Modified gravity is a theoretical framework proposing alterations to general relativity to explain phenomena like galaxy rotation without dark matter.Modified gravity — a class of theories proposing to change the law of gravitation on scales of galaxies and clusters, in order to explain the observed rotation without hypothetical dark matter. In standard cosmology, dark matter adds the missing mass; in alternative approaches, the geometry of spacetime or the potential itself is revised. The best-known model is MOND (Modified Newtonian Dynamics), introducing a threshold acceleration a₀ ≈ 1.2×10⁻¹⁰ m/s², below which the gravitational force transitions to a different regime.
The problem of missing mass was first noticed by Zwicky in 1933 in the Coma Cluster: the velocity dispersion of galaxies indicated a mass many times greater than the visible. In the 1970s, Vera Rubin systematically measured flat rotation curves of spiral galaxies, which were traditionally interpreted as evidence of a massive dark halo. In 1983, Mordehai Milgrom proposed MOND, which reproduced these curves without dark matter, introducing a single constant. Jacob Bekenstein in 2004 developed TeVeS — a relativistic extension of MOND, consistent with lensing and cosmology.
In the Newtonian limit F = G M m / r², but when a ≪ a₀ (typical value for galaxy outskirts) the effective force in MOND becomes F ≈ m a₀ / √(a/a₀) = m √(a₀ a). For a test body at distance r from central mass M, this yields v ≈ (G M a₀)^(1/4) — constant rotation velocity, explaining flat curves. In relativistic versions, the field equations are modified: instead of standard Einstein equations, additional metrics or scalars are introduced. The equivalence principle is usually preserved in the weak field. On galaxy cluster scales, even modified gravity often requires some amount of ordinary dark matter (e.g., sterile neutrinos).
Description of galaxy rotation without adding invisible mass; an alternative model for dark matter.
MOND accurately predicted the rotation curves of many galaxies, but it has not yet succeeded in explaining cluster collisions, as in the case of the Bullet Cluster.
| a_0 | variable | Characteristic acceleration scale — acceleration metre per second squared |
| c | constant | Speed of light in vacuum — speed of light = 299792458 metre per second |
| \ell | constant | A characteristic length scale of the modified gravity theory — length scale metre |
| F | variable | Gravitational force magnitude — force newton |
| M | variable | Mass of the central object — mass kilogram |
| m | variable | Mass of the test particle — mass kilogram |
| r | variable | Distance between masses — distance metre |
| ρ | variable | Energy density of the cosmic fluid — energy density joule per cubic metre |
| w | variable | Equation-of-state parameter of the fluid — equation of state parameter |
| G | constant | Universal constant of gravitation — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |













