Charting the Landscape of Modified Gravity

We explore brane induced gravity on a 3-brane in six locally flat dimensions. To regulate the short distance singularities in the brane core, we resolve the thin brane by a cylindrical 4-brane, with the geometry of 4D Minkowski $\times$ a circle, which has an axion flux to cancel the vacuum pressure in the compact direction. We discover a large diversity of possible solutions controlled by the axion flux, as governed by its boundary conditions. Hence brane induced gravity models really give rise to a {\it landscape} of vacua, at least semiclassically. For sub-critical tensions, the crossover scale, below which gravity may look 4D, and the effective 4D gravitational coupling are sensitive to vacuum energy. This shows how the vacuum energy problem manifests in brane induced gravity: instead of tuning the 4D curvature, generically one must tune the crossover scale. On the other hand, in the near-critical limit, branes live inside very deep throats which efficiently compactify the angular dimension. In there, 4D gravity first changes to $5D$, and only later to $6D$. The crossover scale saturates at the gravitational see-saw scale, independent of the tension. Using the fields of static loops on a wrapped brane, we check the perturbative description of long range gravity below the crossover scale. In sub-critical cases the scalars are strongly coupled already at the crossover scale even in the vacuum, because the brane bending is turned on by the axion flux. Near the critical limit, linearized perturbation theory remains under control below the crossover scale, and we find that linearized gravity around the vacuum looks like a scalar-tensor theory.