Higher dimensional vacuum cosmologies

Higher dimensional cosmological models are reviewed, and, in particular, vacuum solutions of the field equations of generalized Lagrangian theories are investigated. The possibility that the (four-dimensional) properties of matter may be geometric in origin is discussed by studying whether the higher dimensional vacuum field equations reduce (formally) to Einstein's four-dimensional theory with a nonzero energy-momentum tensor (taken to be a cosmological perfect fluid) constituting the material source. In a sense, this generalizes the work done in the five-dimensional Einstein theory by considering that Einstein's four-dimensional theory with matter is embedded in a higher dimensional generalized Lagrangian theory. It is known that the five-dimensional (D = 5) vacuum Einstein field equations (in which the metric is independent of the fifth dimension) give rise to the familiar radiation Friedmann-Robertson-Walker cosmological model (of Einstein's four-dimensional theory with perfect fluid source). However, it is of interest to study models with more general forms of matter, and so we are motivated to study vacuum solutions in higher dimensions (D greater than 5) and in theories more general than Einstein's theory of gravity. A variety of different higher dimensional vacuum solutions with a cosmological metric are found and discussed. In particular, theories in five dimensions (D = 5) with the addition of general quadratic curvature invariants to the usual Einstein-Hilbert action, Einstein's theory in higher dimensions (D greater than 5), and the general D-dimensional Gauss-Bonnet theory are investigated in detail. Many new higher dimensional vacuum cosmological solutions are found. In some cases of particular interest, the general solution is obtained. Approximate solutions are also investigated. The solutions that are obtained then give rise to models with a variety of different forms for the four-dimensional matter; indeed, it is found that models with a wide range of physically acceptable equations of state are possible.