NASA Technical Reports Server (NTRS) 20140005679: Constraints on Lorentz Invariance Violation from Fermi -Large Area Telescope Observations of Gamma-Ray Bursts

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Constraints on Lorentz Invariance Violation from Fermi-Large Area Telescope
Observations of Gamma-Ray Bursts
V. Vasileiou,1,∗ A. Jacholkowska,2,† F. Piron,1 J. Bolmont,2 C. Couturier,2
J. Granot,3 F. W. Stecker,4,5 J. Cohen-Tanugi,1 and F. Longo6,7
1Laboratoire Univers et Particules de Montpellier,
Universit´e Montpellier 2, CNRS/IN2P3, CC 72,
Place Eug`ene Bataillon, F-34095 Montpellier Cedex 5, France
2LPNHE, Universit´e Pierre et Marie Curie Paris 6,
Universit´e Denis Diderot Paris 7, CNRS/IN2P3,
3 4 Place Jussieu, F-75252, Paris Cedex 5, France
1 3Department of Natural Sciences, The Open University of Israel,
0 1 University Road, POB 808, Ra’anana 43537, Israel
2 4Astrophysics Science Division, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, U.S.A.
y 5Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547, U.S.A.
a 6Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34127 Trieste, Italy
M 7Dipartimento di Fisica, Universita` di Trieste, I-34127 Trieste, Italy
WeanalyzetheMeV/GeVemissionfromfourbrightGamma-RayBursts(GRBs)observedbythe
5
Fermi-LargeAreaTelescopetoproducerobust,stringentconstraintsonadependenceofthespeedof
1
lightinvacuoonthephotonenergy(vacuumdispersion),aformofLorentzinvarianceviolation(LIV)

allowed bysome QuantumGravity(QG)theories. First,we usethreedifferentandcomplementary
]
E techniques to constrain the total degree of dispersion observed in the data. Additionally, using a
H maximallyconservativesetofassumptionsonpossiblesource-intrinsicspectral-evolutioneffects,we
constrainanyvacuumdispersionsolelyattributedtoLIV.Wethenderivelimitsonthe“QGenergy
h. scale”(theenergyscalethatLIV-inducingQGeffectsbecomeimportant,EQG)andthecoefficientsof
p theStandardModelExtension. Forthesubluminalcase(wherehighenergyphotonspropagatemore
- slowly than lower energy photons) and without taking intoaccount any source-intrinsic dispersion,
o ourmoststringentlimits(at95%CL)areobtainedfromGRB090510andareEQG,1>7.6timesthe
tr Planckenergy(EPl)andEQG,2>1.3×1011 GeVforlinearandquadraticleadingorderLIV-induced
s vacuumdispersion,respectively. These limitsimprovethelatest constraintsbyFermi andH.E.S.S.
a
[ bya factor of ∼2. Ourresults disfavor any class of models requiringEQG,1(cid:2)EPl.

1 PACSnumbers: 11.30.Cp,04.60.-m,98.70.Rz
v
3
6 I. INTRODUCTION bythesearchforatheoryofQG.Additionalmotivations
4 for testing LI are the need to cut off high-energy (ultra-
3
While general relativity and Quantum Field Theory violet)divergencesin quantum field theoryand the need
.
5 have each enjoyed impressive success so far, their for- for a consistent theory of black holes [2, 3].
0 mulations are currently inconsistent, hence motivating The idea that LI may be only approximate has been
3
searches for unification schemes that can collectively be explored within the context of a wide variety of sug-
1
: subsumed under the name of Quantum Gravity (QG) gested Planck-scale physics scenarios. These include the
v theories. These theories generally predict the existence concepts of deformed relativity, loop quantum gravity,
i
X of a natural scale at which the physics of space-time, non-commutativegeometry,spinfoammodels,andsome
as predicted by relativity theory, is expected to break string theory (M theory) models (for reviews see, e.g.,
r
a down, hence requiring modifications or the creation of a Refs.[4–6]). Thesetheoreticalexplorationsandtheirpos-
new paradigm to avoid singularity problems. This scale, sible consequences, such as observable modifications in
referred to as the“Quantum Gravity energy scale”EQG, the energy-momentumdispersionrelationsforfreeparti-
is in general exp(cid:2)ected to be of the order of the Planck cles and photons, have been discussed under the general
scale[1],EPl ≡ ((cid:2)c5)/G(cid:3)1.22×1019GeV,orinsome heading of“Planck scale phenomenology”.
cases lower (e.g., for some QG scenarios such as loop There is also the motivation of testing LI in order to
quantum gravity). extent or limit its domain of applicability to the highest
Since relativity precludes an invariant length, the in- observable energies. Since the groupof pure LI transfor-
troduction of such a constant scale violates Lorentz In- mations is unbounded at high energies, one should look
variance (LI). Thus, tests of LI are strongly motivated foritsbreakdownathighenergyscales,possiblythrough
effects of Planck scale physics but perhaps through the
effectsofotherunknownphenomena. Toaccomplishsuch
a program, tests of the kinematics of LI violation (LIV)
∗ [email protected] within the contextof physicalinteractiondynamics such
† [email protected] as quantum electrodynamics or standard model physics
2
(e.g.,[7–11])havebeenproposed. Fruitfulframeworksfor PKS2155-304[28,29]. It shouldbe notedthatthe stud-
this kind of analysis, useful for testing the effects of LIV ies above did not assume any dependence of LIV on the
at energies well below the Planck scale, are the Taylor polarizationofthephotons,manifestingasbirefringence.
seriesexpansionoriginallyproposedintheseminalpaper Inthe casethatsuchadependence exists,constraintson
by Amelino-Camelia et al. [12] and the more compre- LIV effects can be produced [10, 30, 31] that are 13 or-
hensive Standard Model Extension (SME) parametriza- dersofmagnitudestrongerthanthe dispersion-onlycon-
tion of Kostelecky´ and collaborators [13, 14]. These straints placed with time-of-flight considerations (as in
phenomenological parameterizations can be viewed as this work). It should be added that there is a class of
low-energy effective field theories, holding at energies theories that allow for photon dispersion without bire-
E (cid:4)EPl and providing an effective frameworkto search fringence that can be directly constrained by our results
for LIV at energies far below the Planck scale. (e.g., [32]).
