================================================================================ GALAXY ROTATION CURVES & DARK MATTER PROBLEM — PUBLISHED DATA REFERENCE ================================================================================ Compiled: 2026-03-18 Purpose: Empirical targets for TLT geometric framework Status: Numbers from published peer-reviewed papers ================================================================================ ############################################################################## 1. THE RADIAL ACCELERATION RELATION (RAR) ############################################################################## SOURCE: McGaugh, Lelli & Schombert (2016) "Radial Acceleration Relation in Rotationally Supported Galaxies" Physical Review Letters, 117, 201101 arXiv: 1609.05917 EXACT FORMULA: g_obs = g_bar / (1 - exp(-sqrt(g_bar / g_dagger))) where g_dagger is the ONLY free parameter. BEST-FIT ACCELERATION SCALE: g_dagger = (1.20 +/- 0.02 (random) +/- 0.24 (systematic)) x 10^-10 m/s^2 DATA SCOPE: - 2693 individual data points - 153 galaxies - Galaxy types: spirals, irregulars — "very different morphologies, masses, sizes, and gas fractions" SCATTER: - Observed scatter: 0.13 dex (total) - Intrinsic scatter: ~0.057 dex (after removing observational errors) - The scatter is DOMINATED by observational uncertainties (distance errors are the primary source) - This is remarkably tight for an astrophysical relation BEHAVIOR: - At high accelerations (g_bar >> g_dagger): g_obs -> g_bar (1:1 line, no dark matter needed, pure Newtonian) - At low accelerations (g_bar << g_dagger): g_obs -> sqrt(g_bar * g_dagger) (the deep-MOND regime) - Transition occurs at g ~ 10^-10 m/s^2 KEY IMPLICATION FOR TLT: The dark matter contribution is FULLY SPECIFIED by the baryonic distribution. Knowing only the visible mass tells you everything about the "dark matter." This is a ONE-PARAMETER relation with essentially zero intrinsic scatter. ------------------------------------------------------------------------------ EXTENDED RAR — Lelli, McGaugh, Schombert & Pawlowski (2017) "One Law to Rule Them All: The Radial Acceleration Relation of Galaxies" Astrophysical Journal, 836, 152 arXiv: 1610.08981 - Extended to 240 galaxies with spatially resolved kinematics - 153 late-type galaxies (spirals + irregulars) - 25 early-type galaxies (ellipticals + lenticulars) - 62 dwarf spheroidals - Spans 9 dex in stellar mass - ALL morphological types follow the SAME relation - Scatter remains <= 0.13 dex - The relation holds over 4 dex in acceleration ############################################################################## 2. BARYONIC TULLY-FISHER RELATION (BTFR) ############################################################################## SOURCE: McGaugh (2012) "The Baryonic Tully-Fisher Relation of Gas Rich Galaxies as a Test of LCDM and MOND" Astronomical Journal, 143, 40 arXiv: 1107.2934 FORMULA: M_bar = A * V_flat^4 NORMALIZATION: A = 47 +/- 6 M_sun km^-4 s^4 SLOPE (from linewidth-based measurements): alpha = 3.41 +/- 0.08 (McGaugh 2012, linewidth) SCATTER: sigma_perp ~ 0.06 dex (total, linewidth-based) This is consistent with being ENTIRELY due to observational errors. MASS RANGE: M_bar ~ 10^7 to 10^11.5 M_sun (over 4.5 dex) ------------------------------------------------------------------------------ REFINED BTFR — Lelli, McGaugh & Schombert (2019) MNRAS, 484, 3267 arXiv: 1901.05966 Using 153 SPARC galaxies with V_flat (velocity along flat part): - Slope: 3.85 +/- 0.09 - Intrinsic orthogonal scatter: 0.026 +/- 0.007 dex - Stellar M/L assumed: Y_star = 0.50 +/- 0.13 M_sun/L_sun at [3.6] CRITICAL: The BTFR is MORE fundamental than the Fall relation (angular momentum-mass relation), which has ~7x more scatter. MOND PREDICTION FOR BTFR: MOND predicts M_bar proportional to V_flat^4 with: A = a0 / (G * constant) Both the slope (exactly 4) and the zero-point are predicted by MOND from a single parameter (a0). WHAT THIS MEANS FOR TLT: The rotation velocity of a galaxy is determined ENTIRELY by its baryonic mass. No free parameters from dark matter halos. A geometric theory must reproduce this v^4 scaling