A Comprehensive Power Spectral Density Analysis of Astronomical Time Series. II. The Swift/BAT Long Gamma-Ray Bursts
DDraft version February 15, 2021
Typeset using L A TEX twocolumn style in AASTeX62
A Comprehensive Power Spectral Density Analysis of Astronomical Time Series. II. The Swift/BAT LongGamma-Ray Bursts
Mariusz Tarnopolski and Volodymyr Marchenko Astronomical Observatory, Jagiellonian University, Orla 171, 30–244, Kraków, Poland
ABSTRACTWe investigated the prompt light curves (LCs) of long gamma-ray bursts (GRBs) from the Swift/BATcatalog. We aimed to characterize their power spectral densities (PSDs), search for quasiperiodic oscil-lations (QPOs), and conduct novel analyses directly in the time domain. We analyzed the PSDs usingLomb-Scargle periodograms, and searched for QPOs using wavelet scalograms. We also attemptedto classify the GRBs using the Hurst exponent, H , and the A − T plane. The PSDs fall into threecategories: power law (PL; P ( f ) ∝ /f β ) with index β ∈ (0 , , PL with a non-negligible Poissonnoise level (PLC) with β ∈ (1 , , and a smoothly broken PL (SBPL; with Poisson noise level) yieldinghigh-frequency index β ∈ (2 , . The latter yields break time scales on the order of 1–100 seconds.The PL and PLC models are broadly consistent with a fully developed turbulence, β = / . For anoverwhelming majority of GRBs (93%), H > . , implying ubiquity of the long-term memory. Wefind no convincing substructure in the A − T plane. Finally, we report on 34 new QPOs: with one ormore constant leading periods, as well as several chirping signals. The presence of breaks and QPOssuggests the existence of characteristic time scales that in at least some GRBs might be related to thedynamical properties of plasma trajectories in the accretion disks powering the relativistic jets.
Keywords: gamma-ray burst: general – methods: data analysis – methods: statistical INTRODUCTIONGamma-ray bursts (GRBs, Klebesadel et al. 1973) aretypically divided into short , coming from double neutronstar (NS-NS) or NS-black hole (BH) mergers (Nakar2007; Berger 2011), and long ones, whose progenitorsare the collapse of massive stars, e.g. Wolf-Rayet orblue supergiants (Woosley & Bloom 2006; Cano et al.2017). The division between the two classes is primarilybased on the bimodal distribution of T (time duringwhich 90% of the GRB’s fluence is accumulated; Kouve-liotou et al. 1993), and the threshold is at T (cid:39) (butcf. Bromberg et al. 2013; Tarnopolski 2015a). GRBs ex-hibit a rich variety of light curve (LC) shapes (Fishmanet al. 1994), which implies complicated mechanisms gov-erning their radiation. The LCs usually exhibit a powerlaw (PL; P ( f ) ∝ /f β ) power spectral density (PSD),with or without a break, and on rare occasions a sign ofa quasiperiodic oscillation (QPO) was noted. [email protected]@oa.uj.edu.pl The first confirmed QPO in a GRB was found in the 5March 1979 event (Barat et al. 1979; Terrell et al. 1980),in which an unambiguous ∼ ∼
20 cycles) followed a 0.2 s outburst. However, subse-quent analyses (Norris et al. 1991; Fenimore et al. 1996)suggested that this transient was actually a soft gammarepeater (SGR). Shortly later, a 4.2 s periodicity was re-ported in a 29 October 1977 event during about 30 s ofits duration (Wood et al. 1981), while Kouveliotou et al.(1988) identified 7 cycles of a 2.2 s quasiperiodicity in along (43 s), hard (extending to 100 MeV) GRB observedon 5 August 1984. Schaefer & Desai (1988) examinedthe significance of periods (in the range 2–18 s) claimedfor 16 GRBs at that time. They confirmed only the ∼ ∼ a r X i v : . [ a s t r o - ph . H E ] F e b Tarnopolski & Marchenko and Suzaku data (Cenko et al. 2010) did not confirm iton a 3 σ significance level, but another reanalysis of theSuzaku LC revealed a 3 σ -significant QPO (Iwakiri et al.2010). Likewise it was found that the Swift LC actuallyexhibits a 3.5 σ significance for the QPO (Ziaeepour &Gardner 2011). Again, like in the 5 March 1979 event,such a periodicity might hint at an SGR nature of thesource, however in this case all other properties ensureits GRB origin. The presumed QPO was speculated tobe caused by magnetorotational instabilities (MRI) ina hyperaccreting disk (Masada et al. 2007), or originatedue to a precessing magnetic field (Ziaeepour & Gard-ner 2011). While an unambiguous conclusion about thisQPO has apparently not been reached, the possibilityof such modulations in the prompt emission of GRBsis fascinating. Moreover, Beskin et al. (2010) discov-ered the first periodic pulsations in the optical promptemission of GRB 080319B, at a period of 8.1 s. Thepresumed (quasi)periodic nature ought to be taken withcaution, though, since the detected period correspondsto only four peaks in the LC.First searches for high-frequency QPOs were unsuc-cesful (Deng & Schaefer 1997; Kruger et al. 2002),but Zhilyaev & Dubinovska (2009) employed a wavelet-based approach to short BATSE GRBs, which yielded aQPO with a leading period of 5.5 ms in one case (triggernumber 207), and a few chirping signals as well. NS-BHmergers are indeed expected to give rise to jet preces-sion triggering the QPO modulation (Stone et al. 2013).However, a subsequent canonical search for QPO fea-tures in PSDs detected no significant signals (Dichiaraet al. 2013b).When it comes to the overall shape of the PSD, Belli(1992) observed PL ( (cid:46) β (cid:46) ) or Lorentzian (i.e.,indicative of an autoregressive process of order 1, i.e.AR(1)) forms in case of 5 long Konus GRBs. Giblinet al. (1998) examined 100 GRBs by computing theirPSDs and fitting a PL. They obtained a wide range of β ,extending up to β ≈ , with 65% of the cases exceedingthe red noise value, i.e. β > .Beloborodov et al. (1998), in turn, constructed theaverage PSD of 214 certainly long GRBs ( T >
20 s ),hence considered the LCs as random realizations of thesame underlying stochastic process, and concluded thatthe obtained β (cid:39) / is consistent with a fully developedturbulence , arising in the internal shock model thatlikely governs the observed variability. A subsequent Through Parseval’s theorem, the energy at frequency f can beexpressed as the Fourier transform of the signal, and the wavenum-ber k ∝ f . Hence it follows that for turbulence the PSD has anexponent of / (cf. Moraghan et al. 2015). analysis with a bigger sample of 514 GRBs confirmed theKolmogorov’s / law for the average PSD (Beloborodovet al. 2000), and showed that dim bursts exhibit steeperPSDs (cf. Ryde et al. 2003; Rong-feng & Li-ming 2003).Also, in lower energy chanels the PSDs are steeper thanat higher energies. Panaitescu et al. (1999), on the otherhand, modeled the PSD based on the internal shockmodel, and found that the / law can be explained byinvoking modulation of the relativistic winds instead ofturbulence. Pozanenko & Loznikov (2000) computed theaverage PSD of 815 long GRBs and fitted it with a PLand an exponential PL with β ∈ ( / , / ) , again roughlyindicative of the Kolmogorov law. Chang & Yi (2000)simulated GRB LCs as a sum of fast-rise-exponential-decay (FRED) pulses, and demonstrated that by adjust-ing the sampling, rise and decay time scales, the / lawcan be recovered. With a Swift sample of GRBs withredshifts, Guidorzi et al. (2012) found that the / lawholds in the rest frame as well (roughly, as β (cid:46) depend-ing on the subsample—for higher redshifts the PSD be-comes shallower), and identified a break in the smoothlybroken PL at time scales ∼
30 s. Dichiara et al. (2013a)arrived at similar results for GRBs observed with Bep-poSAX and Fermi, with a break at ∼ − s, depend-ing on the energy channel. The consistency and persis-tence of the average PSD index of / among differentdata sets, corresponding to different energy bands, in-strument sensitivities etc., strongly suggest that indeedthe collection of GRBs shall be treated as an ensemble,and that each LC is a stochastic realization of the sameunderlying emission mechanism (though possibly gen-erated by values of parameters different from burst toburst, owing to, e.g., different magnetization degrees).Guidorzi et al. (2016) analyzed the individual PSDs ofSwift GRBs, finding that they can be divided into twoclasses: with and without a break. The overall spanof the PL index fell into . (cid:46) β (cid:46) , while the breaktime scales spanned the range ∼ − s, with a loga-rithmic average of 25 s. They also found marginal evi-dence (barely touching the 3 σ level) for QPOs in threecases. Finally, Dichiara et al. (2016) found a statisti-cally significant anticorrelation between the rest-framepeak energy, E restpeak , and the PSD index β , adding tothe long list of correlations between various parametersof the prompt and afterglow phases (Shahmoradi & Ne-miroff 2015; Dainotti & Del Vecchio 2017; Dainotti et al.2018; Dainotti & Amati 2018). The E restpeak − β relationwas discussed on grounds of a few prompt emission mod-els, explaining the overall anticorrelation by invoking thebulk Lorentz factor, Γ , as the key observable connectingthe two quantities. The distribution of the break timescales spanned the range ∼ − s, with a logarithmic POs in Swift/BAT GRBs E restpeak − β relation might be a consequence of the peakenergy–luminosity and duration–luminosity relations.Zhang & Zhang (2014) simulated several LCs withinthe Internal-Collision-induced MAgnetic Reconnectionand Turbulence (ICMART) model, with moderatelyhigh magnetization of the ejected shells. It was foundthat for various reasonable parameter values, the result-ing PSDs were generally consistent with the PL forms ofSwift GRBs. In particular, (cid:46) β (cid:46) for the assumedrange of ratio of the mini-jets and Γ factors (although forsome particular parameters, occasionally steeper PSDswere also arrived at). Additionally, spikier LCs (morevariable on shorter time scales) yielded shallower PSDs.The Hurst exponent, H (a measure of persistence, orself-similarity of a time series), was shown to be able todifferentiate between short and long GRBs (MacLachlanet al. 2013), especially when coupled with other charac-teristics, like duration T and minimum variability timescale (MVTS; Tarnopolski 2015b). H is constrained tothe interval (0 , , and is applicable to both stationaryand nonstationary time series, hence is able to providea universal classification of GRBs. Such a classification,based predominantly on their LCs, was notoriously diffi-cult due to the diverse morphology. Recently, Jespersenet al. (2020) applied a machine learning dimensionalityreduction algorithm, t-distributed stochastic neighbor-hood embedding (t-SNE) to Swift GRBs. t-SNE groupssimilar LCs close together, based on which it was demon-strated that as a result two prominent clusters emerged,each corresponding to the short and long subclasses, re-spectively. This appears to resolve the issue whetherthere are two or three main GRB types (Horváth 2002;Horváth et al. 2008; Tarnopolski 2016a; Horváth et al.2019; Tóth et al. 2019; Tarnopolski 2019a,b,c).The goal of this paper is a possibly comprehensiveanalysis of PSDs of a big sample of GRBs from theSwift catalog, with a particular aim to identify QPOs.In addition, the LCs are investigated directly in the timedomain with the Hurst exponent, and the recently de-veloped A − T plane, which was proven to be capa-ble to classify blazar subtypes based solely on the LCs(Tarnopolski et al. 2020). In Sect. 2 the description ofthe utilized GRB sample is given, and the outline of theemployed analysis methods is provided. In Sect. 3 theresults are presented, and Sect. 4 is devoted to discus-sion. Concluding remarks are gathered in Sect. 5. DATA AND METHODSDescription of the sample is given first. The employedmethods are then briefly described. For a more detailedexplanation, as well as the results of a comprehensive benchmark testing of each method, we refer the readerto Tarnopolski et al. (2020).2.1.
