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Revisiting 2D and 3D Dainotti Correlations for GRBs Using Bayesian Neural Networks

This study employs Bayesian Neural Networks to model-independently calibrate 2D and 3D Dainotti correlations for Gamma-ray bursts using Pantheon+ and OHD data, demonstrating that the Pantheon+ dataset yields tighter constraints and that the 3D correlation offers lower intrinsic scatter than the 2D relation.

Original authors: Nilanjana Bagchi Aurpa, Abha Dev Habib, Nisha Rani

Published 2026-08-10

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CC0 1.0

Original authors: Nilanjana Bagchi Aurpa, Abha Dev Habib, Nisha Rani

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

Imagine the universe as a giant, expanding ocean. To understand how fast it's growing and where it's going, astronomers need to measure distances to faraway islands. For decades, they've used "standard candles"—stars that shine with a predictable brightness, like a specific brand of lightbulb. If you know how bright the bulb is supposed to be, you can tell how far away it is just by how dim it looks to you. The problem is, the brightest lightbulbs we have (Supernovae) only work up to a certain distance. Beyond that, the ocean gets too dark to see them.

Enter Gamma-Ray Bursts (GRBs). These are the universe's most explosive fireworks, visible from so far away that they existed when the universe was just a toddler. They are the perfect "super-bulbs" for the deep cosmos. But there's a catch: we don't know exactly how bright they should be. To use them as rulers, we first have to calibrate them, but doing that usually requires knowing the distance to them already. It's a classic "chicken and egg" problem. Scientists have tried to solve this by finding patterns in the bursts, like how long a burst lasts versus how bright it is, hoping these patterns act as a universal rulebook. However, to find the rulebook, you need to know the distances first, which brings you right back to the chicken-and-egg trap.

This is where a team of researchers from the University of Delhi steps in with a clever new trick. They wanted to break this loop without relying on a pre-set theory of how the universe works. They used a type of artificial intelligence called a Bayesian Neural Network (BNN). Think of this AI not as a robot that just memorizes facts, but as a super-smart detective that learns from clues (data) while keeping a running list of how unsure it is about every guess. They fed this detective two different sets of "known" distance clues: one from the Hubble Data (a collection of local measurements) and another from the Pantheon+ dataset (a massive catalog of supernovae).

The goal was to see if this AI could reconstruct a reliable map of the universe's expansion without assuming a specific shape for the universe first. Once the AI built this map, the team used it to calibrate two different groups of Gamma-Ray Bursts: a "Platinum" group (a small, very carefully selected team of the cleanest bursts) and a "Narendra" group (a much larger, more diverse crowd of bursts). They tested two different rulebooks: a simple 2D rule (linking brightness and time) and a more complex 3D rule (adding a third factor: the peak brightness of the initial explosion).

Here is what they found. The AI did a great job of mapping the universe, and it showed that the "Pantheon+" clues were better than the "Hubble" clues. Why? Because the Pantheon+ data stretched further into the distance, allowing the team to calibrate more Gamma-Ray Bursts. When they used this wider net, the rules they derived became much sharper and more precise.

Most importantly, they confirmed that the 3D rulebook is superior to the 2D one. Just like a 3D model of a building gives you more information than a flat drawing, adding that third piece of data (the peak brightness) made the relationship between the bursts' properties much tighter and less messy. The "intrinsic scatter"—which is like the natural fuzziness or noise in the data—dropped significantly when they used the 3D method. For the Platinum sample, the uncertainty in the 3D relation was around 0.372, compared to 0.559 for the 2D relation.

The study suggests that using this AI approach is a robust way to calibrate these cosmic explosions without getting stuck in circular logic. It proves that the 3D correlation is a stronger tool for measuring the universe. However, the authors are careful to note that the precision is still limited by how many good Gamma-Ray Bursts we have right now. They suggest that future missions, which will spot many more of these bursts, will allow this AI framework to create even sharper maps of the early universe, helping us finally understand the true scale of the cosmos.

Technical Summary: Revisiting 2D and 3D Dainotti Correlations for GRBs Using Bayesian Neural Networks

Problem Statement
Gamma-ray bursts (GRBs) are valuable cosmological probes capable of reaching high redshifts ( $z \sim 9.4$ ), far beyond the reach of Type Ia supernovae. However, their utility as standardizable candles is hindered by the "circularity problem": calibrating GRB luminosity correlations typically requires a fiducial cosmological model, which then biases the inference of cosmological parameters when those same GRBs are used to constrain that model. To establish GRBs as reliable distance indicators, model-independent calibration is necessary. While various methods exist (e.g., interpolation, Gaussian Processes), they often face limitations regarding kernel sensitivity or the lack of inherent uncertainty estimates in standard Artificial Neural Networks (ANNs).