naturally. ############################################################################## 3. MOND (MODIFIED NEWTONIAN DYNAMICS) ############################################################################## SOURCE: Milgrom (1983) "A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis" Astrophysical Journal, 270, 365 FUNDAMENTAL EQUATION: mu(|a|/a0) * a = a_N where: a = true gravitational acceleration a_N = Newtonian acceleration (from visible mass only) a0 = characteristic acceleration scale mu = interpolation function ACCELERATION SCALE: a0 = 1.2 x 10^-10 m/s^2 (Begeman, Broeils & Sanders 1991: a0 ~ 1.2 x 10^-10 m/s^2 optimal fit) INTERPOLATION FUNCTIONS (several empirically acceptable forms): Standard: mu(x) = x / sqrt(1 + x^2) [Milgrom 1983] Simple: mu(x) = x / (1 + x) [Famaey & Binney 2005] RAR form: mu(x) = 1 - exp(-x) [McGaugh et al. 2016] All satisfy: mu(x) -> 1 for x >> 1 (Newtonian regime) mu(x) -> x for x << 1 (deep-MOND regime) where x = |a| / a0 DEEP-MOND LIMIT (a << a0): a = sqrt(a_N * a0) Equivalently: a = sqrt(GM * a0) / r This gives: V_flat = (G * M * a0)^(1/4) [flat rotation curves] Which gives: M = V^4 / (G * a0) [Tully-Fisher, slope exactly 4] NEWTONIAN LIMIT (a >> a0): a = a_N = GM/r^2 [standard Newton] ------------------------------------------------------------------------------ WHAT MOND GETS RIGHT (a priori predictions, confirmed later): 1. Flat rotation curves of isolated disk galaxies (PREDICTED) 2. Baryonic Tully-Fisher relation: M ~ V^4 (PREDICTED, both slope AND normalization) 3. Radial Acceleration Relation with single parameter a0 (PREDICTED) 4. Low surface brightness galaxies have larger mass discrepancies at all radii (PREDICTED before observed) 5. Shape of rotation curves predicted from light distribution alone (PREDICTED — routinely confirmed for individual galaxies) 6. Features in baryonic distribution correspond to features in rotation curve (PREDICTED — Renzo's rule) 7. No dark matter needed in high-surface-brightness regions where a > a0 (CONFIRMED) 8. The mass discrepancy begins precisely where a drops below a0 (CONFIRMED across all galaxy types) ------------------------------------------------------------------------------ WHAT MOND GETS WRONG (known failures): 1. GALAXY CLUSTERS: Factor of 2-3x residual mass discrepancy remains even with MOND. Clusters still need ~2x more mass than visible. The Bullet Cluster offset between gravitational lensing center and X-ray gas center is a challenge (though MOND does reproduce the offset qualitatively). 2. CMB POWER SPECTRUM: Pure MOND cannot reproduce the CMB angular power spectrum without additional mechanisms. Skordis & Zlosnik (2021) proposed a relativistic MOND model compatible with CMB, but it requires multiple extra fields. 3. GRAVITATIONAL LENSING: Even relativistic MOND models struggle to match observed strong gravitational lensing in some systems. 4. ULTRA-FAINT DWARF GALAXIES: MOND underestimates velocity dispersions, sometimes by >10x (observed ~10 km/s vs MOND prediction <1 km/s). However, these may be out of equilibrium. 5. No identified dark matter particle has been found, but MOND also has no complete relativistic theory without extra fields. 6. Solar system constraints: The simple interpolation function predicts corrections that conflict with Mercury's perihelion precession and Cassini tracking data. The standard function provides a sharper transition that avoids this. ------------------------------------------------------------------------------ THE a0 — COSMOLOGY COINCIDENCE: a0 ~ cH0 / (2*pi) ~ c^2 * sqrt(Lambda) / (2*pi) Numerically: cH0 = (3 x 10^8 m/s)(2.2 x 10^-18 s^-1) = 6.6 x 10^-10 m/s^2 cH0 / (2*pi) ~ 1.05 x 10^-10 m/s^2 (compare a0 = 1.2 x 10^-10) This "coincidence" may point to a DEEP connection between: - Local galactic dynamics (a0) - Cosmological expansion (H0) - Dark energy / cosmological constant (Lambda) Verlinde (2016) derived a0 = cH0/6 = 1.1 x 10^-10 m/s^2 from de Sitter space entropy arguments. FOR TLT: If the lattice geometry naturally produces an acceleration scale linked to cosmological parameters, this would be profound. ############################################################################## 