆ ∗ (cid:18) sr (0) ∗ E ∗ A ∗ ( r ∗ (0) , ) − A / ∗ κ ∗ (0) , (cid:19) , (25) e ∗ (0) , = Ω (0) ∗ (cid:18) E ∗ Ω (0) ∗ r / ∗ A / ∗ ( r ∗ (0) , ) − s r / ∗ Ω (0) ∗ r ∗ (0) , κ ∗ (0) , + κ ∗ (0) , (cid:19) , (26)where E ∗ ≡ − a +3 a r (0) ∗ + a (0) s (107 − a ) r / ∗ − − a ) r ∗ + 25 r ∗ . The free parameters κ ∗ (0) , , κ ∗ (0) , , r ∗ (0) , will be fixed from the matching with thetransition timescale expansion to which we turn. TRANSITION TIMESCALE EXPANSION
We consider the expansion in the transition timescale s ≡ η / ( τ − τ ∗ ) (27)around the ISCO crossing time τ ∗ or s ∗ = 0.We define the variables R , ξ and Y as [15, 16] r − r [0] ∗ = η / R ( η, s ) , ℓ − ℓ [0] ∗ = η / ξ ( η, s ) , (28a) e − e [0] ∗ = Ω [0] ∗ h η / Y ( η, s ) + η / ξ ( η, s ) i (28b)where the ISCO values are expanded in powers of η / as r | ∗ = r [0] ∗ + ∞ X i =2 η i/ r [ i ] ∗ , e | ∗ = e [0] ∗ + ∞ X i =4 η i/ e [ i ] ∗ ,ℓ | ∗ = ℓ [0] ∗ + ∞ X i =4 η i/ ℓ [ i ] ∗ , Ω | ∗ = Ω [0] ∗ + ∞ X i =2 η i/ Ω [ i ] ∗ . The values R | ∗ , ξ | ∗ , Y | ∗ encode the shifts of these quan-tities at the ISCO, i.e. R | ∗ = P ∞ i =2 r [ i ] ∗ η i − , . . .In the absence of radial self-force all variables R, ξ, Y, a scale as η in the transition region for standard spins[15, 16, 20, 21]. In the presence of radial self-force, wewill assume the same scaling and show consistency. Wetherefore expand R = ∞ X i =0 η i/ R [ i ] ( s ) , Y = ∞ X i =0 η i/ Y [ i ] ( s ) , (29) ξ = ∞ X i =0 η i/ ξ [ i ] ( s ) , a = a [0] + ∞ X i =1 η i/ a [ i ] ( s ) . (30)The expansion of Ω | ∗ is consistent with Eq. (1). Consis-tently with Eqs. (4) and (6) we have f a = f a [0] ( s ) η + O s ( η / ) , a = t, φ, (31) f r = f r [0] ( s ) η / + O s ( η ) , (32)where O s ( η ) refers to terms of order η at fixed s . Theangular momentum flux-balance law (4) becomes ξ ( s ) = ξ [0] ( s ) + O s ( η / ) with dξ [0] /ds = f φ [0] ( s ). Leading-order transition equations
We now derive the solution to Eqs. (1)-(3)-(4)-(5)-(6) at leading order in the transition timescale expan-sion around the ISCO. The condition (18) together with(1)-(3)-(5) give at leading order in η and at the ISCOthe same quantities a [0] ∗ = a (0) , r [0] ∗ = r (0) ∗ , δ [0] ∗ = 0, e [0] ∗ = e (0) ∗ , ℓ [0] ∗ = ℓ (0) ∗ , Ω [0] ∗ = Ω (0) ∗ as the adiabaticinspiral.From (3) we obtain as algebraic equations a [1] = a [2] = a [3] = 0, while a [4] , a [5] and a [6] are proportional to D | ∗ and therefore vanish as well from (12). This matcheswith the constancy of a (0) , (15), in the adiabatic inspiral.Using (1) and (18) the deviation δ defined in (2) is givenaround the ISCO at leading order as δ = δ [0] η / + O s ( η ) , δ [0] = π ∗ ∆ ∗ R ( s ) − A / ∗ Ω [0] ∗ ∆ ∗ ξ [0] ( s )(33)where π ∗ ≡ a s − a (0) sr (0) ∗ +2(3+ a ) r / ∗ − a (0) sr ∗ .Any quantity X ( r, δ, a ) that is finite at the ISCO can nowbe expanded as X = X [0] ∗ + η / ∂X∂r (cid:12)(cid:12)(cid:12)(cid:12) [0] ∗ R ( s ) + η / ∂X∂δ (cid:12)(cid:12)(cid:12)(cid:12) [0] ∗ δ [0] ( s )+ 12 ∂ X∂r (cid:12)(cid:12)(cid:12)(cid:12) [0] ∗ R ∗ ( s ) ! + O s ( η ) . (34)In particular for η − f φ ( r, δ, a ), comparing with (31) gives f φ [0] ( s ) = − κ ∗ so that ξ [0] ( s ) = − κ ∗ s . It is clear that κ ∗ > f r ( r, δ, a ), comparing with (32) leads to f r [0] ( s ) = ǫ ∗ R ( s ) − ζ ∗ ξ [0] ( s ) , (35) ζ ∗ ≡ A / ∗ Ω [0] ∗ ∆ ∗ ∂f r ∂δ (cid:12)(cid:12)(cid:12)(cid:12) [0] ∗ , ǫ ∗ ≡ ∂ f r ∂r (cid:12)(cid:12)(cid:12)(cid:12) [0] ∗ + π ∗ ∆ ∗ ∂f r ∂δ (cid:12)(cid:12)(cid:12)(cid:12) [0] ∗ . Expanding Eq. (3), the leading-order transition equationsare then given by (cid:18) dR [0] ds (cid:19) = − α ∗ R − β ∗ κ ∗ sR [0] + γ ∗ Y [0] ,d R [0] ds = − ( α ∗ − ǫ ∗ ) R − κ ∗ ( β ∗ − ζ ∗ ) s, (36) dds (cid:18) Y [0] − ǫ ∗ γ ∗ R − κ ∗ ζ ∗ γ ∗ R [0] (cid:19) = 2 κ ∗ β ∗ − ζ ∗ γ ∗ R [0] where the coefficients read as α ∗ ≡ ∂ V geo ∂r (cid:12)(cid:12)(cid:12)(cid:12) [0] ∗ , γ ∗ ≡ ∂V geo ∂ℓ (cid:12)(cid:12)(cid:12)(cid:12) [0] ∗ , (37a) β ∗ ≡ − (cid:18) ∂ V geo ∂r∂ℓ + Ω ∂ V geo ∂r∂e (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) [0] ∗ . (37b)We can assume β ∗ = ζ ∗ and α ∗ = ǫ ∗ since these coeffi-cients are a priori not directly related. Introducing x [0] ≡ ( α ∗ − ǫ ∗ ) / ( β ∗ − ζ ∗ ) − / κ − / ∗ R [0] , y [0] ≡ ( α ∗ − ǫ ∗ ) / ( β ∗ − ζ ∗ ) − / κ − / ∗ γ ∗ ( Y [0] − ǫ ∗ R / (3 γ ∗ ) − κ ∗ ζ ∗ R [0] s/γ ∗ ) and t ≡ (( α ∗ − ǫ ∗ )( β ∗ − ζ ∗ ) κ ∗ ) / s , we obtain the normalizedleading-order transition equations [6, 15, 16] (cid:18) dx [0] dt (cid:19) = − x − x [0] t + y [0] , (38a) d x [0] dt = − x − t, dy [0] dt = 2 x [0] . (38b) The solution x [0] is the Painlev´e transcendent of the firstkind and y [0] is twice its first integral [20]. We thereforeproved that the transition equations (38) are unchangedin the presence of self-force, though the dictionary interms of the orbital quantities is affected through thetwo self-force coefficients ǫ ∗ and ζ ∗ defined in (35). INSPIRAL-TRANSITION MATCHING
The adiabatic inspiral has as range of validity τ < τ ∗ where the bound arises because the expansion be-comes singular at the ISCO. The ISCO is approachedwhen τ ∗ − τ ≪ η − where η − is the radiation-reactiontimescale. The transition solution exists for all s as de-fined in Eq. (27) and approaches the inspiral at earlyproper times with respect to the transition timescale τ ∗ − τ ≫ η − / . The overlapping region between the in-spiral and the transition solution is η − / ≪ τ ∗ − τ ≪ η − with τ < τ ∗ . We will now match the transition solutionas s → −∞ with the inspiral solution as τ → τ ∗ in theoverlapping region.The relevant solution of (38) as t → −∞ is x [0] = √− t + O ( t − ), y [0] = − ( − t ) / + O ( t − ). From (34) and(4), we have ξ [1] ( s ) = 0 and ξ [2] ( s ) = λ ∗ ( − s ) / + O ( s − )where λ ∗ ≡ − η s ( β ∗ − ζ ∗ ) κ ∗ α ∗ − ǫ ∗ ∂f φ ∂r (cid:12)(cid:12)(cid:12)(cid:12) [0] ∗ . (39)Substituting in (28), the dependence in η and τ recom-bines into a dependence in ˜ τ = ητ as s → −∞ as r ( s ) = s ( β ∗ − ζ ∗ ) κ ∗ α ∗ − ǫ ∗ p ˜ τ ∗ − ˜ τ + r ∗ [0] + η / O ( s − )+ O s ( η / ) , (40) ℓ ( s ) = λ ∗ (˜ τ ∗ − ˜ τ ) / + κ ∗ (˜ τ ∗ − ˜ τ )+ ℓ ∗ [0] + η / O ( s − )+ O s ( η / ) . (41)This behavior asymptotically matches with (19) and (21)upon identifying κ ∗ (0) , = κ ∗ , r ∗ (0) , = s ( β ∗ − ζ ∗ ) κ ∗ α ∗ − ǫ ∗ , κ ∗ (0) , = λ ∗ . (42)The three free parameters of the inspiral, namely κ ∗ (0) , , r ∗ (0) , and κ ∗ (0) , are now fixed through the matching interms of the parameters of the transition motion.The deviation δ from quasi-circularity in the transition(33) exactly matches at leading order with the deviationfrom quasi-circularity in the inspiral (7)-(22) thanks tothe equality of coefficients π ∗ = 3 sr ∗ (0) Ω (0) ∗ ∆ ∗ E ∗ / (2 A ∗ ).We also obtain as s → −∞ e ( s ) = e ∗ [0] , (˜ τ ∗ − ˜ τ ) / + κ ∗ Ω ∗ [0] (˜ τ ∗ − ˜ τ ) + e ∗ [0] + η / O ( s − ) + O s ( η / ); (43) e ∗ [0] , Ω − ∗ = λ ∗ + ( r ∗ (0)1 ) (cid:16) ǫ ∗ − α ∗ γ ∗ − ζ ∗ ( α ∗ − ǫ ∗ ) γ ∗ ( β ∗ − ζ ∗ ) (cid:17) . (44)We can also identify (44) with (26) after recognizingequivalent formulae α ∗ = 9 s ∆ ∗ Ω ∗ E ∗ r / ∗ A ∗ , β ∗ = 2 √ A ∗ Ω ∗ r / ∗ , γ ∗ = 2 s ∆ ∗ r ∗ √ A ∗ . (45)This completes the matching. Summing up the formulae(24),(33),(35), the leading self-force in the inspiral nearthe ISCO takes the final form [33] f r (0) = κ ∗ β ∗ α ∗ − ǫ ∗ ) ∂ f r ∂r (cid:12)(cid:12)(cid:12)(cid:12) [0] ∗ (˜ τ ∗ − ˜ τ ) + O η (˜ τ ∗ − ˜ τ ) / , (46)after using β ∗ /α ∗ = ∆ ∗ Ω ∗ A / ∗ /π ∗ . CONCLUSION
We obtained the first exact consistent match of theadiabatic quasi-circular inspiral with the transition solu-tion at leading order in the small mass ratio expansion.We proved that the leading-order transition solution in-cluding all self-force effects is determined in terms of thePainlev´e transcendent of the first kind. This consolidatesprevious partial analyses for equal [6, 7] and small massratios [15, 16, 20, 21].We proved that the adiabatic inspiral needs to takeinto account the secular change of angular velocity in-duced by radiation-reaction in order to match the tran-sition solution, consistently with the 2.5PN radiation-reaction effect occuring in the PN/PM formalism [28, 29].This leads to a leading-order radial self-force in the slowtimescale expansion that was overlooked so far in the self-force formalism. This leading-order self-force vanishes inthe limit of small particle mass, consistently with funda-mental self-force analyses [30, 31].This mathematically self-consistent inspiral-transitionmotion in the small mass ratio expansion, once ex-tended to higher orders and non-perturbatively re-summed, would provide a new tool to further calibrateEOB waveforms [6–14] using self-force theory.
