Asymptotic gravitational wave fluxes from a spinning particle in circular equatorial orbits around a rotating black hole

We present a new computation of the asymptotic gravitational wave energy fluxes emitted by a spinning particle in circular equatorial orbits about a Kerr black hole. The particle dynamics is computed in the pole-dipole approximation, solving the Mathisson-Papapetrou equations with the Tulczyjew spin-supplementary-condition. The fluxes are computed, for the first time, by solving the $2+1$ Teukolsky equation in the time-domain using hyperboloidal and horizon-penetrating coordinates. Denoting by $M$ the black hole mass and by $\mu$ the particle mass, we cover dimensionless background spins $a/M=(0,\pm 0.9)$ and dimensionless particle spins $-0.9\le S/{\mu }^2\le +0.9$. Our results span orbits of Boyer-Lindquist coordinate radii $4\le r/M\le 30$; notably, we investigate the strong-field regime, in some cases even beyond the last-stable-orbit. We compare our numerical results for the gravitational wave fluxes with the 2.5th order accurate post-Newtonian (PN) prediction obtained analytically by Tanaka et al. [Phys. Rev. D 54, 3762 (1996)]: we find an unambiguous trend of the PN-prediction toward the numerical results when $r$ is large. At $r/M=30$ the fractional agreement between the full numerical flux, approximated as the sum over the modes $m=1$, 2, 3, and the PN prediction is $\lesssim 0.5\%$ in all cases tested. This is close to our fractional numerical accuracy ($\sim 0.2\%$). For smaller radii, the agreement between the 2.5PN prediction and the numerical result progressively deteriorates, as expected. Our numerical data will be essential to develop suitably resummed expressions of PN-analytical fluxes in order to improve their accuracy in the strong-field regime.

Figure 1

Comparison of numerical (solid) and analytical PN (dashed) total energy fluxes over $x={\widehat{\mathrm{\Omega }}}^{2/3}$. We consider three values of the background spin, $\widehat{a}=0.0$ (top panel), $\widehat{a}=-0.9$ (middle panel), and $\widehat{a}=+0.9$ (bottom panel). For each value of $\widehat{a}$, we plot the fluxes for a spinning particle with spins $\sigma =-0.9$ (blue, circles), $\sigma =0.0$ (black, stars) with data computed by Hughes [83, 84, 95, 96], see text, and $\sigma =+0.9$ (red, triangles) over $x$. The numerical fluxes are obtained summing up the modes with $m=1$, 2, 3; the PN fluxes are given by Eq. (36), which sums up only the modes considered in [51]. The green filled circles mark the interpolated fluxes at the locations of the LSO, cf. Table 1. At low orbital frequencies $\widehat{\mathrm{\Omega }}\to 0$, or equivalently $x\to 0$ and $\widehat{r}\to \infty$, the numerics are consistent with the PN-prediction, reaching in all cases a fractional difference $\le 0.5\%$ at our outermost data point $\widehat{r}=30$, see also Tables 2, 3, 4.

Figure 2

Dependence of the LO-normalised multipolar fluxes, $\eta \ell m$ on the spin $\sigma$, comparing numerical results (solid) with 2.5PN analytical predictions (dashed), at a weak-field orbit (left) and a strong-field orbit (right), for the three background spins $\widehat{a}=0.0$ (blue, circles), $\widehat{a}=-0.9$ (black, stars), and $\widehat{a}=+0.9$ (red, triangles). The panels show the multipolar energy fluxes in the $S_{-222}$ (top), $S_{-221}$ (top middle), $S_{-233}$ (bottom middle) and $S_{-233}$ (bottom) modes. The small horizontal lines on the $\sigma =0.0$ axis refer to the results for a nonspinning particle from the frequency-domain data of Hughes [84]. Note, for clarity, that these plots show the ${\widehat{\eta }}_{\ell m}$, as obtained from Eqs. (a1)–(a5) by normalizing with the LO-Newtonian expressions in terms of $u\equiv 1/r$ (e.g., for $\ell m=22$ with $32/5u^5$), whereas the values stated in Tables 2, 3, 4 refer to the ${\widehat{F}}_{S_{\ell m}}$ and are thus normalized to the LO-Newtonian expressions in terms of $x\equiv {\widehat{\mathrm{\Omega }}}^{2/3}$ (e.g., for $\ell m=22$ with $32/5x^5$).

Figure 3

Comparison of numerical (solid) and analytical PN (dashed) total energy fluxes over $u\equiv M/r$. Compare with Fig. 1 for further descriptions. In all three panels, i.e. for all considered background spins $\widehat{a}=0.0$ (top panel), $\widehat{a}=-0.9$ (middle panel), and $\widehat{a}=+0.9$ (bottom panel), we observe an unambiguous trend of our numerical data, approximated as the sum of $m=1$, 2, 3-modes, towards the 2.5PN prediction of Tanaka et al. [51] as $u\to 0$.

Figure 4

Cross-check of the energy flux in the $S_{-222}\text{−}=\text{mode}$ as obtained in this study (black solid, circles) against the results of [44] (blue dotted, crosses), at $\widehat{r}=10$ and for $\widehat{a}=0.0$. The 2.5PN prediction is included for completeness (black dashed). The data marked by crosses were extracted visually by inspecting the top right panel in Fig. 4 in [44]. Note that here we show the not Newton-normalized fluxes. The data points of [44] disagree qualitatively with both our numerical data and the 2.5PN accurate analytical flux. Note that we have indications that the underlying dynamics is the same in both cases. Both numerical solutions are consistent with one another at $\sigma =0.0$ and also with the frequency-domain data of [84], which is marked by the small gray cross.