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

The paper, Impact of non-Gaussian likelihood on cosmological constraints from the thermal Sunyaev–Zel’dovich power spectrum: a simulation-based inference analysis, investigates the limitations of standard Gaussian likelihood methods when analyzing data derived from the thermal Sunyaev–Zel’dovich (tSZ) effect, which is known to exhibit highly non-Gaussian statistics.

Motivation and Theoretical Background

The tSZ power spectrum serves as a sensitive probe of cosmology and cluster astrophysics. However, its statistics are non-Gaussian because the signal receives a significant contribution from rare, massive, low-redshift galaxy clusters. This effect is most pronounced on large angular scales (low multipoles). The paper notes that the sampling distribution of the tSZ power spectrum is non-Gaussian and positively skewed at low multipoles, which challenges traditional methods.

The theoretical framework employed utilizes the halo-model formalism. The 1-halo term, which dominates the power spectrum at low multipoles (100), is driven by massive clusters between 10 14 M and 10 15 M at low redshifts (z about 0.3). The paper establishes that the tSZ power spectrum is sensitive to parameters like sigma 8 and m, but the presence of these rare, massive clusters introduces significant non-Gaussianity into the data distribution.

Methodology: Simulation-Based Inference (SBI)

To address the shortcomings of a standard Gaussian likelihood, the authors propose using Simulation-Based Inference (SBI). This approach allows for parameter inference directly from forward-modeled simulations, without requiring an explicit analytic likelihood. The study employs two primary simulation pipelines:

1. The Gaussian Likelihood Simulator: A simplified model where simulated data is generated based on a multivariate Gaussian distribution, serving as a baseline for validation.

2. The Halo-Based Forward Model: A realistic pipeline that generates mock cluster catalogs and sky maps using the halo-based simulations, capturing the true non-Gaussian statistics of the tSZ power spectrum.

The paper implements two SBI techniques: Neural Posterior Estimation (NPE) and Neural Likelihood Estimation (NLE), both utilizing a Masked Autoregressive Flow (MAF) architecture.

Results: Comparison with Real Data (Gaussian Likelihood Setup)

When applying the Gaussian likelihood analysis to real Planck data, the researchers compared the results from the Markov Chain Monte Carlo (MH) reference analysis against their SBI methods:

• NLE vs. MH: The NLE posterior showed excellent agreement with the MH reference posterior, with mean differences being less than 0.05 sigma.

• NPE vs. MH: The NPE posteriors showed poorer agreement, with the largest deviation being about 0.4 sigma for the AIR parameter.

Results: Comparison with Simulated Data (Halo-based Setup)

The halo-based setup, where instrumental noise is neglected, provides a controlled environment to test the non-Gaussian nature of the likelihood:

• Consistency: Both NLE and NPE yielded posterior distributions for the cosmological parameter F in excellent agreement with the results from a Gaussian likelihood analysis.

• Uncertainties: The SBI-based methods resulted in slightly larger uncertainties in all three foreground residual amplitudes compared to the Gaussian likelihood analysis.

Results: Reconstructed Power Spectrum

In a detailed reconstruction of the tSZ signal using 1000 samples drawn from the posterior distribution, the reconstructed tSZ spectrum achieved a mean signal-to-noise ratio of 5.6. This is consistent with constraints on F in Table IV, which corresponds to a fractional uncertainty of around 2.4%.

Validation and Discussion

The study rigorously validated the SBI implementation:

• Convergence: The validation loss function showed that NLE converged after approximately 7,500 simulations, while NPE required more simulations to approach stability (around 17,500).

• Learned Likelihood Accuracy: When comparing the distributions drawn from the learned likelihood against the actual forward-modeled realizations, the learned likelihood successfully reproduces the skewness in the bandpower distribution, whereas Gaussian theory curves failed to capture this highly non-Gaussian behavior.

• Coverage Test: A credible-level coverage test confirmed that, for most parameters, the empirical coverage should agree with the nominal credibility level, indicating that the inferred posteriors are statistically consistent with the true parameters. The bias was found to be less than 0.06 sigma across all parameters.

In conclusion, while a standard Gaussian likelihood approximation is sufficient to provide unbiased cosmological parameter estimates for a Planck-like survey, the authors emphasize that SBI provides a useful validation tool to model non-Gaussian likelihoods beyond analytic approximations.

As a diligent AI researcher, I have analyzed this paper and identified several critical areas where current implementations of Simulation-Based Inference (SBI) can be significantly improved upon the foundational work presented. The following improvements focus on enhancing model fidelity, computational efficiency, and generalization for both Neural Posterior Estimation (NPE) and Neural Likelihood Estimation (NLE).

The paper utilizes Masked Autoregressive Flow (MAF) for density estimation. While effective, MAF is computationally intensive during the forward pass.

Improvement: Implement Hybrid Density Estimators (e.g., Normalizing Flows combined with Mixture of Experts - MoE).

Instead of using a single, complex autoregressive flow to model the entire complex distribution (especially in low-multipole bins where skewness is high), we can use a hybrid architecture. The MoE component handles the highly localized heavy tails and extreme values (the rare massive clusters), while a simpler normalizing flow handles the bulk of the distribution.

What the improved AI system can do:

• Accurately model multi-modal distributions: It will be able to distinguish between different modes of parameter space that contribute to the same observed power spectrum, which is critical when dealing with non-Gaussianity induced by rare clusters.

• Increase training stability and convergence speed: By reducing the complexity of the single flow required for all data, we can accelerate convergence while maintaining fidelity.

The current approach uses standard negative log-likelihood (NLE) or negative log-posterior (NPE) loss functions. This ignores the physical constraints embedded in the halo model itself.

L new = L standard + lambda times (Deviation from physical constraints)

The paper relies on fixed emulators (e.g., CosmoPower). In a large-scale survey (like ACT or SPT), this can be computationally prohibitive and inflexible.

The paper notes that small shifts in foreground amplitude constraints (e.g., 0.5 sigma for AIR) might be due to inference systematics or genuine non-Gaussianity, requiring extensive validation (Sec VIII).

The improved system will enable cosmological inference that is:

1. Physically Grounded: It won't yield physically impossible parameter constraints.

2. Highly Efficient: It can handle massive datasets from future surveys without the computational bottleneck of running full simulations for every point in parameter space.

3. Robust: It will accurately distinguish between genuine physical non-Gaussianity and statistical artifacts, providing a quantitative measure of its own reliability.

The thermal Sunyaev--Zel'dovich (tSZ) power spectrum is a sensitive probe of cosmology and cluster astrophysics, but its statistics are non-Gaussian because the signal receives a significant contribution from rare, massive, low-redshift galaxy clusters. As a result, a Gaussian likelihood fails to describe the statistics of its power spectrum on large scales. We use simulation-based inference (SBI) to test the accuracy of the standard Gaussian power-spectrum likelihood for a Planck-like tSZ analysis. Using halo-based simulations of full-sky Compton- y maps, we train neural posterior and likelihood estimators and compare the resulting constraints with those from a Gaussian likelihood assumption. Using only multipoles < 1000, we find that the Gaussian likelihood assumption gives unbiased cosmological constraints, while the SBI-based inference shows a mild broadening of the posterior distributions for the amplitudes of residual foregrounds. This suggests that the Gaussian likelihood assumption is sufficiently accurate for cosmological inference for a Planck-like tSZ analysis, while SBI provides a useful validation tool to model non-Gaussian likelihoods beyond analytic approximations.