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Figure 1: Estimate of the spatial auto-correlation function, , of temperature fluctuations in the ILC WMAP map. The estimate assumes a simply connected space with , , using 30 000 points randomly chosen from a uniform distribution on the 2-sphere, excluding points falling within the kp2 Galaxy mask. Correlations are shown in against comoving separation in h-1 Gpc ranging from zero separation up to the diameter of the SLS. An approximate conversion from spatial to angular separations on small scales (  h-1 Gpc) can be obtained by setting , i.e. .

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In the text

	Figure 2: Schematic diagram of temperature auto-correlations in a simply connected universe, showing ``typical'' temperature correlations for two pairs of points (i,j)= (1,2), (3,4) that are widely separated on a single copy of the surface of last scattering (SLS). Since the correlation is in general high at low separations and low at high separations, the product is low in both cases.
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In the text

Figure 3: Schematic diagram of temperature cross-correlations in a multiply connected universe (e.g. a Poincaré dodecahedral space, PDS, model), showing ``typical'' temperature correlations for two pairs of points (i,j)= (1,2), (3,4) that are widely separated on a single copy of the SLS, as in Fig. 2. The holonomy transformation g maps points in space to copies of themselves in the covering space. The points x1 and g(x1) are the same physical point, and similarly, x3 and g(x3) are physically identical. Since the comoving separations  d[g(x1),x2] and d[g(x3),x4] are both small, and the correlation is, in general, high at low separations, the product is high in both cases. This should only occur if the holonomy transformation g is physically correct. This is described statistically in Eq. (8).

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In the text

Figure 4: Surface of last scattering and its interior, i.e. the comoving 3-volume of the observable Universe as a subset of S3, embedded in for convenience. All lines appearing straight in this figure correspond to spatial geodesics in S3, i.e. arcs in . The triangle to the left illustrates the geometry of a matched circle. In the embedding, the PDS mapping from one matched circle to another rotates by between circle centres, so an arclength of half this (from the observer to one matched circle centre), i.e. multiplied by the 3-sphere radius, , is shown. The circle radius as measured on the SLS is shown as . The relevant equation describing this triangle is Eq. (15). The right-hand part of the figure shows half the geodesic distance d/2between two points xi, xj lying on the SLS and the angle  subtending this. The relevant equation describing this triangle is Eq. (16). The relations between angles and arclengths of the sides of these triangles are determined by spherical trigonometry.

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In the text

Figure 5: Full sky map showing the optimal orientation of dodecahedral face centres based on 100 000 steps in 10 MCMC chains, using the ILC map and either the kp2 mask ( upper panel) or no mask ( lower panel), showing face centres for which P > 0.5 (see Eq. (25)). The projection is a Lambert azimuthal equal area projection (Lambert 1772) of the full sky, centred on the North Galactic Pole (NGP). The  meridian is the positive vertical axis and galactic longitude increases clockwise. These face centres are derived from the MCMC chains without any constraint on the twist phase .

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In the text

	Figure 6: Full sky map showing the optimal dodecahedral face centres for the ILC map with the kp2 mask ( upper panel) and no mask ( lower panel), as for Fig. 5, but centred on the South Galactic Pole. The  meridian is the negative vertical axis and galactic longitude increases anticlockwise.
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In the text

	Figure 7: Full sky map showing the optimal dodecahedral orientation for the TOH map with the kp2 mask ( upper panel) and no mask ( lower panel), as for Fig. 5, centred on the North Galactic Pole.
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In the text

Figure 8: Distribution of and states in the MCMC chains of the dodecahedral solution used in Table 1. A single point is shown independently of whether the chain spends a long time at that point - implying that it is a highly probable point - or a shorter time at that point, so the density of points does not fully represent the information in the distribution. In addition to the main cluster of points with (see Table 2 for a precise estimate), a secondary weak cluster of points exists for , and an even weaker, tertiary cluster exists for .

