The optimal phase of the generalised Poincaré dodecahedral space hypothesis implied by the spatial cross-correlation function of the WMAP sky maps

¹ Toruń Centre for Astronomy, N. Copernicus University, ul. Gagarina 11, 87100 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: 28 December 2007
Accepted: 13 May 2008

Abstract

Context. Small universe models predicted a cutoff in large-scale power in the cosmic microwave background (CMB). This was detected by the Wilkinson Microwave Anisotropy Probe (WMAP). Several studies have since 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, flat space. Both models assume an FLRW metric and are close to flat with about 30% matter density.

Aims. We study two predictions of the PDS model. (i) For the correct astronomical positioning of the fundamental domain, the spatial two-point cross-correlation function  of temperature fluctuations in the covering space (where the two points in any pair are on different copies of the surface of last scattering (SLS)) should have a similar order of magnitude to the auto-correlation function  on a single copy of the SLS. (ii) Consider a “generalised” PDS model for an arbitrary “twist” phase . The optimal orientation and identified circle radius for a generalised PDS model found by maximising relative to in the WMAP maps should yield one of the two twist angles ±36°.

Methods. Comparison of to extends the identified circles method, using a much larger number of data points. We optimise the ratio of these functions at scales 4.0 h^-1 Gpc using a Markov chain Monte Carlo (MCMC) method over orientation (l, b, θ), circle size α, and twist ϕ.

Results. Both predictions were satisfied: (i) an optimal generalised PDS solution was found for two different foreground-reduced versions of the WMAP 3-year all-sky map, both with and without the kp2 galactic contamination mask. This solution yields a strong cross-correlation between points which would be distant and only weakly correlated according to the simply connected hypothesis. The face centres are {(l,b)} ≈ { (184°, 62°), (305°, 44°), (46°, 49°), (117°, 20°), (176°, -4°), (240°, 13°)} (and their antipodes) to within ≈2°; (ii) this solution has twist ϕ = (+39 ± 2.5)°, in agreement with the PDS model. The chance of this occurring in the simply connected model, assuming a uniform distribution , is about 6-9%.

Conclusions. The PDS model now satisfies several different observational constraints.