Fourth order gravity, scalar-tensor-vector gravity, and galaxy rotation curves

The lambda-CDM model is the best fit to cosmological data and to the observed galactic rotation curves. However, in the absence of a direct detection of dark matter one should explore theories such as modified Newtonian dynamics (MOND), and perhaps also modified gravity theories like fourth order gravity and scalar-tensor-vector gravity (STVG) as possible explanations for the non-Keplerian behavior of galaxy rotation curves. STVG has a modified law for gravitational acceleration which attempts to fit data by fixing two free parameters. We show that, remarkably, the biharmonic equation which we get in the weak field limit of the field equations in a fourth order gravity theory implies a modification of Newtonian acceleration which is precisely of the same repulsive Yukawa form as in the STVG theory, and the corrections could in principle be large enough to try and explain the observed rotation curves. We also explain how our model provides a first principles understanding of MOND. We also show that STVG and fourth order gravity predict an acceleration parameter $a_0$ whose value is of the same order as in MOND.

Figure 1

Correction to the inverse square law as described by Eq. (83): the solid curve depicts the Newtonian falloff given by the first term of (83), whereas the dashed curve depicts the correction given by the second term of (83).

Figure 2

Yukawa-type term in acceleration: plot of the correction factor in Eq. (74).

Figure 3

Plot of the velocity profile corresponding to the acceleration given by Eq. (83). Dashed curve: velocity due to first term, dotted curve velocity due to second term. Thin curve: total velocity.