Black Hole Seeding Prescription

• Black hole seeding prescription defines the criteria, thresholds, and mechanisms for placing initial black hole seeds in simulations, which shape the evolution of supermassive black holes.
• It categorizes seeds into light (Pop III), heavy (DCBH), dynamical/cluster-runaway, primordial, and nuclear cluster delivery channels, each with distinct mass scales and environmental requirements.
• The formulation involves specific physical and numerical parameters that impact observational predictions such as AGN luminosity functions, gravitational-wave event rates, and SMBH–galaxy scaling relations.

A black hole seeding prescription specifies the criteria, physical thresholds, and mechanisms by which the initial black holes (“seeds”) are placed in cosmological simulations and semi-analytic models, forming the foundational step in assembling supermassive black holes (SMBHs) across cosmic time. The chosen seeding blueprint encodes assumptions about the mass scale, formation sites, environmental properties, and timing associated with the early universe’s black hole population, with tangible and lasting impact on SMBH demographics, galaxy-BH scaling relations, the gravitational-wave merger history, and the prospects for distinguishing astrophysical versus exotic (e.g., primordial) formation channels.

1. Classification of Seeding Mechanisms

Black hole seeding prescriptions are broadly categorized by the physical channel responsible for seed formation, the mass scale of the resulting seeds, and their expected abundances and environments.

• Light seeds (Population III remnants): Arise from the collapse of massive, metal-free Population III (Pop III) stars formed at $z\gtrsim 20$ in minihalos ($M_{\rm halo}\sim10^{5-6}\,M_\odot$) with $Z\ll10^{-4}Z_\odot$ and $T_{\rm vir}\gtrsim2\times10^3$ K. Characteristic masses $M_{\rm seed}\sim10^1\text{--}10^3\,M_\odot$; copious ($n\sim0.1\text{--}10\,{\rm cMpc}^{-3}$ at $z\sim10$) (Regan et al., 2024, Chen et al., 3 Sep 2025, Mehta et al., 20 Jan 2026).
• Heavy seeds (direct collapse black holes, DCBH): Form via isothermal collapse of pristine atomic-cooling halos ($M_{\rm halo}\gtrsim10^{7-8}\,M_\odot$, $T_{\rm vir}>10^4$ K, $Z<10^{-5}\text{--}10^{-4}\,Z_\odot$) irradiated by intense Lyman–Werner (LW) flux ($J_{\rm LW}>100\text{--}1000\,J_{21}$), enabling the formation of supermassive stars (SMSs) that collapse directly to BHs (Dijkstra et al., 2014, Agarwal et al., 2012). Seed masses $M_{\rm seed}\sim10^4\text{--}10^6\,M_\odot$, but extremely rare ($n\sim10^{-10}\text{--}10^{-5}\,{\rm cMpc}^{-3}$ at $z\sim10$) (Regan et al., 2024).
• Dynamical/cluster-runaway seeds: Runaway stellar collisions or mergers in extremely dense ($n_\star\gtrsim10^6$ pc$^{-3}$) young star clusters can yield intermediate-mass seeds ($M_{\rm seed}\sim10^3\text{--}10^4\,M_\odot$), with number densities intermediate between Pop III and DCBH channels (Regan et al., 2024).
• Primordial black holes (PBH): Collapse of large density perturbations re-entering the horizon in the early universe seeds BHs with a broad mass function, generally much lower fractional abundance ($f_{\rm PBH}\lesssim10^{-12}$ of the dark matter) (Musco et al., 2020, Dayal, 2024).
• Nuclear star cluster delivery: Black holes embedded in infalling nuclear star clusters can be deposited into galactic centers via dynamical friction, offering a seeding mechanism for (especially) low-mass and bulgeless galaxies (Graham et al., 2021).

These channels form the foundation for classifying seeding prescriptions—and motivate different parameterizations and environmental requirements.

2. Formulation and Parameterization of Seeding Prescriptions

The explicit implementation of a black hole seeding prescription requires the specification of seed mass assignments, host environment thresholds, and event rates, expressed as either deterministic or stochastic rules in semi-analytic or hydrodynamical codes.

2.1. Light-Seed (Pop III) Prescriptions

• First-principles Pop III formation: Seeds form as the endpoint of Pop III stellar evolution, with gas cells eligible for star formation if $\rho>\rho_J=J^2\pi c_s^2/(G\Delta x^2)$ (Jeans criterion), $Z<10^{-4}Z_\odot$, and collapse underway (Mehta et al., 20 Jan 2026).
• IMF sampling: Pop III stars drawn from a top-heavy IMF, typically $$f(\log M)\propto M^{-1.3}\exp[-(M_{\rm char}/M)^{1.6}]$$, $1<M<300\,M_\odot$, $M_{\rm char}\sim20\,M_\odot$ (Mehta et al., 20 Jan 2026, Chen et al., 3 Sep 2025).
• Remnant mapping: BH seed created at stellar death, with $M_{\rm seed}$ assigned as helium core mass for $11<M_*<40\,M_\odot$, full stellar mass for $M_*>40\,M_\odot$ (except for explosive pair-instability SNe) (Mehta et al., 20 Jan 2026, Chen et al., 3 Sep 2025).
• Halo-based “All-Light-Seeds” law: In semi-analytic models, the ALS/CAM25 prescription uses $$M_{\rm seed}=3\times10^6\,h^{-1}M_\odot\,(M_{\rm halo}/10^{10}\,h^{-1}M_\odot)^{1.33}$$ ($M_{\rm halo}$ between $7.9\times10^7$ and $10^{10}\,M_\odot$) (Singh et al., 5 Dec 2025, Cammelli et al., 2024).

