Limiting eccentricity of sub-parsec massive black hole binaries surrounded by self-gravitating gas discs

We study the dynamics of supermassive black hole binaries embedded in circumbinary gaseous discs, with the smoothed particle hydrodynamics code GADGET-2. The subparsec binary (of total mass M and mass ratio q = 1/3) has excavated a gap and transfers its angular momentum to the self-gravitating disc (M-disc = 0.2M). We explore the changes of the binary eccentricity, e, by simulating a sequence of binary models that differ in the initial eccentricity e(0) only. In initially low-eccentric binaries, the eccentricity increases with time, while in high-eccentric binaries e declines, indicating the existence of a limiting eccentricity e(crit) that is found to fall in the interval [0.6, 0.8]. We also present an analytical interpretation for this saturation limit. An important consequence of the existence of e(crit) is the detectability of a significant residual eccentricity e(LISA) by the proposed gravitational wave detector Laser Interferometer Space Antenna (LISA). It is found that at the moment of entering the LISA frequency domain e(LISA) similar to 10(-3)-10(-2), a signature of its earlier coupling with the massive circumbinary disc. We also observe large periodic inflows across the gap, occurring on the binary and disc dynamical time-scales rather than on the viscous time. These periodic changes in the accretion rate (with amplitudes up to similar to 100 per cent, depending on the binary eccentricity) can be considered a fingerprint of eccentric subparsec binaries migrating inside a circumbinary disc.