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

TL;DR

This work tackles how to extract non-Gaussian information from large-scale structure by comparing Minkowski Functionals (MFs) and a directional, weighted morphology statistic called the Conditional Moments of Derivative (CMD) within a likelihood-free, simulation-based inference framework. Using redshift-space halo catalogs from the Big Sobol Sequence (BSQ) simulations at $z=0.5$, the authors train conditional neural density estimators (Neural Spline Flows) to infer cosmological parameters $(\Omega_m,\sigma_8)$ from a set of higher-order statistics, including MF components and CMD (and their combination). The main findings are that CMD yields significantly tighter constraints than MF components alone, with percent improvements up to ~$52\%$ for $\Omega_m$ and ~$45\%$ for $\sigma_8$ for individual components, and a ~27\% gain when combining MF and CMD relative to MF alone; these gains persist across a range of fiducial cosmologies and smoothing scales, indicating complementary anisotropy-sensitive information captured by CMD. The results highlight the value of incorporating directional morphological information in LSS analyses and establish a flexible, likelihood-free framework for comparing high-order statistics in upcoming large surveys.

Abstract

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 $(Ω_{m}=0.3175,\,σ_{8}=0.834)$, and for the smoothing scale $R=15\,h^{-1}$Mpc, we find that the CMD yields tighter forecasts for $(Ω_{m}},\,σ_{8})$ 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.

Figures (9)

• Figure 1: Schematic overview of the simulation-based forecasts pipeline used in this work. Starting from broad priors on cosmological parameters, we explore the parameter space defined by the Big Sobol Sequence (BSQ) simulation suite at $z=0.5$, making use of its redshift-space halo catalogs. From these catalogs, we compute three types of summary statistics: Minkowski functionals (MFs) and the Conditional Moments of Derivative (CMD). These statistics are feed to train neural density estimators that approximate the posterior distribution of parameters. The trained models are evaluated on independent test simulations from the same suite to assess the relative and joint constraining capability of the considered descriptors.
• Figure 2: Distribution of halo number densities across BSQ simulation realizations at redshift $z = 0.5$. The histogram reflects variations in halo abundance due to different cosmological parameter combinations. No minimum halo mass threshold is applied.
• Figure 3: Schematic representation of the neural posterior estimation (NPE) workflow, inspired by Cranmer2020. Starting from a prior over model parameters $\boldsymbol{\theta}$, proposal samples are passed through a simulator to generate mock observations $\mathbfcal{D}$. An unsupervised learning stage maps these data to an approximate posterior, which is evaluated against target data.
• Figure 4: Normalized rank distributions for the five $\Lambda$CDM parameters $\{\boldsymbol{\theta}\}:\{\Omega_{\mathrm{m}}$, $\Omega_{\mathrm{b}}$, $h$, $n_{\mathrm{s}}$, $\sigma_{8}\}$ across the ten summary statistics configurations listed in Table \ref{['tab:summary_stats']}. Each row corresponds to a distinct summary statistics configuration, and each column to one of the cosmological parameters. The horizontal dashed line represents the expectation for perfectly calibrated posteriors (uniform distribution).
• Figure 5: The standard deviation of estimated $\sigma_8$ posteriors across 2000 test simulations (dots) for $\sigma_8\in [0.7-0.9]$. The blue solid line represents the median trend within the red sliding windows. The shaded region encompasses the 16th-84th percentile interval. Here we used the $\bold{MFs}^{(15)}$ as the descriptor.