Emergent Gravity Project

Reproducibility & Method

This page describes how the κ(r) and κ(S) profiles presented in the current paper can be reconstructed from the SPARC dataset.

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1. Data source

The analysis is based on the SPARC database:

https://astroweb.cwru.edu/SPARC/

For each galaxy, SPARC provides radial profiles including:

  • Observed rotation velocity: Vobs(r)
  • Gas contribution: Vgas(r)
  • Disk contribution: Vdisk(r)
  • Bulge contribution (if present): Vbulge(r)

2. Constructing the baryonic velocity

The baryonic contribution is reconstructed as:

Vbar²(r) = Vgas²(r) + Υdisk · Vdisk²(r) + Υbulge · Vbulge²(r)

Typical values used in the main analysis:

  • Υdisk ≈ 0.5
  • Υbulge ≈ 0.7

Robustness tests were also performed under moderate variations of these values.

3. Definition of κ(r)

The inferred compressibility profile is defined as:

κ(r) = Vbar²(r) / Vobs²(r)

This quantity can be computed directly at each radial point provided in the SPARC data.

4. Structural reparametrization

The accumulated structural variable used in the current version of the paper is defined as:

S(r) = ∫₀ʳ σ(r′) dr′

with the minimal phenomenological choice:

σ(r) = | d log Vbar / dr |

In discrete SPARC-like sampling, S(r) is obtained by cumulatively integrating σ(r) along the observed radial points.

5. κ(S) relations

When the inferred compressibility profiles are re-expressed in terms of S, they become significantly more regular and are well described by class-dependent logistic relations:

κ(S) = κ∞ + (1 − κ∞) / [1 + exp((S − Sc) / ΔS)]

The present interpretation is class-dependent rather than universal: the main classes do not collapse into a single law, but into a small family of logistic responses.

6. Practical computation steps

  1. Download SPARC rotation curve files.
  2. Compute Vbar² using gas, disk, and bulge components.
  3. Compute κ(r) at each radius.
  4. Compute σ(r) = |d log Vbar / dr|.
  5. Integrate S(r) cumulatively.
  6. Plot κ(r) and κ(S).
  7. Compare the profiles against the main classes or fit logistic relations where appropriate.

7. Notes on robustness

  • The class structure of κ(r) remains qualitatively stable under moderate Υ variations.
  • The class-dependent κ(S) relations also remain qualitatively stable under these changes.
  • The exact numerical value of Sc depends on the choice of σ(r), so S is not yet a canonical variable.
  • What appears robust is the existence of an accumulated structural variable organizing the response.

8. Interpretation

Within the present framework, S is interpreted as an accumulated structural-information measure. The transition scale Sc is treated as the accumulated structural scale at which the near-baryonic response ceases to be sustained, while ΔS measures how sharply that reorganization occurs.

The present interpretation is deliberately phenomenological. The current version does not claim that S has yet been derived from a microscopic theory, nor that Sc is universal in a strict sense.