2025MNRAS.536.3177A

We present a well-tested, theoretically supported empirical relation that helps decipher the origins, growth, and properties of supermassive black holes (SMBHs). Based on theoretical considerations and analysis of mass (M_BH) versus age (t) distribution of high-redshift (>5.6) SMBHs, we get M_BH = Ms exp [14.6(t - 100)/t(Myr)], which gives the SMBH's seed mass Ms, and its derivative gives the instantaneous mass-accretion rate. It yields seeds of ∼20-420 M_☉ (solar masses) for the recently discovered SMBHs GNz11, CEERS_1019, and UHZ1, and ∼3 × 10⁴ M_☉ for the largest (1.24 × 10¹⁰ M_☉) high-z SMBH. It is applied to 132446 SMBHs at z < 2.4. The resultant seeds are classified based on size and likely formation mechanism: ∼54 per cent are classified as light (<350 M_☉) deemed to be Pop III remnants; ∼40 per cent as intermediate (350-2 × 10³ M_☉) and ∼6 per cent as heavier seeds (2 × 10³-3 × 10⁴ M_☉), both of which formed possibly by mergers of Pop III remnants. The direct collapse black hole (DCBH) mechanism is not required but not excluded. Furthermore, the results show the following. The mass-accretion rate increases exponentially from the seed's inception (z ∼ 30), reaches a broad plateau at z ∼ 8.5-6 coincident with the epoch of reionization, and decreases monotonically thereafter. Sub-Eddington accretion is the norm, except during the first ∼150 Myr SMBHs either experienced super-Eddington accretion or the radiative efficiency was much < 0.1. The largest seed can potentially grow via luminous accretion to (6.6 ± 2.2) × 10¹⁰ M_☉, consistent with a theoretical limit of ∼5 × 10¹⁰ M_☉. The Eddington ratio decreases and radiative efficiency increases as z decreases, consistent with recent findings.