OnemanifestationofLIVistheexistenceofanenergy- The aim ofthis study is to produce a robustandcom-
dependent “maximum attainable velocity” of a particle petitive constraint on the dependence of the velocity of
and its effect onthe thresholds for variousparticle inter- light in vacuo on its energy. Our analysis is performed
actions,particledecays,andneutrinooscillations[7]. As- on a selection of Fermi-LAT [21] GRBs with measured
sumingthatthemassofthephotoniszero,itsmaximum redshifts and bright GeV emission. We first apply three
attainable velocity can be determined by measuring its differentanalysistechniquestoconstrainthetotaldegree
velocity at the highest possible observable energy. This of spectral dispersion observed in the data. Then, using
energy is, of necessity, in the gamma-ray range. Since a set of maximally conservative assumptions on the pos-
we know that LI is accurate at accelerator energies, and sible source-intrinsic spectral evolution (which can mas-
evenat cosmic-rayenergies [15], any deviationof the ve- querade as LIV dispersion), we produce constraints on
locity of a photon from its low energy value, c, must be the degree of LIV-induced spectral dispersion. The lat-
very small at these energies. Thus, a sensitive test of LI ter constraintsare weakerthanthose onthe totaldegree
requires high-energy photons (i.e., gamma rays) and en- of dispersion, yet considerably more robust with respect
tails searching for dependencies of the speed of light in to the presence of a source-intrinsic effects. Finally, we
vacuo on the photon energy. The method used in this convert our constraints to limits on LIV-model-specific
study to search for such an energy dependence consists quantities, such as EQG and the coefficients of the SME.
incomparingthetimeofflightbetweenphotonsofdiffer- The first method used to constrain the degree of dis-
ent energies emitted together by a distant astrophysical persioninthedata,named“PairView”(PV),iscreatedas
source. As will be shown in the next section, the mag- part of this study, and performs a statistical analysis on
nitude of a LIV-induced difference on the time of flight all the pairs of photons in the data to find a common
is predicted to be an increasing function of the photon spectral lag. The second method, named “Sharpness-
energy and the distance of source. Thus, because of the Maximization Method”(SMM), is a modification of ex-
high-energyextendoftheiremission(uptotensofGeV), isting techniques (e.g., DisCan [33]) and is based on the
their large distances (redshifts up to a value of ∼8), and fact that any spectral dispersion will smear the struc-
their rapid (downto ms scale)variabilities,Gamma-Ray ture of the light curve, reducing its sharpness. SMM’s
Bursts(GRBs)areveryeffective probesforsearchingfor best estimate is equal to the negative of a trial degree of
such LIV-induced spectral dispersions [12]. dispersion that, when applied to the actual light curve,
restores its assumed-as-initially-maximal sharpness. Fi-
There have been several searches for LIV applying
nally, the third method employs an unbinned maximum
a variety of analysis techniques on GRB observations.
Some of the pre-Fermi studies are those by Lamon et likelihood(ML)analysistocomparethedataasobserved
atenergieslowenoughfortheLIVdelaystobenegligible
al. [16] using INTEGRAL GRBs; by Bolmont et al. [17]
to the data at higher energies. The three methods were
using HETE-2 GRBs; by Ellis et al. [18] using HETE,
tested using an extensive set of simulations and cross-
BATSE, and Swift GRBs; and by Rodr´ıguez-Mart´ınez
checks, described in severalappendices.
et al. using Swift and Konus-Wind observations of
GRB051221A[19]. Themoststringentconstraints,how- Our constraints apply only to classes of LIV models
ever, have been placed using Fermi observations,mainly that possess the following properties. First, the magni-
thanksto the unprecedentedsensitivityfor detectingthe tude of the LIV-induced time delay depends either lin-
prompt MeV/GeV GRB emission by the Fermi Large early or quadratically on the photon energy. Second,
AreaTelescope(LAT)[20,21]. Theseconstraintsinclude this dependence is deterministic, i.e., the degree of LIV-
those by the Fermi LAT and Gamma-Ray Burst Moni- induced increase or decrease in the photon propagation
tor (GBM) collaborationsusing GRBs 080916C[22] and speed does not have a stochastic (or“fuzzy”) nature as
090510[23],andbyShaoetal.[24]andNemiroffetal.[25] postulated by some of the LI models (see Ref. [34] and
using multiple Fermi GRBs. In addition to these GRB- references therein). Finally, the sign of the effect does
basedstudies,therehavebeensomeresultsusingTeVob- not depend on the photon polarization – the velocities
servationsofbrightActive GalacticNucleiflares,includ- of all photons of the same energy are either increasedor
ing the MAGIC analysis of the flares of Mrk 501 [26, 27] decreased due to LIV by the same exact amount.
and the H.E.S.S. analysis on the exceptional flare of In Sec. II we describe the LIV formalism and nota-
3
tion used in the paper, in Sec. III we describe the data parameter”τn astheratioofthisdelayoverEhn−Eln[37]:
sets used for the analysis, in Sec. IV we describe the
Δt (1+n) 1
tinhtroeeaacncoaulynstispmosestihboledsinatnrdintshicesppreoccterdaul-reevwoleuutisoendetoffetcatkse, τn ≡ Ehn−Eln (cid:3)s± 2H0 EQnG ×κn, (3)
in Sec. V we present the results, in Sec. VI we report
where
anddiscusstheirassociatedsystematicuncertaintiesand
caveats,andfinallyinSec.VIIwecompareourresultsto (cid:10)z
(1+z(cid:5))n
previous measurements and discuss their physical impli- κn ≡ (cid:2) dz(cid:5) (4)
cations. WepresentourMonteCarlosimulationsusedfor ΩΛ+ΩM(1+z(cid:5))3
0
verifying PV and SMM in Appendix A, the calibrations
and verification tests of the ML method in Appendix B, is a comoving distance that also depends on the order of
adirectone-to-onecomparisonoftheresultsofthe three LIV (n), z is redshift, H0 is the Hubble constant, and
methodsaftertheirapplicationonacommonsetofsimu- ΩΛ and ΩM are the cosmological constant and the total
lateddatainAppendixC,andcross-checksoftheresults matter density (parameters of the ΛCDM model).
