Exact Inhomogeneous Cosmological Models

We present new exact inhomogeneous vacuum cosmological solutions of Einstein’s equations. They provide new information about the nature of general cosmological solutions to Einstein’s equations. PACS 04.20.Jb, 98.80H, 04.30.-w Inhomogeneous and anisotropic solutions of Einstein’s equations are of great interest because of the subtleties of their non-linearity and the clues they offer about the generic behaviour of cosmological models near spacetime singularities and at late times. They provide testing grounds for high-energy physics in the early universe and guide the development of quantum cosmology. Here, we present a new class of inhomogeneous vacuum solutions of Einstein’s equations which generalise the Einstein-Rosen solutions [1]. They represent inhomogeneous counterparts of the most general Bianchi-type vacuum universes and may form a leading-order approximation to part of a general solution of the vacuum Einstein equations. Moreover, anisotropic universes containing perfect fluids with pressure less than the energy density behave in general like vacuum universes at early times. Consider the generalized Einstein-Rosen metrics: ds = e (−dξ + dz) + gab(ξ, z)dxdx, (1) where ξ is a timelike coordinate; xa, a = 1, 2, and z are spacelike coordinates. These metrics admit an abelian group of isometries, G2. Spatially-homogeneous cosmologies of Bianchi types I-VII and the axisymmetric cases of Bianchi types VIII and IX, admit such a G2 and are particular cases [2]. Properties of (1) depend on whether Bμ ≡ ∂μ(det gab) 1 2 is spacelike, timelike or null (Greek indices run 0 → 3). The cases with a globally null or spacelike Bμ correspond to plane or cylindrical gravitational waves, respectively [2]. Metrics where the sign of BμB μ varies throughout the spacetime describe colliding gravitational waves [3] or cosmologies with timelike and spacelike singularities [4]. Metrics with a globally timelike Bμ describe cosmological models with spacelike singularities. If the spacelike hypersurfaces are compact then the allowed spatial topologies [5] are a 3-torus, S1 ⊗ S1 ⊗ S1, for (detgab) 1 2 = ξ; a hypertorus, S1 ⊗ S2, or a 3-sphere, S3, for 1 (detgab) 1 2 = sin z sin ξ with 0 ≤ z, ξ ≤ π. We shall present solutions for the globally timelike case. Metrics of this form have been studied approximately by Belinskii and Khalatnikov [6] who argued that they can provide the leading approximation to a general solution of the vacuum Einstein equations near the initial singularity. The metric gab in (1) can be written as [6] gab = R ( eα cosh β sinhβ sinhβ e−α cosh β ) (2) where α = α(ξ, z), β = β(ξ, z), and R = R(ξ, z), so Bμ ≡ R,μ. The vacuum Einstein equations are R̈−R = 0, (3) α̈+ Ṙ R α̇− α − R ′ R α = 2(αβ − α̇β̇) tanhβ, (4) β̈ + Ṙ R β̇ − β − R ′ R β = 1 2 (α̇ − α) sinh 2β, (5) ψ̈ − ψ + 1 2 (β̇ − β) + 1 2 (α̇ − α) cosh β + 1 2 