Shifted-geodesic approximation for spinning-body gravitational wave fluxes

Abstract

We present a shifted-geodesic framework for computing gravitational-wave fluxes from spinning test bodies moving on bound orbits of Kerr black holes. The method provides a simple and efficient means of evaluating energy and angular momentum fluxes incorporating the leading effect of the smaller body's spin. Because post-adiabatic corrections, including secondary spin contributions, are subdominant to the leading adiabatic terms, this approximation is well justified. In particular, we find that oscillatory spin terms typically contribute very little to fluxes, but their contribution to the description of orbits is computationally expensive, making such terms a natural target for approximation. In our framework, orbital frequencies and integrals of the motion are perturbed to include spin effects, while the trajectory retains the global structure of geodesic motion. This simplifies the computation of gravitational radiation. The shifted-geodesic approximation is most reliable for orbits with lower eccentricity, lower inclination, and larger semi-latus recta. The approximation becomes less reliable as we approach the separatrix between stable and unstable orbits; fortunately, many inspirals spend less time in this region of parameter space. A diagnostic inspiral evolution shows very small dephasing due to use of the shifted-geodesic approximation ($\approx10^{-2}$ radians over 1 year), confirming that spin-induced flux corrections can be accurately included using this simple modification to a geodesic trajectory. This approximation provides a rapid and convenient way to compute spinning-body orbits, but is not intended to replace more accurate treatments. We propose it as a pragmatic alternative when speed and simplicity are prioritized, enabling efficient EMRI/IMRI flux calculations and supporting parameter-space studies for LISA. (Abridged)

Figures (12)

• Figure 1: Schematic illustration of the progression from a reference geodesic trajectory to the full spinning-body trajectory for the radial coordinate. The trajectories shown are representative cartoons rather than physical trajectories; for realisic parameter choices, the spin-induced shifts too small to be visible over a small number of cycles. The top left panel shows the reference geodesic orbit (red dashed), with unshifted radial frequency $\Upsilon_r$. The top right panel (shaded gray) depicts the shifted geodesic approximation (solid blue), where the radial frequency is modified by the secular shift $\delta \Upsilon_r^S$, capturing the dominant long-term effect of spin. The middle left panel includes both the secular frequency shift $\delta \Upsilon_r^S$ and the purely oscillatory correction $\delta \chi_r^S$, introducing periodic deviations around the shifted geodesic. The middle right panel shows the full spinning-body trajectory, incorporating all $\mathcal{O}(\sigma)$ corrections: the secular shift $\delta \Upsilon_r^S$, the oscillatory term $\delta \chi_r^S$ and the libration correction $\delta \mathcalligra{r}^S$, which accounts for coupling between radial and polar motion. In the bottom panel, the reference geodesic trajectory (red dashed) and the full spinning-body trajectory (solid blue) are shown, with the shifted geodesic approximation (black dotted) overlaid for direct comparison. This figure highlights how secular and oscillatory corrections successively modify the trajectory, with the shifted geodesic approximation capturing the key secular behavior relevant for gravitational-wave flux computations.
• Figure 2: Convergence of the exact spin-induced energy flux as a function of $n$, shown as the ratio of the contribution at $n_{\max}$ to the partial sum over all modes with $n \leq n_{\max}$. The left column shows $\left| \delta \mathcal{F}_{n_{\max}} \right| /\left| \sum_{n \leq n_{\max}} \delta \mathcal{F}_n \right|$, summed over $l$, $m$ and $k$, as a function of $p$ and $e$, with the rows corresponding to different values of inclination I. The right column illustrates how the choice of $n_{\max}$ affects the convergence criterion. We plot $\left| \delta \mathcal{F}_n \right| / \left|\sum_{n' \leq n} \delta \mathcal{F}_{n'} \right|$, such that the rightmost point on each curve corresponds to $\left| \delta \mathcal{F}_{n_{\max}} \right| / \left| \sum_{n \leq n_{\max}} \delta \mathcal{F}_n \right|$. Here the rows correspond to different values of semilatus rectum $p$ and the three curves in each panel correspond to different eccentricities $e$. Parameters: $a=0.9$, $n_{max}=5$, $k_{max}=4$, $n^s_{\textrm{max}}=8$, $k^s_{\textrm{max}}=8$.
• Figure 3: Corrections to the gravitational wave energy flux due to the spin of the secondary as a function of $n$. The top left panel shows the correction to the outgoing energy flux at infinity due to the spin of the secondary, $\mathcal{F}^{E,\infty}_{lmnk}$, while the top right panel shows the correction to the ingoing energy flux at the horizon due to the spin of the secondary, $\mathcal{F}^{E,H}_{lmnk}$. The bottom left and bottom right panels depict the corresponding corrections to the angular momentum fluxes, $\mathcal{F}^{L,\infty}_{lmnk}$ and $\mathcal{F}^{L,H}_{lmnk}$, respectively. The red circles and dashed curves represent fluxes computed with the exact trajectory, while the yellow squares and dash-dotted curves correspond to fluxes computed with the shifted geodesic approximation. The shifted geodesic approximation generally provides a close match to the exact results, though discrepancies are visible at higher $n$, particularly in the outgoing flux components. Parameters: $p=12$, $a=0.9$, $e=0.2$, $x_I=\sqrt{3}/2$, $l=2$, $m=2$, $k=0$. For the exact trajectory, as highlighted earlier, we use $n^s_\textrm{max}=8$, $k^s_\textrm{max}=8$.
• Figure 4: Absolute difference between the exact gravitational-wave flux corrections due to the secondary's spin and those computed using the shifted geodesic approximation, shown as a function of $n$ and $k$. The top panels show the exact spin-induced corrections to the energy flux $E$ (left) and angular momentum flux $L_z$ (right), summed over $l$ and $m$. The bottom panels display the corresponding differences between the exact and shifted geodesic fluxes for $E$ (left) and $L_z$ (right). Parameters: $p=12$, $a=0.9$, $e=0.2$, $x_I=\sqrt{3}/2$. For the exact trajectory, $n^s_{\textrm{max}}=8$, $k^s_{\textrm{max}}=8$.
• Figure 5: Corrections to the gravitational wave fluxes due to the spin of the secondary as a function of semilatus rectum, $p$. The four panels correspond to the same flux components as Figure \ref{['fig:fluxcomparen']}, with the top left showing the outgoing energy flux to infinity, the top right showing the ingoing energy flux at the horizon, and the bottom panels representing the corresponding angular momentum fluxes. In each panel, the red dashed curve with circular markers represents the exact trajectory, while the orange dot-dashed curve with square markers corresponds to the shifted geodesic approximation. The blue dotted curve represents the case where frequency shifts due to secondary spin, $\delta \Upsilon^S_i$, are omitted, and the solid black curve corresponds to the case where the corrections to the constants of motion due to secondary spin, $\delta \mathcal{C}_i$, are omitted. The results demonstrate that omitting $\delta C_i$ leads to the largest deviations, while the shifted geodesic approximation provides the best agreement with the exact trajectory. Parameters for all curves: $a=0.9M$, $e=0.2$, $x_I=\sqrt{3}/2$, $l=2$, $m=2$, $n=0$, $k=0$. For the exact trajectory, $n^s_{\textrm{max}}=8$, $k^s_{\textrm{max}}=8$.