Acknowledgments.
We thank L. Blanchet, S. Grallaand A. Pound for their very useful comments on themanuscript. G.C. is Senior Research Associate of theF.R.S.-FNRS and acknowledges support from the FNRSresearch credit J.0036.20F, bilateral Czech conventionPINT-Bilat-M/PGY R.M005.19 and the IISN conven-tion 4.4503.15. L.K. acknowledges support from the ESA Prodex experiment arrangement 4000129178 for theLISA gravitational wave observatory Cosmic Vision L3. [1]
LIGO Scientific, Virgo
Collaboration, B. P. Abbott et al. , “Observation of Gravitational Waves from aBinary Black Hole Merger,”
Phys. Rev. Lett. (2016), no. 6, 061102, .[2]
LISA
Collaboration, H. Audley et al. , “LaserInterferometer Space Antenna,” .[3]
LIGO Scientific, Virgo
Collaboration, R. Abbott etal. , “GWTC-2: Compact Binary Coalescences Observedby LIGO and Virgo During the First Half of the ThirdObserving Run,” .[4] M. Maggiore et al. , “Science Case for the EinsteinTelescope,”
JCAP (2020) 050, .[5] LIGO Scientific, Virgo
Collaboration, R. Abbott etal. , “Tests of General Relativity with Binary BlackHoles from the second LIGO-Virgo Gravitational-WaveTransient Catalog,” .[6] A. Buonanno and T. Damour, “Transition from inspiralto plunge in binary black hole coalescences,”
Phys. Rev.
D62 (2000) 064015, gr-qc/0001013 .[7] A. Buonanno, Y. Chen, and T. Damour, “Transitionfrom inspiral to plunge in precessing binaries ofspinning black holes,”
Phys. Rev.
D74 (2006) 104005, gr-qc/0508067 .[8] T. Damour and A. Nagar, “Faithful effective-one-bodywaveforms of small-mass-ratio coalescing black-holebinaries,”
Phys. Rev.
D76 (2007) 064028, .[9] T. Damour and A. Nagar, “An Improved analyticaldescription of inspiralling and coalescing black-holebinaries,”
Phys. Rev.
D79 (2009) 081503, .[10] Y. Pan, A. Buonanno, A. Taracchini, L. E. Kidder,A. H. Mrou´e, H. P. Pfeiffer, M. A. Scheel, andB. Szil´agyi, “Inspiral-merger-ringdown waveforms ofspinning, precessing black-hole binaries in theeffective-one-body formalism,”
Phys. Rev.