4. SPARC DATABASE ############################################################################## SOURCE: Lelli, McGaugh & Schombert (2016) "SPARC: Mass Models for 175 Disk Galaxies with Spitzer Photometry and Accurate Rotation Curves" Astronomical Journal, 152, 157 arXiv: 1606.09251 Data: https://astroweb.case.edu/SPARC/ DATABASE CONTENTS: - 175 nearby late-type galaxies (spirals and irregulars) - Spitzer 3.6 micron photometry (traces stellar mass) - HI 21cm + H-alpha rotation curves (traces gravitational potential) - Morphologies: S0 to Irr - Luminosity range: ~5 dex (10^7 to 10^12 L_sun) - Surface brightness range: ~4 dex - Stellar mass-to-light ratio: Y_star ~ 0.5 M_sun/L_sun at [3.6] (from stellar population models) - Maximum-disk limit: Y_star ~ 0.7 M_sun/L_sun at [3.6] KEY RESULTS FROM SPARC ANALYSIS: 1. Gas fraction linearly correlates with total luminosity 2. Transition from star-dominated to gas-dominated galaxies corresponds to spiral -> dwarf irregular transition 3. V_bar/V_obs (baryonic-to-observed velocity ratio) varies with luminosity and surface brightness: - High-mass, high-surface-brightness: nearly maximal (V_bar ~ V_obs) - Low-mass, low-surface-brightness: submaximal (V_bar << V_obs) 4. HI mass-radius relation is extremely tight (even though stellar mass-HI mass relation has significant scatter) WHERE DARK MATTER DOMINATES (from SPARC and general observations): - In high-surface-brightness spirals: DM begins to dominate at ~2-3 disk scale lengths (R_d), where accelerations drop below a0 - In low-surface-brightness galaxies: DM dominates at ALL radii (the entire galaxy is in the low-acceleration regime) - The TRANSITION from baryon-dominated to DM-dominated is smooth (the "disk-halo conspiracy") — this smoothness is unexplained in CDM but natural in MOND/geometric theories BIG-SPARC (in development as of 2024): ~4000 galaxies from public archives (APERTIF, ASKAP, ATCA, GMRT, MeerKAT, VLA, WSRT) with WISE near-IR photometry ############################################################################## 5. DARK MATTER PROBLEMS (SMALL-SCALE CHALLENGES TO LCDM) ############################################################################## PRIMARY REVIEW: Bullock & Boylan-Kolchin (2017) "Small-Scale Challenges to the LCDM Paradigm" Annual Review of Astronomy and Astrophysics, 55, 343 arXiv: 1707.04256 Scale of challenges: < 1 Mpc, mass < 10^11 M_sun ------------------------------------------------------------------------------ 5A. CORE-CUSP PROBLEM PREDICTED (NFW simulations): rho(r) = rho_0 / [(r/r_s)(1 + r/r_s)^2] Inner slope: rho ~ r^(-1) ("cusp") Outer slope: rho ~ r^(-3) OBSERVED (dwarf galaxies): rho(r) ~ rho_0 / [1 + (r/r_c)^2] (pseudo-isothermal) Inner slope: rho ~ constant ("core") MAGNITUDE: NFW predicts central densities 3-10x higher than observed in typical dwarf galaxies. The discrepancy is most severe in the lowest-mass systems where baryonic feedback should be weakest. PROPOSED SOLUTIONS: Supernova feedback driving gas outflows that gravitationally heat DM orbits. But this requires very efficient energy coupling — more efficient than most simulations achieve. ------------------------------------------------------------------------------ 5B. MISSING SATELLITES PROBLEM PREDICTED: CDM simulations predict thousands of subhalos within the virial radius of a Milky Way-mass galaxy. Some estimates: > 10^11 subhalos total, hundreds with V_max > 10 km/s OBSERVED: ~60 confirmed satellite galaxies of the Milky Way (as of 2024-2025, growing from ~10 in 1999 to ~60 now) RECENT TWIST: Simulations now predict ~220 satellite galaxies, and new discoveries may push observed count above 500. The problem may be evolving or even reversing. STANDARD EXPLANATION: Most subhalos are too small to form stars (suppressed by UV reionization background after z ~ 6-10). ------------------------------------------------------------------------------ 5C. TOO-BIG-TO-FAIL PROBLEM SOURCE: Boylan-Kolchin, Bullock & Kaplinghat (2011, 2012) PROBLEM: The most massive predicted subhalos of a MW-mass galaxy (V_max ~ 30-70 km/s) are too dense to host any of the observed bright MW satellites. These halos