in Appendix D. IntheSMEframework[14],theslightmodificationsin-
ducedbyLIVeffectsarealsodescribedbyaseriesexpan-
sionwithrespecttopowersofthe photonenergy. Inthis
framework, LIV can also be dependent on the direction
II. FORMALISM
of the source. Including only the single term assumedto
dominate the sum, τn is given by:
InQGscenarios,theLIV-inducedmodificationstothe ⎛ ⎞
pexhpoatonnsiodnispinertshioenforremlationcanbedescribedusingaseries τn (cid:3) H10 ⎝(cid:4)0Yjm(nˆ)c((In)+jm4)⎠×κn, (5)
jm
(cid:3) (cid:5) (cid:6) (cid:7)
(cid:4)∞ n
E2 (cid:3)p2c2× 1− s± E , (1) where nˆ is the directionof the source,0Yjm(nˆ) are spin-
n=1 EQG weightedspherical harmonics,and c(n+4) are coefficients
(I)jm
oftheframeworkthatdescribethestrengthofLIV.Inthe
where c is the constant speed of light (at the limit of SMEcaseofadirection-dependentLIV,weconstrainthe
zero photon energy), s± is the “sign of LIV”, a theory- sum (enclosed in parentheses) as a whole. For the alter-
dependent factor equal to +1 (−1) for a decrease (in- nativepossibilityofdirectionindependence,all(cid:2)theterms
crease)inphotonspeedwithanincreasingphotonenergy in the sum become zero except 0Y00 = Y00 = 1/(4π).
(also referred to as the“subluminal”and“superluminal” (n+4)
In that case, we constrain a single c coefficient.
cases). For E (cid:4) EQG, the lowest order term in the se- The coordinates of nˆ are in a Su(In)0-c0entered celestial
ries not suppressed by theory is expected to dominate
equatorial frame described in Section V of Ref. [14], di-
the sum. In case the n = 1 term is suppressed, some-
rectly related to the equatorialcoordinates of the source
thing thatcanhappenifasymmetrylawis involved,the
suchthatitsfirstcoordinateisequalto90◦−Declination
nexttermn=2 willdominate. We notethat ineffective
field theory n=d−4, where d is the mass dimension of and the second being equal to the Right Ascension. Fi-
(n+4)
the dominantoperator. Therefore,the n=1 term arises nally,thecoefficientsc(I)jm canbeeitherpositiveorneg-
fromadimension5operator[35]. Ithasbeenshownthat ative depending on whether LIV-induced dispersion cor-
odd mass-dimension terms violate CPT [13, 36]. Thus, responds to a decrease or increase in photon speed with
if CPT is preserved, then the n = 2 term is expected to an increasing energy respectively (i.e., the sign of the
dominate. In this study, we only consider the n=1 and SME coefficients plays the role of the s± factor of the
n = 2 cases, since the Fermi results are not sensitive to series-expansionframework).
higher order terms. In the important case of a d = 5 modification of the
Using Eq. 1 and keeping only the lowest-order domi- free photon Lagrangian in effective field theory, Myers
nant term, it can be found that the photon propagation and Pospelov have shown that the only d = 5 (n = 1)
speed uph, given by its group velocity, is operator that preserves both gauge invariance and rota-
tional symmetry implies vacuum birefringence [35]. In
(cid:8) (cid:5) (cid:6) (cid:9)
n such a case, and as was mentioned in the Introduction,
∂E n+1 E
uph(E)= (cid:3)c× 1−s± , (2) significantlystrongerconstraintscanbe placedusingthe
∂p 2 EQG existenceofbirefringencethanwithjustdispersion(asin
thiswork). Forthisreason,inthispaperandwhenwork-
wherec≡ limuph(E). Becauseoftheenergydependence ingwithinthegivenassumptionsoftheSMEframework,
E→0
of uph(E), two photons of different energies Eh > El weproceedassumingthatthed=5termsareeitherzero
emitted by a distant source at the same time and from or dominated by the higher-order terms, and proceed to
the same location will arrive on Earth with a time delay constrain the d = 6 terms, which are not expected to
Δt which depends on their energies. We define the“LIV come with birefringence.
4
OuraimistoconstraintheLIV-relatedparametersin-
TABLE I.Distances of AnalyzedGRBs
volved in the above two parametrizations: the quantum
gravity energy EQG (for n = {1,2} and s± = ±1) and GRB Redshift κ1 κ2
the coefficients of the SME framework (for d = 6 for 080916C 4.35 ± 0.15 [41] 4.44 13.50
both the direction dependent and independent cases) . 090510 0.903 ± 0.003 [42] 1.03 1.50
To accomplish this, we first constrainthe total degree of 090902B 1.822 [43]a 2.07 3.96
spectral dispersion in the data, τn, and then using the 090926A 2.1071 ±0.0001 [44] 2.37 4.85
measureddistanceoftheGRBandthecosmologicalcon-
a ThisGRBhadaspectroscopically-measuredredshift,which
stants, we calculate lower limits on EQG through Eq. 3 impliesanerroratthe10−3 level.
and confidence intervals for the SME coefficients using
Eq. 5. We also produce an additional set of constraints
after accounting for GRB-intrinsic spectral evolution ef- and the LAT boresight), the PSF containment radius is
fects (which can masquerade as LIV). In that case, we calculated on a per-photon basis. In this calculation, we
first treat τn as being the sum of the GRB-intrinsic dis- approximate the (unknown) true off-axis angles and en-
persions τint and the LIV-induced dispersion τLIV, then ergieswiththeirreconstructedvalues,somethingthatin-
we constrain τLIV assuming a model for τint, and finally ducesaslighterroratlowenergies. Below∼100MeV,the
constrain EQG and the SME coefficients using the con- LAT angular reconstruction accuracy deteriorates and
straints on τLIV. the95%containmentradiusbecomesverylarge. Tolimit
We employ the cosmological parameters determined the inclusion of background events due to a very large
using WMAP 7-year data ΩM = 0.272 and ΩΛ = ROI radius and also reject some of the least accurately
0.728 [38], and a value of H0 =73.8±2.4 kms−1Mpc−1 reconstructed events, we limit the ROI radius to be less
as measured by the Hubble Space Telescope [39]. than 12◦. The GRB direction used for the ROI’s center
is obtained by follow-up ground-based observations (see
citations in Tab. I) and can be practically assumed to
III. THE DATA coincide with the true direction of the GRB.
Theabovedatasetarefurthersplitandcutdepending
We analyze the data from the four Fermi-LAT ontherequirementsofeachofthethreeanalysismethods
(asdescribedbelow). Figure1showsthelightcurvesand
GRBs having bright GeV prompt emission and
the event time versus energy scatter plots of the GRBs
measured redshifts, namely GRBs 080916C, 090510,
in our sample, and Tab. I shows the GRB redshifts and
090902B, and 090926A. We analyze events passing the
P7 TRANSIENT V6 selection, optimized to provide in- κ1 and κ2 distances (defined in Eq. 4).
