3.1 Galaxy Rotation Curves: Fitting Without Dark Matter

Home ／ Chapter 3: Macroscopic Universe

Terminology and Conventions: In this section, the outer-disk “extra pull” is treated as the combined outcome of two medium effects caused by Generalized Unstable Particles (GUP) over their finite lifetimes: a smooth, cumulative bias called Statistical Tensor Gravity (STG) and a diffuse, low-coherence background called Tensor Local Noise (TBN) that appears when these particles decompose or annihilate. After these first mentions, we use the full English names without abbreviations. The “energy sea” refers to the surrounding medium in Energy Filament Theory (EFT).

I. Phenomena and the Core Puzzle

Many spiral galaxies keep high, nearly flat rotation speeds well beyond the bright stellar disk, where visible matter is sparse and speeds would normally fall. Two unusually tight regularities accompany this behavior:

• Visible mass and a characteristic outer-disk speed lie close to a single relation with very small scatter.
• At each radius, the total centripetal pull tracks the visible-matter pull in a near one-to-one manner, again with tight scatter.

Rotation-curve shapes differ—cuspy versus core-like centers, plateau radii and heights, and fine “texture.” Environment and event history matter. Yet these systems still obey the same two tight relations, pointing to a shared mechanism. Traditional fits add unseen “envelopes” object by object, often requiring bespoke tuning and leaving the extraordinary tightness of the relations unexplained if formation histories are diverse.

Key idea: the outer-disk “extra pull” can emerge from the medium’s statistical response rather than added unseen matter.

II. Mechanism at a Glance: One Tensor Landscape, Three Contributions

1. Baseline Inner Slope (Visible Matter)
2. Stars and gas pull the energy sea into an inward-sloping tensor landscape, setting basic centripetal guidance. This contribution declines rapidly with radius and cannot maintain a flat outer plateau on its own.
3. Observational handle: the more centrally concentrated the light-to-mass ratio and gas surface density, the sharper the inner rise.
4. Smooth Additive Slope (Statistical Tensor Gravity)
5. Generalized Unstable Particles imprint tiny pulls on the tensor field during their lifetimes. These contributions add up across spacetime into a smooth, persistent bias that declines only slowly with radius.
   • Spatial smoothness: the bias weakens gently with radius yet remains effective in the outer disk, sustaining the plateau.
   • Co-tuning with activity: its strength correlates with star-formation rate, mergers or disturbances, gas cycling, and bar/spiral shear.
   • Self-locking: more supply and stirring raise activity, which strengthens the smooth additive slope and locks in the outer-disk speed scale.
   • Observational handle: surface density of star formation, bar strength, gas inflow/outflow, and merger signatures correlate with the plateau’s height and length.
6. Low-Amplitude Texture (Tensor Local Noise)
7. When Generalized Unstable Particles decompose or annihilate, they inject broad-band, low-coherence wave packets that form a diffuse background. This background adds small undulations and line-width broadening without changing the average plateau level.
8. Observational handle: radio halos/relics, low-contrast diffuse structures, and “graininess” in velocity fields, enhanced along merger axes or in high-shear zones.

Radial zoning (intuition):

• Inner region (R ≲ 2–3 R_d): visible guidance dominates; Statistical Tensor Gravity provides fine-tuning → decides cuspy vs. core-like.
• Transition region: comparable contributions → curve turns from steep to flat; the turning radius drifts with activity and history.
• Outer plateau: Statistical Tensor Gravity takes a larger share → a high, extended plateau with mild texture.

Conclusion: the plateau ≈ visible guidance + Statistical Tensor Gravity; the small outer undulations ≈ Tensor Local Noise.

III. Why the Two “Tight Relations” Emerge

• Mass–Velocity: Near a Single Line
• Visible matter supplies and stirs the medium, setting the overall activity of Generalized Unstable Particles; that activity sets the plateau’s speed scale. Visible mass and the outer-disk speed therefore co-vary from a shared cause, leaving little scatter.
• Radial Total-to-Visible Pull: Near One-to-One
• Total centripetal pull equals visible guidance plus the smooth additive slope from Statistical Tensor Gravity. The inner disk is “visible-dominated,” while the outer disk gains an increasing share from Statistical Tensor Gravity. Radius by radius, this yields a smooth mapping from visible pull to total pull.
• Direct check: at a fixed radius, map dynamical residuals against gas/dust shear and diffuse radio intensity; they should correlate in the same direction.

Key idea: the two relations are projections—“mass vs. speed” and “radius vs. pull”—of a single tensor landscape.

IV. Why Cuspy and Core-Like Centers Coexist

• Flattening (“shaving”) mechanism: long-lived activity—mergers, starbursts, strong shear—softens the local tensor landscape and reduces the inner slope, yielding core-like centers.
• Tightening mechanism: a deep potential well with steady supply and mild disturbance restores or preserves a cuspy center.

