Solution of a Braneworld Big Crunch/Big Bang Cosmology

Abstract

We solve for the cosmological perturbations in a five-dimensional background consisting of two separating or colliding boundary branes, as an expansion in the collision speed V divided by the speed of light c. Our solution permits a detailed check of the validity of four-dimensional effective theory in the vicinity of the event corresponding to the big crunch/big bang singularity. We show that the four-dimensional description fails at the first nontrivial order in (V/c)^2. At this order, there is nontrivial mixing of the two relevant four-dimensional perturbation modes (the growing and decaying modes) as the boundary branes move from the narrowly-separated limit described by Kaluza-Klein theory to the well-separated limit where gravity is confined to the positive-tension brane. We comment on the cosmological significance of the result and compute other quantities of interest in five-dimensional cosmological scenarios.

Figures (8)

• Figure 1: The background brane scale factors $b_\pm$ plotted as a function of the Birkhoff-frame time $T$, where $b_{\pm}$ have been normalized to unity at $T=0$. In these coordinates the bulk is Schwarzschild-AdS: the brane trajectories are then determined by integrating the Israel matching conditions, given in Appendix E. In the limit as $T\tt \infty$, the negative-tension brane asymptotes to the event horizon of the black hole, while the positive-tension brane asymptotes to the boundary of AdS.
• Figure 2: The real values of the Lambert W-function. The solid line indicates the principal solution branch, $W_0(x)$, while the dashed line depicts the $W_{-1}(x)$ branch. The two branches join smoothly at $x=-1/e$ where $W$ attains its negative maximum of $-1$.
• Figure 3: The scale factors $b_{\pm}$ on the positive-tension brane (rising curve) and negative-tension brane (falling curve) as a function of the bulk time parameter $x$, to zeroth order in $y_0$. The continuation of the positive-tension brane scale factor on to the $W_{-1}$ branch of the Lambert W-function is indicated by the dashed line.
• Figure 4: The contours of constant $\omega$ in the ($b$, $x_4$) plane. Working to zeroth order in $y_0$, these are given by $x_4 = \frac{1}{2}(b\omega\pm \sqrt{b^2\omega^2-4(b-1)})$, where we have plotted the positive root using a solid line and the negative root using a dashed line. The negative-tension brane is located at $\omega=-1$ for times $x_4<1$, and the trajectory of the positive-tension brane is given (for all time) by the positive root solution for $\omega=1$. The region delimited by the trajectories of the branes (shaded) then corresponds to the bulk. From the plot we see that, for $0<x_4<1$, the bulk is parameterized by values of $\omega$ in the range $-1\le \omega \le 1$. In contrast, for $x_4>1$, the bulk is parameterized by values of $\omega$ in the range $\omega\ge 1$.
• Figure 5: The three-dimensional scale factor $b$, plotted to zeroth order in $y_0$ as a function of $x_4$ and $\omega$, for $x_4<1$ (left) and $x_4>1$ (right). The positive-tension brane is fixed at $\omega=1$ for all time (note the evolution of its scale factor is smooth and continuous), and for $x_4<1$, the negative-tension brane is located at $\omega=-1$.