The DGP brane world model assumes the 4D Einstein-Hilbert term induced on the brane. This model is known to give a mechanism to realize the late-time accelerating expansion of the Universe without introducing additional matter. Such a solution is called the self-acceleration branch in contrast to the normal branch. The normal branch does not show no cosmic expansion in the limit of vanishing matter energy density, while the self-acceleration branch approaches a constant value of Hubble parameter.
In quantizing the theory, however, we face the issue of strong coupling, which is the problem that the non-linear effect of the scalar gravitational interaction becomes important below a certain large length scale. This strong coupling also plays an important role to recover the usual 4d behavior near gravitational sources. If we do not take into account the non-linear effect, the gravity behaves very differently and is not consistent with the observation. Such a peculiar nature make this model very attractive and mysterious.
Moreover, the existence of a ghost excitation is pointed out in the self-accelerating branch of the DGP brane world model. A ghost is a degree of freedom whose action has an overall wrong sign. Such a degree of freedom is thought to be quantum mechanically unstable. Therefore a model such that contains a ghost degree of freedom cannot be a healthy model. However, effective theory of gravity is in general more or less unhealthy at the quantum level. Therefore it seems too quick to abondon the idea of self-acceleration immediately. What we have to do now is to understand whether the self-accleration is tightly related to the presence of a ghost. Moreover, we are doubting that the concept of ghost in curved space may have a little subtlety. We are now trying to clarify these issues.