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

This paper proposes a unified theoretical framework within the Nexus Paradigm of quantum gravity that derives an evolving Faber-Jackson relation from the baryonic Tully-Fisher relation via a common acceleration scale and cosmic time evolution, successfully explaining observed offsets between galaxy populations as differences in formation epochs and validating these dynamically derived ages with high-precision metallicity data.

The Gist

Imagine the universe as a giant, bustling city. For decades, astronomers have been trying to understand the "rules of the road" that govern how these galactic cities are built. They've noticed two main patterns:

1. The Tully-Fisher Rule: For spinning disk galaxies (like our Milky Way), there's a strict link between how heavy they are and how fast they spin.
2. The Faber-Jackson Rule: For round, puffy galaxies (like giant ellipticals or tiny dwarf blobs), there's a link between their mass and how fast their stars are jiggling around inside them.

For a long time, scientists thought these were just two separate rules. But this paper argues they are actually two sides of the same coin, and that coin is time.

Here is the story of the paper, explained simply.

1. The Big Idea: A "Cosmic Clock"

The authors propose a new theory called the Nexus Paradigm. Think of it like this:

Imagine gravity isn't just a static force, but a river that flows and changes speed over billions of years. In the early universe, the "river" flowed differently than it does today.

The paper suggests that all galaxies follow the same basic rule, but the rule has a "time stamp" on it.

• The Rule: A galaxy's mass is tied to how fast its stars move (either spinning or jiggling).
• The Twist: The exact numbers in this rule change slightly as the universe ages. It's like a recipe that changes slightly every year. If you bake a cake today, it tastes different than a cake baked 10 billion years ago, even if you used the same ingredients.

2. Solving the "Offset" Mystery

For years, astronomers noticed a weird glitch. If you plot big galaxy clusters on a graph, they don't line up with individual galaxies. They sit slightly off to the side, as if they are following a different rule.

The Paper's Solution: They aren't following a different rule; they are just at a different stage of the movie.

• Individual Galaxies: Many formed very early in the universe's history (like ancient, weathered stones).
• Galaxy Clusters: These massive groups formed later (like newer, smoother stones).

Because the "recipe" changes over time, the older galaxies and the younger clusters look different on the graph. The paper shows that if you account for when they were born, they all fall perfectly onto the same single line. The "offset" is just a difference in age.

3. The "Galactic Speedometer"

The most exciting part of this paper is that this theory turns galaxy physics into a time machine.

Usually, to figure out how old a galaxy is, astronomers have to look at the color of its stars or the chemicals inside them (like looking at tree rings). It's slow and tricky.

This paper says: "Just measure how fast the stars are moving, and we can tell you exactly when the galaxy was born."

• The Analogy: Imagine you find a car on the side of the road. Usually, you have to look at the rust or the model year to guess its age. But this new theory says, "If you measure how fast the engine is humming, you can instantly know the exact year it was manufactured."

4. What They Found

The authors tested this "speedometer" on 39 different galaxies, ranging from tiny, faint "dwarf" galaxies to massive, heavy "elliptical" giants.

• The Ancient Ones (Ultra-Faint Dwarfs): These tiny galaxies were found to be 12 to 13 billion years old. They formed very early, right after the Big Bang. This matches perfectly with other telescopes (like the Hubble Space Telescope) that have looked at their stars and confirmed they are ancient.
• The Young Ones (Gas-Rich Dwarfs): These are much younger, forming only a few billion years ago (or even recently!).
• The Connection: The paper found a perfect, straight-line connection between the age of the galaxy and its motion. The older the galaxy, the more it fits the "ancient" version of the rule. The younger ones fit the "modern" version.

5. Why This Matters

This is a big deal for three reasons:

1. It Unifies Everything: It proves that the same physics governs a tiny dwarf galaxy and a massive cluster. They aren't different species; they are just different ages of the same thing.
2. It's a New Tool: Astronomers now have a new, fast way to date galaxies without needing to analyze every single star inside them.
3. It Supports a Deep Theory: The math behind this comes from a theory called "Nexus Paradigm," which tries to connect gravity with quantum mechanics (the physics of the very small). The fact that the math works so well on real galaxies suggests this theory might be on the right track.

The Bottom Line

Think of the universe as a giant library. For a long time, we thought the books (galaxies) were written by different authors using different styles. This paper says, "No, it's all one author." The style just evolved over time.

By measuring how fast the stars are moving, we can now read the "publication date" of any galaxy, proving that the universe has a consistent, evolving rhythm that connects the smallest dwarf galaxies to the largest clusters.

Technical Summary

1. Problem Statement

The paper addresses two fundamental challenges in extragalactic astrophysics:

• The Origin of Scaling Relations: The physical origin of the Baryonic Tully-Fisher Relation (BTFR) for rotation-supported disks and the Faber-Jackson Relation (FJR) for pressure-supported spheroids remains debated. While Modified Newtonian Dynamics (MOND) and $\Lambda$CDM models offer explanations, the deep physical connection between these two distinct scaling laws is not fully unified.
• The "Offset" Problem: Observations reveal a systematic offset (approx. 0.6–0.8 dex in baryonic mass) between galaxy populations (e.g., field galaxies) and galaxy clusters on the BTFR/FJR. It is unclear whether this offset implies distinct physical regimes or a different evolutionary stage.
• Cosmic Chronometry: There is a lack of a unified, dynamical method to determine the formation ages of diverse galaxy populations (from ultra-faint dwarfs to massive ellipticals) that is consistent with independent stellar population age estimates.

