**4. Discussion and Conclusions**

Using a sample of 80 GRBs with observed radio afterglow, we test the use of the EP method on log *E*iso and log *T* ∗ <sup>90</sup> for a subsample 80 GRBs, and on log *L*a,radio and log *T* ∗ a,radio for a subsample of 18 GRBs. We find that when considering log *E*iso and log *T* ∗ <sup>90</sup>, we obtain indices for the parameter of redshift evolution, *δ*, of *δ<sup>T</sup>* ∗ 90 = −0.65 ± 0.27 and *δE*iso = 0.39 ± 0.88, while the values of *δ* for log *L*a,radio and log *T* ∗ a,radio are log *T* ∗ a,radio as *δT* ∗ a,radio = −1.94 ± 0.86 and *δL*a,radio = 3.15 ± 1.65.

For log *T* ∗ <sup>90</sup>, for log *T*a,radio, and log *L*a,radio, we find relatively strong evolution of each variable with redshift. The luminosity presents the strongest correlation with redshift, emphasizing the necessity of correction for these effects before using data in correlation analysis. *E*iso, by contrast, appears to be the most independent from redshift, with the smallest value for |*δ*|.

We find that our values are comparable to values obtained in previous studies. A study by Lloyd-Ronning et al. [46] of the cosmological evolution of isotropic energy *E*iso, burst duration *T* ∗ <sup>90</sup>, jet opening angle *θ<sup>j</sup>* , and luminosity *L<sup>j</sup>* reports *δ* values compatible with our findings *T* ∗ <sup>90</sup> and *L*a,radio within 1 *σ*, and values compatible with our *δE*iso within approximately 2 *σ*. Specifically, they find a value of *δE*iso = 2.3 ± 0.5, which agrees with our value of *δE*iso = 0.39 ± 0.88 within 2.17 *σ*. We also see agreement with *δ<sup>T</sup>* ∗ 90 = −0.8 ± 0.3, with a 0.55 *σ* difference from our value of *δ<sup>T</sup>* ∗ 90 = −0.65 ± 0.27, and in the luminosity with *δL*<sup>j</sup> = 3.5 ± 0.5, a 0.2 *σ* difference from our value of *δL*a,radio = 3.15 ± 1.65.

These results also agree with previous values of *δ* for *L*<sup>a</sup> and *T* ∗ a in X-ray and optical wavelengths. Dainotti et al. [7] conduct a similar analysis of the luminosity and rest-frame end time of the plateau emission using X-ray data. Their value for correction for *δT*<sup>∗</sup> a , reported as *δ<sup>T</sup>* ∗ a,X = −0.85 ± 0.3, is compatible with our value of *δ<sup>T</sup>* ∗ a,radio = −1.94 ± 0.86 within 1.23 *σ*. However, they find a very slow evolution in luminosity, with a value of *δL*a,X = −0.05 ± 0.35, which is a 1.94 *σ* difference from our value of *δL*a,radio = 3.15 ± 1.65. This discrepancy is likely due in part to the small sample size, which may exaggerate the

extent of the evolution present in our sample, but may also be due to differences in the behavior of the X-ray and radio emission.

We have corrected the luminosity and time in X-rays with 222 GRB lightcurves with a given redshift, presenting plateaus according the Willingale et al. [23] model and in optical with 181 GRBs with plateaus taken from Dainotti et al. [50], but with the additional analysis of 80 GRBs found in the literature. For tackling this analysis, we followed the same procedure described in the current paper. The results of this analysis reports *σ*, and *δL*a,X = 2.42 ± 0.58, which is a 0.44 *σ* difference from our value. They also report values in optical of *δ<sup>T</sup>* ∗ a,opt = −2.11 ± 0.49 and *δL*a,opt = 3.96 ± 0.58, which both agree with our result within 1 *σ*.

In general, it can be seen that a larger sample size is preferred when applying the EP method. In our case, for the sample pertinent to *E*iso, and *T* ∗ <sup>90</sup> we choose the limiting values while excluding ≤ 10% of the overall sample. However, for the smaller sample of 18 plateau GRBs, the limiting values are chosen so that we do not exclude any data points due to the small sample size. In addition, this conservative choice would allow us to have smaller error bars on the slope of the evolutionary functions. However, the *δ* values obtained for *L*a,radio and *T* ∗ a,radio are similar to values found in previous studies of larger sample sizes, thus indicating that the EP method is still successful even with a small dataset.

GRB correlations in radio afterglows related to the plateau emission are crucial to understand if the jet break is coincident with the end of the plateau emission. To investigate this point, a multiwavelength analysis not only in optical and X-rays must be performed together with the radio data. Since the evolution of the variables with redshift can change the time at which the break happens, it is crucially important to correct for the redshift evolution. This study is also the preliminary step to the investigation of the intrinsic nature of the plateau emission correlations.

After our analysis on the radio observations both for the prompt emission in relation to the variables of *E*iso and *T* ∗ <sup>90</sup> and for the afterglow emission in relation to *L*a,radio, *T* ∗ a,radio, we can conclude:


**Author Contributions:** Conceptualization: M.D.; Methodology: M.D.; Software: M.D.; Validation: M.D. and D.L.; Formal Analysis: D.L.; Investigation, D.L. and M.D.; Data Curation: P.C.; Writing—Original Draft Preparation: D.L.; Writing—Review and Editing Draft: M.D., N.F. and D.L.; Supervision: M.D. and N.F. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Astronomical Observatory of Japan (NAOJ).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data are taken from Chandra and Frail [80], Zauderer et al. [81], Laskar et al. [82,83], Cucchiara et al. [84], Bright et al. [85], Anderson et al. [86], Singer et al. [87], Laskar et al. [88], Kangas et al. [89], Laskar et al. [90], Chen et al. [91], Bhalerao et al. [92], Hallinan et al. [93], Mooley et al. [94], Resmi et al. [95], Margutti et al. [96], Maity and Chandra [97] and Rhodes et al. [98].

**Acknowledgments:** This work was made possible in part by the United States Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists (WDTS) under the Science Undergraduate Laboratory Internships (SULI) program. We thank Cuellar for managing

the SULI program at Stanford National Accelerator Laboratory. We also acknowledge the National Astronomical Observatory of Japan for their support in making this research possible, as well as Kevin J. Zvonarek for his help in preparing the dataset in this work. NF acknowledges financial support from UNAM-DGAPA-PAPIIT through grant IN106521. D. Levine acknowledges support from NAOJ division of Science.

**Conflicts of Interest:** The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.