2.2. Gas-Based and Heavy-Seed Prescriptions

• Gas mass and metallicity thresholding: Seeds inserted in halos when both $M_{\rm gas}(n_{\rm H}>0.1$ cm$^{-3}$, $Z<10^{-4}Z_\odot) > M \times M_{\rm seed}$ and $M_{\rm halo}>\tilde M_h \times M_{\rm seed}$ (Bhowmick et al., 2024, Bhowmick et al., 2021, Bhowmick et al., 2022).
• Lyman–Werner (LW) flux criterion: For DCBH formation, require the above to be bathed in $J_{\rm LW}>J_{\rm crit}$, typically $J_{\rm crit}=10$–$300\,J_{21}$ (Pop II/III spectrum dependent) (Bhowmick et al., 1 Oct 2025, Dijkstra et al., 2014, Agarwal et al., 2012).
• Seed mass assignment: For DCBH, set $M_{\rm seed}=10^4$–$10^6\,M_\odot$ (atomic cooling regime); for cluster-runaway, $M_{\rm seed}=10^3$–$10^4\,M_\odot$ (Ferrara et al., 2014, Regan et al., 2024).

2.3. Stochastic and Hybrid Prescriptions

• Probability-based insertion: Assign seeds with probability $f_{\rm seed}\in[0.01,1]$ in eligible halos; ALS model uses deterministic assignment but other models modulate seeding by gas spin, metallicity, or stochasticity (DeGraf et al., 2019, Singh et al., 5 Dec 2025).
• Pop III.1 isolation model: BHs seeded in minihalos at $z\gtrsim20$ only if located $d_{\rm iso}>50\text{--}100$ kpc (proper) from any prior seed, with $M_{\rm seed}=10^5\,M_\odot$ (Cammelli et al., 2024).
• Subgrid scaling: In multifidelity simulations, stochastic seeding probabilities are calibrated from high-resolution runs to reproduce the correct descendant mass functions in lower-resolution boxes, preserving the impact of original low-mass seeds (Bhowmick et al., 2024).

2.4. Primordial Black Hole (PBH) Seeding

• Density criterion: For a given comoving curvature spectrum $\mathcal P_\zeta(k)$, compute the “shape parameter” $\alpha$ and the corresponding nonlinear density threshold $\delta_c(\alpha)$ for PBH formation, using explicit analytic fits (Musco et al., 2020).
• Abundance scaling: PBHs are placed in all horizon patches exceeding this threshold; the mass function is linked to $\delta_m-\delta_c$, with $M_{\rm PBH}\propto(\delta_m-\delta_c)^\gamma$ (Musco et al., 2020, Dayal, 2024). The PBH fraction in dark matter must respect $f_{\rm PBH}\le10^{-12}$ to avoid observational limits (Dayal, 2024).

3. Physical and Numerical Thresholds

Black hole seeding prescriptions rely on physically motivated choices for mass, metallicity, density, and radiative environment parameters, often constrained by simulation capabilities and by resolution limits.

Channel	Key Thresholds	Seed Mass [$M_\odot$]
Pop III	$n_{\rm H}>\sim1$ cm$^{-3}$, $Z<10^{-4}Z_\odot$, $T_{\rm vir}\gtrsim2\times10^3$ K	$10^1$–$10^3$
DCBH	$T_{\rm vir}>10^4$ K, $Z<10^{-5}$–$10^{-4}Z_\odot$, $J_{\rm LW}>100$–$1000\,J_{21}$	$10^4$–$10^6$
Cluster run.	$n_\star\gtrsim10^6$ pc$^{-3}$, $$\dot M_{\rm gas}\gtrsim10^{-4}\,M_\odot\,\rm yr^{-1}$$	$10^3$–$10^4$
PBH	$\delta_m>\delta_c(\alpha)$ at horizon re-entry	$<10^{-5}$–$10^{3.75}$

Critical parameters like $J_{\rm crit}$ (for H$_2$ suppression), $Z_{\rm crit}$ (for fragmentation), and minimum gas or halo mass are chosen based on theory and simulation calibration, but typically carry factor-of-few uncertainties (Regan et al., 2024, Dijkstra et al., 2014).

