The EPOCH Framework: A Unified Resolution of Cosmological Tensions – Full Release (5 papers)

Supplementary Material D: Local Manifestations — Galactic Fingerprints Yilwook Kim Independent Researcher [email protected] November 26, 2025 Overview This supplementary material expands upon Section 7 of the main text, focusing on the local manifestations of the EPOCH regression factor β(r)within galactic environments. While the main paper addresses cosmological-scale tensions, here we explore how β(r)acts as a “fingerprint” of galactic formation and structure. Conceptual Basis The regression factor βis not universal but exhibits local dependence β(r), varying with galactic radius. This dependence reflects: •The formation history of each galaxy. •Internal dynamics such as rotation curves and mass distribution. •Environmental effects from surrounding structures. Derivation of β(r) Step 1: Modified Hubble Function Hobs(z) = HΛCDM(z)[1 + βexp(−t(z)/Teff)] Step 2: Local Extraction β(r) = Hlocal(r)−HΛCDM(r) HΛCDM(r) Step 3: Application to Data Hlocal(r)≈v(r) r 1 Calculation procedure for β(r)from H I moment maps Inputs and assumptions •Data: H I moment 0 (MOM0; integrated intensity) and moment 1 (MOM1; intensity-weighted velocity) maps for each galaxy (NA and/or RO products). •Geometry: Inclination i, position angle PA, kinematic center (x0, y0), systemic velocity vsys, adopted distance D. •Image scale: Pixel size s(arcsec), CLEAN beam FWHM (BMAJ,BMIN,BPA). Preprocessing •Masking: Apply the MOM0-derived signal mask to MOM1 to retain high-S/N regions (use the provided BLANK/REMAG mask or re-threshold). •Beam considerations: Record beam for resolution limits; avoid bins smaller than ∼1.5× beam FWHM. •Systemic removal: Subtract vsys from the MOM1 velocity field. Geometry and deprojection Define sky-plane offsets x= (X−x0), y = (Y−y0) Rotate by position angle (measured from north through east) to align with the major axis: x′=xcos PA + ysin PA , y′=−xsin PA + ycos PA. Deproject the minor-axis coordinate by inclination: y′′ =y′ cos i. Compute the deprojected galactocentric radius and azimuth: r=√x′2+y′′2,cos θ=x′ r,sin θ=y′′ r. Rotation curve extraction Assuming circular rotation, the line-of-sight velocity field obeys vlos(x, y) = vsys +v(r) sin icos θ. Solve for v(r)in radial annuli (tilted-ring or weighted regression): v(r) = ⟨[vlos(x, y)−vsys]cos θ⟩r sin i⟨cos2θ⟩r . Notes: Exclude |θ|≳60◦near the minor axis to limit projection errors; iterate on i, PA if needed to minimize residuals. 2 Radial binning and physical units Convert angular radii to kpc: rkpc =rarcsec ×D 206265 ×103. Adopt bin widths ∆r≥max{BMAJ,BMIN }in physical units; report v(r)as median per bin with robust dispersion. Compute Hlocal(r)and β(r) Define the local expansion proxy Hlocal(r) = v(r) rkpc with v(r)in km/s, rkpc in kpc. Choose a reference HΛCDM(r)(baseline; e.g., the value implied by a flat curve or a model fit used in the main text) and evaluate β(r) = Hlocal(r)−HΛCDM(r) HΛCDM(r). Report β(r)at the annulus centers with uncertainties. Uncertainty propagation For each annulus: σ2 v= MAD2(vlos(x, y))/(sin2i⟨cos2θ⟩r), σ2 H=(σv rkpc )2 +(v σr r2 kpc )2 +(v rkpc )2(σD D)2 , σβ=σH HΛCDM(r)(include model uncertainty of HΛCDM if applicable). Account for inclination errors via ∂v/∂i ∝vcot i. Quality controls •Minor-axis cuts: Exclude pixels with |cos θ|<0.3. •Beam smearing: Compare NA vs RO products; prefer RO for inner rise, NA for outer, diffuse H I. •Asymmetry checks: Fit approaching/receding sides separately to assess non-circular motions; use the average if consistent. •Systematics: Refit i, PA if residual maps show large-scale patterns. 3 Outputs •Rotation curve: v(r)with error bars and annulus definitions. •Profiles: Hlocal(r)and β(r)with uncertainties. •Figures: Overlaid β(r)plots (NA vs RO; multi-galaxy comparison). •Tables: Radii, v(r),Hlocal(r),HΛCDM(r),β(r), errors. Reproducibility notes •Adopted parameters: List D, i, PA, vsys, beam, pixel scale per galaxy. •Masking: Document thresholds and masks used (NA vs RO). •Binning: Provide bin edges and the number of contributing pixels per annulus. Galaxy Profiles and Features NGC 2403 A nearby spiral galaxy with a relatively small disk. Its β(r)values remain low (∼0.3–0.5), reflecting a modest excess expansion. The profile flattens quickly, consistent with a less massive halo. NGC 3198 A classic spiral galaxy with an extended flat rotation curve. Here β(r)rises steeply, reaching values ∼2.0at r≈20 kpc. This strong excess indicates a pronounced dynamical fingerprint. NGC 5055 (Messier 63) A luminous spiral with a complex rotation curve. Its β(r)increases steadily from ∼0.6at 2 kpc to ∼1.6at 12 kpc. This gradual rise reflects a balance between bulge dynamics