Simulating AGN wind feedback with variable feedback efficiencies in idealised disc galaxies

1. Problem Statement

Active Galactic Nucleus (AGN) feedback is critical for regulating the growth of massive galaxies, yet the physical mechanisms driving this feedback remain poorly understood. Current cosmological simulations (e.g., EAGLE, IllustrisTNG, COLIBRE) typically implement AGN feedback using phenomenological subgrid models with a constant coupling efficiency ($\eta$). This efficiency represents the fraction of the AGN's bolometric luminosity converted into feedback energy (thermal or kinetic).

However, theoretical models and observations of UV line-driven winds (specifically Ultra-Fast Outflows or UFOs) suggest that the coupling efficiency is not constant but depends on the Eddington ratio ($\dot{m}$, the accretion rate normalized to the Eddington rate). Specifically, numerical models (e.g., Qwind) predict a power-law relationship where $\eta \propto \dot{m}^\alpha$. The lack of this physical dependency in current simulations may lead to inaccurate predictions regarding Black Hole (BH) growth histories, particularly at high redshifts, and the regulation of star formation.

2. Methodology

Simulation Framework

• Code: The authors used SWIFT, a hydrodynamical code utilizing Smoothed Particle Hydrodynamics (SPH) with the SPHENIX density-energy scheme.
• Subgrid Physics: The COLIBRE galaxy formation model was employed, which includes:
  • Radiative cooling/heating (HYBRID-CHIMES).
  • Star formation (Schmidt law) and stellar feedback (SN, radiation pressure, winds).
  • Chemical enrichment and dust evolution.
  • AGN Feedback: Implemented as a thermal mode (injecting energy into the surrounding gas). While the physical driver is a wind, the energy is thermalized on subgrid scales due to shocks, justifying a thermal injection approach in low-resolution cosmological contexts.

New Variable Coupling Efficiency Model

The core innovation is a new subgrid model where the coupling efficiency $\eta$ varies as a power law of the Eddington ratio:
$$\eta = \min\{ N_\eta \dot{m}^{\alpha_\eta}, 1 \}$$

• Parameters: $N_\eta$ (normalization) and $\alpha_\eta$ (slope).
• Fiducial Model: Based on Qwind predictions and observational constraints, the authors set $\alpha_\eta = 2.6$ and normalized such that $\eta = 0.1$ when $\dot{m} = 0.1$, yielding $N_\eta = 39.81$.
• Variations: The study tested variations in $N_\eta$ and $\alpha_\eta$ to explore the sensitivity of results to the specific scaling relation.

Initial Conditions (ICs)

• System: Idealized, isolated Milky Way-mass galaxies within a static Dark Matter (DM) halo.
• Components:
  • Static DM halo (Hernquist profile, $M_{200} \approx 1.37 \times 10^{12} M_\odot$).
  • Exponential stellar disc and cold gas disc.
  • Hot Circumgalactic Medium (CGM): A crucial addition, initialized in hydrostatic equilibrium within the non-spherical potential of the disc. The CGM was allowed to dynamically relax for 3 Gyr before the full simulation began to ensure equilibrium.
• Black Holes: A central SMBH was inserted at $t=0.3$ Gyr with initial masses ranging from $10^6$ to $10^9 M_\odot$.
• Resolution: Gas particle mass $m_g = 10^5 M_\odot$.

3. Key Contributions

1. Implementation of Variable Efficiency: First implementation of a power-law dependent AGN coupling efficiency ($\eta \propto \dot{m}^{2.6}$) in a hydrodynamical simulation of an idealized galaxy, motivated by UV line-driven wind theory.
2. Self-Regulation Analysis: Demonstrated how variable efficiency enhances BH self-regulation, modulating feedback strength dynamically based on accretion rates.
3. CGM Role: Highlighted the critical role of the hot CGM in sustaining long-term AGN activity and star formation by replenishing the gas reservoir, a factor often missing in idealized studies.
4. High-Redshift Implications: Provided a potential mechanism to explain the existence of overmassive BHs at high redshifts (rapid growth due to lower feedback efficiency at high $\dot{m}$) while converging to local scaling relations.

