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In the context of theories of Kaluza-Klein type, with a large extradimension, we study self-similar cosmological models in 5D that arehomogeneous, anisotropic and spatially flat. The "ladder" to go between thephysics in 5D and 4D is provided by Campbell-Maagard's embedding theorems. Weshow that the 5-dimensional field equations $R_{AB} = 0$ determine the form ofthe similarity variable. There are three different possibilities: homothetic,conformal and "wave-like" solutions in 5D. We derive the most generalhomothetic and conformal solutions to the 5D field equations. They require theextra dimension to be spacelike, and are given in terms of one arbitraryfunction of the similarity variable and three parameters. The Riemann tensor in5D is not zero, except in the isotropic limit, which corresponds to the casewhere the parameters are equal to each other. The solutions can be used as 5Dembeddings for a great variety of 4D homogeneous cosmological models, with andwithout matter, including the Kasner universe. Since the extra dimension isspacelike, the 5D solutions are invariant under the exchange of spatialcoordinates. Therefore they also embed a family of spatially {\itinhomogeneous} models in 4D. We show that these models can be interpreted asvacuum solutions in braneworld theory. Our work (I) generalizes the 5Dembeddings used for the FLRW models; (II) shows that anisotropic cosmologiesare, in general, curved in 5D, in contrast with FLRW models which can always beembedded in a 5D Riemann-flat (Minkowski) manifold; (III) reveals thatanisotropic cosmologies can be curved and devoid of matter, both in 5D and 4D,even when the metric in 5D explicitly depends on the extra coordinate, which isquite different from the isotropic case.

Self-similar cosmologies in 5D: spatially flat anisotropic models