D89 (2014),no. 8, 084006, .[11] A. Nagar, T. Damour, and A. Tartaglia, “Binary blackhole merger in the extreme mass ratio limit,”
Class.Quant. Grav. (2007) S109–S124, gr-qc/0612096 .[12] S. Bernuzzi and A. Nagar, “Binary black hole merger inthe extreme-mass-ratio limit: a multipolar analysis,” Phys. Rev. D (2010) 084056, .[13] S. Bernuzzi, A. Nagar, and A. Zenginoglu, “Binaryblack hole coalescence in the extreme-mass-ratio limit:testing and improving the effective-one-body multipolarwaveform,” Phys. Rev. D (2011) 064010, .[14] S. Bernuzzi, A. Nagar, and A. Zenginoglu, “Binaryblack hole coalescence in the large-mass-ratio limit: thehyperboloidal layer method and waveforms at nullinfinity,” Phys. Rev. D (2011) 084026, .[15] A. Ori and K. S. Thorne, “The Transition from inspiralto plunge for a compact body in a circular equatorialorbit around a massive, spinning black hole,” Phys.Rev.
D62 (2000) 124022, gr-qc/0003032 .[16] M. Kesden, “Transition from adiabatic inspiral toplunge into a spinning black hole,”
Phys. Rev.
D83 (2011) 104011, .[17] P. A. Sundararajan, “The Transition from adiabaticinspiral to geodesic plunge for a compact object around a massive Kerr black hole: Generic orbits,”
Phys. Rev.
D77 (2008) 124050, .[18] A. Taracchini, A. Buonanno, G. Khanna, and S. A.Hughes, “Small mass plunging into a Kerr black hole:Anatomy of the inspiral-merger-ringdown waveforms,”
Phys. Rev.
D90 (2014), no. 8, 084025, .[19] A. Apte and S. A. Hughes, “Exciting black hole modesvia misaligned coalescences: I. Inspiral, transition, andplunge trajectories using a generalized Ori-Thorneprocedure,” .[20] G. Comp`ere, K. Fransen, and C. Jonas, “Transitionfrom inspiral to plunge into a highly spinning blackhole,”
Class. Quant. Grav. (2020), no. 9, 095013, .[21] O. Burke, J. R. Gair, and J. Sim´on, “Transition fromInspiral to Plunge: A Complete Near-ExtremalTrajectory and Associated Waveform,” Phys. Rev. D (2020), no. 6, 064026, .[22] T. Hinderer and E. E. Flanagan, “Two timescaleanalysis of extreme mass ratio inspirals in Kerr. I.Orbital Motion,”
Phys. Rev.
D78 (2008) 064028, .[23] K. Glampedakis and D. Kennefick, “Zoom and whirl:Eccentric equatorial orbits around spinning black holesand their evolution under gravitational radiationreaction,”
Phys. Rev. D (2002) 044002, gr-qc/0203086 .[24] E. E. Flanagan and T. Hinderer, “Transient resonances in the inspirals of point particles into black holes,” Phys. Rev. Lett. (2012) 071102, .[25] E. E. Flanagan, S. A. Hughes, and U. Ruangsri,“Resonantly enhanced and diminished strong-fieldgravitational-wave fluxes,”
Phys. Rev. D (2014),no. 8, 084028, .[26] J. Miller and A. Pound, “Two-timescale evolution ofextreme-mass-ratio inspirals: waveform generationscheme for quasicircular orbits in Schwarzschildspacetime,” .[27] A. Pound and B. Wardell, “Black hole perturbationtheory and gravitational self-force,” .[28] B. R. Iyer and C. M. Will, “Post-newtoniangravitational radiation reaction for two-body systems,” Phys. Rev. Lett. (Jan, 1993) 113–116.[29] L. Blanchet, “Gravitational Radiation fromPost-Newtonian Sources and Inspiralling CompactBinaries,” Living Rev. Rel. (2014) 2, .[30] A. Pound, “Self-consistent gravitational self-force,” Phys. Rev. D (2010) 024023, .[31] S. E. Gralla and R. M. Wald, “Derivation ofGravitational Self-Force,” Fundam. Theor. Phys. (2011) 263–270, .[32] We expect f r (0) (˜ τ ) → τ → −∞ . We also expect f r (0) (˜ τ ) → τ → f r (0) → η → η → f r (0) → τ = ητητ