Sample
The mask-weighted, background-subtracted LCs, ina 64-ms binning and covering the total energy range − keV, were downloaded from the Swift/BATcatalog (Lien et al. 2016). The portions of the LCswithin respetive T intervals were extracted. We fo-cus on long GRBs with sufficient number of points toconduct a meaningful time series and PSD analysis. Wetherefore utilized LCs with more than 50 points , i.e.with T > . . We excluded confirmed short GRBswith extended emission. We ended with 1160 GRBs inour sample. 2.2.
PSDs
Lomb-Scargle periodogram
To calculate the PSD of a time series { x k ( t k ) } nk =1 witha constant time interval between consecutive observa-tions, δt = t k +1 − t k ≡
64 ms , the Lomb-Scargle peri-odogram (LSP; Lomb 1976; Scargle 1982; VanderPlas2018) is computed in the standard way as P LS ( ω ) = 12 σ (cid:18) n (cid:80) k =1 ( x k − ¯ x ) cos[ ω ( t k − T )] (cid:19) n (cid:80) k =1 cos [ ω ( t k − T )]+ (cid:18) n (cid:80) k =1 ( x k − ¯ x ) sin[ ω ( t k − T )] (cid:19) n (cid:80) k =1 sin [ ω ( t k − T )] , (1)where ω = 2 πf is the angular frequency, T ≡ T ( ω ) is T ( ω ) = 12 ω arctan n (cid:80) k =1 sin(2 ωt k ) n (cid:80) k =1 cos(2 ωt k ) , (2)and ¯ x and σ are the sample mean and variance.The lower limit for the sampled frequencies is f min =1 / ( t max − t min ) , corresponding to the length of thetime series. Since we are dealing with uniformly sam-pled data, the upper limit is the Nyquist frequency, https://swift.gsfc.nasa.gov/results/batgrbcat/ This is a requirement of the software wavepal used for thewavelet scalograms; cf. Sect. 2.3. https://swift.gsfc.nasa.gov/results/batgrbcat/summary_cflux/summary_GRBlist/GRBlist_short_GRB_with_EE.txt Tarnopolski & Marchenko f max ≡ f Nyq = δt . The total number of sampled fre-quencies is N P = n f max f min , (3)and we employ n = 100 hereinafter.2.2.2. Binning and fitting
For fitting in the log-log space, binning is applied.The values of log f are binned into approximately equal-width bins, with at least two points in a bin, and therepresentative frequencies are computed as the geomet-ric mean in each bin. The PSD value in a bin is taken asthe arithmetic mean of the logarithms of the PSD (Pa-padakis & Lawrence 1993; Isobe et al. 2015). We requirethe binned PSDs to consist of at least seven points forthe fitting, which left us with 1150 GRBs suitable forthe PSD analysis.The following models were fitted to the binned LSPs:1. pure power law (PL): P ( f ) = P norm f − β , (4)2. PL plus Poisson noise (PLC): P ( f ) = P norm f − β + C, (5)3. smoothly broken PL (SBPL; McHardy et al. 2004)plus Poisson noise: P ( f ) = P norm f − β (cid:16) ff break (cid:17) β − β + C, (6)4. SBPL plus Poisson noise, with a fixed β = 0 : P ( f ) = P norm (cid:16) ff break (cid:17) β + C, (7)where the parameter C is an estimate of the Poissonnoise level coming from the uncertainties of individualmeasurements (see further in this Section), β is the PLindex, f break is the break frequency (from which thebreak time scale is calculated as T break = 1 /f break ), and β , β are the low- and high-frequency indices, respec-tively. PL is a case of PLC with C = 0 . The reasonfor considering them separately is that the PLC modeldegenerates when β P LC → , since P ( f ) → P norm + C =const . then, and hence fitting a pure PL diminishes the When β = 2 , this is a Lorentzian (plus Poisson noise), i.e. aPSD of an AR(1) process. parameter uncertainties (cf. Żywucka et al. 2020). Sim-ilarly, SBPL was found to often take advantage of thedegree of freedom provided by the possibility to vary β , and led to overfitting, hence the two variants of theSBPL were considered separately as well. For complete-ness, an SBPL with β = β reduces to a PLC.Fits of different models were compared using the smallsample Akaike information criterion ( AIC c ) given by AIC c = 2 p − L + 2( p + 1)( p + 2) N − p − , (8)where L is the log-likelihood, p is the number of param-eters, and N is the number of points fitted to (Akaike1974; Hurvich & Tsai 1989; Burnham & Anderson 2004).For a regression problem, L = − N ln RSS N , (9)where RSS is the residual sum of squares; p is an implicitvariable in L . A preferred model is one that minimizes AIC c . This criterion is a trade-off between the goodnessof fit and the complexity of the model, expressed via thenumber of parameters p . What is essential in assesingthe relative goodness of a fit in the AIC c method isthe difference, ∆ i = AIC c,i − AIC c, min , between the AIC c of the i th model and the one with the minimal AIC c . If ∆ i < , then there is substantial support forthe i th model (or the evidence against it is worth onlya bare mention), and the proposition that it is a properdescription is highly probable. If < ∆ i < , then thereis strong support for the i th model. When < ∆ i < , there is considerably less support, and models with ∆ i > have essentially no support.The MVTS, τ , is defined herein as the time scale (cor-responding to a frequency f = 1 /τ ) at which the Pois-son noise level dominates over the PL/SBPL component.E.g., for the PLC case it is obtained by solving the equa-tion P norm f − β = C , and similarly for the SBPL case(which, however, does not yield a closed-form solution,hence is obtained numerically). The standard errors of f are estimated via bootstrapping: 1000 random real-izations of the best-fit PSD were generated by varyingthe parameters within their uncertainties, and the stan-dard deviation of the resulting sample was calculated.We record only cases with ∆ τ < τ .The Poisson noise level, coming from the statisticalnoise due to uncertainties in the LC’s observations, ∆ x k ,is the mean squared error, with a normalization suitablefor LSP: P Poisson = 12 σ n n (cid:88) k =1 ∆ x k . (10)2.3. QPOs
POs in Swift/BAT GRBs ψ ( t ) is a wave packet, i.e. its location andinstantaneous frequency are well constrained. We usethe Morlet wavelet hereinafter, ψ ( t ) = 1 π / (cid:20) exp ( i ω t ) − exp (cid:18) − ω (cid:19)(cid:21) exp (cid:18) − t (cid:19) , (11)with ω = 10 − to ensure a good frequency resolu-tion. The mother wavelet gives rise to the dictionary,or child wavelets, ψ s,l ( t ) , into which the analyzed signalis decomposed: x ( t ) = (cid:80) s,l W ( s, l ) ψ s,l ( t ) . The coeffi-cients of such a decomposition, W ( s, l ) , depend on thelocation, l ∈ R , and the scale, s ∈ R + . The scalogramtherefore allows not only to identify the frequency, butlocalize it within the time series in the temporal domainas well, and visualize it in the time-frequency space. Forthis purpose, we utilize the method implemented in thepackage wavepal (Lenoir & Crucifix 2018a,b). To testthe significance of the detected features, they are testedagainst a continuous-time autoregressive moving average(CARMA) stochastic model (Kelly et al. 2014). This isa more general family of noise than the easily testedwhite noise, or commonly considered colored noise. Weaim to detect QPOs at the level of at least σ (99.73%confidence level).2.4. Hurst exponents
