Single-pulse Gamma-Ray Bursts Have Prevalent Hard-to-soft Spectral Evolution

The sample of bursts analyzed in this work was collected from the time-resolved spectra catalog produced by Yu et al. (2016). Of the 81 bursts in the catalog, 62 were utilized in this study. Yu et al. (2016) used CSPEC data for 15 of the bursts, which they note has lower temporal resolution with higher spectral resolution and a duration of around 8000 s. The other bursts in the catalog were created using time-tagged event (TTE) data. For the creation of light curves in this study, TTE data were used. TTE has a duration of around 300 s (Meegan et al. 2009). Yu et al. (2016) fit each burst on multiple time intervals. Four different spectral models were used: BAND, COMP, SBPL, and power law (Yu et al. 2016). The Band function (BAND. Band et al. 1993) is a four-parameter piecewise function with an exponentially smooth transition between two power laws (Yu et al. 2016). The smoothly broken power law (SBPL; Ryde & Svensson 1999; Kaneko et al. 2006) is also a smooth transition between two asymptotic power-law behaviors. In general, SBPL has a parameter that controls the length and smoothness of the transition. In the Yu et al. (2016) catalog, this parameter is fixed at 0.3. The Comptonized model (COMP) is a three-parameter power-law fit with an exponential cutoff. Finally, the power-law fit is a simple two-parameter power law. Additional details about the fits used can be found in Yu et al. (2016) and the references therein. For a given interval of a given burst, Yu et al. (2016) provide the best functional fit in their catalog. Thus, a burst may have different functional fits for different time intervals. Peak energy was of principal interest for this work; thus, any spectral fit that did not include a peak-energy value was omitted. In particular, the power-law fits have no defined peak energy, and therefore, any fit using the power law could not be used. All bursts with at least four spectral fits that included well-defined peak-energy values were included. Of the 19 excluded, 16 were excluded due to having too few spectra with peak-energy values. Due to the difference in duration for data used, some GRBs were excluded as a majority of spectra occurred after or before the TTE data. The identification numbers of excluded bursts and the reason for their exclusion are reported in Table 1.

While we used the spectral fits from Yu et al. (2016), we reaccumulated burst light curves to ensure consistency and a format suitable for our horizontal line technique (see Section 2.2 below). Light-curve TTE data were collected from the HEASARC BROWSE Gamma-ray Burst Monitor (GBM) Burst Catalog (Gruber et al. 2014; von Kienlin et al. 2014; Bhat et al. 2016; von Kienlin et al. 2020). ¹ From the 12 NaI detectors, the two with the brightest signals were chosen. For each burst we define a region of interest as

$$\begin{eqnarray}&&\max ({T}_{90\mathrm{start}}-0.5{T}_{90},{T}_{{\rm{dstart}}})\leqslant t\leqslant \min ({T}_{90\mathrm{start}}+2{T}_{90},{T}_{\mathrm{end}}),\end{eqnarray} \tag{ 1 }$$

where T_90start is the start of the T₉₀ period, T_dstart is the earliest available TTE data, and T_end is the end of the TTE data. If the burst is sufficiently short, the region of interest lasts 2.5T₉₀. Unfortunately, not all bursts have TTE data coverage over the entire 2.5T₉₀ period. TTE includes data from 30 s prior to the trigger time and lasts for around 300 s (Meegan et al. 2009). In the case that the T_90start − 0.5T₉₀ occurred before 30 s prior to trigger time, the earliest TTE data were used as the beginning of the region of interest. Similarly, if T_90start + 2T₉₀ occurred after the 300 s duration, the last TTE time was used as the end of the region of interest. For all bursts, the region of interest was divided into 120 equal-length bins. Broadly, the length of each bin was the length of the region of interest for the burst divided by 120. For bursts where the 2.5T₉₀ fell within the TTE data, bin duration was given simply by ${T}_{\mathrm{bin}}=\tfrac{{T}_{90}}{48}$. This method ensured that the primary burst behavior was well described in the light curve.

In order to classify the light curves and time-resolved spectra, we developed two independent techniques. The first is an exercise in citizen science, in which individuals from the public were asked to classify light curves and spectra. The second, instead, is a set of computer algorithms that automatically classify burst behavior.

In the citizen science project, 24 participants were asked to rank all 62 light curves and spectra. Light curves and spectra were unlabeled and given to participants separately to minimize bias. Participants are classified on a ternary system. Each light curve was classified as fast rise, exponential decay (FRED); unsure; or not an FRED. ² Similarly, the spectra were classified as "hard-to-soft," unsure, or not "hard-to-soft." Prior to classification, participants were given an instruction sheet that gave brief descriptions of the meanings of FRED and hard-to-soft, as well as example light curves from outside the data set. For both sets of data, a score of "yes" was assigned a value of 1, "unsure" a value of 0.5, and "no" a value of 0. The final score for a burst was the average score across all 24 participants. The uncertainty on the scores was taken using the standard error of the mean.

