Hierarchical summaries for primordial non-Gaussianities

Technical Summary: Hierarchical Summaries for Primordial Non-Gaussianities

Problem Statement
The advent of Stage IV galaxy redshift surveys (e.g., DESI, Euclid) offers the potential to detect primordial non-Gaussianities (PNG), specifically the local type parameterized by $f_{\text{NL}}$. A detection with precision $\sigma_{f_{\text{NL}}} \sim 1$ would distinguish between single-field and multi-field inflation scenarios. However, current constraints from Large Scale Structure (LSS) and the Cosmic Microwave Background (CMB) remain at $\sigma_{f_{\text{NL}}} \sim 5–9$. Improving these constraints requires novel data compression strategies that can extract information beyond standard $n$-point statistics (power spectrum and bispectrum) without incurring prohibitive computational costs associated with full field-level machine learning analyses.

Methodology
The authors propose a hybrid estimator that hierarchically combines standard global $n$-point statistics with a local, field-level neural summary. This approach is motivated by theoretical work suggesting that primordial non-Gaussianity generates non-local correlations, while late-time gravitational evolution produces only local effects; thus, combining global statistics with local field analysis can disentangle these signals.

The methodology employs the following components:

1. Simulations: The study utilizes the Quijote-PNG suite of N-body simulations, specifically the local PNG dataset. This includes 1,000 Latin hypercube simulations with $f_{\text{NL}} \in (-300, 300)$ for training, 5,000 realizations at the fiducial cosmology ($f_{\text{NL}}=0$) for covariance estimation, and 1,000 simulations with displaced $f_{\text{NL}}$ values ($\pm 50$) for numerical derivative calculations. The analysis focuses on Friend-of-Friend (FoF) halo catalogs within a $1 \text{ Gpc}^3 h^{-3}$ volume across redshifts $z = 0, 0.5, 1, 2, 3$.
2. Standard Statistics: The power spectrum ($P(k)$) and bispectrum ($B(k)$) are computed from the halo number density field. The power spectrum is measured up to $k_{\text{max}} \sim 0.8 \, h \text{ Mpc}^{-1}$, while the bispectrum is measured up to $k_{\text{max}} \sim 0.5 \, h \text{ Mpc}^{-1}$.
3. Field-Level Compression (PatchNet): Instead of analyzing the full simulation volume, the halo density field is divided into non-overlapping sub-volumes ("patches"). A Convolutional Neural Network (CNN), termed PatchNet, processes these patches to compress the field-level information into a single scalar value per patch.
   • Architecture: The network consists of three blocks of 3D convolutional and average pooling layers (kernel size $3\times3\times3$), followed by three dense layers. It outputs a single value representing the estimated $f_{\text{NL}}$ for the patch.
   • Training: The network is trained to minimize a modified mean squared error loss, aggregating predictions across all patches within a simulation box.
   • Hybrid Estimator: The final summary statistic is a concatenation of the global $P(k)$ (and optionally $B(k)$) with the aggregated PatchNet output (the mean of the patch predictions).
4. Fisher Information Analysis: The constraining power of the hybrid estimator is evaluated using the Fisher information matrix. The covariance matrix and derivatives with respect to $f_{\text{NL}}$ are estimated numerically using the simulation suites.

Key Results
The analysis demonstrates that the inclusion of patch-based field-level compression consistently enhances constraints on $f_{\text{NL}}$ compared to standard summary statistics alone:

• Magnitude of Improvement: The hybrid estimator yields gains of 30–45% in constraining power (reduction in $\sigma_{f_{\text{NL}}}$) at low $k_{\text{max}}$ ($\sim 0.1 \, h \text{ Mpc}^{-1}$) for the standard statistics.
• Complementarity: The patch information captures information beyond the bispectrum. Even when combining the power spectrum, bispectrum, and patches, the result is tighter than the power spectrum plus patches alone, indicating that the bispectrum retains large-scale squeezed triangle information that the limited patch size cannot access. Conversely, the patches provide small-scale information not captured by the bispectrum.
• Redshift and Mass Dependence: The improvement is observed across all tested redshifts ($z=0$ to $3$) and halo mass cuts. Notably, the relative improvement from patches is greater for higher mass cuts (more biased objects), which are relevant for future survey targets like quasars.
• Efficiency: The patch-based approach allows for effective field-level analysis with a significantly reduced computational budget compared to full-volume neural networks, as dividing the volume into patches increases the effective training set size.

Significance and Claims
The paper claims that this hierarchical combination offers a computationally efficient and scalable pathway to tighten PNG constraints from forthcoming survey data. The authors assert that their method approximates a global field-based analysis by leveraging the theoretical principle that locality protects primordial signals from late-time contamination. By optimally using $n$-point statistics for large scales and machine learning for small scales, the approach captures essential information content more efficiently than computing progressively higher-order correlation functions. The results suggest that patch-based compression provides additional information relative to standard statistics, particularly in the non-linear regime where modeling $P(k)$ and $B(k)$ becomes increasingly complex.

The authors note that while the halo mass function is highly informative, the network learns a relationship between the density contrast and $f_{\text{NL}}$ that extends beyond the mass function alone. Future work is identified as necessary to test other PNG shapes, marginalize over cosmological parameters, and address observational effects such as survey masks and lightcone geometry.