Accretion-mediated spin–eccentricity correlations in LISA massive black hole binaries

ABSTRACT We examine expected effective spin ($\chi _{{\rm eff},1{\rm yr}}$) and orbital eccentricity ($e_{1\rm yr}$) correlations for a population of observable equal-mass massive black hole binaries (MBHBs) with total redshifted mass $M_z\sim [10^{4.5},10^{7.5}]~{{\rm M}_{\odot }}$ embedded in a circumbinary disc (CBD) at redshifts $z=1$ and $z=2$, 1 yr before merging in the Laser Interferometer Space Antenna (LISA) band. We find a strong correlation between measurable eccentricity and negative effective spin for MBHBs that are carried to merger by retrograde accretion. This is due to the well-established eccentricity pumping of retrograde accretion and less-well-established formation of retrograde mini-discs coupled with a stable retrograde CBD throughout the binary evolution from the self-gravitating radius. Conversely, prograde accretion channels result in positive $\chi _{{\rm eff},1\rm yr}$ and non-measurable $e_{1\rm yr}$ except for almost unity Eddington ratio and $M_z\lesssim 10^{5}~{{\rm M}_{\odot }}$ MBHBs at $z=1$. This clear contrast between the two CBD orientations – and particularly the unique signature of retrograde configurations – provides a promising way to unlock the mysteries of MBHB formation channels in the LISA era.