2Flow2SN: Dusty Supernova Cosmology with Photometric Redshifts and Contamination using Neural Simulation-Based Inference

Detailed Research Summary of FlowSN2 (FlowSN2 / Flow2SN)

As a meticulous researcher, I have thoroughly analyzed the provided text excerpts from the paper concerning FlowSN2 (also referred to as Flow2SN). This framework represents a significant methodological advancement in photometric Type Ia Supernova (SN Ia) cosmology, moving beyond traditional regression techniques to a sophisticated joint inference approach.

Here is a comprehensive and detailed summary synthesizing the core concepts, methodology, and results described:

1. Core Methodology: Reframing the Inference Problem

The central innovation of FlowSN2 lies in fundamentally reframing photometric SN cosmology. Instead of treating the problem as a simple regression task—predicting distance from light curve summaries given a spectroscopic redshift—FlowSN2 treats it as a joint inference problem. This joint inference explicitly incorporates **photometric redshifts ** as an additional, uncertain data dimension.

• Joint Inference: The framework simultaneously infers the cosmological parameters (theta) and the photometric redshifts of the SNe.

• Amortization over the Hubble Diagram: A key technical achievement is that FlowSN2 achieves amortization over the Hubble diagram. This means that instead of retraining a separate model for every single SN observation, it learns a general form of data distribution (via neural likelihood estimation) that can be reused across different cosmological models and redshift bins. This drastically reduces computational overhead, allowing for inference across alternative cosmological parameterizations (e.g., w 0w a CDM) away from the fiducial training cosmology without requiring a full retraining cycle.

• Neural Likelihood Estimation: The framework employs normalising flows to approximate the complex data distributions of selected SN objects. This allows for flexible and powerful likelihood modeling that is transferable across different cosmological contexts.

2. Key Components and Enhancements

FlowSN2 integrates several sophisticated statistical components to handle real-world observational challenges:

• Contamination Treatment: A crucial feature is the principled Bayesian resampling approach, inspired by Autenrieth et al. (2026). This modular method is used to disentangle intrinsic SN variations from extrinsic host-galaxy dust effects and explicitly addresses potential photometric contamination from non-Ia objects that might have misspecified likelihoods. This avoids the need for an explicit, potentially flawed, contaminant likelihood model.

• SN Standardisation Upgrade: The framework upgrades the standardisation model to the Simple-BayeSN (Mandel et al. 2017) model. This allows for a clear and explicit separation of:

• Extrinsic SN parameters (e.g., dust).

• Intrinsic SN parameters.

• Host-galaxy mass steps.

• Joint Modeling Structure: The combined data likelihood is defined as the product of the individual observed SN data likelihoods: P(obs, obs I obs, theta) = product s=1 N obs P(s, s I s = 1, theta). This structure assumes conditional independence of individual SN data given the underlying model parameters.

3. Advantages Over Existing Methodologies

The paper repeatedly emphasizes the unique advantages of FlowSN2 in the context of evolving dark energy debates:

• Model Comparison Efficiency: The framework enables Bayesian model comparison at negligible additional computational cost.

• Robustness to Selection Effects and Uncertainties: It naturally allows for comparing posterior constraints across different cosmological models (e.g., w 0w a CDM) under the influence of:

• Selection effects.

• Photometric redshift uncertainties.

• Contamination.

The authors note that alternative methods used to handle selection effects can converge to different cosmological posteriors than those derived using FlowSN2's probabilistic prescription, lending strong support to its statistical rigor.

4. Application and Validation (Experimental Context)

The framework is designed for high-throughput applications, specifically targeting upcoming wide-field photometric surveys:

• Scalability: FlowSN2 is capable of scaling to over a hundred thousand SNe without requiring retraining, making it highly efficient for large datasets.

• Cosmological Constraints (Experimental Context): The framework has been tested across five cosmological parameter sets. In specific experimental contexts (C1-C3), a CMB prior is added to constrain the w CDM model parameters. For other experiments (C4 and C5), a BAO prior is included to constrain the w 0w a CDM model.

• Performance Metrics: The results show that FlowSN2 successfully recovers true cosmological and population parameters, with posterior distributions showing close agreement with both tractable analytical solutions and pure-Ia baselines.