4. Seeding Prescriptions in Hydrodynamical and Semi-Analytic Models

Implementation in simulation codes or semi-analytic frameworks follows modular workflows:

Hydrodynamical Codes:

• Gas-based seeding: At each coarse timestep, scan all halos; if above both minimum halo and star-forming metal-poor gas mass, spawn $M_{\rm seed}$ at the potential minimum (Bhowmick et al., 2024, Bhowmick et al., 2021).
• Fine time evolution: High-resolution runs directly resolve Pop III star formation and seed emergence at the grid-cell level, naturally linking stellar evolution and collapse to BH creation (Mehta et al., 20 Jan 2026).

Semi-Analytic Models (SAMs):

• Merger-tree models assign seeds according to deterministic, stochastic, or environmental rules, propagate them down trees, and evolve them with subsequent BH accretion and mergers.
• Example: PINOCCHIO+GAEA implementations utilize ALS (All-Light-Seeds), HMT (halo-mass threshold), or Pop III.1 isolation models, with parameterized assignment rules for occupation fraction and seed mass (Cammelli et al., 2024, Singh et al., 5 Dec 2025).
• DCBH seeding requires tracking local radiative fields for LW flux, metal enrichment by supernova events, and, optionally, clustering information (Dijkstra et al., 2014, Agarwal et al., 2012).

5. Observational Consequences and Discriminants

Different seeding prescriptions imprint lasting signatures on SMBH and galaxy properties and the expected event rates for future surveys:

• Low-Mass End Occupation: Light-seed prescriptions generically yield near-unity SMBH occupation down to low galaxy masses; heavy-seed/threshold models show a turnover or dearth at $M_*\sim10^9$–$10^{10}\,M_\odot$ (Ricarte et al., 2018, DeGraf et al., 2019).
• High-Redshift AGN Luminosity Function: The abundance of faint AGN at $z\gtrsim7$ scales with seed abundance; faint-end normalization and redshift evolution in Lynx or JWST deep surveys directly probe the seeding channel (Ricarte et al., 2018, Bhowmick et al., 2024, Chen et al., 3 Sep 2025).
• Gravitational-Wave Merger Statistics: The mass and redshift distribution of LISA-detectable BH mergers are highly sensitive to seed abundance and mass. Light-seed models yield more events/year, especially in the IMBH regime ($10^3$–$10^5\,M_\odot$), and produce higher merger rates at high $z$ (Singh et al., 5 Dec 2025, Chen et al., 3 Sep 2025, Bhowmick et al., 2024, Bhowmick et al., 1 Oct 2025).
• Scaling Relations: The $M_{\rm BH}$–$M_*$ relation at high $z$ depends on the initial seeding but converges at the high-mass end due to hierarchical mergers and accretion. Seed models primarily affect the scatter and normalization at the low-mass end and at early times (Bhowmick et al., 2021, Ricarte et al., 2018).
• Metal Enrichment and Host Properties: Heavy-seed channels rely on the delay of metal enrichment and suppression of H$_2$ cooling, making their environmental requirements rare and spatially clustered (Dijkstra et al., 2014, Agarwal et al., 2012, Regan et al., 2024).

6. Special Cases and Alternative Channels

• Nuclear Star Cluster Delivery: Infalling nuclear star clusters containing intermediate-mass black holes can spiral into galactic centers via dynamical friction, as directly observed in NGC 4424 (Nikhuli cluster), with clear timescales and tidal constraints, offering a non-cosmological, in-situ assembly route (Graham et al., 2021).
• Primordial Black Hole Driven Structure: PBH prescriptions formalize the relation between initial curvature perturbations and the PBH mass function, providing analytic recipes for embedding PBH seeds in cosmological codes. PBH models are constrained by both gravitational-wave backgrounds and high-$z$ AGN properties (Musco et al., 2020, Dayal, 2024).

7. Implications for Future Studies and Observational Campaigns

The choice of black hole seeding prescription dictates the initial black hole mass function, the redshift and host-mass dependence of occupation fractions, and the ultimate gravitational-wave and AGN event rates observable by next-generation facilities (LISA, LGWA, Lynx, JWST). Distinguishing among competing prescriptions requires:

• Joint EM+GW diagnostics: Utilizing both electromagnetic AGN census and gravitational-wave detected mergers to distinguish between seed abundance and mass scenarios (Ricarte et al., 2018, Bhowmick et al., 2021).
• Targeted environmental probes: Surveys for SMS or low-mass AGN in high-$z$ galaxies can constrain heavy seed and direct-collapse models (Agarwal et al., 2012).
• High-resolution cosmological simulations: Essential for resolving the pathway from Pop III stars to SMBHs and for reproducing the local and global scaling relations over cosmic time (Mehta et al., 20 Jan 2026, Bhowmick et al., 2024, Chen et al., 3 Sep 2025).

Systematic exploration and calibration of seeding prescriptions against increasingly stringent observational data will progressively clarify the channels dominating the early growth of SMBHs and help break degeneracies inherent in post-seeding accretion and feedback models.