on the $\ell$-$k$ plane that can in principle be used to separate contaminants from the cosmological signal.

To elucidate the contrast in these signatures, suppose we discard the assumption (from previous derivations) of translationally invariant statistics. In general, the two-point correlator will cease to exhibit the diagonal form given by Eq. . As a concrete example of this, consider a random temperature field that is statistically isotropic but not homogeneous. In the radial direction, suppose this field has some fixed (non-random and angular position-independent) radial dependence. Such a field would be an appropriate description for a (hypothetical) population of unresolved point sources. Under these assumptions, Eq. reduces to $$\overline{T}_{\ell m} (k) = a_{\ell m} q_\ell (k),$$ where $$\label{eq:qellk}
q_\ell (k) \equiv \sqrt{\frac{2}{\pi}} \int_0^\infty dr r^2 j_\ell (kr) \kappa (\nu_r),$$ with $\kappa (\nu_r)$ specifying the spectral (and therefore radial) dependence of our hypothetical sky as it appears in our data. The two-point correlator then becomes $$\langle \overline{T}_{\ell m} (k) \overline{T}_{\ell^\prime m^\prime}^* (k^\prime) \rangle  = C_\ell q_\ell(k) q_\ell (k^\prime) \delta_{\ell \ell^\prime} \delta_{m m^\prime},$$ where statistical rotation invariance of the field allows us to invoke relation $\langle a_{\ell m} a_{\ell^\prime m^\prime}^* \rangle \equiv C_\ell \delta_{\ell \ell^\prime} \delta_{m m^\prime}$, with $C_\ell$ signifying the angular 