The U-Cell Model and eROSITA Galaxy Clusters: Isothermal Profiles, Beta Parameters, and the MDAR Transition

The U-Cell Model (UCM) predicts isothermal dark matter halo profiles ρ ∝ r⁻² from its substrate action. We compare three UCM predictions against eROSITA All-Sky Survey galaxy cluster data. First, for gas in hydrostatic equilibrium in an isothermal potential, the UCM predicts a Cavaliere–Fusco-Femiano beta parameter β_UCM = μ m_p v_flat²/(3 k_B T) ≈ 0.5–0.7 for typical cluster temperatures and velocity dispersions, consistent with the empirically measured range β ≈ 0.55–0.75 from eROSITA and XMM-Newton profiles. Second, the UCM acceleration scale g† = f_cold⁴ c H₀ = 1.207×10⁻¹⁰ m/s² predicts that the MDAR transition occurs at r ~ 0.3–0.5 Mpc in σ ~ 800 km/s clusters — the precise regime probed by Li et al. (2024) for 22 eROSITA clusters. Third, the UCM isothermal envelope extends to r_virial ≈ 8.02 × r_200; Lyskova et al. (2023) detect stacked emission to 2×r_200, consistent with the inner 25% of the UCM envelope. Crucially, while MOND fails at cluster scales by a factor 2–4 (no dark matter), the UCM has no such problem: it produces the same MDAR transition as observed in galaxies while simultaneously providing the substrate dark matter needed for cluster masses — unifying MOND's galactic successes with ΛCDM's cluster successes in a single parameter-free framework. The steeper-than-self-similar L_X–T relation from 3061 eROSITA clusters (Ramos-Ceja et al. 2025) is attributable to AGN feedback on top of the UCM isothermal gravitational background. The temperature profile is predicted to be flat at intermediate radii (0.2 r_500 < r < 0.8 r_500), with a mandatory decline toward r_200 due to accretion shocks.