Effect of redshift bin mismatch on cross correlation between DESI Legacy Imaging Survey and Planck CMB lensing potential

We study the importance of precise modelling of the photometric redshift error distributions on the estimation of parameters from cross correlation measurements and present a working example of the scattering matrix formalism to correct for the redshift bin mismatch of objects in tomographic cross correlation analysis. We measured the angular galaxy auto-power spectrum and cross-power spectrum in four tomographic bins with redshift intervals $z = [0.0,0.3,0.45,0.6,0.8]$ from the cross correlation of Planck Cosmic Microwave Background lensing potential and photometric galaxy catalogue from the Dark Energy Spectroscopic Instrument Legacy Imaging Survey Data Release 8. We estimated galaxy linear bias and amplitude of cross correlation using maximum likelihood estimation to put constraints on the $\sigma_{8}$ parameter. We show that the modified Lorentzian function used to fit the photometric redshift error distribution performs well only near the peaks of the distribution. We adopt a sum of Gaussians model to capture the broad tails of the error distribution. Our sum of Gaussians model yields $\sim 2-5\,\sigma$ smaller values of cross correlation amplitude compared to the $\Lambda$CDM expectations. We compute the $\sigma_{8}$ parameter after correcting for the redshift bin mismatch of objects following the scattering matrix approach. The $\sigma_{8}$ parameter becomes consistent with $\Lambda$CDM model in the last tomographic bin but shows $\sim 1-3\,\sigma$ tension in other redshift bins.