Light speed variation from gamma ray bursts: criteria for low energy photons
aa r X i v : . [ a s t r o - ph . H E ] O c t Eur. Phys. J. C manuscript No. (will be inserted by the editor)
Light speed variation from gamma ray bursts: criteria for low energyphotons
Yue Liu , Bo-Qiang Ma a,1,2,3,4 School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing, China Center for High Energy Physics, Peking University, Beijing 100871, China Center for History and Philosophy of Science, Peking University, Beijing 100871, ChinaReceived: date / Accepted: date
Abstract
We examine a method to detect the light speedvariation from gamma ray burst data observed by the FermiGamma-ray Space Telescope (FGST). We suggest new cri-teria to determine the characteristic time for low energy pho-tons by the energy curve and the average energy curve re-spectively, and obtain similar results compared with thosefrom the light curve. We offer a new criterion with both thelight curve and the average energy curve to determine thecharacteristic time for low energy photons. We then applythe new criteria to the GBM NaI data, the GBM BGO data,and the LAT LLE data, and obtain consistent results for threedifferent sets of low energy photons from different FERMIdetectors.
According to Einstein’s relativity, the speed of light is a con-stant c in free space. However, it is speculated that the ef-fect of quantum gravity may bring a tiny correction to thelight speed of the order E / E Pl , where E is the photon en-ergy and E Pl = p ¯ hc / G ≈ . × GeV is the Planckenergy. The matter effect of the universe may also cause amodification to the light speed in the cosmological space.It is very difficult to measure the light speed variation byordinary experiments on Earth because such a variation of c is extremely small. One approach to solve this problemis to focus on photons from far away astrophysical objects.Amelino-Camelia et al. first suggested detecting light speedvariation due to the Lorentz invariance violation (LV) fromgamma-ray bursts (GRBs) [1, 2]. Gamma-ray bursts are ex-tremely energetic and rather quick processes in the universe.During a GRB, photons with different energies are emittedfrom the source and these photons travel through the cosmo-logical space to reach the detectors. Some of the GRBs are a e-mail: [email protected] so energetic that they emit high energy photons with energy E high , src &
30 GeV at the source (src). Meanwhile, a GRBalso emits numerous low energy photons: E low , src . E high , src and E low , src are still very small comparedto the Planck energy, the little difference in the light speedcan be accumulated during the very long distance of travel.The tiny light speed variation leads to the different time oftravel, so it contributes to the difference between the arrivaltimes of high energy photons and low energy photons.Ellis et al. first analyzed the GRB data of photons aimingto detect quantum gravity induced light speed variance [3],and they also developed a robust method to collect high en-ergy photon data from different observations and analyzedthem collectively [4]. In later work, refs. [5, 6] combinedhigh energy photons from different GRBs of the Fermi tele-scope, and analyzed them collectively according to the methodprescribed in ref. [4]. Reference [7] used the first main peakin the low energy light curve rather than the trigger time asthe low energy characteristic time. Some progress has beenmade in References [5–10] on detecting the light speed vari-ation from analysis of energetic photon events detected bythe Fermi Gamma-ray Space Telescope (FGST) [11, 12].By analyzing the time lags between energetic photon eventsand the corresponding low energy photon signals for sev-eral GRBs with known redshifts, a regularity was found forthe time lags between photons of different energies. Such aregularity suggests a tiny light speed variation of the form v ( E ) = c ( − E / E LV ) , where E LV ≃ . × GeV. In thiswork we check this method carefully, mainly focusing onthe determination of a remarkable low energy photon signalfor each GRB in the analysis. Besides light curves, we applymore criteria on choosing a low energy characteristic timefrom the photon energy released per unit time and averageenergy of photons. We thus offer a new criterion to includethe average energy curve in addition to the light curve to de-termine the characteristic time for low energy photons. Such a criterion not only includes the widely used light curve butalso takes into account the changes of GRB energy distribu-tion. We then apply the new criteria to the GBM NaI data,the GBM BGO data, and the LAT LLE data, and obtain simi-lar results for three different sets of low energy photons fromdifferent FERMI detectors. With the new analysis, we arriveat the results that are consistent with each other.In sect. 2, we review the basic method to detect the lightspeed variation from GRBs. The usual criterion is based onthe light curve of a GRB (sect. 2). In sect. 3, we providemore options on the criterion for the characteristic time oflow energy photons. We use more criteria considering dif-ferent aspects of intrinsic nature of the source. In sect. 3.1,we focus on the energy released. In sect. 3.2, we use theaverage energy per photon of an energy band. In sect. 3.3,we combine the criterion of the light curve in sect. 2 andthe criterion of the average energy in sect. 3.2 and give ourrecommendation of the characteristic times. In sect. 4, weapply the new criteria to the GBM NaI data, the GBM BGOdata and the LAT LLE data, and obtain consistent resultsfor three different sets of low energy photons. In this sectionwe also provide the estimation on uncertainties. Section 5serves as a summary and conclusion. E high , src for the high energy photonand E low , src for the low energy photon, in the source frame.These two photons need not to be emitted at the same time,so we use ∆ t in to represent the intrinsic time lag between thehigh energy photon and the low energy photon when theyare emitted, i.e., ∆ t in = t high , src − t low , src , where t high ( low ) , src is the emitting time of the high (low) energy photon in thesource reference frame. For each photon, it travels throughthe cosmological space and is captured by a detector, and itsobserved energy can be written as E obs = E src / ( + z ) , (1)where z is the redshift of the GRB, and E src is the energy ofphoton when it is emitted from the source. So the energiesof the two photons can be written as E high , obs = E high , src / ( + z ) , (2) and E low , obs = E low , src / ( + z ) . (3)The observed time lag between the high energy photon andthe low energy photon is caused by two factors: the intrin-sic time lag ( ∆ t in ) at the source, and the time lag caused bythe difference of light speed (e.g., the time lag caused bythe LV effect or by matter effect of the universe), which isrepresented by ∆ t LV here. Thus we have ∆ t obs = ∆ t LV + ∆ t in ( + z ) . (4)Now we consider the general form of dispersion relation ofa photon. For a photon with energy E , if E ≪ E Pl , the LVeffect leads to a modified form of dispersion relation E = p c (cid:20) − s n (cid:18) pcE LV , n (cid:19) n (cid:21) , (5)where s n = ± s n = −
1) or slower ( s n = +
