Modified Gravity

Ever since it has been established in the 1930s that Newton’s or Einstein’s theory cannot account for the orbital velocities stars in the outer regions of a spiral galaxy, there has been an interest in modifying gravity theory.

If successful, a modified gravity theory would do away with the need for dark matter. Instead, the anomalous velocities of stars far from a galactic center would be accounted for by a deviation from Newton’s or Einstein’s predictions at large distances.

A successful modified gravity theory has to clear many hurdles, among them:

• Stability: A gravity theory is not very useful if it cannot account for the stability of stars and planets, planetary systems, or galaxies.
• Solar system tests: Gravity theory has been tested to high precision in terrestrial laboratories and within the solar system.
• Galaxy rotations: Galaxy rotations follow a simple empirical relationship known as the Tully-Fisher law. To do away with dark matter, a modified gravity theory must be able to predict this relationship.
• Galaxy cluster masses: The motion of galaxies in gravitationally bound galaxy clusters must be accounted for by a cluster mass profile that does not require matter in excess of the known amount of baryonic matter.
• Gravitational lensing: The lensing profile of galaxy clusters has to be accounted for in a manner consistent with the visible distribution of matter (stars and galaxies, X-ray emitting gas).
• Cosmology: Precision measurements of the cosmic microwave background, measurements of the mass power spectrum through the distribution of galaxies, and the luminosity-distance relationship of Type Ia supernovae are three important observational tests of cosmology. The standard model of cosmology can account for these, but a modified gravity theory may have difficulties.

Yes, Einstein had it easy a hundred years ago. Today, with all the observational data at hand, the bar is set much higher. Why, we might ask, would one even bother with a modified gravity theory then, if the standard model of cosmology works so well? Simple: in the standard model, 96% of the universe is invisible, undetectable, or both. Surely that is a good enough reason.

Without further ado, there is a modified gravity theory that successfully meets these challenges (at least as far as we know.) It is called MOG, and it grew out of research by Moffat into nonsymmetric gravitational theory. MOG is a field theory. Its basic postulate is a field theory Lagrangian that, in addition to the standard Einstein-Hilbert term, incorporates a massive vector field. This massive vector field results in a repulsive gravitational force that cancels out gravity at short range. In other words, rather than making gravity stronger at large distances, MOG does the opposite: it assumes that gravity was stronger all along, but at shorter distances (galactic, subgalactic, or smaller scales) much of that strength is canceled out by the repulsive force.

The mass of the MOG vector field and its strength are governed by two constants. In MOG, these two constants, as well as Newton’s gravitational constant G are running; they are promoted to scalar fields. Thus the full MOG Lagrangian consists of the Einstein-Hilbert tensor term describing ordinary gravity; the massive vector term describing the repulsive force; three massless scalar fields; and potentials associated with the vector field and the three scalar fields. (For this reason, MOG has also been known by the acronym STVG, which stands for Scalar-Tensor-Vector Gravity theory.)

The result is a theory that is in good agreement with observation at scales ranging from the laboratory all the way to cosmology. In the solar system, and up to the scale of star clusters, MOG predicts no observable deviation from Einstein gravity. On the scale of galaxies, MOG correctly predicts the Tully Fisher law of galaxy rotation curves. On even larger scales, MOG is consistent with the lensing and mass distribution of galaxy clusters. MOG is fitted to the precision acoustic power spectrum of the cosmic microwave background, it is consistent with the luminosity-distance relationship of Type Ia supernovae, and it correctly reproduces the matter power spectrum of galaxy distributions. The latter provides an especially notable possible test of the theory: at soon-to-be-achieved data resolutions, the presence or absence of baryonic oscillations in the matter power spectrum can once and for all determine whether or not collisionless dark matter dominates the universe.

