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 nonresonant. 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 $⟨\dot{\mathcal{Q}}⟩$, but these nonvanishing 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.

Figure 1

Inclined eccentric geodesics around a Kerr black hole with spin $a/M=0.9$ and mass $M=1$. The plot on the left is a nonresonant geodesic with orbital parameters $(p,e,x)=(4.700,0.5,\cos \pi /4)$. The plot on the right, in contrast, is a 1:2 $r\theta$-resonant geodesic with orbital parameters $(p,e,x)\simeq (4.607,0.5,\cos \pi /4)$. The semilatus rectum $p$ is truncated at four significant digits and specifically chosen so that the particle undergoes one radial libration for every two polar librations. The solid (blue) curves trace out the three-dimensional motion of the two geodesics in Boyer-Lindquist coordinates $(r,\theta ,\phi )$. The solid (grey) curves show various two-dimensional projections of the three-dimensional orbit.

Figure 2

Projected motion for two $1:2$ $r\theta$-resonant geodesics around a Kerr black hole with spin $a/M=0.9$ and mass $M=1$. The left plot suppresses the azimuthal motion and depicts the radial and polar motion of both geodesics in the poloidal plane. Three-dimensional projections of these two geodesics are mapped separately in the center and right plots. Both geodesics share the orbital parameters $(p,e,x)\approx (4.607,0.5,\cos \pi /4)$ and the same $r\theta$-resonant frequencies. While they both start at $r_p(0)=r_{\min }$, they differ in their initial polar positions and polar velocities. The solid (blue) line in the left plot and the center plot trace out a geodesic with the initial offset ${\lambda }_0^{(\theta )}=0$ ($q_{\theta 0}={\overline{q}}_0=0$), while the dashed (green) line in the left plot and the far right plot represent a geodesic with the initial offset ${\lambda }_0^{(\theta )}=-3{\mathrm{\Lambda }}_{\theta }/8$ ($q_{\theta 0}={\beta }_{\theta }{\overline{q}}_0=-3\pi /4$). The motion for each plot is shown from $\lambda =0$ to $\lambda =47$.

Figure 3

Poloidal motion depicted on the torus for orbits with parameters $(a/M,p,e,x)=(0.9,6,0.5,\cos \pi /4)$ (left panel) and $(a/M,p,e,x)=(0.9,4.607,0.5,\cos \pi /4)$ (right panel). In the right plot parameters are chosen to generate a $1:2$ $r\theta$ resonance. The orbit in the left panel is nonresonant. There the solid (blue) line follows the path of a geodesic with fiducial initial conditions $q_{(a)0}=(t_0,q_{r0},q_{\theta 0},{\phi }_0)=(0,0,0,0)$, while the dashed (red) line has initial conditions $q_{(a)0}=(0,\pi ,\pi /2,0)$. (Both paths are plotted over the Mino time interval $\lambda \in [0,4]$, with $M=1$.) Given sufficient time, both paths will fill the entire torus and completely overlap. In the right panel the solid (blue) line follows the path of a geodesic with the initial resonant phase ${\overline{q}}_0=0$ on the Mino time interval $\lambda \in [0,\mathrm{\Lambda }]$, where $\mathrm{\Lambda }\simeq 4.494$ is the (net) resonant Mino time period. The dashed (red) line follows a geodesic with initial phase ${\overline{q}}_0=\pi /2$ (${\theta }_p(\lambda =0)=\pi -{\theta }_{\min }$) on the same time interval. Unlike the nonresonant paths, the resonant geodesic flows in the right panel will never overlap with one another.

Figure 4

Convergence of the SSF $l$ modes for resonant models listed in Table 1. The dashed and dotted lines depict comparative power-law rates of convergence for ${\overline{F}}_{\phi }^{\mathrm{res}}(\overline{q}=5\pi /16;{\overline{q}}_0=5\pi /32/{\beta }_{\theta })$ as more regularization terms are incorporated. The (black) squares represent individual $l$ modes of the unregularized SSF, the sum of which clearly diverges. The (red) triangles are the residuals from subtracting $A_{\phi }$ and $B_{\phi }$. The (blue) diamonds represent the residuals after subtracting $D_{\phi ,2}$, obtained through numerical fitting. The (purple) circles represent the inclusion of $D_{\phi ,4}$, also approximated via a numerical fit.

Figure 5

Radial component of the SSF as a function of the resonant angle variable $\overline{q}$, i.e., ${\overline{F}}_r^{\mathrm{res}}(\overline{q};{\overline{q}}_0)$, for the six resonant geodesics listed in Table 1. The SSF is weighted by the cube of the pericentric radius $r_{\min }^3$, so that all six orbits are of comparable magnitude. The dot-dashed (black) line represents the SSF for a resonant geodesic with an initial resonant phase of ${\beta }_{\theta }{\overline{q}}_0=q_{\theta 0}=0$, while the solid (red) line represents the SSF for a resonance with the same orbital parameters but an initial resonant phase of ${\beta }_{\theta }{\overline{q}}_0=q_{\theta 0}=-\pi /2$. The shaded grey region represents all of the SSF values produced by varying the initial phase parameter ${\overline{q}}_0$ from 0 to $2\pi$.