1) than the lowenergy photon, and E LV , n denotes the n th-order Lorentz in-variance violation scale to be determined.We assume n = v = ∂ E / ∂ p , and obtain v ( E ) = c (cid:20) − s (cid:18) EE LV (cid:19)(cid:21) , (6)where E LV represents E LV , .Once we obtain the relationship between speed and en-ergy, we can apply it to the two photons mentioned above.To calculate the time lag, ∆ t LV , we use the Λ CDM Uni-verse model that the universe consists of matter and dark en-ergy (cosmological constant). It can be proved that ∆ t LV (inthe observer reference frame) can be written as [13, 14] ∆ t LV = ( + z ) KE LV , (7)with K = s E high , obs − E low , obs H ( + z ) Z z ( + z ′ ) d z ′ p Ω m ( + z ′ ) + Ω Λ , (8)where H = . ± . − Mpc − is the present day Hub-ble constant, and Ω m = . + . − . and Ω Λ = . + . − . are the matter density and dark energy density [15].In the derivation above, ∆ t in and E LV are unknown butother parameters: z , E low , obs , E high , obs and ∆ t obs are knownin principle. What we should do next is to obtain E LV basedon these parameters and some reasonable assumptions.In refs. [5, 6], a method to look for the light speed vari-ation from GRBs was introduced. According to eq. 7, thetime lag caused by the light speed variation ( ∆ t LV ) can beextracted by K , a factor that depends on the low and high photon energies and the redshift of the GRB. What we canobtain from data is the observed time lag between high andlow energy photons ∆ t obs = t high , obs − t low , obs , (9)which is the sum of ∆ t LV and the intrinsic emission timelag ( ∆ t in ) in eq. 4. Then we get ∆ t obs + z = KE LV + ∆ t in . (10)If LV really exists, we can expect that there is a relationshipbetween K and ∆ t obs due to the light speed variation, whichinfers that E LV is of certain value, so our method is to plot theevents in a ∆ t obs / ( + z ) versus K plot and find out whetherthere is a correlation between K and ∆ t obs / ( + z ) .What we do next is based on the data from the FERMItelescope. The FERMI telescope consists of the Fermi LargeArea Telescope (LAT) [11, 16, 17] and the Gamma-RayBurst Monitor (GBM) [12, 18] detectors. LAT is mainlyused to record high energy events while GBM is used torecord low energy events. GBM consists of 12 Sodium Io-dide (NaI) detectors and 2 Bismuth Germanate (BGO) de-tectors. The energy range of NaI detectors is about 8 − µ s relative to spacecraft time [20,21]. Therefore, we can treat these two instruments as well-synchronized. We choose the photons whose intrinsic ener-gies are higher than 30 GeV at source as high energy eventsand the photons detected by the GBM NaI detectors as thelow energy events. From the LAT telescope, the energies andthe observed arrival time of the high energy events can beread directly. Although the LAT data trace the direction ofthe photon events, the GBM detectors do not record the di-rections. Thus, the background contamination can be big.Fortunately, we focus on the characteristic time when thelight curve deviation from the background is significant. Sincethe background is expected to be stable over time, it merelyadds a constant pedestal to the light curve without inducingany time-varying features. We therefore do not perform thebackground subtraction below because the background ac-tually does not influence the determination of t low , as shownlater in sect. 4.2.We search the high energy photons in all the GRBs de-tected by the FERMI telescope before 2016.12.31 and listthem in table 1. The latest Pass 8 data [22, 23] are usedto read the energies and arrival times of photons. Here wechoose those photons whose energies are higher than 30 GeVwhen they are emitted at the source (after they are emit-ted, the redshift effect reduces their observed energies). Informer works, the selection rule of high energy photons isover 10 GeV in the observer frame in refs. [5–8] and it ischanged to over 40 GeV in the source frame in ref. [9]. We here choose photons with energies over 30 GeV at source asan optional choice with some subsidiary information.As the number of photons with such a high energy froma single GRB is quite small, we need to combine differentGRBs together. As the sample GRBs we choose must haveknown redshifts and photons with enough high energies, theGRBs that meet these restrictions are: GRB 080916C, GRB090510, GRB 090902B, GRB 090926A, GRB 100414A,GRB 130427A, GRB 140619B and GRB 160509A (the red-shift of GRB 140619B is obtained from ref. [24] and otherredshifts are obtained from ref. [25], also see refs. [26–31]).For high energy events, each of GRB 080919C and GRB130427A has 4 events, and GRB 090902B has 6 events.Each of the other GRBs has only one event. Therefore, thereare 19 high energy events that meet the requirements above.The energies of GBM (NaI) photons are less than 2 MeV,and thus are negligible when calculating K , because E low ≪ E high . Therefore, K can be written approximately as K = s E high , obs H ( + z ) Z z ( + z ′ ) d z ′ p Ω m ( + z ′ ) + Ω Λ . (11)The discussion above adopts a situation consisting of a highenergy photon and a low energy photon. For each GRB, itis reasonable to use the arrival time of a single high energyphoton to mark a high energy process, but it is more compli-cated to mark the time of low energy process because thereare plenty of low energy photons emitted during the wholeGRB process. We need to form a low energy event for eachGRB, considering all the low energy photons at all time, tofit the situation above. The approximate expression of K tellsus that the energy of a low energy event does not matter, sowe do not need to focus on the exact value of energy of lowenergy photons. What remains unclear is the time of a lowenergy event ( t low , obs ), which does not have a clear definitionbecause while t high , obs is a property of one single high energyphoton, t low , obs can only be obtained from a set of low en-ergy photons with different energies and arrival times.It is reasonable to assume that the low energy processhas effect on the number of photons emitted per unit of time,so the characteristic time of a low energy event in the ob-server reference frame, i.e., t low , obs , must correlate with acharacteristic point in the light curve. In refs. [5, 6], t low , obs ischosen as the trigger time of GBM detector, while in refs. [7,8], t low , obs is the time of the first main pulse in the lightcurve, i.e., t peak , obs . As the trigger time is strongly affectedby the performance of GBM detectors and the distance ofthe source, we choose the time of the first main peak in thelight curve as the signal time of low energy photons. As thepeaks of a light curve mark the moments with the largestdensities of photons, the first main peak can serve as a sig-nificant benchmark that represents the intrinsic property ofthe GRB objectively. We choose the highest point of this Table 1: The data of high energy ( >
30 GeV) photon events from GRBs with known redshifts, read from Pass-8 data [22] ofthe FERMI telescope. t high , obs refers to the observed arrival time since trigger.GRB z t high , obs (s) E high , obs (GeV) E high , src (GeV) K ( s · GeV ) ± .
15 40.509 27.4 146.7 9.87080916C(2) 4.35 ± .
15 16.545 12.4 66.5 4.47080916C(3) 4.35 ± .
15 43.999 5.71 30.5 2.05080916C(4) 4.35 ± .
15 28.210 6.72 36.0 2.42090510 0.903 ± .
003 0.828 29.9 56.9 7.21090902B(1) 1.822 81.746 39.9 112.5 12.9090902B(2) 1.822 26.168 18.1 51.1 5.85090902B(3) 1.822 45.608 15.4 43.5 4.97090902B(4) 1.822 14.167 14.2 40.1 4.59090902B(5) 1.822 42.374 12.7 35.7 4.10090902B(6) 1.822 11.671 11.9 33.5 3.84090926A 2.1071 ± . ± . ± . ± . ± . ± .
