MOND and Dark Matter Are Not Tested by the Same Data — Epoche C2
It is common to hear that modified Newtonian dynamics and cold dark matter are two answers to one question, and that the evidence has already chosen. That summary is wrong in a specific and instructive way. The two proposals are not evaluated against the same observations. Each is strongest exactly where the other has least to say, and the honest comparison has to be made scale by scale. Units here follow the astrophysical literature: distances in kiloparsecs (kpc) and megaparsecs (Mpc), masses in solar masses $M_\odot$, speeds in km s$^{-1}$. Accelerations are quoted in SI, in m s$^{-2}$, as Famaey and McGaugh do. What MOND actually delivers Milgrom's 1983 proposal is one sentence long: below a universal acceleration scale $a_0$, the acceleration $a$ felt by a test particle is no longer equal to the Newtonian value $a_N = GM/r^2$ but satisfies $a^2/a_0 = a_N$ in the deep low-acceleration limit. The consequence follows in two lines. Setting the centripetal acceleration $v^2/r$ equal to $a = \sqrt{GMa_0}/r$ gives $$v^4 = G M_b a_0,$$ where $v$ is the asymptotic circular speed, $M_b$ the total baryonic mass and $G$ Newton's constant. The radius has cancelled: the rotation curve is flat, and its height depends only on the mass of the stars and gas. This is the baryonic Tully-Fisher relation, and MOND predicts it with no scatter at all. Put numbers in. The Milky Way's baryonic mass is about $6\times10^{10}\,M_\odot$, which is $6\times10^{10}\times1.989\times10^{30} = 1.19\times10^{41}$ kg. With $G = 6.674\times10^{-11}$ m$^3$ kg$^{-1}$ s$^{-2}$ and $a_0 = 1.2\times10^{-10}$ m s$^{-2}$, we get $v^4 = 7.97\times10^{30}\times1.2\times10^{-10} = 9.56\times10^{20}$ m$^4$ s$^{-4}$, so $v = 1.76\times10^{5}$ m s$^{-1} = 176$ km s$^{-1}$. The observed outer circular speed of our Galaxy is of order 180 km s$^{-1}$. One number, no fitting. The strongest version of this evidence is the radial acceleration relation. Across 153 rotating galaxies and some 2,700 independent radii, the measured centripetal acceleration is a tight single-valued function of the acceleration computed from the visible matter alone, with a root-mean-square scatter of about 0.13 dex — a factor $10^{0.13} = 1.35$, or 35%, which is close to what the observational errors alone would produce. Consider what that means for the halo picture. A dark halo is fitted to each galaxy with roughly two free parameters, on top of a stellar mass-to-light ratio; MOND uses the same mass-to-light ratio plus one universal constant. For 153 galaxies the counts are about $3\times153 = 459$ against $153+1 = 154$, a factor of three. Worse for the halo picture, nothing in it explains why the fitted halo should conspire with the baryons it is supposed to dominate. One further coincidence is hard to dismiss. Taking $H_0 = 70$ km s$^{-1}$ Mpc$^{-1}$ $= 2.27\times10^{-18}$ s$^{-1}$, the combination $cH_0/2\pi = (3.00\times10^{8}\times2.27\times10^{-18})/6.28 = 1.08\times10^{-10}$ m s$^{-2}$, which differs from $a_0$ by 11%. Whether this is a clue or an accident is unresolved. What dark matter is required by Now change scale, and the evidence changes character entirely. The cosmic microwave background records acoustic oscillations in the photon-baryon fluid before recombination. Odd-numbered peaks are compressions, even-numbered ones rarefactions, and the relative heights measure how much gravitating matter does not couple to photons. In a baryon-only universe the decaying gravitational potentials drive the oscillations and successive peaks are strongly damped; a pressureless component holds the wells open and keeps the third peak high. Planck's fit gives $\Omega_b h^2 = 0.0224$ for baryons and $\Omega_c h^2 = 0.120$ for cold dark matter — a ratio of $0.120/0.0224 = 5.4$. The total, $\Omega_m h^2 = 0.143$ with $h = 0.674$, gives $\Omega_m = 0.143/0.674^2 = 0.315$. This is a measurement of a mass component at redshift 1100, where MOND, as a modification of non-relativistic dynamics, makes no prediction at all. Clusters of galaxies are the second front, and here MOND does not merely fall silent — it fails on its own terms. Applying MOND to cluster dynamics reduces the missing mass but does not remove it: a residual discrepancy of roughly a factor of two remains, which MOND must attribute to some undetected matter. A theory built to abolish dark matter needs dark matter in clusters. The Bullet Cluster sharpens this. Two clusters have passed through each other; the hot gas, which carries most of the baryonic mass and is visible in X-rays, was slowed by ram pressure and now sits between the two galaxy concentrations, while the weak-lensing mass peaks track the galaxies. Lensing measures the total gravitating mass along the line of sight, and it peaks where the baryons are not. Observation MOND Cold dark matter Flat rotation curves, radial acceleration relation predicted from one constant fitted per galaxy Third acoustic peak of the CMB no prediction without a relativistic theory predicted, and fitted to sub-percent accuracy Cluster masses factor $\sim2$ still missing accounted for Bullet Cluster offset requires a collisionless component natural Laboratory detection of the particle not applicable none, after four decades Where the comparison is genuinely undecided Two qualifications keep this from being a rout. First, the particle has not been found. Direct-detection experiments have pushed the spin-independent cross-section for a weakly interacting massive particle scattering off a nucleon below $10^{-47}$ cm$^2$ near 40 GeV$/c^2$ — the astro-particle convention quotes cross-sections in cm$^2$ and masses in GeV$/c^2$ — which excludes most of the parameter space that originally motivated the candidate. Second, the Bullet Cluster's inferred collision speed is uncomfortably high for structure formation in a cold dark matter universe, and at least one merging cluster, Abell 520, shows a lensing peak sitting on the gas rather than the galaxies. Neither point overturns t