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

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 Faber-Jackson relation 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}σ^4 $, where $σ$ is the velocity dispersion and $ H(t)$ is the time varying Hubble parameter. Applying this framework to a sample of 39 galaxies spanning five orders of magnitude in baryonic mass, from ultra-faint dwarfs to massive cluster ellipticals, we demonstrate 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 (formation redshift $z\sim 3-5$, 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. Independent validation using metallicity-based stellar population ages reveals a Pearson correlation coefficient of $ r=0.961$ with our dynamically derived ages, providing strong empirical support for the framework. The evolving Faber-Jackson relation unifies pressure-supported systems across all mass scales and establishes galaxy scaling relations as precise cosmic chronometers.