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Keywords:

  • cosmic microwave background;
  • cosmological parameters;
  • cosmology: observations;
  • cosmology: theory;
  • distance scale

ABSTRACT

The cosmological principle, promoting the view that the Universe is homogeneous and isotropic, is embodied within the mathematical structure of the Robertson–Walker (RW) metric. The equations derived from an application of this metric to the Einstein Field Equations describe the expansion of the Universe in terms of comoving coordinates, from which physical distances may be derived using a time-dependent expansion factor. These coordinates, however, do not explicitly reveal the properties of the cosmic space–time manifested in Birkhoff's theorem and its corollary. In this paper, we compare two forms of the metric – written in (the traditional) comoving coordinates, and a set of observer-dependent coordinates – first for the well-known de Sitter universe containing only dark energy, and then for a newly derived form of the RW metric, for a universe with dark energy and matter. We show that Rindler's event horizon – evident in the comoving system – coincides with what one might call the ‘curvature horizon’ appearing in the observer-dependent frame. The advantage of this dual prescription of the cosmic space–time is that with the latest Wilkinson Microwave Anisotropy Probe results, we now have a much better determination of the Universe's mass-energy content, which permits us to calculate this curvature with unprecedented accuracy. We use it here to demonstrate that our observations have probed the limit beyond which the cosmic curvature prevents any signal from having ever reached us. In the case of de Sitter, where the mass-energy density is a constant, this limit is fixed for all time. For a universe with a changing density, this horizon expands until de Sitter is reached asymptotically, and then it too ceases to change.