$\overline{T}_{\ell m} (k)$ to $P(k)$, we combine Eqs. , , and to obtain $$\label{eq:YetAnotherTellm}
\overline{T}_{\ell m} (k) = \sqrt{\frac{2}{\pi}} \int \! \frac{d^3 k^\prime}{(2 \pi)^3} \widetilde{T} (\mathbf{k}^\prime) \int \! d^3 r\, j_\ell (kr) Y_{\ell m}^*({\hat{\mathbf{r}}}) e^{i \mathbf{k}^\prime \cdot \mathbf{r}}.$$ To simplify this, we expand $e^{i \mathbf{k}^\prime \cdot \mathbf{r}}$ in spherical harmonics using the identity $$\label{eq:PlaneWaveSphericalHarmonicExpansion}
e^{i \mathbf{k} \cdot \mathbf{r}} = 4\pi \sum_{\ell m} i^\ell j_\ell (kr) Y_{\ell m}^* ({\hat{\mathbf{k}}}) Y_{\ell m} ({\hat{\mathbf{r}}}),$$ which leads to $$\label{eq:TlmTkConversion}
\overline{T}_{\ell m} (k) = \frac{i^\ell}{(2\pi)^{\frac{3}{2}}} \int \frac{d^3 k^\prime}{k k^\prime} Y_{\ell m}^* ({\hat{\mathbf{k}}}^\prime) \delta^D (k - k^\prime) \widetilde{T} (\mathbf{k}^\prime).$$ This provides a link between the temperature field as expressed in our $(k,\ell, m)$ basis, and the same field in the rectilinear Fourier basis. Taking a cue from Eq. , where the power spectrum is closely related to the two-point correlation between different rectilinear Fourier modes, we may form a two-point correlator between different modes in our spherical Fourier-Bessel basis, giving $$\begin{aligned}
\label{eq:CurvedPspecDef}
\langle \overline{T}_{\ell m} (k) \overline{T}_{\ell^\prime m^\prime}^* (k^\prime) \rangle && = \frac{i^\ell (-i)^{\ell^\prime}}{(2\pi)^3} \!\! \int \frac{d^3 k_1}{k k_1} \frac{d^3 k_2}{k^\prime k_2}  \nonumber \\
&& \qquad \times Y_{\ell m}^* ({\hat{\mathbf{k}}}_1) Y_{\ell^\prime m^\prime} ({\hat{\mathbf{k}}}_2) \langle \widetilde{T} (\mathbf{k}_1) \widetilde{T} (\mathbf{k}_2)^* \rangle\nonumber \\
&& \qquad \times \delta^D (k - 