37 0.613 22.7 83.5 7.96160509A 1.17 76.506 51.9 112.6 14.23peak as t low , obs , and the details are discussed in the next sub-section.As this method depends on the criterion for choosinglow energy events, we mainly focus on the determinationof low energy characteristic times from different viewpointsand different data in the following sections. We try to pro-vide a more comprehensive criterion and test it with moredata.2.2 Determination of low energy characteristic time fromlight curvesFrom discussion in the previous section, the arrival time ofa high energy photon can be read directly from the data. Wenow need to find a proper way to determine t low , obs . Follow-ing refs. [7, 8], we choose the first peak in the light curve oflow energy photons as t low , obs to mark an intrinsic low en-ergy process of GRB. It is considered to be a better choicethan the trigger time [6, 7], because the latter is also sen-sitive to the performance of detectors besides the proper-ties of the GRB itself. In ref. [7], photons ranging from 8-260 keV are used to plot the low energy light curve, but herewe choose another energy band. The number of recordedphotons is determined not only by the numbers of emittedphotons, but also by the efficiency of the detector. Due to the materials in front of NaI detectors in GBM and otherfactors, the effective area of the detector depends on energy(see fig. 11 in ref. [12]). NaI detectors can record photonsranging from 8-1000 keV, but the effective area of the de-tector drops quickly when the energy is lower than 20 keVand higher than 200 keV. In the band of 20-200 keV, theeffective area is nearly the same. If we choose low energyphotons regardless of the dependence of effective area onenergy, we actually assume implicitly that the light curvesfor each low energy band are of the same shape. However,the light curves for different energy bands are not necessar-ily the same. In ref. [7], the low energy band is set to be8-260 keV, but in that work, the variance of effective areais not considered. We use another low energy band here, 20-200 keV, in which the effective area is nearly constant. Thus,we take all the photons in 20-200 keV band from GBM datainto account, and in this band, the dependence of effectivearea on energy is much smaller than that of 8-260 keV (wecall the 20-200 keV band as Band-Obs later).Following former works [5–8], we choose the time of thefirst peak in the light curve in Band-Obs as the low energyphoton arrival time t low , obs and put it into eq. 9 to calculate ∆ t obs for each corresponding high energy event.In order to combine different GRBs, a more plausibleway to analyze the data is to set the time axis to t low , obs / ( + z ) , because that refers to the time at source. Thus, to be spe-cific, we use the data from two triggered NaI detectors withmost detected photons and bin their events in 0.5 second tofind the first main peak and choose the highest point in thelight curve binned in 32 ms around this peak as the charac-teristic time for low energy photons . In this way, with bothwide bins (0.5 s) and narrow bins (0.032 s), we can avoidstochastic fluctuations of the light curve when the GRB isnot very bright (see fig. 1 and table 2).Table 2: t low , obs / ( + z ) chosen from light curves in Band-Obs.GRB z t low , obs / ( + z ) (s)080916C 4.35 0.480090510 0.903 0.288090902B 1.822 3.456090926A 2.1071 1.376100414A 1.368 0.096130427A 0.3399 0.384140619B 2.67 -0.032160509A 1.17 6.432Combining t low , obs in table 2 with t high , obs and K in ta-ble 1, we can obtain K and ∆ t obs / ( + z ) using eqs. 9 and 11for each energetic photon event, and then draw the ∆ t obs / ( + z ) - K plot as shown in fig. 2. From this figure, we noticethat if we ignore 130427(2), 130427(3) and 130427(4), therest events roughly fall on three parallel lines (dashed linesin fig. 2). 9 events (080916C(2), 090902B(1), 090902B(2),090902B(4), 090902B(6), 090926A, 100414A, 130427A(1)and 160509A) fall surprisingly on the middle line, which iscalled the “mainline" in refs. [7, 8]. Besides, 4 events (080916C(3),080916C(4), 090902(3) and 090902(5)) fall on the upperline and 3 events (080916C(1), 090510, and 140619B) fallon the lower line. It is noticed in ref. [8] that the events onthe lower line have relatively higher energies (in the sourceframe) while the events on the upper line roughly have lowerenergies. An assumption may be made that photons with dif-ferent intrinsic energies have different intrinsic time lag sta-tistically, and therefore we introduce three lines to fit higher,medium, and lower energy photons. That is to say, for eachline, it is assumed that the events on this line may share asame intrinsic time lag, and the slope of the line represents1 / E LV according to sect. 2.1. Different parallel lines implya same E LV with different intrinsic time lags.However, in our analysis, three events, 130427(2), 130427(3)and 130427(4), which were not included in former analy-ses in refs. [7, 8], fall far from the three lines suggested by Remark: In this subsection, we use the peaks chosen in refs. [7, 8].More options will be discussed later. those works. Thus, the 3 parallel lines seem less plausible,though all the long bursts have events (including 130427(1))on the mainline. As there is no convincing reason to ignorethe three events off three lines, we fit all points with a singleline and obtain the slope ( ± ) × − GeV − , from whichwe obtain a lower bound on E LV with | E LV | ≥ × GeV.
One remaining question is how to select the characteristictime for low energy photons from more convincing view-points. In the discussion above, we select the first main peakof the light curve. That is to say, we choose the time whenthe density of photons is sharp. We need to offer more op-tions on the selection of t low to explore whether t low deter-mined above represents a characteristic mechanism. If themechanism at t low is significant, we should expect that itnot only results in notable changes in photon numbers butalso changes in other aspects such as the energy distribution.If the characteristic times determined from different view-points are consistent with each other, we may assume thatthere is an important process at around t low which results insignificant phenomena in different aspects.In this section, we offer other criteria to determine thecharacteristic time for low energy photons. As before, ourdata is obtained from two GBM NaI detectors with mostdetected photons.In consideration of the fact that the light curve shows thenumber of photons but not the energy released from source,our next method is to plot the whole energy received per binwithin a low energy band versus time, and then choose thefirst main peak of this “energy curve" as the characteristictime.Another method is to calculate the average energy perphoton in the low energy band and draw the average energyversus time plot. This method is based on a speculation thatthe energy distribution may change severely during a char-acteristic process so that the average energy per photon alsochanges a lot. In this way, the first significant peak (or dip)of the average energy curve (the plot of average energy perphoton versus time) can represent the change point of the en-ergy distribution and thus a characteristic time of the GRB.At last, we offer a new criterion based on both light curveand average energy curve in subsection 3.3.3.1 Criterion 1: Energy received in a certain bandIn this subsection, we focus on the total energy received perbin within the low energy band. We choose the band, Band-Obs, and select all the photons in this energy band. Then thetotal energy of selected photons in each time bin is calcu-lated. The energy resolution of NaI detectors varies with the Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB 080916C (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16) (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (cid:17)(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23) (Band-Obs) binned in 32msbinned in 0.5s
Fig. 1: The light curves in Band-Obs of 8 GRBs. The x -axis refers to the time t obs / ( + z ) . The thin (blue) curves are thelight curves binned in 0.032 s and the thick (red) curves are those binned in 0.5 s. The y -axis is the normalized counts, i.e.,the counts per bin divided by the maximum counts per bin of the corresponding light curve. The vertical dashed lines (black)refer to the peaks we choose for each GRB. K (10 s · GeV) ∆ t ob s / ( + z ) ( s ) -10010203040506070 130427A(2)130427A(3)130427A(4) 140619B090510 Fig. 2: The ∆ t obs / ( + z ) - K plot for all the events in table 1.Two hollow triangles refer to two events from short bursts,while the others come from long bursts. We obtain this plotby analyzing the light curves in Band-Obs. The hollow cir-cles refer to events of GRB 080916C. The solid circles referto events of 090902B. The plus signs refer to events of GRB130427A. The cross refers to the event 090926A. The solidtriangle refers to 100414A and the star refers to 160509A.The three parallel dotted lines are suggested in refs. [7, 8].The dashed line is the fit of all points.energy. Within Band-Obs, the energy resolution is ∼ t low , obs / ( + z ) .Table 3: The t low , obs / ( + z ) determined by the energy crite-rion (Criterion 1).GRB z t low , obs / ( + z ) t low , obs / ( + z ) determined by this energy criterion (seeTable 3) are similar to those determined by light curves. Itis easy to understand, as more photons are often associatedwith more energy received. However, as the energy distribu-tion of photons may change over time, the energy receivedis not necessarily proportional to the number of photons.This criterion focuses on the tensity of the GRB. The fitof points on the plots of ∆ t obs / ( + z ) - K is shown in fig. 4,whose slope is still ( ± ) × − GeV − , showing a con-sistency with fig. 2.3.2 Criterion 2: Average energy per photon for low energyphotonsIt makes sense to assume that the energy distribution of GRBphotons is different from that of the background photons be-cause of the special mechanism in the GRB source. There-fore, the occurrence of a GRB can cause a significant changeto the average energy curve (average energy per photon dur-ing a bin versus time). Still, we select the photons in a certainenergy band from data of two GBM NaI detectors with mostdetected photons. The average energy of these photons ineach time bin is calculated and used for the average energycurve.As the sharp change of the average energy curve ought tobe related with some characteristic process, we choose thefirst significant change of the average energy curve as thecharacteristic time, t low , obs / ( + z ) . As t low , obs / ( + z ) oughtto be able to represent the intrinsic nature of low energyphotons, a low energy band is still needed. Average energycurves in Band-Obs are plotted in figs. 5. We bin the curvesin 0.5 s and 0.032 s. We first find the first significant peakof the 0.5 s binned curve and then choose the highest pointaround this peak in the 0.032 s binned curve as the charac-teristic time.The photons detected can be divided into two parts: back-ground photons and GRB photons. The background photonsdominate before trigger, so the average energy curve beforetrigger represents the average energy of background pho-tons. After the GRB begins, the average energy curve rep-resents the average energy of both background photons andGRB photons. When the number of GRB photons is muchlarger than that of background photons (such as 130427A,090902B, 090926A and 160509A), the average energy curvecan approximately represent the average energy of GRB pho-tons.The average energy curves are quite different from thelight curves. As the number of background photons is small,the fluctuation of these curves is quite large, so the 0.5 sbinned curves are useful for finding the peak.The relative changes of average energy can be clearlyseen. One example is 130427A, which is quite interesting in Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d e n e r gy p e r b i n ( % ) GRB 080916C (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d e n e r gy p e r b i n ( % ) GRB (cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29) B (Band-Obs) binned in 32msbinned in 0.5s Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d e n e r gy p e r b i n ( % ) GRB (cid:30) (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d e n e r gy p e r b i n ( % ) GRB (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d e n e r gy p e r b i n ( % ) GRB (cid:31) !" (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d e n e r gy p e r b i n ( % ) GRB %&’()*+ (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d e n e r gy p e r b i n ( % ) GRB (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d e n e r gy p e r b i n ( % ) GRB ,-./234 (Band-Obs) binned in 32msbinned in 0.5s
Fig. 3: The energy curves (energy received per bin) in Band-Obs of 8 GRBs. The x -axis is ∆ t obs / ( + z ) . The thin curves (blue)are the energy curves binned in 0.032 s and the thick (red) curves are those binned in 0.5 s. The y -axis is the normalizedenergy received, i.e., the energy received in Band-Obs per bin divided by the maximum energy received per bin of thecorresponding energy curve. The vertical dashed lines (black) refer to the first main peaks we choose for each GRB. K (10 s · GeV) ∆ t ob s / ( + z ) ( s ) -10010203040506070 Fig. 4: The ∆ t obs / ( + z ) - K plot for Band-Obs using Crite-rion 1. The dashed line marks the fit of all points, while thedotted lines were suggested in refs. [7, 8].many aspects [32]. This indicates that the GRB process isnot a stable process: the energy distribution changes greatlyat some time but remains quite stable during other periodsof time. By comparison, the average energy of 100414A ismore stable during the GRB, and this implies that the energydistribution may not change much during the whole GRBprocess of 100414A. It is interesting that even if the firstpeak of the light curve of 130427A is lower than the fol-lowing “platform", the average energies of the peak and the“platform" are similar in the average energy curve. It showsthat though the first peak in the light curve corresponds to aless intensive process, this process may share similar proper-ties with the following intensive process, because they sharea similar average energy.In this subsection, we choose the first significant peak (ordip) of the average energy curve as the characteristic time,and the results are different from those of sect. 2 in somecases. These t low , obs / ( + z ) corresponding to each GRB foreach band are listed in table 4.Table 4: t low , obs / ( + z ) determined by the average energycriterion (Criterion 2).GRB z t low , obs / ( + z ) (s)080916C 4.35 0.256090510 0.903 0.352090902B 1.822 1.856090926A 2.1071 0.992100414A 1.368 0.800130427A 0.3399 0.128140619B 2.67 0.032160509A 1.17 4.928 The ∆ t obs / ( + z ) - K plot for this criterion (fig. 6) is dif-ferent from those in former discussion, as shown in figs. 2and 4. The points near the “mainline" become a little morescattered mainly because four events of GRB 090902B movesup along the y direction compared to figs. 2 and 4. The fit ofall points gives a slope ( ± ) × − GeV − , which is stillconsistent with previous results.In this subsection, we offer an alternative viewpoint onthe signal for low energy photons. We focus on the changeof energy distribution but not the tensity of GRB. Howeverthe results are similar.3.3 Criterion 3: Combine photon numbers and their averageenergyIn the following discussion, we compare the results fromCriterion 2 (criterion of the average energy curve) with thatfrom sect. 2. Short bursts only have one significant peak intheir light curves, energy curves and average energy curves;therefore the characteristic times, t low , obs / ( + z ) , for the 3criteria are nearly the same. Therefore we only discuss longbursts below. For long bursts, the results of sect. 3.1 andsect. 2 show that the obtained t low , obs / ( + z ) are almost thesame for the light curve criterion (we call it Criterion 0,and the same below) and Criterion 1 (criterion of the energycurve). The peaks chosen by Criterion 0 and Criterion 1 areof nearly the same position. Therefore, we mainly comparethe peaks chosen by Criterion 0 and Criterion 2.We notice that for GRB 080916C, GRB 090926A andGRB 130427A, the peaks chosen by Criterion 0 and Crite-rion 2 are very close to each other, as shown in figs. 7, 8and 9. Their t low , obs / ( + z ) refer to significant peaks of boththe 0.5 s binned light curve and 0.5 s binned average energycurve. This shows that the characteristic times for these 3GRBs are not only the times when the number of photonsper bin sharply changes but also the times when the averageenergy per bin significantly changes. This strongly impliesthat there exists some special physical mechanisms aroundthese t low , obs / ( + z ) . Before now, we only know that these t low , obs / ( + z ) match the time with maximum number ofphotons. Now with the help of the average energy curve, wecan distinguish the peak that also matches the time of thechange of energy distribution from other peaks that do notmatch this mechanism. This helps to reduce artificial factorsespecially when choosing the first significant peak for theGRBs with multiple peaks. As the inner process of a GRBis very complex, one may want to choose the characteristiclow energy event from more perspectives. The main peak ina light curve refers to a process with many photons emit-ted. One can be more confident that this is a characteristicprocess, if the energy distribution also changes significantly.Therefore, we offer a balanced criterion that considers bothlight curve and average energy curve: Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB 080916C (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB B (Band-Obs) binned in 32msbinned in 0.5s Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB ; (Band-Obs) binned in 32msbinned in 0.5s Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB <=>?@B (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB
CDEFGHI (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB (Band-Obs) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB
JKLMNOP (Band-Obs) binned in 32msbinned in 0.5s
Fig. 5: The average energy curves in Band-Obs (average energy per photon in Band-Obs) of 8 GRBs. The x -axis is ∆ t obs / ( + z ) . The thin curves (blue) are the energy curves binned in 0.032 s and the thick (red) curves are those binned in 0.5 s. The y -axis is the average energy per photon. The vertical dashed lines (black) refer to the significant peaks we choose for eachGRB. K (10 s · GeV) ∆ t ob s / ( + z ) ( s ) -10010203040506070 Fig. 6: The ∆ t obs / ( + z ) - K plot for Band-Obs using Crite-rion 2. The dashed line marks the fit of all points, while thedotted lines were suggested in refs. [7, 8].Criterion 3: choose the first significant peak of thelight curve in Band-Obs that also matches a signifi-cant change in the average energy curvewhere we have considered the fact that the results for differ-ent bands do not vary much.Now we offer Criterion 3 to the other long bursts (GRB090902B, GRB 100414A and GRB 160509A).For GRB 100414A (see fig. 10), the peak chosen withthe criterion in sect. 2 (and in refs. [7, 8]) does not matcha peak in the average energy curve. Using Criterion 3, wesuggest a new characteristic time marked by the dash-dottedline (red) in fig. 10 ( t low , obs / ( + z ) = .
