A Dynamical-Photometric Phase Space for Spiral Galaxies: Probing the Local Coupling Between Light and Gravity

The paper introduces a dynamical–photometric phase space for spiral galaxies, defined by the kinematic variable X(R) = V(R)/R (the angular frequency of circular motion, probing local mean mass density) and the photometric variable Y(R) = d ln I/d ln R (the logarithmic surface brightness gradient). This framework places the local gravitational scale and the logarithmic surface brightness gradient into direct pointwise correspondence at each galactocentric radius R, without requiring knowledge of the stellar mass-to-light ratio.

The baryon-dominated inner disc is characterized by large negative Y, while the dark-matter-dominated outer region approaches Y → 0. This two-regime behaviour is described by the smooth sigmoid relation Y = [a ln X + b]/(1 + exp[k(X − Xtrans)]), which reduces to the logarithmic coupling Y = a ln X + b in the baryonic zone. The logarithmic coupling is physically motivated: it emerges approximately from the combination of an exponential stellar disc (Freeman 1970) with a self-gravitating Freeman-disc rotation curve (Binney & Tremaine 2008), via elimination of the shared radial scale R/Rd under a leading-order Taylor expansion.

The framework is applied to 136 late-type galaxies from the SPARC database, spanning inclinations 20°–89°, distances 1–130 Mpc, and five decades in stellar mass. The median coefficient of determination is R2 = 0.930, with 75 per cent of galaxies achieving R2 > 0.80. The median RMS scatter is 0.22 in dimensionless units of Y. Statistical validation includes eight independent tests (chi-square goodness-of-fit with p > 0.05, reduced chi-square χ2ν ∈ [0.5, 2.0], F-test with p 0.05, Kolmogorov–Smirnov test with p > 0.05, and Pearson and Spearman rank correlation tests each requiring p > 0.05) together with 5-fold cross-validation. The transition parameter Xtrans identifies the onset of dark-matter dominance, corresponding to a median transition radius Rtrans = 5.40 kpc across the sample, ranging from 0.5 kpc in low-mass dwarf irregulars to 50 kpc in the most extended massive spirals.

Detailed results are presented for ten representative galaxies spanning i = 20°–89°, Hubble types Im to Scd, and distances 9.6–66.4 Mpc. For these ten galaxies, the sigmoid achieves R2 > 0.97 in every case (median 0.997), and 5-fold cross-validation returns R2 within 0.001 of the full-data fit. The transition radius Rtrans ranges from 2.31 kpc (NGC 4085) to 14.89 kpc (NGC 4088), with a median of 6.34 kpc. All eight statistical tests pass for all ten representative galaxies. The paper cautions that galaxies with N ≤ 8 phase-space points have large parameter uncertainties (e.g., σa > 3), and results for galaxies with N ≥ 10 and σa < 1 should be considered the most physically reliable.

The phase-space slope a shows a highly statistically significant variation with inclination: a Kruskal–Wallis test comparing three inclination zones — low (i < 50°, n = 32, ã = 0.60), intermediate (50° ≤ i < 75°, n = 53, ã = 0.75), and high (i ≥ 75°, n = 34, ã = 1.11) — returns H = 15.61, p = 0.0004. A Mann–Whitney pairwise comparison between the low- and high-inclination groups gives p = 0.0005. The binned medians show a monotonic increase of ã with inclination, attributed to projection effects (non-axisymmetric structure at low inclination, line-of-sight integration at high inclination).

The two-regime structure is interpreted as a signature of dark-matter halo dominance: in the inner disc, baryons dominate and the logarithmic coupling holds; in the outer disc, Y → 0 while X remains large because the extended dark-matter halo sustains circular motion, which cannot be reproduced by a stellar disc alone (which would require Keplerian fall-off reducing X → 0). The transition radius Rtrans = Vflat/Xtrans is an empirical transition scale, associated with but not identical to the dark-matter onset radius from full mass decompositions.

The phase space is conceptually distinct from the radial acceleration relation (RAR): the RAR compares two centripetal acceleration amplitudes integrated over enclosed mass, while the present framework compares the local kinematic scale X = V/R to the structural gradient of the light Y = d ln I/d ln R. Because Y is a local derivative, it is more sensitive to localised features — bars, rings, bulge–disc transitions — than the RAR.

The phase-space parameters connect to global disc fraction: galaxies with larger stellar disc fractions are expected to exhibit larger Rtrans and more extended Zone 1 loci, while systems with lower disc fraction should transition to Y ≈ 0 at smaller radii, consistent with the observed variation of Rtrans across galaxy types and masses. The most compact transition radii are found in low-surface-brightness and dwarf irregular galaxies, which are known to be dark-matter-dominated at all observed radii.

Sharp discontinuities in Y(R) visible in some galaxies arise from physical structural transitions: bar–disc interfaces, ring resonances at Lindblad resonances, and the bulge–disc transition. These are physical properties of the galaxies and not artefacts of the smoothing.

The main conclusions are: (i) the logarithmic coupling Y = a ln X + b is physically motivated, emerging approximately from the elimination of the shared radial scale Rd between an exponential photometric profile and a Freeman-disc rotation curve; (ii) the sigmoid model accurately describes the full 136-galaxy sample with median R2 = 0.930, all eight statistical tests passing for all ten representative galaxies, and 5-fold cross-validation confirming excellent generalisation; (iii) the two-regime structure is consistent with dark-matter halo dominance; (iv) the empirical transition scale Rtrans = Vflat/Xtrans is measurable with a median of 5.40 kpc across 136 galaxies; (v) the phase-space slope varies significantly with inclination (Kruskal–Wallis H = 15.61, p = 0.0004; Mann–Whitney p = 0.0005); (vi) the phase space is complementary to the RAR, probing the structural gradient rather than the amplitude of the gravitational field; (vii) the phase-space parameters connect to global disc fraction.

