Search for Lorentz Invariance Violation from stacked Gamma-Ray Burst spectral lag data
PPrepared for submission to JCAP
Search for Lorentz InvarianceViolation from stacked Gamma-RayBurst spectral lag data
Rajdeep Agrawal, a Haveesh Singirikonda, a and Shantanu Desai a, a Department of Physics, Indian Institute of Technology, Hyderabad, Telangana-502285, IndiaE-mail: [email protected], [email protected] ,[email protected]
Abstract.
A number of works have found evidence for a turn-over in the spectral lag datafor individual Gamma-Ray Bursts (GRBs), caused by an energy-dependent speed of light,which could be a possible manifestation of Lorentz invariance violation (LIV). Here, we stackspectral lag data from a total of 37 GRBs (with a total of 91 measurements), to verify if thecombined data is consistent with a unified model consisting of intrinsic astrophysical emission,along with another contribution due to LIV. We then carry out Bayesian model comparisonto ascertain if this combined spectral lag data shows a preference for an energy-dependentspeed of light, as compared to an intrinsic astrophysical emission mechanism. We do not finda decisive evidence for such an energy-dependent speed of light for two different models ofLIV. Furthermore, none of the models considered provide a good fit to the stacked spectrallag data. Corresponding author. a r X i v : . [ a s t r o - ph . H E ] F e b ontents h ( z ) using Gaussian Process Regression 6 In special relativity, the speed of light, c , is a Lorentz invariant quantity. However, this ansatz is not valid in many theories beyond the Standard Model of Particle Physics and also variousquantum gravity and string theory models (See [1–3] for reviews of such models). In thesemodels, Lorentz invariance is expected to be broken at very high energies close to the Planckscale ( E pl ∼ GeV), and the speed of light is a function of the energy of the associatedphoton [4]. Alternately, one can think of the vacuum refractive index as been different fromunity in such models [5]. The energy-dependent speed of light can be written as [6]: v ( E ) = c (cid:20) − s ± n + 12 (cid:18) EE QG (cid:19) n (cid:21) , (1.1)where s ± = ± denotes the sign of the LIV, corresponding to sub-luminal ( s ± = +1 ) orsuper-luminal ( s ± = − ); E QG denotes the quantum gravity scale where LIV effects kick in,and n is a model-dependent term and is usually equal to one or two, corresponding to linearor quadratic LIV. The values of n for different LIV models can be found in [7].A plethora of searches for Lorentz invariance violation (LIV) have been carried out usingphotons, neutrinos, and gravitational waves, by looking for the energy dependent speed oflight as given by Eq. 1.1, with s ± = +1 . The astrophysical sources used for searches of LIVwith photons include pulsars [8, 9], AGNs [10–13], and Gamma-Ray Bursts (GRBs, hereafter)[5–7, 14–29]. Reviews of all these astrophysical searches for LIV with these sources can befound in [11, 30, 31]. The corresponding results on LIV with neutrinos can be found in [32–37]. A constraint on LIV using the first gravitational wave event GW150914 has also beenobtained [38].The observable used in almost all the studies of LIV with GRBs consists of spectrallags, which can be defined as the difference in arrival times between high energy and lowenergy photons, and is positive if the high energy photons arrive earlier than the low energyones. The first such systematic study with a large GRB sample was carried out by [5], who– 1 –onsidered a sample of 35 GRBs in the redshift range z = 0 . − . from HETE, BATSE,and SWIFT. The spectral lags were modeled as sum of a constant intrinsic lag togetherwith another contribution due to an energy-dependent speed of light. They also found a σ evidence for the higher energy photons to arrive earlier than the lower energy ones, which atface value points to evidence for LIV [5]. However, when an additional systematic offset wasadded to make the χ /DOF equal to one, the statistical significance for LIV reduced to about σ . Subsequently, they set a lower limit of E QG ≥ (0 . − . × GeV at 95% c.l.The first convincing case for a spectral lag turnover from positive to negative lags, usingmultiple spectral lag data from a single GRB (GRB 160625B) was made in [39]. This workmodeled the time-lag data as a sum of intrinsic astrophysical time-lag and lag due to theenergy-dependent speed of light from LIV. The intrinsic time delay proposed in this work wasa phenomenological model, parameterized as function of the energy. All previous works priorto [39] had assumed a constant intrinsic lag in the source frame. They found that the spectrallag for this GRB shows a turnover at around 8 MeV, indicating a transition from positive tonegative lags. They argued that this transition could be a signature of LIV, which kicks in athigh energies. Their best-fit value for E QG was log( E QG /GeV ) are . +0 . − . and . +0 . − . for linear and quadratic LIV models, respectively [39]. The statistical significance