Testing the cosmological principle on gigaparsec scales

Recent observational analyses have suggested possible evidence of hemisphere asymmetry in cosmological datasets. Parameterizations of this kind place observers in a privileged position-specifically on the plane that divides the two hemispheres. To quantify potential deviations from the cosmological principle without presuming a special location, we develop a stochastic framework that parametrizes departures from statistical homogeneity and isotropy. The near-uniform temperature of the cosmic microwave background indicates that anisotropy is negligible (at the $\lesssim 10^{-5}$ level) on the last scattering surface. This serves as a zero boundary condition, enabling the construction of an orthogonal basis of functions below the recombination redshift. Within this basis, we expand the relative deviation from the Hubble diagram of isotropic models (such as $\Lambda$CDM or $w_0w_a$CDM) in a hierarchy of increasing resolution. Applying this approach, we test the cosmological principle using Type Ia supernovae, strong lensing time delays, and gravitational-wave standard sirens. For the class of large-scale anisotropies and low-order radial variations described by this framework, the current datasets are found to be consistent with statistical homogeneity and isotropy on gigaparsec scales.