creased statistics for signal-limited analyses [40]. Its
maindifferencefromtheearlierP6 V3 TRANSIENTse-
lection used to produce previous Fermi constraints on Time Interval Selection
LIV[22,23]consistsinimprovementsintheclassification
algorithms,whichbroughtanincreaseintheinstrument’s The analyzed time intervals are chosen to correspond
acceptance mostly below ∼300 MeV1. to the period with the highest temporal variability, fo-
We reject events with reconstructedenergies less than cusing on the brightest pulse of each GRB. This choice
30 MeV because of their limited energy and angular re- is dictated by the fact that GRB emission typically ex-
construction accuracy. We do not apply a maximum- hibitsspectralvariability,whichcanpotentiallymanifest
energy cut. In the case of GRB 080916C, however, we as a LIV-dispersion effect (see discussion in Sec. VI for
removed an 106 GeV event detected during the prompt details on GRB spectral variability). By focusing on a
emission, since detailed analyses by the LAT collabora- narrow snapshot of the burst’s emission, we aim to ob-
tion2 showedthatitwasactuallyacosmic-rayeventmis- tain constraints that are affected as little as possible by
classified as a photon. such GRB-intrinsic effects. Starting from this require-
We keep events reconstructed within a circular region ment, we select the time intervals to analyze, hereafter
ofinterest(ROI)centeredontheGRBdirectionandofa referred to as the “default”time intervals, using a pro-
radiuslargeenoughtoaccept95%oftheGRBeventsac- cedure we devised a priori and applied identically on all
cordingtotheLAT instrumentresponsefunctions,i.e.,a four GRBs.
radius equal to the 95% containment radius of the LAT We start by characterizing the brightest pulse in each
point spread function (PSF). Because the LAT PSF is GRB by fitting its time profile with the flexible model
a function of the true photon energy and off-axis angle used by Norris et al. [45] to successfully fit more than
(the angle between the photon true incoming direction 400 pulses of bright BATSE bursts:
(cid:15)
Aexp[−(|t−tmax|/σr)v] t<tmax
I(t)= ‘ (6)
Aexp[−(|t−tmax|/σd)v] t≥tmax,
1 For a detailed list of differences see http://fermi.gsfc.nasa.gov/
ssc/data/analysis/documentation/Pass7 usage.html where tmax is the time of the pulse’s maximum intensity
2 notpublished A, v is a parameter that controls the shape of the pulse,
5
105 105 105 105
104 104 104 104
V) V) V) V)
e e e e
M M M M
y ( y ( y ( y (
g g g g
er er er er
En103 En103 En103 En103
102 102 102 102
24
25
16 22 14
GRB080916C GRB090510 GRB090902B GRB090926A
20
14
18 12 20
ec12 ec16 ec ec
s s s10 s
0 5 0 0
0.310 0.014 0.3 0.315
er er 12 er 8 er
p p p p
ents 8 ents 10 ents 6 ents 10
Ev 6 Ev 8 Ev Ev
6 4
4
5
4
2 2
2
0 0 0 0
-5 0 5 10 15 20 25 30 -2 0 2 4 6 8 10 12 14 0 20 40 60 80 0 5 10 15 20
Time after trigger (sec) Time after trigger (sec) Time after trigger (sec) Time after trigger (sec)
FIG.1. TimeandenergyprofilesofthedetectedeventsfromthefourGRBsinoursample. Eachcolumnshowsaneventenergy
versuseventtimescatterplot(top)andalightcurve(bottom). Theverticallinesdenotethetimeintervalsanalyzed(solidline
forn=1anddashedlineforn=2),thechoiceofwhichisdescribedinSec.IV.Ifadashedlineisnotvisible,itapproximately
coincides with thesolid one.
and σr and σd are the rise and decay time constants. stringent and robust limits obtained by Fermi [23] and
For v = {1,2} the equation describes a two-sided expo- H.E.S.S.[28,29]. Theintervalresultingfromthisexpan-
nential or Gaussian function respectively. We use the sionis the one chosenfor the analysis (hereafter referred
best fit parameters (as obtained from a maximum likeli- to as the “default” interval). The main reason for ex-
hoodanalysis)todefinea“pulseinterval”extendingfrom tending the interval is to avoid constraining the possible
the time instant that the pulse height rises to 5% of its emissiontimeofthehighest-energyphotonsintheinitial
amplitude to the time instant that it fells to 15% of its “pulse interval”to a degree that would imply an artifi-
amplitude. We choose such an asymmetric cut because cially small level of dispersion.
of the long falling-side tails of GRB pulses.
We then expand this initial “pulse interval” until no
photons thatwere generatedoutsideof it (atthe source)
could have been detected inside of it (at the Earth) due
to LIV dispersion, and also until no photons that were
generated inside of it (at the source) could have been
detected outside of it (at the Earth) due to LIV disper- The choiceof time intervalforGRB 090510andn=1
sion. We use conservative values of EQG,1 = 0.5×EPl is demonstrated in Fig. 2. The (default) time intervals
and EQG,2 = 1.5×1010 GeV for the maximum degree for all GRBs are shown in Fig. 1 with the vertical solid
ofLIVdispersionconsideredinextendingthe timeinter- (n = 1) and dashed lines (n = 2), and are also reported
val, values which correspond to roughly one half of the in Tab. II.
6
inducinganon-zerowidthtothedistributionofli,j. Sim-
35 ilarlytothepreviousidealcasehowever,theli,j distribu-
tion will be peaked at approximately τn. For a realistic
30
lightcurveconsistingofoneormorepeakssuperimposed
c
se 25 on a smoothly varying emission, the distribution of li,j
0.06 20 will be composedof a signalpeak centeredat ∼τn (con-
er sistingofli,j valuescreatedprimarilybyeventsi,j emit-
s p 15 ted temporally close and with not too similar energies)
vent 10 and a smoother underlying wide background (consisting
E of the rest of the li,j values).
