Amati, D'Agostino, Luongo, Muccino & Tantalo (2019) overcame the circularity problem of using gamma-ray bursts (GRBs) as distance indicators by calibrating the EpE_{\mathrm{p}}–EisoE_{\mathrm{iso}} (Amati) correlation through a model-independent Bézier reconstruction of H(z)H(z) from observational Hubble data, producing 193 GRB distance moduli for cosmological fits. Later literature has already delivered part of the source paper's own future-work agenda: applications to dynamical dark energy (Montiel et al. 2020; Khadka et al. 2021; Alfano et al. 2024; Zhu et al. 2026) and curvature-agnostic calibration from baryon acoustic oscillation (BAO) observables (Alfano et al. 2024; Cao & Ratra 2024). Two elements, however, remain unaddressed in the follow-up literature: a genuine multilevel (hierarchical) Bayesian formulation of the calibration with sampling beyond random-walk Metropolis–Hastings, and model-independent calibration of the SN Ia Phillips relation within the same framework, enabling homogeneous joint GRB+SN Ia cosmological fits. This problem asks for the completion of that program as one coherent objective: a joint, curvature-agnostic, hierarchical calibration whose uncertainty propagation is explicit and auditable, validated by application to flat and non-flat dynamical dark-energy models and by comparison with previous model-independent calibrations.

Background

Gamma-ray bursts (GRBs) are the most luminous electromagnetic explosions in the Universe and are detected out to redshift z≳8z\gtrsim 8, making them attractive distance indicators bridging the gap between Type Ia supernovae (SN Ia, observed to z≈2z\approx 2) and the early Universe. Long GRBs obey an empirical power-law relation, the 'Amati correlation', between the photon energy EpE_{\mathrm{p}} at which the prompt νFν\nu F_\nu spectrum peaks and the isotropic-equivalent radiated energy EisoE_{\mathrm{iso}}: log⁡(Ep/1 keV)=q+m [log⁡(Eiso/1 erg)−52]\log(E_{\mathrm{p}}/1\,\mathrm{keV}) = q + m\,[\log(E_{\mathrm{iso}}/1\,\mathrm{erg}) - 52]. Because EisoE_{\mathrm{iso}} is inferred from the measured fluence through the luminosity distance dL(z)d_L(z), which itself depends on the cosmological parameters one wishes to constrain, calibrating the correlation and constraining cosmology with the same GRBs biases both — the 'circularity problem'. Its resolution requires a calibration that does not assume a cosmological model.

Amati, D'Agostino, Luongo, Muccino & Tantalo (2019) proposed such a model-independent calibration. They fitted the expansion history H(z)H(z) with a second-order Bézier curve (a linear combination of Bernstein basis polynomials) through 31 observational Hubble data (OHD) points — direct H(z)H(z) measurements obtained by the differential-age (cosmic-chronometer) method applied to pairs of passively evolving galaxies, which require no assumed cosmology — integrated the result to obtain dL(z)d_L(z) under a spatially flat geometry, calibrated the EpE_{\mathrm{p}}–EisoE_{\mathrm{iso}} relation (q=2.06±0.03q=2.06\pm0.03, m=0.50±0.02m=0.50\pm0.02, intrinsic scatter σex≈0.20\sigma_{\mathrm{ex}}\approx 0.20 dex), and converted 193 GRBs into distance moduli for Markov Chain Monte Carlo (MCMC) cosmological fits combined with the SN Ia JLA (Joint Light-curve Analysis) sample, obtaining Ωm=0.34−0.15+0.13\Omega_m=0.34^{+0.13}_{-0.15} and w=−0.86−0.38+0.36w=-0.86^{+0.36}_{-0.38} at 2σ2\sigma for the flat wwCDM model and favouring Λ\LambdaCDM by Akaike and Deviance Information criteria. Their MCMC used a Metropolis–Hastings (MH) random-walk sampler. The paper itself listed four items of unfinished business: (a) the flatness assumption only approximately removes the calibration's curvature dependence, so a fully curvature-sensitive treatment remains open should the curvature density Ωk\Omega_k prove non-negligible; (b) the MH random walk may be suboptimal for strongly correlated models, and a proper multilevel ('hierarchical') Bayesian treatment is left for future work; (c) the framework is proposed, but not carried out, for calibrating the SN Ia Phillips relation — the empirical correlation between light-curve decline rate and peak luminosity that makes SNe Ia standardizable candles — so that GRBs and SNe could be combined homogeneously; and (d) the technique still needs application to dynamical dark-energy models and comparison with previous model-independent calibrations. Two later results sharpen the stakes: Kumar et al. (2023) found evidence for redshift evolution of the Amati parameters from a model-independent OHD calibration, and Cao & Ratra (2024) found a >2σ>2\sigma tension in Ωm\Omega_m between GRB-calibrated and H(z)H(z)+BAO constraints — complications that a rigorous joint treatment must address.