Computational methods were also used to determine if given light curves and time-resolved spectra were single-peaked and hard-to-soft, respectively. By definition, a set of time-resolved spectra being hard-to-soft means that the peak frequency for a given time is not larger than the peak frequency for all previous time steps. Let _peak,i be the peak energy at some time index i, and let σ_,i be the error in peak energy at time index i. Let j be some time step before i. To determine if a given peak frequency is not larger than the previous points, the difference in peak energy should be negative within one uncertainty. In other words,

$$\begin{eqnarray}&&{\epsilon }_{{\rm{peak}},i}-{\epsilon }_{{\rm{peak}},j}-\sqrt{{\sigma }_{{\epsilon }_{i}}^{2}+{\sigma }_{{\epsilon }_{j}}^{2}}\lt 0,\quad j\lt i\end{eqnarray} \tag{ 2 }$$

should be true for all peak energies of index j, where j < i. A code was produced that systematically verified that Equation (2) was true for every time value i, for all j before i. While testing a given burst, each time a spectra of the burst fails this test, a point is added to the running total for that burst. After running the test on all peak-energy values for that burst, the hard-to-soft score is the total number of times the burst violates Equation (2). This score, however, grows with the number of peak-energy values. The total number of tests ran for N peak-energy values is 1 + 2 + ... + (N − 1), or $\tfrac{N(N-1)}{2}$. To normalize for the availability of peak energies, the score for a given burst is thus divided by $\tfrac{N(N-1)}{2}$. Finally, to place the test on a more meaningful scale, the normalized score was subtracted from one so that a hard-to-soft burst would correspond to a score close to one.

To test for single-peak behavior, we developed a test in which a series of horizontal lines was used. To understand the logic of the test, consider a burst with a single peak. If a horizontal line is placed along the burst, all the points of the burst that are above the line should be in sequence. In other words, a single peak should only have one upward crossing and one downward crossing. Taking advantage of this fact, a set of 16 equally spaced horizontal lines was placed on each burst. The highest line was placed two count-rate standard errors below the peak count rate. Photon counts follow a Poisson distribution; thus, the standard error for a count N is $\sqrt{N}$. The average background count was determined by taking the median of all counts outside the region defined by T_90start < t < T_90start + T₉₀. The lowest line was placed two uncertainties above the background. This lowest line was excluded from the test to ensure that any unusual background behavior would not affect the test. In total, 15 lines affected the score in our test. For a given line, we determined the set of points that were at least one uncertainty above the test line. The test then determines if, between the first and last crossing, any points fell at least one uncertainty below the line. See Figure 1 for an example line and a few light-curve cases that yield similar scores. For every point that fell at least one uncertainty below the line, a tally was added to the running total for that burst. This test was run on all 15 lines. To normalize the scores, they were divided by 1800. This comes from the fact that there were 15 lines and 120 bins on the region of interest. If there were perfect delta spikes at both ends of the region of interest, the score would then be 15 x 120, giving 1800. For a more general version, a test consisting of m lines with N_bins bins on the region of interest would be normalized by m · N_bins. Finally, the normalized score was subtracted from one. This places the scores on a scale from zero to one, where a score of one means that the burst was perfectly single-peaked; no points violated the horizontal line test. Conversely, a score of zero represents a burst with multiple extremely well-separated narrow peaks. The virtue of using multiple lines is best seen in Figure 1. A single line may miss some behaviors, such as in Figure 1(C) where only one line catches the shallow valley behavior. This is similarly helpful for narrow, deep valleys, in which a single line would only show one point as violating the test, but the deep peak is caught by multiple lines, increasing the score as seen in Figure 1(B). This test is versatile in its ability to catch multiple, different multipeaked behaviors through the multiple lines. It is also powerful in not requiring the pulse to have any specific analytical description. An alternative method is found in Guidorzi (2015; see also Guidorzi et al. 2024).

