COLIBRE Galaxy Formation Model

• The COLIBRE Galaxy Formation Model is a comprehensive simulation framework that models galaxy evolution over a wide range of mass scales and redshifts with detailed multiphase ISM and dust physics.
• It integrates advanced non-equilibrium chemistry, stochastic star formation, and calibrated feedback from stars, supernovae, and AGN to match observed galaxy properties.
• The model employs high-resolution techniques and self-consistent calibration to achieve excellent numerical convergence and robust predictions for key low-redshift observables.

The COLIBRE Galaxy Formation Model is a state-of-the-art, physically motivated, and extensively calibrated suite of cosmological hydrodynamical simulations designed to model galaxy formation and evolution across a broad range of mass scales and redshifts. It explicitly resolves the multiphase interstellar medium (ISM), captures non-equilibrium chemistry, incorporates detailed dust evolution, and provides self-consistent stellar and AGN feedback recipes, representing a significant advancement over previous models. The model is calibrated to reproduce observed galaxy stellar mass functions, size-mass relations, and black hole mass scaling at low redshift, and achieves very good numerical convergence and excellent agreement with diverse observational data (Schaye et al., 28 Aug 2025).

1. Multiphase ISM, Non-Equilibrium Chemistry, and Radiative Cooling

COLIBRE directly models the cold (down to $\sim$10 K), warm ($\sim$10$^4$ K), and hot ISM phases without imposing an artificial pressure floor, unlike many older large-volume simulations. Hydrogen and helium abundances are evolved in non-equilibrium, crucial for accurately determining the free electron density that controls metal-line cooling rates. Metal cooling is treated using equilibrium tables, but with a correction factor based on the non-equilibrium electron density:

$$\Lambda_{\rm metal} = \Lambda_{\rm metal}(\rho, T, Z) \times \left(\frac{n_e^{\rm NE}}{n_e^{\rm eq}}\right)$$

Cooling rates account for both a redshift-dependent metagalactic UV background and a local ISRF, as well as self-shielding by gas and dust (typically estimated locally using the Jeans length) (Schaye et al., 28 Aug 2025).

The chemical network, based on "hybrid-chimes," tracks approximately 157 ions and molecules, ensures non-equilibrium treatment for H and He, and includes molecule formation on grain surfaces (e.g., H$_2$, CO). Dust is critical for accurately modeling the ISM and molecular cloud physics, modulating cooling, shielding, and chemical pathways.

2. Dust Evolution and Chemical Coupling

Dust grain evolution is explicitly followed, with three species (carbonaceous/graphite and two silicates—olivine forms forsterite, fayalite) and two size bins (small, $r_{\rm s} \approx 0.01\ \mu$m; large, $r_{\rm l} \approx 0.1\ \mu$m). Formation occurs via gas-phase accretion and grain coagulation, enhanced by a subgrid clumping factor for unresolved molecular clouds:

$$\tau_{\rm acc} \propto \left( \frac{r_{\rm g}}{0.1\,\mu{\rm m}} \right) \left( \frac{n_{\rm H}'}{10\,{\rm cm}^{-3}} \right)^{-1} \left( \frac{T}{10\,{\rm K}} \right)^{-1/2}$$

Destruction processes include sputtering in hot gas and astration (removal by star formation). The dust model directly couples to the chemical network and cooling, controlling formation rates of molecules like H$_2$ and CO (Schaye et al., 28 Aug 2025).

3. Star Formation, Early Feedback, and Turbulent Diffusion

Star formation is stochastic; gas is eligible if locally gravitationally unstable as determined by a dimensionless stability parameter:

$$\alpha \equiv \frac{ \sigma_{\rm th}^2 + \sigma_{\rm turb}^2 } { G\, \langle N_{\rm ngb}\rangle^{2/3}\,m_{\rm g}^{2/3}\,\rho_{\rm g}^{1/3} } < \alpha_{\rm crit}$$

with $\alpha_{\rm crit}=1$. SFR density follows a Schmidt law:

$$\dot{\rho}_\star = \varepsilon\, \frac{\rho_{\rm g}}{ t_{\rm ff} } = \varepsilon \left( \frac{32 G}{3\pi} \right)^{1/2} \rho_{\rm g}^{3/2}$$

where $\varepsilon\sim0.01$ (per free-fall time). At higher resolution, star formation naturally proceeds in denser and colder gas (Schaye et al., 28 Aug 2025).

Early feedback from massive stars includes mechanical winds, radiative pressure, and ionization producing H II regions. Local attenuation by dust/gas is modeled, and gas in H II regions is kept at %%%%13$\alpha_{\rm crit}=1$14%%%% K and excluded from star formation, dispersing natal clouds prior to supernova onset.

Turbulent mixing is implemented as explicit diffusion between SPH particles:

$\frac{dX}{dt} = \nabla\cdot(D\,\nabla X)$

where $D$ is proportional to the local velocity shear. This is essential to distribute metals and dust, mitigating inhomogeneities arising from discrete enrichment events.

