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In this paper, we explore the establishment and evolution of the empirical correlation between black hole mass ( M  BH ) and velocity dispersion (σ) with redshift. We trace the growth and accretion history of massive black holes (MBHs) starting from high‐redshift seeds that are planted via physically motivated prescriptions. Two seeding models are explored in this work: ‘light seeds’, derived from Population III remnants, and ‘heavy seeds’, derived from direct gas collapse. Even though the seeds themselves do not satisfy the   M  BH –σ  relation initially, we find that the relation can be established and maintained at all times if self‐regulating accretion episodes are associated with major mergers. The massive end of the   M  BH –σ  relation is established early, and lower mass MBHs migrate on to it as hierarchical merging proceeds. How MBHs migrate towards the relation depends critically on the seeding prescription. Light seeds initially lie well below the   M  BH –σ  relation, and MBHs can grow via steady accretion episodes unhindered by self‐regulation. In contrast, for the heavy seeding model, MBHs are initially over‐massive compared to the empirical correlation, and the host haloes assemble prior to kick‐starting the growth of the MBH. We find that the existence of the   M  BH –σ  correlation is purely a reflection of the merging hierarchy of massive dark matter haloes. The slope and scatter of the relation however appear to be a consequence of the seeding mechanism and the self‐regulation prescription. We expect flux limited active galactic nucleus surveys to select MBHs that have already migrated on to the   M  BH –σ  relation. Similarly, the Laser Interferometer Space Antenna (LISA) is also likely to be biased towards detecting merging MBHs that preferentially inhabit the   M  BH –σ . These results are a consequence of major mergers being more common at high redshift for the most massive, biased, galaxies that host MBHs which have already migrated on to the   M  BH –σ  relation. We also predict the existence of a large population of low‐mass ‘hidden’ MBHs at high redshift which can easily escape detection. Additionally, we find that if MBH seeds are massive,  ∼10 5  M ⊙  , the low‐mass end of the   M  BH –σ  flattens towards an asymptotic value, creating a characteristic ‘plume’.

Journey to the M BH –σ relation: the fate of low‐mass black holes in the Universe