A roadmap to cosmological parameter analysis with third-order shear statistics

Context. Weak gravitational lensing is a powerful probe of cosmology, with second-order shear statistics commonly used to constrain parameters such as the matter density Ω m and the clustering amplitude S 8 . However, degeneracies between parameters persist and can be broken by including higher-order statistics, such as the third-order aperture mass. To jointly analyse second- and third-order statistics, an accurate model of their cross-covariance is essential. Aims. This work derives and validates a non-tomographic analytical model for the cross-covariance between second- and third-order aperture mass statistics. Analytical models are computationally efficient and enable cosmological parameter inference across a range of models, in contrast to numerical covariances derived from simulations or resampling methods, which are either costly or biased. Methods. We derived the cross-covariance from real-space estimators of the aperture mass. Substituting the Halofit power spectrum, BiHalofit bispectrum, and a halo-model-based tetraspectrum, the model was validated against numerical covariances from the N -body Scinet LIghtCone Simulations (SLICS) using both shear catalogues and convergence maps. We performed a Markov chain Monte Carlo parameter analysis using both analytical and numerical covariances for several filter scale combinations. Results. The cross-covariance separates into three terms governed by the power spectrum, bispectrum, and tetraspectrum, with the latter dominating. While the analytical model qualitatively reproduces simulation results, differences arise due to modelling approximations and numerical evaluation issues. The analytical contours are systematically tighter, with a combined figure of merit that is 72% that of the numerical case, increasing to 80% when small-scale information is excluded. These differences largely stem from an underprediction of the second-order covariance. Conclusions. This work completes the analytical covariance framework for second- and third-order aperture mass statistics, enabling joint parameter inference without the need for large simulation suites. While further refinement is needed to improve quantitative accuracy, the model represents a key step towards simulation-independent cosmic shear analyses.