Simulation-based tension quantification of the cosmic dipole

The cosmic dipole measured in surveys of cosmologically distant sources consistently exceeds the expectation derived from the cosmic microwave background, posing a significant challenge to the standard $\Lambda$CDM cosmology. In the era of precision cosmology, quantifying robust tensions is constrained by our ability to model complex, non-linear effects that often result in intractable likelihood functions. In this paper, we present a flexible simulation-based inference (SBI) architecture for measuring the cosmic dipole tension using Neural Ratio Estimators (NREs). We design and train an ensemble of NREs to evaluate the log Bayesian evidence ratio and measure $N\sigma$ tension. We validate our approach against nested sampling, demonstrating that it accurately recovers the ground-truth. Under the kinematic interpretation of the dipole, we apply our approach to Planck, the NRAO VLA Sky Survey (NVSS), the Rapid ASKAP Continuum Survey (RACS), and the Wide-field Infrared Survey Explorer catalogue (CatWISE). Here, leveraging SBI, we measure a $\approx5.7\sigma$ tension between CatWISE and Planck. We demonstrate the extensibility of our approach by applying it to forward-modelled simulations of the CatWISE Eddington bias, revealing a $\approx6.7\sigma$ tension. The methodology proposed here enables robust tension quantification as we enter the era of LSST, Euclid, and the SKA.