# On Constraining the Growth History of Massive Black Holes via Their Distribution on the Spin-Mass Plane

## TL;DR

This study models the spin evolution of massive black holes through a two-phase accretion process, revealing how their spin and radiative efficiency distributions relate to their growth history and mass, consistent with observed data.

## Contribution

It introduces a two-phase accretion model to explain MBH spin evolution and constrains the chaotic accretion episodes and radiative efficiencies based on spin measurements.

## Key findings

- MBHs undergo significant chaotic accretion with small mass episodes.
- High-mass MBHs tend to have intermediate-to-high spins.
- The average radiative efficiency of active MBHs is around 0.09-0.15.

## Abstract

The spin distribution of massive black holes (MBHs) contains rich information on the MBH growth history. In this paper, we investigate the spin evolution of MBHs by assuming that each MBH experiences two-phase accretion, with an initial phase of coherent-accretion via either the standard thin disc or super-Eddington disc, followed by a chaotic-accretion phase composed of many episodes with different disc orientations. If the chaotic-phase is significant to the growth of an MBH, the MBH spin quickly reaches the maximum value because of the initial coherent-accretion, then changes to a quasi-equilibrium state and fluctuates around a value mainly determined by the mean ratio of the disc to the MBH mass ($M_{\bullet}$) in the chaotic-accretion episodes, and further declines due to late chaotic-accretion if $M_\bullet \gtrsim (1-3) \times 10^8 M_\odot$. The turning point to this decline is determined by the equality of the disc warp radius and disc size. By matching the currently available spin measurements with mock samples generated from the two-phase model(s) on the spin-mass plane, we find that MBHs must experience significant chaotic-accretion phase with many episodes and the mass accreted in each episode is roughly 1-2 percent of M_bh or less. MBHs with $M_{\bullet}\gtrsim 10^8 M_{\odot}$ appear to have intermediate-to-high spins ($\sim 0.5-1$), while lighter MBHs have higher spins ($\gtrsim 0.8$). The best matches also infer that (1) the radiative efficiencies ($\eta$) of those active MBHs appear to slightly decrease with $M_{\bullet}$; however, the correlation between $\eta$ and $M_{\bullet}$, if any, is weak; (2) the mean radiative efficiency of active MBHs is $<\eta> \sim 0.09-0.15$, consistent with the global constraints.