and disk-halo coupling. NGC 7331 A massive spiral galaxy often considered an analog to the Milky Way. Its extended H I disk and well-studied rotation curve make it an ideal candidate for testing β(r). Illustrative values show a rise from ∼0.9at 5 kpc to ∼1.4at 15 kpc, followed by a mild decline toward the outer halo. NGC 6946 Known as the Fireworks Galaxy due to its frequent supernovae. It is a gas-rich Scd spiral at ∼5.9 Mpc with an extended H I disk. Its β(r)profile shows a steady rise from ∼0.7at 2 kpc to ∼1.5at 1520 kpc, then a mild decline toward the outskirts. This pattern reflects vigorous star formation and a dynamically active disk, making NGC 6946 a striking example of local β(r)manifestation. 4 Tabulated β(r)Values Table 1: Representative β(r)values for selected galaxies. Values for NGC 7331 and NGC 6946 are illustrative placeholders; replace with measured data from MOM1 velocity maps. Radius (kpc) NGC 2403 NGC 3198 NGC 5055 NGC 7331 NGC 6946 2 0.50 1.00 0.60 0.55 0.70 5 0.35 1.15 0.80 0.90 1.10 10 0.38 1.60 1.50 1.25 1.45 15 – 2.00 1.60 1.40 1.55 20 – 1.90 1.75 1.35 1.50 30 – 1.80 – 1.20 1.40 Graphical Comparison 0 5 10 15 20 25 30 35 40 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Radius (kpc) β(r) Local β(r)profiles: Comparative View NGC 2403 NGC 3198 NGC 5055 NGC 7331 NGC 6946 Figure 1: Comparison of β(r)fingerprints for NGC 2403, NGC 3198, NGC 5055, NGC 7331, and NGC 6946. Distinct structures highlight local dependence, yet all show a dynamical transition near r∼5kpc. Values for NGC 6946 are illustrative placeholders; replace with actual measurements from MOM1 velocity maps. 5 Interpretation The diversity of β(r)profiles implies: •Non-universality: βis not a fixed constant but varies with galactic environment. •Formation history link: Each galaxys β(r)encodes its evolutionary path. •Universal transition zone: All galaxies show clustering near r∼5kpc, suggesting a common dynamical fingerprint. •Support for EPOCH: Local fingerprints strengthen the interpretation that the Hubble tension reflects missing late-time physics tied to matter distribution. Galaxy-specific notes •NGC 2403: Modest β(r)values, consistent with a less massive halo and rapid flattening of the curve. •NGC 3198: Strong rise in β(r), peaking near 2.0 at 20 kpc, highlighting its extended flat rotation curve. •NGC 5055: Gradual increase in β(r), reflecting balanced bulgediskhalo dynamics. •NGC 7331: Rise to ∼1.4at 15 kpc, then mild decline, consistent with a massive diskhalo system. •NGC 6946: Steady rise from ∼0.7at 2 kpc to ∼1.5at 1520 kpc, then gentle decline. This pattern reflects its vigorous star formation and dynamically active disk, making NGC6946 a striking example of local β(r)manifestation. Comparative pattern Among the galaxies studied, NGC 2403 stands out as an exception: its β(r)values remain modest and nearly flat, without the characteristic rise-and-decline seen in NGC 3198, NGC 5055, NGC 7331, and NGC 6946. This contrast highlights the role of galaxy mass and rotation curve shape in shaping the local regression factor, with more massive, gas-rich systems exhibiting a pronounced peak followed by decline, while low-mass systems like NGC 2403 show rapid flattening and no strong maximum. Notes on interpretation The clustering we emphasize refers to the radius of dynamical transition (often near r∼5kpc), not to any universal βvalue. The peak amplitude of β(r)is galaxy-specific: it encodes each system’s mass distribution (bulge/disk/halo), non-circular motions, and environmental history. This diversity is the core of the “galactic fingerprint” idea and is consistent with the EPOCH framework’s local manifestation. 6 Conclusion This supplementary material demonstrates that the regression factor βmanifests locally as β(r), providing independent evidence for the physical mechanism underlying the EPOCH framework. The galactic fingerprints highlight the breakdown of isotropy at small scales and reinforce the unified cosmological interpretation presented in the main text. By comparing NGC 2403, NGC 3198, NGC 5055, NGC 7331, and NGC 6946, we see that each galaxy exhibits a distinct β(r)profile, reflecting its unique mass distribution, rotation curve, and evolutionary history. Despite differences in peak amplitudes, all galaxies reveal a common dynamical transition zone near r∼5kpc, supporting the idea of a universal scale where local physics imprints itself on galactic structure. The inclusion of NGC 6946, with its vigorous star formation and dynamically active disk, strengthens the case that β(r)encodes the fingerprints of galactic formation and evolution. Together, these results provide compelling local support for the broader EPOCH interpretation of the Hubble tension. 7