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, 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, that is, the source-to-solution operator, in the metamaterial device are equivalent to measurements in a universe given by $(\mathbb{R}_1\times M,−dt^2+g)$, where $(M,g)$ is a closed, orientable, $C^\infty$-smooth, 3-manifold. Thus the obtained construction could be used to simulate cosmological models using metamaterial devices.