From Nonparametric Distance Reconstruction to Testing the Etherington Relation and Cosmic Curvature Using 2D and 3D BAO Measurements

We present a joint test of cosmic curvature, $\Omega_{k0}$, and the cosmic distance-duality relation (CDDR) using the Etherington relation, which connects the luminosity and angular diameter distances at the same redshift. In this work, we combine the angular diameter distance measurements from recent Baryon Acoustic Oscillation (BAO) observations with luminosity distances reconstructed from Cosmic Chronometers data of Hubble parameter $H(z)$ using a non-parametric technique, Gaussian Process. A key part of our analysis is the systematic comparison of different BAO measurements (2D BAO, 3D BAO, and 3D DESI BAO) to determine whether any potential tension between angular and anisotropic BAO data affects constraints on the distance duality parameter $\eta(z)$ and $\Omega_{k0}$. We adopt four representative parameterizations of $\eta(z)$ to examine the correlation between $\eta(z)$ and $\Omega_{k0}$. Our results show no evidence for violation of the CDDR, with $\eta(z)$ consistent with unity at the 99\% confidence level for all BAO datasets and parameterizations. In all scenarios, the best-fit values of $\Omega_{k0}$ mildly favor a non-flat universe, although a spatially flat universe remains compatible at the 95\% confidence level. The constraints on $\eta_1$ and $\Omega_{k0}$ indicate slight variations across different BAO datasets, but the discrepancies between the 2D and 3D BAO measurements do not introduce any significant bias, and no statistically meaningful tension is observed. Our work provides robust constraints on cosmic curvature and the validity of the CDDR based on non-parametric distance reconstruction.