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The human and computer rankings generally agreed with each other. Comparisons of the human classifications and computer classifications for both hard-to-soft and single-peaked metrics can be seen in Figures 2 and 3, respectively. In both Figures 2 and 3, there is a positive trend suggesting the tests agree. It is important to note that while both the human and computational tests are on a zero-to-one scale, they do not hold the same meaning. For instance, it is common for the human single-peak test scores to be near zero, as this simply means a majority of the participants stated "no" for the classification. However, on the computational single-peaked test, a score of zero would mean that the burst has well-separated delta-function-like peaks. This is an ideal multipeaked burst, and as such, most bursts do not have scores close to zero even with multiple peaks. Similarly, scores for the computational hard-to-soft test would have to be strictly monotonically increasing, which again is unlikely. Thus a score of zero is again quite unlikely. This means while a score of 0.8 for the human single-peak test indicates the burst is single-peaked, a similar score on the computational single-peak test does not. This holds similarly true for the hard-to-soft test. For both the computational single-peak and hard-to-soft tests, a score must be much closer to one for the burst to be considered single-peaked or hard-to-soft than in the human tests.

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In order to quantitatively compare the results, both Pearson and Spearman tests were run to test for linear correlation and monotonic correlation, respectively. Table 2 shows the Pearson and Spearman scores for both the single-peak and hard-to-soft tests as well as the probability of a set of random data showing a correlation as strong or stronger.

For both sets of tests, there was both a fairly strong Spearman and Pearson correlation with a high probability. This shows that the tests generally agreed with one another; a human scoring of hard-to-soft typically corresponded to a computational scoring of hard-to-soft and similarly for the single-peak test.

There are several outliers that can be seen in Figures 2 and 3. The most noticeable outlier for the single-peak test is GRB110407998, whose light curve and spectra are shown in Figure 4. GRB110507998 was ranked as more single-peaked than other bursts with similar computational scores. The light curve seems to show a generally single-peaked behavior aside from a small secondary peak at 10 s. The horizontal line test picked up on this second peak, whereas the human eye seemed to generally classify this as background. The second peak rises up nearly 1000 photons s⁻¹ above the background, well above the uncertainty.

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The most noticeable outlier in the hard-to-soft tests is GRB110721200, whose light curve and spectra can be seen in Figure 5. GRB110721200 was generally considered much more hard-to-soft than other bursts with similar computational rankings. The spectra of GRB110721200 decrease until around 3 s before increasing slightly and then decreasing further. The computational hard-to-soft test picked up several failure points in the section of increasing peak energy around 3 s. Humans ranked this as hard-to-soft.

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Except for these outliers, the computational and human results generally agreed. It is difficult to directly compare the results given the different scales for the tests. For instance, only a handful of bursts scored a single-peaked human score greater than 0.8, whereas almost all bursts had computational single-peak scores above 0.8. That being said, from Figure 2, the human test seemed to have more bursts sitting in the range around 0.5, or the "unsure" range when the computational test would consider them multipeaked. This may indicate that the computational test is stricter in its classification of bursts as single-peaked.

The hard-to-soft scores and single-peaked scores were directly compared for both the computational and human methods. We will first consider the unbinned results. Figures 6 and 7 show the human and computational results, respectively. To test for potential correlations, the Pearson and Spearman tests were again run. Scores and probability p-values can be seen in Table 3. These tests are readily applicable as they do not require uncertainty, and for neither the human nor computational tests are there well-defined uncertainties.

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On both the Pearson and Spearman tests, the human scores showed significant positive correlations. This indicates that bursts that were more single-peaked were similarly more hard-to-soft. The computational tests were less significant with respect to the Pearson tests, indicating they are not a linear correlation. However, the Spearman test was far more significant, which indicates that there is a monotonic correlation between the single-peaked and hard-to-soft scores. This demonstrates that a more single-peaked burst is more hard-to-soft, albeit not linearly.

To further analyze the data, both the human and computational scores were binned with respect to the single-peak scores. The hard-to-soft scores for each burst in a given bin were averaged, and the standard error was calculated on the hard-to-soft scores. The human scores were binned into two bins corresponding to single-peaked and multipeaked scores. Single-peaked scores were any burst with a score greater than 0.667; multipeak bursts were the remaining bursts. The computational rankings were binned unevenly. Looking at the unnormalized single-peak scores for the horizontal line test, there was a clumping of bursts with unnormalized scores of three or fewer. This corresponds to final scores of at least 0.9983. Any burst with a computational single-peak score less than this was considered multipeaked.

The number of combined uncertainties between the two bins were calculated for both the human and computational tests. The results are shown in the last column of Table 3. For both the human and computational tests, there is a significant difference between the hard-to-soft scores for the single-peaked and multipeaked bursts. In particular, in both cases, the single-peaked bursts had significantly larger hard-to-soft scores as compared to multipeaked bursts. Along with the results of the Pearson and Spearman scores, this associates single-peaked behavior with hard-to-soft behavior.