184 s). It marks asignificant peak of the light curve and it also falls around thepeak of the average energy curve. The 0.5 s binned curvesare used to show the main processes of GRB, and from fig. 10,we notice that the new t low , obs / ( + z ) coincides with the sig-nificant peaks in both light curve and average energy curvebinned in 0.5 s. On the contrast, the former t low , obs / ( + z ) (marked by the vertical solid line) does not mark a signif-icant peak in the average energy curve. Actually the former t low , obs / ( + z ) does not mark a significant peak in the 0.5 sbinned light curve. This suggests that the new t low , obs / ( + z ) chosen by Criterion 3 may reflect a physical mechanism andthus a more convincing characteristic time for low energyphotons. Therefore, we may suggest that t low , obs / ( + z ) = .
184 s be used in later discussion. Besides, we can see thatother peaks in the light curves (marked by vertical dottedlines) do not match the significant peaks in the average en-ergy curve, and this means that these peaks may not be re-lated with a special mechanism that changes the average en-ergy. Therefore, these peaks are not chosen as the character-istic time. We can see that by considering the average energycurve, Criterion 3 helps to restrict the possible peaks chosen from the light curve, and this helps to reduce the arbitrari-ness in choosing t low , obs / ( + z ) . We therefore recommendto use Criterion 3 for the choice of t low , obs / ( + z ) as thecharacteristic time for low energy photons.Now we come to GRB 160509A. 160509A [8] is con-sidered to be a strong evidence to support the prediction ofref. [7], so it is important to check whether t low , obs / ( + z ) for 160509A still holds for the new Criterion 3. In the 0.5 sbinned light curve of 160509A (see fig. 11), there is a smallpeak at around 1.120 s (marked by dotted line). We do notchoose this peak because it is too small compared to the nextpeak in sect. 2. Now we can give more evidence by usingCriterion 3. From the average energy curve of 160509A, wefind that the bigger peak in the light curve (marked by solidline, which we choose as t low , obs / ( + z ) ) can match the mainpeak in the 0.5 s binned average energy curve. However thesmaller peak in the light curve (at around 1.120 s) does notmatch the main peak in the average energy curve. There isa smaller peak but this peak is also too small compared tothe next peak. Therefore, we do not treat the smaller peakas the first significant peak and suppose that the next peakrepresents more reasonable significant time for low energyphotons.It becomes more complicated to apply the criterion toGRB 090902B (see fig. 12). The 0.5 s binned light curve has3 main peaks while the 0.5 s binned average energy curveonly has two. It seems that the 3 peaks of the light curve donot match the significant peak in the average energy curve.We notice that the first peak of the light curve is significantand is the nearest to the first main peak of the average energycurve. Therefore, we assume that the first peak of the lightcurve is related with the physical mechanism shown by theaverage energy curve.Table 5: The low energy characteristic times we recommend(using Criterion 3) for 8 GRBs.GRB z t low , obs / ( + z ) (s) t low , obs (s)080916C 4.35 0.480 2.568090510 0.903 0.288 0.548090902B 1.822 3.456 9.753090926A 2.1071 1.376 4.275100414A 1.368 1.184 2.804130427A 0.3399 0.384 0.515140619B 2.67 -0.032 -0.117160509A 1.17 6.432 13.957In summary, Criterion 3 combines the effect of max-imum number of photons and the effect of changing theenergy distribution. We recommend this criterion becausemore aspects of the GRB are considered and thus arbitrari- Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB 080916C (Band-Obs) binned in 32msbinned in 0.5s
The light curve of GRB 080916C.
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB 080916C (Band-Obs) binned in 32msbinned in 0.5s
The average energy curve of GRB 080916C.Fig. 7: The light curve and average energy curve of GRB 080916C in Band-Obs. The x -axis refers to ∆ t obs / ( + z ) . Thevertical solid lines refer to t obs / ( + z ) chosen by Criterion 0 (the first main peak of the light curve) and the vertical dashedlines refer to t obs / ( + z ) chosen by Criterion 2 (the significant peak of the average energy curve). Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB
QRSTUVW (Band-Obs) binned in 32msbinned in 0.5s
The light curve of GRB 090926A.
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB
XYZ[\]^ (Band-Obs) binned in 32msbinned in 0.5s
The average energy curve of GRB 090926A.Fig. 8: The light curve and average energy curve of GRB 090926A in Band-Obs. The vertical solid lines refer to t obs / ( + z ) of Criterion 0 (the first main peak of the light curve) and the vertical dashed lines refer to t obs / ( + z ) of Criterion 2 (thesignificant peak of the average energy curve).ness is reduced in this way. Considering the fact that dif-ferent bands for low energy photons do not affect the resultsmuch, we recommend the method of determining t low , obs / ( + z ) below:1. First bin the light curve and the average energy curve forBand-Obs in 0.5 s.2. Then find the first significant peak of the light curve thatalso matches a significant change of the average energycurve.3. Use the time of the highest point around this peak in the32 ms binned light curve as the characteristic time forlow energy photons t low , obs / ( + z ) . To conclude this section, we list t low , obs / ( + z ) for 8 GRBswe recommend (using the method above) in table 5 and drawthe ∆ t obs / ( + z ) - K plot in fig. 13. By fitting all the points,we obtain the slope ( ± ) × − GeV − , still consistentwith previous results. Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (Band-Obs) binned in 32msbinned in 0.5s
The light curve of GRB 130427A.
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB (Band-Obs) binned in 32msbinned in 0.5s
The average energy curve of GRB 130427A.Fig. 9: The light curve and average energy curve of GRB 130427A in Band-Obs. The vertical solid lines refer to t obs / ( + z ) of Criterion 0 (the first main peak of the light curve) and the vertical dashed lines refer to t obs / ( + z ) of Criterion 2 (thesignificant peak of the average energy curve). Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB _‘abcde (Band-Obs) binned in 32msbinned in 0.5s
The light curve of GRB 100414A.
