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We examine expected effective spin (₄₅₅, 1 yr) and orbital eccentricity (e₁ ₘₑ) correlations for a population of observable equal-mass massive black hole binaries (MBHBs) with total redshifted mass Mᵦ10^4. 5, 10^7. 5~ M_ embedded in a circumbinary disc (CBD), one-year before merging in the 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 the formation of retrograde CBD-aligned mini-discs, as observed in hydrodynamical simulations. Conversely, prograde accretion channels result in positive ₄₅₅, 1 yr and non-measurable e₁ ₘₑ. 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.
Garg et al. (Tue,) studied this question.