The M–σ Relation in Galaxy Formation

Definition and Physical Meaning

In the context of the M–σ relation in astrophysics, σ (sigma) refers to the stellar velocity dispersion. This term describes the spread in the velocities of stars—essentially, how quickly stars are moving around—in the central region (usually the bulge) of a galaxy.

Key Parameters

σ = velocity dispersion:

• A statistical measure (standard deviation) of the speeds at which stars are moving
• Typically measured in kilometers per second (km/s)
• Higher σ indicates faster and more chaotic stellar motion in that region

M = black hole mass:

• Mass of the supermassive black hole at the center of the galaxy
• Usually expressed in solar masses (\(M_\odot\))

The Empirical Relation

The M–σ relation is expressed as:

\[ M_{\rm BH} \propto \sigma^p \]

where:

• \(M_{\rm BH}\) is the supermassive black hole mass
• \(\sigma\) is the stellar velocity dispersion in the galactic bulge
• \(p\) is an empirical constant (typically between 4 and 5)

This empirical law links the mass of a galaxy’s central black hole to the velocity dispersion of the stars in the galactic bulge.

Significance in Cosmology

The M–σ relation is one of the most important scaling relations in galaxy formation theory, suggesting a fundamental connection between:

• Central black hole growth
• Bulge formation processes
• Overall galaxy evolution

Connection to Decoherence Framework

In the context of the Decoherence as First Principle framework, the M–σ relation may reflect:

• Co-evolution through decoherence: Both SMBH growth and stellar velocity dispersion could emerge from the same early gravitational decoherence events
• Gravitational pointer states: The relation might represent the stabilization of gravitational configurations through decoherence processes
• Scale-invariant decoherence: The power-law form suggests underlying scale-invariant decoherence mechanisms

This provides an alternative to traditional feedback models, where the correlation arises naturally from shared decoherence history rather than causal feedback loops.

Note: This relation is central to understanding the co-evolution of supermassive black holes and their host galaxies, and may provide observational constraints for decoherence-based cosmological models.