Generalized perturbation equations in bouncing cosmologies

We consider linear perturbation equations for long-wavelength scalar metric perturbations in generalized gravity, applicable to nonsingular cosmological models including a bounce from collapse to expansion in the very early universe. We present the general form for the perturbation equations, which follows from requiring that the inhomogeneous universe on large scales obeys the same local equations as the homogeneous Friedmann-Robertson-Walker background cosmology (the separate universes approach). In a pseudolongitudinal gauge this becomes a homogeneous second-order differential equation for adiabatic perturbations, which reduces to the usual equation for the longitudinal gauge metric perturbation in general relativity with vanishing anisotropic stress. As an application we show that the scale-invariant spectrum of perturbations in the longitudinal gauge generated during an ekpyrotic collapse are not transferred to the growing mode of adiabatic density perturbations in the expanding phase in a simple bounce model.