A Theoretically Innovative MOG Theory

Canadian physicist John Moffat's MOG modified gravity theory is a long standing tensor, vector, scalar modification of General Relativity (GR). As the link explains:

Scalar–tensor–vector gravity theory, also known as MOdified Gravity (MOG), is based on an action principle and postulates the existence of a vector field, while elevating the three constants of the theory to scalar fields. In the weak-field approximation, STVG produces a Yukawa-like modification of the gravitational force due to a point source. Intuitively, this result can be described as follows: far from a source gravity is stronger than the Newtonian prediction, but at shorter distances, it is counteracted by a repulsive fifth force due to the vector field.

STVG has been used successfully to explain galaxy rotation curves, the mass profiles of galaxy clusters, gravitational lensing in the Bullet Cluster, and cosmological observations without the need for dark matter. On a smaller scale, in the Solar System, STVG predicts no observable deviation from general relativity. The theory may also offer an explanation for the origin of inertia.

Yukawa forces are forces carried by massive mediator bosons (in contrast to the massless mediator boson of electromagnetism, the photon, which has an infinite range as a result), whose range is a function of the mediator mass. 

The most familiar example of a Yukawa force is the nuclear binding force (sometimes called the residual strong force) that holds photons and neutrons in atomic nuclei together, which is mediated by like composite mesons, especially pions (neutral pions have a mass of about 135 MeV, while charged ones have a mass of about 140 MeV) that have an effective range on the order of femtometers, which is similar to the size of an atomic nucleus.

In contrast, GR without a cosmological constant (including Deur's approach to explaining dark matter phenomena as gravitional) is a tensor theory, and GR with a cosmological constant is a tensor-scalar theory. Newtonian gravity is a scalar theory. Several of the main relativistic generalizations of MOND are also tensor, vector, scalar theories.

MOG, while not the subject of as much scholarship as MOND (Israeli physicist Mordehai Milgrom's 1983 non-relativistic toy model modification of Newtonian gravity that does a good job of replicating dark matter phenomena is almost near equilibrium systems of galaxy size or smaller), MOG is still one of the older modified gravity theories, has received considerable investigation from scientists other than its inventor, is relativistic, is more easily generalized to cosmology scale problems, and unlike MOND, models galaxy cluster phenomena often attributed to dark matter more successfully, at the cost of being somewhat less intuitive to understand.

Moffat's latest short paper formulates his MOG theory in a manner, that while essentially identical to the original, is easier to apply to cosmology scale questions.

We develop a Stueckelberg gauge-invariant formulation of modified gravity (MOG). 

The massive vector field is made gauge-invariant by introducing a compensating scalar field, without requiring a Higgs field, spontaneous symmetry breaking, or a vacuum expectation value to fix the effective Newtonian gravitational coupling. This separates the gauge-invariant origin of the vector mass from the cosmological evolution of the gravitational coupling. 

The formulation preserves the finite-range vector interaction of MOG, while allowing the effective gravitational coupling to be treated as an independent scalar or scale-dependent quantity. This distinction is important for cosmological tests, since early-universe constraints and late-time large-scale gravitational phenomena need not be tied to a symmetry-breaking vacuum. The Stueckelberg formulation provides a gauge-invariant framework for comparing MOG with nucleosynthesis, cosmic microwave background, large-scale structure, lensing, and distance data.

John W. Moffat, "Stueckelberg Gauge Invariant Formulation of MOG" arXiv:2606.26427 (June 4, 2026).

Another new MOG paper constrains the value of one of that theory's physical constants (to a value inconsistent with the range in the previous literature on the topic):

The scalar-tensor-vector-gravity (STVG), a prototype of modified gravity developed by Moffat, can correctly explain galaxy rotation curves, cluster dynamics, Bullet Cluster phenomena and cosmological data without invoking the observationally elusive general relativistic (GR) dark matter. Further, recent observations of neutron star masses are shown to defy some GR predictions, whereas STVG turns out to be more consistent with those observations. These successes indicate that STVG could be a potential candidate for a new theory of gravity. 

However, an important question concerns the possible range of values of the STVG dimensionless parameter α imposed by various physical scenarios. In the literature, the range 0.03 < α < 2.47 corresponding to different central source masses has been suggested. We show here that the α can be considerably constrained into the range 0 < α < 10^−5 assuming that the updated GPS fluctuation does not exceed the α-dependent correction to the terrestrial Sagnac delay.

R. Kh. Karimov, R. N. Izmailov, K. K. Nandi, "Terrestrial Sagnac delay in scalar-tensor-vector-gravity" arXiv:2606.27033 (June 25, 2026).

A footnote on f(R) gravity

Probably the other modified gravity theory with significant scholarship from multiple astrophysicists that is most often used to explain dark matter phenomena gravitationally is f(R) gravity (the image below is from this link), which like GR with a cosmological constant, and unlike MOG or some relativistic generalizations of MOND, is a tensor-scalar theory. The way f(R) gravity modifies GR is not with an extra vector field, but with a higher order derivative term. The standard Ricci scalar R in the Einstein-Hilbert action is replaced by a general function of R (e.g., R + (alpha)*R^2). Mathematically and dynamically, this higher-order derivative theory is exactly equivalent to standard General Relativity coupled to a single, dynamical scalar field (known as the scalaron), rather than only having the static scalar dark energy field that is equivalent to the cosmological constant.

Like MOG, f(R) gravity has a Yukawa correction to the gravitational potential, which (at least in part, it also has a time and scale dependent gravitational constant) is how it can explain some or all dark matter phenomena without dark matter particles.