should have formed stars — they are "too big to fail" at galaxy formation — yet nothing with those properties is observed. The observed bright satellites have V_max ~ 12-25 km/s, systematically below the predicted V_max of the most massive subhalos. ------------------------------------------------------------------------------ 5D. DIVERSITY PROBLEM SOURCE: Oman et al. (2015) MNRAS, 452, 3650 PROBLEM: Galaxies with SIMILAR maximum rotation velocities (V_max) show VERY DIFFERENT inner rotation curve shapes. CDM PREDICTION: At fixed V_max, the inner mass profile should be nearly universal (low scatter) — dark matter halos at fixed mass are self-similar. OBSERVATION: Some dwarfs at fixed V_max have steeply rising rotation curves (consistent with cuspy NFW), others have slowly rising curves (cores). The observed diversity far exceeds CDM predictions. MAGNITUDE: Inner enclosed masses vary by factor ~4-5 at fixed V_max, while CDM predicts factor ~1.5 variation. ------------------------------------------------------------------------------ 5E. THE "CONSPIRACY" PROBLEM (DISK-HALO CONSPIRACY) PROBLEM: In standard CDM, the dark matter halo and the baryonic disk form through different physical processes (hierarchical merging vs. gas cooling). There is NO REASON for the halo properties to correlate tightly with disk properties. YET: The transition from disk-dominated to halo-dominated rotation is remarkably smooth. The dark halo "knows about" the disk — its properties adjust to match the luminous component regardless of the galaxy's merger history. QUANTITATIVELY: The scatter in the disk-halo connection is much smaller than expected from the stochastic merger histories of CDM halos. FOR TLT: This is perhaps the STRONGEST argument for a geometric origin. If there is no dark matter — only geometry — there IS no conspiracy. The "halo" IS the geometry of the baryonic distribution. ------------------------------------------------------------------------------ 5F. RENZO'S RULE (SANCISI'S LAW) STATEMENT: "For any feature in a galaxy's luminosity profile, there is a corresponding feature in the rotation curve, and vice versa." CLASSIC EXAMPLE: NGC 1560 — a sizeable "kink" in the gas distribution between 4-6 kpc is mirrored in the total rotation curve, despite dark matter supposedly dominating at that radius. If DM dominates, baryonic features should be smoothed out. RECENT UPDATE (2025): A statistical analysis of SPARC galaxies finds the evidence for Renzo's rule is weaker than previously claimed — excess features in rotation curves lack clear baryonic counterparts in some cases. The rule may hold for individual striking cases but not universally in a statistical sense. FOR TLT: Even if not universal, the EXISTENCE of cases where baryonic features directly imprint on the total rotation curve is hard to explain if DM is a separate component. ############################################################################## 6. ANGULAR MOMENTUM AND ROTATION ############################################################################## SOURCE: Lelli, McGaugh, Schombert, Desmond & Katz (2018) "The angular momentum-mass relation: a fundamental law from dwarf irregulars to massive spirals" Astronomy & Astrophysics, 2018 arXiv: 1804.04663 THE FALL RELATION (j* - M* relation): j_star proportional to M_star^alpha EXPONENT: alpha = 0.55 +/- 0.02 SCATTER: sigma_perp = 0.17 +/- 0.01 dex (orthogonal intrinsic) [Note: some studies report sigma = 0.15 dex] - Holds as a SINGLE, UNBROKEN power law from dwarf irregulars to massive spirals (7 < log M_star/M_sun < 11.5) - Residuals correlate with morphology: earlier types below, later types above - NO EVOLUTION in the Fall relation from z=0 to z=1 COMPARISON WITH BTFR: - BTFR intrinsic scatter: 0.026 dex - Fall relation scatter: 0.17 dex (7x larger) - The BTFR is MORE fundamental than the Fall relation - Angular momentum is NOT the primary driver of rotation curves HALO SPIN PARAMETER: lambda = J * |E|^(1/2) / (G * M^(5/2)) From N-body simulations: - Median: ~ 0.035 - Distribution: log-normal with sigma(log10 lambda) ~ 0.25 CRITICAL FINDING: Galaxy spin (lambda_gal) and host halo spin (lambda_halo) are