Cosmological Perturbations in a Big Crunch/Big Bang Space-time

TL;DR

This work provides a concrete, gauge-invariant framework to propagate cosmological perturbations through big crunch/big bang singularities in ekpyrotic/cyclic brane-world models. By mapping the five-dimensional brane dynamics to a four-dimensional moduli space theory and employing a Milne-like gauge, the authors derive a precise, universal matching prescription across the singularity. They show that long-wavelength, scale-invariant growing perturbations generated before the bounce persist into the post-bounce hot big bang phase, with quantitative dependence on brane tensions, radiation densities, and collision velocity. The results bolster the plausibility of ekpyrotic/cyclic cosmologies and illuminate how five-dimensional physics imprints observable late-time perturbations, while outlining pathways to address nonlinear effects near the singularity.

Abstract

A prescription is developed for matching general relativistic perturbations across singularities of the type encountered in the ekpyrotic and cyclic scenarios i.e. a collision between orbifold planes. We show that there exists a gauge in which the evolution of perturbations is locally identical to that in a model space-time (compactified Milne mod Z_2) where the matching of modes across the singularity can be treated using a prescription previously introduced by two of us. Using this approach, we show that long wavelength, scale-invariant, growing-mode perturbations in the incoming state pass through the collision and become scale-invariant growing-mode perturbations in the expanding hot big bang phase.

Figures (4)

• Figure 1: Locally, the collision of two branes may be embedded in Minkowski space-time. The usual Minkowski space-time coordinates $T$ and $Y$ are are expressed as $T=t\, {\rm cosh} y$ and $Y=t\, {\rm sinh} y$, where the Lorentz-invariant coordinate $t$ is constant on the dashed lines. The collision event is constructed in two steps. First the $y$ coordinate is compactified by identifying $y$ with $y+2y_0$, to produce the double-conical space-time shown at the right. Second, the circular sections of these cones are orbifolded by the $Z_2$ symmetry $y \rightarrow 2y_0-y$. The two fixed points of the $Z_2$ symmetry are two tensionless branes moving at a relative speed of ${\rm tanh} y_0$, which collide and pass through one another at $t=0$.
• Figure 2: The definition of a space-time manifold is that when viewed 'up close' (left figure), it should appear to be locally flat. We define singular space-times of the type we are interested in here as space-times for which there exists a single coordinate system covering the neighborhood of the singularity in both the incoming and outgoing space-times, within which the collision event appears locally identical to the idealized situation of tensionless $Z_2$ branes colliding in Minkowski space-time (right figure).
• Figure 3: Continuation of left and right moving modes. A free field propagating in the lower quadrant may be decomposed into left and right movers as it approaches the past light cone of the origin $T=Y=0$. The left movers are regular across $Y=T<0$ and may be continued into the left quadrant $Y<0, |T|<|Y|$. The right movers are regular across the right segment $Y=-T>0$ and may be continued into the right quadrant $Y>0, |T|<|Y|$. If we impose vanishing boundary conditions at large Lorentz-invariant separation from the origin in the left and right quadrants, then once we know the left mover in the left quadrant, the right mover on the null segment $Y=-T<0$ is uniquely determined, and similarly the left mover on $Y=T>0$. One thereby obtains a unique matching rule from the incoming, lower quadrant to the outgoing, upper one.
• Figure 4: The worldlines of the positive and negative tension branes are plotted for some fixed value of the uncompactified coordinates $\vec{x}$. The four-dimensional effective theory is used to predict the intrinsic geometries of the positive and negative tension branes, i.e. their space-time metrics $g_{\mu \nu}^+$ and $g_{\mu \nu}^-$, according to equation (\ref{['eq:branemets']}). The four-dimensional effective theory is used to describe the incoming and outgoing perturbed branes far to the past or future of the collision event. The brane metrics also provide boundary data for the five-dimensional bulk metric which we solve for as a power series expansion in time about the collision event.