The structure of these functions is readily apparent from geometrical considerations. A single plane wave contributes to a range of angular scales from at to larger angles as , where (see Fig. 1). The power in of a single plane wave shown in Fig. 3(a) (top panel) drops to zero , has a concentration of power around and an extended low amplitude tail to .
Now if the plane wave is multiplied by an intrinsic angular dependence, the projected power changes. The key to understanding this effect is to note that the intrinsic angular behavior is related to power in as
Similarly, the structures of , and are apparent from the intrinsic angular dependences of the G11, G21 and G22 sources,
There are two interesting consequences of this behavior. The sharpness of the radial function around quantifies how faithfully features in the k-space spectrum are preserved in -space. If all else is equal, this faithfulness increases with |m| for G|m|m due to aliasing suppression from On the other hand, features in G|m|+1m are washed out in comparison due projection suppression from the factor.
Secondly, even if there are no contributions from long wavelength sources with , there will still be large angle anisotropies at which scale as
The same arguments apply to the spin-2 functions with the added complication of the appearance of two radial functions and . The addition of spin-2 angular momenta introduces a -contribution from except for m=0. For , the -contribution strongly dominates over the -contributions; whereas for , -contributions are slightly larger than -contributions (see Fig. 4). The ratios reach the asymptotic values of
Now let us consider the low tail of the spin-2 radial functions. Unlike the spin-0 projection, the spin-2 projection allows increasingly more power at and/or ,i.e. , as |m| increases (see Table 1 and note the factors of ). In this limit, the power in a plane wave fluctuation goes as
Finally it is interesting to consider the cross power between spin- and spin-2 sources because it will be used to represent the temperature-polarization cross correlation. Again interesting geometric effects can be identified (see Fig. 5). For m=0, the power in correlates (Fig. 5, top panel solid line, positive definite); for m= 1, oscillates (short dashed line) and for m=2, anticorrelates (long dashed line, negative definite). The cross power involves only due to the opposite parity of the modes.
These properties will become important in § III and IVB and translates into cross power contributions with opposite sign between the scalar monopole temperature cross polarization sources and tensor quadrupole temperature cross polarization sources . Vector dipole temperature and polarization sources do not contribute strongly to the cross power since correlations and anticorrelations in will cancel when modes are superimposed. The same is true of the scalar dipole temperature cross polarization as is apparent from Figs. 3 and 4. The vector cross power is dominated by quadrupole temperature and polarization sources (Fig. 5 lower panel).