10.3 Cascading brane-world model

This is a 6D extension of the DGP model [111Jump To The Next Citation Point, 112Jump To The Next Citation Point]. The action is given by
1 ∫ ∘ ------ 1 ∫ ∘ ------ S = --2- d6x − (6)g(6)R + --2- d5x − (5)g(5)R 2κ6 ∫ 2κ5 4−brane -1-- 4 ∘ --(4)-[(4) ] + 2κ2 d x − g R + Lmatter . (448 ) 5 3−brane

As in the 5D DGP model, we can define the cross over scale. In this model, there are two cross over scales:

(5) M 35 (6) M 36 rc = --2, rc = --2. (449 ) M 4 M 5
Assuming that (6)rc ≫ (5)rc, the gravitational potential on the 3-brane cascades from a 1∕r (4D gravity) regime at short scales, to a 1∕r2 (5D gravity) regime at intermediate distance and finally a 1∕r3 (6D gravity) regime at large distances.

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 [162]. On the other hand, the graviton propagator on the 3-brane in this model behaves like G(p) → log (p(5)rc) in the M5 → 0 limit where p is 4D momentum. Thus the cross-over scale (5)rc 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 [243]. 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, 112Jump To The Next Citation Point, 113]. More precisely, there is a critical tension

2(6) −2 2 λcrit = 3- rc M 4 (450 )
and above the critical tension λ > λcrit, the model is free of ghosts. It is still unclear what is the meaning of having the critical tension for the existence of the ghost. For example, what happens if there is a phase transition on the 3-brane and the tension changes from λ > λcrit to λ < λcrit? This remains a very interesting open question. This model also provides an interesting insight into the “degravitation” mechanism by which the cosmological constant does not gravitate on the 3-brane [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 [1]. The simplest de Sitter solutions have been obtained [324]. 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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