Poincaré dodecahedral space parameter estimates

¹ Toruń Centre for Astronomy, Nicolaus Copernicus University, ul. Gagarina 11, 87-100 Toruń, Poland e-mail: boud@astro.uni.torun.pl
² École nationale supérieure de physique de Strasbourg, Université Louis Pasteur, Bd. Sébastien Brant, BP 10413, 67412 Illkirch Cedex, France

Received: 26 July 2008
Accepted: 25 October 2008

Abstract

Context. Several studies have proposed that the preferred model of the comoving spatial 3-hypersurface of the Universe may be a Poincaré dodecahedral space (PDS) rather than a simply connected, infinite, flat space.

Aims. Here, we aim to improve the surface of last scattering (SLS) optimal cross-correlation method and apply this to observational data and simulations.

Methods. For a given “generalised” PDS orientation, we analytically derive the formulae required to exclude points on the sky that cannot be members of close SLS-SLS cross-pairs. These enable more efficient pair selection without sacrificing the uniformity of the underlying selection process. For a sufficiently small matched circle size α and a fixed number of randomly placed points selected for a cross-correlation estimate, the calculation time is decreased and the number of pairs per separation bin is increased. Using this faster method, and including the smallest separation bin when testing correlations, (i) we recalculate Monte Carlo Markov Chains (MCMC) on the five-year Wilkinson Microwave Anisotropy Probe (WMAP) data; and (ii) we seek PDS solutions in a small number of Gaussian random fluctuation (GRF) simulations in order to further explore the statistical significance of the PDS hypothesis.

Results. For , a calculation speed-up of 3–10 is obtained. (i) The best estimates of the PDS parameters for the five-year WMAP data are similar to those for the three-year data; (ii) comparison of the optimal solutions found by the MCMC chains in the observational map to those found in the simulated maps yields a slightly stronger rejection of the simply connected model using α rather than the twist angle ϕ. The best estimate of α implies that, given a large-scale auto-correlation as weak as that observed, the PDS-like cross-correlation signal in the WMAP data is expected with a probability of less than about 10%. The expected distribution of ϕ from the GRF simulations is not uniform on .

Conclusions. Using this faster algorithm, we find that the previous PDS parameter estimates are stable to the update to five-year WMAP data. Moreover, for an infinite, flat, cosmic concordance model with Gaussian random fluctuations, the chance of finding both (a) a large-scale auto-correlation as weak as observed; and (b) a PDS-like signal similar to what is observed is less than about 0.015% to 1.25%.