Does the Amati Correlation Exhibit Redshift-Driven Heterogeneity in Long GRBs?

redInvestigating the evolution of GRBs with redshift requires large datasets of long GRBs covering a wide range of redshifts. In a previous study, Singh2024 used 162 long GRBs compiled by Demianski17 to explore differences between low and high-redshift GRB populations. However, Dirirsa2019 and Khadka2021 have raised concerns about the suitability of this dataset for cosmological applications, noting that the constraints derived from these GRBs show inconsistencies with those obtained from other cosmological probes such as Type Ia supernovae, BAO, H(z) measurements, and CMB anisotropy data. To address these concerns and validate the results of Singh2024, a larger dataset with more accurate measurements of GRB observables is required. In the current study, we utilize a more comprehensive and recent dataset compiled by jia2022, containing 221 long GRBs spanning a redshift range of $0.034\leq z\leq 8.2$. While many GRBs in this new dataset overlap with the previous 162-GRB sample, they benefit from updated values of key parameters, including isotropic equivalent energy and peak energy. This dataset primarily derives from the joint observations conducted by Swift and Fermi. In some cases, $E_{p,i}$ is provided directly by Swift/BAT. The Swift satellite has significantly contributed to detecting several GRBs with their redshifts. Swift is equipped with three instruments: the Burst Alert Telescope (BAT) for GRB detection, the X-ray Telescope (XRT), and the Ultra-Violet and Optical Telescope (UVOT). BAT, with its large field of view, detects GRBs in the energy range of 15 keV to 150 keV. Upon detection, the satellite slews to the burst direction for XRT and UVOT observations. The energy range of XRT is 0.2-10 keV, while the UVOT is capable of observations in the $170-600$ nm wavelength range (Gehrels2004).

The Fermi Gamma-Ray Space Telescope (FGST), launched by NASA in 2008, has become a cornerstone in the study of high-energy gamma rays (Atwood2009). The primary instrument onboard FGST is the Large Area Telescope (LAT), a high-energy gamma-ray imaging telescope with a wide field of view, operating in the energy range from below 20 MeV to more than 300 GeV. Additionally, Fermi is equipped with the Gamma-ray Burst Monitor (GBM) and the Anticoincidence Detector (ACD), which complement the observations made by the LAT. The LAT provides unprecedented sensitivity across a broad energy range, spanning approximately 20 MeV to 300 GeV, making Fermi an invaluable tool in high-energy astrophysics.

In this study, we employed two approaches to obtain the best-fit values of the Amati parameters: (1) a single linear regression applied over the entire redshift range, and (2) separate linear regressions for different redshift ranges. To compare the performance of these approaches, we utilized the Akaike Information Criterion (AIC) (akaike1974new; akaike1981modern; akaike1987factor; greenwood2024aic), a widely used tool for model selection. For the single linear regression across the full redshift range, the AIC is defined as

where $k$ represents the number of model parameters, and $L_{full}$ is the maximum likelihood of the linear model for the complete dataset. \textcolormagentaIt is important to note that the chi-squared statistic, $\chi^{2}$, introduced in Eq. 2, is additive. The likelihood function $L$ is proportional to $\exp{(-\chi^{2}/2)}$, and for independent observations, the total likelihood is obtained by multiplying the individual likelihoods. Accordingly, to compute the AIC for the subgroup analysis, we multiply the likelihoods for each redshift group, assuming the independence of observations for individual GRBs. The AIC for the grouped analysis is given by

where $L_{low-z}$ and $L_{high-z}$ are the likelihoods of the linear models for the low-redshift and high-redshift samples, respectively. Lower AIC values indicate a better-fitting model by penalizing models with excessive complexity, thereby reducing the risk of overfitting. To quantify the difference between the models, we define $\Delta AIC$ (li2023testing; greenwood2024aic) as

The $\Delta AIC$ value helps determine whether the compared models differ significantly in terms of their fit to the data. If $\Delta AIC\geq 2$, the difference is considered significant, with the model having the lower AIC value being preferred. A $\Delta AIC\geq 10$ suggests that the model with the higher AIC has substantially less support and is highly unlikely to be the best model for explaining the data (PhysRevD.106.123523; greenwood2024aic).

The optimal parameters of the Amati relation are derived through a Bayesian approach, yielding the best-fit values and their associated $1\sigma$ uncertainties for the comprehensive dataset G, as summarized in Table 2. Notably, the intercept ($a$) of the linear Amati relation exhibits a negative value, while its slope demonstrates a positive trend. The intrinsic scatter remains consistently below unity across all instances. Figure 3 visually represents the probability distribution of the three parameters, with vertical lines denoting the $1\sigma$ confidence level.

