Amati Relation in GRBs: Calibration & Cosmology

• The Amati relation is an empirical power-law correlation between the rest-frame spectral peak energy (Eₚ,ᵢ) and the isotropic-equivalent radiated energy (E_iso) of long GRBs, serving as a cosmological probe.
• Calibration strategies like cosmography and local regression reduce circularity by independently anchoring luminosity distances, enhancing its reliability as a standard candle.
• Ongoing research addresses redshift evolution and selection effects, refining statistical models and theoretical interpretations of GRB prompt emission physics.

The Amati relation is an empirical, power-law correlation between the rest-frame spectral peak energy ($E_{p,i}$) and the isotropic-equivalent radiated energy ($E_{\mathrm{iso}}$) of long gamma-ray bursts (GRBs). Originally established in the early 2000s, the Amati relation has become a cornerstone of attempts to use GRBs as cosmological probes, especially at high redshifts inaccessible to Type Ia supernovae. The calibration, cosmological dependence, statistical robustness, and physical origin of the Amati relation are active topics of research and debate.

1. Formal Definition and Mathematical Framework

The Amati relation is conventionally expressed in the following logarithmic form: $$\log_{10} E_{\mathrm{iso}} = a + b\,\log_{10} E_{p,i}$$ where $E_{\mathrm{iso}}$ is the isotropic-equivalent energy radiated by the GRB in erg, and $E_{p,i}$ is the intrinsic (rest-frame) spectral peak energy, frequently expressed in keV. Given the observed spectral peak $E_{p,obs}$ and redshift $z$, $E_{p,i} = (1+z)E_{p,obs}$. The value of $E_{\mathrm{iso}}$ is computed as: $$E_{\mathrm{iso}} = \frac{4\pi d_L^2 S_{\mathrm{bol}}}{1+z}$$ where $S_{\mathrm{bol}}$ is the bolometric fluence and $d_L$ the luminosity distance, which, unless model-independently anchored, depends on the adopted cosmological parameters.

Best-fit values for $(a,b)$ in the relation differ by sample and calibration method. For example, a cosmographic calibration yields $a = 49.154 \pm 0.306$, $b = 1.444 \pm 0.117$ (Capozziello et al., 2010), while a Bayesian approach on a larger sample produces $a \simeq 1.52$, $b \simeq 52.67$ (interpreted within the notation $$\log(E_{\mathrm{iso}}/1~\mathrm{erg}) = b + a \log[(E_{p,i}(1+z)/300~\mathrm{keV})]$$) (Demianski et al., 2011). Intrinsic scatter, usually denoted $\sigma_{\mathrm{int}}$, is included to encapsulate physical and measurement dispersion ($\sim0.41$ dex (Demianski et al., 2011) or as low as 0.15 when using favored anchor datasets (Govindaraj et al., 2022)).

2. Calibration Strategies and Circularity

A significant challenge in using the Amati relation for cosmology is the “circularity problem”: $d_L$ depends on the cosmological model, but the Amati relation is used to probe $\cosmoparams$. Several calibration methods have been developed to mitigate this issue:

• Cosmographic approaches expand $d_L(z)$ as a Taylor series in $z$ or $y = z/(1+z)$, parameterized by cosmokinematic parameters ($q_0$, $j_0$, $s_0$). These are directly fit to SNe Ia Hubble diagrams, and then used to compute $d_L$ for GRB redshifts independent of any dynamical cosmological model (Capozziello et al., 2010).
• Local regression anchoring utilizes a non-parametric fit to a low-$z$ anchor dataset (e.g., SNe Ia (Demianski et al., 2011), galaxy clusters (Govindaraj et al., 2022), Hubble parameter data (Kumar et al., 2022), or quasars (Dai et al., 2021)) to determine $d_L(z)$ at overlapping GRB redshifts. GRBs with $z$ below the anchor limit (e.g., $z < 1.55$ for Union2 SNe Ia) are used to fit $(a,b)$, and the calibration is then extrapolated to high-$z$ GRBs.
• Simultaneous cosmology–correlation fitting incorporates $(a,b,\sigma_{\mathrm{int}})$ and cosmological parameters into a joint likelihood, optimizing over all, thus avoiding external calibration and reducing circularity (Khadka et al., 2020, Cao et al., 10 Apr 2024).

