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...

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Complex metrics are a double-edged sword: they allow one to replace singular spacetimes, such as those containing a big bang, with regular metrics, yet they can also describe unphysical solutions in which quantum transitions may be more probable than ordinary classical evolution. In the cosmological context, we investigate a criterion proposed by Witten (based on works of Kontsevich & Segal and of Louko & Sorkin) to decide whether a complex metric is allowable or not. Because of the freedom to deform complex metrics using Cauchy’s theorem, deciding whether a metric is allowable in general requires solving a complicated optimisation problem. We describe a method that allows one to quickly determine the allowability of minisuperspace metrics. This enables us to study the off-shell structure of minisuperspace path integrals, which we investigate for various boundary conditions. Classical transitions always reside on the boundary of the domain of allowable metrics, and care must be taken in defining appropriate integration contours for the corresponding gravitational path integral. Perhaps more surprisingly, we find that proposed quantum (‘tunnelling’) transitions from a contracting to an expanding universe violate the allowability criterion and may thus be unphysical. No-boundary solutions, by contrast, are found to be allowable, and moreover we demonstrate that with an initial momentum condition an integration contour over allowable metrics may be explicitly described in arbitrary spacetime dimensions.