The Evolving Faber-Jackson Relation: A Unifying Framework for Galaxy Ages and the Baryonic Tully-Fisher Connection

TL;DR

This paper presents a unified theoretical framework explaining the origins and evolution of the baryonic Tully-Fisher and Faber-Jackson relations, linking galaxy kinematics to cosmic time and galaxy formation history.

Contribution

It introduces a common acceleration scale and exponential evolution model derived from quantum gravity, unifying galaxy scaling laws and explaining observed offsets.

Findings

The framework accurately predicts galaxy ages from dwarf to elliptical scales.

Observed offsets between galaxy populations are explained by differences in formation epochs.

Strong empirical correlation (r=0.961) between stellar population ages and dynamical ages supports the model.

Abstract

The baryonic Tully-Fisher relation (BTFR) and Faber-Jackson relation (FJR) represent fundamental scaling laws linking the baryonic mass of galaxies to their kinematics, yet their physical origin and apparent offsets between different galaxy populations have remained enigmatic. Here we present a unified theoretical framework demonstrating that both relations emerge from a common acceleration scale of order $10^{-10}m/s^2$ and evolve with cosmic time through a common exponential kernel. We derive the evolving FJR directly from the evolving BTFR within the Nexus Paradigm of quantum gravity, showing that the normalization scales as $M_b \propto e^{-4\int H(t)\,dt}\sigma^4 $, where $\sigma$ is the velocity dispersion and $ H(t)$ is the time varying Hubble parameter. Using this framework on a sample of 39 galaxies ranging from ultra-faint dwarfs to massive cluster ellipticals, we show that the observed offset between galaxy populations arises naturally from differences in their formation epochs. Ultrafaint dwarf galaxies yield ages of $ 12\pm0.8$ Gyr, in excellent agreement with independent Hubble Space Telescope stellar population ages showing synchronization within $\sim 1$ Gyr. Later-type dwarfs show systematically younger ages of $3.5-6.0$ Gyr. The dynamical age reported by the evolving FJR measures the time since a galaxy last achieved virial equilibrium. Dark matter-deficient galaxies exhibit ancient stellar populations but very young dynamical ages, consistent with a recent violent collision. Independent validation using metallicity-based stellar population ages reveals a Pearson correlation coefficient of $r=0.961$ with our dynamically derived ages for undisturbed systems, providing strong empirical support for the framework. The evolving FJR links pressure-supported systems at every mass scale, making galaxy scaling relations accurate markers of both formation and disturbance events.