The Hurst exponent H measures the statistical self-similarity of a time series x ( t ) . It is said that x ( t ) isself-similar (or self-affine) if it satisfies x ( t ) . = λ − H x ( λt ) , (12)where λ > and . = denotes equality in distribution.The meaning of H can be understood as follows: for apersistent stochastic process, if some measured quan-tity grows on average (over some time periods), thesystem prefers to maintain their growth. The processis, however, probabilistic, and hence at some point theobserved quantity will eventually start to decrease (onaverage). But the process still has long-term memory(which is a global feature), therefore it prefers to de-crease for some time until the transition occurs ran-domly again. In other words, the process prefers tosustain its most recent behavior (in a statistical sense).In case of H < . , the process is anti-persistent, andit possesses short-term memory, meaning that the ob-served values frequently switch from relatively high torelatively low (with respect to a stationary mean), andthere is no preference among the increments. This is https://github.com/guillaumelenoir/WAVEPAL a so-called mean-reverting process. A PSD in form ofa PL is indicative of a self-similar process. There isa (piecewise) linear relation between H and the index β of a PL PSD: H = ( β + 1) / for β ∈ ( − , , and H = ( β − / for β ∈ (1 , , with β = 0 (white noise)and β = 2 (red noise) both yielding H = 0 . . The case β = 1 (pink noise) is at the border, with no precise H value assigned.We utilize three algorithms for extracting H : the de-trended fluctuation analysis (DFA), and two wavelet-based methods: the discrete wavelet transform (DWT)with the Haar wavelet as the basis, and the averagedwavelet coefficient (AWC) method.2.4.1. Detrended fluctuation analysis—DFA
In the DFA algorithm (Peng et al. 1994, 1995) onestarts with calculating the accumulative sum X ( t ) = t (cid:88) k =1 (cid:0) x k − ¯ x (cid:1) , (13)which is next partitioned into non-overlapping segmentsof length ς each. In each segment, the correspondingpart of the time series X ( t ) is replaced with its linearfit, resulting in a piecewise-linear approximation of thewhole X ( t ) , denoted by X lin ( t ; ς ) . The fluctuation as afunction of the segment length ς is defined as F ( ς ) = (cid:34) N N (cid:88) t =1 ( X ( t ) − X lin ( t ; ς )) (cid:35) / . (14)The slope a of the linear regression of log F ( ς ) versus log ς is an estimate for H : H = a if a ∈ (0 , , and H = a − if a ∈ (1 , .2.4.2. Averaged wavelet coefficient—AWC
The AWC method (Simonsen et al. 1998) relies di-rectly on the scaling in Eq. (12) and employs the con-tinuous wavelet transform, which leads to W ( λs, λl ) = λ H +1 / W ( s, l ) . (15)The AWC is defined as the standard arithmetic meanover the locations l at a given scale s : W ( s ) = (cid:104)| W ( s, l ) |(cid:105) l . (16)By a linear regression of log W ( s ) vs. log s , an estimateof H can be obtained from the slope a via H = a − / if a ∈ ( / , / ) , and H = a + / if a ∈ ( − / , / ) . The
Mathematica implementations are avail-able at https://github.com/mariusz-tarnopolski/Hurst-exponent-and-A-T-plane.
Tarnopolski & Marchenko
Discrete wavelet transform—DWT H can be obtained with the DWT using, e.g., the Haarwavelet as the basis (Veitch & Abry 1999; Knight et al.2017). The relation between the variance of the wavelettransform coefficients d j,k (where j corresponds to thescale s , and k · j to the location l ) and the scale j canbe written as log var( d j,k ) = a · j + const . (17)The slope a is obtained by fitting a line to the linear partof the log var( d j,k ) vs. j relation, and H is obtained as H = ( a − / when a ∈ (1 , , and H = ( a + 1) / when a ∈ ( − , . 2.5. The
A − T plane
The
A − T plane was initially designed to providea fast and simple estimate of the Hurst exponent(Tarnopolski 2016b). It is also well suited to differ-entiate between types of colored noise, Eq. (4), charac-terized by different PL indices β (Zunino et al. 2017),and to discriminate between regular and chaotic dynam-ics (Zhao & Morales 2018). It comprises of the fractionof turning points, T , and the Abbe value, A .2.5.1. Turning points
Consider three consecutive data points, x k − , x k , x k +1 .They can be arranged in six ways; in four of them, theywill create a peak or a trough, i.e., a turning point(Kendall & Stuart 1973; Brockwell & Davis 1996). Theprobability of finding a turning point in such a sub-set is hence / . Let T denote the fraction of turningpoints in a time series comprised of n points. Therefore T ∈ [0 , , and is asymptotically equal to / for a purelyuncorrelated time series (white noise). A process with T > / (i.e., with raggedness exceeding that of a whitenoise) will be more noisy than white noise. Similarly,a process with T < / will be ragged less than whitenoise. All of these cases can be realized for variousstochastic processes (e.g., PL or autoregressive movingaverage) as well as real-world instances (Bandt & Shiha2007; Tarnopolski 2019d).2.5.2. Abbe value
The Abbe value is defined as half the ratio of themean-square successive difference to the variance (vonNeumann 1941a,b; Kendall 1971): A = n − n − (cid:80) i =1 ( x i +1 − x i ) n n (cid:80) i =1 ( x i − ¯ x ) ≡
12 var ( dX )var ( X ) , (18)where dX denotes the increments (consecutive differ-ences) of process X . A quantifies the smoothness ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● β C Figure 1.
Locations in the
A − T plane of the PL plusPoisson noise PSDs of the form P ( f ) ∝ /f β + C , with β ∈ { , . , . . . , } . For each PSD, 100 realizations of thetime series were generated, and the displayed points are themean locations of them. The error bars depict the standarddeviation of A and T over these 100 realizations. The case β = 0 is a pure white noise, with ( A , T ) = (1 , / ) . Thegeneric PL case ( C = 0 ) is the lowest curve (red); with anincreasing level of the Poisson noise C , the curves are raisedand shortened, as the white noise component starts to dom-inate over the PL part. (raggedness) of a time series by comparing the sumof the squared differences between two successive mea-surements (the variance of the differenced process dX )with the variance of the whole time series X . It ap-proaches zero for time series displaying a high degree ofsmoothness, while the normalization factor ensures that A tends to unity for a white noise process (Williams1941). In astronomy it has been rarely utilized, withsome recent, nonextensive examples (Shin et al. 2009;Mowlavi 2014; Sokolovsky et al. 2017; Pérez-Ortiz et al.2017; but see also Lafler & Kinman 1965). In particular,it was demonstrated that blazar subclasses, observed in γ -rays, are separated in the A − T plane (Tarnopolskiet al. 2020), as are optically observed blazar candidates(Żywucka et al. 2020) behind the Magellanic Clouds (Ży-wucka et al. 2018).In Figure 1 the locations in the
A − T plane of PLprocesses are shown, as well as PLC ones. In effect ofintroducing Poisson noise C , the location ( A , T ) of theotherwise PL-type signal is being dragged closer to thepoint (1 , / ) corresponding to white noise. Therefore,the region of availability of time series with PSDs ofthe PLC type is two-dimensional (bounded from belowby the PL limit, and from above by the line T = / ),allowing for nontrivial relations between A and T .2.5.3. Coarse graining
The so-called coarse-grained sequences are calculatedaccording to Zunino et al. (2017). They are obtained
POs in Swift/BAT GRBs { x k } into nonoverlapping segmentsof length d , and each segment is averaged, resulting insmoothed sequences { y dj } : y dj = jd (cid:88) k =( j − d +1 x k (19)for j ∈ { , . . . , (cid:98) n/d (cid:99)} . The A − T plane, constructedas a function of the temporal scale factor d , allows toinvestigate various temporal resolutions given only onerealization of the process, i.e., an LC in only one binning(preferably a small one). This approach was appliedto a periodically driven thermostat (Zhao & Morales2018), and shown to successfully differentiate betweenregular, chaotic and stochastic realizations of time series.However, very long time series, with n = 200 , , wereutilized to demonstrate such phenomenon. RESULTSThe results are discussed in the following Sections. Ta-ble 1 describes the contents of the accompanying online-only file containing all results.3.1.
