Prospects to Explore High-redshift Black Hole Formation with Multi-band Gravitational Waves Observatories

The assembly of massive black holes in the early universe remains a poorly constrained open question in astrophysics. The merger and accretion of light seeds (remnants of Population III stars with mass below $\sim 1000M_{\odot}$) or heavy seeds (in the mass range $10^4-10^6 M_{\odot}$) could both explain the formation of massive black holes, but the abundance of seeds and their merging mechanism are highly uncertain. In the next decades, the gravitational-wave observatories coming online are expected to observe very high-redshift mergers, shedding light on the seeding of the first black holes. In this Letter we explore the potential and limitations for LISA, Cosmic Explorer and Einstein Telescope to constrain the mixture ratio of light and heavy seeds as well as the probability that central black holes in merging galaxies merge as well. Since the third generation ground-based gravitational-wave detectors will only observe light seed mergers, we demonstrate two scenarios in which the inference of the seed mixture ratio and merging probability can be limited. The synergy of multi-band gravitational-wave observations and electromagnetic observations will likely be necessary in order to fully characterize the process of high-redshift black hole formation.