Resonant self-force effects in extreme-mass-ratio binaries: A scalar model

Extreme-mass-ratio inspirals (EMRIs), compact binaries with small mass-ratios $\epsilon\ll 1$, will be important sources for low-frequency gravitational wave detectors. Almost all EMRIs will evolve through important transient orbital $r\theta$-resonances, which will enhance or diminish their gravitational wave flux, thereby affecting the phase evolution of the waveforms at $O(\epsilon^{1/2})$ relative to leading order. While modeling the local gravitational self-force (GSF) during resonances is essential for generating accurate EMRI waveforms, so far the full GSF has not been calculated for an $r\theta$-resonant orbit owing to computational demands of the problem. As a first step we employ a simpler model, calculating the scalar self-force (SSF) along $r\theta$-resonant geodesics in Kerr spacetime. We demonstrate two ways of calculating the $r\theta$-resonant SSF (and likely GSF), with one method leaving the radial and polar motions initially independent as if the geodesic is non-resonant. We illustrate results by calculating the SSF along geodesics defined by three $r\theta$-resonant ratios (1:3, 1:2, 2:3). We show how the SSF and averaged evolution of the orbital constants vary with the initial phase at which an EMRI enters resonance. We then use our SSF data to test a previously-proposed integrability conjecture, which argues that conservative effects vanish at adiabatic order during resonances. We find prominent contributions from the conservative SSF to the secular evolution of the Carter constant, $\langle \dot{\mathcal{Q}}\rangle$, but these non-vanishing contributions are on the order of, or less than, the estimated uncertainties of our self-force results. The uncertainties come from residual, incomplete removal of the singular field in the regularization process. Higher order regularization parameters, once available, will allow definitive tests of the integrability conjecture.