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Direct imaging (DI) surveys suggest that gas giants beyond 20 AU are rarearound FGK stars. However, it is not clear what this means for the formationfrequency of Gravitational Instability (GI) protoplanets due to uncertaintiesin gap opening and migration efficiency. Here we combine state-of-the-artcalculations of homogeneous planet contraction with a population synthesiscode. We find DI constraints to be satisfied if protoplanet formation by GIoccurs in tens of percent of systems if protoplanets `super migrate' to smallseparations. In contrast, GI may occur in only a few percent of systems ifprotoplanets remain stranded at wide orbits because their migration is`quenched' by efficient gap opening. We then use the frequency of massivegiants in radial velocity surveys inside 5 AU to break this degeneracy -observations recently showed that this population does not correlate with thehost star metallicity and is therefore suspected to have formed via GI followedby inward migration. We find that only the super-migration scenario cansufficiently explain this population whilst simultaneously satisfying the DIconstraints and producing the right mass spectrum of planets inside 5 AU. Ifmassive gas-giants inside 5 AU formed via GI, then our models imply thatmigration must be efficient and that the formation of GI protoplanets occurs inat least a tens of percent of systems.

Constraining the initial planetary population in the gravitational instability model