5 Following the above picture, the estimator τˆn of τn is
taken as the location of the most prominent peak in the
0
-1 -0.5 0 T0im.5e a1fter 1t.r5igge2r (s2e.c5) 3 3.5 4 li,j distribution. This peak becomes taller andnarrower,
thus more easily detectable, as the variability time scale
decreasesandasthewidthoftheenergyrangeincreases.
FIG. 2. Demonstration of the calculation of the default time Searching for the peak using a histogram of the li,j
intervalforGRB090510andlinearLIV.Thedatapointsshow valueswouldrequireus tofirstbinthe data,a procedure
theeventrateforenergiesgreaterthan30MeVfocusingonits that would include choosing a bin width fine enough to
main pulse, the thick curve shows the fit on the pulse using allow for identifying the peak with good sensitivity but
Eq. 6, the dashed vertical lines denote the“pulse interval”,
also wide enough to allow for good statistical accuracy
and the solid vertical lines denotetheextended time interval
in the bin contents. We decided not to use a histogram
(thedefault interval) chosen for theanalysis.
to avoid the subjective choice of bin width. Instead, we
useakerneldensityestimate(KDE),asitprovidesaway
to perform peak finding on unbinned data, and as it is
IV. DATA ANALYSIS
readilyimplementedineasytouse toolswiththe ROOT
TKDE method3. We use a Gaussiankernelfor the KDE
A. PairView and Sharpness-MaximizationMethods and a bandwidth chosen such as to minimize the Mean
Integrated Squared Error calculated between the KDE
Because the way we calculate confidence intervals is and a very finely binned histogram of the photon-pair
identical between PV and SMM, we first describe how lags.
the best estimate of the LIV parameter is calculated by
eachofthese two methods, andthen proceedto describe
their common confidence-interval calculation procedure. Best-Estimate Calculation: Sharpness-Maximization Method
SMM is based on the fact that the application of any
Best-Estimate Calculation: PairView formofspectraldispersiontothedata(e.g.,byLIV)will
smear the light curve decreasing its sharpness. Based
The PV method calculates the spectral lags li,j be- on this, SMM tries to identify the degree of dispersion
tween all pairs of photons in a data set and uses the dis- thatwhenremovedfromthedata(i.e.,whenthenegative
tribution of their values to estimate the LIV parameter. value of it is applied to the data) maximizes its sharp-
Specifically, for a data set consisting of N photons with ness. Thisapproachis similartothe“DispersionCancel-
detection times t1...N and energies E1...N, the method lation”(DisCan)technique[33],the“MinimalDispersion”
startsbycalculatingtheN×(N−1)/2photon-pairspec- method [46], and the “Energy Cost Function” method
tral lags li,j for each i>j: [26, 46]. The most important difference between these
approaches is the way the sharpness of the light curve is
li,j ≡ Etni−−tEjn (7) meWaseusretadr.t the application of SMM by analyzing a data
i j
set consisting of photons with detection times ti and en-
(where n is the order of LIV), and creates a distribution ergies Ei to produce a collection of“inversely smeared”
of their values. data sets, each corresponding to a trial LIV parameter
Let us examine how the distribution of li,j values de- τn, by subtracting Ein×τn from the detection times ti.
For each of the resulting data sets, the modified pho-
pends on the properties of the data and LIV disper-
sion. For a light curve comprising at emission a single ton detection times are first sorted to create a new set
t(cid:5), and then the sharpness of its light curve is measured
δ-function pulse and for a dispersion τn, the li,j distri- i
bution will consist of a single δ-function peak at a value
of exactly τn. For a light curve comprising (at emission)
a finite-width pulse, the now non-zero time differences
betweenthe emissiontimes ofthe eventsbehaveasnoise 3 http://root.cern.ch/root/html/TKDE.html
7
using t(cid:5)i and Ei. After this procedure has been applied Inthatcase,theconstraintsprovidedbySMMwillbeon
on a range of trial LIV parameters,we find the inversely the aggregateeffect.
smeareddatasetwiththesharpestlightcurve,andselect
the trial τn value usedto produce it as the best estimate
ρ=60
of τˆn.
In their analysis of the data from a flare of the blazar m.)
or ρ=40
Mrk 501, the MAGIC collaboration [26] quantified the b. n
sharpnessofthe lightcurveusingan“EnergyCostFunc- ar
S ( ρ=20
tion”, which was essentially the sum of the photon ener- e
ur
gies detected in some predefined time interval chosen to as ρ=10
e
correspondtothemostactivepartoftheflare. Scargleet M
s
al.[33]exploredarangeofdifferentcostfunctionstomea- es ρ=5
n
p
surethe sharpnessofthe lightcurve,including Shannon, ar
h
Renyi, and Fisher information, variance, total variation, S ρ=2
and self-entropy, finding that the Shannon information
is the most sensitive. In this study, we use a function S ρ=1
thatissimilartothe Shannoninformationandisdefined -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
as: τ1(s/GeV)
(cid:16) (cid:17)
N(cid:4)−ρ FIG. 3. Curves of SMM’s sharpness measure S(τn) versus
S(τn)= i=1 log t(cid:5)i+ρρ−t(cid:5)i , (8) tahdeitffreiarelnvtalρuevaolfuteh.eTLhIVesepacruarmveestewreτr1e, egaecnherpartoedduucesdingustinhge
GRB090510-inspired datasetdescribedinAppendixAafter
where ρ is a configurable parameter of the method. the application of a dispersion equal to +0.04 s/GeV (value
denotedwiththeverticalline). Thecirclesdenotethemaxima
Differentvaluesofρwilltunethealgorithmtoevaluate
thesharpnessofthelightcurvefocusingonintervalscon- of the curves, the positions of which are used to produce τˆ1.
