The size of the Universe according to the Poincaré dodecahedral space hypothesis

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

Received: 3 June 2011
Accepted: 20 July 2011

Abstract

Context. One of the Friedmann-Lemaître-Robertson-Walker (FLRW) models that best fits the Wilkinson microwave anisotropy probe (WMAP) sky maps of the cosmic microwave background is that whose comoving space is the Poincaré dodecahedral space. The optimal fit of this model to WMAP data was recently found using an optimal cross-correlation method. For geometrical reasons, there was concern that systematic error in the estimate of the matched-circle (observer-centred) angular radius α, or equivalently, the (comoving) size of the Universe 2r_inj (twice the injectivity radius), might be much higher than the random error.

Aims. In order to increase the falsifiability of the model, especially by multiple imaging of collapsed objects, it would be useful to reduce the uncertainty in this estimate and to estimate the fraction of the sky where multiply imaged gravitationally bound objects should potentially be detectable.

Methods. A corollary of the matched circles principle – the existence of matched discs – is introduced in order to describe a useful subset of multiply imaged objects. The cross-correlation method at  ≲ 1 h^-1 Gpc is applied to WMAP 7-year data in order to improve the estimate of α.

Results. The improved matched-circle radius estimate is α = 23  ±  1.4°, where the uncertainty represents systematic error dependent on the choices of galactic mask and all-sky map. This is equivalent to 2r_inj = 18.2  ±  0.5 h^-1 Gpc for matter density parameter Ω_m = 0.28  ±  0.02. The lowest redshift of multiply imaged objects is z = 106  ±  18. Multiply imaged high overdensity (rare) peaks visible during 200 > z > 106 should be present in matched discs of radius 14.8  ±  2.3°.

Conclusions. The accuracy in the matched circle radius estimate is considerably improved by using the higher resolution signal. The predicted matched discs (over 200 > z > 106) project to about 20% of the full sky. Since any object located exactly in the discs would be multiply imaged at equal redshifts, evolutionary effects would be small for objects that are nearly located in the discs.