Renormalized second post-Newtonian spin contributions to the accumulated orbital phase for LISA sources

Abstract

We give here a new third post-Newtonisn (3PN) spin-spin contribution (in the PN parameter ε) to the accumulated orbital phase of a compact binary, arising from the spin-orbit precessional motion of the spins. In the equal mass case this contribution vanishes, but LISA sources of merging supermassive binary black holes have typically a mass ratio of 1:10. For such non-equal masses this 3PN correction is periodic in time, with period approximately ε −1 times larger than the period of gravitational waves. We derive a renormalized and simpler expression of the spin-spin coefficient at 2PN, as an average over the time-scale of this period of the combined 2PN and 3PN contribution. We also find that for LISA sources the quadrupole-monopole contribution to the phase dominates over the spin-spin contribution, while the self-spin contribution is negligible even for the dominant spin. Finally we define a renormalized total spin coefficient σ to be employed in the search for gravitational waves emitted by LISA sources.

Numerical simulations of black-hole binaries and gravitational wave emission

“…The impact of different SMBH assembly models on the mass and mass ratio distribution of detectable binaries has been discussed by various authors. The general consensus is that mass ratios q 1/10 (and down to q ≈ 10 −4 ) should be common [31,216,217]. In contrast with the case of stellar-mass BHs, there is by now some observational evidence for SMBH binaries [218,219,220,221,222].…”

Renormalized spin coefficients in the accumulated orbital phase for unequal mass black hole binaries

“…The renormalized spin-spin coefficient at 2PN thus is adjusted by the average of the 3PN quasi-precessional contribution, taken over the period of change induced by these quasi-precessions (a time-scale one PN order lower than the orbital period). We have checked that the renormalized spin-spin coefficient gives closer results to the numerical estimates for compact binaries with the mass ratio of order 1/10 [14].…”

“…We remark that according to [14], the mass quadrupole -mass monopole coupling for maximally rotating black holes (with quadrupole moment originating entirely in their rotation) gives rise to contributions in the accumulated orbital phase larger by a half to one order of magnitude than the spin-spin contribution, at least for the typical unequal mass ratios. Therefore for a successful detection this qualifies as the second most important finite mass contribution to be considered, after the spin-orbit effect.…”

“…This coefficient arises by modifying the 2PN contribution with the average of some of the 3PN contributions. By using this renormalized coefficient in the context of the 2PN accurate dynamics, one can bring closer the 2PN analytical prediction to the numerical results [14]. ‡ Thus, in the unequal mass case we propose to replace…”

“…From among the quasi-precessional evolutions, we have recently integrated [14] the SO evolution (averaged over one orbit) of the angle ∆ψ, finding…”