Methodology
This study employs Bayesian Neural Networks (BNNs) to perform model-independent calibration of GRB luminosity correlations. The methodology proceeds as follows:

Reconstruction of Luminosity Distance: Two independent BNN models were trained to reconstruct luminosity distance ( $d_L$ ) from observational data without assuming a specific cosmological model:
- BNN-OHD: Trained on the updated Observational Hubble Data (OHD), comprising 32 data points ( $0.07 < z < 1.965$ ).
- BNN-Pantheon: Trained on the Pantheon+ Type Ia Supernova sample, comprising 1701 light curves extending to $z \sim 2.261$ .
- The BNN framework utilizes Bayesian inference (via the No-U-Turn Sampler) to marginalize over network parameters, providing both aleatoric and epistemic uncertainties. Hyperparameters were optimized using the Widely Applicable Information Criterion (WAIC).
GRB Sample Selection: Two distinct GRB datasets were calibrated using the reconstructed distances from the BNN models:
- Platinum Sample: A highly selected subset of 50 long GRBs ( $0.553 \le z \le 5.0$ ) with well-defined X-ray plateaus, designed to minimize selection biases.
- Narendra et al. Sample: A larger, more homogeneous compilation of 246 long GRBs ( $0.0331 \le z \le 8.26$ ) derived primarily from Swift observations.
- Due to redshift overlap constraints, subsets of these samples (ranging from 19 to 147 GRBs depending on the calibration source) were used for analysis.
Correlation Analysis: The calibrated luminosity distances were used to constrain the parameters of:
- The 2D Dainotti Relation, linking rest-frame plateau luminosity ( $L_X$ ) and duration ( $T^*_a$ ).
- The 3D Fundamental Plane, which adds the prompt peak luminosity ( $L_{peak}$ ) to the relation.
- Bayesian MCMC analysis was performed with weakly informative uniform priors to determine the best-fit parameters ( $a, b, C_0$ ) and intrinsic scatter ( $\sigma_{int}$ ).

Key Results

Calibration Source Impact: The BNN-Pantheon calibration yielded tighter constraints on Dainotti parameters compared to BNN-OHD. This is attributed to the Pantheon+ dataset's extension to higher redshifts ( $z \lesssim 2.26$ ), which allows for a larger effective calibration sample compared to the OHD limit ( $z \lesssim 2.0$ ). For the Platinum sample, the uncertainty on the 2D slope ( $a$ ) decreased from $\pm 0.203$ (OHD) to $\pm 0.163$ (Pantheon+).
2D vs. 3D Correlations: The 3D Fundamental Plane consistently exhibited lower intrinsic scatter than the 2D relation for both GRB samples. For the Platinum sample, $\sigma_{int}$ decreased from $0.559$ (2D) to $0.372$ (3D) when using Pantheon+ calibration. This confirms that including prompt-emission peak luminosity provides additional physical information that tightens the standardization relation.
Sample Dependence: The inferred correlation parameters differed between the Platinum and Narendra samples, reflecting their distinct selection criteria. The Platinum sample (stricter selection) generally showed lower scatter.
Consistency: The results for the Platinum sample using Pantheon+ calibration are in good agreement with previous model-independent studies using Gaussian Processes (Favale et al.) and ANNs (Mukherjee et al.), despite differences in reconstruction techniques.

Significance and Claims
The paper claims that Bayesian Neural Networks provide a robust framework for the model-independent calibration of GRB luminosity correlations, offering reliable uncertainty propagation that addresses limitations of previous machine learning approaches. The study demonstrates that the choice of underlying distance probe is a critical factor in model-independent calibration, directly influencing the size of the usable GRB sample and the precision of the resulting constraints.

The authors conclude that while BNNs offer a consistent and rigorous method for calibration, the ultimate precision of the Dainotti relations remains limited by the size and quality of currently available GRB samples. They note that future missions (SVOM, Einstein Probe, THESEUS) are expected to provide larger datasets with accurate plateau properties, which, combined with the BNN framework, will enable even tighter calibrations for probing the high-redshift universe.

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