Ascan beseen, too small or too large values of ρ correspond
sistingofdifferentnumbersofevents(i.e.,ofρ events)or
toareducedaccuracyformeasuringthepositionofthepeak.
equivalentlyfocusingondifferenttimescales. Asaresult,
thechoiceofρaffectstheperformanceofthealgorithmin
two ways. For a smallvalue ofρ (up to ∼3),some ofthe
durations in the denominator of S(τn) can become rela- Confidence-Interval Calculation
tively verysmall,makingsome ofthe 1/(t(cid:5) −t(cid:5)) terms
i+ρ i
very large. In this case, S(τn) can fluctuate significantly
The PV and SMM methods produce a confidence in-
asafunctionofthetriallag,decreasingtheaccuracywith
tervalonthe bestLIVparameterbymeansofarandom-
whichthe best LIV parametercanbe measured. For too
ization analysis.
large values of ρ the algorithm essentially tries to mini-
Westartbyproducingonehundredthousandrandom-
mizethetotaldurationofthe analyzeddata,focusingon
ized data sets by shuffling the association between ener-
time scales larger than the variability time scale, ending
gies and times of the detected events. Because the total
up with a diminished sensitivity (in practice the peak of
S becomes flatter). These effects are demonstrated in number of events and the distributions of energies and
timesareidenticalbetweentheactuallydetectedandthe
Fig. 3.
randomized data sets, their statistical power (i.e., their
Tochoosethevalue ofρ wefirstgeneratea largenum-
ability to constrain the dispersion) is similar. However,
ber of simulated data sets inspired by the GRB under
because of the randomization,any dispersionpotentially
study, we then apply the method using a series of dif-
present in the actual data is lost. After the set of ran-
ferent ρ values, and finally we choose the ρ value that
domizeddatasetsisconstructed,thebestLIVparameter
producesthemostconstrainingmedianupperlimitonτn
is measured on each one of them and the measurements
(fors± =+1). Thesesimulateddatasetsareconstructed
are used to create a (normalized to unity) distribution
similarly to the procedure described in Appendix A us-
ing a light-curve template produced by a KDE of the fr.
actual light curve, with the same statistics as the data, We then define the measurement error on τn (for the
a spectrum similar to that in the data, and without any generalcaseofanyτn)asE =τˆn−τn andtheprobability
spectral dispersion applied. distribution function (PDF) of E as PE((cid:9)), where (cid:9) is a
Finally, it should be noted that the method’s descrip- randomrealizationofE. We assume thatPE hasa negli-
gibledependenceonτn (atleastfortherangeofvaluesof
tion above was for the case of zero source-intrinsic spec-
tral evolution effects since the light curve of the GRB τn expected to be present in the data) and approximate:
mission at the source was treated as being maximally PE((cid:9))(cid:3)PE((cid:9)|τn =0). (9)
sharp. This picture is equivalent to assuming that there
is an initial (imaginary) maximally-sharp signal that is The PDF PE for the case of a zero τn, PE((cid:9)|τn =0), can
first distortedby GRB-intrinsiceffects and then by LIV. be identified as the normalized distribution fr produced
8
using our randomizationsimulations. Thus, where F(Et,t|τn) is the model for the photonflux which
is incident on the LAT at the photon (true) energy Et
PE((cid:9))(cid:3)fr((cid:9)). (10) andtime t, whereasAeff(Et)4 andD(Et,E)aretheLAT
effectiveareaandenergyredistributionfunctions,respec-
Since E is a quantity with a known PDF and since it tively. As the energy resolution with the LAT is better
dependsonthe unknownparameterτn,itcanbe usedas than15%above100MeV [40]we canneglectanyenergy
a pivotal quantity to construct a two-sided Confidence mis-reconstruction effects. Assuming no spectral vari-
Interval (CI) of Confidence Level (CL) for τn as: ability and that the flux spectrum follows a power law
with possible attenuation at the highest energies, then:
CL=Pr(q(1−CL)/2 <E <q(1+CL)/2) (11)
=Pr(q(1−CL)/2 <τˆn−τn <q(1+CL)/2) F(E,t|τn)=φ0 E−Γ e−E/Ef f(t−τnEn), (14)
=Pr(τˆn−q(1+CL)/2 <τn <τˆn−q(1−CL)/2)
whereΓisthetime-independentspectralindex,Ef isthe
=Pr(LL<τn <UL), cutoff energy, and the function f(t) is the time profile
twhheelroewLeLra=ndτˆnup−pqe(r1+liCmLit)/s2daenfidniUngLth=eτˆCnI−,aqn(1d−qC(1L−)/C2La)/r2e ocafstehoefeamniussllioLnIVth-iantduwcoeudldlabgeτnrEecne.ivWedebeyxptlhaeinLfuArTtheinr
andq(1+CL)/2arethe(1−CL)/2and(1+CL)/2quantiles below how the function f(t) is derived from the data in
practice. Finally, defining the observed spectral profile
of fr. −Γ −E/E
To produce a lower limit on EQG for the subluminal as Λ(E)=φ0 E e f Aeff(E), we obtain:
owritthheitssulopwerelrumorinuaplpcearslei,mwite, ruessepeecqt.iv3elys,ubasntditusotlivnegfτonr P(E,t|τn)=Λ(E)f(t−τnEn)/Npred (15)
EQG. . Thus, the ML estimator τˆn of the LIV parameter τn
satisfies:
(cid:3) (cid:7)
B. Likelihood Method (cid:4)N ∂logf(ti−τnEin) − Nfit ∂Npred =0.
∂τn Npred ∂τn
The ML fit procedure used in this work has been de- i=1 τn=τˆn
veloped and applied by Martinez and Errando [27] to (16)
MAGIC data for the 2005 flare of Mkn 501 and by For the brightest LAT-detected GRBs, Nfit (cid:3) 50 typi-
Abramowski et al. [29] to H.E.S.S. data for the gigantic cally(see Tab.II) thus agoodestimate ofthe sensitivity
flareofPKS2155−304in2006. Thissectiondescribesits offered by the estimator τˆn can be obtained by consid-
key aspects, its underlying assumptions, and the details ering the ideal case of the large sample limit. In this
of its application to GRB data. regime, τˆn is unbiased and efficient like any ML estima-
TheMLmethodconsistsincomparingthearrivaltime tor. Namely,itsvariancereachestheCram´er-Raobound,
ofeachdetectedphotonwithatemplatelightcurvewhich e.g., given by Eq. (9.34) page 217 of [47]:
iqsuashdirfatteidcailnlytoimnethbeyeavnenatm’soeunnetrgdye.peFnodrinagfilixneedarnlyumor- (cid:3) (cid:10) Emax(cid:10) t2 1 (cid:5)∂P (cid:6)2 (cid:7)−1
b{Eeri,otfi}iin=d1e,NpendobensetrvevedenitnstNhefitenweirtghyeannedrgtieims eanindtetrimvaelss V[τˆn]= Nfit Ecut t1 P ∂τn dE dt . (17)
fit
[Ecut,Emax]and[t1,t2],theunbinnedlikelihoodfunction Asthetimeprofilecanbemeasureduptoverylargetimes
is: in case of large photon statistics, one can show that the
N(cid:18)fit standard deviation of τˆn is simply given by:
L= P(Ei,ti|τn), (12) 1
i=1 σ[τˆn]= (cid:2) , (18)
Nfit(cid:7)g2(cid:8)(cid:7)E2n(cid:8)h
where P is the PDF of observing one event at en- (cid:10)
+∞
ergy E and time t, given τn. For an astrophysi- where (cid:7)g2(cid:8) = f(cid:5)(t)2/f(t) dt (= 1/σ2 for a Gaus-
cal source observed by a gamma-ray telescope, it is
−∞
P(Ei,ti|τn)=R(Ei,ti|τn)/Npred, where R is the ex- (cid:10)sian time profile of standard deviation(cid:10)σ), (cid:7)Em×n(cid:8)h =
pecteddi(cid:10)fferenti(cid:10)alcountrateatenergyE andtimetand Emax Emax
Emax t2 Em×n Λ(E) dE / Λh and Λh = Λ(E) dE.