For most scenarios, the complicated field equations of MOG can be reduced to a relatively simple acceleration law that reads:

$$\frac{d^2r}{dt^2}=-\frac{GM}{r^2}+\frac{G-G_N}{r^2}M\left(1+\frac{r}{r_0}\right)e^{-r/r_0},$$

where \(G_N\) is the Newtonian acceleration, while the parameters \(G\) and \(r_0\) are given by:

\begin{align}
G&=G_N+(G_\infty-G_N)\frac{M}{(\sqrt{M}+E)^2},\qquad(G_\infty\simeq 20G_N, ~E\simeq 25000~M_\odot^{1/2}),\\
r_0&=\frac{\sqrt{M}}{D},\qquad(D\simeq 6250~M_\odot^{1/2}{\rm kpc}^{-1}),
\end{align}

and can be plotted as follows:

MOG and its physical consequences are worked out in a series of papers by Moffat and others, dating back to 1979. Some of the more important and current papers are:

• Nonsymmetric Gravitational Theory.
  J. W. Moffat, Phys. Lett. B, Vol 355, No 3-4, 447-452 (1995).
  NGT was the original version of the theory.
• Galaxy Rotation Curves Without Non-Baryonic Dark Matter.
  J. R. Brownstein and J. W. Moffat, The Astrophysical Journal, Volume 636, Issue 2, pp. 721-741.
  This paper describes the immediate predecessor of STVG, MSTG (Metric-Skew-Tensor-Gravity), and applies its observational consequences (which are the same as those of STVG) to galaxy rotation, finding good agreement with the data of over 100 galaxies.
• Galaxy cluster masses without non-baryonic dark matter.
  J. R. Brownstein and J. W. Moffat. Monthly Notices of the Royal Astronomical Society, Volume 367, Issue 2, pp. 527-540.
  Still using MSTG, this paper derives the mass profile of galaxy clusters, finding good agreement with data from over 100 clusters.
• Scalar tensor vector gravity theory.
  J. W. Moffat. Journal of Cosmology and Astroparticle Physics, Issue 03, pp. 004 (2006).
  This paper introduces the STVG version of MOG.
• The Bullet Cluster 1E0657-558 evidence shows modified gravity in the absence of dark matter.
  J. R. Brownstein and J. W. Moffat. MNRAS 382 (2007)  29-47.
  This paper, which earned a press release by the Royal Astronomical Society, offers detailed calculations that demonstrate that the so-called Bullet Cluster, supposedly a “smoking gun” proof that dark matter exists, in fact can be easily accounted for using MOG.
• Modified Gravity: Cosmology without dark matter or Einstein’s cosmological constant.
  J. W. Moffat and V. T. Toth. eprint arXiv:0710.0364.
  In this paper, the cosmological consequences of MOG are explored. MOG is found to be consistent with the acoustic power spectrum of the cosmic microwave background, the luminosity-distance relationship of Type Ia supernovae, and the matter power spectrum.
• Fundamental parameter-free solutions in Modified Gravity.
  J. W. Moffat and V. T. Toth. Class. Quant. Grav. 26 (2009) 085002.
  This paper offers a solution of the MOG field equations that no longer relies on fitted parameters, putting MOG on the route towards becoming a theory with real predictive power. This is a key paper that contains our most important results.
  
• The bending of light and lensing in modified gravity.
  J. W. Moffat and V. T. Toth. eprint arXiv:0805.4774. MNRAS 397 (2009) 1885-1992.
  This paper explores in detail the relativistic bending of light in a MOG field. It takes a few tentative steps towards deriving a MOG solution for an extended distribution of matter.
• Modified gravity and the origin of inertia.
  J. W. Moffat and V. T. Toth. MNRAS Letters 395 (2009) L25-L28.
  One curious consequence of MOG is that the theory violates Birkhoff’s theorem: a test particle inside a spherically symmetric shell of matter experiences a force. Can this force account for inertia, in line with Mach’s principle? This paper explores this possibility and finds a possible violation of the equivalence principle at very small accelerations that may be experimentally testable.