As in the 5D DGP model, we can define the cross over scale. In this model, there are two cross over scales:
This model addresses several fundamental issues in induced gravity models in 6D spacetime. Without the induced gravity term on the 4-brane, the 6D graviton propagator diverges logarithmically near the position of the 3-brane . On the other hand, the graviton propagator on the 3-brane in this model behaves like in the limit where is 4D momentum. Thus the cross-over scale acts as a cut-off for the propagator and it remains finite even at the position of the 3-brane.
A more serious issue in the induced gravity model is that most constructions seem to be plagued by ghost instabilities [128, 152]. Usually, the regularization of the brane is necessary and it depends on the regularization scheme whether there appears a ghost or not . In the cascading model, there is still a ghost if the 3-brane has no tension. However, it was shown that by adding a tension to the 3-brane, this ghost disappears [111, 112, 113]. More precisely, there is a critical tension[132, 112].
Cosmological solutions in the cascading brane model are again notoriously difficult to find because it is necessary to find 6-dimensional solutions that depend on time and two extra-coordinates . The simplest de Sitter solutions have been obtained . Interestingly, there exists a self-accelerating solution for the 3-brane even when the solution for the 4-brane is in the normal branch. It is still not clear whether this self-accelerating solution is stable or not and it is crucial to check the stability of this new self-accelerating solution.
A similar class of models includes intersecting branes [217, 102, 103]. In this model, we have two 4-branes that intersect and a 3-brane sits at the intersection. Again there are self-accelerating de Sitter solutions and cosmology has been studied by considering a motion of one of the 4-branes. A model without a 4-brane has been studied by regularizing a 3-brane by promoting it to a 5D ring brane [219, 218].
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