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This paper is motivated by the need to improve the post-Newtonian (PN)amplitude accuracy of waveforms for gravitational waves generated byinspiralling compact binaries, both for use in data analysis and in thecomparison between post-Newtonian approximations and numerical relativitycomputations. It presents: (i) the non-linear couplings between multipolemoments of general post-Newtonian matter sources up to order 3.5PN, includingall contributions from tails, tails-of-tails and the non-linear memory effect;and (ii) the source mass-type octupole moment of (non-spinning) compactbinaries up to order 3PN, which permits to complete the expressions of theoctupole modes (3,3) and (3,1) of the gravitational waveform to order 3.5PN. Atthis occasion we reconfirm by means of independent calculations our earlierresults concerning the source mass-type quadrupole moment to order 3PN. Relateddiscussions on factorized resummed waveforms and the occurence of logarithmiccontributions to high order are also included.

Non-linear multipole interactions and gravitational-wave octupole modes for inspiralling compact binaries to third-and-a-half post-Newtonian order