Colliding Branes in Heterotic M-theory

TL;DR

This work models the big-bang as a collision of two end-of-the-world branes in heterotic M-theory, showing that demanding bounded curvature selects a unique near-collision geometry. Through brane-comoving and static-bulk analyses, the authors derive a complete, global solution in which the collision region is locally $((2d\text{ Milne})/\mathbb{Z}_2)\times \mathbb{R}_3\times \mathrm{CY}$, with the five-dimensional spacetime remaining regular away from the collision. A key result is that the negative-tension brane undergoes a mild, regulator-friendly bounce off a bulk warp-factor zero, allowing a smooth continuation through the collision and into the expanding phase. The eleven-dimensional lift reveals how the Calabi–Yau volume and inter-brane separation evolve, suggesting a plausible bridge to four-dimensional cosmology once moduli are stabilized and suitable perturbations are considered.

Abstract

We study the collision of two flat, parallel end-of-the-world branes in heterotic M-theory. By insisting that there is no divergence in the Riemann curvature as the collision approaches, we are able to single out a unique solution possessing the local geometry of (2d compactified Milne)/Z_2 x R_3, times a finite-volume Calabi-Yau manifold in the vicinity of the collision. At a finite time before and after the collision, a second type of singularity appears momentarily on the negative-tension brane, representing its bouncing off a zero of the bulk warp factor. We find this singularity to be remarkably mild and easily regularised. The various different cosmological solutions to heterotic M-theory previously found by other authors are shown to merely represent different portions of a unique flat cosmological solution to heterotic M-theory.

Figures (4)

• Figure 1: A Kruskal plot of the entire solution, in the spirit of Chen et al. Gary. The exact trajectories of the positive- and negative-tension branes are plotted in red and green respectively, while the naked singularity is indicated by a thick black line. Representative orbits of the bulk Killing vector field are shown with dashed lines, while the solid straight lines indicate the Boyer axes. The bounces of the negative-tension brane off the naked singularity, as well as the collision of the branes themselves, are shown at a magnified scale in the inset. In this plot, we have chosen the relative rapidity of the brane collision to be $2y_0 = 1$. Analogous plots for greater collision rapidities may be found in Figure \ref{['doubleK']} of Appendix C.
• Figure 2: Continuation of the scaling solution (accurate to leading order in $y_0$) beyond the bounce of the negative-tension brane at $\alpha y_0\tau=1/4$, using the absolute value prescription. Contours of constant $\omega$ are plotted against the fourth power of the three-dimensional scale factor $b$ and the brane conformal time $\tau$. The contours shown are, from left to right, $\omega = -1$, $-0.75$, $-0.5$, $0$, $0.5$, and $1$. Thus the left-most and right-most trajectories correspond to the negative- and positive-tension branes respectively (for which $b_\pm^4 = |1\pm 4 \alpha y_0 \tau|$), with the region in between representing the bulk.
• Figure 3: $UV$ plotted as a function of $\bar{r}$, according to (\ref{['UVrel']}) (taking $\lambda=1$). Note in particular the smooth continuation to $\bar{r}>1$ generated by the sign flip in $\sigma$. The surfaces of constant $\bar{r}$ correspond to the hyperbolae $UV=\mathrm{constant}$, with this constant being positive for $0\le \bar{r} < 1$, and negative for $\bar{r} >1$.
• Figure 4: Kruskal diagrams illustrating the brane trajectories for different collision rapidities. In the upper plot $y_0=1$, and in the lower plot $y_0=1.5$. The corresponding plot for $y_0=0.5$ appeared previously in Figure \ref{['Kruskal']}.