Impact of non-Gaussian likelihood on cosmological constraints from the thermal Sunyaev--Zel'dovich power spectrum: a simulation-based inference analysis

Technical Summary: Impact of Non-Gaussian Likelihood on Cosmological Constraints from the tSZ Power Spectrum

Problem Statement
The thermal Sunyaev–Zel'dovich (tSZ) power spectrum is a primary probe for cosmological parameters (specifically $\sigma_8$ and $\Omega_m$) and cluster astrophysics. However, its statistical properties are inherently non-Gaussian. This non-Gaussianity arises because the signal on large angular scales is dominated by rare, massive, low-redshift galaxy clusters, leading to a positively skewed sampling distribution. Standard cosmological analyses, such as those by the Planck Collaboration, typically assume a Gaussian likelihood for the power spectrum. While this approximation may hold at high multipoles where contributions are distributed over many clusters, it fails to capture the skewness at low multipoles ($\ell < 1000$). The lack of an accurate analytic likelihood model that captures these non-Gaussian statistics motivates the need for alternative inference methods to ensure unbiased cosmological constraints.

Methodology
The authors employ Simulation-Based Inference (SBI) to bypass the need for an explicit analytic likelihood function. The workflow involves:

1. Forward Modeling:
   • Gaussian Simulator: A baseline simulator generates power spectra using a multivariate Gaussian distribution with a fixed covariance matrix (including trispectrum contributions) to validate the SBI pipeline against standard methods.
   • Halo-Based Simulator: A more realistic forward model generates full-sky Compton-$y$ maps. This involves Poisson sampling of halo masses and redshifts from the Tinker et al. (2008) halo mass function, assigning them generalized Navarro–Frenk–White (GNFW) pressure profiles (Arnaud et al. 2010), and convolving them with a beam. Foreground residuals (CIB, IR, RS) and correlated noise are added as Gaussian random fields.
2. Neural Density Estimation:
   • The authors train deep neural networks using the sbi package (PyTorch) to learn either the posterior distribution directly (Neural Posterior Estimation, NPE) or the likelihood function (Neural Likelihood Estimation, NLE).
   • A Masked Autoregressive Flow (MAF) architecture is used. To mitigate training stochasticity, an ensemble of 10 independently trained networks is employed to form a robust posterior estimate.
3. Validation:
   • The SBI framework is first validated against a standard Gaussian likelihood analysis using real Planck data.
   • It is then applied to a mock dataset generated by the halo-based forward model to assess the impact of non-Gaussianity.
   • Convergence tests are performed regarding the number of simulations, and coverage tests are conducted to ensure the inferred posteriors are statistically consistent with true parameters.

Key Contributions

• First SBI Application to tSZ: This work represents the first application of SBI to the analysis of the tSZ power spectrum.
• Validation of Gaussian Approximation: The study provides a rigorous test of the standard Gaussian likelihood assumption for Planck-like surveys.
• Non-Gaussian Likelihood Modeling: It demonstrates the capability of NLE to accurately learn and reproduce the highly skewed, non-Gaussian sampling distribution of the tSZ power spectrum at low multipoles, which Gaussian approximations fail to capture.
• Robustness Analysis: Through extensive ablation studies, the authors distinguish between genuine physical differences in the likelihood and potential artifacts arising from neural network training or finite simulation sizes.

Results

• Cosmological Constraints: Using only multipoles $\ell < 1000$, the Gaussian likelihood assumption yields unbiased cosmological constraints. The SBI-based inference (both NPE and NLE) produces results for the cosmological parameter combination $F = \sigma_8(\Omega_m/B)^{0.40}h^{-0.21}$ that are in excellent agreement with the Gaussian likelihood results, despite the underlying non-Gaussianity.
• Foreground Constraints: While cosmological constraints remain stable, the SBI-based inference shows a mild broadening of the posterior distributions for the amplitudes of residual foregrounds compared to the Gaussian likelihood. Specifically, the NLE-based constraint on the infrared source amplitude ($A_{IR}$) is approximately $0.5\sigma$ lower than the Gaussian likelihood result.
• Likelihood Learning: The NLE method successfully learns the non-Gaussian structure of the likelihood, reproducing the positive skewness observed in the forward-modeled simulations, whereas the Gaussian approximation fails at low multipoles.
• Coverage: Coverage tests confirm that the SBI-inferred posteriors are well-calibrated, with biases less than $0.06\sigma$ across all parameters.

Significance and Claims
The paper claims that for a Planck-like tSZ analysis (full-sky, specific noise levels), the standard Gaussian likelihood assumption is sufficiently accurate for cosmological inference, as it yields unbiased constraints on key parameters. However, the authors emphasize that SBI serves as a crucial validation tool. It allows researchers to model non-Gaussian likelihoods beyond analytic approximations, revealing subtle differences in foreground constraints that might otherwise be missed.

The authors note that their conclusion regarding the sufficiency of the Gaussian likelihood may not automatically extend to other surveys like ACT or the South Pole Telescope (SPT), which observe smaller sky fractions and thus experience higher sample variance and stronger non-Gaussianity at higher multipoles. Furthermore, they highlight that SBI offers a flexible framework for future applications, such as incorporating uncertainties in pressure profiles, combining tSZ power spectra with cluster number counts, and analyzing higher-order moments, which are difficult to handle with traditional analytic likelihoods. The work suggests that while the Gaussian approximation is robust for current Planck-like constraints, SBI provides a necessary pathway for handling the complex statistics of future, higher-resolution CMB surveys.