The Vainshtein mechanism in the Decoupling Limit of massive gravity

Abstract

We investigate static spherically symmetric solutions of nonlinear massive gravities. We first identify, in an ansatz appropriate to the study of those solutions, the analog of the decoupling limit (DL) that has been used in the Goldstone picture description. We show that the system of equations left over in the DL has regular solutions featuring a Vainshtein-like recovery of solutions of General Relativity (GR). Hence, the singularities found to arise integrating the full nonlinear system of equations are not present in the DL, despite the fact those singularities are usually thought to be due to a negative energy mode also seen in this limit. Moreover, we show that the scaling conjectured by Vainshtein at small radius is only a limiting case in an infinite family of non singular solutions each showing a Vainshtein recovery of GR solutions below the Vainshtein radius but a different common scaling at small distances. This new scaling is shown to be associated with a zero mode of the nonlinearities left over in the DL. We also show that, in the DL, this scaling allows for a recovery of GR solutions even for potentials where the original Vainshtein mechanism is not working. Our results imply either that the DL misses some important features of nonlinear massive gravities or that important features of the solutions of the full nonlinear theory have been overlooked. They could also have interesting outcomes for the DGP model and related proposals.

Figures (7)

• Figure 1: Solution for $w$ in the case of the BD potential. The numerical solution is shown by solid thick (blue) line. For distances much larger than the Vainshtein radius, $\xi\gg 1$, the solution is well approximated by the asymptotic (\ref{['largexibis']}), shown by dashed thin (black) line. Close to the source, $\xi\ll 1$, the numerical solution approaches the $Q$-scaling asymptotic, Eq. (\ref{['QSCALING']}), shown by dotted thin (black) line.
• Figure 2: Plot of the numerical solution (solid black curve), and the series expansion (\ref{['w-DL']}) given up to order $\xi$ (dotted red curve) and up to order $\xi^{13}$ (dashed blue curve). The expansion up to $\xi^{13}$ approximates the numerical solution with a precision better that $99\%$ in the range $0\leqslant \xi \leqslant 0.95$.
• Figure 3: Plot of the numerical solutions for $w$ in the case of the AGS potential, starting from $\xi_{i}=2.5$. For $\xi\gg 1$ all the solutions approach the asymptotic (\ref{['largexibis']}), shown by dashed thin (black) line. At small distances, $\xi\ll 1$ the solutions pick up different asymptotic regimes. The solid thick (blue) line corresponds to the Vainshtein scaling $w\sim 1/\sqrt{\xi}$. The dash-dotted thick (green) line corresponds to a solution with the $Q$-scaling (\ref{['QSCALING']}). The dashed thick (red) line corresponds to a solution which first follows the Vainshtein scaling and then finally picks up the $Q$-scaling (\ref{['QSCALING']}).
• Figure 4: Plot of the numerical solution for $w$ in the case of the BD potential in the presence of a source of radius $\xi_{\odot}=0.01$ (depicted above by a light blue area), starting from $\xi_{i}=6$. The blue thick line corresponds to the $Q$-scaling of Eq. (\ref{['diverging solution at small distance']}), which is picked up by the asymptotic behaviour at infinity. Note that the solution is almost not affected by the presence of the source.
• Figure 5: Plot of the numerical solutions for $w$ in the case of the AGS potential in the presence of a source of radius $\xi_{\odot}=0.01$ (depicted above by a light blue area), starting from $\xi_{i}=3.5$. The solid thick (blue) line corresponds to the solution having the behaviour Eq. (\ref{['constant solution at small distance']}) inside the source. The thick dash-dotted (red) line illustrates the $Q$-scaling, Eq. (\ref{['diverging solution at small distance']}), inside the source.