PSDs
We were able to obtain meaningful fits of Eqs. (4)–(7)to the PSDs of 1132 GRBs. The best fit was chosenbased on the
AIC c . As a result, 207 PSDs were mod-eled best by a pure PL, 548 by a PLC, and 377 yieldedSBPL (among which 277 had fixed β = 0 ). 831 MVTSwith ∆ τ < τ were recorded among the PLC and SBPLcases. Exemplary fits are shown in Fig. 2. The distribu-tions of the β , β , β indices, time scales τ and T break ,as well as the scatter plots illustrating the relations be-tween the SBPL parameters, are displayed in Fig. 3.When the PL model is considered, many GRBs exhibitflat PSDs, | β P L | (cid:46) . , i.e. closely resembling whitenoise (Fig. 3(a)), owing to the weakness of the burstand significant Poisson noise contamination. The val-ues extend to β P L ∼ , while β P LC concentrates around β P LC ∼ , with the bulk of it spanning the range ∼ β ∼ and β ∼ , although there areheavy tails in both distributions, extending to β < − and β > (Fig. 3(d) and (e)). Very steep PSDs (i.e.,with high value of β ) essentially imply no variability onthe associated timescales, because the power drops dras-tically from the conventional PL at lower frequencies tothe Poisson noise level at higher frequencies. This meansthat in these instances there is a sharp cut-off at T break below which variability on shorter timescales is wipedout (excluding the region of Poisson noise domination).A prominent turnover (i.e., β being very negative)sometimes leads to QPO-like features as in Fig. 2(d). Often it is a break incorporating just a handful of pointsin the binned PSD, though. The break time scale T break falls mostly in the range 1–100 s, about an order ofmagnitude greater than the MVTS (Fig. 3(c) and (f)).There are moderate or weak correlations between theparameters of the SBPL model (i.e., β , β , and T break ;Fig. 3(g)–(k)).It should be emphasized that the Poisson noise lev-els C obtained by fitting Eqs. (5)–(7) are extremely wellcorrelated ( r = 0 . ; 95% CI: (0 . , . ) with the es-timates inferred from the LC uncertainties via Eq. (10).This means that the statistical fluctuations are the pre-dominant origin of MVTS in the Swift sample, and henceit does not carry physical interpretation regarding theGRB progenitors. In other words, MVTS is the timescale above which the actual signal present in the GRBbreaks above the (background or instrumental) noiselevel. 3.2. QPOs
We searched for QPOs with leading periods
T
Table 1.
Contents of the table with time series and PSD properties of the GRBs.Column Column name Symbol Description1
Number
Number Consecutive number of the GRB in the sample (reverse chronological order)2
GRB
GRB name Identifying name of the GRB, according to the Swift catalog3
T90 T Duration of the GRB, in seconds4 betaPL β PL Exponent β of the pure PL fit5 e_betaPL ∆ β PL Uncertainty of the exponent β of the pure PL fit6 betaPLC β PLC
Exponent β of the PL plus Poisson noise (PLC) fit7 e_betaPLC ∆ β PLC
Uncertainty of the exponent β of the PL plus Poisson noise (PLC) fit8 beta1SBPL β Low-frequency exponent β of the SBPL fit9 e_beta1SBPL ∆ β Uncertainty of the low-frequency exponent β of the SBPL fit10 beta2SBPL β High-frequency exponent β of the SBPL fit11 e_beta2SBPL ∆ β Uncertainty of the high-frequency exponent β of the SBPL fit12 Tbreak T break Break time scale of the SBPL fit, in seconds13 e_Tbreak ∆ T break Uncertainty of the break time scale of the SBPL fit, in seconds14
MVTS τ Minimum variability time scale, in seconds15 e_MVTS ∆ τ Uncertainty of the minimum variability time scale, in seconds16 H H Hurst exponent17 e_H ∆ H Uncertainty of the Hurst exponent18
HPL H PL Hurst exponent inferred from the index β PL e_HPL ∆ H PL Uncertainty of the Hurst exponent inferred from the index β PL z z Redshift21
Epeak E peak Peak energy of the spectral model, in keV22 e_Epeak ∆ E peak Uncertainty of the peak energy of the spectral model, in keV23 logLiso log L iso Logarithm of the peak isotropic luminosity ( L iso in erg s − )24 e_logLiso ∆ log L iso Uncertainty of the logarithm of the peak isotropic luminosity
Note —This table is available in its entirety in machine-readable form. and obtain a leading period of 8.02 s, persistently span-ning almost 100 s of the LC. Moreover, the scalogramreveals another QPO, with a slightly higher period of ∼ (or ) resonance—a moderate order. Ziaeepour &Gardner (2011) showed that invoking a precession of astrong external magnetic field (present in case of starsbelieved to be progenitors of long GRBs) might lead toan oscillating behavior in the prompt LC. The overallnature of the two QPOs present in this GRB is unclear,though. We obtained a very similar picture for GRB120116A, which has the same overall FRED-like shapewith an 8 s QPO overlaid (Fig. 4(b)). The QPOs inboth GRBs are thence likely to be a result of the samemechanism and conditions at the emission site.Fig. 5 shows examples of novel detections of a slightlychirping signal (Fig. 5(a)), another constant leading pe-riod (Fig. 5(b)), and a case of harmonics remaining inan apparent resonance (Fig. 5(c)).3.3. Hurst exponents
To estimate H , the algorithms from Sect. 2.4 (DFA,DWT, AWC) were utilized. To obtain robust estimates,first were selected only those GRBs for which all three H estimates were consistent with each other, within thestandard errors. Next, a time evolution of the three H values were investigated, and only cases with no suddenjumps between H ∼ and H ∼ were kept. Givena time series with length n , it was divided into slidingwindows of size (cid:98) n/ (cid:99) , resulting in (cid:100) n/ (cid:101) such chunks.To each, the three algorithms were applied, and henceprovided the time evolution of H . Examples of this pro-cedure are shown in Fig. 6(a) and (b). This eventuallyled to 335 estimates of H , whose distributions are dis-played in Fig. 6(f). Over 90% of GRBs are character-ized by H > . , meaning they possess long-term mem-ory. This is highly consistent with the overall shape ofmost LCs, which are comprised of one or more FRED-like pulses. Overall, a pulse by itself is persistent: theinitial rise lasts for a prolonged period of time (longerthan the sampling time step), and is followed by a pro-longed decay, i.e. a trend is present in an LC, lead-ing to H > . . Therefore, when on the rising side ofthe pulse, one can expect that the rise will continue, POs in Swift/BAT GRBs ]2101234 l o g P ( f ) (a) Best fit: PL
PLPLCSBPLSBPL flat 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0log f [s ]10123456 l o g P ( f ) (b) Best fit: PLC
PLPLCSBPLSBPL flat3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0log f [s ]101234 l o g P ( f ) (c) Best fit: SBPL
PLPLCSBPLSBPL flat 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0log f [s ]0.50.00.51.01.52.02.53.0 l o g P ( f ) (d) Best fit: SBPL
PLPLCSBPLSBPL flat
Figure 2.
Representative PSD forms. (a) GRB 130907A exhibits a pure PL; (b) GRB 120116A—PLC; (c) GRB 070318—SBPL;(d) GRB 190821A—a PSD dominated by a QPO shape. The horizontal gray dashed lines mark the Poisson noise levels inferredfrom the measurements’ errors. The vertical solid lines denote f : blue—PLC; red—SBPL; black—SBPL with fixed β = 0 .Vertical dashed lines mark f break of: red—SBPL; black—SBPL with fixed β = 0 . Widths of the shaded regions symbolize thestandard errors of f and f break . and when on the decaying part, one shall expect it tofurther continue its decay. A low signal-to-noise ratiocan, however, allow the statistical fluctuations in formof white or otherwise anticorrelated noise to dominate,hence leading to H < . (short-term memory). Onthe other hand, some of the few cases yielding H < . also exhibit pronounced pulses. H can be applied toboth stationary and nonstationary processes, and bothtypes can exhibit short- and long-term memory. In manyGRBs the variability is clearly nonstationary, but thegoverning process may as well be antipersistent (i.e., becharacterized by H < . ). It therefore follows that thedistinction H ≶ . is not trivially connected merelywith the shape of the pulses or the signal-to-noise ratioof the LCs.Additionally, since there is a theoretical linear rela-tion between H and the PL index β P L , for the 187 GRBs with a PL PSD the H were extracted directlyfrom the β P L values. Their distribution is displayed inFig. 6(g), while Fig. 6(h) demonstrates that these val-ues are consistent with the H obtained with the otherthree algorithms. Indeed, these latter H exhibit a veryhigh correlation with β P L , in perfect agreement with thetheoretical predictions ( r = 0 . , 95% CI: (0 . , . ;Fig. 6(i)). This is, however, not the case when simi-lar inference is performed using the indices β P LC fromthe PLC model (Fig. 6(j)): there is a weak anticorrela-tion between the two, and the obtained H values do notfollow the theoretical predictions at all. This seems tobe the fault of the Poisson noise contaminating the sig-nal, as the H extraction algorithms (DFA, DWT, AWC)treat the time series as a whole, so the random fluctu-ations obscure the self-affinity that the algorithms rely0 Tarnopolski & Marchenko β PL P D F ( a ) β PLC ( b ) - τ [ sec ]( c )- - - β P D F ( d ) β ( e ) T break [ sec ]( f )- - - β β , SBP L ( g ) - - - β T b r ea k [ s e c ] ( h ) β T b r ea k [ s e c ] ( i ) β β , SBP L ( j ) β T b r ea k [ s e c ] ( k ) Figure 3.
