20 december, Holger Then: Maass waveforms and an application to cosmology

Abstract. A brief outline on the numerical computation of Maass waveforms in hyperbolic three-space is given. The results are then applied to cosmology. In a linear perturbed Robertson-Walker universe the Einstein equations of general relativity can be separated, if one knows the eigenvalues and eigenfunctions of the Laplacian. Assuming the universe to be a finite, but non-compact hyperbolic manifold, the metric perturbations can be expanded in terms of Maass waveforms. This allows to compute the temperature fluctuations in the cosmic microwave background (CMB). The assumption of a finite hyperbolic universe seems rather strange, but comparing the results with astronomical observations yields evidence for the assumption to be true. The assumption of a non-compact universe is made for technical reasons that allow to introduce Maass waveforms. Since this assumption has no inluence on the CMB, it could also be dropped.