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB fghijkl (Band-Obs) binned in 32msbinned in 0.5s
The average energy curve of GRB 100414A.Fig. 10: The light curve and average energy curve of GRB 100414A in Band-Obs. The vertical solid lines refer to t obs / ( + z ) of Criterion 0 (the first main peak of the light curve) and the vertical dashed lines refer to t obs / ( + z ) of Criterion 2 (thesignificant peak of the average energy curve). The vertical dash-dotted line (red) refers to t obs / ( + z ) of Criterion 3 and thedotted lines refer to other peaks in the light curve. t low , obs / ( + z ) (the first main peak). Inorder to use the FERMI data comprehensively, we now ap-ply the method and Criterion 3 to other low energy photondetectors of the FERMI telescope. The FERMI telescope hastwo parts: GBM and LAT. GBM consists of 12 NaI detec- tors and 2 BGO detectors whose effective energy bands areapproximately 8-1000 keV and 150 keV - 30 MeV respec-tively, while LAT approximately covers the photon energyrange 20 MeV - 300 GeV. To make full use of the FERMIdata, we use the following methods to plot three sets of lightcurves and average energy curves for the 8 GRBs discussedabove:1. Set-NaI: For each GRB, draw the light curve and aver-age energy curve by using all the photon events of thetwo GBM NaI detectors with most number of detectedphotons. Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB mnopqrs (Band-Obs) binned in 32msbinned in 0.5s
The light curve of GRB 160509A.
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB tuvwxyz (Band-Obs) binned in 32msbinned in 0.5s
The average energy curve of GRB 160509A.Fig. 11: The light curve and average energy curve of GRB 160509A in Band-Obs. The vertical solid lines refer to t obs / ( + z ) of Criterion 0 (the first main peak of the light curve) and the vertical dashed lines refer to t obs / ( + z ) of Criterion 2 (thesignificant peak of the average energy curve). The dotted lines refer to a smaller peak in the light curve. Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB {|}~(cid:127)(cid:128)(cid:129) (Band-Obs) binned in 32msbinned in 0.5s
The light curve of GRB 090902B.
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB (cid:130)(cid:131)(cid:132)(cid:133)(cid:134)(cid:135)(cid:136) (Band-Obs) binned in 32msbinned in 0.5s
The average energy curve of GRB 090902B.Fig. 12: The light curve and average energy curve of GRB 090902B in Band-Obs. The vertical solid lines refer to t obs / ( + z ) of Criterion 0 (the first main peak of the light curve) and the vertical dashed lines refer to t obs / ( + z ) of Criterion 2 (thesignificant peak of the average energy curve). The dotted lines refer to the other peaks in the light curve.2. Set-BGO: For each GRB, draw the light curve and av-erage energy curve by using all the photon events of thetwo GBM BGO detectors.3. Set-LLE: For each GRB, draw the light curve by usingthe LAT Low-Energy (LLE) events [16] and draw theaverage energy curve by using the LLE events with en-ergies less than 1 GeV. The number of photons with en-ergies larger than 1 GeV is quite small and thus causeslittle influence to the light curve. However, as the countper bin is also small, these high-energy events lead to no-table changes of the average energy curves. Therefore,we cut off the high energy events in the average energycurves. Since the 1- σ angular resolution of LAT is about 3 . ◦ [21], a 12 ◦ region of interest (ROI) is chosen as inref. [8] in order to include more photons from the GRBsource.As the energies for most LLE events are less than ∼ >
30 GeV), the LV effect of LLE (and alsoBGO, NaI) photons, if really exists, must produce tiny ar-rival time shift compared to that of high energy events. Thusthe LLE events can still be treated as low energy photonsand we can ignore the LV effect on them. Because it is im-portant to test whether the t low we choose really represent K (10 s · GeV) ∆ t ob s / ( + z ) ( s ) -10010203040506070 Fig. 13: The ∆ t obs / ( + z ) - K plot for Band-Obs using Cri-terion 3. Two triangles refer to two events from short GRBsand the other points are events from long bursts. The dashedline marks the fit of all points, while the dotted lines weresuggested in refs. [7, 8].significant low energy process, we do this analysis to lookfor evidence from data of other detectors.All the curves are binned in 0.5 s and 0.032 s in thesource frame ( t obs / ( + z ) ) as in the previous section. Foreach set of curves, we find the first main peak in the 0.5s-binned curves for each GRB and choose the highest pointaround this peak as the characteristic time. The light curvesand average energy curves of NaI, BGO and LLE data areplotted respectively in figs. 17-18 (Set-NaI), figs. 19-20 (Set-BGO), and figs. 21-22 (Set-LLE). The characteristic timeswe choose for 8 GRBs for each curve set are listed in table 6.For all these figures, the left column refers to the light curvesand the right column refers to the average energy curves ofthe corresponding GRB. The x -axes refer to the time sincetrigger in the source frame, i.e. t obs / ( + z ) . The thin curves(blue) are binned in 32 ms and the thick curves (red) arebinned in 0.5 s. The y -axes of the light curves refer to thenormalized counts, i.e., the counts per bin divided by themaximum counts per bin of the corresponding light curve.The y -axes of the average energy curves refer to the aver-age energy per photon (measured in the observer referenceframe). The vertical lines (black) refer to the characteristictimes we choose for each GRB.In former sections, for each GRB, the characteristic peakin the light curve is required to match a peak in the corre-sponding average energy curve, but this time, the peak inthe light curve matches a dip in the average energy curvefor NaI and BGO data. This suggests that, within a broad We download the data from https://fermi.gsfc.nasa.gov/ssc/observations/types/grbs/lat_grbs/table.php .We find that the LLE (LAT) data for GRB 100414A are not available.Therefore, we do not plot the LLE curves for GRB 100414A. band, the average energy of GRB photons is lower than thatof background photons.To clarify this, energy distributions of Set-NaI of GRB090902B before and after trigger are plotted in fig. 14. Inprevious sections, Band-Obs (marked by red dashed lines) isapplied. However, in this section, all the photons (Set-NaI)are used. As shown in fig. 14, if we only consider photonswithin Band-Obs, obviously the average energy increase af-ter trigger (i.e. approximately after GRB begins). However,if we focus on photons from the whole band of NaI detec-tors, relatively more lower energy photons makes the aver-age energy of whole band after trigger smaller. This explainswhy there is a peak when we choose Band-Obs and whythere is a dip if we use Set-NaI.As mentioned before, the energy distribution within Band-Obs can be treated as the real energy distribution of GRBand background photons, while the energy distribution out-side Band-Obs is distorted by the performance of detectors.However, as we focus on the changes of energy distribution,Set-NaI is used in this section.For Set-NaI and Set-BGO, the peaks we choose in thelight curves still match significant changes (the sharp de-creasing of average energy per photon) in the correspond-ing average energy curves, especially for long bursts (e.g.080916C, 090926A, 130427A, 160509A, etc.). This indi-cates that, around the characteristic time, there may be somecertain mechanism that causes significant changes in aspectsof