**1. Introduction**

Gamma-ray bursts (GRBs) can be observed at extremely high redshift, making them a valuable tool for studies of the early Universe. The ability to observe these highly energetic objects at redshifts out to *z* = 9.4 [1–11] has created significant interest in using them as standardizable candles, similar to Type Ia supernovae. However, observations of GRBs have shown a very diverse population with few common characteristics.

Phenomenologically, GRBs are characterized by the main event, called the prompt emission, which is usually observed in gamma-rays, hard X-rays and sometimes in optical, while the afterglow is the counterpart in soft X-rays (≈66% of observed GRBs), in optical (≈38% of observed GRBs) and sometimes in radio (≈6.6% of observed GRBs). GRB radio afterglows are very difficult to observe, indeed, similar to the X-ray observations which are characterized by the detector limits, and additional difficulties rise due to the limited allocated time for the follow-up observations in the radio band after the GRB trigger. Bursts are classified following the duration of the prompt episode (*T*90). The population of short GRBs (sGRBs) usually has harder spectra and a duration of less than 2 s. In contrast, the population of long GRBs (lGRBs) has softer spectra and a duration larger than 2 s [12].

**Citation:** Dainotti, M.; Levine, D.; Fraija, N.; Chandra, P. Accounting for Selection Bias and Redshift Evolution in GRB Radio Afterglow Data. *Galaxies* **2021**, *9*, 95. https://doi.org/ 10.3390/galaxies9040095

Academic Editors: Elena Moretti and Francesco Longo

Received: 30 September 2021 Accepted: 4 November 2021 Published: 7 November 2021

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However, this classification is still in debate, and so in some bursts, it is not clear if GRBs with intermediate features belong to short or long GRBs [13,14]. LGRBs are associated with the core collapses of dying massive stars [15,16] and sGRBs by merging two compact objects; a black hole (BH) with a neutron star (NS) and two NSs [17–19].

A crucial breakthrough in the analysis of GRB features is the discovery of the plateau emission by the Neils Gehrels Swift Observatory [20]. The plateau emission is a flat part of the lightcurves which follows the decay phase of the prompt emission [21–24]. In the current paper we focus on properties resulting from this plateau emission, as well as both the prompt and afterglow emission. In general, attempts have been made to find standardizable properties, such as a plateau of GRBs or through prompt and afterglow correlation studies. We here mention a few of them: Yonetoku et al. [25], Dai and Wang [26], Ghirlanda [27], Dainotti et al. [28], Amati et al. [29], Dainotti et al. [30–34], Dainotti [35].

However, it is clear from past and current studies [36–39] that observations of GRBs are further susceptible to selection bias and evolutionary effects, which may change the results of any subsequent analysis and can substantially impact the results related to cosmological application of GRB relations [40]. In GRB studies, it is, therefore, crucial to know whether the studied correlations are intrinsic or artificially created as a result of observational biases and redshift evolution. "Redshift evolution" is the dependence of the variable of interest on redshift, and thus "independent of redshift" indicates the absence of such evolution.

In the study of GRB correlations, all variables must be computed in the rest-frame, as we are comparing objects at different epochs. This introduces another source of redshift dependence included in the definition of luminosity distance:

$$D\_L = (1+z)\frac{c}{H\_0} \int\_0^z \frac{dz'}{\sqrt{\Omega\_M (1+z')^3 + \Omega\_\Lambda}} \,\,\,\tag{1}$$

where *H*<sup>0</sup> is the Hubble constant at the present day and Ω*<sup>M</sup>* is the matter density in a flat Universe assuming the equation of state parameter w = −1. Indeed, usually one of the variables in the correlation is either a luminosity or energy which, by definition, depends on the luminosity distance. Ideally, all correlations we use must be corrected for redshift evolution, if any, requiring the removal of any existing redshift dependence.

There do exist statistical techniques that are capable of correcting for these effects, as well as correcting for data truncation from detector limits [41–43]. Among the methods to remove evolution, we consider here the Efron–Petrosian (EP) [41] method. The EP method is a well-established example of these kinds of techniques, and has been used to recover intrinsic relationships in many correlations in the past [7,40,44–48].

Lloyd et al. [44] discuss the correlation between *Ep*, or the peak of the *νF<sup>ν</sup>* spectrum, with flux and fluence in GRBs, later investigated in the rest-frame and known as the *E<sup>p</sup>* − *E*iso relation [49]. A further modification of this relation is the one discovered by Yonetoku et al. [25] in which *E<sup>p</sup>* is correlated with the prompt isotropic luminosity, *L*iso. The EP method provides an explanation on how to perform analysis on truncated data, and in Yonetoku et al. [25], Amati et al. [49] it is illustrated that the method is capable of recovering the correlation present in the original "parent" sample with the truncated data.

This technique has been further explored regarding the luminosity function and formation rate of sGRBs in a recent study by Dainotti et al. [48]. They look at the intrinsic distributions of these variables using the EP method, and introduce a method of accounting for incompleteness of redshift data with the Kolmogorov–Smirnov (KS) test (this is described in more detail in Section 2.2). They find a strong evolution of luminosity with redshift, emphasizing the necessity of this correction. The analysis presented in Dainotti et al. [48] is also relevant, as it emphasizes that both sGRBs and lGRBs undergo strong redshift evolution.

It should also be noted that though this method is mainly applied to GRB correlation studies, it has been also successfully applied in studies of Active Galactic Nuclei as well [45].

