High-redshift cosmography with a possible cosmic distance duality relation violation

In this study, we used geometric distances at high redshifts (both luminosity and angular) to perform a cosmographic analysis with the Pad\'e method, which stabilizes the behaviour of the cosmographic series in this redshift regime. However, in our analyses, we did not assume the validity of the Cosmic Distance Duality Relation (CDDR), but allowed for potential violations, such as $d_L(z) = \eta(z)(1+z)^2d_A(z)$, where three different functional forms of $\eta(z)$ are considered. By incorporating updated data from supernovae (SN), baryon acoustic oscillations (BAO), and cosmic chronometers (CC), we obtained observational constraints on cosmographic models alongside possible CDDR violations. Interestingly, we found that potential CDDR violations introduce new statistical correlations among cosmographic parameters such as $H_0$, $q_0$, and $j_0$. Nonetheless, within this framework, we did not observe significant deviations from the CDDR, and our results remain consistent with the predictions of the $\Lambda$CDM model. In the same time, this work provides a novel and straightforward method for testing the CDDR by fixing the background evolution through cosmographic techniques, paving the way for new geometric observational tests of possible deviations from standard cosmology.