Mistral AGN Feedback Model

• The paper introduces a physically motivated AGN feedback model that uses kinetic momentum deposition to simulate fast, bipolar outflows.
• It implements two variants—continuous and stochastic—to replicate observed BAL quasar wind properties and improve simulation realism.
• Numerical results show that MistralS effectively regulates supermassive black hole growth, suppressing star formation and matching observed galaxy scaling relations.

The Mistral AGN feedback model is a physically motivated, numerically robust subgrid prescription for radiatively efficient active galactic nucleus (AGN) winds, developed for galaxy formation simulations using the Arepo moving-mesh code. Mistral is inspired by observations of broad absorption line (BAL) quasar outflows with velocities $v \sim 10^3$–$10^4\,\mathrm{km\,s^{-1}}$, and aims to provide a more realistic kinetic-momentum driven coupling of AGN feedback to the surrounding interstellar and circumgalactic media (ISM, CGM) relative to standard thermal feedback models. Two complementary variants—Mistral-continuous (MistralC) and Mistral-stochastic (MistralS)—are implemented and tested across idealized and cosmological settings, demonstrating significant improvements in reproducing key observed properties of galaxy and supermassive black hole (SMBH) populations (Farcy et al., 10 Apr 2025).

1. Physical Motivation and Model Variants

Observations of bright, radiatively efficient AGN, notably BAL quasars exhibiting outflows at $v \sim 10^3$–$10^4\,\mathrm{km\,s^{-1}}$ (e.g., Tombesi et al. 2011), strongly indicate a feedback mechanism dominated by kinetic momentum deposition rather than purely thermal energy injection. The Mistral model operationalizes this by coupling AGN wind momentum to gas near the SMBH through two distinct schemes:

• Mistral-continuous (MistralC): Delivers continuous, radially directed momentum to all gas cells within the SMBH's kernel volume, with directionality weighted to favor a bipolar structure aligned with local angular momentum. This mimics a persistent, gentle wind that pressurizes both ISM and CGM.
• Mistral-stochastic (MistralS): Deposits momentum stochastically by imparting fixed-velocity ($v_w = 10^4\,\mathrm{km\,s^{-1}}$) kicks to probabilistically selected gas cells within the kernel. The kicks are aligned parallel/anti-parallel to the kernel’s angular momentum, producing bursty, collimated, bipolar outflows akin to those observed in BAL quasars.

Figure 1 in the referenced work compares these approaches to both the IllustrisTNG Isotropic Thermal and Random Wind (kinetic) AGN feedback prescriptions. MistralS, in particular, is designed to match the morphology and energetics of AGN-driven winds inferred from observations (Farcy et al., 10 Apr 2025).

2. Mathematical Formulation and Key Equations

Let $\dot M_{\rm BH,inf}$ be the Bondi-limited inflow rate onto the SMBH, capped at $\dot M_{\mathrm{Edd}}$. The actual accretion accounting for wind mass loss is

$$\dot M_{\rm BH} = \frac{\dot M_{\rm BH,inf}}{1 + \psi}$$

where the mass-loading parameter $\psi$ quantifies the ejected wind mass per unit of SMBH accretion: $$\psi \equiv \frac{\dot M_{\rm BH,wind}}{\dot M_{\rm BH}} = \frac{2\varepsilon_w c^2}{v_w^2}$$ with $\varepsilon_w$ the wind coupling efficiency and $v_w$ the wind velocity.

The resulting wind properties are: $$\dot M_{\rm BH,wind} = \frac{\psi}{1+\psi} \dot M_{\rm BH,inf}$$

$$\dot E_{\rm BH} = \varepsilon_w \left(\frac{1}{1+\psi}\right) \dot M_{\rm BH,inf} c^2$$

$$\dot p_{\rm BH} = \frac{\psi}{1+\psi}\dot M_{\rm BH,inf} v_w$$

The wind momentum flux can alternatively be written as

$$\dot p_{\rm wind} = \eta_p \left(\frac{L_{\rm AGN}}{c}\right) = \eta_p \left(\frac{\varepsilon_r \dot M_{\rm BH} c^2}{c}\right)$$

where $\varepsilon_r$ is the radiative efficiency ($0.1$–$0.2$), and $\eta_p$ is the dimensionless momentum-loading factor. For fiducial values $\varepsilon_w=10^{-3}$ and $v_w=10^4\,\mathrm{km\,s^{-1}}$, $\eta_p \approx 0.3$ (Farcy et al., 10 Apr 2025).

3. Numerical Implementation in Arepo

Mistral is implemented by treating SMBHs as sink particles with a smoothing volume comprising $n_{\rm BH,ngb}$ gas cells (e.g., 512 for TNG100 zooms). Gas properties such as density, sound speed, and total angular momentum are kernel-averaged. At each timestep $\Delta t$:

• The Bondi inflow rate $\dot M_{\rm BH,inf}$ is computed.
• The wind and accreted mass are subtracted from neighboring gas cells, weighted by kernel values.
• Feedback energy and wind mass for the timestep are determined:

  $$\Delta E_{\rm BH} = \varepsilon_w \dot M_{\rm BH} c^2 \Delta t$$

  $$\Delta M_{\rm wind} = \frac{\psi}{1 + \psi} \dot M_{\rm BH,inf} \Delta t$$

For MistralC, all kernel cells receive simultaneous momentum kicks to conserve $\Delta E_{\rm BH}$, distributed radially and weighted to favor a bipolar configuration. For MistralS, a "mass bucket" $M_{\rm bucket}$ is tracked; each kernel cell is considered in turn, and—based on a probabilistic criterion—selected cells are ejected at fixed $v_w$ along $\pm j_{\rm tot}$, reducing $M_{\rm bucket}$ accordingly. This method minimizes excessive dilution of feedback energy, producing bursty, high-velocity outflows (Farcy et al., 10 Apr 2025).

