### 10.2 Cosmology in 6D brane-world models

It is much harder to obtain cosmological solutions in 6D models than in 5-dimensional models. In 5D
brane-world models, cosmological solutions can be obtained by considering a moving brane in a static bulk
spacetime. This is because the motion of the homogeneous and isotropic brane does not change the bulk
spacetime thanks to Birkoff’s theorem. However, this is no longer the case in 6D spacetime. Thus we need to
find time dependent bulk solutions that are coupled to a motion of a brane, which requires us to solve
non-linear partial differential equations numerically. Moreover, as is mentioned above, there appears
a curvature singularity if one considers an infinitely thin brane and puts matter other than
tension on the brane [419, 420]. Thus we need some regularization scheme to find cosmological
solutions.
One of the popular ways to regularize a brane is to promote a brane that is a point-like object
in two extra-dimensions to a 5D ring [353]. The ring is a codimension one object and it is
possible to consider a motion of this ring. However, a problem is that this ring brane is not
homogeneous as one of dimensions on the brane is compact and this breaks Birkhoff’s theorem.
It has been shown that cosmology obtained by the motion of a brane in a static bulk shows
pathological behaviour [351, 326]. This indicates that we need to find fully time-dependent bulk
solutions.

At the moment, the only accessible way is to solve the 6D bulk spacetime using the gradient expansion
methods assuming physical scales on a brane are much lower than mass scales in the bulk [150, 16, 233]. It
is still an open question what are high energy effects in 6D models.

As in 5D models, there are many generalizations such as the inclusion of the induced gravity term on a
brane and the Gauss–Bonnet term in the bulk [43, 222, 101, 84, 349, 239, 83, 82].