We study the dynamics of supermassive black hole binaries embedded in circumbinary gaseous discs , with the SPH code Gadget-2 .
The sub-parsec 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 } = 0.2 M ) .
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 LISA .
It is found that at the moment of entering the LISA frequency domain e _ { LISA } \sim 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 \sim 100 \% , depending on the binary eccentricity ) can be considered a fingerprint of eccentric sub-parsec binaries migrating inside a circumbinary disc .