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Strong Lensing Cosmology with Population-level Calibrated Neural Ratio Estimation

This paper demonstrates that Neural Ratio Estimation, enhanced by a post hoc calibration procedure to correct model overconfidence, offers a scalable and efficient method for inferring cosmological parameters like the dark energy equation of state ( $w$ ) and matter density ( $\Omega_m$ ) from large populations of strong gravitational lensing systems.

Original authors: Sreevani Jarugula, Brian D. Nord, Aleksandra Ćiprijanović, Shubhendu Trivedi

Published 2026-08-25

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CC BY 4.0

Original authors: Sreevani Jarugula, Brian D. Nord, Aleksandra Ćiprijanović, Shubhendu Trivedi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe is not just expanding; it is accelerating, a cosmic speed-up driven by a mysterious force known as dark energy. To understand why this is happening, astronomers must measure two fundamental ingredients of the cosmos: the total amount of matter, which includes invisible dark matter, and the specific nature of dark energy itself. These measurements are difficult because the universe is vast and the signals are faint. For decades, scientists have relied on a technique called strong gravitational lensing, where a massive galaxy in the foreground bends the light from a distant background galaxy, creating distorted, arc-like images. By studying how much the light bends, researchers can map the mass of the foreground galaxy and, crucially, measure the distances between these objects. These distances hold the key to calculating the rate of cosmic expansion and the properties of dark energy. However, the sheer volume of data expected from upcoming sky surveys will be overwhelming, requiring new methods to extract precise answers from millions of lensing systems without getting lost in the complexity.

A team of researchers has developed a new approach to tackle this challenge, using artificial intelligence to learn directly from simulated images of the universe. Instead of trying to fit complex mathematical formulas to every single lens, which is computationally expensive and often impossible for such high-dimensional data, they trained a neural network to recognize the statistical relationship between the images and the underlying cosmological parameters. The scientists generated millions of simulated lens images, each created with different values for the amount of matter and the nature of dark energy. They taught the computer to distinguish between images generated by one set of cosmic rules versus another, effectively learning to calculate a likelihood ratio that tells how probable a specific image is under different cosmological scenarios. This method allows them to combine information from many individual lenses into a single, powerful population-level estimate.

The study revealed that looking at a single lensing system is rarely enough to pin down the nature of dark energy with high precision. While a single galaxy can offer a hint about the total amount of matter, the data is too noisy to constrain the equation of state for dark energy, which describes how its pressure relates to its density. However, when the researchers combined the data from a population of one hundred lenses, the picture sharpened significantly. The artificial intelligence model was able to narrow down the possible values for the total matter density to within about 2.9 percent and the dark energy parameter to within 22.8 percent. This demonstrates that by pooling data from many objects, the method can overcome the limitations of individual measurements and provide robust constraints on the universe's composition.

A critical part of this work involved addressing a common flaw in artificial intelligence models: overconfidence. Neural networks often produce answers that look very certain but are actually too narrow, excluding the true values of the parameters they are trying to measure. To fix this, the researchers introduced a post-hoc calibration step. After the model made its initial predictions, they adjusted the results using a statistical technique that broadened the uncertainty ranges just enough to ensure the true values were captured within the predicted intervals. This calibration was essential; without it, the model would have claimed to know more than it actually did, potentially leading scientists to reject valid theories or misinterpret errors as new physics. With the calibration applied, the model's confidence became reliable, and the results matched the performance of more traditional, slower analytical methods while using far fewer data points.

The researchers tested their system on various regions of the parameter space to ensure it worked consistently everywhere. While the method performed exceptionally well across the entire test range, they found that in some specific sub-regions, the calibration was less uniform, suggesting that the model still needs refinement to handle every possible cosmic scenario equally well. They also noted that their simulations assumed ideal conditions, such as perfect knowledge of the galaxies' redshifts and velocity dispersions, and the removal of the foreground galaxy's light, which are challenges in real-world observations. Despite these limitations, the study serves as a proof of concept, showing that neural ratio estimation can efficiently handle the massive datasets expected from future telescopes. By turning the problem of analyzing millions of lenses into a manageable learning task, this approach offers a scalable path toward understanding the dark energy that drives the accelerating universe.

Technical Summary: Strong Lensing Cosmology with Population-level Calibrated Neural Ratio Estimation

Problem Statement
Strong gravitational lensing offers a pathway to constrain cosmological parameters, specifically the dark energy equation of state ( $w$ ) and the total matter density ( $\Omega_m$ ). However, upcoming surveys (e.g., LSST, Euclid, Roman, CSST) are expected to yield $\mathcal{O}(10^5)$ galaxy-galaxy lensing systems. Traditional parameter inference methods face significant computational challenges when fitting high-dimensional, analytically intractable likelihoods for such large populations. Furthermore, existing Simulation-based Inference (SBI) techniques, particularly Neural Posterior Estimation (NPE), are often unsuitable for population-level inference because combining individual posteriors is not formally permissible. Additionally, SBI models frequently exhibit "overconfidence," producing posteriors that are artificially narrow and exclude true physical parameters, posing a risk of false model rejection or misinterpretation of systematics as new physics.