2. Methodology and Theoretical Framework

The authors propose a unified framework based on the Nexus Paradigm (NP) of quantum gravity, which posits that spacetime is described by quantized Ricci solitons (De Sitter and Anti-De Sitter).

A. Theoretical Derivation

1. From BTFR to BFJR: The authors derive the evolving Baryonic Faber-Jackson Relation (BFJR) directly from the evolving BTFR.
   • They start with a metric equation derived from the quantized spacetime soliton model, leading to a modified gravity equation where the acceleration scale $a_0 \sim 10^{-10} \, \text{m/s}^2$ emerges naturally.
   • By applying the Virial Theorem to pressure-supported systems and combining it with the universal acceleration scale, they derive the relation:
     $$M_b = K_0 (1+z_{form})^{-4} \sigma^4 (\alpha_\infty - 2\beta)^2$$
     Where $M_b$ is baryonic mass, $\sigma$ is velocity dispersion, $z_{form}$ is formation redshift, and $\beta$ is the velocity anisotropy parameter.
2. Time Evolution: The normalization of the relation is not static but evolves with cosmic time via an exponential kernel involving the Hubble parameter $H(t)$:
   $$M_b \propto e^{-4 \int_{t_{form}}^{t_0} H(t) dt} \sigma^4$$
   This implies that the apparent offset between galaxy populations is a function of their formation epoch ($z_{form}$), not a difference in fundamental physics.

B. Data and Analysis

• Sample: A compilation of 39 galaxies spanning five orders of magnitude in baryonic mass, including:
  • Ultra-faint dwarfs (UFDs) and Classical dwarf spheroidals (dSphs).
  • Gas-rich dwarf irregulars (THINGS survey).
  • Starbursting dwarfs (Lelli et al. sample).
  • A newly discovered gas-rich UFD (KK153).
• Method:
  • For rotation-supported systems, velocity dispersion ($\sigma$) was derived from rotation velocities ($V_{rot}$) using the Toomre Stability Relation.
  • For pressure-supported systems, observed $\sigma$ and $M_b$ were used to invert the evolving BFJR equation to solve for the formation lookback time ($t_{age}$) and redshift ($z_{form}$).
  • Validation: Dynamically derived ages were cross-validated against independent metallicity-based ages (from isochrone fitting and stellar population synthesis).

3. Key Contributions

• Unification of Scaling Laws: The paper demonstrates that the BTFR and FJR are manifestations of the same underlying physical law, differing only by the dynamical support mechanism (rotation vs. pressure) and the time-evolution kernel.
• Resolution of the Offset: The observed mass offsets between different galaxy populations (e.g., UFDs vs. clusters) are reinterpreted as a consequence of different formation epochs. Older systems (UFDs) lie on the relation at high redshift, while younger systems lie near $z=0$.
• New Cosmic Chronometer: The authors establish the evolving BFJR as a precise "dynamical chronometer" capable of dating galaxies without relying solely on stellar population modeling.
• Quantum Gravity Application: The work applies the Nexus Paradigm of quantum gravity to astrophysical scaling relations, linking the acceleration scale $a_0$ to the quantum states of spacetime solitons.

4. Key Results

• Age Determination:
  • Ultra-faint Dwarfs (UFDs): Derived mean age of 12.2 ± 0.8 Gyr ($z_{form} \sim 3-5$). This aligns perfectly with independent Hubble Space Telescope (HST) observations showing UFDs are as old as the globular cluster M92 and synchronized within ~1 Gyr.
  • Classical dSphs: Ages range from 10–13 Gyr ($z_{form} \sim 1-3$).
  • Gas-rich/Starburst Dwarfs: Systematically younger ages of 0–3 Gyr ($z_{form} \approx 0$), indicating ongoing or recent formation.
• Statistical Validation:
  • A comparison between BFJR-derived ages and metallicity-based ages yields a Pearson correlation coefficient of $r = 0.961$.
  • The Orthogonal Distance Regression (ODR) slope is $1.00 \pm 0.06$, consistent with a 1:1 relationship, with a low RMS scatter of 0.29 Gyr.
  • The separation between early-forming (UFDs) and late-forming (dwarf irregulars) populations is statistically significant at >6$\sigma$.
• The Evolving Diagram: The data fits theoretical tracks of $M_b \propto (1+z)^{-4}\sigma^4$, confirming that the slope remains invariant (approx. 4) while the normalization shifts with cosmic time.

5. Significance and Implications

• Cosmological Consistency: The results support the hierarchical assembly model of $\Lambda$CDM, where the smallest structures (UFDs) formed first. The synchronization of UFD ages suggests their star formation was truncated by a global event (likely reionization), consistent with the framework's predictions.
• Fundamental Physics: The persistence of the universal acceleration scale $a_0 \sim 10^{-10} \, \text{m/s}^2$ over 13 Gyr places tight constraints on modified gravity theories, suggesting $a_0$ is constant to within measurement uncertainties.
• Paradigm Shift: The paper transforms galaxy scaling relations from static empirical correlations into dynamic tools for probing cosmic history. It suggests that "offsets" in scaling relations are not errors or distinct physics but signatures of the universe's expansion history imprinted on galaxy formation epochs.
• Future Applications: The framework provides a roadmap for using JWST and future surveys (Euclid, Rubin) to measure the evolution of these relations at high redshift ($z > 6$) and refine the understanding of the earliest galaxies.

In conclusion, Marongwe and Kauffman provide a robust theoretical and empirical unification of galaxy scaling relations, demonstrating that the Faber-Jackson and Tully-Fisher relations are evolving cosmic chronometers driven by a fundamental acceleration scale and the expansion history of the universe.