Reviving stochasticity: uncertainty in SMBH binary eccentricity is unavoidable

We study supermassive black hole (SMBH) binary eccentricity of equal-mass galaxy mergers in $N$-body simulations with the KETJU code, which combines the GAGDET-4 fast multipole gravity solver with accurate regularised integration and post-Newtonian corrections around SMBHs. In simulations with realistic, high eccentricity galactic merger orbits, the hard binary eccentricity is found to be a non-linear function of the deflection angle in the SMBH orbit during the final, nearly radial close encounter between the SMBHs before they form a bound binary. This mapping between the deflection angle and the binary eccentricity has no apparent resolution dependence in our simulations spanning the resolution range of $1\times10^5$-$8\times10^6$ particles per galaxy. The mapping is also captured using a simple model with an analytic potential, indicating that it is driven by the interplay between a smooth asymmetric stellar background potential and dynamical friction acting on the SMBHs. Due to the non-linearity of this mapping, in eccentric major merger configurations small, parsec-scale variations in the merger orbit can result in binary eccentricities varying in nearly the full possible range between $e=0$ and $e=1$. In idealised simulations, such variations are caused by finite resolution effects, and convergence of the binary eccentricity can be achieved with increasing resolution. However, in real galaxies, other mechanisms such as nuclear gas and substructure that perturb the merger orbit are likely to be significant enough for the binary eccentricity to be effectively random. Our results indicate that the distribution of these effectively random eccentricities can be studied using even moderate resolution simulations.