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.

The 3-torus (T 3 ) Friedmann-Lemaître-Robertson-Walker (FLRW) model better fits the nearly zero large-scale two-point auto-correlation of the Wilkinson Microwave Anisotropy Probe (WMAP) cosmic microwave background (CMB) sky maps than the infinite flat model. The T 3 model's parameters, recently found using an optimal cross-correlation method on WMAP data, imply approximately equal-redshift topological lensing at redshifts z ∼ 6, the redshift range of the upcoming generation of new instruments and telescopes. We investigate observational strategies that can reject the T 3 solution for a given region of parameter space of physical assumptions, or provide good candidate topologically lensed galaxy pairs for detailed spectroscopic followup. T 3 holonomies are applied to (i) existing z ∼ 6 observations and (ii) simulated observations, creating multiply connected catalogues. Corresponding simply connected catalogues are generated. The simulated observational strategies are motivated by the matched discs principle. Each catalogue is analysed using a successive filter method and collecting matched quadruples. Quadruple statistics between the multiply and simply connected catalogues are compared. The expected rejection of the hypothesis, or detection of candidate topologically lensed galaxies, is possible at a significance of 5% for a pair of T 3 axis-centred northern and southern surveys if photometric redshift accuracy is σ(z phot ) < ∼ 0.01 for a pair of nearly complete 100 deg 2 surveys with a total of > ∼ 500 galaxies over 4.3 < z < 6.6, or for a pair of 196 deg 2 surveys with > ∼ 400 galaxies and σ(z phot ) < ∼ 0.02 over 4 < z < 7. Dropping the maximum time interval in a pair from ∆t = 1 h −1 Gyr to ∆t = 0.1 h −1 Gyr requires σ(z phot ) < ∼ 0.005 or σ(z phot ) < ∼ 0.01, respectively. Millions of z ∼ 6 galaxies will be observed over fields of these sizes during the coming decades, implying much stronger constraints. The question is not if the hypothesis will be rejected or confirmed, it is when.

The backreaction of inhomogeneities describes the effect of inhomogeneous structure on average properties of the Universe. We investigate this approach by testing the consistency of cosmological N -body simulations as non-linear structure evolves. Using the Delaunay Tessellation Field Estimator (DTFE), we calculate the kinematical backreaction Q from simulations on different scales in order to measure how much N -body simulations should be corrected for this effect. This is the first step towards creating fully relativistic and inhomogeneous N -body simulations. In this paper we compare the interpolation techniques available in DTFE and illustrate the statistical dependence of Q as a function of length scale.

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