Gravitational-wave tails of memory

Gravitational-wave tails are linear waves that backscatter on the curvature of space-time generated by total mass-energy source. The non-linear memory effect arises from gravitational sourced stress-energy distribution themselves. These two effects due to quadratic multipolar interactions (mass-quadrupole and quadrupole-quadrupole) well known. Also known themselves (cubic "tails-of-tails") tails-of-tails or vice-versa (quartic "tails-of-tails-of-tails"). In this work, we focus cubic "tails-of-memory" corresponding mass-quadrupole-quadrupole interaction, as "spin-quadrupole tails", which interaction between mass, angular momentum quadrupole. tails-of-memory spin-quadrupole contribute asymptotic waveform at fourth-post-Newtonian (4PN) order beyond quadrupolar radiation.