Problem Statement

Complete, as one coherent program, the model-independent standardization framework proposed in Amati et al. (2019), whose common objective is an auditable, circularity-free calibration pipeline spanning the supernova and GRB redshift ranges:

  1. Formulate the calibration of the EpE_{\mathrm{p}}–EisoE_{\mathrm{iso}} correlation as a genuine multilevel (hierarchical) Bayesian model in which the model-independent, curvature-agnostic reconstruction of the expansion history (Bézier/Bernstein over OHD, BAO-observable-based reconstruction as in Alfano et al. 2024, or an equivalent model-independent basis), the measurement errors on EpE_{\mathrm{p}} and fluence (an errors-in-variables treatment, i.e. regression that accounts for measurement error in both variables), the intrinsic scatter of the correlation, and redshift evolution of its parameters (estimated and tested within the joint fit rather than assumed away) are treated jointly rather than sequentially; and sample the resulting strongly correlated posterior with methods robust beyond random-walk Metropolis–Hastings.

  2. Extend the same model-independent calibration to the SN Ia Phillips relation, so that GRBs and SNe Ia are standardized within one homogeneous framework and can be combined in a single cosmological fit without any model-dependent intermediate step.

  3. Validate the completed framework by deriving constraints on dynamical dark-energy models (e.g. the Chevallier–Polarski–Linder (CPL) or other w0waw_0w_a parametrizations, in which the dark-energy equation-of-state parameter w(z)w(z) varies with redshift) in both flat and non-flat geometries, and by comparing the results against previous model-independent calibrations of the same correlation.

All three parts are required together: the hierarchical treatment supplies the uncertainty propagation, the SN Ia calibration supplies the homogeneous low-redshift anchor, and the dynamical dark-energy application and comparison establish that the method's behavior extends beyond the originally tested flat Λ\LambdaCDM and wwCDM cases.

References

  1. Lorenzo Amati, Rocco D'Agostino, Orlando Luongo, Marco Muccino, Maria Tantalo, 'Addressing the circularity problem in the Ep−Eiso correlation of gamma-ray bursts', Monthly Notices of the Royal Astronomical Society: Letters 486, L46–L51 (2019), DOI 10.1093/mnrasl/slz056, https://arxiv.org/abs/1811.08934
  2. Ariadna Montiel, J. I. Cabrera, Juan Carlos Hidalgo, 'Improving sampling and calibration of GRBs as distance indicators', Monthly Notices of the Royal Astronomical Society (2020), DOI 10.1093/mnras/staa3926, https://arxiv.org/abs/2003.03387
  3. Narayan Khadka, Orlando Luongo, Marco Muccino, Bharat Ratra, 'Do gamma-ray burst measurements provide a useful test of cosmological models?', Journal of Cosmology and Astroparticle Physics 09 (2021) 042, DOI 10.1088/1475-7516/2021/09/042, https://doi.org/10.1088/1475-7516/2021/09/042
  4. Anna Chiara Alfano, Orlando Luongo, Marco Muccino, 'Cosmological constraints from calibrated Ep−Eiso gamma-ray burst correlation by using DESI 2024 data release', Journal of Cosmology and Astroparticle Physics 12 (2024) 055, DOI 10.1088/1475-7516/2024/12/055, https://arxiv.org/abs/2408.02536
  5. Shulei Cao, Bharat Ratra, 'Testing the standardizability of, and deriving cosmological constraints from, a new Amati-correlated gamma-ray burst data compilation', Journal of Cosmology and Astroparticle Physics 10 (2024) 093, DOI 10.1088/1475-7516/2024/10/093, https://doi.org/10.1088/1475-7516/2024/10/093
  6. Darshan Kumar, Nisha Rani, Deepak Jain, Shobhit Mahajan, Amitabha Mukherjee, 'Gamma rays bursts: a viable cosmological probe?', Journal of Cosmology and Astroparticle Physics 07 (2023) 021, DOI 10.1088/1475-7516/2023/07/021, https://doi.org/10.1088/1475-7516/2023/07/021
  7. Zhen Huang, Zhiguo Xiong, Xin Luo, Guangzhen Wang, Yu Liu, Nan Liang, 'Gamma-Ray Bursts Calibrated from the Observational H(z) Data in Artificial Neural Network Framework', Journal of High Energy Astrophysics 47, 100377 (2025), DOI 10.1016/j.jheap.2025.100377, https://arxiv.org/abs/2502.10037
  8. Nilanjana Bagchi Aurpa, Abha Dev Habib, Nisha Rani, 'Reconstructing Gamma Ray Burst Energy Relations with Observational H(z) data in Neural Network Framework', arXiv:2601.08550 (2026), https://arxiv.org/abs/2601.08550
  9. Ziyan Zhu, Qingquan Jiang, Yu Liu, Puxun Wu, Nan Liang, 'Cosmological Constraints on the Phenomenological Interacting Dark Energy Model with Fermi Gamma-Ray Bursts and DESI DR2', Journal of High Energy Astrophysics 51, 100534 (2026), DOI 10.1016/j.jheap.2025.100534, https://arxiv.org/abs/2511.16032
  10. Marco Muccino, Orlando Luongo, 'Generalizing the Hubble constant determination including curvature and dynamical dark energy', Open Astronomy (2026), DOI 10.1515/astro-2025-0022, https://doi.org/10.1515/astro-2025-0022
  11. Nan Liang, Zihao Li, Xiaoyao Xie, Puxun Wu, 'Calibrating Gamma-Ray Bursts by Using a Gaussian Process with Type Ia Supernovae', The Astrophysical Journal 941, 84 (2022), DOI 10.3847/1538-4357/aca08a, https://doi.org/10.3847/1538-4357/aca08a