Derivation

The temperature and polarization distribution of the radiation is in general a function of both spatial position $\vec{x}$ and angle $\vec{n}$ defining the propagation direction. In flat space, we know that plane waves form a complete basis for the spatial dependence. Thus a spin- field like the temperature may be expanded in  

$$G_\ell^m = (-i)^{\ell} \sqrt{ 4\pi \over 2\ell+1} Y_\ell^m(\hat{n}) \exp(i \vec{k} \cdot \vec{x}) \, ,$$ (8)

where the normalization is chosen to agree with the standard Legendre polynomial conventions for m=0. Likewise a spin-2 field like the polarization may be expanded in  

$${}_{\pm 2}G_\ell^m = (-i)^\ell \sqrt{ {4\pi \over 2\ell+1}}... ...tom{m}} Y_{\ell}^{m}(\hat{n})] \exp(i\vec{k} \cdot \vec{x})\, .$$ (9)

The plane wave itself also carries an angular dependence of course,  

$$\exp( i \vec{k} \cdot \vec{x} ) = \sum_\ell (-i)^\ell \sqrt{4\pi(2\ell+1)} j_\ell(kr) Y_\ell^0(\hat{n}) \, ,$$ (10)

where $\hat{e}_3 = \hat{k}$ and $\vec{x} = - r \hat{n}$ (see Fig. 2). The sign convention for the direction is opposite to direction on the sky to be in accord with the direction of propagation of the radiation to the observer. Thus the extra factor of $(-1)^\ell$ comes from the parity relation Eqn. (6).

  

Figure 2: Projection effects. A plane wave $\exp(i \vec{k} \cdot \vec{x})$ can be decomposed into $j_\ell(kr) Y_\ell^0$ and hence carries an ``orbital'' angular dependence. A plane wave source at distance r thus contributes angular power to $\ell \approx kr$ at $\theta=\pi/2$ but also to larger angles $\ell \ll kr$ at $\theta=0$ which is encapsulated into the structure of $j_\ell$(see Fig. 3). If the source has an intrinsic angular dependence, the distribution of power is altered. For an aligned dipole $Y_1^0 \propto \cos\theta$ (`figure 8's) power at $\theta=\pi/2$ or $\ell \approx kr$ is suppressed. These arguments are generalized for other intrinsic angular dependences in the text.

The separation of the mode functions into an intrinsic angular dependence and plane-wave spatial dependence is essentially a division into spin ($\, {}_{s}^{\vphantom{m}} Y_{\ell'}^{m}$) and orbital ($Y_{\ell}^0$) angular momentum. Since only the total angular dependence is observable, it is instructive to employ the Clebsch-Gordan relation of Eqn. (8) to add the angular momenta. In general this couples the states between $\vert\ell -\ell'\vert$ and $\ell+\ell'$. Correspondingly a state of definite total $\ell$ will correspond to a weighted sum of $j_{\vert\ell-\ell'\vert}$ to $j_{\ell + \ell'}$ in its radial dependence. This can be reexpressed in terms of the $j_\ell$ using the recursion relations of spherical Bessel functions,

$$\begin{array} {rcl}\displaystyle{}{j_\ell(x) \over x} &=& {1... ...\ell j_{\ell-1}(x) - (\ell +1) j_{\ell+1}(x) ] \,.\end{array}$$

We can then rewrite  

$$G_{\ell'}^{m} = \sum_\ell (-i)^\ell \sqrt{4\pi(2\ell+1)} \, j_\ell^{(\ell' m)}(kr) \, Y_\ell^{m}(\hat {n}) \, ,$$ (11)

where the lowest $(\ell',m)$ radial functions are  

$$\begin{array} {lll} j_\ell^{(00)}(x) = j_\ell(x) \, ,\qquad ... ...!}} \, {j_\ell(x) \over x^2} } \vphantom{\Bigg[},\end{array}$$ (12)

with primes representing derivatives with respect to the argument of the radial function x=kr. These modes are shown in Fig. 3.

  

Figure 3: Radial spin-0 (temperature) modes. The angular power in a plane wave (left panel, top) is modified due to the intrinsic angular structure of the source as discussed in the text. The left panel corresponds to the power in scalar (m=0) monopole G₀⁰, dipole G₁⁰, and quadrupole G₂⁰ sources (top to bottom); the right panel to that in vector (m=1) dipole $G_1^{\pm 1}$and quadrupole $G_2^{\pm 1}$sources and a tensor (m=2) quadrupole $G_2^{\pm 2}$ source (top to bottom). Note the differences in how sharply peaked the power is at $\ell \approx kr$ and how fast power falls as $\ell \ll kr$.The argument of the radial functions kr=100 here.

Postscript: (a)(b)

Similarly for the spin $\pm 2$ functions with m>0 (see Fig. 4),  

$${}_{\pm 2} G_2^m = \sum_\ell (-i)^\ell \sqrt{4\pi(2\ell+1)... ...)] \, \, {}_{\pm 2}^{\vphantom{m}} Y_{\ell}^{m}(\hat{n}) \, ,$$ (13)

where 

$$\begin{array} {rcl}\displaystyle{}\epsilon^{(0)}_\ell(x) & =... ...(x) \over x^2} + 4{j_\ell'(x) \over x} \right] \, ,\end{array}$$

which corresponds to the $\ell'=\ell,\ell\pm 2$ coupling and 

$$\begin{array} {rcl}\displaystyle{}\beta^{(0)}_\ell(x) &=& 0\... ...ft[ j_\ell'(x) + 2 {j_\ell(x) \over x} \right] \, ,\end{array}$$

which corresponds to the $\ell' = \ell \pm 1$ coupling. The corresponding relation for negative m involves a reversal in sign of the $\beta$-functions

$$\begin{array} {rcl}\displaystyle{}\epsilon_\ell^{(-m)} & = &... ...number\\ \beta_\ell^{(-m)} & = & -\beta_\ell^{(m)} .\end{array}$$

These functions are plotted in Fig. 4. Note that $\epsilon^{(0)}_\ell = j^{(2)}_\ell$ is displayed in Fig. 3.

  

Figure 4: Radial spin-2 (polarization) modes. Displayed is the angular power in a plane-wave spin-2 source. The top panel shows that vector (m=1, upper panel) sources are dominated by B-parity contributions, whereas tensor (m=2, lower panel) sources have comparable but less power in the B-parity. Note that the power is strongly peaked at $\ell = kr$ for the B-parity vectors and E-parity tensors. The argument of the radial functions kr=100 here.