Gravitational waves from parabolic encounters: A study of linear and nonlinear memory

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 $\textit{linear memory}$ adheres to the source properties, the $\textit{non-linear memory}$ 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). The exactly parabolic case is studied here primarily as an idealized separatrix between bound and unbound motion, and our analysis highlights some of the key challenges and salient aspects of GW memory in this regime.