Model-independent standardization of gamma-ray burst plateau distances for decisive discrimination of one- versus two-component dark energy

Gamma-ray bursts (GRBs) are the only point-like distance indicators observed out to redshift $z\approx9$ and could fill the redshift gap between Type Ia supernovae (SNe Ia, $z\lesssim2.9$ ) and the cosmic microwave background. Li et al. (2024, A&A 689, A165) standardized plateau GRBs through the two-parameter correlation between rest-frame plateau luminosity $L_{b,z}$ and plateau end time $T_{b,z}$ , and its tighter three-parameter extension including the prompt peak energy $E_{p,i}$ ; they calibrated these relations model-independently by Gaussian-process reconstruction of 36 $H(z)$ measurements ( $z\leq2.36$ ), and combined the calibrated X-ray (31 GRBs) and optical (50 GRBs) samples with SNe Ia to constrain the two-component dark energy model $X_1X_2$ CDM. The two dark-energy equation-of-state parameters $\omega_1$ and $\omega_2$ came out consistent within $1\sigma$ , and information criteria favored the simpler one-component $\omega$ CDM model, but the analysis is limited by the model-independent calibration ceiling $z\lesssim2.5$ , large calibrated intrinsic dispersions, and a strong $\omega_1$ - $\omega_2$ -density degeneracy. The open problem is whether GRBs can be developed into a distance probe precise enough to decide, model-independently, whether dark energy comprises one or two components: extending calibration beyond the anchor ceiling, reducing intrinsic scatter with larger physically homogeneous plateau samples, and converting the current inconclusive statistical preference into a decisive, robust discrimination.

Background

Gamma-ray bursts (GRBs) are the most luminous electromagnetic transients in the Universe, detected as point sources out to redshift $z\approx9.4$ ; a long GRB afterglow light curve often shows a shallow plateau phase followed by a power-law decay, attributed to energy injection from the central engine (e.g., a newborn magnetar). Because SNe Ia reach only $z\lesssim2.9$ and the cosmic microwave background (CMB) probes a single epoch at $z\approx1100$ , GRBs are a candidate distance indicator for the otherwise unprobed interval $2.9\lesssim z\lesssim9$ . A GRB luminosity correlation must be calibrated before use, and calibrating it with distances computed in an assumed cosmological model would make any resulting cosmological constraint circular (the circularity problem). The standard workaround reconstructs the Hubble parameter $H(z)$ — measured without reference to dark-energy models via cosmic chronometers, differential galaxy ages that yield $H(z)$ directly — into a smooth function by Gaussian-process (GP) regression, a non-parametric Bayesian interpolation method, and integrates it to obtain luminosity distances. In the source analysis this calibration was anchored by 36 $H(z)$ points spanning $0.07<z<2.36$ , so calibrated GRB distances are model-independent only for $z\lesssim2.5$ . The luminosity correlations used are the two-dimensional Dainotti relation, between the rest-frame luminosity at the end of the plateau $L_{b,z}$ and the rest-frame plateau end time $T_{b,z}$ , and its three-dimensional extension, the fundamental plane adding the rest-frame prompt peak energy $E_{p,i}$ ; each relation carries an intrinsic scatter $\sigma_{\rm int}$ reflecting genuine GRB diversity on top of measurement errors. On the theory side, $\omega$ CDM is the one-component dark energy model with constant equation-of-state (EoS) parameter $\omega=p/\rho$ ( $\omega=-1$ is a cosmological constant); $X_1X_2$ CDM (following Gong & Chen 2007) splits dark energy into two non-interacting fluids with constant EoS $\omega_1$ and $\omega_2$ and present-day density parameters $\Omega_{x1}$ and $\Omega_{x2}$ , giving five free parameters in the flat case ( $\Omega_{m0}$ , $\Omega_{x1}$ , $\omega_1$ , $\omega_2$ , $H_0$ ). Li et al. (2024) screened GRBs with plateau-plus-decay morphology (decay index $\approx-2$ ) that plausibly share one physical origin: 31 X-ray GRBs from Wang et al. (2022) and 50 optical GRBs from Si et al. (2018). Fitting the calibrated correlations by Markov-chain Monte Carlo (MCMC) and constraining both cosmologies with SNe Ia plus the GRB samples, they found $\omega_1$ and $\omega_2$ consistent at $1\sigma$ with large error bars (e.g., $\omega_1=-1.20^{+0.41}_{-0.54}$ , $\omega_2$ similarly imprecise), concluding that the composition of dark energy seems to prefer a single component, while the Akaike and Bayesian information criteria (AIC/BIC, penalized likelihood-ratio statistics for model comparison) favored the model with fewer free parameters. Model-independent GRB calibration was previously established on the Platinum sample (50 plateau GRBs, $0.553\leq z\leq5$ ) by Cao, Dainotti & Ratra (2022), whose GRB-only constraints were consistent with but less precise than $H(z)$ +BAO (baryon acoustic oscillation) data, and Dainotti et al. (2022) showed the optical fundamental plane is as effective as the X-ray one but that 142-284 well-measured optical GRBs would be needed to match SNe Ia-era precision on $\Omega_M$ .

Problem Statement

Can gamma-ray bursts be standardized, through the plateau fundamental-plane correlations $L_{b,z}$ - $T_{b,z}$ and $L_{b,z}$ - $T_{b,z}$ - $E_{p,i}$ , into a model-independent distance probe capable of decisively discriminating the two-component dark energy model $X_1X_2$ CDM from the one-component $\omega$ CDM model? Answering this requires three jointly necessary and scientifically linked parts of one objective: (i) a model-independent calibration of the GRB luminosity relations that extends beyond the current low-redshift anchor ceiling — $z\lesssim2.5$ set by the $H(z)$ data used in the source analysis, $z\lesssim2.9$ set by the largest SNe Ia compilation used in later calibrations — without assuming any dark-energy model in the distance scale; (ii) calibrated intrinsic dispersions small enough, through larger physically homogeneous and well-measured plateau GRB samples and improved prompt/afterglow measurements, that GRB distance-modulus uncertainties no longer dominate the cosmological constraints on ( $\Omega_{m0}$ , $\Omega_{x1}$ , $\omega_1$ , $\omega_2$ ); and (iii) a determination of whether the EoS parameters of the two dark-energy components actually differ — breaking the current $\omega_1$ - $\omega_2$ - $\Omega_{x}$ degeneracy — and whether model-comparison statistics (AIC/BIC or Bayesian evidence) then robustly favor one model over the other, rather than the present inconclusive $1\sigma$ consistency of $\omega_1$ and $\omega_2$ . A quantitative demonstration that no attainable combination of sample size, measurement precision, and intrinsic scatter can achieve this discrimination is an admissible resolution of equal standing.

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