Modified gravity and cosmology with two extra dimensions

In this thesis, we investigate the gravitational consequences of
theories in which the four spacetime dimensions of our universe are
augmented by two spatial extra dimensions. More specifically, the
focus is on braneworld scenarios, where our world is confined on a
hypersurface in the higher-dimensional bulk, allowing the extra
dimensions to be large or even infinite. Our main motivation for
studying such models is that they could in principle be able to
solve the cosmological constant (CC) problem via degravitation: the
CC only curves the extra space, leaving the brane geometry flat. A
major difference to the simpler case of a codimension-one brane is
that here, gravitational waves can be emitted into the bulk, even
at the 3D homogeneous and isotropic level, as is relevant for
cosmology. Therefore, we first analyze the question how an outgoing
wave boundary condition can be implemented, which is necessary in
order to obtain a closed set of modified Friedmann equations
predicting the cosmological on-brane evolution. We find that a
potential tool from the literature, provided by a certain
decomposition of the Weyl tensor - while being applicable to plane
gravitational waves - fails for cylindrical waves. This failure is
related to the fact that it is already impossible to locally
separate incoming from outgoing linear cylindrical waves (on flat
spacetime), as we demonstrate by explicitly deriving the
corresponding nonreflecting boundary condition, which is nonlocal
in time. We then consider a generalization of the
Dvali-Gabadadze-Porrati (DGP) model, containing an additional
compact on-brane dimension on top of the one infinite codimension.
Since here the 3D maximally symmetric brane emits plane waves, the
Weyl tensor criterion can be used to exclude incoming bulk waves,
and we derive the resulting Friedmann equations. If the compact
dimension is stabilized, DGP cosmology is recovered, but we find
indications that the stabilization should break down when the CC
starts to dominate, which would lead to additional, potentially
interesting late time modifications. If, on the other hand, the
compact direction is allowed to expand freely, there are
dynamically degravitating solutions - which, however, lack a 4D
regime and are thus ruled out, as we demonstrate by fitting to
supernova data. Next, we turn to the codimension-two version of the
DGP model. By numerically solving the full nonlinear coupled
bulk-brane system for cosmological symmetries on the (regularized)
brane, we show that in some region of parameter space, a CC - but
also any other fluid component - gets degravitated dynamically, and
a static geometry is approached via the emission of Einstein-Rosen
waves. For other model parameters, pathological super-accelerating
solutions are encountered. The origin of this unstable behavior is
traced back to a tachyonic ghost mode which is identified in this
parameter region by studying linear metric perturbations around a
nontrivial pure tension background. While confirming the ghost
result on Minkowski from the literature, we gain the important
insight that the ghost disappears if the brane tension is large
enough, thereby reconciling the model with the physical expectation
of a healthy low energy effective theory. Unfortunately, the
healthy region is again incompatible with an appropriate 4D gravity
regime, and therefore ruled out phenomenologically. The preceding
analysis only covered sub-critical brane tensions, meaning that the
deficit angle of the exterior conical geometry is less than 2π. In
the following chapter, we investigate super-critical tensions
(first in 4D), and find that the (regularized) static solution is
no longer stable. Instead, the axial direction expands at an
asymptotically constant rate, and the exterior geometry (which is
necessarily compact) takes the form of a growing cigar. We are able
to derive an analytic relation between the expansion rate and the
tension, which - when adapted to the 6D setup - only yields a
(small) constant shift in the CC, and can therefore not help with
the CC problem. Finally, the case of two finite codimensions is
analyzed within the model of supersymmetric large extra dimensions
(SLED). First, we show that - contrary to recent claims in the
literature - a brane-localized flux cannot help avoiding the
fine-tuning (which is here imposed by flux quantization) in order
to obtain 4D flat solutions, basically because only scale invariant
brane couplings ensure a flat brane. Next, we ask if a more
realistic model with a finite brane width and scale invariance
breaking couplings could still be successful by predicting a small
enough (albeit nonzero) 4D curvature, but find a negative answer:
If the extra-dimensional volume is within its currently allowed
range, both effects give way too large contributions to the
curvature, unless the brane width were many orders of magnitude
below the bulk Planck length, and again some sort of fine-tuning
were invoked.