Figure 6

Time component of the SSF as a function of the resonant angle variable $\overline{q}$, i.e., ${\overline{F}}_t^{\mathrm{res}}(\overline{q};{\overline{q}}_0)$, for the six resonant geodesics listed in Table 1. The dot-dashed (black) line represents the SSF for a resonant geodesic with an initial resonant phase of ${\beta }_{\theta }{\overline{q}}_0=q_{\theta 0}=0$, the solid (red) line represents an initial resonant phase of ${\beta }_{\theta }{\overline{q}}_0=q_{\theta 0}=-\pi /2$, and the shaded grey region represents all of the SSF values produced by varying ${\overline{q}}_0$ from 0 to $2\pi$.

Figure 7

Polar component of the SSF as a function of the resonant angle variable $\overline{q}$, i.e., ${\overline{F}}_{\theta }^{\mathrm{res}}(\overline{q};{\overline{q}}_0)$, for the six resonant geodesics listed in Table 1. The dot-dashed (black) line represents the SSF for a resonant geodesic with an initial resonant phase of ${\beta }_{\theta }{\overline{q}}_0=q_{\theta 0}=0$, the solid (red) line represents an initial resonant phase of ${\beta }_{\theta }{\overline{q}}_0=q_{\theta 0}=-\pi /2$, and the shaded grey region represents all of the SSF values produced by varying ${\overline{q}}_0$ from 0 to $2\pi$.

Figure 8

Azimuthal component of the SSF as a function of the resonant angle variable $\overline{q}$, i.e., ${\overline{F}}_{\phi }^{\mathrm{res}}(\overline{q};{\overline{q}}_0)$, for the six resonant geodesics listed in Table 1. The dot-dashed (black) line represents the SSF for a resonant geodesic with an initial resonant phase of ${\beta }_{\theta }{\overline{q}}_0=q_{\theta 0}=0$, the solid (red) line represents an initial resonant phase of ${\beta }_{\theta }{\overline{q}}_0=q_{\theta 0}=-\pi /2$, and the shaded grey region represents all of the SSF values produced by varying ${\overline{q}}_0$ from 0 to $2\pi$.

Figure 9

Time component of the SSF ${\widehat{F}}_t$ projected on the poloidal motion two-torus for the six sources listed in Table 1. The SSF is normalized by the cube of each source’s pericenter distance. Colors correspond to values of the self-force (see colorbar). The self-force is constant along each (solid) contour line with tic labels in the colorbar giving the values on those contours. The dot-dashed lines depict the resonant motion for fiducial initial conditions (${\overline{q}}_0=0$).

Figure 10

Radial component of the scalar self-force ${\widehat{F}}_r$ for the six orbits listed in Table 1.

Figure 11

Polar component of the scalar self-force ${\widehat{F}}_{\theta }$ for the six orbits listed in Table 1.

Figure 12

Azimuthal component of the scalar self-force ${\widehat{F}}_{\phi }$ for the six orbits listed in Table 1.

Figure 13

Residual variations in the energy (top panel) and angular momentum (bottom panel) fluxes as a function of the initial phase $q_{\theta 0}={\beta }_{\theta }{\overline{q}}_0$ for the $e02.23$ (solid black curves), $e05.23$ (dot-dashed blue curves), and $e05.12$ (dotted red curves) orbits in in Table 1.

Figure 14

Separate contributions from the dissipative and conservative components of the self-force to the residual variations $⟨\delta \dot{\mathcal{E}}⟩$ in energy (top row) and $⟨\delta {\dot{\mathcal{L}}}_z⟩$ in angular momentum (bottom row) for the $e02.12$ (left), $e02.23$ (middle), and $e05.23$ (right) orbits plotted as functions of the resonant-orbit phase. Contributions from the dissipative self-force are given by the solid (black) curves, while contributions from the conservative self-force are depicted by the dashed (red) curves. The (grey) shaded region represents the formal uncertainty in our calculation of the conservative SSF, $⟨\delta \dot{\mathcal{Q}}{⟩}^{\mathrm{cons}}\pm {\sigma }_{\mathrm{cons}}$, due to fitting for higher-order regularization parameters. The calculation of ${\sigma }_{\mathrm{cons}}$ is discussed below (in Sec. 6b). As expected, the residual variations in energy and angular momentum arising from the conservative SSF (dashed/red curves) are consistent with zero.

Figure 15

Contributions from the dissipative and conservative components of the self-force to $⟨\delta \dot{\mathcal{Q}}⟩$ for the $e02.12$ (left), $e02.23$ (middle), and $e05.23$ (right) orbits. Contributions from the dissipative self-force are given by the solid (black) curves, while contributions from the conservative self-force are depicted by the dashed (red) curves. The (grey) shaded region represents the formal uncertainty in our calculation of the conservative SSF, $⟨\delta \dot{\mathcal{Q}}{⟩}^{\mathrm{cons}}\pm {\sigma }_{\mathrm{cons}}$, due to fitting for higher-order regularization parameters. The calculation of ${\sigma }_{\mathrm{cons}}$ is discussed in detail in Sec. 6b.