BARELY CORRELATED, especially at z >= 1. The null correlation reflects an anticorrelation between angular momentum retention factor and lambda_halo (from mergers and "wet compaction" phases). SPIRALS VS ELLIPTICALS: - Fall (1983): spirals and ellipticals follow parallel tracks in the j-M plane with identical slopes but different normalizations - Ellipticals have ~5x LESS specific angular momentum than spirals at the same stellar mass FOR TLT ON ANGULAR MOMENTUM: The BTFR being tighter than the Fall relation is important. It means that MASS (not angular momentum) is the primary quantity that determines rotation velocity. However, angular momentum does correlate with morphology and may encode geometric information about HOW the mass is distributed rather than how much there is. The spin parameter distribution (lambda ~ 0.035) is universal and emerges from tidal torques during structure formation — this could have a geometric interpretation in TLT. ############################################################################## 7. KEY NUMBERS SUMMARY ############################################################################## ACCELERATION SCALES: a0 (MOND) = 1.20 x 10^-10 m/s^2 cH0 = 6.6 x 10^-10 m/s^2 cH0/(2*pi) = 1.05 x 10^-10 m/s^2 Verlinde's derivation = cH0/6 = 1.1 x 10^-10 m/s^2 All within factor ~1.1 of each other COSMOLOGICAL MASS RATIOS (Planck 2018): Omega_b * h^2 = 0.0224 +/- 0.0001 (baryons) Omega_c * h^2 = 0.120 +/- 0.001 (cold dark matter) Omega_m = 0.315 +/- 0.007 (total matter) DM-to-baryon ratio = 0.120/0.0224 = 5.36:1 Visible mass fraction = ~16% of total matter DM fraction = ~84% of total matter (Or ~27% of total energy density, with ~68% dark energy) NFW PROFILE (Milky Way typical): rho(r) = rho_0 / [(r/r_s)(1 + r/r_s)^2] Milky Way: Virial mass: M_vir ~ 1.3 +/- 0.3 x 10^12 M_sun DM within 20 kpc: ~1.37 x 10^11 M_sun Concentration: c = r_vir/r_s ~ 10-15 Scale radius: r_s ~ 15-25 kpc Virial radius (r_200): ~200-250 kpc General concentration-mass relation: c ~ 10 for M ~ 10^12 M_sun (MW-like) c ~ 15 for M ~ 10^11 M_sun c ~ 5-8 for M ~ 10^14 M_sun (clusters) Higher mass -> lower concentration (less concentrated) DISK SCALE LENGTHS (typical spirals): Milky Way R_d ~ 2.5-3.5 kpc Range for spirals ~ 1-10 kpc DM starts dominating ~ 2-3 R_d (where a drops below a0) In low-surface-brightness galaxies: DM dominates at ALL radii BARYONIC TULLY-FISHER: M_bar = A * V_flat^4 A = 47 +/- 6 M_sun km^-4 s^4 Slope = 3.85 +/- 0.09 (empirical, Lelli 2019) MOND prediction: exactly 4 Intrinsic scatter: 0.026 dex RADIAL ACCELERATION RELATION: g_obs = g_bar / (1 - exp(-sqrt(g_bar/g_dagger))) g_dagger = 1.20 x 10^-10 m/s^2 Total scatter: 0.13 dex Intrinsic scatter: ~0.057 dex FALL RELATION (angular momentum): j_star proportional to M_star^0.55 Scatter: 0.17 dex Halo spin parameter: lambda ~ 0.035 (median) ############################################################################## 8. VERLINDE'S EMERGENT GRAVITY (2016) ############################################################################## SOURCE: Verlinde (2016) "Emergent Gravity and the Dark Universe" arXiv: 1611.02269 CORE IDEA: Gravity is emergent from quantum information (holographic principle). In de Sitter space, dark energy creates an entropy displacement when matter is present. The "elastic response" to this displacement produces an ADDITIONAL gravitational force — the "apparent dark matter." KEY DERIVATION: For spherical symmetry, the apparent dark matter enclosed mass M_D at radius r around baryonic mass M_B satisfies: M_D(r)^2 = (c * H0 / 6G) * M_B * r^2 Equivalently, the extra gravitational acceleration: g_D = sqrt(a0_V * g_N / 6) where a0_V = cH0 = 6.6 x 10^-10 m/s^2 This reproduces MOND in the deep acceleration regime. ACCELERATION SCALE DERIVED: a_M = cH0/6 = 1.1 x 10^-10 m/s^2 (Compare empirical a0 = 1.2 x 10^-10 m/s^2 — within 10%) WHAT IT REPRODUCES: - MOND phenomenology for spherically symmetric systems - The RAR functional form - The BTFR - Connection of a0 to cosmological parameters (H0, Lambda) WHAT IT DOES NOT DO: - No complete treatment for non-spherical (disk) systems - Galaxy cluster residuals still present - No