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Interpretation

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 Thus factors of in the intrinsic angular dependence suppress power at (aliasing suppression''), whereas factors of suppress power at (projection suppression''). Let us consider first a m=0 dipole contribution (see Fig. 2). The dependence suppresses power in at the peak in the plane-wave spectrum (compare Fig. 3(a) top and middle panels). The remaining power is broadly distributed for . The same reasoning applies for Y20 quadrupole sources which have an intrinsic angular dependence of . Now the minimum falls at causing the double peaked form of the power in shown in Fig. 3(a) (bottom panel). This series can be continued to higher and such techniques have been used in the free streaming limit for temperature anisotropies [11].

Similarly, the structures of , and are apparent from the intrinsic angular dependences of the G11, G21 and G22 sources,
 (14)
respectively. The factors imply that as m increases, low power in the source decreases (compare Fig. 3(a,b) top panels). G21 suffers a further suppression at from its factor.

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
 (15)
This scaling puts an upper bound on how steeply the power can rise with that increases with |m| and hence a lower bound on the amount of large relative to small angle power that decreases with |m|.

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
 (16)
for fixed . These considerations are closely related to the parity of the multipole expansion. Although the orbital angular momentum does not mix states of different spin, it does mix states of different parity since the plane wave itself does not have definite parity. A state with electric'' parity in the intrinsic angular dependence (see Eqn. 9) becomes  Thus the addition of angular momentum of the plane wave generates magnetic'' B-type parity of amplitude out of an intrinsically electric'' E-type source as well as E-type parity of amplitude .Thus the behavior of the two radial functions has significant consequences for the polarization calculation in §IIIC and implies that B-parity polarization is absent for scalars, dominant for vectors, and comparable to but slightly smaller than the E-parity for tensors.

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
 (17)
Comparing these expressions with Eqn. (22), we note that the spin- and spin-2 functions have an opposite dependence on m. The consequence is that the relative power in large vs. small angle polarization tends to decrease from the m=2 tensors to the m=0 scalars.

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.

Figure 5: Spin-0 Spin-2 (temperature polarization) modes. Displayed is the cross angular power in plane wave spin-0 and spin-2 sources. The top panel shows that a scalar monopole (m=0) source correlates with a scalar spin-2 (polarization quadrupole) source whereas the tensor quadrupole (m=2) anticorrelates with a tensor spin-2 source. Vector dipole (m=1) sources oscillate in their correlation with vector spin-2 sources and contribute negligible once modes are superimposed. One must go to vector quadrupole sources (lower panel) for a strong correlation. The argument of the radial functions kr=100 here.

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 [9]. 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).

Next: Perturbation Classification Up: Radial Modes Previous: Derivation
Wayne Hu
9/9/1997