Radial Functions

  It often useful to represent the eigenmodes in a spherical coordinate system $(\chi,\theta,\phi)$ where $\chi$is the radial coordinate scaled to the curvature radius. Here we explicitly write down the forms and properties of the radial modes in an open geometry and describe the modifications necessary to treat closed geometries.

By separation of variables in the Laplacian, we can write  

$$\, {}_{s}^{\vphantom{m}} {G}_{j}^{m} = \sum_\ell (-i)^\ell ... ...u) \, \, {}_{s}^{\vphantom{m}} {Y}_{\ell}^{m}({\hat{n}}) \, ,$$ (50)

and the goal is to find explicit expressions for $\, {}_{s}^{\vphantom{(j m)}} {\alpha}_{\ell}^{(j m)}$. Here the $\ell$-weights are set to reproduce the flat space conventions of spherical Bessel functions (see also [1]). We proceed by analyzing the lowest $j = {\rm min}(\vert s\vert,\vert m\vert)$ harmonic 

$$\begin{array} {rcl}\, {}_{0}^{\vphantom{m}} G_{j}^{m} &=& n^... ...t{m}_1 \pm i \hat{m}_2)^{i_2} Q_{i_1 i_2}^{(m)} \,,\end{array}$$

where $\hat{m}_1$ and $\hat{m}_2$ form a right-handed orthonormal basis with $\hat{n}$.We can now determine $\, {}_{s}^{\vphantom{(j m)}} {\alpha}_{\ell}^{(j m)}$ from the radial representation of ${\bf Q}^{(m)}$ [18] 

$$\begin{array} {rcl}\phi_{\ell}^{(00)}(\chi,\nu) & = & \Phi... ...u^2+1)}} {\rm csch}^2\chi\, \Phi_\ell^\nu(\chi) \,,\end{array}$$

for $\, {}_{0}^{\vphantom{(m m)}} {\alpha}_{\ell}^{(m m)}=\phi_\ell^{(m m)}$;similarly for $\, {}_{\pm 2}^{\vphantom{(2 m)}} {\alpha}_{\ell}^{(2 m)}=\epsilon_\ell^{(m)}\pm i\beta_\ell^{(m)}$, 

$$\begin{array} {rcl}\epsilon_{\ell}^{(0)}(\chi,\nu) & = & \sq... ...coth}^2 \chi \right) \Phi_\ell^\nu(\chi) \right]\,,\end{array}$$

and  

$$\begin{array} {rcl}\beta_{\ell}^{(0)}(\chi,\nu) &=& 0 \,, \n... ...(\chi)+2{\rm coth}\chi\Phi_\ell^\nu(\chi) \right]\,,\end{array}$$

for m > 0. For m<0, $\beta_\ell^{(-m)} = -\beta_\ell^{(m)}$ while the other two functions remain the same. Here $\Phi_\ell^\nu(\chi)$ is the hyperspherical Bessel function whose properties are discussed extensively by [6].

The overall normalization of the modes here has been altered from those of [6,18] in the case of vector and tensor temperature modes such that

$$\, {}_{s}^{\vphantom{(j m)}} {\alpha}_{\ell}^{(j m)}(0,\nu) = {1 \over 2\ell+1}\delta_{\ell,j} \, ,$$ (51)

where the difference lies in the lack of curvature dependence in the relation. Our choice simplifies the equations since it preserves the flat space form of the equations locally around the origin. It also defines the normalization of the polarization modes with respect to Q_ij^(m) through Eq. (B2).

The properties of the hyperspherical Bessel functions imply useful properties for the radial functions. For our purposes, the important relations they obey are:

$$\begin{array} {rcl}{d \over d\chi}\Phi_\ell^\nu& = & {1 \ove... ...sqrt{\nu^2 + (\ell+1)^2} \Phi_{\ell+1}^\nu\right]\,,\end{array}$$

which define the series in terms of its first member

$$\Phi_0^\nu= {\sin{\nu\chi} \over \nu\sinh{\chi}}.$$ (52)

Notice that $\lim_{K \rightarrow 0} \Phi_\ell^\nu(\chi) = j_\ell(kr)$.

From the recursion relations of $\Phi_\ell^\nu$ one establishes the corresponding relations for the radial function  

$${d \over d\chi} [\, {}_{s}^{\vphantom{(j m)}} {\alpha}_{\ell... ...\, {}_{s}^{\vphantom{(j m)}} {\alpha}_{\ell}^{(j m)}\right]\, ,$$ (53)

for the lowest j, where recall

$$\, {}_{s}^{\vphantom{m}} {\kappa}_{\ell}^{m} =\sqrt{\left[ ... ...)\over \ell^2}\right] \left[ 1+{\ell^2\over\nu^2}\right] }\, .$$ (54)

The construction of the higher $\, {}_{s}^{\vphantom{m}} {G}_{\ell}^{m}$ via the recursion relation of Eq. (25) also returns the higher radial harmonics. A few useful ones are 

$$\begin{array} {rcl}\phi_{\ell}^{(10)}(\chi,\nu) & = & \sqrt{... ...eft[ {\rm csch}\chi \Phi_\ell^\nu(\chi) \right]' \,.\end{array}$$

Furthermore, the recursion relation obeyed by the higher radial harmonics is the same as Eq. (B9), by virtue of Eq. (C5) and explicit substitution of the radial form Eq. (C3). This j-independence of the recursion relation implies that $\phi_\ell^{(j m)}$ is a solution to the temperature hierarchy Eq. (33) for any j and aids in the construction of the integral solutions in §IIID.

Finally, the radial functions for a closed geometry follow by replacing all $\nu^2 + n$, where n is integer, with $\nu^2 - n$ and trigonometric functions with hyperbolic trigonometric functions (see [6,18] for details).