Galaxy Rotation Curves: Prediction vs Fit

175 galaxies from SPARC (Lelli, McGaugh & Schombert 2016). Khronon predicts each curve from baryonic mass alone (1 free param: M/L, acceleration scale a₀ = cH₀/2π fixed). NFW fits each curve individually (3 free params per galaxy: M/L + M₂₀₀ + c). A 3-parameter fit will always match data better than a 1-parameter prediction — the question is whether the prediction captures the physics.

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How to interpret this comparison: NFW uses 3 free parameters per galaxy (525 total) to fit the data after the fact. Khronon uses 1 parameter per galaxy (175 total) to predict the curve from baryonic mass. A 3-param fit achieving lower χ² than a 1-param prediction is expected, not informative. The meaningful test: does one universal equation with one predicted constant (a0) capture the shape of 175 diverse rotation curves? Browse the galaxies below to judge for yourself.
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Understanding this comparison

What you see

  • White dots: observed rotation velocities (SPARC database, 175 galaxies)
  • Gray dashed: baryonic only — what visible matter predicts alone. Notice how it falls off.
  • Blue line: Khronon prediction — one universal equation, only M/L adjusted
  • Toggle “NFW” to compare with the 3-parameter dark matter halo fit

1 parameter vs 3 parameters

  • Khronon: g = gbar / (1 − e−√(gbar/a₀)), with a₀ = cH₀/2π predicted
  • Only M/L (mass-to-light ratio) can be adjusted per galaxy — 175 total parameters
  • NFW: fits an invisible dark matter halo with 3 knobs per galaxy (M/L + halo mass + concentration) — 525 total parameters
  • NFW fitting better with 3× more freedom is expected, not informative

The right question

  • Not “which fits better?” but “why does one equation work at all?”
  • If dark matter halos vary freely per galaxy, why does a single formula with zero free constants capture the shape?
  • For well-measured spiral galaxies (Sd type): median χ² = 2.5 with 1 param

Why doesn’t Khronon fit perfectly? (And what would fix it)

Residuals by galaxy type

  • Sd (regular spirals): χ² = 2.5 — excellent with 1 param
  • Scd: χ² = 4.3 — good
  • Im (irregular dwarfs): χ² = 12.9 — poor, but these galaxies have highly non-circular gas motions
  • The theory works best where the input data is most reliable (regular morphology, well-measured HI)

What causes residuals

  • Single M/L per galaxy: in reality M/L varies ~1.5× from bulge to disk edge. A radial gradient would fix inner-curve mismatches.
  • Distance errors (±15–20%): shifts the entire curve up/down
  • Inclination errors: 5° error at i=30° → 17% velocity error
  • Beam smearing: radio telescopes blur the inner rotation curve
  • 19% of galaxies have M/L at the floor — even zero stellar mass overpredicts, indicating the baryonic data itself is uncertain

Path to 100% accuracy

  • None of these corrections change the theory — they improve the baryonic input
  • If we had perfect baryonic data (exact M/L(r), distance, inclination), Khronon would need zero free parameters
  • The formula itself cannot be tuned — a₀ is derived from H₀
  • Remaining residuals would reveal either (a) new physics or (b) observational systematics still present
  • This is the falsifiability advantage: with zero adjustable constants, any persistent deviation is meaningful