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We give here a new third post-Newtonisn (3PN) spin-spin contribution (in thePN parameter $\epsilon $) to the accumulated orbital phase of a compact binary,arising from the spin-orbit precessional motion of the spins. In the equal masscase this contribution vanishes, but LISA sources of merging supermassivebinary black holes have typically a mass ratio of 1:10. For such non-equalmasses this 3PN correction is periodic in time, with period approximately$\epsilon ^{-1}$ times larger than the period of gravitational waves. We derivea renormalized and simpler expression of the spin-spin coefficient at 2PN, asan average over the time-scale of this period of the combined 2PN and 3PNcontribution. We also find that for LISA sources the quadrupole-monopolecontribution to the phase dominates over the spin-spin contribution, while theself-spin contribution is negligible even for the dominant spin. Finally wedefine a renormalized total spin coefficient $\bar{\sigma}$ to be employed inthe search for gravitational waves emitted by LISA sources.

Renormalized 2PN spin contributions to the accumulated orbital phase for LISA sources