Climbing up the memory staircase: Equatorial zoom-whirl orbits

The nonlinear, or null gravitational, memory effect for zoom-whirl orbits around fast spinning Kerr black holes for extreme mass ratio systems has a staircase structure. The latter is characterized by a periodic fast increase in the memory during the whirl phases of the orbit and near constant levels when the orbit is near apoapsis. We study the contribution of different spherical harmonic modes to this effect and discuss the relative importance, thereof. Because the hereditary memory effect increases in magnitude linearly in the number of periapsis passages on a dynamical time scale, while the oscillatory parent gravitational waves are sourced by an orbit that evolves on a much longer radiation reaction time scale, the magnitude of the former relative to the latter increases with time. We also consider orphan memory and the memory effect for extreme mass ratio inspirals. We then discuss the prospect of detection with the Laser Interferometer Space Antenna.