for thisspectral lag transition corresponds to a Z -score of between . − . σ (using frequentisttechniques), and ∆ AIC/BIC > for the quadratic LIV model [40]. The information theorytechniques theorefore point to decisive evidence for the quadratic LIV model.Subsequently, [7] stacked the spectral lag data for GRB 160625B along with the datafor 35 GRBs from [5] and argued that this provides a more robust estimate of the intrinsictime lag. Their best-fit estimates for log( E QG /GeV ) are . +0 . − . and . ± . forlinear and quadratic LIV models, respectively. Most recently, a similar spectral lag transitionfrom positive to negative lags (similar to GRB 1606025B) was detected in GRB 1901114Cat ∼ > , pointing to decisiveevidence using Jeffreys scale. Similar to [7, 39], they obtained best-fit values for linearand quadratic models and their 2 σ bounds are given by log( E QG /GeV ) = . +0 . − . and . ± . , respectively. Furthermore, they also constrained the parameters of LIV StandardModel Extension models [41]. They also computed the χ /DOF for the null hypothesis(consisting of only intrinsic emission) as well as all the LIV models considered. They showedthat all the LIV models have χ /DOF are less than or close to one, indicating a good fit tothe LIV hypothesis [27]. The MAGIC collaboration however failed to find a similar evidencefor an energy-dependent speed of light in the TeV gamma ray data for the same GRB. Usingconservative assumptions on spectral and temporal evolution, the MAGIC collaboration seta lower limit of O (10 ) and O (10 ) GeV for linear and quadratic models, respectively [28].We note that even though some of the above aformentioned analyses using spectral lagdata, obtained bound σ confidence intervals for E QG , which is less than the Planck energyscale, the conclusions of these works only reported one-sided lower limits on E QG , which issame as the central estimate [7, 27, 39]. This does not adhere to the formal way of calculatingone-sided lower (or upper) limits recommended by the PDG [42].Furthermore, as pointed out in [40], the estimated LIV energy scale estimated in [39]would contradict previous limits by approximately 3-4 orders of magnitude [15, 17]. Thispoint would also apply to the recent results in [7, 27]. Still, given the tantalizing hints forLIV in each of the aforementiond works [5, 7, 27, 39], if the spectral lag data for all the GRBs– 2 –an be described by a unified model, one would expect the net statistical significance of theLIV to get enhanced, if we stack the spectral lag data from all the GRBs, and analyze themuniformly with the same model. In this work, we therefore stack the data from all the threeworks which found these hints for LIV signatures in the spectral lag data [5, 7, 27]. We thendo a Bayesian model comparison of the hypothesis that the combined spectral lag data is acombination of both an intrinsic and a LIV-induced lag, as compared to only an intrinsic lagdue to astrophysical emission. The model assumed for the intrinsic astrophysical emission issame as that proposed in [39], which has also been used in [7]. The outline of this paper isas follows. We briefly discuss the model comparison technique in Sec. 2. The datasets usedfor this analysis are described in Sec. 3. The analysis procedure used to analyze the GRBspectral lag data is discussed in Sec. 4. The reconstruction of the cosmic expansion historyusing chronometers is described in Sec. 5. Our results are presented in Sec. 6 and we concludein Sec. 7. Since a central theme of this work is model selection, we give a brief primer on these tech-niques. There are three main methods used for model comparison: frequentist, informationtheory, and Bayesian. A comparison and contrast of these methods can be found in [43–50]. In our previous works, we have applied all these techniques to a large number of modelselection problems in Astrophysics and Cosmology [40, 51–59]. In this work, we shall onlyuse the Bayesian method for model comparison, as this is the most robust among the varioustechniques and does not involve any assumptions [46]. We provide a brief prelude to theBayesian model comparison technique. For more details, the reader can refer to [46–48, 50](and references therein).Bayesian Model comparison is based on the Bayes Theorem in probability, which states: P ( M | D ) = P ( D | M ) P ( M ) P ( D ) (2.1)for a model M with respect to data D . Here, P ( M | D ) represents the posterior probabilityand P ( D | M ) is the marginal likelihood, also known as the Bayesian Evidence. This can bedefined as: P ( D | M ) = E ( M ) = (cid:90) P ( D | M, θ ) P ( θ | M ) dθ (2.2)where θ is the vector of parameters associated with the model M , P ( θ | M ) is the prior onthe the parameter vector ( θ ) for that model. To perform model comparison between twomodels M and M , we calculate the Posterior odds ratio, which is the ratio of their posteriorprobabilities. So, the odds ratio for model M over M is given by: O = P ( M | D ) P ( M | D ) (2.3)Using Eq. 2.1 and 