Adjudicating cosmographic-expansion bias and testing $\Lambda$CDM at the deceleration–acceleration transition with model-independently calibrated gamma-ray burst correlations

Gamma-ray burst (GRB) correlations, calibrated without assuming a cosmological model, can constrain the redshift $z_t$ of the transition from decelerated to accelerated expansion and the jerk parameter $j_t$ there. The source analysis (Alfano et al. 2024) found that two standard cosmographic expansions about $z_t$ — the direct Hubble-rate expansion (DHE) and the direct deceleration-parameter expansion (DDPE) — give systematically different $h_0$ and $z_t$ from the same calibrated data, with the DDPE results discordant with the concordance $\Lambda$CDM paradigm, and that the Dainotti $L_X$–$T_X$–$L_p$ correlation carries the largest extrinsic scatter. The open problem is to make calibrated-GRB transition-epoch cosmography reliable enough to test $\Lambda$CDM: determine which expansion (if either) yields unbiased parameters, reduce or explain the Dainotti-correlation scatter, extend the calibration beyond OHD-only Bézier interpolation and spatial flatness, and establish whether realistically improved data can detect small deviations from $\Lambda$CDM at the transition epoch.

Background

Long gamma-ray bursts (GRBs) are visible to redshift $z\approx 9$, far beyond type Ia supernovae (SNe Ia, $z\lesssim 3$), so several empirical correlations between prompt or afterglow luminosities and observable spectral–temporal quantities have been proposed to standardize them as distance indicators: the Amati correlation between peak energy $E_p$ and isotropic equivalent energy $E_{iso}$; the Yonetoku correlation between peak luminosity $L_p$ and $E_p$; the Combo relation between the luminosity $L_0$ at the end of the prompt phase, $E_p$, and the prompt duration $T$; and the Dainotti (3D) correlation between the X-ray luminosity $L_X$ at the end of the plateau phase, its rest-frame duration $T_X$, and the peak luminosity $L_p$. Because these correlations involve luminosities computed through a luminosity distance $d_L(z)$, fitting them and the cosmology simultaneously is circular (the circularity problem); a standard remedy is to calibrate the correlation coefficients model-independently, e.g. by interpolating observational Hubble data (OHD — direct $H(z)$ measurements, largely from the cosmic-chronometer technique, which infers $H(z)$ from the age spread of passively evolving galaxies) with Bézier polynomials, and then use the calibrated GRBs as standardizable candles at high redshift.

Cosmography infers the kinematics of the expansion without postulating a dynamical model by expanding the Hubble rate $H(z)$ or the deceleration parameter $q(z)=-\ddot a/(aH^2)$ in Taylor series, where $a(t)$ is the scale factor; the jerk $j$ (the third derivative of $a$ in units of $H^3$) measures deviations from constant acceleration, and $\Lambda$CDM predicts $j=1$ everywhere. The deceleration–acceleration transition occurs at $q(z_t)=0$, at $z_t\approx 0.6$–$0.8$ in $\Lambda$CDM. Two expansions about the transition are in use: the direct Hubble expansion (DHE), a Taylor series for $H(z)$ around $z=z_t$, and the direct deceleration-parameter expansion (DDPE), a Taylor series for $q(z)$ around $z_t$ from which $H(z)$ is then reconstructed. A generic dark-energy contribution can be encoded in a function $F(z)$ normalized so that a cosmological constant corresponds to $F\equiv 1$ with $F'=F''=0$; the source analysis reports $F_t=F(z_t)$ and its first two derivatives, plus the matter density $\Omega_m$ at $z_t$.

Alfano et al. (2024) calibrated the four correlations above via Bézier interpolation of a 33-point OHD catalog (assuming spatial flatness to compute $d_L$), then jointly fit each calibrated GRB set with SNe Ia and baryonic-acoustic-oscillation (BAO) data through Metropolis–Hastings Monte Carlo to obtain $h_0$, $z_t$, $j_t$, $F_t$, $F_t'$, $F_t''$ and $\Omega_m$ under both expansions. Their DHE results are broadly consistent with $\Lambda$CDM, but the DDPE results differ systematically and are discordant with the concordance paradigm; the Dainotti correlation shows the largest intrinsic extrascatter; and under the DHE fits every inferred $h_0$ is compatible with the Planck CMB value while incompatible at more than $2\sigma$ with local distance-ladder (Cepheid-calibrated SNe Ia) values (under DDPE the picture is mixed, with only the $L_X$–$T_X$–$L_p$ correlation similarly discrepant) — a facet of the still-unresolved Hubble tension. Broader work confirms that the choice and order of cosmographic expansion materially bias high-redshift constraints (Hu & Wang 2022 compared $z$-, $y$-, $E(y)$-, logarithmic, and Padé expansions; later analyses by the same group moved to Padé approximants), so the DHE/DDPE discordance is a specific instance of a recognized methodological hazard rather than a settled question.

Problem Statement

The overarching objective is to make model-independently calibrated GRB cosmography of the deceleration–acceleration transition epoch reliable enough to test $\Lambda$CDM at $z\simeq z_t$ and at higher redshifts. For the Amati ($E_p$–$E_{iso}$), Yonetoku ($L_p$–$E_p$), Combo ($L_0$–$E_p$–$T$) and Dainotti ($L_X$–$T_X$–$L_p$) correlations calibrated without assuming a cosmological model, the required results are:

1. Expansion unbiasedness. Determine whether the direct Hubble-rate expansion (DHE) or the direct deceleration-parameter expansion (DDPE) about $z_t$ — or neither — yields unbiased estimates of $h_0$, $z_t$ and $j_t$ at current data precision, e.g. by mock-data recovery where the true cosmographic parameters are known, and state the origin of the systematic DHE/DDPE discordance (truncation bias, calibration noise, data coverage).

2. Dainotti-correlation scatter. Reduce and/or physically explain the large intrinsic extrinsic scatter of the $L_X$–$T_X$–$L_p$ correlation after model-independent calibration (selection biases, redshift evolution, plateau-parameter systematics) for the full calibrated sample — not only for a preselected subsample — so that this correlation constrains rather than degrades the transition-epoch parameters.

3. Calibration robustness. Assess the sensitivity of the transition-epoch constraints to the calibration assumptions — Bézier interpolation of a 33-point OHD catalog and imposed spatial flatness — by extending the calibration to BAO data (e.g. DESI volume-averaged distances) and to both zero- and non-zero-curvature universes, as proposed in the source work.

4. Detectability of $\Lambda$CDM deviations. With the above systematics controlled, establish whether improved OHD/BAO calibration data, larger and cleaner GRB samples, and reduced GRB systematic errors allow detection of small deviations from $\Lambda$CDM at the transition epoch, as quantified by $F_t$, $F_t'$, $F_t''$ and $\Omega_m$ at $z_t$ (a negative result — demonstrating the required precision is unreachable — is also admissible), and confront the resulting $h_0$ with both CMB-inferred and local distance-ladder values.

These four results jointly support one objective: a decisive, bias-controlled GRB-based test of the concordance paradigm at the transition epoch. Deliverables 1–3 are the systematics controls that make deliverable 4 (or its principled negative) scientifically meaningful.

References

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