The curious case of parabolic encounters: gravitational waves with linear & non-linear memory

TL;DR

The paper investigates gravitational-wave memory in parabolic encounters ($e=1$), showing that naive extrapolations from eccentric/hyperbolic cases can misestimate memory. It introduces a dedicated parabolic parameterization and an effective field theory framework to compute GW signals and energy spectra, revealing that linear memory vanishes for parabolic orbits while non-linear memory produces jumps tied to strong radiation near periapsis. The work highlights a distinctive zero-frequency limit structure and demonstrates how radiation can alter orbital parameters, potentially leading to capture into bound orbits. These insights establish a robust theoretical foundation for studying memory in marginally bound binaries and motivate future detector capabilities to probe such bursts. The results integrate soft-theorem intuition with stress-tensor-based EFT calculations to characterize both linear and non-linear memory in a regime where traditional extrapolations fail.

Abstract

The memory effect is known to introduce a permanent displacement in the gravitational wave (GW) detectors after the passage of a GW signal. While the linear memory adheres to the source properties, the non-linear memory is a secondary effect sourced by the GW itself. In the present work, we discuss GW signals with both these kinds of memory effects, while focusing on the parabolic limit of an encounter. This special case is theoretically intriguing and emerges as a limiting situation for both eccentric and hyperbolic events. However, in this paper, we argue that a simple extrapolation of memory calculations for eccentric or hyperbolic cases to the parabolic case may lead to incorrect estimations. Therefore, we treat the parabola as a special case and use an intrinsic parameterization, with which we calculate gravitational wave signals and their energy spectrum via an effective field theory formalism. Unlike the hyperbolic case, which is known to have linear memory, we notice that parabolic encounters bring out new features in the zero frequency limit (ZFL). Our work highlights some of the key challenges and salient aspects of these encounters, and paves the way to study such binary evolution with nonzero memory.

Figures (7)

• Figure 1: Schematic of a parabolic trajectory for a binary system involving masses $m_1$ and $m_2$. Inset: we focus on the region near the periapsis, which will be our main interest.
• Figure 2: Behaviour of $\frac{1}{\nu^{1/3}}$ pole in the time domain, with the transient spike at $t=0$, obtained through numerical fourier transform.
• Figure 3: Power Spectrum for different $r_{min}$ values.
• Figure 4: In the left panel, the blue point denotes $r_{min} = 5.1 M$, below which, the points are neglected (grey). The black line shows physically relevant cases from which 3 cases ($r_{min} = 8 M, 10 M, 12 M$), as shown with red points, are selected for analysis. Right panel shows the variation of $r_{min}$ with $\xi$ for different instantaneous captures.
• Figure 5: Variation of $L_z$ (left) and $E_{in}$ (right) near the peripasis with respect to $\xi$, respectively, for different instantaneous captures. Different colors in the plot have same meaning as in Fig. \ref{['fig:rmin_subfigures']}.