§3 Cosmology and topology.

§ 3.1. Random point symmetry and multipole selection rules.

In addition to homotopy, the random nature of the CMBR suggests to demand point symmetry of the source. Homotopy together with point symmetry imply sharp selection rules for the multipole distribution of CMBR.

Given a spherical 3-manifold {\cal M} of polyhedral shape, its points serve as the model domain for amplitudes of the CMBR. One common assumption made in such models is randomness. Given a random function f^{{\rm random}} on {\cal M}, we may compare it with its transform obtained by a rotation which is a point symmetry of {\cal M}. It is argued in [41] that the random function f^{{\rm random}} should not depend on this rotation, in other words that it be invariant under point symmetry operations of {\cal M}. This assumption has observable consequences on the low multipole expansion of f^{{\rm random}}. In Table 3 taken from [41] we give as a function of the polyhedral point group M the multiplicity of the lowest six multipole orders. It can be seen from this Table that, depending on the symmetry group, the lowest multipole order l>0 occurs at a value up to l=6 for the dodecahedron and icosahedron.

\begin{array}[]{|l||l|l|l|l|l|l|l|l|l|l|l|}\hline l&{\cal C}_{2}&D_{2}&{\cal C}_{3}&D_{3}&{\cal C}_{4}&D_{4}&{\cal C}_{6}&D_{6}&T&O&{\cal J}\\
\hline 0&1&1&1&1&1&1&1&1&1&1&1\\
\hline 1&1&0&1&0&1&0&1&0&0&0&0\\
\hline 2&3&2&1&1&1&1&1&1&0&0&0\\
\hline 3&3&1&3&1&1&0&1&0&1&0&0\\
\hline 4&5&3&3&2&3&2&1&1&1&1&0\\
\hline 5&5&2&3&1&3&1&1&0&0&0&0\\
\hline 6&7&4&5&3&3&2&3&2&2&1&1\\
\hline\end{array} (19)
Table 3. The multiplicity m(\Gamma _{1}\downarrow l) for the values l,\: 0\leq l\leq 6 of the multipole order and selected point groups M in the notation of [42], assuming a function invariant under the point group.

The last three columns of this Table apply with T to the Platonic tetrahedron, with O to the cube and octahedron, and with {\cal J} to the dodecahedron and icosahedron respectively. The assumption of point symmetry for example excludes the quadrupole order l=2 for the regular tetrahedron, cube, octahedron, dodecahedron and icosahedron. To include quadrupole contributions one must relax the point symmetry while keeping the homotopic boundary conditions, or turn to polyhedra of lower point symmetry.