Histograms of the β indices of the (a) PL and (b) PLC, (c) the MVTS ( >
64 ms ) of all applicable cases, the (d) low-and (e) high-frequency indices of the SBPL model, and (f) the break time scales of the SBPL model. The relations betweenSBPL parameters: (g) β − β , and T break vs. (h) β and (i) β . Vertical dashed lines in panels (g) and (h) highlight the regionswithin which correlation coefficients were calculated; (j) and (k) are the magnifications of the indicated regions. The correlationcoefficients and 95% confidence intervals (CIs) within the regions are (i) r = − . (95% CI: ( − . , − . ), (j) r = 0 . (95%CI: (0 . , . ), and (k) r = 0 . (95% CI: ( − . , . ). on. A meaningful inference of H from any signal is hencea subtle matter.Finally, we note there is nearly no correlation be-tween H and the parameters of the SBPL model( β , β , T break ). Recall that no conditions were imposedon the signal-to-noise ratio of the GRBs; we aimed toanalyze as much of the Swift catalog as was technicallypossible. 3.4. The
A − T plane
The
A − T plane is linked with the Hurst exponents,as well as the PSD form. In Fig. 7(a) displayed are the ( A , T ) locations of the 1150 GRBs in the 64 ms binning.The gray area in the background is the region of avail-ability of the PLC, i.e. above the pure PL line (redpoints in Fig. 1) and below the line T = / (highlightedwith a gray dashed line in Fig. 7(a)). The size of thepoint is proportional to the logarithm of the number POs in Swift/BAT GRBs Table 2.
Identified QPOs.
Number GRB name Period (sec) Comment6 GRB200107B . ± . ; . ± . harmonics,
34 GRB190821A . → . up-chirp75 GRB190103B . ± . constant102 GRB180823A . ± . constant122 GRB180626A . ± . ; . ± . harmonics,
190 GRB170823A . → . down-chirp212 GRB170524B . → . down-chirp232 GRB170205A . ± . constant250 GRB161202A . → . up-chirp251 GRB161129A . → . up-chirp252 GRB161117B . ± . constant272 GRB160824A . ± . ; . ± . ; . ± . harmonics, ∗
455 GRB140730A . ± . constant462 GRB140709B . ± . ; . ± . harmonics,
470 GRB140619A . ± . ; . ± . ; . ± . harmonics, ∗
496 GRB140323A . ± . ; . ± . harmonics,
551 GRB130812A . ± . constant618 GRB121209A . → . up-chirp622 GRB121125A . ± . ; . ± . harmonics,
632 GRB121014A . ± . constant701 GRB120116A . ± . constant756 GRB110422A . → . up-chirp777 GRB110207A . ± . constant783 GRB110107A . → . up-chirp805 GRB100924A . → . up-chirp914 GRB090709A . ± . ; . ± . harmonics,
945 GRB090404 . ± . constant963 GRB090102 . ± . constant1007 GRB080810 . ± . ; . ± . ; . ± . harmonics, . ± . ; . ± . harmonics, . ± . ; . ± . harmonics, . ± . constant1324 GRB050418 . → . up-chirp1335 GRB050306 . ± . constant Note —Approximately constant leading periods are given with corresponding uncertainties(indicated with the ’ ± ’ sign). Period ranges of the chirping signals are indicated with arrows,’ → ’, showing the direction of period evolution. For the harmonics, the closest integer ratiosare provided. ∗ These high-order ratios might as well be spurious, or be obscured due to uncertainties. of measurements n in the LC. There is essentially noprominent correlation between n and both A and T .Figures 7(b)–(d) display the A − T plane as well, butwith the size of the points indicating the index β of thebest-fit PSD. In case of pure PL (Fig. 7(b)) the relationis consistent with Fig. 1, i.e. steeper PSDs are locatedat lower values of A and—to some extent—lower val-ues of T as well. However, in case of PLC and SBPLmodels (Fig. 7(c) and (d), respectively) the situationseems to be reverted, with the steepest PSDs crowd-ing near the white noise point (1 , / ) . Note that very steep PSDs (i.e., with β (cid:38) − ) imply virtually no,or very little, variability on the associated time scales(cf. Sect. 3.1). The Poisson noise dominates such PLCcases, and a combination of white noises at two differentpower levels (at time scales < ∼ > T break , since T break ≈ τ insuch instances) occurs in SBPL, especially when β ≈ .After excluding these extremely steep instances, thereis no correlation between β and both A and T . Fi-nally, there are strong anticorrelations between A and log f ( r = − . for PLC; 95% CI: ( − . , − . , and r = − . for SBPL; 95% CI: ( − . , − . ), confirm-2 Tarnopolski & Marchenko
00 2020 4040 6060 8080 100100 120120 140140 160160Time (sec)1.02.04.08.016.0 P e r i o d ( s e c ) ) 0.00.891.772.663.554.445.326.217.17.99 P o w e r ( F l u x ) c o un t / s GRB 090709A (a)
00 1010 2020 3030 4040 5050Time (sec)1.02.04.08.0 P e r i o d ( s e c ) ) 0.00.220.440.660.881.11.321.541.76>=1.98 P o w e r ( F l u x ) c o un t / s GRB 120116A (b)
Figure 4. (a) Wavelet scalogram of GRB 090709A. There is a significant (>3 σ ) QPO, with a leading period ∼ sec, persistentthrough most of the LC. Another, slightly shorter component is visible at a period ∼ . sec. (b) GRB 120116A exhibits verysimilar features. ing that the level of Poisson noise contaminating theLCs primarily determines their locations in the A − T plane.To circumvent this, the LCs were binned accordingto the MVTS (effectively smoothing the LCs), and theresulting ( A , T ) locations are shown in Fig. 7(e). Thispicture is completely random, since many binned LCsturned out to contain only a handful of points. There-fore, in Figs. 7(f)–(h) are shown only those binned LCswith at least 50, 100, and 200 points, respectively. Thelonger the binned LCs, the more consistent their loca-tions are with the region of availability of PLC models.It is therefore crucial to highlight the importanceof sufficient length of a time series for calculating itslocation in the A − T plane robustly. While the val-ues of ( A , T ) in principal can be computed and used to characterize any experimental time series (evenextremely short ones), when dealing with stochasticprocesses, very short realizations will give essentiallya random outcome. Consider, e.g., a realization ofwhite noise with n values. It has an expected num-ber of turning points µ T = / ( n − , and standarddeviation σ T = (cid:112) (16 n − / (Kendall & Stuart1973). The distribution of T (for a fixed n ) willtend to a Gaussian parametrized by ( µ T , σ T ) . For n ∈ { , , , } , these are (rounded to the nearestinteger): ( µ T , σ T ) ∈ { (3 , , (15 , , (32 , , (665 , } .Translating to µ T = µ T /n , σ T = σ T /n , one gets { (0 . , . , (0 . , . , (0 . , . , (0 . , . } . Inother words, the expected value of T and its standarddeviation are asymptotically equal to / and zero, re-spectively, but for short time series σ T (or σ T ) can POs in Swift/BAT GRBs
00 1010 2020 3030 4040Time (sec)1.02.04.08.016.0 P e r i o d ( s e c ) ) 0.00.521.031.542.062.573.083.64.114.62 P o w e r ( F l u x ) c o un t / s GRB 121209A (a)
00 55 1010 1515 2020 2525 3030 3535Time (sec)1.02.04.08.0 P e r i o d ( s e c ) ) 0.00.180.370.550.730.911.11.281.46>=1.64 P o w e r ( F l u x ) c o un t / s GRB 090102 (b)
00 2020 4040 6060 8080 100100 120120Time (sec)1.02.04.08.016.0 P e r i o d ( s e c ) ) 0.00.761.522.283.053.814.575.336.096.85 P o w e r ( F l u x ) c o un t / s GRB 080810 (c)
Figure 5.
