iResum: a new paradigm for resumming gravitational wave amplitudes

We introduce a new, resummed, analytical form of the post-Newtonian (PN), factorized, multipolar amplitude corrections $f_{\ell m}$ of the effective-one-body (EOB) gravitational waveform of spinning, nonprecessing, circularized, coalescing black hole binaries (BBHs). This stems from the following two-step paradigm: (i) the factorization of the orbital (spin-independent) terms in $f_{\ell m}$; (ii) the resummation of the residual spin (or orbital) factors. We find that resumming the residual spin factor by taking its inverse resummed (iResum) is an efficient way to obtain amplitudes that are more accurate in the strong-field, fast-velocity regime. The performance of the method is illustrated on the $\ell=2$ and $m=(1,2)$ waveform multipoles, both for a test-mass orbiting around a Kerr black hole and for comparable-mass BBHs. In the first case, the iResum $f_{\ell m}$'s are much closer to the corresponding "exact" functions (obtained solving numerically the Teukolsky equation) up to the light-ring, than the nonresummed ones, especially when the black-hole spin is nearly extremal. The iResum paradigm is also more efficient than including higher post-Newtonian terms (up to 20PN order): the resummed 5PN information yields per se a rather good numerical/analytical agreement at the last-stable-orbit, and a well-controlled behavior up to the light-ring. For comparable mass binaries (including the highest PN-order information available, 3.5PN), comparing EOB with Numerical Relativity (NR) data shows that the analytical/numerical fractional disagreement at merger, without NR-calibration of the EOB waveform, is generically reduced by iResum, from a $40\%$ of the usual approach to just a few percents. This suggests that EOBNR waveform models for coalescing BBHs may be improved using iResum amplitudes.Comment: 6 pages, 7 figures. Improved discussion for the comparable-mass cas

Linear-in-mass-ratio contribution to spin precession and tidal invariants in Schwarzschild spacetime at very high post-Newtonian order

Using black hole perturbation theory and arbitrary-precision computer algebra, we obtain the post-Newtonian (pN) expansions of the linear-in-mass-ratio corrections to the spin-precession angle and tidal invariants for a particle in circular orbit around a Schwarzschild black hole. We extract coefficients up to 20pN order from numerical results that are calculated with an accuracy greater than 1 part in $10^{500}$. These results can be used to calibrate parameters in effective-one-body models of compact binaries, specifically the spin-orbit part of the effective Hamiltonian and the dynamically significant tidal part of the main radial potential of the effective metric. Our calculations are performed in a radiation gauge, which is known to be singular away from the particle. To overcome this irregularity, we define suitable Detweiler-Whiting singular and regular fields in this gauge, and we devise a rigorous mode-sum regularization method to compute the invariants constructed from the regular field

Raising and Lowering operators of spin-weighted spheroidal harmonics

In this paper we generalize the spin-raising and lowering operators of spin-weighted spherical harmonics to linear-in-$\gamma$ spin-weighted spheroidal harmonics where $\gamma$ is an additional parameter present in the second order ordinary differential equation governing these harmonics. One can then generalize these operators to higher powers in $\gamma$. Constructing these operators required calculating the $\ell$-, $s$- and $m$-raising and lowering operators (and various combinations of them) of spin-weighted spherical harmonics which have been calculated and shown explicitly in this paper

Finding high-order analytic post-Newtonian parameters from a high-precision numerical self-force calculation

We present a novel analytic extraction of high-order post-Newtonian (pN) parameters that govern quasi-circular binary systems. Coefficients in the pN expansion of the energy of a binary system can be found from corresponding coefficients in an extreme-mass-ratio inspiral (EMRI) computation of the change $\Delta U$ in the redshift factor of a circular orbit at fixed angular velocity. Remarkably, by computing this essentially gauge-invariant quantity to accuracy greater than one part in $10^{225}$, and by assuming that a subset of pN coefficients are rational numbers or products of $\pi$ and a rational, we obtain the exact analytic coefficients. We find the previously unexpected result that the post-Newtonian expansion of $\Delta U$ (and of the change $\Delta\Omega$ in the angular velocity at fixed redshift factor) have conservative terms at half-integral pN order beginning with a 5.5 pN term. This implies the existence of a corresponding 5.5 pN term in the expansion of the energy of a binary system. Coefficients in the pN series that do not belong to the subset just described are obtained to accuracy better than 1 part in $10^{265-23n}$ at $n$th pN order. We work in a radiation gauge, finding the radiative part of the metric perturbation from the gauge-invariant Weyl scalar $\psi_0$ via a Hertz potential. We use mode-sum renormalization, and find high-order renormalization coefficients by matching a series in $L=\ell+1/2$ to the large-$L$ behavior of the expression for $\Delta U$. The non-radiative parts of the perturbed metric associated with changes in mass and angular momentum are calculated in the Schwarzschild gauge

Self-force as a cosmic censor in the Kerr overspinning problem

It is known that a near-extremal Kerr black hole can be spun up beyond its extremal limit by capturing a test particle. Here we show that overspinning is always averted once back-reaction from the particle's own gravity is properly taken into account. We focus on nonspinning, uncharged, massive particles thrown in along the equatorial plane, and work in the first-order self-force approximation (i.e., we include all relevant corrections to the particle's acceleration through linear order in the ratio, assumed small, between the particle's energy and the black hole's mass). Our calculation is a numerical implementation of a recent analysis by two of us [Phys.\ Rev.\ D {\bf 91}, 104024 (2015)], in which a necessary and sufficient "censorship" condition was formulated for the capture scenario, involving certain self-force quantities calculated on the one-parameter family of unstable circular geodesics in the extremal limit. The self-force information accounts both for radiative losses and for the finite-mass correction to the critical value of the impact parameter. Here we obtain the required self-force data, and present strong evidence to suggest that captured particles never drive the black hole beyond its extremal limit. We show, however, that, within our first-order self-force approximation, it is possible to reach the extremal limit with a suitable choice of initial orbital parameters. To rule out such a possibility would require (currently unavailable) information about higher-order self-force corrections.Comment: 13 pages, 3 figure

EMRI corrections to the angular velocity and redshift factor of a mass in circular orbit about a Kerr black hole

This is the first of two papers on computing the self-force in a radiation gauge for a particle moving in circular, equatorial orbit about a Kerr black hole. In the EMRI (extreme-mass-ratio inspiral) framework, with mode-sum renormalization, we compute the renormalized value of the quantity $h_{\alpha\beta}u^\alpha u^\beta$, gauge-invariant under gauge transformations generated by a helically symmetric gauge vector; and we find the related order $\frak{m}$ correction to the particle's angular velocity at fixed renormalized redshift (and to its redshift at fixed angular velocity). The radiative part of the perturbed metric is constructed from the Hertz potential which is extracted from the Weyl scalar by an algebraic inversion\cite{sf2}. We then write the spin-weighted spheroidal harmonics as a sum over spin-weighted spherical harmonics and use mode-sum renormalization to find the renormalization coefficients by matching a series in $L=\ell+1/2$ to the large-$L$ behavior of the expression for $H := \frac12 h_{\alpha\beta}u^\alpha u^\beta$. The non-radiative parts of the perturbed metric associated with changes in mass and angular momentum are calculated in the Kerr gauge