Hubble tension tomography: BAO vs SN Ia distance tension

We investigate the redshift dependence of the Hubble tension by comparing the luminosity distances obtained using an up-to-date BAO dataset (including the latest DESI data) calibrated with the CMB-inferred sound horizon and the $\mathrm{Pantheon}+\text{SN}$ Ia distances calibrated with Cepheids. Using a redshift tomography method. We find: (1) The BAO-inferred distances are discrepant with the $\mathrm{Pantheon}+\text{SN}$ Ia distances across all redshift bins considered, with the discrepancy level varying with redshift. (2) The distance discrepancy is more pronounced at lower redshifts ($z\ensuremath{\in}[0.1,0.8]$) compared to higher redshifts ($z\ensuremath{\in}[0.8,2.3]$). The consistency of $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ best fit parameters obtained in high and low redshift bins of both BAO and SN Ia samples is investigated, and we confirm that the tension reduces at high redshifts. Also a mild tension between the redshift bins is identified at higher redshifts for both the BAO and $\mathrm{Pantheon}+\text{data}$ with respect to the best fit value of ${H}_{0}$ in agreement with previous studies which find hints for an ``evolution'' of ${H}_{0}$ in the context of $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$. These results confirm that the low redshift BAO and SN Ia distances can only become consistent through a reevaluation of the distance calibration methods. An $H(z)$ expansion rate deformation alone is insufficient to resolve the tension. Our findings also hint at a possible deviation of the expansion rate from the Planck18/$\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model at high redshifts $z\ensuremath{\gtrsim}2$. We show that such a deformation is well described by a high redshift transition of $H(z)$ like the one expressed by ${\mathrm{\ensuremath{\Lambda}}}_{\mathrm{s}}\mathrm{CDM}$ even though this alone cannot fully resolve the Hubble tension due to its tension with intermediate/low $z$ BAO data.