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A low-acceleration modification associated with Mordehai Milgrom. For an isolated spherical system, $g\mu (g/a_0)=g_N$, with $\mu (x)\to 1$ at high $x$ and $\mu (x)\to x$ at low $x$. An isolated point-like baryonic distribution then gives $v_c^4=G{M}_b{a}_0$ and a flat asymptotic galaxy rotation curve. The algebraic spherical relation is not a complete relativistic field theory and should not be assumed to hold vectorially for arbitrary mass distributions.^[ref]

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 $\mu (x)\to 1$ at high acceleration and $\mu (x)\to x$ at low acceleration. For an isolated baryonic mass $M_b$ in the latter regime,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.^[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.^[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.^[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. 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.