Digital Object Identiﬁer (DOI) https://doi.org/10.1007/s00220-017-3029-0
Commun. Math. Phys. 357, 569–595 (2018)
On the Inverse Problem of Finding Cosmic Strings
and Other Topological Defects
, Lauri Oksanen
, Plamen Stefanov
, Gunther Uhlmann
Department of Mathematics and Statistics, University of Helsinki, Box 68, Helsinki 00014, Finland.
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK.
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA.
Department of Mathematics, University of Washington, Box 354350 Seattle, Washington 98195, USA.
Institute for Advanced Study, Hong Kong University of Science and Technology, Clear Water Bay, Hong
Received: 13 May 2015 / Accepted: 1 March 2016
Published online: 16 November 2017 – © Springer-Verlag GmbH Germany, part of Springer Nature 2017
Abstract: We consider how microlocal methods developed for tomographic problems
can be used to detect singularities of the Lorentzian metric of the Universe using mea-
surements of the Cosmic Microwave Background radiation. The physical model we
study is mathematically rigorous but highly idealized.
We study the detection of singularities of the Lorenzian metric of the Universe from
Cosmic Microwave Background (CMB) radiation measurements. The singularities are
considered in the sense of the wave front set that describes where the metric is non-smooth
in the spacetime and also in which direction the singularity occurs. The direction of the
singularity is characterized by using the Fourier transform of the metric, see Deﬁnition
A singularity in the metric could be caused for example by a cosmic string [2,39].
A cosmic string is a singularity in the stress energy tensor that is supported on a two-
dimensional timelike surface in the spacetime. The existence of cosmic strings ﬁnds
support in super-string theories ; however, there is no direct connection between
string theory and the theory of cosmic strings. We refer to [23,26,32] regarding the
existence (or inexistence) of cosmic strings in view of CMB measurements collected by
the Planck Surveyor mission in 2013.
The singularities of which potential detectability is interesting to study include cosmic
stings, monopoles, cosmic walls and black holes. There is a vast physical literature
concerning the effects of particular types of singularities or topological defects on the
CMB measurements, see e.g. [4,5,7] and references therein. The contribution of the
present paper is to adapt techniques from the mathematical study of inverse problems to
CMB measurements. These techniques allow us to detect singularities without a priori
knowledge of their geometry. Hence it might be also possible to detect singularities that
are not predicted by the current physical knowledge. Furthermore, the techniques allow