both observed counts and energy distribution of photons.Moreover, this mechanism affects the photons within theNaI and BGO bands (about 8 keV - 30 MeV), which aremuch broader than Band-Obs.The average energy curves of Set-LLE are different fromthose of Set-NaI and Set-BGO. For some GRBs, it is diffi-cult to choose characteristic times from the LLE average en-ergy curves to match the peaks in the light curves. One pos-sible reason is that, for LLE data, the energy distribution ofGRB photons is similar to that of background photons. Forexample, we plot the energy distribution histogram from theLLE data of GRB 160509A, using the events: (1) within 4-10 s (in the source frame) since trigger and (2) before trigger(see fig. 15). From the light curve, Case 1 covers an inten-sive GRB process and Case 2 covers the background pho-tons. Thus Case 1 can approximately show the energy dis-tribution of GRB photons while Case 2 reflects the energydistribution of background photons (as shown in fig. 16).From the BGO energy distributions, the GRB photons tendto have lower energies than the background photons, as themiddle part of the GRB energy distribution (Case 1) is de-pressed compared to the background distribution. Howeverthe LLE average energy curves show that the difference ofdistributions between the two cases is comparably smaller.It leads to difficulty in determining a characteristic time insome LLE average energy curves. Table 6: The characteristic times (since trigger) for low energy photons from NaI, BGO and LLE data. As we have mentioned,we do not have t low for GRB 100414A from LLE data because of the lack of data.GRB z t low , obs / ( + z ) (s) t low , obs ( s ) NaI BGO LLE NaI BGO LLE080916C 4.35 0.480 0.864 0.864 2.568 4.622 4.622090510 0.903 0.288 0.288 0.384 0.548 0.548 0.731090902B 1.822 3.456 3.456 2.624 9.753 9.753 7.405090926A 2.1071 1.376 1.344 1.920 4.275 4.176 5.966100414A 1.368 1.184 1.184 - 2.804 2.804 -130427A 0.3399 0.352 0.224 -0.032 0.472 0.300 -0.043140619B 2.67 0.096 0.000 0.096 0.352 0.000 0.352160509A 1.17 6.432 5.536 5.504 13.957 12.013 11.944
Energy (keV) C oun t s Energy Distribution for GRB 090902B, before trigger
Energy (keV) C oun t s Energy Distribution for GRB 090902B, after trigger
Fig. 14: The energy distributions of Set-NaI photons of GRB 090902B within [-20,0]s and [0,20]s since trigger. The reddashed lines mark Band-Obs.Although the LLE average energy curves do not helpmuch, the peaks in the corresponding light curves are sharpand clear. From figs. 17, 18, 19, 20, 21 and 22, we find thecharacteristic times using the Criterion 3 for curves in Set-NaI and Set-BGO and the criterion in ref. [7] (first main peakin the light curve) for Set-LLE. The characteristic times, t low , obs / ( + z ) , for 3 sets of curves are listed in table 6. Fromthe table, the 3 sets of t low , obs / ( + z ) are consistent witheach other, as the maximum difference of t low , obs / ( + z ) for each GRB is less than 1 s (about two 0.5 s bins). Asthe peaks are chosen according to the 0.5 s-binned curves,a 1 s time interval can be assumed to be related with thesame energetic process. Besides, for each GRB, the uncer-tainty in choosing the highest point in the 0.032 s-binnedcurve may also lead to a difference about one or two largebins (0.5 s) among three t low , obs / ( + z ) . Therefore, the three t low , obs / ( + z ) may reflect the same intrinsic process foreach GRB. The consistence of three sets of t low , obs / ( + z ) also suggests that the mechanism around the characteristictime influences both low energy ( ∼
10 keV) and medium en- ergy ( ∼
100 MeV) photons. Thus, the first main peak foreach GRB may represent an important process with widespectrum of energy, and this process is treated as the lowenergy event in sect. 2.1.Also the results are consistent with t low , obs / ( + z ) in for-mer sections, and the ∆ t obs / ( + z ) - K plots for three sets (asshown in fig. 23) are remarkably similar to the plots in for-mer sections. All the fittings arrive at a slope ( ± ) × − GeV − ,showing consistency with figs. 2, 4, 6, and 13.4.2 Estimation of UncertaintiesAccording to the method in sect. 2.1, the uncertainty of thefinal E LV comes from the uncertainties of the times and en-ergies of high and low energy events, the uncertainty of red-shifts and the error from the linear fit. In former sections,we fit the middle 9 events with linear function and obtainthe mainline. The uncertainty there only contains the errorcaused by the fitting. With this method, we obviously over- Case 1
Energy (MeV) C oun t s GRB 160509A (LLE, 4 ~ 10 s (src) since trigger)
Case 2
Energy (MeV) C oun t s GRB 160509A (LLE, before trigger)
Fig. 15: The energy distribution histograms for GRB160509A from LLE data. The upper one uses the photonswithin 4-15 s (with t obs / ( + z ) ) since trigger. The lowerone uses the photons before trigger. The y -axis refers to thecounts of certain rectangle. We can see that most of LLEphotons are .
100 MeV.estimate the accuracy of E LV . Now we try to estimate theuncertainty in consideration of all uncertainties caused bythe method. As the accurate value of E LV is now less im-portant than its order of magnitude, we simply do a roughestimation.To estimate the uncertainty of the slope of the main-line, we need to obtain the uncertainties of K and ∆ t obs / ( + z ) for each point in the ∆ t obs / ( + z ) - K plot. According tosect. 2.1, the error of K for each high energy event is de-cided by the error of energy and the error of redshfit. Forhigh energy photons, the energy resolution is a function ofincidence angle and energy. For simplicity, we take energyresolution as ∆ E / E ≃
10% (see fig. 18 in ref. [11]). As thatthe error of redshift is absent for some GRB, and that formost GRB on the mainline, the uncertainty of the redshiftis less than 5% as shown in table 1, we ignore the uncer- Case 1
Energy (keV) C oun t s × GRB 160509A (BGO, 4 ~ 10 s (src) since trigger)
Case 2
Energy (keV) C oun t s GRB 160509A (BGO, -20 ~ 0 s (src) since trigger)
Fig. 16: The energy distribution histograms for GRB160509A from BGO data. The upper one uses the photonswithin 4-15 s (with t obs / ( + z ) ) since trigger. The lower oneuses the photons within − t obs / ( + z ) ) sincetrigger. The y -axis refers to the counts of certain rectangle.The bounds of the rectangles are chosen according to theenergy channels of the instrument.tainty of redshift and simply apply a 10% error in K for eachevents.The error in the ∆ t obs / ( + z ) = t high , obs / ( + z ) − t low , obs / ( + z ) mainly comes from the determination of t low , obs / ( + z ) because the time resolution of high energy events [11] isnegligible compared to the uncertainty of t low , obs / ( + z ) .These t low , obs / ( + z ) are read mainly from GBM data andthe background photons are non-negligible as shown in thelight curves before. However, the background does not influ-ence the determination of low energy characteristic time, be-cause that the background counts for all discussed GRBs arequite stable and that we choose a peak relative to the back-ground. For example, for BGO light curve of GRB 080916Cwhose background photons occupy a large part, we do abackground subtraction using the method in the commonlyused tool, GTBurst [33], i.e., fitting the background before and after GRB with a polynomial function, and plot its lightcurve in fig. 24. The low energy photon characteristic timedetermined from this background subtracted curve is also t low , obs / ( + z ) = .
