The critical tension in the Cascading DGP model
  We study the behaviour of weak gravitational fields in the 6D Cascading DGP
model using a bulk-based approach. To deal with the ambiguity in the thin limit
of branes of codimension higher than one, we consider a specific regularization
of the internal structure of the branes where the 5D brane can be considered
thin with respect to the 4D one. We consider the solutions corresponding to
pure tension sources on the 4D brane, and study perturbations at first order
around these background solutions. We adopt a 4D scalar-vector-tensor
decomposition, and focus on the scalar sector of perturbations. We show that,
in a suitable 4D limit, the trace part of the 4D metric perturbations obeys a
decoupled equation which suggests that it is a ghost for background tensions
smaller than a critical tension, while it is a healthy field otherwise. We give
a geometrical interpretation of the existence of the critical tension and of
the reason why the relevant field is a ghost or not depending on the background
tension. We however find a value of the critical tension which is different
from the one already found in the literature. Differently from the results in
the literature, our analysis implies that, choosing the background tension
suitably, we can construct ghost-free models for any value of the free
parameters of the theory. We suggest that the difference lies in the procedure
used to evaluate the pillbox integration across the codimension-2 brane. We
confirm the validity of our analysis by performing numerically the integration
in a particular case where the solution inside the thick cod-2 brane is known
exactly. We stress that the singular structure of the perturbation fields in
the nested branes set-ups is very subtle, and that great care has to be taken
when deriving the codimension-2 junction conditions.
