Semi-Analytic Seed Model Framework

• Semi-analytic seed models are computational frameworks that use merger trees and seeded initial conditions to simulate the hierarchical assembly of galaxies, black holes, and planets.
• They apply analytic prescriptions for processes like star formation, feedback, and accretion, enabling efficient exploration of diverse physical scenarios.
• Variations in merger tree construction and seed selection critically impact evolutionary outcomes, observable scaling relations, and feedback regimes.

A semi-analytic seed model is a computational framework that couples analytic or parametrized prescriptions for physical processes to the hierarchical growth of structure traced by merger trees or similar constructs. In galaxy formation and related domains, these models are designed to capture the assembly and evolution of galaxies, stars, planets, and black holes by “seeding” objects with initial conditions and following their subsequent growth via accretion, mergers, and feedback. The semi-analytic approach enables efficient exploration of parameter space and physical scenarios that would be prohibitive for full numerical simulations, but introduces sensitivity to seed choices and input algorithms.

1. Role and Construction of Merger Trees

Merger trees form the backbone of semi-analytic models (SAMs) for galaxy and black hole seeding. They encapsulate the hierarchical growth of dark matter haloes through cosmic time and provide the framework for baryonic evolution by cataloguing when and how haloes merge, accrete, or lose mass (Lee et al., 2014). The construction of merger trees is non-trivial and depends on algorithmic choices:

• Particle ID matching algorithms (MergerTree, VELOCIraptor, ySAMtm) track halo membership across simulation snapshots to link progenitors and descendants.
• Trajectory-based and catalogue-modifying algorithms (Consistent Trees, HBT) can patch missing links and improve subhalo continuity.

The differences between these algorithms propagate into variations in tree topology, e.g., the number of identified subhaloes, the length and continuity of branches, and the ability to capture the histories of satellite vs. central galaxies.

2. Seed Insertion and Initial Conditions

In semi-analytic seed models, the initial population of galaxies, black holes, or planets is defined by explicit “seeding” criteria:

• Galaxies are seeded in newly collapsed haloes above a minimum mass. Physical triggers can include the ability to cool gas (e.g., via H$_2$ or atomic cooling), abundance of gas mass, or other environmental criteria (Hegde et al., 2023).
• Black holes are seeded at galaxy formation, often with a fixed or probabilistic seed mass. For example, in ν²GC models, a black hole is seeded with $10^3\,M_\odot$ or a random value in $10^{3-5}\,M_\odot$ when the host halo forms a galaxy (Shirakata et al., 2016).
• Planetary systems are seeded by initializing a set of planetary embryos (Earth–Super-Earth mass) with specified masses, semi-major axes, and small eccentricities following disk migration or gas-disk dispersal (Kimura et al., 26 May 2025).

Key model sensitivity arises because the seed mass and physical environment provide the initial conditions for subsequent growth and assembly, especially at low-mass/high-redshift regimes where feedback and accretion efficacy are weak.

3. Physical Prescriptions and Model Calibration

To propagate the seeded populations forward, semi-analytic models use analytic or empirical formulas for key physical processes:

• Star formation: $$\dot{m}_* = \alpha \frac{m_{\rm cold}}{t_{\rm dyn,gal}}$$, where $\alpha$ is the star formation efficiency. This rate can be tuned to adjust for different merger tree growth rates (Lee et al., 2014).
• Supernova feedback: $$\dot{m}_{\rm rh} = \epsilon^{\rm SN}_0 \left(\frac{150\,{\rm km\,s}^{-1}}{V_{\rm disc}}\right)^{\alpha_{\rm rh}} \dot{m}_*$$; efficiency parameters are calibrated to suppress star formation where needed (Lee et al., 2014).
• AGN feedback: $$\dot{m}_{\rm BH,R} = \kappa_{\rm AGN} \left(\frac{m_{\rm BH}}{10^8\,M_\odot}\right) \left(\frac{f_{\rm hot}}{0.1}\right) \left(\frac{V_{\rm vir}}{150\,{\rm km/s}}\right)^3$$, with $\kappa_{\rm AGN}$ reflecting the strength of feedback necessary for massive haloes (Lee et al., 2014).
• Black hole growth: $dM_{\rm BH}/dt = (1-\eta)\,\dot{M}_{\rm acc}$, with observed $M_{\rm BH}$–$M_{\rm bulge}$ scaling relations serving as constraints (Shirakata et al., 2016).
• Planetary orbital evolution: Secular perturbation theory governs eccentricity evolution: $e_i(t) = \sqrt{h_i(t)^2 + k_i(t)^2}$, with collision and scattering events determined by analytic estimates of crossing timescales and probability (Kimura et al., 26 May 2025).

Models can be independently calibrated for each tree or seed prescription to recover observed global properties (e.g., mass functions, scaling relations). This reduces discrepancies in end-state observables but amplifies diversity in evolutionary tracks and feedback regimes.

