Transition from inspiral to plunge into a highly spinning black hole

Abstract

We extend the Ori-Thorne-Kesden procedure to consistently describe the non-quasi-circular transition around the ISCO from inspiral to plunge into a black hole of arbitrary spin, including nearextremal. We identify that for moderate or high spins the transition is governed by the Painlevé transcendent equation of the first kind while for extremely high spins it is governed by a self-similar solution to the Korteweg-de Vries equation. We match the transition solution at leading order in the high spin limit with the analytical quasi-circular inspiral in the near-horizon region. We also show that the central black hole of an extreme mass ratio binary has a near-extremality parameter that scales at least as the mass ratio due to superradiant gravitational wave emission, which excludes extremely high spins.A theoretical modeling effort is required in order to produce a database of accurate and faithful templates for extreme mass ratio inspirals (EMRIs) for the LISA mission [1,2] or its proposed extension [3]. In addition to the main science objectives of the mission, intermediate mass ratio coalescences (IMRACs) which consist in intermediate mass black holes plunging into supermassive black holes constitute a potential source for LISA [4]. Such sources require an accurate modeling of the transition from inspiral to plunge since the number of cycles spent in that latter phase is observationally significant.The transition from inspiral to plunge was modeled for quasi-circular inspirals in the equatorial plane by Ori and Thorne [5] within black hole perturbation theory under some simplifying assumptions (see also the EOB framework [6] and [7-14] for extensions). Such a transition occurs around the innermost stable circular orbit (ISCO). One of the original assumptions of [5] was that the orbit is quasi-circular around the transition. However, such an assumption was shown to lead to mathematical inconsistencies by Kesden [11], though its analysis has been overlooked in the subsequent literature. It is therefore required to relax that hypothesis and consider a non-circular transition motion. The Ori-Thorne-Kesden model applies for all moderate spins but becomes inconsistent in the high spin regime where λ ≡ 1 − J 2 /M 4 → 0 [11]. The main purpose of this paper is to complete the Ori-Thorne-Kesden analysis to cover the high spin regime where new qualitative features arise.Geometrically thin disks allow to spin up black holes only up to the Thorne bound λ ≥ 0.06 [15]. Other accretion models might however by-pass this bound since no fundamental limitation exists on how fast accretion can spin up a black hole [16]. More fundamentally, the high spin limit λ → 0 can be viewed as the leading order result of a perturbative expansion for small λ. In the high spin regime, the ISCO lies within the near-horizon region of Kerr which is described by the near-horizon FIG. 1: Evolution of the near-horizon radius R N in terms of proper time τ during the inspiral (solid black line) and transition to plunge (dotted-dashed red line) for a bina...

Near-horizon geodesics of high spin black holes

“…Such geodesics reach the spatial boundary of the near-horizon region at infinite past proper time and therefore physically reach the asymptotically flat Kerr region once the near-horizon is glued back to the exterior Kerr region. It turns out that bounded geodesics in the near-horizon Kerr region also arise in the study of gravitational waves since they correspond to the end-point of the transition motion [45]. Timelike outgoing geodesics originating from the white hole horizon and reaching the near-horizon boundary are also relevant for particle emission within the near-horizon region [42].…”

Transition from inspiral to plunge: A complete near-extremal trajectory and associated waveform

“…and Γ in (B6) defined through equation (31). Notice that none of the coefficients in our transition equation of motion depend on the extremality parameter .…”

“…The final regime concerns very rapidly rotating black holes, where η. Using the results in appendix A, one can identify the a priori dominant contributions to the master equation (30) and (31) to be (ignoring coefficients of O(1)) dR dτ…”

Location of the last stable orbit in Kerr spacetime

“…Once the secondary is captured its orbit will decay through gravitational wave emission until it reaches the separatrix and plunges into the massive black hole. Consequently, knowledge of the location of the separatrix is a key ingredient in models of these binaries [4][5][6][7][8][9][10][11]. The region of parameter space near the separatrix is also interesting as it is here that the well known relativistic orbital precession is taken to the extreme, with arbitrary large precession possible when approaching the separatrix [12].…”

“…The upper/lower signs in Eqs. (10) and (11) are to be chosen when the particle is outgoing/ingoing in the radial equation, or downgoing/upgoing in the polar equation. A sign flip occurs in an equation when the particle passes a turning point of the radial or polar motion.…”