Frequency space derivation of linear and non-linear memory gravitational wave signals from eccentric binary orbits

The memory effect in gravitational wave (GW) signals is the phenomenon, wherein the relative position of two inertial GW detectors undergoes a permanent displacement owing to the passage of GWs through them. Measurement of the memory signal is an important target for future observations as it establishes a connection between observations with field-theoretic results like the soft-graviton theorems. Theoretically, the memory signal is predicted at the leading order quadrupole formula for sources like binaries in hyperbolic orbits. This can be in the realm of observations by Advanced LIGO, Einstein-Telescope, or LISA for black-holes with masses $\sim$ $O(10^3 \, M_\odot$) scattered by the super-massive black-hole at the galactic center. Apart from the direct memory component there is a non-linear memory signal in the secondary GW emitted from the primary GW chirp-signals emitted by coalescing binaries. In this paper, we compute the gravitational wave signals and their energy spectrum using the field-theoretic method by computing the scattering amplitudes for eccentric elliptical and hyperbolic binary orbits. The field theoretic calculation gives us the gravitational waveforms of linear and non-linear memory signals directly in the frequency space. The frequency domain templates are useful for extracting signals from the data. We compare our results with other calculations of linear and non-linear memory signals in literature and point out novel features we find in our calculations like the presence of $\log(\omega)$ terms in the linear memory from hyperbolic orbits.