Npred = R(E,t|τn)dE dt is the total number Ecut Ecut
Ecut t1
ofevents predictedby the model. Fora point-like source
observed by the Fermi-LAT:
(cid:10) ∞ 4 Theeffectiveareaalsodepends onthedirectionofthesourcein
R(E,t|τn)= F(Et,t|τn)Aeff(Et)D(Et,E)dEt, instrument coordinates, a typically continuously varying quan-
tity. We can drop the time dependence by approximating
0
(13) Aeff(Et,t)withitsaveragedovertheobservationvalueAeff(Et).
9
The above expression for σ[τˆn] is a good approximation forbetter statisticsandbecausethe calculationsneedan
(within a factor 2 to 3) of the actual standard deviation estimate of the GRB flux at times that are also external
of τˆn, and it gives a useful estimate of the expected sen- to the default time intervals.
sitivity. However,our final results are based on a proper We then proceed with calculating the likeli-
derivationofconfidenceintervalsasdescribedfurtherbe- hood function L for a series of trial values of
low in this section. the LIV parameter τn, and plotting the curve of
The spectral profile Λ(E) is constant with time since −2Δln(L)=−2ln[L(τn)/L(τˆn)] as a function of τn. We
Γ is assumed to be constant during the considered time first produce a best estimate of τn, τˆn, equal to the
interval(see further discussionson possible spectralevo- location of the minimum of the −2Δln(L) curve. We
lution effects in Sec. VI). The spectral profile is also in- also produce a CI on τn for an approximately two-sided
dependent of the LIV parameter, and is only used as CL (90%, 99%) using the two values of τn around the
a weighting function in the PDF normalization Npred. global minimum at τˆn for which the curve reaches a
For these reasons, we approximate the spectral profile values of 2.71 and 6.63, respectively5. Hereafter we
by a power-law function (with a fixed attenuation when refer to these CIs as being obtained“directly from the
needed): data”. In addition, we produce a set of“calibrated”CIs
on τn using Monte Carlo simulations and as described
Λ(E)∝E−γ e−E/Ef. (19) in Appendix B. The calibrated CIs take into account
intrinsic uncertainties arising from the ML technique
Thespectralindexγ isobtainedfromthefitoftheabove (e.g., due to biases from the finite size of the event sam-
function to the time-integratedspectrum S(E) observed ple or from an imperfect characterization of the GRB’s
by the LAT: light curve), and are, most importantly, constructed to
(cid:10) have proper coverage. Our final constraints on the LIV
t2 parameter and the LIV energy scale are produced using
S(E)= F(E,t|τn)Aeff(E)dt. (20)
the calibrated CIs.
t1
As a final note, we would like to stress that the time
In practice, we define a Ecut ∈ [100,150]MeV and we shift τn(cid:7)En(cid:8)l in Eq. (22) has been set to zero following
fit the spectrum S(E) above Ecut (see Fig. 7 for an ex- Refs.[27,29]. Thisimpliesthatthetimecorrectionofany
ample) in order to obtain a fairly good estimate of the evententeringthelikelihoodfunctionisoverestimatedby
spectral index, namely γ (cid:3) Γ within errors (see discus- a factor 1/ηn, with ηn = 1−(cid:7)En(cid:8)l/En ∈ [0.5,1.0] for
sion in Sec. VI regarding this approximation). For the E ∈ [0.1,30]GeV, n = 1 and, e.g., (cid:7)E(cid:8)l = 50MeV. In
case of GRB 090926A,we use a power-lawfunction that principle, ignoringthis time shift would thus produce an
hasanexponentialbreak,inaccordancewiththefindings additionaluncertainty τˆn−τn which is negativeonaver-
of Ackermann et al. [48]. age. This would also slightly distort the likelihood func-
Knowledgeofthetimeprofilef(t)iscrucialfortheML tionsince ηn varieswith photonenergy,possiblycausing
analysis. Typically,Ecut dividestheLATdatasetintwo a reduction in sensitivity. In the large sample limit, one
samples of roughly equal statistics. The ML analysis is can show that the bias of the estimator takes the form
performedusingeventswithenergiesaboveEcut,whereas bn (cid:3)−τn(cid:7)En(cid:8)l(cid:7)En(cid:8)h/(cid:7)E2n(cid:8)h, namely the fractionalbias
the fitofthe lightcurveC(t)observedby the LAT below bn/τn is negativeand decreaseswith increasinghardness
Ecut is used to derive the time profile: of the spectrum. In practice, it ranges from ∼0.5% to
(cid:10) ∼8%forspectralhardnessessimilartothe onesofbursts
C(t)= EcutΛ(E)f(t−τnEn)dE (cid:3)Λl f[t−τn(cid:7)En(cid:8)l], wsteataisntaiclsyzaevda.ilaIbnleaidndoituiorna,ndaluyesistoanthdetolimthiteedrelpahtiovteolny
Emin
(cid:10) (21) small values of τn likely to be present in the data, the
Ecut ratio bn/σ[τˆn] is also negligible (a few percent at most).
whereEmin =30MeV,Λl = Λ(E)dEand(cid:7)En(cid:8)l = One should, however, keep this effect in mind for future
(cid:10) Emin analyses of much brighter sources and/or in case of sig-
Ecut
En Λ(E) dE / Λl. The Taylor expansion used in nificant detections of LIV effects.
