## 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.

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.