A hint of Poincaré dodecahedral topology in the WMAP first year sky map

¹ Toruń Centre for Astronomy, N. Copernicus University, ul. Gagarina 11, 87-100 Toruń, Poland e-mail: boud@astro.uni.torun.pl
² Nicolas Copernicus Astronomy Centre, ul. Bartycka 18, 00-716 Warsaw, Poland

Received: 25 February 2004
Accepted: 26 May 2004

Abstract

It has recently been suggested by Luminet et al. ([CITE]) that the WMAP data are better matched by a geometry in which the topology is that of a Poincaré dodecahedral model and the curvature is “slightly” spherical, rather than by an (effectively) infinite flat model. A general back-to-back matched circles analysis by Cornish et al. ([CITE]) for angular radii in the range , using a correlation statistic for signal detection, failed to support this. In this paper, a matched circles analysis specifically designed to detect dodecahedral patterns of matched circles is performed over angular radii in the range on the one-year WMAP data. Signal detection is attempted via a correlation statistic and an rms difference statistic. Extreme value distributions of these statistics are calculated for one orientation of the 36° “screw motion” (Clifford translation) when matching circles, for the opposite screw motion, and for a zero (unphysical) rotation. The most correlated circles appear for circle radii of  ± , for the left-handed screw motion, but not for the right-handed one, nor for the zero rotation. The favoured six dodecahedral face centres in galactic coordinates are   and their opposites. The six pairs of circles independently each favour a circle angular radius of 11 ± . The temperature fluctuations along the matched circles are plotted and are clearly highly correlated. Whether or not these six circle pairs centred on dodecahedral faces match via a rotation only due to unexpected statistical properties of the WMAP ILC map, or whether they match due to global geometry, it is clear that the WMAP ILC map has some unusual statistical properties which mimic a potentially interesting cosmological signal.