Correlation Dimension Analysis of SDSS Galaxies: Quantifying Geometry and Clustering Contributions to the Homogeneity Scale

This paper presents a correlation dimension analysis of 387,643 galaxies from the Sloan Digital Sky Survey (SDSS) Main Galaxy Sample, addressing the long-standing question of cosmic homogeneity. The correlation dimension D2(r) measures how galaxy clustering varies with scale. For a homogeneous distribution, D2 equals 3; deviations below 3 indicate clustering. Previous studies found that SDSS galaxies plateau at D2 approximately equal to 2.5 at large scales (80-150 Mpc), well below the homogeneous value. This has fueled debate about whether the universe truly transitions to homogeneity. We resolve this debate by decomposing the observed D2 suppression into two components using a novel shuffled-redshift control test. This test preserves the exact SDSS angular footprint while eliminating spatial clustering, isolating the pure geometry effect. Key findings: 1. Suppression budget: The total suppression from D2=3 to D2=2.5 at 80-150 Mpc is fully explained by survey geometry (62 percent, contributing 0.31) and galaxy clustering (38 percent, contributing 0.19). 2. Geometry validation: A spatially uniform distribution observed through the SDSS footprint would show D2 approximately equal to 2.7, not 3.0. This confirms that partial sky coverage is the dominant effect. 3. Full-sky estimate: After applying an empirical geometry correction, the full-sky D2 estimate rises to 2.81, within 0.17 of the LCDM prediction (2.98). 4. LCDM consistency: The remaining 0.17 gap is expected given modeling simplifications including linear galaxy bias and neglect of redshift-space distortions. We conclude that the SDSS galaxy distribution is consistent with LCDM and the cosmological principle of large-scale homogeneity. The observed D2 plateau below 3 is primarily a geometric artifact of partial sky coverage, not evidence for cosmic inhomogeneity. The shuffled-redshift methodology developed here provides a template for geometry correction applicable to future surveys including DESI, Euclid, and the Nancy Grace Roman Space Telescope. Data products include correlation dimension measurements, bootstrap uncertainties, covariance matrices, shuffled-redshift validation results, and publication-quality figures. Version 2UPDATE (December 29th, 2025): Independent Validation with BOSS DR12 LOWZ We have extended this analysis to the BOSS DR12 LOWZ sample (361,762 luminous red galaxies at 0.15 < z < 0.43) for independent cross-validation. The LOWZ sample differs from MGS in galaxy type (color-selected LRGs vs. flux-limited), redshift range (extending to ~1650 Mpc vs. ~550 Mpc), and sky footprint (BOSS DR12 vs. SDSS Legacy). Key validation findings: Different observed D2: LOWZ shows D2 = 2.76 at 80-150 Mpc, higher than MGS (2.50), due to less geometry suppression in the optimized BOSS footprint. Official random catalogs: Using BOSS random catalogs (which encode the full survey selection function) provides more rigorous geometry characterization than shuffled redshifts. The randoms yield D2 = 2.87, indicating minimal geometry suppression. Different suppression budget: LOWZ shows 56% geometry / 44% clustering, compared to 62% / 38% for MGS—different footprints yield different geometry fractions, as expected. Higher galaxy bias: LOWZ best-fit bias b = 1.69, approximately twice the MGS value (b = 0.80), consistent with LRGs being more strongly clustered tracers. Cross-sample consistency: Despite different observed values, both samples converge to consistent full-sky estimates after geometry correction: D2 = 2.89 (LOWZ) and 2.81 (MGS). Excellent ΛCDM agreement: The LOWZ modeling gap is only 0.01 (full-sky estimate 2.89 vs. ΛCDM prediction 2.90), indicating near-perfect agreement after geometry correction. This cross-sample validation—achieved across different galaxy populations, redshift ranges, and survey footprints—confirms that the geometry correction methodology is robust and generalizable. The convergence of full-sky D2 estimates from independent samples strengthens our conclusion that the cosmological principle of large-scale homogeneity is upheld. Updated data products include LOWZ D2 measurements, BOSS random catalog analysis, combined comparison figures, and summary tables. Version 3 Add public git repo with all analysis tools under active development with MIT License.