We study the evolution of black holes ( BHs ) on the M _ { BH } - \sigma and M _ { { BH } } - M _ { bulge } planes as a function of time in disk galaxies undergoing mergers .
We begin the simulations with the progenitor black hole masses being initially below ( \Delta \log M _ { BH,i } \sim - 2 ) , on ( \Delta \log M _ { BH,i } \sim 0 ) and above ( \Delta \log M _ { BH,i } \sim 0.5 ) the observed local relations .
The final relations are rapidly established after the final coalescence of the galaxies and their BHs .
Progenitors with low initial gas fractions ( f _ { gas } = 0.2 ) starting below the relations evolve onto the relations ( \Delta \log M _ { BH,f } \sim - 0.18 ) , progenitors on the relations stay there ( \Delta \log M _ { BH,f } \sim 0 ) and finally progenitors above the relations evolve towards the relations , but still remaining above them ( \Delta \log M _ { BH,f } \sim 0.35 ) .
Mergers in which the progenitors have high initial gas fractions ( f _ { gas } = 0.8 ) evolve above the relations in all cases ( \Delta \log M _ { BH,f } \sim 0.5 ) .
We find that the initial gas fraction is the prime source of scatter in the observed relations , dominating over the scatter arising from the evolutionary stage of the merger remnants .
The fact that BHs starting above the relations do not evolve onto the relations , indicates that our simulations rule out the scenario in which overmassive BHs evolve onto the relations through gas-rich mergers .
By implication our simulations thus disfavor the picture in which supermassive BHs develop significantly before their parent bulges .