The primordial curvature perturbation in the ekpyrotic Universe

In the ekpyrotic scenario the Universe is initially collapsing, the energy density coming from a scalar field with a negative exponential potential. On the basis of a calculation ignoring the gravitational back-reaction the authors of the scenario claim that during collapse the vacuum fluctuation creates a perturbation in the comoving curvature, which has a flat spectrum in accordance with observation. In this note the back-reaction is included, and it is found that the spectrum during collapse is strongly scale-dependent with negligible magnitude. The spectrum is continuous across the bounce if the spacetime is smooth, making it unlikely that the ekpyrotic scenario can be compatible with observation. Introduction In the ekpyrotic Universe [1, 2] (called in a slightly different version [3] the pyrotechnic Universe) the Big Bang originates when a brane moving in an extra dimension collides with a fixed one. Except around the time of the collision, the extra dimension can be integrated out to give a four-dimensional field theory with Einstein gravity and flat space. Before the collision, the Universe is collapsing, with the energy density provided by the field φ which defines the position of the moving brane. In this phase the Hubble parameter H , the energy density ρ and the negative potential V are related by the Friedmann equation ρ = 3M P H = 1 2 φ̇ + V (φ) . (1) The Universe is supposed initially to be almost static with negligible energy density. The energy density increases as the Universe collapses, while remaining negligible compared with the potential. When the moving brane collides with the stationary one, the energy density is converted to radiation, and at the same time the Universe starts to expand. This ekpyrotic scenario is similar to the pre-big-bang scenario [4, 6, 5]. In both cases the 4-D field theory breaks down at the bounce, whose description would in principle require a string theory calculation (though the calculation may be more tractable in the ekpyrotic case since the bounce can take place far below the string scale). The main