Improvements to AI Systems:

1. Hybrid Physics-Informed Regression with Adaptive Sigmoid Fitting

• Implement a neural network or Gaussian process that explicitly incorporates the sigmoid relation Y = [a ln X + b]/(1 + exp[k(X − Xtrans)]) as a prior or inductive bias.

• The AI can automatically detect the baryon-dominated (logarithmic) and dark-matter-dominated (flat Y) regimes from raw galaxy rotation curves and surface brightness profiles, without needing mass-to-light ratios.

• Output: per-galaxy parameters (a, b, k, Xtrans) with uncertainty estimates, plus a confidence flag for galaxies with N ≥ 10 and σa < 1.

2. Automated Transition Radius (Rtrans) Detection

• Train a model to identify the onset of dark-matter dominance directly from the phase-space curve, using the sigmoid inflection point.

• The AI can output Rtrans for any spiral galaxy with sparse or noisy data, and flag cases where Rtrans is unreliable (e.g., N ≤ 8).

• This enables large-scale automated dark-matter mapping across surveys without full mass decomposition.

3. Inclination-Aware Correction Module

• Use the observed monotonic increase of slope a with inclination (ã = 0.60 → 1.11) to build a correction layer.

• The AI can de-project the phase-space slope to an intrinsic (face-on) value, reducing systematic bias in galaxy classification and dark-matter fraction estimates.

• This is critical for comparing galaxies observed at different orientations in large sky surveys.

4. Structural Feature Detector (Bars, Rings, Bulge–Disc Transitions)

• Since Y(R) is a local derivative, it is sensitive to sharp discontinuities (e.g., bar–disc interfaces, Lindblad resonances).

• Train a transformer or convolutional model on Y(R) to classify and localize these substructures automatically, distinguishing physical features from noise.

• The AI can then provide a structural decomposition of a galaxy (bulge, bar, disc, ring) purely from photometric and kinematic profiles.

5. Galaxy Disc Fraction Estimator

• Use the phase-space parameters (especially Rtrans and the extent of the logarithmic zone) to predict the stellar-to-total mass fraction.

• The AI can infer disc fraction from rotation curve + surface brightness data alone, bypassing expensive stellar population synthesis models.

• This enables rapid, homogeneous disc fraction measurements for thousands of galaxies.

6. Complementary RAR-Phase Space Joint Model

• Build a multi-task AI that simultaneously predicts the Radial Acceleration Relation (RAR) and the new phase-space relation, sharing latent features.

• The AI can exploit the complementary sensitivities (RAR for amplitude, phase space for gradient) to better constrain dark matter profiles and test modified gravity theories.

• Output: joint posterior distributions of halo parameters (e.g., NFW scale radius, concentration) with reduced degeneracies.

7. Outlier and Artifact Detection for Low-Quality Data

• Use the eight statistical tests (chi-square, Durbin–Watson, runs test, KS, etc.) as a composite loss or validation metric during training.

• The AI can automatically reject or down-weight galaxies with poor fits (R2 3) and flag them for manual inspection, improving sample purity in automated pipelines.

8. Predictive Scaling Laws for Galaxy Evolution

• Train a generative model on the phase-space parameters (a, b, k, Xtrans) across the 136-galaxy sample, spanning five decades in stellar mass.

• The AI can predict how Rtrans and the slope a evolve with mass, Hubble type, and surface brightness, providing empirical scaling relations for galaxy formation simulations.

• This can be used to generate synthetic galaxy catalogs for testing dark matter models.

What the Improved AI System Can Do:

• Automatically map dark-matter dominance radii for thousands of spiral galaxies from existing surveys (e.g., SPARC, DESI, Euclid) with minimal human intervention.

• Provide inclination-corrected, physically interpretable parameters (a, b, Rtrans) for each galaxy, enabling unbiased comparisons across populations.

• Detect and catalog non-axisymmetric structures (bars, rings) and bulge–disc transitions in a fully automated manner.

• Estimate disc fractions and dark-matter content without stellar population synthesis, accelerating large-scale cosmological analyses.

• Jointly constrain gravity theories by combining RAR and phase-space diagnostics, with explicit uncertainty quantification.

• Flag low-quality or ambiguous data for follow-up, improving the reliability of downstream scientific conclusions.

The interplay between luminous matter distribution and the local gravitational field within disc galaxies encodes physical information beyond that captured by global scaling relations. We introduce a dynamical--photometric phase space defined by the kinematic variable X(R)=V(R)/R and the photometric variable Y(R)= d I/d R, placing the local gravitational scale and the logarithmic surface brightness gradient into direct pointwise correspondence at each galactocentric radius R. The quantity X=V/R= omega represents the angular frequency of circular motion and acts as a probe of the local mean mass density, while Y measures the radial steepness of the stellar light distribution. The baryon-dominated inner disc is characterized by large negative Y, whereas the dark-matter-dominated outer region approaches Y to0. This two-regime behaviour is described by the smooth sigmoid relation Y=[a X+b]/(1+ [k(X-X trans)]), which reduces to the logarithmic coupling Y=a X+b in the baryonic zone. We apply this framework to 136 late-type galaxies from the SPARC database, spanning inclinations 20 -- 89, distances 1 -- 130,Mpc, and five decades in stellar mass. The median coefficient of determination is R squared=0.930. Statistical validation includes eight independent tests together with 5-fold cross-validation. The transition parameter X trans identifies the onset of dark-matter dominance, corresponding to a median transition radius R trans=5.40,kpc across the sample.