These methods enable model-independent or weakly model-dependent calibration of the Amati relation, extending its cosmological utility. Model-independence of $(a,b,\sigma_{\mathrm{int}})$ is routinely verified against multiple cosmological scenarios (Khadka et al., 2020).

3. Redshift Evolution, Population Heterogeneity, and Selection Effects

The fundamental assumption underlying most calibrations is the universality of the Amati relation; i.e., that the correlation parameters do not evolve with redshift. Multiple analyses challenge or support this view:

• Several studies report significant (e.g., $>3\sigma$) differences between the parameters $(a,b)$ calibrated on low- and high-redshift GRBs (Lin et al., 2015, Singh et al., 2022, Singh et al., 23 Jun 2024). These effects persist across different cosmological models, suggesting that they are not mere artifacts of the cosmological background.
• Other research using debiased samples and more robust calibration datasets finds that, after accounting for selection effects—particularly Malmquist bias and $E_{\mathrm{iso}}$-dependent detection thresholds—the tension between low-$z$ and high-$z$ Amati relation parameters largely vanishes, casting doubt on the reality of redshift evolution of the correlation (Huang et al., 2020). Nonetheless, an intrinsic $E_{\mathrm{iso}}$-dependence (energy scaling) remains.
• Bayesian and pulse-wise analyses find no statistically significant redshift evolution within current sample sizes (typically $<1\sigma$), especially when using model-independent anchor calibration at low redshift (Basak et al., 2013, Dai et al., 2021, Han et al., 24 Aug 2024).
• Selection effects—arising from instrumental sensitivity, the ability to measure $E_{p,i}$, and redshift determination probability—strongly constrain the observed population. These effects can create apparent correlations even when none exist in the underlying population or accentuate intrinsic population boundaries. However, careful modeling can separate physical boundaries from selection-induced ones (Collazzi et al., 2011, Heussaff et al., 2013).

The current consensus is nuanced: while clear evidence exists for redshift-driven heterogeneity in some samples (Singh et al., 23 Jun 2024), properly debiased or model-independently calibrated data can mask, reduce, or eliminate the statistical significance of such evolution (Huang et al., 2020, Han et al., 24 Aug 2024). Intrinsic evolution cannot be entirely ruled out.

4. Physical Origin and Theoretical Basis

The Amati relation's origin has been derived analytically from the prompt emission physics of GRBs, in particular from internal shock models with synchrotron emission in a relativistic outflow (“fireball” model):

• For on-axis observers, it is demonstrated that $E_{p} \propto \gamma$ and $E_{\mathrm{iso}} \propto \gamma^2$, where $\gamma$ is the bulk Lorentz factor. Eliminating $\gamma$ yields the canonical relation $E_{p} \propto E_{\mathrm{iso}}^{1/2}$ (Xu et al., 2022). This theoretical expectation matches the exponent seen in empirical fits.
• Off-axis emission, or other significant viewing angle effects, give rise to a flatter relation ($E_{p} \propto E_{\mathrm{iso}}^{0.25-0.31}$), providing a natural explanation for observed outliers.
• Pulse-wise analyses extend the correlation to individual temporal components of bursts, often strengthening the intrinsic connection and possibly reducing the scatter (Basak et al., 2013).

This supports the interpretation of the Amati relation as a consequence of the prompt emission physics, modified in the observed sample by the spread in microphysical parameters (e.g., electron distribution, magnetic field) and geometric effects.

