= Solution
The inference of <dark matter> uses a gravitational law to turn observed motions and lensing into a <mass> distribution. <modified gravity> asks whether some discrepancies instead reveal a breakdown of that law. The strongest motivation at galactic scales is the regular relation between baryonic structure and <galaxy rotation curves>, including approximately flat outer rotation speeds and the <baryonic Tully-Fisher relation>.
In <modified Newtonian dynamics>, the low-acceleration spherical relation is $g\,\mu(g/a_0)=g_N$, with $\mu(x)\to1$ at high acceleration and $\mu(x)\to x$ at low acceleration. For an isolated baryonic <mass> $M_b$ in the latter regime,
$$
\frac{v_c^4}{r^2a_0}=\frac{GM_b}{r^2},
\qquad
\boxed{v_c^4=GM_ba_0}.
$$
Thus an asymptotically flat rotation curve and a fourth-power <mass>–speed relation arise from a common acceleration scale, without assigning an independent halo profile to every <galaxy>. This is a substantive phenomenological motivation. Distances, stellar <mass-to-light ratios>, gas distributions and the external gravitational environment must still be controlled.<A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis by Mordehai Milgrom>[Astrophysical Journal 270, pp. 365–370, low-acceleration law and asymptotic rotation-speed prediction]{ref}
A nonrelativistic acceleration prescription does not itself specify light bending or cosmology. A relativistic completion must determine the <metric> followed by photons and matter. <tensor-vector-scalar gravity> illustrates the need for additional dynamical fields and a specified physical <metric>; lensing predictions then have to be calculated within that theory rather than imported unchanged from <general relativity>.<Relativistic gravitation theory for the MOND paradigm by Jacob Bekenstein>[Abstract and relativistic field construction, physical metric and lensing]{ref}
Several observations challenge a simple replacement of <dark matter> by a single galactic acceleration law. <galaxy clusters> retain substantial <mass> discrepancies even after such a modification. In the <bullet Cluster dark-matter separation>, the dominant X-ray-emitting baryonic gas is displaced from the galaxies, whereas weak-lensing <mass> concentrations lie nearer the galaxies. Collisionless <dark matter> accompanying the galaxies provides a natural account of this separation. The inference uses a lensing model and does not constitute a theorem excluding every conceivable modified-gravity theory, but a successful alternative must reproduce the displaced lensing signal quantitatively.<A direct empirical proof of the existence of dark matter by Douglas Clowe and others>[L109–L113, comparison of optical galaxies, X-ray gas and weak-lensing mass map]{ref}
The <cosmic microwave background>, acoustic structure, expansion history and growth of cosmic structure provide additional simultaneous tests. A theory that fits <galaxy rotation curves> must also supply the appropriate cosmological perturbation evolution and lensing. Adding extra clustering fields can help, but may reintroduce an effective unseen gravitating component. Solar-system and binary-pulsar agreement with <general relativity> constrain alternatives in their tested regimes, although a modification designed to act only at very low acceleration is not automatically ruled out by a high-acceleration test.
A discriminating programme compares dynamics, which probes the potential accelerating nonrelativistic matter, with lensing, which probes the sum of the two scalar <metric> potentials. Scale-dependent discrepancies, <gravitational slip> and environmental screening can then be tested with controlled <galaxy bias> and selection effects. Galaxy-scale tensions in a dark-matter model must likewise be separated from uncertain baryonic astrophysics. \b[Regular galactic scaling relations motivate <modified gravity>; clusters and the combined cosmological evidence strongly challenge its simplest dark-matter-free forms.] Failure to detect a particular <WIMP> is not positive evidence for a particular modified-gravity theory.
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