Correlation pattern of CMB skies in
Compact Hyperbolic Universe

In the Universe with topologically compact space, small enough to fit within the sphere of last scattering, the CMB temperature is expected to be identical along the pairs of circles if temperature fluctuations are dominated by the surface terms at the last-scattering sphere and no smoothing or pixelization is done (Cornish, Spergel, Starkman, 1996). We show the matched pairs of circles in two ("small" and "large") CH models superimposed on random realizations of the theoretical sky generated from computed pixel-pixel correlation matrix in these models (Bond, Pogosyan, Souradeep, 1997). The physical value of the curvature radius dc (or ) in both cases is such that the volume of the space VD is close to the volume of the last-scattering sphere VLs.

Small m004(-5,1) space with =0.9 has VD=0.98, VD/VLs=0.85. The Dirichlet domain of this space is quite symmetric. It has 16 closest neighbours, 12 of which are at the proper distance R=1.1 dc and the rest four - at R=1.2 dc.

Large v3543(2,3) space with =0.6 has VD=6.45, VD/VLs=0.34. The Dirichlet domain of this space is rather squashed. It has 38 closest neighbours, separated by the proper distance varying from R=1.8 dc to R=2.5 dc.

Small space with =0.9 has 8 pairs of circles, shown in matching colours. The angular size of the 12 larger circles is 60 degrees, while 4 smaller ones span 37 degrees on the sky. Points marked by triangles and diamonds on one circle in a pair are identified with corresponding points on the second circle. CMB temperature fluctuations for =0.9 are dominated by the surface effect at last-scattering sphere and Even after smoothing and pixelization as was done for the map below, the repetition of the pattern as one follows the circles from triangles towards the diamonds and beyond is quite pronounced.

CMB temperature along the pairs of correlated circles in the "small" model

The CH model with small 3-volume and =0.9. dT/T in theoretical realization is plotted at pixels (level 5 pixelization) nearest to the circles and traversed in the direction from diamonds to triangles. Each box contains the temperature cross-correlation coefficient between pairs of circles. back

Large space with =0.6 has 35 pairs of correlated circles out of which 8 pairs of the largest size are shown superimposed on a theoretical sky realization. The angular size of these large circles range from 74 down to 57 degrees. Points marked by triangles and diamonds on one circle in a pair are identified with corresponding points on the second circle. The pattern as one follows the circles from triangles towards the diamonds comes below.
The CH model with large 3-volume and =0.6 dT/T in theoretical realization is plotted at pixels (level 5 pixelization) nearest to the circles and traversed in the direction from diamonds to triangles. Each box contains the temperature cross-correlation coefficient between pairs of circles.