4. Stellar Mass Loss, Supernova Feedback, and Outflows

Stellar mass loss and metal enrichment use updated yields for AGB stars, core-collapse SN, Type Ia SN, and r-process sources. Feedback energy from CCSNe is parameterized:

$$\Delta E_{\rm CCSN} = 10^{51}\,{\rm erg} \; f_E \, m_* \int_{m_{\rm d}(t+\Delta t)}^{m_{\rm d}(t)} \Phi(m)\;dm$$

with $f_E$ possibly dependent on local birth pressure $P_{\rm birth}$. SN energy injects both stochastic thermal (target $\Delta T_{\rm SN}$, density-dependent) and kinetic (10% of energy, driving turbulence, $\sim$50 km/s) channels. Energy accumulates until adequate for the target temperature jump, reducing over-cooling in dense gas (Schaye et al., 28 Aug 2025).

A distinctive feature is the parameterized variation of SN feedback strength with ambient conditions:

$$f_E(P_{\rm birth}) = f_{E,\mathrm{min}} + \frac{ f_{E,\mathrm{max}} - f_{E,\mathrm{min}} } {1+\exp\left[ -\frac{ \log_{10}P_{\rm birth} - \log_{10}P_{E,\mathrm{pivot}} }{ \sigma_P } \right] }$$

5. BH Growth, AGN Spin-Dependent Feedback, and Subgrid Calibration

Supermassive black holes are seeded in halos above a defined mass threshold. Accretion follows a modified Bondi-Hoyle-Lyttleton rate:

$$\dot{m}_{\rm BHL} = 4\pi G^2 \frac{ \rho_{\rm g} }{ (c_s^2 + v^2)^{3/2} } m_{\rm BH}^2$$

with modifications for turbulence ($f_{\rm turb}$) and vorticity ($f_{\rm ang}$). The Eddington limit is not strictly enforced (up to 100x the classic Eddington rate is allowed).

Fiducial AGN feedback is purely thermal: energy accumulates and is injected stochastically once enough is stored to heat nearby gas by $\Delta T_{\rm AGN}$, with sampling adjusted for BH mass.

A subset of simulations uses a hybrid jet/thermal AGN model with spin-dependent jet formation (Blandford-Znajek-like mechanism). BH spin evolves with gas accretion, mergers, and torques. Depending on the accretion-disc state (thin, slim, thick), the feedback splits into bipolar kinetic jets ($v_{\rm jet} \propto m_{\rm BH}^{1/2}$) and thermal energy:

$$\dot{m}_{\rm BH} = (1-\epsilon_r-\epsilon_{\rm jet}-\epsilon_{\rm wind})\epsilon_{\rm accr}\dot{m}_{\rm accr}$$

This adds a physically motivated channel for regulating massive galaxy and BH growth (Schaye et al., 28 Aug 2025).

Subgrid feedback parameters are calibrated via machine-learning emulators trained on moderate-volume runs ($\sim$50 cMpc, $m_7$), targeting agreement with $z\sim0$ scaling relations (stellar mass function, galaxy size-mass, BH–stellar mass relation).

6. Simulation Suite, Convergence, and Observational Comparisons

The COLIBRE suite spans three resolutions: particle masses of $\sim10^5$, $10^6$, $10^7\,M_\odot$, in volumes of up to 50, 200, and 400 cMpc. The largest runs contain 136 billion ($5\times3008^3$) particles. Dark matter is supersampled by a factor of 4, ensuring nearly equal baryonic and dark matter particle masses, suppressing spurious energy transfer.

The subgrid model is calibrated to reproduce observed low-redshift galaxy properties:

• Stellar mass function
• Galaxy size-mass relation
• Black hole mass in massive galaxies

A broad array of other observables are matched, including quenched fractions, specific SFRs, gas fractions of H I and H$_2$, dust masses, metallicities, and circumgalactic X-ray luminosity. Numerical convergence is very good over the observed range. Agreement with data is described as "excellent" for these diagnostics (Schaye et al., 28 Aug 2025).

7. Limitations and Prospects

While COLIBRE represents significant progress, limitations are acknowledged:

• Inability to resolve internal molecular cloud structure directly; subgrid clumping factors are required for dust and molecule growth rates.
• AGN feedback efficiency remains sensitive to numerical resolution because of injection in unresolved inner regions.
• Physical parameters (feedback strength, energy coupling efficiency, etc.) must be empirically calibrated, limiting strict ab initio predictive power.
• Physics such as magnetic fields, fully-coupled radiation transport, and cosmic ray pressure are omitted.

A plausible implication is that further increases in resolution and expansion of physical ingredients (e.g., full radiation hydrodynamics) may further improve model fidelity and predictive power.

Concluding Perspective

The COLIBRE galaxy formation model synthesizes advanced physical prescriptions for an explicitly multiphase ISM, live dust chemistry, and sophisticated feedback with a rigorous calibration methodology. Its ability to capture chemical, thermal, and dynamical evolutionary pathways—validated against a wide span of observational data—marks it as a reference hydrodynamical model for galaxy formation applications at low redshift and sets a robust platform for further testing and development in cosmological simulations (Schaye et al., 28 Aug 2025).