Exemplary wavelet scalograms of (a) an up-chirp in GRB 121209A, (b) a constant leading period in GRB 090102,and (c) a resonance in GRB 080810. Tarnopolski & Marchenko H DFA H DWT H AWC H ( a ) H DFA H DWT H AWC ( b ) H DFA P D F %( c ) H DWT %( d ) H AWC %( e ) H %( f ) H PL P D F %( g ) H H P L ( h ) β PL H ( i ) β PLC ( j ) Figure 6. (a)–(b) Time evolution of H . (a) The estimates for GRB 191001B are consistent with each other. The global valueof H = 0 . is indicated with a horizontal black line, and the gray region around it marks the uncertainty of ∆ H = 0 . . (b)For GRB 180111A no unambiguous estimate can be obtained. (c)–(f) Distributions of the Hurst exponents: (c)–(e) obtainedwith different methods (DFA, DWT, AWC, respectively), and (f) the final values, H , being the mean of the three. Vertical redlines denote H = 0 . , and the percentage of cases H > . is indicated in each panel. (g) Distribution of the Hurst exponentsobtained directly from the values of β PL . (h) Relation between the H obtained with other methods, and H PL obtained fromthe values of β PL . Diagonal dashed line marks the identity relation. (i)–(j) Hurst exponents obtained directly from the PLindices in case of a pure PL and PLC, respectively. Inclined dashed lines depict the theoretical relations. In (i), the overallcorrelation is r = 0 . (along the theoretical line when considered continuous; 95% CI: (0 . , . ), while in (j) r = − . (95%CI: ( − . , − . ), and does not follow well the predicted values. POs in Swift/BAT GRBs ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●●●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●●●● ●●●● ●●●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●●●● ●● ●● ●●●● ●● ●●●● ●●●●●● ●● ●●●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●●●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●●●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●●●● ●●●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●●●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●●●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●●●● ●● ●● ●● ●● ●●●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●●●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●●●● ●● ●● ●●●● ●● ●● ●● ●● ●●●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●●●●●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● PLPLCSBPL mean errorN = ( a ) ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●●●●●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●●●● ●● ●●●● ●● ●●●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● β PL = ( b ) ●● ●●●● ●● ●●●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●●●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●●●● ●● ●● ●● ●●●●●● ●● ●●●● ●● ●● ●●●● ●● ●●●● ●●●● ●● ●● ●●●● ●● ●●●● ●● ●● ●●●●●●●● ●● ●● ●● ●● ●●●●●● ●●●● ●● ●● ●●●● ●●●● ●● ●● ●●●● ●● ●●●● ●● ●● ●● ●●●●●● ●●●●●● ●●●● ●●●●●● ●●●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●●●● ●● ●●●● ●● ●●●● ●● ●● ●● ●●●●●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●●●● ●● ●● ●● ●● ●●●● ●●●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●●●● ●● ●●●●●●●● ●● ●● ●● ●●●● ●●●● ●●●● ●● ●● ●● ●● ●●●● ●●●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●●●● ●●●● ●● ●●●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●●●● ●● ●● ●●●● ●● ●●●●●● ●● ●● ●●●●●● ●● ●● ●●●● ●● ●● ●●●● ●●●●●●●● ●● ●●●●●● ●● ●● ●● ●● ●● ●● ●●●●●● ●● ●●●● ●●●●●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●●●● ●●●●●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●●●● ●●●● ●● ●● ●●●●●● ●● ●●●● ●●●●●●●● ●● ●● ●● ●● ●● ●●●●●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●●●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●●●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●●●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●●●● ●●●● ●● ●● ●●●● ●● ●●●● ●● ●● ●●●● ●● ●● ●● ●●●● ●●●● ●● ●●●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●●●●●● ●● ●● ●● ●● ●●●● ●●●●●● ●● ●● ●● ●●●● ●● ●●●●●● ●● ●● ●● ●●●● ●● ●●●● ●● ●●●● ●● ●● ●●●● ●● ●● ●● ●● ●● β PLC = ( c ) ●● ●● ●●●●●● ●● ●● ●●●●●● ●● ●● ●● ●●●● ●●●● ●●●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●●●●●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●●●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●●●● ●●●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●●●●●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●●●● ●●●● ●● ●● ●● ●● ●●●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●●●● ●● ●● ●● ●● ●● ●● ●●●●●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●●●● ●● ●● ●●●●●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●●●● ●● ●● ●●●●●● ●●●● ●● ●● ●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●●●●●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●●●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● β = ( d ) ( e ) ( f ) ( g ) ( h ) Figure 7.
The
A − T plane. The mean errors are represented in the upper parts of the plots. (a) The point size is proportionalto the logarithms of the length of the time series, and indicated in the lower right corner. The gray area is the region of availabilityfor PLC type of PSDs (cf. Fig. 1). (b)–(d) The sizes of the points indicate the values of index β , as indicated in the lower rightcorner of each panel. (e)–(h) The ( A , T ) locations of the LCs binned according to the MVTS values. In panel (e) locations ofall such binned LCs are represented, while in (f), (g), and (h) displayed are binned LCs with at least 50, 100, and 200 points,respectively. Tarnopolski & Marchenko constitute a substantial fraction of µ T (or µ T ). For in-stance, a time series with only n = 6 will have 33%, 42%,and 17% chance of yielding T = 2 , , , respectively.Such trend is indeed observed in Figs. 7(e)–(h): thelonger the binned LCs, the more constrained to the re-gion of availability they are. Therefore, the Poissonnoise is a serious obstacle in analyzing the variabilityof astronomical time series, GRBs in particular. Recallthat, according to Sect. 3.1, the MVTS does not bearany physical meaning (at least for the Swift sample in-vestigated herein), as it very strongly depends on thePoisson noise level inferred from the individual measure-ments’ uncertainties. Finally, given all the above con-siderations, the A − T plane, while potentially usefulin classifying LCs (cf. Żywucka et al. 2020; Tarnopol-ski et al. 2020), does not hint at any clustering of longGRBs into more than one group (other than an over-concentration of flat PSDs at (1 , / ) , likely owing tolow signal-to-noise ratio), consistent with the findingsof Jespersen et al. (2020).Finally, the coarse graining (Eq. (19)) was applied tothe 64 ms binned LCs, using d ∈ { , . . . , } , with theintent to possibly obtain separated clusters for some par-ticular value of d . Such an approach was successful incase of economic and physiological data (Zunino et al.2017) with small values of d . For the GRBs herein,though, we mostly observe variations of Fig. 7(a) forsmall d , and higher d (leading to relatively short coarse-grained sequences) resembling Fig. 7(e) when the result-ing time series are too short. We therefore again do notobtain any clustering of long GRBs in the A − T plane.3.5.
The E restpeak − β relation Dichiara et al. (2016) studied 123 GRBs observed byvarious instruments. They found a statistically signifi-cant anticorrelation ( r = − . , 95% CI: ( − . , − . ,using their published data) between the rest-frame peakenergy ( log E restpeak ) and the PL index (denoted by themwith α ; we continue using the symbol β hereinafter).We gathered the E peak values of the Band , Comp ,and
Sbpl spectral fits (Kaneko et al. 2006; Gruberet al. 2014) from the Fermi/GBM catalog (Gruberet al. 2014; von Kienlin et al. 2014; von Kienlin et al.2020; Narayana Bhat et al. 2016) by cross-matching thespatio-temporal localizations of the Swift and FermiGRBs, and complemented them with redshift mea-surements when available. The specific E peak val- Note this
Sbpl is in a different context than the PSD fromEq. (6). https://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html ◼◼ ◼◼◼◼ ◼◼ ◼ ◼◼◼ ◼◼◼◼ ◼ ◼◼◼ ◼◼▲ ▲▲▲ ▲ ▲ ▲▲▲ ▲ ▲▲▲ ▲▲ ▲▲▲▲ β PL , β PLC , β l og E pea k r e s t [ k e V ] r = - r = - r = - r = - r = - r = - Figure 8.
The relation between E restpeak and the β indices ob-tained from PL (blue circles), PLC (red squares), and SBPLmodels (green triangles), and correlation coefficients high-lighted with respective colors, where purple refers to the jointset β = β PL ∪ β PLC , dirty yellow to β = β PLC ∪ β , and blackto β = β PL ∪ β PLC ∪ β . ues for the common GRBs were chosen based on the flnc_best_fitting_model entry from the Fermi cat-alog. We consider β P L , β P LC , and β . Eventually, weend with 12, 22, and 20 entries, respectively. They aredisplayed in Fig. 8.An overall anti-correlation between the log E restpeak and β values can be seen. It is strongest in case of thePLC fits, r = − . (95% CI: ( − . , − . )—evenstronger than in Dichiara et al. (2016). It is weaker forthe pure PL case ( r = − . ; 95% CI: ( − . , . —consistent with a lack of correlation), and similar whenthe set β P L ∪ β P LC is considered ( r = − . , 95% CI: ( − . , − . ). The weakest correlation ( r = − . ,95% CI: ( − . , . —consistent with a lack of corre-lation) is attained for β , and for the whole set of β itis a moderate r = − . (95% CI: ( − . , . —barelyconsistent with a lack of correlation). We note, how-ever, that (i) our sample is ∼ E restpeak values there are GRBs significantlycontaminated by the Poisson noise component as well,which likely affects the log E restpeak − β relation. Since the95% CIs for r contain (at least marginally) the value de-scribing the sample of Dichiara et al. (2016), we do notreject the existence of such correlation (although somesubsamples of our β allow a lack of correlation, too);however, a bigger sample is definitely required to con-strain the relation further, which is outside the scope ofthis paper. POs in Swift/BAT GRBs
The L iso − f relation The peak isotropic luminosity is computed as (Schae-fer 2007) L iso = 4 πd L ( z ) P (cid:82) E / (1+ z ) E / (1+ z ) EN ( E ) dE (cid:82) E max E min EN ( E ) dE , (20)where d L ( z ) is the luminosity distance to a sourceat redshift z , calculated using the latest cosmologi-cal parameters within a flat Λ CDM model (PlanckCollaboration et al. 2020): H = 67 . − Mpc − , Ω m = 0 . , Ω Λ = 0 . ; P is the energy flux(in units of erg cm − s − ) over the time range ofthe peak flux of the GRB ; and N ( E ) is the spec-tral model over the time range of the peak flux (ex-pressed in units of ph cm − s − keV − ): Plaw , Band , Comp , or
Sbpl , chosen for each GRB according to the pflx_best_fitting_model entry from the Fermi/GBMcatalog, and with parameters from therein (Kanekoet al. 2006; Gruber et al. 2014). The integrationlimits are set using { E , E } = { , } keV , and { E min , E max } = { , } keV is the observing band-width of Fermi/GBM (cf. Sect. 3.5). The uncertaintiesof log L iso are obtained by bootstrapping the parametersof N ( E ) (Ukwatta et al. 2010).We obtain 81 GRBs with L iso estimates. Hereinafterwe employ 25 GRBs that have PSDs best fitted by aPLC, and 26 with an SBPL form. We compare ourresults (Fig. 9) with those of Ukwatta et al. (2011) re-garding the relation between redshift-corrected charac-teristic frequency, (1+ z ) f , and L iso . Our sample yields r = 0 . (95% CI: ( − . , . ) and r = 0 . (95% CI: (0 . , . ) for the PLC and all 51 cases, respectively,while Ukwatta et al. (2011) obtained r = 0 . (95% CI: (0 . , . ) with a sample of 58 GRBs. Our correlationis marginally significant, although weaker and slightlyinconsistent with that of Ukwatta et al. (2011).The discrepancy lies in (i) different models employed:Ukwatta et al. (2011) fitted the PSDs with a piecewiselinear function, with a nonzero slope for f < f , anda constant level for f > f , while Eq. (5) describes asmooth transition. Therefore, our f has a slightly dif-ferent meaning than the f of Ukwatta et al. (2011).Moreover, (ii) we did not impose any constraints on theinitial sample, as we aimed to analyze the whole Swiftcatalog, so that more noisy GRBs might be adding vari-ance to the L iso − f relation. Note that our sample is The pflx_xxxx_ergflux entries from the Fermi/GBM cata-log were employed, where xxxx stands for
Band , Comp , PLaw ,or
Sbpl . ▲ ▲▲ ▲▲▲▲▲▲ ▲▲▲▲ ▲▲▲▲ ▲▲▲▲▲▲▲ ▲▲◼ ◼◼ ◼◼ ◼◼ ◼ ◼◼◼ ◼ ◼ ◼◼◼◼◼◼ ◼ ◼◼◼ ◼◼ - - - f ( + z ) [ / s ] l og L i s o [ e r g / s ] r = r = r = Figure 9.
