Signatures of Circumbinary Disk Dynamics in Multi-Messenger Population Studies of Massive Black Hole Binaries

We investigate the effect of cutting-edge circumbinary disk (CBD) evolution models on massive black hole binary (MBHB) populations and the gravitational wave background (GWB). We show that CBD-driven evolution leaves a tell-tale signature in MBHB populations, by driving binaries towards an equilibrium eccentricity that depends on binary mass ratio. We find high orbital eccentricities ($e_{\rm b} \sim 0.5$) as MBHBs enter multi-messenger observable frequency bands. The CBD-induced eccentricity distribution of MBHB populations in observable bands is independent of the initial eccentricity distribution at binary formation, erasing any memory of eccentricities induced in the large-scale dynamics of merging galaxies. Our results suggest that eccentric MBHBs are the rule rather than the exception in upcoming transient surveys, provided that CBDs regularly form in MBHB systems. We show that the GWB amplitude is sensitive to CBD-driven preferential accretion onto the secondary, resulting in an increase in GWB amplitude $A_{\rm yr^{-1}}$ by over 100\% with just 10\% Eddington accretion. As we self consistently allow for binary hardening and softening, we show that CBD-driven orbital expansion does not diminish the GWB amplitude, and instead increases the amplitude by a small amount. We further present detection rates and population statistics of MBHBs with $M_{\rm b} \gtrsim 10^6 \, M_{\odot}$ in LISA, showing that most binaries have equal mass ratios and can retain residual eccentricities up to $e_{\rm b} \sim 10^{-3}$ due to CBD-driven evolution.