5. Statistical Robustness and Practical Cosmological Applications

The application of the Amati relation as a standard candle depends on:

• Precision and scatter: The intrinsic scatter is typically in the range $0.15 - 0.7$ dex, depending on sample selection and calibration method. Lower scatter (e.g., when anchoring with well-characterized galaxy clusters) increases practical utility (Govindaraj et al., 2022).
• Calibration method: Simultaneous fitting and copula-based improved correlations permit better handling of the covariance structure and the possible redshift dependence, but the gain in constraining power is often offset by increased parameter degeneracies (Liu et al., 2022, Liu et al., 2022).
• Sample limitations: The size, quality, and selection of the dataset strongly affect the statistical power and cosmological consistency. Some compilations yield $>2\sigma$ tension in the matter density parameter $\Omega_{m,0}$ when compared to BAO and $H(z)$ constraints (Cao et al., 10 Apr 2024, Cao et al., 12 Feb 2025).
• Classification: The Amati relation most strictly holds for “Amati-type” GRBs, which are largely long-duration, high-energy events. Short-duration GRBs and “non-Amati-type” events are clear outliers in the $(E_{p}, E_{\mathrm{iso}})$ plane (Qin et al., 2013). Classification based on deviation from the empirical relation offers a more robust separation than duration-based schemes.

Despite scatter, model calibration dependencies, and selection biases, calibrated Amati relations have been used to generate Hubble diagrams up to $z\sim9$ and to place bounds on cosmological parameters. However, the strength and utility of these constraints remain limited compared to better-established probes.

6. Open Issues, Controversies, and Future Prospects

Several open issues challenge both the universality and cosmological utility of the Amati relation:

• Selection and instrument-related biases: Differing detector thresholds, energy response, and fluence sensitivities for Swift, Fermi, BATSE, Suzaku, and Konus generate sample-dependent distributions in the $(E_{p,\mathrm{obs}}, S_{\mathrm{bolo}})$ plane. A significant fraction of bursts are “violators” of the Amati limit in diverse instrument samples, indicating the imprint of selection bias (Collazzi et al., 2011).
• Redshift evolution and energy scale dependence: Persistent $>2\sigma$ differences between low- and high-$z$ calibration in multiple large samples, unless fully debiased, raise the need for more systematic modeling or population evolution (Lin et al., 2015, Singh et al., 2022, Singh et al., 23 Jun 2024).
• Intrinsic versus extrinsic evolution: Simulations and debiasing analyses indicate that at least a significant component of the observed evolution is due to Malmquist bias and $E_{\mathrm{iso}}$-dependent selection (Huang et al., 2020). However, simulation studies also show that genuine evolution in $a$ and $b$ is plausible and may reflect underlying astrophysical changes.
• Improved statistical modeling: Use of copula methods, advanced multivariate regressions, and MCMC/Bayesian approaches can improve robustness and account for covariance and evolutionary effects, though increased complexity does not always outperform the simpler linear form by objective selection criteria (AIC, BIC) (Liu et al., 2022, Liu et al., 2022, Han et al., 24 Aug 2024).
• Sample expansion and next-generation observations: Forthcoming missions such as THESEUS and eXTP, with higher sensitivity and larger uniform samples, are expected to clarify the nature and evolutionary behavior of the Amati relation (Singh et al., 23 Jun 2024).

7. Summary Table: Calibration Approaches and Evolutionary Claims

Calibration Technique	Evolution Detected?	Key References
Cosmography (SNe Ia anchor)	No ($<1\sigma$)	(1003.53191104.5614)
Local SNe Ia regression	No ($<1\sigma$)	(1104.56142408.13466)
Quasar Hubble diagram anchor	No ($<1\sigma$)	(Dai et al., 2021)
Galaxy cluster distance anchor (low-$z$)	No (limited $z$)	(Govindaraj et al., 2022)
Direct high vs low-$z$ sample division	Yes ($>2\sigma$)	(Lin et al., 2015Singh et al., 2022Singh et al., 23 Jun 2024)
Pulse-wise spectral analysis	No	(Basak et al., 2013)
Debiased via $E_{\mathrm{iso}}$ matching	No	(Huang et al., 2020)

The calibration, universality, and physical meaning of the Amati relation for GRBs remain complex. Its application as a cosmological standard candle is conditional on sample selection, calibration methodology, mitigation of selection effects, and proper handling of potential population evolution. While current evidence supports partial model-independence and general robustness, particularly with carefully selected low-redshift anchors, redshift-driven heterogeneity and selection-induced artifacts must be rigorously controlled. Forthcoming high-precision GRB datasets are expected to further refine or modify the empirical and theoretical understanding of this relation.