Cosmological perturbations in a big-crunch–big-bang space-time

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.