CMB lensing tomography with clustering estimation of lens redshift distributions

We develop a clustering-based redshift estimation approach for CMB lensing tomography, focusing on the kernel function of the lensing galaxies. Within a linear galaxy bias framework, we derive estimators for this kernel from two-point cross-correlations between lens mass and reference samples. The reconstructed kernel then enables a theoretical prediction for the angular cross-power spectrum $C_{g\kappa }$ between CMB lensing convergence and lens galaxies. As a proof of concept, we measure $C_{g\kappa }$ by correlating the \emph{Planck} PR4 convergence map with NVSS+SUMSS radio galaxies ($0\lesssim z\lesssim 3$). We estimate the radio-galaxy kernel by collectively cross-correlating their distribution with spectroscopic and photometric surveys (2MPZ, LOWZ-CMASS, eBOSS DR16 LRGs, and Gaia-unWISE QSOs). From the measured $C_{g\kappa }$, we obtain ${\sigma }_8=0.86_{-0.09}^{+0.12}$ when the density parameter is set to the {\it Planck} value of ${\mathrm{\Omega }}_m=0.315$; this is in good agreement with the \emph{Planck} normalisation of ${\sigma }_8=0.812$.