Self-forced inspirals with spin-orbit precession

We develop the first model for extreme mass-ratio inspirals with misaligned angular momentum and primary spin, and zero eccentricity—also known as quasispherical inspirals—evolving under the influence of the first-order in mass-ratio gravitational self-force. The forcing terms are provided by an efficient spectral interpolation of the first-order gravitational self-force in the outgoing radiation gauge. In order to speed up the calculation of the inspiral, we apply a near-identity (averaging) transformation to eliminate all dependence of the orbital phases from the equations of motion while maintaining all secular effects of the first-order gravitational self-force at postadiabatic order. The resulting solutions are defined with respect to “Mino time”; thus, we perform a second averaging transformation, so the inspiral is parametrized in terms of Boyer-Lindquist time, which is more convent for LISA data analysis. We also perform a similar analysis using the two-timescale expansion and find that using either approach yields self-forced inspirals that can be evolved to subradian accuracy in less than a second. The dominant contribution to the inspiral phase comes from the adiabatic contributions, so we further refine our self-force model using information from gravitational wave flux calculations. The significant dephasing we observe between the lower and higher accuracy models highlights the importance of accurately capturing adiabatic contributions to the phase evolution.

Figure 1

Relative error in the rate of energy (a) and angular momentum (b) loss between the interpolated fluxes and the fluxes calculated with a grid of verification points that were not used for the interpolation. The white dots indicate the locations of the data points that were interpolated to produce the model, and the green crosses indicate the positions of verification data. The relative error is always $<{10}^{-5}$ with the exception of orbits that are very close to the ISSO.

Figure 2

Relative error in the rate of energy (a) and angular momentum (b) loss between the asymptotic fluxes and the interpolated GSF model. The white dots indicate the locations of the data points the model is interpolating. The green crosses indicate the locations of the flux calculations and thus where the comparisons are made. The relative error only rises to $\sim {10}^{-2}$ when close to the ISSO and is otherwise typically of the order $\sim {10}^{-3}–{10}^{-5}$.

Figure 3

Trajectory through $(r_p,{\theta }_{\mathrm{inc}})$ space for an inspiral with $\epsilon ={10}^{-2}$, $a=0.9M$, and initial conditions $r_p(0)=7.75M,x(0)=0.75$. We use such a large mass ratio to highlight the orbital timescale oscillations one encounters when using the OG equations. Using the NIT or TTE equations of motion averages out these oscillations and results in almost identical inspirals. We also see that the TTE equations of motion break down further from the ISSO than the OG or NIT equations of motion.

Figure 4

Difference in the orbital phases for a quasispherical Kerr inspiral with $M={10}^6{M}_{\odot }$, $\epsilon ={10}^{-5}$, and $a=0.9M$ when using the OG equations versus the NIT or TTE equations of motion. In both cases, the difference stays small throughout the inspiral, only becoming large as the secondary approaches the ISSO when the adiabatic assumption breaks down.

Figure 5

Absolute difference in the quantities of a prograde inspiral with $a=0.9M$ and an initial inclination of $x_0=0.75$ after evolving from $r_p=5M$ to $r_p=4M$ when using different evolution equations. The solid lines show the difference between OG and NIT equations of motion, and the dashed lines show the difference between using the OG and TTE equations of motion. We see that the differences generally follow the black $\epsilon$ curve, so they converge linearly with the mass ratio until we reach mass ratios $\le {10}^{-3}$ where the error in the phases becomes dominated by interpolation and numerical error.

Figure 6

“Semirelativistic” quadrupole waveform strains for an EMRI with $M={10}^6{M}_{\odot }$, $\epsilon ={10}^{-5}$, $r_p(0)=7.75M$, and $x(0)=0.75$ as viewed edge on (i.e., with the detector at a latitude of $\mathrm{\Theta }=\pi /2$ with respect to the source’s frame), sampled once every 10 s. The waveform strain is normalized by the luminosity distance from the source to the detector $D$ and the mass of the secondary $\mu$. Panel (a) shows the first 3.5 hours, and panel (b) shows the last 3.5 hours of this year-long waveform. The blue curve is the waveform generated from the OG inspiral, the dashed yellow curve is the waveform generated from the NIT inspiral, and the purple dotted curve is the waveform generated from the TTE inspiral.

Figure 7

Mismatch between year-long OG and NIT/TTE waveforms as a function of orbital inclination and mass ratio. We do not see a substantial difference in accuracy between using either NIT or TTE waveforms. We also see that the mismatch remains smaller than 0.03 for mass ratios as large as $\epsilon \approx 0.5$. (a) Mismatch between OG and NIT waveforms. (b) Mismatch between OG and TTE waveforms.

Figure 8

Absolute difference in ${\phi }_{\varphi }$ for a NIT inspiral evolved from a point in the parameter space to the ISSO with a typical EMRI mass ratio of $\epsilon ={10}^{-5}$, depending on whether one incorporates high precision fluxes or not. Using the high precision fluxes results in a more faithful inspiral, with phase differences with respect to the lower accuracy model in the range $\sim 10–{10}^4$ radians. As such, interpolating the adiabatic contributions to the averaged equations of motion to high precision is vitally important for accurate adiabatic and postadiabatic EMRI waveforms.

Figure 9

The blue curves show the adiabatic trajectories through $\{r_p,{\theta }_{\mathrm{inc}}\}$ space, while the orange contours denote the final value of ${\phi }_{\varphi }^{(1)}$ when evolved from that point in the parameter space to the ISSO.