Do the Amati and Yonetoku Relations Evolve with Redshift for Swift GRBs?

By analyzing a sample of 241 Swift long gamma-ray bursts, this study demonstrates that the Amati and Yonetoku relations exhibit no significant redshift evolution and actually yield better fits at high redshifts, confirming their reliability as cosmological probes.

The Gist

Imagine the universe is a giant, dark ocean, and Gamma-Ray Bursts (GRBs) are massive, sudden flashes of light from exploding stars. Astronomers love these flashes because they are so bright they can be seen from the very edge of the universe, helping us map out the history of space and time.

However, there's a problem: these flashes aren't all the same size. Some are like a tiny sparkler; others are like a nuclear bomb. If you just see a flash, you don't know if it's a small explosion nearby or a giant one far away. To use them as "mile markers" to measure the universe, scientists need to figure out how to make them standard candles—objects where, if you know how bright they should be, you can calculate exactly how far away they are.

To do this, scientists discovered two "rules of thumb" (called the Amati and Yonetoku relations) that link how bright a burst looks to how much energy it actually has. Think of it like a rule that says, "If a firework has a certain color, it must have a specific amount of gunpowder."

The Big Question: Do the Rules Change Over Time?

The universe is expanding, and light from distant objects gets stretched (redshifted). The big question this paper asks is: Do these "rules of thumb" stay the same for explosions that happened billions of years ago (high redshift) compared to ones that happened more recently (low redshift)?

If the rules change as we look further back in time, then we can't trust them to measure the universe accurately. It would be like using a ruler that shrinks every time you walk a mile; your measurements would be wrong.

How They Tested It

The authors, Ali and Walid, acted like detectives. They gathered a massive list of 241 long-lasting explosions (GRBs) caught by the Swift satellite.

They used two main methods to check if the rules were breaking:

1. The "Slicing" Method: They cut their data into slices based on how far away the explosions were (low, medium, and high distance). Then, they checked if the "rules" looked different in each slice.
2. The "Split" Method: They split the data into two big piles: "Nearby" and "Far Away." Then they compared the rules for the two piles to see if they matched.

They also created a "filtered" list, removing the dimmer, harder-to-see explosions to see if that changed the results.

What They Found

Here is the verdict, translated into plain English:

1. The Rules Are Sturdy (No Evolution)
The good news is that the rules did not change. Whether the explosion happened 1 billion years ago or 10 billion years ago, the relationship between the color and the energy remained consistent. The "ruler" didn't shrink. This means scientists can confidently use these bursts to measure the universe's expansion history.

2. High-Definition vs. Low-Definition
While the rules didn't change, they worked much better for the distant explosions (high redshift) than the nearby ones.

• The Analogy: Imagine trying to hear a song. The distant explosions are like a clear, high-quality radio broadcast. The nearby explosions are like listening to the same song through a wall with static.
• The Result: The "distant" data fit the rules perfectly. The "nearby" data was messy and had a lot of "static" (errors).

3. Which Rule is Better?
They found that the Yonetoku relation (linking peak brightness to energy) was a much more reliable "ruler" than the Amati relation. It was less messy and gave more consistent results, especially for the distant bursts.

Why Are Nearby Bursts Messy?

The authors suggest a fascinating reason for the "static" in the nearby data.

• The Theory: The distant bursts are likely all caused by the same thing: massive stars collapsing.
• The Twist: The nearby bursts might be a mix of different things. Some are collapsing stars, but others might be caused by colliding black holes or neutron stars. Because they have different origins, they don't follow the same "rule of thumb" as neatly as the distant ones do. It's like trying to measure the speed of all cars, but your sample includes some bicycles and motorcycles mixed in—the data gets messy.

The Bottom Line

This paper confirms that Gamma-Ray Bursts are excellent tools for exploring the early universe. The "rules" connecting their light and energy hold true across billions of years. However, to get the cleanest, most accurate measurements, astronomers should focus on the distant, high-redshift bursts, as the nearby ones are a bit too "noisy" and might be coming from different types of cosmic events.

In short: The cosmic ruler works, but you get the best measurements when you look at the farthest stars.

Technical Summary

Here is a detailed technical summary of the paper "Do the Amati and Yonetoku Relations Evolve with Redshift for Swift GRBs?" by Hasan and Azzam (2026).

1. Problem Statement

Gamma-ray bursts (GRBs) are powerful stellar explosions observable at high redshifts ($z > 9$), making them potential probes for the early universe. However, they are not standard candles. To utilize them for cosmology, they must be "standardized" using intrinsic correlations between observed and intrinsic parameters. Two primary correlations are:

• The Amati Relation: A power-law correlation between the equivalent isotropic energy ($E_{iso}$) and the intrinsic peak spectral energy ($E_{p,i}$).
• The Yonetoku Relation: A power-law correlation between the peak isotropic luminosity ($L_{iso}$) and $E_{p,i}$.

A critical prerequisite for using these relations as cosmological tools is determining whether they are immune to redshift evolution. Previous studies have yielded conflicting results: some suggest the relations are static, while others (notably Singh et al.) claim evidence of evolution, particularly when splitting data by redshift cutoffs. These discrepancies often stem from small sample sizes or dataset incompleteness. This study aims to resolve these conflicts using a large, systematic dataset to determine if the Amati and Yonetoku relations evolve with redshift.