Table 2 also presents the optimal parameters, namely $a$ and $b$, for the distinct subgroups $\rm{G_{1}}$ and $\rm{G_{2}}$ (with the threshold at $z=1.5$). The intercept is still negative and the slope is positive in both subgroups. Figure 4 illustrates the probability distributions associated with these subgroups, delineating the distinctive nature of $\rm{G_{1}}$ and $\rm{G_{2}}$. Vertical lines are employed to mark the $2\sigma$ confidence level. Notably, the values of $a$ and $b$ exhibit a discrepancy exceeding the $2\sigma$ threshold, indicating a discernible distinction in the GRB populations at low and high redshifts. Despite a smaller intrinsic scatter observed at high redshifts, the disparity is not statistically significant.

Table 3 presents the optimal values of $a$ and $b$, for the subgroups $\rm{G^{\prime}_{1}}$ and $\rm{G^{\prime}_{2}}$ (with the threshold at $z=2$). Their $1\sigma$ errors have also been shown in the respective columns. The continuously negative intercept and positive slope observed in both subgroups underscore a consistent trend. The corresponding posterior probability distributions for these parameters have been shown in Figure 5. The vertical lines have been drawn at $1\sigma$ level. The significance is comparatively smaller in this case however, it is still at more than $1\sigma$ for both $a$ and $b$. In this case, the intrinsic scatter is also significantly different for the two subgroups.

red\textcolormagenta To illustrate the differences in the $E_{p,i}-E_{iso}$ relations between low-$z$ and high-$z$ GRBs, we present a comparative analysis in figures 6 and 7. Figure 6 shows the linear fits for an older dataset of 162 GRBs, divided into two subgroups at the redshift threshold of $z=1.5$. In contrast, figures 7 display the fits for the main dataset, comprising 221 GRBs. The solid line represents the fit for low-$z$ GRBs in all these figures, while the dashed line corresponds to high-$z$ GRBs. The results indicate that the linear fits for low-$z$ and high-$z$ GRBs are distinct, with no overlap between the two groups.
\textcolormagenta To further validate our findings, we computed the AIC for the single linear fit applied across the entire redshift range and the fits for the individual subgroups. The AIC values for both approaches are presented in table 4. As shown in the table, the AIC for the subgroup fits is lower, and the difference in AIC values, as defined in Eq. 11, exceeds the threshold for significance. Therefore, the subgroup-based fit is preferred, as it provides a better representation of the data compared to the single linear fit over the full redshift range.

Possible biases in each instrument could arise due to their design, calibration, and operational characteristics which may be partly responsible for the observed differences in GRB populations. BAT (Barthelmy_2005) may exhibit biases related to its coded aperture imaging technique, potentially leading to systematic errors in position determination. It can affect the identification of the host galaxy and hence the redshift measurement. Another issue with the BAT instrument is its capability to detect energies up to 150 keV only (Gehrels2004), which is less than the average peak energy of GRBs (Kaneko_2006). Thus, for several Swift observed GRBs, it is not possible to directly determine the fluence and $E_{p,obs}$. FGST’s LAT may face biases due to background noise, energy calibration uncertainties, and instrument response variations across its wide energy range. GBM’s biases could include sensitivity variations across its energy range and potential systematic errors in event classification. While these instruments provide invaluable data, understanding and mitigating biases are crucial for accurate gamma-ray observations.

As mentioned in 3.1, we observe significant differences in best-fit values of Amati parameters for low and high-$z$ GRB subgroups. Below we analyse our results in the context of differences in the low-$z$ and high-$z$ GRB host galaxies.

In a detailed study (Palla2020) and (Grieco_2014) claim that at high $z$, long GRBs host galaxies include all morphological types including early-type galaxies. This could be because at high $z$ the early-type galaxies can be observed during their active star formation period. In contrast, star formation in the early-type galaxies at low $z$ is often quenched.

High-$z$ GRB hosts are younger and thus have low metallicities. On the other hand, low-$z$ hosts, especially the early-type ones, are older and have higher metallicities (Palla2020). These differences underscore significant evolutionary changes in galaxy properties and environments from high to low redshifts, affecting the nature and identification of long GRB host galaxies. At a given redshift, dust-obscured GRB hosts are more massive than optically bright galaxies, with massive GRB hosts becoming fairly common at $z>1.5$ (Perley_2013). The authors also link the smaller GRB production at low redshift to metallicity. Using the optically unbiassed GRB host (TOUGH) Survey (Schulze_2015) claims that GRB hosts at high $z$ are faint and low in metallicity (2015ApJ...804...51C) The studies based on Damped Lyman $\alpha$ systems at high $z$ support the existence of two GRB progenitor channels, with one being mildly influenced by metallicity (2015ApJ...804...51C).