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

¹ Center for Physical Sciences and Technology Saulėtekio al. 3 Vilnius LT-10257, Lithuania
² Astronomical Observatory, Vilnius University Saulėtekio al. 3 Vilnius LT-10257, Lithuania

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Received: 11 July 2025
Accepted: 29 October 2025

Abstract

Context. The tight empirical M − σ relation between the mass of a supermassive black hole (SMBH) and the velocity dispersion of the host galaxy bulge is often interpreted as the result of self-regulation via active galactic nucleus (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. It is unclear whether this paradigm is applicable under more realistic conditions.

Aims. 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.

Methods. We ran 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 ≥1 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.

Results. We find that AGNs with significantly sub-Eddington luminosities cannot suppress SMBH feeding, while luminosities exceeding ∼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.

Conclusions. Active galactic nucleus 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.