Effect of redshift bin mismatch on the cross correlation between the DESI Legacy Imaging Survey and the <i>Planck</i> CMB lensing potential

Aims . We study the importance of precise modelling of the photometric redshift error distributions when estimating parameters from cross-correlation measurements. We present a working example of the scattering matrix formalism to correct for the effects of galaxies ending in wrong redshift bins due to their photometric redshift errors. Methods . We measured the angular galaxy auto-power spectrum and cross-power spectrum in four tomographic bins with the redshift intervals z = [0.0, 0.3, 0.45, 0.6, 0.8] from the cross-correlation of the Planck cosmic microwave background lensing potential and the photometric galaxy catalogue from the Dark Energy Spectroscopic Instrument Legacy Imaging Survey Data Release 8. We estimated the galaxy linear bias and the amplitude of cross correlation using maximum likelihood estimation to put constraints on the σ8 parameter. Results . 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 values of the cross-correlation amplitude that are ∼2–5 σ smaller than those expected based on the Λ cold dark matter (ΛCDM) model. We compute the σ 8 parameter after correcting for the redshift bin mismatch of objects following the scattering matrix approach. The σ 8 parameter becomes consistent with ΛCDM model in the last tomographic bin but shows a tension of ∼1–3 σ in other redshift bins.