Dark Sage SAM: Galaxy Evolution Model

• Dark Sage SAM is a high-resolution galaxy evolution framework that discretizes galactic discs into annuli defined by specific angular momentum.
• It computes star formation, feedback, and mass transfer locally, integrating secular, environmental, and AGN-driven processes.
• The model reproduces observed scaling relations and morphological transitions through careful calibration against mass, metallicity, and gas fraction data.

The Dark Sage Semi-Analytic Model (SAM) is a galaxy formation and evolution framework that introduces a physically motivated, multidimensional treatment of baryonic structure and dynamics, with a focus on disc angular momentum. Built as a heavily modified successor to the SAGE code, Dark Sage tracks the evolution of gas and stars within galactic discs using resolved annuli in specific angular momentum (j), allowing for local computation of key physical processes. This approach enables rigorous predictions of disc structure, the mass–angular momentum relation, and the impact of secular and environmental mechanisms in a cosmological context (Stevens et al., 2016, Stevens et al., 2023).

1. Model Architecture and Innovations

Dark Sage operates atop dark-matter halo merger trees derived from cosmological N-body simulations (e.g., Millennium, MillenniumTNG). Its distinctive feature is the discretization of both gaseous and stellar discs into typically 30 concentric annuli, each defined by a fixed interval in j rather than radius. Within each annulus, mass, energy, metals, angular momentum, stellar age composition, and kinematics evolve under local physics. The model tracks:

• The interstellar medium (ISM), parsed into annuli by j
• The stellar disc, resolved both radially (j) and into multiple stellar-age bins per annulus (in recent versions)
• Spheroid components (instability-driven bulge, merger-driven bulge, intrahalo stars, intergalactic stars)
• A multi-reservoir circumgalactic medium (CGM), with "hot," "fountain" (reheated but bound), and "outflowing" (unbound) components
• Supermassive black holes (SMBHs) and their respective accreted mass channels

Each galaxy evolves through sequential sub-steps involving mass exchange, angular-momentum transport, chemical enrichment, and kinematic heating, with all computations performed at the annulus or annulus-age-bin level (Stevens et al., 2016, Stevens et al., 2023).

A summary of the core annulus machinery is provided below.

Quantity	Description	Annulus Representation
$j$	Specific angular momentum	30 logarithmic bins
Gas/Stellar Mass	Baryons per bin	Resolved radially ($j$)
Metallicity, Age, $\sigma$	Tracked per annulus (and age bin for stars)	30xN_age grid (new versions)

2. Angular Momentum Discretization and Disc Structure

Specific angular momentum in Dark Sage is defined as $j = J / M$; for circular orbits in axisymmetric potentials, $j(r) = r \, v_{\rm circ}(r)$. The disc is divided logarithmically in $j$:

$j_i = j_1 \, f_{\rm step}^{i-1}, \quad i=1...30,$

with $j_1$ and the step factor $f_{\rm step}$ (typically 1.4) chosen to encompass the physical $j$ range. Each annulus thus covers $[j_{i-1}, j_i]$. Evolution of mass between annuli strictly conserves total $J$. The corresponding physical radius is determined dynamically at each timestep by solving

$j = r \, v_{\rm circ}(r) = \sqrt{G M(<r) r},$

with $M(<r)$ including both dark matter and baryons.

Cooling, star formation, feedback, and external processes (e.g., ram-pressure stripping, minor mergers) are all computed per annulus. The ISM is further subdivided into molecular, atomic, and ionized phases using local metallicity and pressure-dependent prescriptions (Stevens et al., 2016, Stevens et al., 2023, Stevens et al., 2018).

3. In-Situ Physical Processes: Star Formation, Feedback, and Instabilities

Star Formation and Molecular Partitioning

In each gas annulus, star formation follows an H$_2$-regulated law:

$$\Sigma_{\rm SFR}(r) = \epsilon_{\rm SF} \, \Sigma_{\rm H_2}(r),$$

where $\epsilon_{\rm SF}$ is a calibrated efficiency parameter. The ratio of H$_2$ to H I is a function of either local pressure (Blitz–Rosolowsky law) or metallicity and surface density (Krumholz–McKee–Tumlinson/Blitz–Rosolowsky, selectable as model options).

Feedback

Supernova and stellar feedback are computed per annulus, with a mass-loading factor that typically scales inversely with local gas surface density. In recent versions, Dark Sage employs an energy-conserving, time-dependent formulation: energy from stellar feedback heats (and can eventually eject) cold gas from annuli, with the division between bound "fountain" gas and unbound outflows set by energy conservation arguments.

AGN feedback operates in radio and quasar modes. Radio mode prevents cooling by heating the CGM in proportion to $M_{\rm BH}$ and $M_{\rm hot}$; quasar mode, triggered during rapid SMBH growth in mergers or instabilities, ejects ISM gas on an annulus-by-annulus basis (Stevens et al., 2016, Stevens et al., 2023).

Disc Instabilities and Mass Transfer

Stability is computed using local Toomre Q parameters:

• For stars: $$Q_*(r) = \kappa(r)\sigma_*(r) / (3.36\,G\,\Sigma_*(r))$$
• For gas: $$Q_{\rm gas}(r) = \kappa(r) c_s / (\pi\,G\,\Sigma_{\rm gas}(r))$$

The two-component total $Q$ adopts the Romeo-Wiegert (2011) criterion. Annuli with $Q<1$ are unstable: a fraction $f_{\rm move}$ of mass is redistributed radially to stabilize the disc, with low-j mass potentially feeding the bulge or central SMBH.

Disc and stellar-velocity dispersions, as well as the effects of kinematic heating and radial migration, are included in the latest implementations via stellar age bins and explicit $\sigma$ evolution (Stevens et al., 2023).

