Improved null tests of $\Lambda$CDM and FLRW in light of DESI DR2

The DESI DR2 BAO data, in combination with CMB and different SNIa datasets, exclude the flat $\Lambda$CDM model at more than 2.5$\sigma$ when analyzed through the $w_0w_a$CDM parametrization for evolving dark energy. This simple parametrization may not accurately capture the behavior of the entire redshift range at late times, which may introduce bias in the results. We use null tests that probe for deviations from flat $\Lambda$CDM at late times, independent of any specific dark energy parametrization. We provide several diagnostics for null tests and discuss their advantages and disadvantages. In particular, we derive diagnostics that improve on previous ones, such as the popular $O_{\rm m}$ diagnostic. The diagnostics are derived from both background and perturbed quantities. Using the combination of DESI DR2 BAO and supernova data, with or without CMB data, we find that deviations from flat $\Lambda$CDM are at $\sim1\sigma$ confidence level in most of the redshift range (more than 1$\sigma$ for a few small redshift intervals in a few cases). When considering SDSS BAO data instead of DESI BAO data, in combination with PantheonPlus, with or without CMB data, we find even smaller deviations. Since spatial curvature can potentially modify the results, we also test for curvature in the general $\Lambda$CDM model and the general FLRW model. While there is slight evidence for nonzero cosmic curvature at lower redshifts in a general $\Lambda$CDM model, there is no statistically significant evidence in a general FLRW model.