4. Parameter Choices and Calibration

Key parameters are set and calibrated as follows:

• Wind velocity $v_w = 10^4\,\mathrm{km\,s^{-1}}$, reflecting observed BAL/quasar outflows (Tombesi et al. 2011; Matzeu et al. 2023).
• Coupling efficiency $\varepsilon_w$ is tuned through calibration; $\varepsilon_w = 10^{-3}$ yields realistic stellar and SMBH masses at $z = 0$ and $z = 2$ for test halos, with lower values causing over-quenching, and higher values impeding the growth of SMBHs and galaxies.
• Radiative efficiency $\varepsilon_r$ is set to $0.1$ for the idealized setups and $0.2$ in cosmological zoom runs, consistent with TNG100 conventions (Farcy et al., 10 Apr 2025).

These settings enable MistralS to match observed scaling relations and gas fractions in a range of halos without the need for a BH-mass–dependent feedback switch.

5. Simulation Setups

Mistral is validated in two classes of simulations:

• Idealized Milky Way–mass disk: An m12 disk galaxy is placed in a $1.5 \times 10^{12} \,M_\odot$ NFW halo, with $M_{\rm DM} = 5 \times 10^5\,M_\odot$, $m_{\rm baryon} = 8 \times 10^4\,M_\odot$, and softening $\sim 200\,\mathrm{pc}$. Star formation and BH accretion are triggered after $600\,\mathrm{Myr}$, with total evolution over $1.2\,\mathrm{Gyr}$.
• Cosmological zoom-in simulations: 15 halos with $M_{\rm halo}(z=2) = 10^{12}$–$3 \times 10^{13}\,M_\odot$ from the TNG100 volume, run at TNG100-equivalent resolution ($M_{\rm DM} = 7.5 \times 10^6\,M_\odot$, $m_{\rm baryon} = 1.6 \times 10^6\,M_\odot$, $\epsilon_{\rm DM} = 740\,\mathrm{pc}$ comoving). Evolution proceeds from $z_{\rm init} \approx 20$ to $z=2$ (three halos to $z=0$).

6. Results and Impact on Galaxy Evolution

MistralS demonstrates distinct advantages in regulating galaxy and SMBH co-evolution:

• Idealized galaxy: MistralC drives short-lived, cold, dense bipolar fountains that rapidly recycle to the disk, thus increasing both star formation rate (SFR) and BH accretion. By contrast, MistralS produces hot, low-density bipolar outflows exceeding $50\,\mathrm{kpc}$ at $v \gtrsim 1000\,\mathrm{km\,s^{-1}}$, suppressing SFR by $\sim 50\%$ and BH growth by a factor $>2$. The IllustrisTNG quasar mode fails to launch correspondingly fast winds at this scale, while its kinetic mode (Random Wind) requires operating in the low–Eddington regime (Farcy et al., 10 Apr 2025).
• Cosmological zooms ($z=2$): MistralS quenches $\sim 60\%$ of massive galaxies (with sSFR $<10^{-10}\,\mathrm{yr}^{-1}$), accurately reproducing the empirical stellar-to-halo mass (SMHM) relation and matching observed scatter in the star-forming main sequence. MistralS aligns the BH–stellar mass relation with local and high-$z$ constraints, and effectively suppresses BH accretion rates in more massive halos ($M_{\rm halo} > 10^{12.5}\,M_\odot$). Cold gas fractions within $R_{\rm vir}/10$ fall by $>80\%$ in massive systems, congruent with CO observations at $z \approx 2$; hot gas fractions within $R_{500}$ better match X-ray group/cluster data than TNG or MistralC. MistralS uniquely maintains large-scale outflows at galaxy and halo scales while preventing gas inflows (ejective plus preventive feedback). MistralC and standard TNG fail to achieve such regime-spanning regulation (Farcy et al., 10 Apr 2025).

7. Comparison with IllustrisTNG Thermal AGN Feedback and Broader Implications

The IllustrisTNG Isotropic Thermal mode injects energy that is rapidly radiated away at high ISM densities, limiting wind launching efficiency. Its Random Wind kinetic mode launches effective outflows, but only upon transitioning to low-Eddington accretion and invoking explicit BH-mass–dependent switching. In contrast, MistralS operates across all Eddington ratios, obviates parameter-tuned switching, and systematically generates winds with velocity, mass-loading, and bipolar geometry in line with observed BAL outflows.

Across the mass range $10^{12}$–$3 \times 10^{13}\,M_\odot$, MistralS recovers key scaling relations, suppresses cold gas reservoirs, and yields hot gas fractions consistent with X-ray observations, all without additional tuning or multimode switches. By providing a radiatively efficient, momentum-driven AGN wind prescription applicable self-consistently to a wide set of galaxy–halo environments, Mistral (especially MistralS) offers a promising avenue for interpreting high-redshift JWST populations and the co-evolution of galaxies and SMBHs. Its Arepo-based implementation supports incorporation into other moving-mesh cosmological codes for further studies of quasar-mode feedback in galaxy evolution (Farcy et al., 10 Apr 2025).