Quantifying Weighted Morphological Content of Large-Scale Structures via Simulation-Based Inference

In this work, we perform a simulation-based forecasting analysis to compare the constraining power of two higher-order summary statistics of the large-scale structure (LSS), the Minkowski Functionals (MFs) and the Conditional Moments of Derivative (CMD), with a particular focus on their sensitivity to nonlinear and anisotropic features in redshift-space. Our analysis relies on halo catalogs from the Big Sobol Sequence(BSQ) simulations at redshift $z=0.5$, employing a likelihood-free inference framework implemented via neural posterior estimation. At the fiducial cosmology of the Quijote simulations $(\Omega_{m}=0.3175,\,\sigma_{8}=0.834)$, and for the smoothing scale $R=15\,h^{-1}$Mpc, we find that the CMD yields tighter forecasts for $(\Omega_{m}},\,\sigma_{8})$ than the zeroth- to third-order MFs components, improving the constraint precision by ${\sim}(44\%,\,52\%)$, ${\sim}(30\%,\,45\%)$, ${\sim}(27\%,\,17\%)$, and ${\sim}(26\%,\,17\%)$, respectively. A joint configuration combining the MFs and CMD further enhances the precision by approximately ${\sim}27\%$ compared to the standard MFs alone, highlighting the complementary anisotropy-sensitive information captured by the CMD in contrast to the scalar morphological content encapsulated by the MFs. We further extend the forecasting analysis to a continuous range of cosmological parameter values and multiple smoothing scales. Our results show that, although the absolute forecast uncertainty for each component of summary statistics depends on the underlying parameter values and the adopted smoothing scale, the relative constraining power among the summary statistics remains nearly constant throughout.