Methodology
The authors propose a framework using Neural Ratio Estimation (NRE) to perform population-level inference. The approach involves three primary stages:

Data Generation and Simulation:
- Simulated lens images ( $32 \times 32$ pixels, $g$ -band) are generated using deeplenstronomy based on DES Data Release 1 characteristics.
- The simulation assumes idealized conditions: perfect lens light subtraction, no external shear, and perfectly known astrophysical parameters (lens/source redshifts $z_l, z_s$ and stellar velocity dispersion $\sigma_v$ ).
- Training data consists of $\sim 2$ million lens images with cosmological parameters ( $w, \Omega_m$ ) drawn from uniform priors ( $w \in [-1.8, -0.35]$ , $\Omega_m \in [0.1, 0.9]$ ).
Neural Ratio Estimation (NRE) Model:
- Instead of estimating the posterior directly (NPE) or the likelihood (NLE), the model estimates the likelihood-to-evidence ratio (LR), $r(x|\eta) = p(x, \eta) / [p(x)p(\eta)]$ .
- A ResNet architecture processes the lens images, while separate dense layers embed the known astrophysical parameters ( $\zeta$ ) and cosmological parameters ( $w, \Omega_m$ ).
- The network is trained as a binary classifier to distinguish between joint distribution samples (class $y=1$ ) and marginal distribution samples (class $y=0$ ).
- For a population of $N$ lenses, the population-level log-LR is obtained by summing the individual log-LRs, assuming independent and identically distributed (i.i.d.) observations.
- Markov Chain Monte Carlo (MCMC) is used to sample the posterior $p(w, \Omega_m | \{x, \zeta\})$ from the combined log-LR.
Post Hoc Calibration:
- To address the tendency of SBI models to be overconfident, the authors introduce a post hoc calibration procedure.
- The calibrated log-LR is defined as $\ln \hat{r} = \Gamma \sum \ln r + \Delta$ , where $\Gamma$ acts as a temperature-like scaling factor to adjust posterior width, and $\Delta$ is a normalization offset.
- Optimal values for $\Gamma$ and $\Delta$ are determined via a three-round Sobol sequence search to minimize the Earth Mover's Distance (EMD) between the posterior rank histograms and a uniform distribution (Simulation-based Calibration).

Key Results

Individual vs. Population Inference: Individual lenses provide limited constraints on $w$ and only modest constraints on $\Omega_m$ . However, as the population size increases, the constraints tighten significantly.
Performance Metrics: For a population of 100 lenses in a standard $\Lambda$ CDM universe, the calibrated NRE model achieves a median fractional uncertainty of 22.8% in $w$ and 2.9% in $\Omega_m$ .
Calibration Efficacy:
- Uncalibrated models produce $\cup$ -shaped rank histograms, indicating overconfidence.
- The post hoc calibration successfully broadens the posteriors, making the rank histograms closer to uniform and improving posterior coverage (the fraction of true values falling within confidence intervals).
- The calibrated model reduces the EMD between the posterior and the uniform distribution (e.g., from 0.69 to 0.41 for $w$ in the Entire Test Region).
Comparison to Analytic Methods: The calibrated NRE approach yields constraints comparable to analytic hierarchical Bayesian methods (e.g., Li et al. 2024) but requires a sample size 100 times smaller (100 lenses vs. 10,000 mock observations).
Training Set Size: Increasing the training set from 500k to 2M images did not significantly reduce overconfidence, suggesting that post hoc calibration is a necessary component of the framework at these scales.

Significance and Limitations
The paper presents a proof of concept demonstrating that NRE, combined with a novel post hoc calibration procedure, offers a scalable and efficient approach for cosmological parameter inference from large populations of galaxy-scale lenses. The method successfully mitigates the overconfidence typically found in neural density estimation, providing more trustworthy uncertainty estimates.

The authors explicitly note several limitations and caveats:

Idealized Data: The study assumes perfect lens light subtraction, perfect deblending, and perfectly known astrophysical parameters ( $z_l, z_s, \sigma_v$ ), ignoring observational uncertainties.
Single Band: Analysis is limited to $g$ -band imaging; multi-band photometry was not included.
Calibration Consistency: While calibration improves performance globally, its efficacy varies across different sub-regions of the parameter space. Some sub-regions remain slightly overconfident or underconfident, and the causes of this variability are not yet fully understood.
Scalability: The current calibration search (Sobol sequence with MCMC) is computationally expensive, and future work may require more efficient optimization strategies or secondary neural networks to predict scaling factors.

The study concludes that while the method shows promise for future surveys (Rubin, Euclid, Roman), bridging the gap between these idealized simulations and real observational data remains a critical next step.

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