Experimental mathematics meets gravitational self-force

Abstract

It is now possible to compute linear in mass-ratio terms in the post-Newtonian (PN) expansion for compact binaries to very high orders using black hole perturbation theory applied to various invariants. For instance, a computation of the redshift invariant of a point particle in a circular orbit about a black hole in linear perturbation theory gives the linear-in-mass-ratio portion of the binding energy of a circular binary with arbitrary mass ratio. This binding energy, in turn, encodes the system's conservative dynamics. We give a method for extracting the analytic forms of these PN coefficients from high-accuracy numerical data using experimental mathematics techniques, notably an integer relation algorithm. Such methods should be particularly important when the calculations progress to the considerably more difficult case of perturbations of the Kerr metric. As an example, we apply this method to the redshift invariant in Schwarzschild. Here we obtain analytic coefficients to 12.5PN, and higher-order terms in mixed analytic-numerical form to 21.5PN, including analytic forms for the complete 13.5PN coefficient, and all the logarithmic terms at 13PN. At these high orders, an individual coefficient can have over 30 terms, including a wide variety of transcendental numbers, when written out in full. We are still able to obtain analytic forms for such coefficients from the numerical data through a careful study of the structure of the expansion. The structure we find also allows us to predict certain "leading logarithm"-type contributions to all orders. The additional terms in the expansion we obtain improve the accuracy of the PN series for the redshift observable, even in the very strong-field regime inside the innermost stable circular orbit, particularly when combined with exponential resummation.

25 pages, 3 figures; version accepted by PRD, including various small additions and corrections

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Cited by in corpus (32)

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• Metric perturbations produced by eccentric equatorial orbits around a Kerr black hole
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• Scalar self-force for highly eccentric equatorial orbits in Kerr spacetime
• Octupolar invariants for compact binaries on quasi-circular orbits
• Determination of new coefficients in the angular momentum and energy fluxes at infinity to 9PN for eccentric Schwarzschild extreme-mass-ratio inspirals using mode-by-mode fitting
• Conservative second-order gravitational self-force on circular orbits and the effective one-body formalism
• Gravitational self-force corrections to gyroscope precession along circular orbits in the Kerr spacetime
• Self-force correction to geodetic spin precession in Kerr spacetime
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• Detweiler's redshift invariant for spinning particles along circular orbits on a Schwarzschild background
• Note on Accuracy of the Post-Newtonian Approximation for Extreme-Mass Ratio Inspirals: Retrograde Orbits
• Post-Newtonian expansion of gravitational energy and angular momentum fluxes: inclined spherical orbits about a Kerr black hole
• Multipolar Particles in Helically Symmetric Spacetimes
• Electromagnetic self-force in the five dimensional Myers-Perry space time
• Post-Newtonian expansion of fluxes from a scalar charge on an inclined-spherical orbit about a Kerr black hole
• Estimating effective higher order terms in the post-Newtonian binding energy and gravitational-wave flux: Non-spinning compact binary inspiral
• Self-force and fluid resonances