2. Methodology

Data Sample:

• Source: Swift GRB catalog (as of July 2025).
• Selection Criteria:
  • Long GRBs only ($T_{90} > 2$ s).
  • Single, reported redshift.
  • Availability of fluence ($S_{obs}$), 1-sec peak photon flux ($F_\gamma$), and spectral parameters (specifically cutoff power-law fits to derive $E_p$).
  • Exclusion of bursts with large errors (>500% in fluence, flux, or energy).
• Final Samples:
  • Unfiltered Dataset: 241 Long GRBs.
  • Filtered Dataset: 87 Long GRBs (subset with $F_\gamma \geq 2.6$ photons/cm²/s) to test for completeness bias.

Analytical Approaches:
The authors employed two distinct methods to test for redshift evolution:

1. Binning Approach: The datasets were sorted by redshift and split into 3, 4, and 5 bins with equal numbers of GRBs. Each bin was fitted independently to the Amati and Yonetoku relations.
2. Redshift Cutoff Approach: The datasets were split into low-redshift ($z < z_c$) and high-redshift ($z \geq z_c$) groups using cutoffs of $z_c = 1.5$ and $z_c = 2.0$.

Fitting Procedure:

• The relations were linearized in logarithmic form: $\log(E_{iso}/L_{iso}) = A \log(E_{p,i}) + B$.
• Optimization: A least $\chi^2$ method was used to determine the best-fit parameters ($A$, $B$) and the intrinsic scattering parameter ($\sigma_{int}$).
• Error Handling: Total error included observational errors and an intrinsic scattering term ($\sigma_{int}$) to account for unmodeled physics or measurement uncertainties.
• Model Comparison: The Akaike Information Criterion (AIC) was calculated to compare the quality of fits across different bins and datasets.

Cosmological Model:
Calculations assumed a flat $\Lambda$CDM model with $H_0 = 70.8$ km/s/Mpc, $\Omega_M = 0.27$, and $\Omega_\Lambda = 0.73$. $k$-corrections were applied to transform observed energies to the rest-frame band [1–104] keV.

3. Key Contributions

1. Large-Scale Systematic Analysis: The study utilizes a significantly larger sample (241 Swift LGRBs) than many previous investigations, allowing for more robust statistical binning and splitting.
2. Dual-Dataset Validation: By comparing an "unfiltered" sample against a "flux-limited" (filtered) sample, the authors explicitly address the issue of dataset completeness and selection bias.
3. Resolution of Evolution Debate: The study provides a definitive test of whether the Amati and Yonetoku relations evolve, addressing conflicting literature by using consistent methodology across multiple binning strategies.
4. Identification of Redshift-Dependent Reliability: The paper introduces the finding that while the relations do not evolve (i.e., the slope parameters remain statistically consistent), their reliability (goodness of fit) is strongly dependent on redshift.

4. Results

Redshift Evolution:

• No Systematic Evolution: Both the Amati and Yonetoku relations show no significant redshift evolution. The fitting parameters ($A$ and $B$) for all redshift bins (3, 4, and 5 bins) and for the low/high redshift split groups are consistent with the global fit within $2\sigma$.
• Consistency: The results from the binning method and the cutoff method are mutually consistent, reinforcing the conclusion that the intrinsic correlations do not change with cosmic time.

Goodness of Fit and Reliability:

• High-Redshift Superiority: A critical finding is that high-redshift bins ($z > z_{cutoff}$) consistently yield better fits than low-redshift bins.
  • $\sigma_{int}$ Trends: The intrinsic scattering parameter ($\sigma_{int}$) is significantly lower for high-redshift groups, indicating the data points cluster more tightly around the correlation line.
  • AIC Values: The Akaike Information Criterion values are lower for high-redshift subsets, confirming a better model fit.
• Low-Redshift Issues: Low-redshift bins exhibit larger errors and higher $\sigma_{int}$.
  • For the Amati relation, the low-redshift fit is particularly poor and sensitive to dataset completeness (the filtered vs. unfiltered datasets yielded different Amati slopes).
  • For the Yonetoku relation, the results are more consistent across both datasets, though the high-redshift superiority remains.

Dataset Completeness:

• The Amati relation appears sensitive to the completeness of the dataset (flux limits), whereas the Yonetoku relation is more robust.
• The "low-redshift excess" of GRBs (relative to the star-formation rate) suggests that low-redshift GRBs may have different progenitors (e.g., compact object mergers) or suffer from different selection effects, leading to the poorer fits observed at low $z$.

5. Significance

• Cosmological Probes: The study validates the Amati and Yonetoku relations as reliable tools for cosmology, specifically for high-redshift GRBs. The lack of redshift evolution confirms they can be used to constrain cosmological parameters without needing complex evolution corrections.
• Strategic Application: The authors recommend that future cosmological studies utilizing these relations should focus exclusively on high-redshift GRBs ($z > 1.5$ or $2.0$) to maximize accuracy and minimize scatter.
• Progenitor Insights: The discrepancy in fit quality between low and high redshifts supports the hypothesis that low-redshift GRBs may include a population with different physical origins (e.g., mergers) compared to the high-redshift population, which is dominated by massive star collapse.
• Methodological Guidance: The paper highlights the necessity of accounting for dataset truncation (e.g., using the Efron-Petrosian method) when analyzing high-redshift GRBs to avoid selection bias, as simple flux cuts can remove critical high-redshift data.

In conclusion, while the Amati and Yonetoku relations do not evolve with redshift, their utility as standard candles is maximized at high redshifts where the correlations are tighter and less affected by selection biases or potential progenitor diversity.