The relation between L iso and the redshift-corrected frequency (1 + z ) f . Red squares denote the 25GRBs whose PSDs are best fitted with a PLC; green trianglescorrespond to the remaining 26 SBPL fits. The correspond-ing correlation coefficients r are indicated with respectivecolors. about he same size (51 vs. 58) as that of Ukwatta et al.(2011). DISCUSSION4.1.
PSDs
The PSDs of Swift GRBs examined herein come inthree shapes: a pure PL, a PL with Poisson noise, and anSBPL. The PL case includes flat PSDs, i.e. white noise,which are quite abundant in our sample. They are char-acteristic of GRBs which are dim, i.e. have a low signal-to-noise ratio, and hence are dominated by the Pois-son statistics. The PSDs which are colored noise haveindices β (cid:46) , so are generally flatter than red noise.When the Poisson noise component becomes significant,the β indices in the PLC case rise as well, falling in therange (cid:46) β (cid:46) . There is also a non-negligible fractionof steeper PSDs, with (cid:46) β (cid:46) , and a few instances of β > . The latter basically implies no variability on thecorresponding time scales, since it means that for everydecade in frequency there is a > orders of magnitudechange in the power. Such steep PLC models ought tobe considered artifacts, unless proven otherwise, sincewe observed they occur when the binned PSD exhibitsjust one or two points at low frequencies (reflecting thelength of the LC), greatly above the Poisson noise level,and hence the fitting results become severely biased.The mode of the β distribution in the PLC model isat 1.8, somewhat close to the value / expected in theturbulence model. Finally, the SBPL model has its low-frequency index β gathered around zero, and not ex-ceeding 2, while the high-frequency index β mostly fallsin the range (cid:46) β (cid:46) . Such steepness is more reliablethan in the PLC case, since it occurs at time scales lo-8 Tarnopolski & Marchenko cated between the region of low-frequency PL part (wellabove the Poisson noise level, so detected confidently),and the high-frequency region of Poisson noise domi-nance. This shows that there is a characteristic timescale, locating the break T break between the two PL partswith β and β , on the order of − s, and hence im-plies there are either two dominant processes workingin the progenitors at the emission sight, or—when β is very steep—the variability at the intermediate timescales is essentially wiped out. The latter can be indica-tive of a sharp cut-off corresponding to, e.g., the inneredge of the accretion disk.4.2. QPOs
Another feature that was uncovered in some numberof GRBs are QPOs—either with an approximately con-stant leading period, or in the form of up- or down-chirps. As noted in Sect. 1, there have not been many re-ports on QPOs in GRBs, hence the instances gathered inTable 2 are remarkably numerous. The already proposedgeneration mechanisms (MRI (Masada et al. 2007); pre-cessing magnetic field (Ziaeepour & Gardner 2011)) canbe complemented with some models employed for activegalactic nuclei (AGNs), since both GRBs and AGNs of-ten exhibit striking similarities (Wang et al. 2014; Wuet al. 2016; Deng et al. 2016). In the simplest scenario,association of the break time scale with the viscous timescale of an accretion disk, coupled with the Keplerianmotion on a circular orbit around the newly formingBH (Mohan & Mangalam 2014; Żywucka et al. 2020)can explain the PSD breaks in GRBs as well. Sincea relativistic two-body problem (in both Schwarzschildand Kerr metrics) allows for an inspiral (which is im-possible in the Newtonian framework), occuring in a fi-nite time (Levin & Perez-Giz 2008), a fragmented ac-cretion disk could result in QPOs, lasting several cy-cles, and possibly chirping signals as well. The orbitalperiod at the innermost stable circular orbit (ISCO) is T ISCO = 12 π √ GM • /c ≈ . · − M • /M (cid:12) [s] (Har-tle 2003), giving for a typical stellar-mass BH with M • = 10 M (cid:12) a period of T ISCO = 4 . —an order ofmagnitude smaller than the employed binning (64 ms),and smaller than the detected QPO time scales. Theaccretion disk might actually be truncated, with an in-ner edge at a radius r = kr ISCO ( k (cid:62) ), in case of For a Kerr BH with dimensionless spin a > , T ISCO iseven smaller (Bardeen et al. 1972), up to a factor of √ for amaximally rotating BH. However, a period on the order of miliseconds is comparableto the QPO discovered in a short BATSE GRB (Zhilyaev & Du-binovska 2009), hence an inspiral inside the ISCO might, at leastin some cases, lead to a QPO. which the period T = k / T ISCO . E.g., k = 20 changesthe 4.5 ms period to 0.4 s—still a few times shorter thanthe shortest QPO reported in Table 2, which would re-quire k ≈ . A plausible range of the cut-off can extendup to k ∼ , giving T = 4 . , consistent with some ofthe QPOs in Table 2. While some QPOs might thereforebe indeed due to a truncated disk, it seems unlikely to bea universal explanation. Several more sophisticated or-bital models (oscillatory modes in accretion disks, boththin and thick; relativistic precession; tidal disruption(TD) models; warped disk; etc.) were considered in thecontext of X-ray binaries and microquasars (Török et al.2011; Kotrlová et al. 2020, and references therein), andpredict the existence of resonant QPOs of a wide rangeof frequency ratios (cf. fig. 3 in Kotrlová et al. 2020).However, since M • M (cid:12) f U Hz (cid:38) (cf. fig. 2 in Kotrlová et al.2020), where f U is the higher frequency forming the res-onant ratio f U /f L > , and the ratios in Table 2 are ofthe order of unity, the time scales of the QPOs are ofthe order of 0.01 s. The TD model, in turn, predictsthat inhomogeneities with density ρ in the disk will bestretched and disrupted at the Roche limit, and eventu-ally lead to modulation with a period T T D ∼ ( Gρ ) − / (Čadež et al. 2008; Kostić et al. 2009; Török et al. 2011).Assuming rocky material (planetary/cometary debris)with ρ = 5500 kg m − (Earth’s density), T T D = 1650 s .Lower ρ gives higher T T D . To match the 10 s QPOperiod, ρ ∼ kg m − , which is an unlikely possibility.The relativistic motion around Kerr BHs can lead toeven more complicated, three dimensional orbits, giv-ing rise to breaks as well as QPOs (Rana & Mangalam2019, 2020). The low-frequency QPOs, with time scales −
10 s , in fact arise naturally in this setup, hence ap-pear to be a probable description for the QPOs in GRBs(cf. table 10 in Rana & Mangalam 2020), and accountfor resonant QPOs as well. Finally, the Lense-Thirringprecession leads to frequencies matching the QPOs whenthe disk is truncated at k (cid:38) (cf. fig. 5 in Ingram et al.2009), an order of magnitude smaller than in the aboveorbital models.A detailed picture was painted with the use of magne-tohydrodynamical simulations of a forced perturbancewithin a magnetized accretion disk (Pétri 2005). Thisscenario seems plausible, since among the 10 chirps weidentified, 8 are up-chirps, i.e. with a decreasing pe-riod. The two down-chirps would require a differentmechanism, though. An appealing one might be dueto a helical jet, which when applied to AGNs, results indown-chirps (Mohan & Mangalam 2015). On the otherhand, accretion flows in which MRI dominates do not ex-hibit QPOs, since MRI turbulence destroys coherence,hence nullifies QPOs, while in a magnetically choked ac- POs in Swift/BAT GRBs ∼ GM • /c ≈ .