Emin
Eq. (21) is justified as LIV-induced lags are effectively
negligible for low-energy events, and it yields the time C. Estimating the Systematic Uncertainty due to
profile: Intrinsic Spectral Evolution
f(t)=C[t+τn(cid:7)En(cid:8)l]/Λl (cid:3)C(t)/Λl. (22) So far we haveconcentratedon characterizingthe sta-
tisticaluncertaintiesofourmeasurements. However,sys-
In practice, we fit the light curve C(t) with a function
comprising up to three Gaussian functions (see for ex-
ample Fig. 6). The fit is performed on events detected
in a time interval somewhat wider than the default time 5 These two values correspond to the (90%, 99%) CL quantile of
intervals (defined in the beginning of Sec. IV) to allow aχ21 distribution.
10
tematic uncertainties can also be very important and value of τint (for a particular observation) is not likely
should be taken into account, if possible. Here, we de- larger than the width of the allowed range of τn, as de-
scribe how we model the dominant systematic uncer- scribed by its CIs7. Thus, we start with the working
tainty in our data, namely the intrinsic spectral evolu- assumptionthatthe widthofthe possiblerangeofτint is
tion observed in GRB prompt-emission light curves. A equal to the possible range of τn (as inferred by our CIs
detaileddiscussionofthisphenomenonisgiveninSec.VI. on it).
For simplicity, in our introduction of LIV formalism Second, we assume that the τint has a zero value on
(Sec. II), we implicitly ignored the presence of GRB- average. This is a reasonable assumption given that we
intrinsic effects (which can in general masquerade as a analyze in this study only cases where there is no clear
LIV-induceddispersion),andinsteadjustusedthequan- detection of a spectral lag signal (i.e., τn is consistent
tity τn to describe the degree of dispersion in the data. with zero within the uncertainty of its measured value).
However, τn describes the total degree of dispersion and Moreover, this also avoids the need for introducing by
is, in general, the sum of the LIV-induced degree of dis- hand a preferred sign for (cid:7)τint(cid:8).
persion, described by a parameter τLIV, and the GRB- In principle, there are infinite choices for a particular
intrinsic degree of dispersion, described by a parameter shapeofτint givenourconstraintforitswidthand(zero)
τint, i.e., mean value. We choose the one that produces the least
stringent (the most conservative) overall constraints on
τn =τint+τLIV. τLIV, by modeling τint so that it reproduces the allowed
range of possibilities of τn. For example, if our mea-
Ourmethodsdonotdifferentiatebetweenthedifferent surements imply that the data are compatible with (i.e.,
sourcesofdispersion. Instead,theydirectlymeasureand they cannot exclude) a positive τn, then we appropri-
constrainttheirsumτn. We caneitherignoreanyintrin- ately adjust τint to match (explain) this possibility. This
sic effects (i.e., assume τint (cid:3) 0) and proceed directly to approachleads to confidence intervals on τLIV that have
constrainLIVusingtheobtainedCIonτn orwecanfirst the largest possible width. Other choices for modeling
assume a model for τint, proceed to constrain τLIV, and τint can produce intervals more stringent either at their
finally constrain LIV using the CI on τLIV. The second lowerortheirupperedge,buttheycannotproducemore
approachis moreappropriateforconstrainingLIV,since stringentoverall(i.e.,whenconsideringboththeiredges)
its results are more robust with respect to the presence constraints. The implementation of our model for τint,
of GRB-intrinsic effects6. defined as Pτint(τ˜int) with τ˜int being a random realiza-
In principle, one could try to model τint using some tionof τint, depends onthe particularmethod PV/SMM
knowledge of the physical processes generating the de- versus ML, and is described separately below.
tectedGRBemissionorpossiblyusingphenomenological For constructing CIs on τLIV with PV and SMM, we
modelsconstructedfromlargesetsofMeV/GeVobserva- use a similar approachas in Eq. 11. However, instead of
tions of GRBs. Unfortunately, because of the scarcity of using as a pivotal quantity E =τˆn−τn, we now use
GRBobservationsatLATenergies,neitherapproachhas
reached a mature enough stage to produce trustworthy E(cid:5) =τˆn−τLIV
androbustpredictionsof GRB spectrallags(at suchen- =τˆn−τn+τint
ergies). Thus, any attempts to model τint would, at this =E +τint. (23)
point, likely end up producing unreliable constraints on
LIV. However,a more robust and conservativeapproach IfwedefinethePDFofE(cid:5)asPE(cid:2)((cid:9)(cid:5)),where(cid:9)(cid:5)isarandom
can be adopted, as follows. realization of E(cid:5), and if q(cid:5) and q(cid:5) are its
(1−CL)/2 (1+CL)/2
Since we do not have a model for τint that reliably (1−CL)/2and(1+CL)/2quantiles,thenstartingfrom
predictsGRB-intrinsiclags,weinsteadchoosetomodelit CL=Pr(q(cid:5) <E(cid:5) <q(cid:5) ),wederiveaCIon
(1−CL)/2 (1+CL)/2
ina waythatproducesthe mostreasonablyconservative τLIV of confidence level CL:
constraints on τLIV.
One of the main considerations behind modeling τint CL=Pr(LL(cid:5) <τLIV <UL(cid:5)) (24)
is the reasonable assumption that our measurements of =Pr(τˆn−q((cid:5)1+CL)/2 <τLIV <τˆn−q((cid:5)1−CL)/2).
τn are dominated by GRB intrinsic effects or in other
words that our constraints on τn also apply to τint. We Similarly to the CI on τn which depends on the quan-
startwiththefactthatwealreadyhaveobtainedacoarse tilesofPE (approximatedbyfr),theCIonτLIV depends
measurement of the possible magnitude of τint, provided
by our constraints on τn. Specifically, we know that the
7 Thealternativecaseofalargeτintbeingapproximatelycanceled
by an oppositely large τLIV seems extremely unlikely since it
wouldrequiretheimprobablecoincidence ofLIVactuallyexist-
6 SincethemajorityofpreviouslypublishedLIVconstraintshave ing,thatthesignofthedispersionduetoLIVbeingoppositeof
not taken into account GRB-intrinsic effects, limits of the first the sign of the dispersion due to intrinsic effects, and that the
approach are still useful for comparing experimental results magnitudesofthetwoeffectsbecomparableforeachofthefour
acrossdifferentstudies. GRBs(a“conspiracyofNature”).