Observations of the population of cold Jupiter planets ($r>$1 AU) show that nearly all of these planets orbit their host star on eccentric orbits. For planets up to a few Jupiter masses, eccentric orbits are thought to be the outcome of planet-planet scattering events taking place after gas dispersal. We simulate the growth of planets via pebble and gas accretion as well as the migration of multiple planetary embryos in their gas disc. We then follow the long-term dynamical evolution of our formed planetary system up to 100 Myr after gas disc dispersal. We investigate the importance of the initial number of protoplanetary embryos and different damping rates of eccentricity and inclination during the gas phase for the final configuration of our planetary systems. We constrain our model by comparing the final dynamical structure of our simulated planetary systems to that of observed exoplanet systems. Our results show that the initial number of planetary embryos has only a minor impact on the final orbital eccentricity distribution of the giant planets, as long as damping of eccentricity and inclination is efficient. If damping is inefficient (slow), systems with a larger initial number of embryos harbor larger average eccentricities. In addition, for slow damping rates, we observe that scattering events already during the gas disc phase are common and that the giant planets formed in these simulations match the observed giant planet eccentricity distribution best. These simulations also show that massive giant planets (above Jupiter mass) on eccentric orbits are less likely to host inner super-Earths as these get lost during the scattering phase, while systems with less massive giant planets on nearly circular orbits should harbor systems of inner super-Earths. Finally, our simulations predict that giant planets are on average not single, but live in multi-planet systems.
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