Conclusion: cusps and cores are two end states of the same tensor network under different event histories and environments.

V. Putting Multi-Band Observations on One Tensor Map (How-To)

Co-map these quantities:

• Height and radial extent of the rotation-curve plateau.
• Stretch and central offset of weak/strong lensing convergence (kappa, κ) contours.
• Shear stripes and non-Gaussian wings in gas velocity fields.
• Diffuse intensity and orientation of radio halos/relics.
• Direction of polarization/magnetic-field lines as tracers of long-term shear.

Co-mapping criteria:

• Spatial alignment: the above features co-locate and co-orient along merger axes, bar axes, or tangents to spiral arms.
• Epoch consistency: during active phases, the diffuse background rises first (Tensor Local Noise), followed—over tens to hundreds of millions of years—by deeper or longer plateaus (Statistical Tensor Gravity). Quiet phases reverse this sequence.
• Cross-band coherence: after accounting for medium-dependent dispersion, the directions of plateaus and residuals agree across bands because the tensor landscape sets them.

VI. Testable Predictions (Operationalized for Observation and Fitting)

1. P1 | Noise Before Lift (Temporal Order)
2. Prediction: after a starburst or merger, the diffuse radio background rises first due to Tensor Local Noise. Over tens to hundreds of millions of years, the plateau’s height and radius increase as Statistical Tensor Gravity strengthens.
3. Observation strategy: perform joint multi-epoch, multi-ring fitting to measure the lag between the diffuse rise and the plateau’s deepening or extension.
4. P2 | Environmental Dependence (Spatial Pattern)
5. Prediction: along high-shear directions or merger axes, plateaus extend farther and sit higher, with stronger “graininess” in velocity fields.
6. Observation strategy: extract sectoral rotation curves and diffuse-background profiles along bar and merger axes and compare.
7. P3 | Co-mapped Cross-Checks (Multi-Modal)
8. Prediction: major axes of κ contours, peaks of velocity shear, radio streaks, and principal polarization directions align.
9. Observation strategy: register four maps on one coordinate system and compute cosine similarity between their vectors.
10. P4 | Outer-Disk Spectral Shape
11. Prediction: the power spectrum of outer-disk velocity residuals shows a gentle slope in the mid- to low-frequency range, matching the broad-band, low-coherence character of Tensor Local Noise.
12. Observation strategy: compare the peak and tilt of the residual spectrum with those of the diffuse radio background.
13. P5 | Fitting Workflow (Parameter Economy)
14. Steps:
    • Use photometry and gas to set priors for the baseline inner slope from visible matter.
    • Use star-formation rate, merger indicators, bar strength, and shear to set priors for the amplitude and scale of Statistical Tensor Gravity.
    • Use diffuse radio intensity and texture to set priors for the broadening caused by Tensor Local Noise.
    • Fit the full rotation curve with a small shared parameter set, then verify by co-mapping with lensing and velocity fields.
    • Goal: one parameter set for multiple data modes, instead of object-specific envelope tuning.

VII. An Intuitive Analogy

A convoy in a tailwind. The engines represent visible guidance. The tailwind represents Statistical Tensor Gravity, which declines slowly with distance yet sustains speed. Small bumps represent Tensor Local Noise, which adds slight “graininess” to the speed curve. What to manage: throttle (supply), “road” maintenance (shear/activity), and tailwind sustainment (amplitude of the smooth additive slope).

VIII. Relation to Conventional Interpretations

• A different route to explanation: instead of attributing “extra pull” to added unseen matter, we reframe it as the medium’s statistical response: a smooth additive slope from Statistical Tensor Gravity plus low-amplitude texture from Tensor Local Noise.
• Fewer degrees of freedom: three co-sourced drivers—visible supply, long-term stirring, and the resulting tensor bias—govern outcomes and reduce object-specific tuning.
• One map, many projections: rotation curves, lensing, gas kinematics, and polarization are different projections of the same tensor landscape.
• Inclusive rather than adversarial: a future discovery of a new component could fit as a microscopic source; for the main features of rotation curves, the medium’s statistical effects already provide a unified fit.

IX. Conclusion

A single tensor landscape explains the flat outer rotation, the two tight relations, the coexistence of cuspy and core-like centers, and small-scale texture:

• Visible matter shapes the baseline inner slope.
• Statistical Tensor Gravity lays a smooth, persistent, slowly declining additive slope that sustains the outer-disk speed and locks the speed scale to visible mass.
• Tensor Local Noise adds low-amplitude “graininess” without changing the overall plateau.

In summary: the rotation-curve question shifts from “How much unseen matter should we add?” to “How is the same tensor landscape continuously reshaped?” Under this unified, medium-based mechanism, plateaus, tight relations, central morphologies, and environmental dependencies appear as facets of one physical process rather than separate puzzles.