5. Specific Results Highlight (Experimental Context B)

The provided excerpt from Section B details the application of FlowSN2 in constraining dark energy models:

• Prior Incorporation: The combined posterior likelihood incorporates priors derived from CMB data (for C1-C3) and BAO data (for C4 and C5).

• Constraint Results:

• For cosmology C4, the results yield w 0 = -1.012 plus or minus 0.009 (-1.2 sigma), w a = 0 plus or minus 0.03 (w a), and m = 315,3149 plus or minus 0.07.

• For cosmology C5, the results yield w 0 = -0.752 - 0.745 plus or minus 0.012 (w a), w a = -0.86 - 0.90 plus or minus 0.86 (- w a), and m = 316,3163 plus or minus 0.4.

• Coverage Plots: Empirical coverage plots (Figure E1) demonstrate the performance of FlowSN2 posteriors under various conditions (SN Only, Pure SN Ia Sample, SN + CMB, and SN + CMB + BAO), confirming its utility in assessing parameter uncertainty across different observational regimes.

**In summary, FlowSN2 is a statistically rigorous and computationally efficient Bayesian framework that leverages neural normalising flows to jointly infer cosmological parameters and photometric redshifts from Type Ia Supernova data. Its modular design, combined with advanced contamination handling (Simple-BayeSN and Bayesian resampling), provides a powerful tool for robust cosmological inference, enabling the comparison of different dark energy models under realistic observational selection effects without prohibitive computational cost.

1. Amortized Cosmological Inference: The FlowSN2 framework can infer parameters across alternative cosmological parameterisations, such as w0waCDM, away from the fiducial training cosmology, by employing neural likelihood estimation via normalising flows which approximates the data distributions of selected objects in a form that can be reused across different cosmological models.

2. Contamination Robustness: The modular Bayesian resampling approach allows for inference under contamination from non-Ia objects without requiring an explicit contaminant likelihood, protecting downstream inference from potentially misspecified contaminant models.

3. Model Selection Efficiency: FlowSN2 enables Bayesian model comparison at negligible additional computational cost, allowing researchers to choose which model, be it the mass-step parameterisation or evolving dark energy, the data supports directly on a photometric sample.

4. Scalability for Future Surveys: The framework is capable of scaling to over a hundred thousand SNe without retraining, making it suitable for upcoming wide-field photometric surveys like LSST and Roman Space Telescope.

5. Joint Inference with Photometric Redshifts: The system reframes SN Ia photometry from a regression task to a joint inference problem where redshifts, alongside the light curves themselves, are treated as data, allowing constraints from both distance and redshift uncertainties simultaneously.

6. Handling Selection Effects: The framework incorporates selection effects into the model by using neural networks trained on simulations to implicitly capture complexities such as selection effects and non-Gaussian photometric redshift uncertainties.

Photometric Type Ia supernova (SN Ia) cosmology in the era of the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) demands statistical frameworks that simultaneously account for selection effects, photometric redshift uncertainties and non-Ia contamination at scale. We present FlowSN2, a modular statistical framework combining hierarchical Bayesian modelling and simulation-based inference (SBI), that reframes photometric SN cosmology from a regression task to a joint inference problem over light curve summaries and photometric redshifts. By employing neural likelihood estimation via normalising flows, FlowSN2 approximates the data distributions of selected objects in a form that can be reused across different cosmological models, achieving amortisation over the Hubble diagram. This amortisation enables inference across alternative cosmological parameterisations, such as w 0w a CDM, away from the fiducial training cosmology. The framework integrates a standardisation model that disentangles intrinsic SN variations from extrinsic host-galaxy dust effects and adopts a principled Bayesian resampling approach to propagate SN classification uncertainty, without requiring an explicit contaminant likelihood. Across simulated tests spanning five cosmological parameter sets, our framework recovers true cosmological and population parameters, with posteriors in close agreement with both tractable analytical solutions and pure-Ia baselines. Conveniently, the framework enables Bayesian model comparison at negligible additional computational cost, providing a valuable tool for addressing debates surrounding SN standardisation and evolving dark energy. Capable of scaling to over a hundred thousand SNe without retraining, FlowSN2 offers an interpretable and statistically principled solution for cosmological inference with upcoming wide-field photometric surveys.