The MOG weak field approximation and observational test of galaxy
  rotation curves
  As an alternative to dark matter models, MOdified Gravity (MOG) theory can
compensate for dark matter by a covariant modification of Einstein gravity. The
theory introduces two additional scalar fields and one vector field. The aim is
to explain the dynamics of astronomical systems based only on their baryonic
matter. The effect of the vector field in the theory resembles a Lorentz force
where each mass has a charge proportional to the inertial mass. In this work,
we obtain the weak field approximation of MOG by perturbing the metric and the
fields around Minkowski space-time. We derive an effective gravitational
potential which yields the Newtonian attractive force plus a repulsive Yukawa
force. This potential, in addition to the Newtonian gravitational constant,
$G_N$, has two additional constant parameters $\alpha$ and $\mu$. We use the
THINGS catalog of galaxies and fix the two parameters $\alpha$ and $\mu$ of the
theory to be $\alpha =8.89 \pm 0.34$ and $\mu =0.04 \pm 0.004 {\rm kpc}^{-1}$.
We then apply the effective potential with the fixed universal parameters to
the Ursa-Major catalog of galaxies and obtain good fits to galaxy rotation
curve data with an average value of $\bar{\chi^2} = 1.07 $. In the fitting
process, only the stellar mass-to-light ratio $(M/L)$ of the galaxies is a free
parameter. As predictions of MOG, our derived $M/L$ is shown to be correlated
with the color of galaxies, and we fit the Tully-Fisher relation for galaxies.
As an alternative to dark matter, introducing an effective weak field potential
for MOG opens a new window to the astrophysical applications of the theory.
