Well-proportioned universes suppress the cosmic microwave background quadrupole

A widespread myth asserts that all small universe models suppress the cosmic microwave background (CMB) quadrupole. In actual fact, some models suppress the quadrupole while others elevate it, according to whether their low-order modes are weak or strong relative to their high-order modes. Elementary geometrical reasoning shows that a model's largest dimension determines the rough value l(min) at which the CMB power spectrum l(l + 1) C-l/2pi effectively begins; for cosmologically relevant models, l(min) less than or equal to3. More surprisingly, elementary geometrical reasoning shows that further reduction of a model's smaller dimensions - with its largest dimension held fixed - serves to elevate modes in the neighbourhood of l(min) relative to the high-l portion of the spectrum, rather than suppressing them as one might naively expect. Thus among the models whose largest dimension is comparable to or less than the horizon diameter, the low-order C-l tend to be relatively weak in well-proportioned spaces (spaces whose dimensions are approximately equal in all directions) but relatively strong in oddly proportioned spaces (spaces that are significantly longer in some directions and shorter in others). We illustrate this principle in detail for the special cases of rectangular 3-tori and spherical spaces. We conclude that well-proportioned spaces make the best candidates for a topological explanation of the low CMB quadrupole observed by COBE and WMAP.