4. Impact of Model Choices on Evolution and Observables

The construction of merger trees and choice of seed prescription directly impacts galaxy and black hole assembly:

• Differences between tree algorithms yield variations in satellite abundance, merger rates, and star formation histories. For example, HBT trees yield more satellites and earlier star formation compared to JMerge (Lee et al., 2014).
• Seed mass selection for black holes affects the $M_{\rm BH}$–$M_{\rm bulge}$ relation; high constant seed masses ($\sim10^5\,M_\odot$) result in over-massive SMBHs for dwarfs, while low or randomized seeds yield relations consistent with empirical data (Shirakata et al., 2016).
• Feedback parameter adjustment produces divergent evolutionary histories; tuning for agreement at $z=0$ leads to significant variations in feedback-dominated epochs, AGN/supernova suppression regimes, and the fraction of stars formed in situ versus accreted.
• Planetary system evolution depends strongly on orbital distance and stellar mass; accurate representation of both collision and scattering outcomes across compact systems requires careful modification of separation metrics and collision probability formulas (Kimura et al., 26 May 2025).

These sensitivities complicate the interpretation of the assembly history and necessitate robust uncertainty quantification.

5. Applications and Utility of Semi-Analytic Seed Models

Semi-analytic seed models enable a broad range of applications across astrophysics:

• Galaxy formation: Analysis of the impact of tree-building algorithms on galaxy assembly and feedback; efficient parameter exploration for cosmic star formation rates and mass functions (Lee et al., 2014).
• Black hole growth: Evaluation of seeding scenarios and their observational consequences, e.g., tracing SMBH–bulge scaling relations, and implications for AGN populations over cosmic time (Shirakata et al., 2016, Ricarte et al., 2017).
• Early star formation: Modeling the threshold conditions for Population III star formation in minihalos and subsequent feedback, with predictions for global 21-cm signals and observable high-redshift SNe (Hegde et al., 2023).
• Planetary system statistics: Fast synthesis of terrestrial/super-Earth populations, enabling statistical comparison with exoplanet surveys and integration with disk evolution, migration, and atmospheric physics modules (Kimura et al., 26 May 2025).
• Gravitational wave backgrounds: Linking seed black hole populations and binary evolution in galaxy formation models to the multi-band stochastic gravitational wave background detectable by SKA, LISA, and ET (Li et al., 2023).

The computational efficiencies, combined with physical transparency of analytic prescriptions, render the semi-analytic approach indispensable for large-scale population synthesis and parameter constraint studies.

6. Limitations, Controversies, and Uncertainty Budget

The semi-analytic seed model, while powerful, is subject to several limitations:

• Uncertainties in physical prescriptions: Accretion and feedback rates depend on empirically motivated parameters that may not capture complex physics, especially at low-mass/high-redshift domains.
• Algorithmic arbitrariness: Differences between merger tree builders and seed placement criteria can lead to divergent evolutionary scenarios that are often indistinguishable within observational error budgets (Lee et al., 2014).
• Degeneracies: Parameter adjustments in feedback, star formation efficiencies, and seed masses can trade off in matching a given observable, but leave the history unconstrained and affect predictions for e.g., SN rates, AGN duty cycles, and GW backgrounds.
• Built-in simplifications: Treatments of binary black hole mergers, delay times, and planetary collision outcomes are idealized relative to simulation-based approaches, potentially missing rare but consequential events (Kimura et al., 26 May 2025, Ricarte et al., 2017).

Given these factors, the interpretation of semi-analytic model outputs requires careful consideration of uncertainty sources, comparative studies with alternate algorithms, and calibration against a suite of observables.

7. Summary and Future Directions

The semi-analytic seed model occupies a critical role in astrophysical modeling by enabling the connection between initial conditions, hierarchical structure formation, and the assembly of galaxies, black holes, stars, and planets. The sensitivity of evolutionary histories to tree algorithms, seed prescriptions, and parameter choices highlights the need for rigorous uncertainty quantification and multi-observable calibration. Advances in model transparency, integration with hydrodynamic simulations, and joint constraints from the electromagnetic and gravitational wave domains will further refine the predictive power of semi-analytic approaches.

Key equations fundamental to these models include:

$$\dot{m}_* = \alpha \frac{m_{\rm cold}}{t_{\rm dyn, gal}}$$

$$\dot{m}_{\rm rh} = \epsilon^{\rm SN}_0 \left(\frac{150\,{\rm km\,s}^{-1}}{V_{\rm disc}}\right)^{\alpha_{\rm rh}} \dot{m}_*$$

$$\dot{m}_{\rm BH,R} = \kappa_{\rm AGN} \left(\frac{m_{\rm BH}}{10^{8}\,M_\odot}\right) \left(\frac{f_{\rm hot}}{0.1}\right) \left(\frac{V_{\rm vir}}{150\,{\rm km/s}}\right)^3$$

$\log(M_{\rm BH}) = a + b \log(M_{\rm bulge})$

$e_i(t) = \sqrt{h_i(t)^2 + k_i(t)^2}$

The continuous development and multi-disciplinary integration of semi-analytic seed models will remain vital to advancing theoretical predictions and interpretation in astrophysics.