EDP Sciences
Free Access
Volume 439, Number 2, August IV 2005
Page(s) 479 - 485
Section Cosmology (including clusters of galaxies)
DOI https://doi.org/10.1051/0004-6361:20042081

A&A 439, 479-485 (2005)
DOI: 10.1051/0004-6361:20042081

A constraint on any topological lensing hypothesis in the spherical case: it must be a root of the identity

B. F. Roukema

Torun Centre for Astronomy, N. Copernicus University, ul. Gagarina 11, 87-100 Torun, Poland
    e-mail: boud@astro.uni.torun.pl

(Received 28 September 2004 / Accepted 9 May 2005)

Three-dimensional catalogues of objects at cosmological distances can potentially yield candidate topologically lensed pairs of sets of objects, which would be a sign of the global topology of the Universe. In the spherical case (i.e. if curvature is positive), a necessary condition, which does not exist for either null or negative curvature, can be used to falsify such hypotheses, without needing to loop through a list of individual spherical 3-manifolds. This condition is that the isometry between the two sets of objects must be a root of the identity isometry in the covering space S3. This enables numerical falsification of topological lensing hypotheses without needing to assume any particular spherical 3-manifold. By embedding S3 in euclidean 4-space, $\mathbb{R} ^4$, this condition can be expressed as the requirement that Mn = I for an integer n, where M is the matrix representation of the hypothesised topological lensing isometry and I is the identity. Moreover, this test becomes even simpler with the requirement that the two rotation angles, $\theta,\phi$, corresponding to the given isometry, satisfy $ {2\pi \over \theta}, {2\pi \over \phi} \in {\mathbb{Z} }$. The calculation of this test involves finding the two eigenplanes of the matrix M. A GNU General Public Licence numerical package, called eigenplane, is made available for finding the rotation angles and eigenplanes of an arbitrary isometry M of S3.

Key words: cosmology: observations -- cosmological parameters -- cosmic microwave background -- quasars: general

© ESO 2005