864 s, which is consistent with table 6.As the background subtraction does not influence the deter-mination of first main peak, we do not apply backgroundsubtraction in all other figures.
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) -40-20020406080100 GRB 080916C (in src, BGO)
Fig. 24: BGO light curves for GRB 080916C with and with-out background subtraction. The upper two curves refer tothe light curves without background subtraction (also shownin fig. 19). After subtracting the background we obtain thelower two curves. The time t low , obs / ( + z ) determined fromthe light curves with and without background subtraction arethe same, marked by the vertical line.As a rough estimation, the uncertainty of t low , obs / ( + z ) is chosen as the maximum difference of t low , obs / ( + z ) un-der different criteria, i.e., about 1 s. The fluctuations of thebackground may lead to uncertainty in selecting the highestpoint around the first main peak, but the peak in 0.5 s-binnedcurve is clear so the uncertainty caused by the backgroundfluctuation should be ∼ t low , obs / ( + z ) for a certain GRB actually contains the uncertainty ofchoosing the highest point around the peak.These uncertainties do not significantly change the fit-tings, from which we arrive at the lower bound on E LV with | E LV | ≥ × GeV.
We aim to check whether the speed of light changes with thephoton energy by using the gamma-ray burst data from theFERMI telescope (FGST). We review the work in refs. [7, 8]and also suggest two other criteria on determining the char-acteristic time t low for low energy photons. The first is tochoose the first main peak of received energy curve. It re-sults in nearly the same ∆ t obs / ( + z ) - K plot as from thelight curve. The second is to choose the first significant peak(or dip) of the average energy curve, which reflects the changeof energy distribution. We obtain a slightly different ∆ t obs / ( + z ) - K plot, but the results are still consistent with those inrefs. [7, 8]. Finally, we offer a new criterion that uses the av-erage energy curve in addition to the light curve to determinethe characteristic time for low energy photons. We apply thenew criteria to three different sets of GBM NaI, GBM BGO,and LAT LLE data of low energy photons from FERMI de-tectors. It is remarkable that we obtain consistent resultsfor the characteristic time of three different sets of low en-ergy photons. The regularity that several high energy photonevents from different GRBs fall on a same line [7, 8] stillpersists under different situations. With this more compre-hensive analysis, we arrive at consistent results of the lowerbound on E LV with | E LV | ≥ × GeV. A recent workstudied the LV energy scale from statistical estimators[34],with lower bound exceeding either 8 . × GeV or 2 . × GeV, which is consistent with our results.
This work is supported by National Natural Science Founda-tion of China (Grant No. 11475006). It is also supported byHui-Chun Chin and Tsung-Dao Lee Chinese UndergraduateResearch Endowment (CURE).
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Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (cid:142)(cid:143)(cid:144)(cid:145)(cid:146)(cid:147) (cid:148)(cid:149)(cid:150)(cid:151)(cid:152) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (cid:153)(cid:154)(cid:155)(cid:156)(cid:157)(cid:158)(cid:159) (cid:160)¡¢£⁄ binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB ¥ƒ§¤'“« ‹›fifl(cid:176) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB 080916C –†‡·(cid:181) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB ¶•‚„”» …‰(cid:190)¿(cid:192) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB `´ˆ˜¯˘˙ ¨(cid:201)˚¸(cid:204) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB ˝˛ˇ—(cid:209)(cid:210)(cid:211) (cid:212)(cid:213)(cid:214)(cid:215)(cid:216) binned in 32msbinned in 0.5s
Fig. 17: Curves in Set-NaI (part 1): light curves and average energy curves of GBM NaI data. Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (cid:217)(cid:218)(cid:219)(cid:220)(cid:221)(cid:222)(cid:223) (cid:224)Æ(cid:226)ª(cid:228) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (cid:229)(cid:230)(cid:231)ŁØ binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB
μ(cid:236)(cid:237)(cid:238) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (cid:239)(cid:240)æ(cid:242)(cid:243)(cid:244)ı (cid:246)(cid:247)łøœ binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB ß(cid:252)(cid:253)(cid:254)(cid:255)1(cid:0) ((cid:1)(cid:2)(cid:3)(cid:4) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB (cid:5)(cid:6)(cid:7)(cid:8)(cid:9) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB (cid:10)(cid:11)(cid:12)(cid:13)(cid:14) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB (cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:21) (cid:22)(cid:23)(cid:24)(cid:25)(cid:26) binned in 32msbinned in 0.5s
Fig. 18: Curves in Set-NaI (part 2): light curves and average energy curves of GBM NaI data. Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB 080916C (BGO) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (BGO) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB !" (BGO) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB ’)*+,-. (BGO) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB 080916C (BGO) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB /23456 (BGO) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB (BGO) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB >?@ABCD (BGO) binned in 32msbinned in 0.5s
Fig. 19: Curves in Set-BGO (part 1): Light curves and average energy curves of GBM BGO data. Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB
EFGHIJK (BGO) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (BGO) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (BGO) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (BGO) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB
LMNOPQR (BGO) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB (BGO) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB
STUVWXY (BGO) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( k e V ) GRB (BGO) binned in 32msbinned in 0.5s
Fig. 20: Curves in Set-BGO (part 2): Light curves and average energy curves of GBM BGO data. Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB 080916C (LLE, with ROI = 12 deg) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB 090510 (LLE, with ROI = 12 deg) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB 090902B (LLE, with ROI = 12 deg) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB
Z[\]^_‘ (LLE, with ROI = 12 deg) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( M e V ) GRB (LLE, with ROI = 12 deg) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( M e V ) GRB 090510 (LLE, with ROI = 12 deg) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( M e V ) GRB 090902B (LLE, with ROI = 12 deg) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( M e V ) GRB abcdefg (LLE, with ROI = 12 deg) binned in 32msbinned in 0.5s
Fig. 21: Curves in Set-LLE (part 1): Light curves and average energy curves of LLE data. Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB (LLE, with ROI = 12 deg) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB hijklmn (LLE, with ROI = 12 deg) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 N o r m a li ze d c oun t s p e r b i n ( % ) GRB opqrstu (LLE, with ROI = 12 deg) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( M e V ) GRB (LLE, with ROI = 12 deg) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( M e V ) GRB vwxyz{| (LLE, with ROI = 12 deg) binned in 32msbinned in 0.5s
Time since trigger (s) -5 0 5 10 15 20 25 A v e r a g e e n e r gy p e r pho t on ( M e V ) GRB }~(cid:127)(cid:128)(cid:129)(cid:130)(cid:131) (LLE, with ROI = 12 deg) binned in 32msbinned in 0.5s
Fig. 22: Curves in Set-LLE (part 2): Light curves and average energy curves of LLE data. K (10 s · GeV) ∆ t ob s / ( + z ) ( s ) -10010203040506070 K (10 s · GeV) ∆ t ob s / ( + z ) ( s ) -10010203040506070 K (10 s · GeV) ∆ t ob s / ( + z ) ( s ) -10010203040506070 Fig. 23: The ∆ t obs / ( + z ) - K plots from NaI, BGO and LLE data. The dots refer to events from long bursts and the trianglesrefer to the events from short bursts. The middle dotted line in each plot refers to the mainline. For each plot, 9 events outof 17 fall on the mainline. As the LLE data for GRB 100414A seem not available in the website, we apply the characteristictime from NaI and BGO data (with t low , obs / ( + z ) = .