4. Integrated Galaxy Scaling Relations and Morphological Evolution

Dark Sage reproduces the integrated mass–specific angular momentum ($M_*$–$j_*$) relation for discs. The model yields $$j_{*,{\rm disc}}\propto m_{*,{\rm disc}}^{0.6 - 0.7}$$ at $z=0$, in close agreement with observed galaxy samples (Stevens et al., 2016). The inclusion of disc instabilities is critical to match the observed normalization and scatter: disabling Q instabilities reduces $j_*$ by $\sim$0.4 dex at high mass and increases scatter.

With annular resolution, Dark Sage predicts the morphological sequence at fixed stellar mass as an emergent property: disc-dominated galaxies have the highest stellar-disc $j_*$, but including gas, bulge-dominated systems (with extended high-j cold gas) can exhibit the highest total disc $j$ (Porras-Valverde et al., 2019). The model robustly demonstrates that at high masses halo mass, not halo spin, primarily sorts galaxies into bulge- or disc-dominated types, with disk-dominated galaxies residing in halos $\sim$1/10th the mass of their bulge-dominated counterparts at fixed $M_*$ (Porras-Valverde et al., 2019).

The model naturally produces realistic surface-density profiles: exponential in the outer disc, with central cusps ascribed to pseudobulge formation driven by instabilities. Assigning mass within 0.2 $r_d$ to pseudobulges improves agreement with observed bulge/disk mass functions (Stevens et al., 2016).

5. Environmental and Secular Regulation: Gas Content and Quenching

Dark Sage directly models environmental effects such as ram-pressure stripping of cold gas and strangulation (hot gas stripping) using local j-based criteria:

• Cold-gas stripping occurs annulus-by-annulus if

$$\rho_{\rm hot,cen}(R_{\rm sat}) v_{\rm sat}^2 \geq 2\pi G \Sigma_{\rm gas}(r)[\Sigma_{\rm gas}(r) + \Sigma_*(r)]$$

• Satellite galaxies cease to accrete new hot gas upon infall.
• Strangulation is implemented proportional to subhalo mass loss, optionally with ram-pressure-based stripping radius (Stevens et al., 2017).

The model successfully reproduces the relative impact of these processes on H I fractions and quenching for central and satellite galaxies as a function of halo mass and environment, with most environmental suppression of cold gas attributable to ram-pressure stripping (Stevens et al., 2017). However, satellites in Dark Sage remain too gas-poor compared to observations, a limitation ascribed to the complete suppression of hot-gas accretion and overly efficient central cold-gas stripping due to simplified inner disc models.

The model's dimensional approach enables detailed exploration of the relation between global disc stability ($q$ parameter) and atomic gas fraction $f_{\rm atm}$, closely following analytic predictions for exponential discs and recovering observed trends and their scatter (Stevens et al., 2018).

6. AGN Feedback and Stellar Mass–Halo Mass Scatter

Comparative studies demonstrate that standard semi-analytic AGN feedback in Dark Sage, which acts as smooth, cumulative heating of spherically-averaged gas reservoirs, leads to higher scatter ($\sigma\sim0.5-0.6$ dex near $M_{\rm vir}\sim10^{12}\,M_\odot$) in the stellar mass–halo mass relation (SMHMR) than observed or predicted by hydrodynamic simulations (TNG, $\sim$0.15 dex) (Porras-Valverde et al., 2023). Introducing sharp cooling shutdown models (e.g., turning off cooling above $m_{\rm BH}=10^8\,M_\odot$) can bring the low-mass end scatter into agreement with TNG/observations, but does not eliminate excess scatter for the most massive halos, suggesting fundamental differences in quenching implementation (Porras-Valverde et al., 2023).

Proposed improvements include threshold-driven kinetic-mode feedback, explicit multiphase outflow modeling, and more direct coupling of AGN feedback energy to ISM removal.

7. Parameter Calibration and Predictive Power

Dark Sage has evolved to employ a minimal set of free parameters (three in the newest release: $f_{\rm move}^{\rm gas}$ for instability-driven migration, the SN feedback energy scale $\mathcal{S}_{\rm SN}$, and AGN radiative efficiency $\epsilon_{\rm AGN}$), calibrated via particle-swarm optimization against the $z=0$ stellar mass function, H I mass function, and cosmic SFRD (Stevens et al., 2023). This parsimonious approach, combined with energy-conserving feedback and explicit disc structure, positions Dark Sage as a model capable of matching hydrodynamic codes in predictive detail for galaxy structure, metallicity gradients, Tully–Fisher zero points, resolved star formation histories, and more, while retaining semi-analytical flexibility (Stevens et al., 2023).

A comprehensive assessment of the model's calibration and constraints is summarized:

Calibration Target	Agreement Level (dex)	Notes
Stellar Mass Function	$\lesssim$0.03	By construction (PSO-fit)
H I Mass Function	$\lesssim$0.1	Sat/Cen offset excess
Baryonic Tully–Fisher	$\lesssim$0.03	SPARC data
Metallicity Gradients	Compatible	MaNGA/CALIFA/SAMI
SMHMR Scatter	Excess at high mass	∼0.5–0.6 dex vs. 0.15 dex

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Dark Sage is a highly resolved, physically grounded semi-analytic model that provides a unique framework for dissecting the interplay between angular momentum, structure, and evolution in galaxies over cosmic time. Its multidimensional annulus-based architecture enables both global scaling relation predictions and nuanced assessments of secular, feedback, and environmental processes, making it a key tool for the theoretical interpretation of current and upcoming large-scale galaxy surveys (Stevens et al., 2016, Stevens et al., 2023, Porras-Valverde et al., 2019, Stevens et al., 2018, Stevens et al., 2017, Porras-Valverde et al., 2023).