derivation of CMB power spectrum - Needs more development for cosmological structure formation CONNECTION TO TLT: Verlinde's approach shares the philosophy that "dark matter" is not a substance but a geometric/entropic effect. The key difference is: Verlinde uses holographic entanglement entropy, while TLT would use lattice geometry. If TLT can derive the same a0 from its geometric structure, this would be a major result. ############################################################################## 9. WHAT TLT MUST REPRODUCE TO BE VIABLE ############################################################################## MINIMUM REQUIREMENTS (the empirical facts any theory must match): [ ] The RAR: g_obs = g_bar / (1 - exp(-sqrt(g_bar/a0))) with a0 ~ 1.2 x 10^-10 m/s^2 and scatter < 0.13 dex [ ] The BTFR: M_bar proportional to V_flat^4 (or close to 4) with intrinsic scatter < 0.03 dex [ ] Flat rotation curves at large radii for isolated disk galaxies [ ] The transition acceleration scale a0 ~ cH0/(2*pi) connects local dynamics to cosmology [ ] Low-surface-brightness galaxies are MORE dark-matter-dominated (in geometric terms: the geometric correction is larger) [ ] Features in baryonic distribution appear in rotation curves [ ] The diversity of rotation curve shapes at fixed V_max [ ] Galaxy cluster dynamics (MOND fails here — TLT must do better or explain why clusters are different) [ ] CMB angular power spectrum (the acoustic peaks require either dark matter or its geometric equivalent at z ~ 1100) [ ] Gravitational lensing by galaxies and clusters STRONG ADVANTAGES OF A GEOMETRIC APPROACH: - The "conspiracy" between disk and halo dissolves: there IS no halo - The RAR being a one-parameter relation with zero intrinsic scatter is NATURAL if there is only one thing (geometry) not two (baryons + dark matter) - The a0-cosmology connection suggests the acceleration scale is SET by the large-scale geometry (de Sitter radius, Hubble radius) - Renzo's rule is automatic: baryonic features MUST appear in the rotation curve because baryons ARE the only source POTENTIAL CHALLENGES FOR GEOMETRIC APPROACH: - Must explain galaxy clusters (where even MOND needs extra mass) - Must explain CMB power spectrum (baryonic acoustic oscillations require dark matter or geometric equivalent to suppress oscillations) - Must explain Bullet Cluster lensing offset from gas - Must be consistent with Big Bang nucleosynthesis Omega_b - Must explain structure formation timeline ############################################################################## 10. SOURCE BIBLIOGRAPHY ############################################################################## McGaugh, Lelli & Schombert (2016) — RAR Phys. Rev. Lett. 117, 201101 | arXiv:1609.05917 Lelli, McGaugh, Schombert & Pawlowski (2017) — Extended RAR ApJ 836, 152 | arXiv:1610.08981 Lelli, McGaugh & Schombert (2016) — SPARC database AJ 152, 157 | arXiv:1606.09251 Lelli, McGaugh & Schombert (2019) — BTFR velocity definitions MNRAS 484, 3267 | arXiv:1901.05966 Lelli, McGaugh, Schombert, Desmond & Katz (2018) — Fall relation A&A | arXiv:1804.04663 McGaugh (2012) — BTFR gas-rich galaxies AJ 143, 40 | arXiv:1107.2934 Milgrom (1983) — MOND original ApJ 270, 365 Milgrom (2020) — a0-cosmology connection arXiv:2001.09729 Verlinde (2016) — Emergent gravity arXiv:1611.02269 Bullock & Boylan-Kolchin (2017) — Small-scale challenges ARA&A 55, 343 | arXiv:1707.04256 Oman et al. (2015) — Diversity problem MNRAS 452, 3650 Boylan-Kolchin, Bullock & Kaplinghat (2011) — Too-big-to-fail MNRAS 415, L40 Navarro, Frenk & White (1996) — NFW profile ApJ 462, 563 Planck Collaboration (2018) — Cosmological parameters A&A 641, A6 | arXiv:1807.06209 Skordis & Zlosnik (2021) — Relativistic MOND + CMB Phys. Rev. Lett. 127, 161302 Fall (1983) — Angular momentum classification IAU Symposium 100 Famaey & McGaugh (2012) — MOND review Living Reviews in Relativity 15, 10 Renzo's Rule revisited (2025) — Statistical test MNRAS | arXiv:2508.03569 ================================================================================ END OF DATA COMPILATION ================================================================================