The rotation curve of a point particle in stringy gravity

Double Field Theory suggests to view the whole massless sector of closed strings as the gravitational unity. The fundamental symmetries therein, including the $\mathbf{O}(D,D)$ covariance, can determine unambiguously how the Standard Model as well as a relativistic point particle should couple to the closed string massless sector. The theory also refines the notion of singularity. We consider the most general, spherically symmetric, asymptotically flat, static vacuum solution to ${D=4}$ Double Field Theory, which contains three free parameters and consequently generalizes the Schwarzschild geometry. Analyzing the circular geodesic of a point particle in string frame, we obtain the orbital velocity as a function of $R/(M_{\scriptscriptstyle{\infty}}G)$ which is the dimensionless radial variable normalized by mass. The rotation curve generically features a maximum and thus non-Keplerian over a finite range, while becoming asymptotically Keplerian at infinity, $R/(M_{\scriptscriptstyle{\infty}}G)\rightarrow \infty$. The adoption of the string frame rather than Einstein frame is the consequence of the fundamental symmetry principle. Our result opens up a new scheme to solve the dark matter/energy problems by modifying General Relativity at `short' range of $R/(M_{\scriptscriptstyle{\infty}}G)$.