Post-Newtonian corrections to the gravitational-wave memory for quasicircular, inspiralling compact binaries

The Christodoulou memory is a nonlinear contribution to the gravitational-wave field that is sourced by the gravitational-wave stress-energy tensor. For quasicircular, inspiralling binaries, the Christodoulou memory produces a growing, nonoscillatory change in the gravitational-wave plus polarization, resulting in the permanent displacement of a pair of freely-falling test masses after the wave has passed. In addition to its nonoscillatory behavior, the Christodoulou memory is interesting because even though it originates from 2.5 post-Newtonian (PN) order multipole interactions, it affects the waveform at leading (Newtonian/quadrupole) order. The memory is also potentially detectable in binary black-hole mergers. While the oscillatory pieces of the gravitational-wave polarizations for quasicircular, inspiralling compact binaries have been computed to 3PN order, the memory contribution to the polarizations has only been calculated to leading order (the next-to-leading order 0.5PN term has previously been shown to vanish). Here the calculation of the memory for quasicircular, inspiralling binaries is extended to 3PN order. While the angular dependence of the memory remains qualitatively unchanged, the PN correction terms tend to reduce the memorys magnitude. Explicit expressions are given for the memory contributions to the plus polarization and the spin-weighted spherical-harmonic modes of the metric and curvature perturbations. Combined with the recent results of Blanchet et al.(2008), this completes the waveform to 3PN order. This paper also discusses: (i) the difficulties in extracting the memory from numerical simulations, (ii) other nonoscillatory effects that enter the waveform at high PN orders, and (iii) issues concerning the observability of the memory.