4. Key Results

A. Impact of Varying Coupling Efficiency Parameters

• Modest Galaxy Changes: For a fixed BH mass, varying the normalization ($N_\eta$) or slope ($\alpha_\eta$) of the coupling efficiency resulted in only modest variations in global galaxy properties (Star Formation Rate, stellar mass, cold gas mass).
• Self-Regulation: The system exhibits strong self-regulation. Higher coupling efficiencies lead to stronger feedback, which expels gas, lowers the accretion rate ($\dot{m}$), and consequently reduces $\eta$ (since $\eta \propto \dot{m}^{2.6}$). This feedback loop stabilizes the total AGN energy injection ($E_{AGN}$) across different parameter choices.
• BH Growth: Lower coupling efficiencies (e.g., at high $\dot{m}$) lead to faster BH growth and weaker outflows, as less energy is injected to suppress accretion.

B. Impact of Black Hole Mass ($M_{BH}$)

• Primary Driver: $M_{BH}$ is the dominant factor shaping galaxy evolution, far outweighing the effects of varying coupling efficiency parameters.
• High Mass ($M_{BH} \ge 10^8 M_\odot$):
  • Constant Model: Strong feedback rapidly expels central gas, leading to a sharp drop in accretion and early quenching of star formation.
  • Variable Model: Feedback is weaker initially (due to lower $\eta$ at lower $\dot{m}$), allowing the BH to grow faster. However, as the BH grows and $\dot{m}$ drops, $\eta$ drops further, preventing the "runaway" quenching seen in the constant model. This results in better self-regulation and sustained, albeit reduced, activity.
• Low Mass ($M_{BH} \le 10^7 M_\odot$): AGN feedback is weak; star formation continues largely unimpeded, and the galaxy remains star-forming.

C. Outflow Behavior

• Geometry: Outflows are primarily bipolar (polar direction), driven by AGN feedback, while equatorial regions are dominated by inflows (galactic fountain).
• Efficiency Comparison: The fiducial variable model produces lower mass and energy outflow rates at CGM scales (50 kpc) at early times compared to the constant model. However, at late times, the variable model sustains outflows closer to the galactic center due to less gas depletion.
• Thermal Dominance: The outflow power at 50 kpc is dominated by thermal energy (~70-80%), even though the driver is a wind, confirming that kinetic energy is thermalized on subgrid scales.

D. Role of the CGM

• Simulations without a CGM showed that AGN feedback rapidly depletes the central gas disc, leading to a cessation of star formation and BH accretion after ~1.7 Gyr.
• Simulations with a CGM showed that cooling and inflow from the halo continuously replenish the disc, sustaining both star formation and AGN activity over the full 3.3 Gyr simulation.

5. Significance and Implications

• High-Redshift SMBHs: The variable coupling efficiency model offers a plausible explanation for the "overmassive" BHs observed at high redshifts ($z > 6$) by JWST. In this model, high accretion rates ($\dot{m}$) lead to lower coupling efficiencies ($\eta$), allowing BHs to grow rapidly without being immediately quenched by strong feedback.
• Model Robustness: The study suggests that while the details of the coupling efficiency matter for BH growth histories, the global galaxy properties (stellar mass, SFR) are robustly regulated by the BH mass itself, provided the feedback mechanism allows for self-regulation.
• Future Directions: The authors note that cosmological simulations are required to test if this trend holds in a full hierarchical structure formation context, where the Eddington ratio distribution differs significantly from idealized isolated discs.

In conclusion, the paper demonstrates that implementing a physically motivated, variable AGN coupling efficiency improves the realism of BH self-regulation in simulations, leading to faster BH growth at high accretion rates while maintaining global galaxy properties similar to those produced by constant efficiency models.