Global Deep-MOND Parameter as a Theory Discriminant

Different formulations of modified Newtonian dynamics predict somewhat different rotation curves for the same mass distribution. Here I consider a global attribute of the rotation curve that might provide a convenient discriminant between theories when applied to isolated, pure-disk galaxies that are everywhere deep in the modified Newtonian dynamics regime. This parameter is $\mathcal{Q}\equiv ⟨V^2⟩/V_{\infty }^2$, where $⟨V^2⟩\equiv M^{-1}\int 2\pi r\Sigma (r)V^2(r)dr$, with $\Sigma (r)$ the disk’s surface density, $M$ its total mass, and $V_{\infty }$ the asymptotic (constant) rotational speed. The comparison between the observed and predicted values of $\mathcal{Q}$ is oblivious to the distance, the inclination, the mass, and the size of the disk, and to the form of the interpolating function. For the known modified-gravity theories $\mathcal{Q}$ is predicted to be a universal constant [independent of $\Sigma (r)$]: $\mathcal{Q}=2/3$. The predicted $\mathcal{Q}$ value for modified-inertia theories does depend on the form of $\Sigma$. However, surprisingly, I find here that it varies only little among a very wide range of mass distributions, $\mathcal{Q}\approx 0.73\pm 0.01$. While the difference between the theories amounts to only about 5% in the predicted rms velocity, a good enough sample of galaxies may provide the first discerning test between the two classes of theories.