Gravitational radiation from hyperbolic orbits: comparison between self-force, post-Minkowskian, post-Newtonian, and numerical relativity results

TL;DR

This paper develops a frequency-domain Regge-Wheeler-Zerilli self-force calculation for gravitational radiation from a small body on hyperbolic/parabolic orbits around a Schwarzschild black hole and validates the results against state-of-the-art post-Minkowskian and post-Newtonian predictions, as well as numerical relativity. The method yields the radiated energy to infinity and to the horizon, and demonstrates agreement with the latest PM expansions up to high velocity ($v_\infty$ near 0.7), while also assessing PN limits and proposing PN-PM hybrids for improved intermediate-velocity accuracy. A first comparison with NR data confirms consistency away from the strong-field critical orbit, supporting the reliability of SF and PM results in the scattering regime. The work lays groundwork for extending these comparisons to the radiated angular momentum, Kerr perturbations, and memory effects, with implications for high-energy scattering models and EMRI modeling in future gravitational-wave observations.

Abstract

In this work I use a frequency-domain Regge-Wheeler-Zerilli approach to compute the gravitational wave energy radiated by a compact body moving along a hyperbolic or parabolic geodesic of a Schwarzschild black hole. I compare my results with the latest post-Minkowskian (PM) calculations for the radiated energy and find agreement for hyperbolic orbits with large impact parameters and characterized by a velocity at infinity, $v_\infty$, as large as $v_\infty/c=0.7$. I also find agreement between my results and the leading-order PM expansion for the radiation absorbed by the black hole. I make further comparisons with post-Newtonian (PN) theory and show the effectiveness of a simple PN-PM hybrid model. Finally, I make a first comparison of the radiated energy between self-force and numerical relativity.

Figures (8)

• Figure 1: Comparison of the $\mathcal{O}(\nu^2)$ contribution to the radiated energy between PM (solid curves) and PM-3PN (dashed curves) results. The left plot shows the comparison in the relatively strong field with $b/M=15$. The right plot shows the comparison in the weak field with $b/M=100$ (this value is more typical of the values explored in this work). In both cases it is notable how well the PN expansion at each PM order agrees with the full PM result, even for quite large values of $v_\infty$. In both plots the lowest value of $v_\infty$ shown corresponds to the critical velocity for that value of $b$ (recall below the critical velocity the geodesic orbit plunges into the black hole). Recall also the weak-field criteria in Eq. \ref{['eq:weakfield_SF']} means the PM and PM-3PN results shown here will not be accurate for trajectories near the critical orbit when compared to the full numerical result.
• Figure 2: Convergence of the weighting coefficient for the up solution for the $l=2, m=2$ mode with $\omega=0.01$ and $b=70$, $v_\infty=0.7$. Although the integral appears to converge quite rapidly, the inset shows that the integrand is oscillating rapidly. Resolving all these oscillations is a significant computational bottleneck. I use the magnitude of the oscillations near the largest value of $r_p/m_1$ as an estimate on the error in $C^\pm_{\ell m\omega}$.
• Figure 3: (Left panel) The spectrum of the total radiated energy for each value of $\omega m_1$ for a variety of $\l$-modes (summed over $m$) for a hyperbolic orbit with $b/m_1=95$ and $v_\infty=0.7$. (Inset) The contribution to the $\ell$-modes of $\Delta E^\infty$ (after integrating over $\omega$). For large $\ell$ the contribution drops off exponentially. The red, dashed line shows an exponential fit to the modes between $\ell=8$ and $\ell=12$. (Right panel) Contribution to the total energy absorbed by the horizon for each value of $\omega m_1$ for a variety of ${\ell m}$-modes for a hyperbolic orbit with $b/m_1=90$ and $v_\infty=0.35$. For the weak-field orbits considered in this work the $\ell=2$ modes dominate so much that all the other modes can be ignored.
• Figure 4: Comparison of the total radiated energy from my SF calculation with PM and PN results for $r_{\rm min} = 100m_1$. The PN results agree well with the SF data at low velocities but the agreement rapidly worsens for larger values of $v_\infty$. The PM results agree well at high velocities but less well at low velocities. This is because for these fixed periastron orbits as $v_\infty$ decreases the weak-field criteria \ref{['eq:weakfield_SF']} is less well satisfied. The PM results improved at low velocities by adding additional PN information using the PN-PM hybrid described in Sec. \ref{['sec:hybrid']}. I find using just 1PN information (up to 7PM) the hybrid significantly out performs the pure 4PM result. By including 2PN information (up to 7PM) the 4PM hybrid is better than the 5PM result across the range of $v_\infty$ plotted.
• Figure 5: (Left panel) Comparison of the total radiated energy between SF and PM for $v_\infty/c=0.35$. After subtracting each PM order I find the residual has the expected scaling. In particular, after subtracting all highest known term (5PM) the residuals scales as $b^{-6}$. For a reference $b^{-6}$ curve I plot the 6PM-3PN result of Cho. Although the coefficient of this curve is close to the residual with 5PM, the residual with the 6PM-3PN result (not shown) does not scale as $b^{-7}$ across the range of $b$ values plotted. (Right panel) The same as the left panel but for $v_\infty/c=0.7$. There is some noise in the residual with the 5PM result but the scaling follows the $b^{-6}$ reference curve. This noise stems from the need to include many more $(\ell,m)$-modes for this high velocity orbit, and the high-$\ell$ modes have significantly more noise in their frequency spectrum -- see the left panel of Fig. \ref{['fig:spectrum']}.