  We study how statistical properties of supermassive black holes depend on the
frequency and conditions for massive seed formation in cosmological simulations
of structure formation. We develop a novel method to recalculate detailed
growth histories and merger trees of black holes within the framework of the
Illustris simulation for several seed formation models, including a physically
motivated model where black hole seeds only form in progenitor galaxies that
conform to the conditions for direct collapse black hole formation. While all
seed models considered here are in a broad agreement with present observational
constraints on black hole populations from optical, UV and X-ray studies, we
find that they lead to widely different black hole number densities and halo
occupation fractions which are currently observationally unconstrained. In
terms of future electromagnetic spectrum observations, the faint-end quasar
luminosity function and the low-mass-end black hole-host galaxy scaling
relations are very sensitive to the specific massive seed prescription.
Specifically, the direct collapse model exhibits a seeding efficiency which
decreases rapidly with cosmic time and produces much fewer black holes in low
mass galaxies, in contrast to the original Illustris simulation. We further
find that the total black hole merger rate varies by more than one order of
magnitude for different seed models, with the redshift evolution of the chirp
mass changing as well. Supermassive black hole merger detections with LISA and
International Pulsar Timing Array may hence provide the most direct means of
constraining massive black hole seed formation in the early Universe.
