Tightening Cosmological Constraints Within and Beyond $Λ$CDM Using Gamma-Ray Bursts Calibrated with Type Ia Supernovae

Abstract

Context. Gamma-ray bursts (GRBs) reach redshifts beyond Type Ia supernovae (SNe Ia) and can extend distance measurements into the early Universe, but their use as distance indicators is limited by the circularity problem in calibrating empirical luminosity relations. Aims. We present a model-independent methodology to overcome this circularity by combining Pantheon$+$ SNe Ia, a distance reconstruction based on artificial neural networks (ANNs), and two GRB correlations (Amati and Combo) into a distance ladder from low to high redshift, with the goal of constraining cosmological parameters in $Λ\mathrm{CDM}$ and $w_0 w_a \mathrm{CDM}$. Methods. We use the ReFANN to reconstruct the luminosity distance $d_L(z)$ and distance modulus $μ(z)$ from the Pantheon$+$ dataset, with hyperparameters optimized via approximate Bayesian computation rejection and a risk function. This model-independent reconstruction calibrates the Amati and Combo relations using a low-redshift ($z<1$) GRB sample from Fermi GBM and Swift-XRT. The calibrated relations then provide distance estimates for GRBs at $z \geq 1$. Finally, a joint Bayesian analysis simultaneously constrains the cosmological and GRB correlation parameters, ensuring self-consistent uncertainty propagation. Results. We obtain consistent cosmological constraints from two independent GRB correlations. The Hubble constant $H_0$ agrees with SNe Ia values, though potentially influenced by Pantheon$+$ dataset. High-redshift GRBs favour a higher matter density $Ω_m$ than the Pantheon$+$ and hint at possible dark energy evolution.Conclusions. We present a framework that mitigates GRB cosmology's circularity problem, extending the distance ladder to $z \sim 9$ and establishing GRBs as a high-redshift probe.

Figures (15)

• Figure 1: The reconstructed distance modulus $\mu(z)$ using ReFANN with different network hyperparameters. Each line corresponds to a different combination of the number of hidden layers ($N_{\rm hl}$) and the number of nodes per layer ($N_{\mathrm{node}}$). The observed data with error bars are shown as orange points, and the shaded regions represent the $1\sigma$ uncertainty of the ANN reconstructions. The corresponding $d_L(z)$ reconstruction is shown in Appendix Fig. \ref{['figANN-dL']}.
• Figure 2: The risk heatmap for luminosity distance $d_L(z)$ (top panel) and distance modulus $\mu(z)$ (bottom panel). The heatmaps show the log of the risk values for different combinations of hidden layers and mid-node counts in the ANN hyperparameters. The color bar represents the range of $\log_{10}$ risk values, with purple indicating lower risk and green indicating higher risk.
• Figure 3: Comparison between the Pantheon$+$ SNe Ia distance modulus $\mu(z)$ reconstructed with the ANN and the low-redshift GRB calibration subset. The solid line represents the reconstructed relation, with the shaded band indicating the 68% reconstruction uncertainty. The SNe Ia data points are shown with their observational error bars, while the low-redshift GRBs ($z\leq 1$) are overplotted to demonstrate consistency between the GRB calibration anchor and the SNe distance ladder. The corresponding $d_L(z)$ comparison is shown in Appendix Fig. \ref{['figc-dL']}.
• Figure 4: Summary and comparison of our $H_0$ and $w_0w_a$ constraints, comparing other results with the results from Amati and Combo. The top panel shows $H_0$ constraints in flat $\Lambda$CDM for Planck18 2020AA...641A...6P, DESI+Planck18 2025JCAP...02..021A, ACT DR6 2025JCAP...11..062L2025JCAP...11..063C2024ApJ...962..113M, PantheonPlus+SH0ES 2022ApJ...938..110B, and GRBs, with thick and thin bars respectively indicating $68\%$ and $95\%$ credible intervals and markers indicating medians. The bottom panel shows $(w_0, w_a)$ constraints, with filled contours for combined constraints in $68\%$ and $95\%$ credible intervals and dashed contours for individual constraints. The black star marks the $\Lambda$CDM point $(-1,0)$. Amati and Combo contours are shown as lightly shaded backgrounds for comparison.
• Figure 5: Empirical correlations standardizing GRBs as distance indicators. Both panels share the same symbol and color scheme to distinguish between low-redshift and high-redshift GRBs, plotted alongside their measurement uncertainties. The top panel illustrates the Amati relation between the isotropic equivalent energy, $E_{\text{iso }}$, and the intrinsic peak energy, $E_{\mathrm{p}, i}$. The bottom panel presents the Combo correlation in the parameter space of $\log L_0$ versus $\log \left[E_{\mathrm{p}, i} / \mathrm{keV}\right]-1.46 \log [(\tau / \mathrm{s}) /|1+\alpha|]$. In each panel, the solid line indicates the best-fit correlation, with the grey shaded bands marking the corresponding $1 \sigma$ and $3 \sigma$ intrinsic scatter regions.