Simple relations from complex outflows: How the $M-σ$ relation emerges in a multi-phase environment

TL;DR

This work tests whether the $M-\sigma$ relation can emerge from AGN feedback in a turbulent, multi-phase bulge with realistic cooling. Using purpose-built hydrodynamical simulations and a grid-based wind injection, the authors explore SMBH fueling across a range of sub- to super-Eddington luminosities over $\ge 1$ Myr. They find that dense gas is predominantly affected by wind momentum, while energy resides in hot diffuse gas that escapes through low-density channels; a critical luminosity around $0.7\,L_{\rm Edd}$ (for a mass set by $M_{\sigma}$) efficiently suppresses SMBH growth, aligning with momentum-driven outflow expectations. These results support the momentum-driven mechanism as a robust route to establishing the $M-\sigma$ relation even in realistic, clumpy ISMs and across cosmic time.

Abstract

The tight empirical $M-σ$ relation between the mass of a SMBH) and the velocity dispersion of the host galaxy bulge is often interpreted as the result of self-regulation by AGN feedback. This picture is motivated by analytical and semi-analytical models in which momentum-driven AGN winds can expel the gas once the SMBH reaches a critical mass. However, these models typically assume idealised conditions: smooth gas distributions, spherical symmetry, and very efficient cooling of the shocked AGN wind. We checked whether AGN outflows can establish the $M-σ$ relation in a multi-phase and turbulent galactic bulge subject to realistic radiative cooling while conserving the shocked AGN wind energy. We calculated a suite of purpose-built hydrodynamical simulations of AGN outflows in turbulent gas shells, covering a wide range of constant AGN luminosities. We tracked the outflow evolution over the course of $\geq1$~Myr. We analysed the effect of AGN outflow on the cold dense gas and SMBH feeding, estimating the luminosity threshold for removing most of the cold gas from the central regions. We find that AGNs with significantly sub-Eddington luminosities cannot suppress SMBH feeding, while luminosities exceeding $\sim 0.7$ times Eddington clear out both the diffuse hot gas and the cold clumps, consistent with the momentum-driven outflow formalism. We also show that dense gas clusters are affected almost exclusively by the AGN wind momentum, while the shocked wind energy escapes through low-density channels and inflates large bubbles of diffuse gas. Therefore, AGN wind-driven energy-conserving feedback in a turbulent multi-phase medium affects the dense gas only via the wind momentum. Thus, the momentum-driven outflow paradigm is applicable for explaining the $M-σ$ relation even in realistic systems.

Figures (10)

• Figure 1: Gas density integrated through a $20^\circ$ wedge around the mid-plane ($z=0$), defined by $\left|z\right| < 0.18\left(x^2+y^2\right)$, in simulations L0.3 (top row), L1.0 (middle), and L1.7 (bottom). Columns show $t=0.5$, $0.75$, and $1.0$ Myr from left to right. Brighter colours indicate higher gas densities, in the range $\sim10^{-25}$ g cm$^{-3} < \rho <10^{-19}$ g cm$^{-3}$.
• Figure 2: Kinematic phase diagrams at $t=0.5$ Myr in simulations L0.5 (left), L1.0 (middle), and L2.0 (right). In each panel, the top 2D histogram shows radial velocity against density, and the bottom the radial velocity against temperature. 1D histograms at the top and right show distributions of individual gas properties. Grey contours show corresponding distributions in the control simulation. Vertical dotted lines show $v_{\rm r} = 0$ and $v_{\rm r} = -\sigma_{\rm b}$; horizontal dotted line shows $T = 3\times10^4$ K.
• Figure 3: Mass of cold rapidly infalling gas in radial $3.3$-pc-wide bins at $0.25$, $0.5,$ and $0.75$ Myr (solid red, blue, and green lines and shading, respectively) in simulations L0.5, L1.0, and L2.0 (top, middle, and bottom panels, respectively). Solid lines show the mean of the four stochastically different simulations; shading encompasses their full range. Dotted lines in each panel show equivalent results from the control simulation.
• Figure 4: Top: Total mass of cold rapidly infalling gas within the central $500$ pc as a function of time in the control (red), L0.5 (blue), L1.0 (green), and L2.0 (magenta) simulations. We added the mass accreted by the SMBH particle to this total. Solid lines show the mean of the four stochastically different simulations; shading encompasses their full range. Squares mark the times when the mean mass reaches its minimum value. Bottom: Minimum value of cold rapidly infalling gas mass in all simulations (circles) as a function of luminosity, with colours denoting the time when the minimum is reached. The solid red line is the best linear fit between $\log{M}$ and $L_{\rm AGN}$ of all simulations except the control and L0.1; the dashed lines indicate $95\%$ confidence limits on the line parameters, estimated via bootstrapping.
• Figure 5: Top: Clusters (coloured by gas density) detected at $t = 0.75$ Myr in simulation L1.0, plotted over the gas density map (grey scale). Middle panel: Momentum loading factor ($f_{\rm p}$) distributions (Eq. \ref{['eq:fp']}; box and whisker plots) in simulations with different AGN luminosities. Circles highlight outliers beyond 1.5 times the interquartile range covered by the whiskers. The horizontal dashed line highlights $f_{\rm p} = 1$, which corresponds to pure momentum driving, for ease of comparison. Bottom panels: Momentum loading factor ($f_{\rm p}$) as a function of $d_{\rm cl}$ (left) and mean gas density (right) in simulations L0.5 (red), L1.0 (green), and L2.0 (blue).