004 s for M • = 10 M (cid:12) (see McKinney et al. 2012 for details), i.e.again too low to match the QPOs detected in GRBs.Finally, oscillations of the shock front can give rise toQPOs with Hz and sub-Hz frequencies, i.e. could pos-sibly explain some of the QPOs in GRBs as well (Iyeret al. 2015; Palit et al. 2019, and references therein).GRB emission is of synchrotron nature (Burgess et al.2020; Ghisellini et al. 2020), and comes from the shocksin relativistic jets. We discussed possibilities of gener-ating QPOs of appropriate periods in the surroundingdisk (except for the shock front oscillations), assumingthey transfer with a one-to-one correspondence to the jetvia disk-jet coupling. This might not be entirely true,and/or the observed QPOs might as well arise due to acombination of more than one effect.4.3. Persistence
A conservative methodology used to estimate theHurst exponents revealed that 93% of GRBs are char-acterized by
H > . , meaning they exhibit long-termmemory, or persistence. Recall that the value of H canbe attributed to both stationary (e.g., colored noise with β < or fractional Gaussian noise) and nonstation-ary (e.g., PL with β > or fractional Brownian mo-tion), and hence the notion of smoothness it quantifiesis broader that (non)stationarity. We therefore showedthat the autocorrelations in the GRBs’ variability per-sist throughout the LCs, hence—so to speak—the ran-dom component embedded in the γ -ray signal is struc-tured on a fundamental level.4.4. A − T plane
We attempted also to classify the GRB prompt LCsin the recently developed
A − T plane. We consideredhere only long GRBs, with T > . , and since thedichotomy between short and long GRBs is well estab-lished, we expected to verify with yet another approachthe existence of the presumed third, intermediate classof GRBs. We indeed observed hints of clustering in tworegions of the A − T plane. However, one of the groupstends to gather around the point (1 , / ) , i.e. the loca-tion of white noise processes. When examined in de-pendence on the β indices of the PSDs, it turned outthat area is occupied by PLC and SBPL models withvery high values of β —i.e., those GRBs predominantlyexhibitng white noise PSDs. As noted above, such casesare either due to the objects being dim, spurious fits,or a nontrivial coexistence of two white noise processesrepresented by components with different powers. Wetherefore conclude there are no unambiguous signs of a sub-classification of long GRBs’ LCs. While this is nota definite proof of the non-existence of the third class onits own, it is consistent with other works tackling this is-sue more directly (Tarnopolski 2016a, 2019a,c; Jespersenet al. 2020). 4.5. Correlations
Finally, we critically revisited the E restpeak − β and L iso − f relations, confirming their existence in thewhole Swift catalog. Comparing with the correspond-ing results of Dichiara et al. (2016) and Ukwatta et al.(2011), we obtained slightly stronger and significantlyweaker, respectively, correlations in the appropriate re-lations. In case of the L iso − f relation, the overallpositive correlation might in fact be a simple luminosityeffect: the brighter the source, the higher its signal-to-noise ratio, hence the lower the contamination of thesignal with Poisson noise. This then implies that thelocation of the critical frequency, f , is shifted to higherfrequencies, making the white noise component less sig-nificant. This is backed up also by a very strong correla-tion ( r = 0 . ) between the Poisson noise levels obtainedfrom fits and directly from the uncertainties of the LCmeasurements.The E restpeak − β relation, in turn, connects the spectralenergetic properties of a GRB with a characteristic of anLC. An anticorrelation between the two might be a signof a common, physical parameter governing the valuesof E restpeak and β , e.g. the Γ factor. SUMMARY1. The PSDs with PL and PLC shapes are broadlyconsistent with the / Kolmogorov law expectedin a fully developed turbulence. Several cases of anSBPL model were also obtained, with break timescales on the order of 1–100 s.2. We reported on QPOs detected in the waveletscalograms of 34 GRBs: 10 chirping signals (8up-chirps and 2 down-chirps), and 24 QPOs withconstant leading periods (13 with a single QPO,8 with two coexisting QPOs, and 3 triple QPOs).In particular, we confirmed a persistent QPO withan ∼ H > . , i.e.they express long-term memory in the promptLCs, not connected trivially with the PSD fea-tures.0 Tarnopolski & Marchenko
4. The
A − T plane did not reveal any meaningfulseparation of long GRBs into subclasses.5. The E restpeak − β and L iso − f relations were con-firmed, though the latter seems to be a straight-forward result of the luminosity effect. We thank Agnieszka Janiuk for useful comments onthe draft, and the anonymous reviewer for helpful sug-gestions. M.T. acknowledges support by the Polish Na-tional Science Center (NSC) through the OPUS grant2017/25/B/ST9/01208. V.M. was supported by theNSC grant 2016/22/E/ST9/00061. Software:
Mathematica (v10.4; Wolfram Re-search 2016),
SciPy (v1.1.0; Virtanen et al. 2020), wavepal (Lenoir & Crucifix 2018a,b) .REFERENCES
Akaike, H. 1974, IEEE Transactions on Automatic Control,19, 716Bandt, C., & Shiha, F. 2007, Journal of Time SeriesAnalysis, 28, 646, doi: 10.1111/j.1467-9892.2007.00528.xBarat, C., Chambon, G., Hurley, K., et al. 1979, A&A, 79,L24Bardeen, J. M., Press, W. H., & Teukolsky, S. A. 1972,ApJ, 178, 347Belli, B. M. 1992, ApJ, 393, 266, doi: 10.1086/171503Beloborodov, A. M., Stern, B. E., & Svensson, R. 1998,ApJL, 508, L25, doi: 10.1086/311710—. 2000, ApJ, 535, 158, doi: 10.1086/308836Berger, E. 2011, NewAR, 55, 1,doi: 10.1016/j.newar.2010.10.001Beskin, G., Karpov, S., Bondar, S., et al. 2010, ApJL, 719,L10, doi: 10.1088/2041-8205/719/1/L10Boçi, S., & Hafizi, M. 2018, Mem. Soc. Astron. Italiana, 89,289. https://arxiv.org/abs/1802.01695Brockwell, P. J., & Davis, R. A. 1996, Time Series: Theoryand Methods, 2nd ed. (Springer-Verlag New York)Bromberg, O., Nakar, E., Piran, T., & Sari, R. 2013, ApJ,764, 179, doi: 10.1088/0004-637X/764/2/179Burgess, J. M., Bégué, D., Greiner, J., et al. 2020, NatureAstronomy, 4, 174, doi: 10.1038/s41550-019-0911-zBurnham, K. P., & Anderson, D. R. 2004, SociologicalMethods & Research, 33, 261Cano, Z., Wang, S.-W., Dai, Z.-G., & Wu, X.-F. 2017,Advances in Astronomy, 2017, 8929054Cenko, S. B., Butler, N. R., Ofek, E. O., et al. 2010, AJ,140, 224, doi: 10.1088/0004-6256/140/1/224Chang, H.-Y., & Yi, I. 2000, ApJL, 542, L17,doi: 10.1086/312915Dainotti, M. G., & Amati, L. 2018, PASP, 130, 051001,doi: 10.1088/1538-3873/aaa8d7Dainotti, M. G., & Del Vecchio, R. 2017, NewAR, 77, 23,doi: 10.1016/j.newar.2017.04.001 Dainotti, M. G., Del Vecchio, R., & Tarnopolski, M. 2018,Advances in Astronomy, 2018, 4969503,doi: 10.1155/2018/4969503de Luca, A., Esposito, P., Israel, G. L., et al. 2010, MNRAS,402, 1870, doi: 10.1111/j.1365-2966.2009.16012.xDeng, M., & Schaefer, B. E. 1997, ApJ, 491, 720,doi: 10.1086/305001Deng, W., Zhang, H., Zhang, B., & Li, H. 2016, ApJL, 821,L12, doi: 10.3847/2041-8205/821/1/L12Dichiara, S., Guidorzi, C., Amati, L., & Frontera, F. 2013a,MNRAS, 431, 3608, doi: 10.1093/mnras/stt445Dichiara, S., Guidorzi, C., Amati, L., Frontera, F., &Margutti, R. 2016, A&A, 589, A97,doi: 10.1051/0004-6361/201527635Dichiara, S., Guidorzi, C., Frontera, F., & Amati, L. 2013b,ApJ, 777, 132, doi: 10.1088/0004-637X/777/2/132Fenimore, E. E., Klebesadel, R. W., & Laros, J. G. 1996,ApJ, 460, 964, doi: 10.1086/177024Fishman, G. J., Meegan, C. A., Wilson, R. B., et al. 1994,ApJS, 92, 229, doi: 10.1086/191968Ghisellini, G., Ghirlanda, G., Oganesyan, G., et al. 2020,A&A, 636, A82, doi: 10.1051/0004-6361/201937244Giblin, T. W., Kouveliotou, C., & van Paradijs, J. 1998, inAmerican Institute of Physics Conference Series, Vol.428, Gamma-Ray Bursts, 4th Hunstville Symposium, ed.C. A. Meegan, R. D. Preece, & T. M. Koshut, 241–245Golenetskii, S., Aptekar, R., Mazets, E., et al. 2009, GRBCoordinates Network, 9647, 1Gotz, D., Mereghetti, S., von Kienlin, A., & Beck, M. 2009,GRB Coordinates Network, 9649, 1Gruber, D., Goldstein, A., Weller von Ahlefeld, V., et al.2014, ApJS, 211, 12, doi: 10.1088/0067-0049/211/1/12Guidorzi, C., Dichiara, S., & Amati, L. 2016, A&A, 589,A98, doi: 10.1051/0004-6361/201527642Guidorzi, C., Margutti, R., Amati, L., et al. 2012, MNRAS,422, 1785, doi: 10.1111/j.1365-2966.2012.20758.xHartle, J. B. 2003, Gravity: An Introduction to Einstein’sGeneral Relativity (Addison-Wesley)