Boundary Terms and Junction Conditions for the DGP Pi-Lagrangian and Galileon

In the decoupling limit of DGP, Pi describes the brane-bending degree of freedom. It obeys second order equations of motion, yet it is governed by a higher derivative Lagrangian. We show that, analogously to the Einstein-Hilbert action for GR, the Pi-Lagrangian requires Gibbons-Hawking-York type boundary terms to render the variational principle well-posed. These terms are important if there are other boundaries present besides the DGP brane, such as in higher dimensional cascading DGP models. We derive the necessary boundary terms in two ways. First, we derive them directly from the brane-localized Pi-Lagrangian by demanding well-posedness of the action. Second, we calculate them directly from the bulk, taking into account the Gibbons-Hawking-York terms in the bulk Einstein-Hilbert action. As an application, we use the new boundary terms to derive Israel junction conditions for Pi across a sheet-like source. In addition, we calculate boundary terms and junction conditions for the galileons which generalize the DGP Pi-lagrangian, showing that the boundary term for the n-th order galileon is the (n-1)-th order galileon.Comment: 23 pages, 1 figure. Extended the analysis to the general galileon field. Version to appear in JHE

Classical Duals of Derivatively Self-Coupled Theories

Solutions to scalar theories with derivative self-couplings often have regions where non-linearities are important. Given a classical source, there is usually a region, demarcated by the Vainshtein radius, inside of which the classical non-linearities are dominant, while quantum effects are still negligible. If perturbation theory is used to find such solutions, the expansion generally breaks down as the Vainshtein radius is approached from the outside. Here we show that it is possible, by integrating in certain auxiliary fields, to reformulate these theories in such a way that non-linearities become small inside the Vainshtein radius, and large outside it. This provides a complementary, or classically dual, description of the same theory -- one in which non-perturbative regions become accessible perturbatively. We consider a few examples of classical solutions with various symmetries, and find that in all the cases the dual formulation makes it rather simple to study regimes in which the original perturbation theory fails to work. As an illustration, we reproduce by perturbative calculations some of the already known non-perturbative results, for a point-like source, cosmic string, and domain wall, and derive a new one. The dual formulation may be useful for developing the PPN formalism in the theories of modified gravity that give rise to such scalar theories.Comment: 20 pages. v2 refs adde

Multi-field galileons and higher co-dimension branes

In the decoupling limit, the DGP model reduces to the theory of a scalar field pi, with interactions including a specific cubic self-interaction - the galileon term. This term, and its quartic and quintic generalizations, can be thought of as arising from a probe 3-brane in a 5-dimensional bulk with Lovelock terms on the brane and in the bulk. We study multi-field generalizations of the galileon, and extend this probe brane view to higher co-dimensions. We derive an extremely restrictive theory of multiple galileon fields, interacting through a quartic term controlled by a single coupling, and trace its origin to the induced brane terms coming from Lovelock invariants in the higher co-dimension bulk. We explore some properties of this theory, finding de Sitter like self accelerating solutions. These solutions have ghosts if and only if the flat space theory does not have ghosts. Finally, we prove a general non-renormalization theorem: multi-field galileons are not renormalized quantum mechanically to any loop in perturbation theory.Comment: 34 pages, 2 figures. v2 typos corrected, comments added, version appearing in PR