2.2, this can be further written as: O = E ( M ) P ( M ) E ( M ) P ( M ) = B P ( M ) P ( M ) (2.4)where the term B is the Bayes Factor, given by the ratio of Bayesian Evidences of the twomodels. If we were to assume equal apriori probabilities for both models, the Odds Ratio is– 3 –he same as Bayes Factor. Therefore, we obtain: O = B = (cid:82) P ( D | M , θ ) P ( θ | M ) dθ (cid:82) P ( D | M , θ ) P ( θ | M ) dθ (2.5)The Bayes factor is then used for Bayesian model comparison. Note that unlike other modelselection techniques, Bayesian model comparison does not involve the computation of best-fitparameters for a given model.The model with the larger value of Bayesian evidence will be considered as the favoredmodel. We then use the Jeffrey’s scale to qualitatively assess the significance of the favoredmodel [48]. According to this scale, a Bayes Factor < indicates negative support for themodel in the numerator ( M ), thereby favouring the model M . A value exceeding 10 impliesstrong evidence for M , while a value greater than 100 indicates decisive evidence. Therefore,a smoking gun evidence for model M over M requires a Bayes factor greater than 100. We briefly describe each of the datasets analyzed in [5, 27, 39] that are used for our stackedanalysis. More details can be found in the original papers and references therein.Ellis et al [5] considered a sample of 35 different GRBs (with one spectral lag per GRB)in the redshift range . < z < . , detected by three different telescopes, viz. BATSE,HETE, and Neil Gehrels SWIFT. For BATSE data, the spectral lags were measured in the115-320 keV band relative to the 25-55 keV band. The Neil Gehrels SWIFT and HETE datawere also normalized to the same energy bands. The spectral time-lag data along with theerrors for all the 35 GRBs can be found in Table 1 of [5]. The lower and higher energyintervals corresponding to these time lags were assumed to be the centers of the energy bins,viz. 40 keV and 217.5 keV, respectively. The error in energy difference was the quadraturesum of the half-widths for the two energy bins.The second dataset used in this work is the spectral lag data for GRB 160625B analyzedin [39]. This GRB is located at a redshift of z = 1 . . This GRB was detected by bothFermi-GBM and Fermi-LAT. The light curve for this GRB contained three isolated sub-bursts, lasting about 770 seconds. This long duration facilitated the measurement of 37spectral lags in the 15-350 keV intervals, with respect to a fixed energy of 11.34 keV. Thespectral lag data along with their error bars can be found in Table 1 of [39]. The error inthe energy difference was equal to the half-width of each energy bin.The final dataset we used consists of 19 spectral lag measurements of GRB190114C,located at a redshift of z = 0 . [27]. This GRB was detected by the Neil Gehrels SWIFTtelescope with T ranging equal to 362 and 116 seconds in 15-350 keV and 50-300 keV,respectively. This GRB was also detected by Fermi-LAT at MeV energies and by the MAGICtelescope up to TeV energies [60]. Using the Fermi-GBM light curves, [27] constructed thespectral lag data from 15 keV to 5000 keV, compared to the lowest energy value of 12.5 keV.These spectral lags along with their errors can be found in Table 1 of [27]. Similar to GRB160625B, we used the half-widths of each energy bin to characterize the error for energy value.Finally, we combine all the aforementioned spectral lag measurements. We therefore geta total of 91 spectral lag measurements from 37 GRBs, located at different redshifts, coveringa range from 0.25 to 6.29. Since all the GRB redshifts were spectroscopic, we do not considerthe error in their redshifts for our analysis, as they are expected to be negligible.– 4 – Model used for time lags
The observed spectral time lag ( ∆ t obs ) for a given GRB for an energy interval ∆ E can bemodeled as a sum of two delays: ∆ t obs = ∆ t obsint + ∆ t LIV (4.1)where ∆ t obs is the observed spectral lag; ∆ t obsint is the intrinsic time delay due to astrophysicalemission in the observer frame, and ∆ t LIV due to LIV.Many initial works on searches for LIV assumed a constant intrinsic time delay in thesource frame [5, 20–24, 29, 61]. We shall use the following model for the intrinsic time emissionin the source frame [39], ∆ t int = τ (cid:104)(cid:16) EKeV (cid:17) α − (cid:16) E KeV (cid:17) α (cid:105) (4.2)This model for the intrinsic emission in the source frame (first introduced in [39]) wasobtained from the statistical properties of 50 single-pulsed GRBs [61]. This parametric formfor the intrinsic emission has been used for LIV searches using spectral lags in a number ofworks [7, 24, 39, 40, 62]. Du et al [27] used a slight variant of Eq. 4.2, where α was replacedby − α and τ by - τ . For this analysis, we shall choose broad priors on α and τ to encompassboth these variants. We note however that the detailed emission mechanism of GRBs isstill unknown, and there could be other astrophysical parameters characterizing the intrinsicmechanism. Ref. [63] has