Transformation Optics for the Modelling of Waves in a Universe with Nontrivial Topology

We consider how transformation optics and invisibility cloaking can be used to construct models in subsets $\mathbb{R}^3$ with a varying metric, where the time-harmonic waves for a given angular wavenumber $k$, are equivalent to the waves in some closed orientable manifold. The obtained models could in principle be physically implemented using a device built from metamaterials. In particular the measurements in the metamaterial device given by the Helmholtz source-to-solution operator are equivalent to Helmholtz source-to-solution measurements in a universe given by $(\mathbb{R}_+\times M, -dt^2 +g)$, where $(M,g)$ is a closed, orientable, $C^\infty$-smooth, 3-dimensional Riemannian manifold. Thus the obtained construction could be used to simulate cosmological models using metamaterial devices.