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In the text

	Figure 9: Schematic diagram showing the approximately parallel positioning of two ``approximately matched annuli'' or ``approximately matched discs'' in two copies of the SLS, as per Fig. 4, when the matched circle angular radius  is relatively small. The centres of the two matched circles projected onto their respective spheres (copies of the SLS), i.e. P and P', are separated by .
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In the text

Figure 10: Full sky map showing the optimal dodecahedral orientation for the ILC map with the kp2 mask, as for Fig. 5, centred on the North Galactic Pole, from an MCMC chain starting at the PDS orientation and circle size suggested in Roukema et al. (2004), for an initial twist of . The optimal orientation is clearly very close to what is found from arbitrary initial positions, shown in the previous figures.

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In the text

	Figure 11: As for Fig. 1, together with an estimate of the spatial cross-correlation function, (thick curve), made assuming the Poincaré dodecahedral space with the best estimate parameters listed in Tables 1 and 2, corrected to an exact dodecahedral solution: in degrees. Values of are meaningful up to , half the distance from the centre of one face of the fundamental domain to the matching face.
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In the text

Figure 12: Matched circle pair 1: this and the following figures show matched circles for the best PDS model found here, corrected to an exact PDS solution as indicated in Fig. 11. Temperature fluctuations are from the ILC 3-year WMAP map using the kp2 galactic contamination mask, shown in mK against comoving distance along a circle in h-1 Gpc. The coordinates of the two dodecahedral face centres (l,b)i are indicated. Solid lines show the northern galactic member of a pair; dashed lines show the southern member. The pixels used for these plots contributed almost nothing to the method used for finding the optimal cross-correlation, since pairs of points at nearly zero implied spatial separation are rare. (See Sect. 4.7.)

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In the text

	Figure 13: Matched circle pair 2, as in Fig. 12.
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In the text

	Figure 14: Matched circle pair 3, as in Fig. 12. The kp2 galactic contamination mask cuts severely into these two circles.
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In the text

	Figure 15: Matched circle pair 4, as in Fig. 12. As in Fig. 14, the Galaxy cut is severe.
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In the text

	Figure 16: Matched circle pair 5, as in Fig. 12.
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In the text

	Figure 17: Matched circle pair 6, as in Fig. 12.
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In the text

	Figure 18: Full sky map showing the optimal dodecahedral orientation for the ILC map with the kp0 galactic contamination mask, as for Fig. 5, centred on the North Galactic Pole.
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In the text

	Figure 19: Full sky map showing the optimal dodecahedral orientation for the TOH map with the kp0 galactic contamination mask, as for Fig. 5, centred on the North Galactic Pole.
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In the text

	Figure 20: Full sky map showing the optimal dodecahedral orientation for , as in Fig. 5, centred on the North Galactic Pole ( upper panel), and the corresponding values in degrees (lower panel) for the WMAP Q frequency band map, in which the Galaxy very strongly dominates, with no galactic mask. Clearly, the signal is very different from what is found in the foreground corrected maps.
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In the text

Figure 21: Schematic derivation of the preferred twist of a matched circles pair implied by a map dominated by the Galactic Plane (GP). The SLS is shown by a circle. The GP is shown as a horizontal edge-on, solid disc. The sky map can be thought of as a binary map, zero everywhere except at the GP where it is positively valued. One copy, AB, of a matched circle is shown to the left, at an arbitrary angle with respect to the GP, in projection, containing point A, far from the GP, and B, intersecting with the GP. The circle intersects with the GP at only two points, B and another point behind B in this projection. Dashed arrows indicate the translation (with no twist) from one side of the sky to the other, so that A maps to A', B maps to B'. See Sect. 5.1.1.

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In the text

	Figure 22: Comparison of simulated ``generalised'' PDS maps with random input twists , to the estimated (output) twists  , both in degrees. The two correlate well, with no sign of any tendency to favour . The rms difference between and is . See Sect. 5.3 for details.
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In the text