Self-force gravitational waveforms for extreme and intermediate mass ratio inspirals. III. Spin-orbit coupling revisited

The first- and second-order dissipative self-force and the first-order conservative self-force are applied together with spin-orbit coupling to the quasicircular motion of a test mass in the spacetime of a Schwarzschild black hole, for extreme or intermediate mass ratios. The partial dephasing of the gravitational waveform (at the order that is independent of the system’s mass ratio) due to the self-force is compared with that of spin-orbit coupling. We find that accurate waveforms for parameter estimation need to include both effects. Specifically, we find a particular value for the spin parameter such that the spin-orbit effect cancels out the self-force effect on the waveform. Exclusion of dephasing effects that are independent of the mass ratio therefore might lead to a nonperturbative error in the estimation of the system’s parameters.

Figure 1

Schematic representation (not to scale) of our physical model: a small compact mass $\mu$ with spin angular momentum $S^{\alpha }=s{\mu }^2/r{\delta }_{\theta }^{\alpha }$ which is aligned (as in this drawing) or antialigned with the orbital angular momentum, orbits with three-velocity $\overrightarrow{v}$ a Schwarzschild black hole of mass $M$ in a quasicircular orbit. The latter decays because of the combined effects of radiation reaction and spin-orbit coupling, a trajectory represented by the solid spiral. The ISCO is represented by a dotted circle.

Figure 2

The polarization state $h_{22}^{+}$ of the gravitational waveforms as a function of the time. Shown are the cases $s=0$ (solid curve), $s=-1$ (dash-dotted curve), $s=+1$ (dashed curve), and ${\mathrm{\Phi }}^{(1)}\equiv 0$ (dotted curve). All these cases are in phase at $t=0$.

Figure 3

Same as Fig. 2 for the $h_{22}^{\times }$ polarization state.

Figure 4

The waveforms for the two polarization states $h_{22}^{+,\times }$ for the case $s=0.281$ [solid (plus) and dashed (cross) curves], and for the ${\mathrm{\Phi }}^{(1)}\equiv 0$ case [circle (plus) and square (cross) as functions of the time $t$]. For each polarization state, the two waveforms are in phase at $t=0$.

Figure 5

The overlap integral $C_{\max }$ for either polarization state $h_{22}^{+,\times }$ for the case $s=0.281$ (represented by plus and cross correspondingly) with the corresponding ${\mathrm{\Phi }}^{(1)}\equiv 0$ case as functions of the length of the interval of the segment being integrated. The inset shows the same for $h_{22}^{\times }$ (represented by circle) for the case ${\mathrm{\Phi }}^{(1)}\equiv 0$ and $h_{22}^{+}$ for the case $s=0.281$.

Figure 6

The evolution of the relative phase of the gravitational waveform between the second-order waveform with $s\ne 0$ and the case $s=0$ [solid (blue) curves] in increments of $\mathrm{\Delta }s=0.2$, and between the first-order waveform [${\mathrm{\Phi }}^{(1)}\equiv 0$] and the second-order waveform with $s=0$ [dashed (red) curve]. The inset shows the same between the first-order waveform [${\mathrm{\Phi }}^{(1)}\equiv 0$] and the second-order waveform with $s=0.281$. On this scale the two polarization states look indistinguishable.

Figure 7

The relative phase of the gravitational waveform between the second-order waveform with $s\ne 0$ and the case $s=0$ [circle (blue) and the corresponding best fit curve], and between the first-order waveform [${\mathrm{\Phi }}^{(1)}\equiv 0$] and the second-order waveform with $s=0$ [cross (red)]. The values of $s$ shown are between $-1$ and $+1$ in increments of 0.2. All data are taken at $ϵt/M=86.40$. The fit function is the parabola $\mathrm{\Delta }\varphi =0.33817s^2+2.7367s+9.1375\times 10^{-4}$. On this scale the two polarization states look indistinguishable.

Figure 8

The value of $C_{\max }$ as a function of the duration of the integration interval for the polarization state $h_{22}^{+}$. Shown are the cases $s=+1$ (circle, blue) and $s=-1$ (square, red). In both cases the overlap integral is calculated with the waveform for $s=0$.

Figure 9

The length of the window for which the overlap integral $C_{\max }=0.96$ as a function of the spin parameter $s$ for the polarization state $h_{22}^{+}$.