Among GRB correlations in particular we focus our attention to the rest-frame time at the end of the plateau emission, *T* ∗ a,radio, and its correspondent luminosity *L*a,radio, this is

an extension in radio of the so-called 2D Dainotti relation in X-rays [29–31] and optical [50]. For the very recent analysis on the 2D Dainotti relation in radio see Levine et al. (2021) in preparation. For a review of the subject of GRB correlations and selection biases in the prompt and afterglow see Dai and Wang [26], Ghirlanda [27], Dainotti [35], Dainotti and Del Vecchio [51], Dainotti et al. [52], Dainotti and Amati [53].

One of the main problems in the application of GRB relationships as theoretical model discriminators and as cosmological tools is the fact that correlations must be intrinsic to the physics and not induced by biases. There are several examples of how the correlations are used to interpret theoretical models both in the prompt and afterglow emission. The photospheric emission and the Comptonization models [54–62] are the two main models used to test the *Epeak*–*E*iso and the Yonetoku et al. [25] correlations, the latter between *Epeak* and the isotropic energy in the prompt emission. Otherwise, the parameter space pinpointed by those correlations can be the effect of selection biases and not of the true underlying physics. To this end, it is necessary to apply these correlations as model discriminators only after correction for such biases. Indeed, for example the plateau emission in X-rays and optical, which reconciles with the existence of the 2D Dainotti relation, can be derived through the equations of a fast rotating NS, the so-called magnetar model [17–19,63–65]. In Rowlinson et al. [66], Rea et al. [67], Stratta et al. [68] the derivation of the parameter space of the magnetic field and spin period have been computed accounting for selection biases and redshift evolution. The current status in the literature is that only a few correlations have been corrected for selection biases and evolutionary effects through the EP method, such as Dainotti et al. [7,48,69,70,71].

Specifically, Dainotti et al. [7] examine this correlation in X-ray for a sample of 101 GRBs that present a plateau, or flattening, in their light curves. After correction for evolutionary effects using the EP method, they conclude that the observed correlation is intrinsic at the 12 *σ* level. In mimicking the evolution of each variable with redshift, they tested both a simple and more complex model, finding similar results in both cases. Dainotti et al. [40] further examine the importance of these corrections when studying the cosmological properties of GRBs, applying the EP method to a simulated correlation between luminosity and time at the end of the plateau emission for 101 GRBs and testing whether a 5 *σ* deviation from the intrinsic values strongly changes the cosmological results. They demonstrated that their results change with this deviation by 13% regarding the values of Ω*M*, emphasizing the necessity of applying such corrections. The problem of evolution of the variables and their correction is not only important for GRB-cosmology studies, but also for more general cosmological studies. Indeed, in Dainotti et al. [72] it has been shown that there is indication of a possible evolution even of the Hubble constant. If this is not due to selection biases, a new physical cosmological model which relies on alternative theories, such as the modified theory of gravity, must be accounted for.

Regarding the correlations in GRB afterglows, Lloyd-Ronning et al. [46,73] discussed the correlation of not only luminosity with redshift, but also isotropic energy, *E*iso, *T* ∗ 90, and the jet opening angle, *θ<sup>j</sup>* , for a sample of 376 GRBs. They emphasize the difficulty of obtaining intrinsic values for these quantities due to inherent biases in observation methods, and additional truncation from detector limits that can introduce false correlations in the data [74]. They find strong evolution with redshift for each of these variables, indicating that achromatic properties of GRBs are also susceptible to selection bias. A further study by Lloyd-Ronning et al. [47] discusses the evolution of *θ<sup>j</sup>* with redshift in greater detail, using the EP method to recover the intrinsic behavior of the jet opening angle.

In this study, we seek to determine whether the strong evolution of *E*iso and *T* ∗ <sup>90</sup> vs. redshift initially found by Lloyd-Ronning et al. [46,73] is still the same for GRBs with observed radio afterglow. In addition to the isotropic energy, we apply the EP method to the luminosity, and break time in radio wavelengths to determine if these variables are strongly affected by inherent bias and evolutionary effects.

We here point out that we are aware that the plateau sample is a subsample of a more extended population of plateaus that we cannot see. We have fixed the issue of the biases related to the redshift evolution and due to the selection threshold with the Efron–Petrosian method; however, we cannot account for the missing population of GRBs for which the follow-up has not been tackled. Nonetheless, it is crucial to discuss the *L*a,radio versus the redshift, since this correlation has been studied in X-rays and optical extensively and it is important to investigate if this correlation holds true in radio with comparable slopes to optical and X-ray. The first step to investigate the correlation is to determine if the variables involved are subjected to redshift evolution and selection biases.

In summary, the main point of the paper is the study of the redshift evolution and the removal of selection biases through the EP method. The analysis of the true correlations can be done only if we first determine the evolution among the variables. The plateau emission has been extensively investigated in X-rays and optical, but so far there has not been a statistical analysis of the existence of the plateau in radio. The radio observations of the plateau emission can cast a light on whether or not the end time of the plateau is indeed a jet break. This point can be revealed only through such a study. The evaluation of the jet break allows one to better understand the evolution of the GRB and its physics in relation to the standard fireball model [75,76] or other scenarios.

This paper is organized as follows: in Section 2, we discuss the selection of our sample, as well as the formulation of the EP method and its application to our sample. In Section 3, we present the results of this analysis, and in Section 4, we discuss the implications of our study, as well as a comparison to previous studies, and present our conclusions.
