The consistency of codimension-2 braneworlds and their cosmology

TL;DR

This work shows that distributional codimension-2 branes in six-dimensional Lovelock gravity are self-consistent when the bulk action includes the Gauss–Bonnet term, with topological matching conditions describing pure codimension-2 defects. Two regularisation schemes are analyzed: smooth interiors (which permit energy exchange with the bulk via a time-varying deficit angle $\beta$) and ring regularisation (which yields a closed, energy-conserving brane system). The cosmology on the brane acquires modifications to the Friedmann equations through deficit-angle dynamics and extrinsic-curvature terms, allowing geometric self-acceleration and potential self-tuning mechanisms. The results clarify when purely codimension-2 junction conditions suffice and when codimension-1 contributions must be included, and they point to observable signatures in the early and late-time evolution that could constrain such higher-dimensional theories.

Abstract

We study axially symmetric codimension-2 cosmology for a distributional braneworld fueled by a localised 4D perfect fluid, in a 6D Lovelock theory. We argue that only the matching conditions (dubbed topological) where the extrinsic curvature on the brane has no jump describe a pure codimension-2 brane. If there is discontinuity in the extrinsic curvature on the brane, this induces inevitably codimension-1 distributional terms. We study these topological matching conditions, together with constraints from the bulk equations evaluated at the brane position, for two cases of regularisation of the codimension-2 defect. First, for an arbitrary smooth regularisation of the defect and second for a ring regularisation which has a cusp in the angular part of the metric. For a cosmological ansatz, we see that in the first case the coupled system is not closed and requires input from the bulk equations away from the brane. The relevant bulk function, which is a time-dependent angular deficit, describes the energy exchange between the brane and the 6D bulk. On the other hand, for the ring regularisation case, the system is closed and there is no leakage of energy in the bulk. We demonstrate that the full set of matching conditions and field equations evaluated at the brane position are consistent, correcting some previous claim in the literature which used rather restrictive assumptions for the form of geometrical quantities close to the codimension-2 brane. We analyse the modified Friedmann equation and we see that there are certain corrections coming from the non-zero extrinsic curvature on the brane. We establish the presence of geometric self-acceleration and a possible curvature domination wedged in between the period of matter and self-acceleration eras as signatures of codimension-2 cosmology.

Figures (5)

• Figure 1: On the left, the conical singularity is regularised by a family of regular interiors $C_\lambda$ smoothly connected to the exterior solution of the cone at the boundary $\partial C$. On the right, a ring regularisation is chosen where the internal space function $L$ is not smooth, but has a cusp across the ring at $r=\epsilon$.
• Figure 2: The conical singularity regularised by a cap glued to the bulk at a ring intersurface at $r=\epsilon$. Both $K_{\mu\nu}$ and $L$ have jumps across at the ring.
• Figure 3: The grouping of equations according to the various functions that are to be determined. On the left table, the first column has the codimension-2 junction conditions, while the other columns have the various orders in $r$ of the Einstein equations. The ${\cal O}(1/\epsilon)$ component of the $(\theta \theta)$ Einstein equation vanishes identically.
• Figure 4: The functional dependence on the scale factor of the energy density and the extrinsic curvature correction to the Friedmann equation for different equations of state of the brane matter in the self-accelerating case. From top to bottom we have the equations of state for cosmological constant, domain walls, cosmic strings, dust, radiation and stiff matter.
• Figure 5: The log-log evolution of the energy densities as a function of the scale factor. With solid lines, there are the contributions of the standard matter and the geometrically induced $1/(2|\alpha|)$ cosmological constant. With dashed lines are the evolutions of the extra component ($\propto A^2$) in the Friedmann equation due to the extrinsic curvature of the brane. The dependence of this extra component on the scale factor is noted. This dependence changes whenever a new matter component becomes dominant, i.e. at matter domination at $a_{eq}$ and at the cosmological constant domination at $a_{acc}$.