$O_k$ null test with multi-task Gaussian processes: cosmic curvature and data compatibility

The $O_k$ null test can not only assess whether the cosmic curvature is zero, therefore if true reducing degeneracies between cosmic curvature and other cosmological parameters, but also provide a model-independent check of compatibility between different data sets. However, traditional implementations often require absolute distance data from Type Ia supernovae (SNe Ia) or baryon acoustic oscillation (BAO) measurements, limiting their applicability because such absolute distance data usually are not accessible. The BAO Alcock Paczynski (AP) parameter $F_{AP}$ is a measurement of a distance ratio, making the Dark Energy Spectroscopic Instrument (DESI) AP measurements particularly well suited for the $O_k$ null test because no absolute distance measurements are required. We propose a novel null test of cosmic curvature tailored to DESI BAO data that combines $F_{AP}$ with ratios such as $D_V'/D_V$ or $D_M'/D_M$. Crucially, this construction eliminates the need for absolute distance measurements. We further develop multi-task Gaussian processes to perform the null test. This approach can also be applied to a joint DESI BAO and SNe Ia dataset, and we find that DESI BAO and SNe Ia data are compatible. Although there is $\sim 2\sigma$ evidence of nonzero curvature at low redshift $z\lesssim 0.5$, this result is not conclusive largely due to the lack of observational data in the corresponding redshift range.