found a correlation between spectral and spectral evolution. In thiswork however, use Eq. 4.2, since it has also been used in the earlier works.The intrinsic time delay in the observer frame is given by ∆ t obsint = (1 + z )∆ t int , (4.3)where the (1 + z ) term accounts for the cosmological time dilation. For the null hypothesis,the spectral lags will be directly fitted to only Eq. 4.3.The time delay due to linear and quadratic LIV models can be obtained from n = 1 and n = 2 , respectively of the following equation [64], ∆ t LIV = t low − t high = − (cid:18) n H (cid:19) (cid:32) E n − E n E nQG,n (cid:33) (cid:90) z (1 + z (cid:48) ) n h ( z (cid:48) ) dz (cid:48) (4.4)where E QG,n is the Lorentz-violating or quantum gravity scale, above which Lorentz violationis turned on. In Eq. 4.4, n = 1 represents linear LIV and n = 2 quadratic LIV. Finally, h ( z ) ≡ H ( z ) H is the dimensionless Hubble parameter as a function of redshift. For the currentstandard Λ CDM model [65], h ( z ) = (cid:112) Ω M (1 + z (cid:48) ) + Ω Λ and this parametric form has beenused for the analysis in [5, 24, 27, 39, 40]. Here, we reconstruct h ( z ) in a non-parametricmanner using Gaussian Process Regression (GPR) similar to the analysis in [7]. The integralin Eq. 4.4 encompassed in the parameter K ( z ) defined as K ( z ) = (cid:90) z dz (cid:48) (1 + z (cid:48) ) n h ( z (cid:48) ) (4.5)The estimation of K ( z ) in a model-independent fashion using GPR similar will be discussedin Sec. 5. We note that ∆ t LIV for other LIV models such as Lorentz Violating Standard– 5 –odel Extension [41] has also been evaluated in [27]. However, we shall not analyze thesemodels as they have a large number of free parameters.The last ingredient which we need for our analysis is the total error ( σ tot ) in the observ-ables. This includes the error in the observed spectral lag ( σ t )) as well as the error in theenergy ( σ E ). We use the prescription in [66] to combine these errors and the total error σ tot is given as: σ tot = σ t + (cid:16) ∂f∂E (cid:17) σ E . (4.6)where f is the particular model been tested. In this section, we shall discuss the calculation of Eq. 4.5 in a model-independent fashionusing cosmic chronometers, which is agnostic to any particular Cosmology.
In Equation 4.5, K ( z ) is generally obtained from the underlying cosmological model. UsingGPR, we obviate this requirement and reconstruct h ( z ) directly from the data. We use Hubbleparameter measurements from cosmic chronometers (CC) [67], which is a model independentmethod for measuring the Hubble parameter using the redshifts and relative ages of galaxies.For this work, we use the use the same chronometer dataset, as that in [55], which consistsof 31 data points covering the redshift range . < z < . [68].The cosmic chronometer technique uses the following way defining the Hubble parame-ter [67]: H ( z ) = −
11 + z dzdt (5.1)For small changes in z and t , this relation can be generalized to H ( z ) = −
11 + z ∆ z ∆ t (5.2)Therefore, through the measurements of redshifts and ages from spectroscopic analysis ofgalaxies in this formula, the value of the Hubble parameter ( H ( z ) ) can be estimated at aparticular redshift z . The added advantage here is that only the relative ages of the galaxiesare needed for this measurement. Once we obtain sufficient measurements of H ( z ) at differentredshifts, one can interpolate (or extrapolate) between these measurements to get H ( z ) atany input redshift. Therefore, no underlying cosmological model is used to reconstruct H ( z ) .We now discuss the technique used for reconstructing the expansion history at any redshiftusing these chronometers. h ( z ) using Gaussian Process Regression The GPR technique has seen increased usage in Astronomy literature as it has proved to bea useful tool for model-independent analyses (see [55] and references therein). Simply put,the purpose of GPR is to reconstruct a quantity from a data set without the assumptionof a parametric model. The popular choices for model-independent analyses are to assumea parameterization, such as different kinds of polynomials, for a function. This choice ofparameterization can be arbitrary sometimes and there’s no rule written in stone about whichchoice is better. The advantage of using GPR is that it completely removes the need for– 6 –hoosing a parametric form for a given function, and can reconstruct the quantity despitethis. The premise for this technique lies in the idea of Gaussian distribution extended tofunctions. One requirement for this method is that the errors in the data used must beGaussian. A covariance function is used to connect the points at which the data is availableto the other points in space. This covariance function will help in predicting the value atthis point from the information available from the data set. The most common choice for thecovariance function is the squared exponential function, which is given by k ( x, ˜ x ) = σ exp (cid:18) − ( x − ˜ x ) l (cid:19) (5.3)This function is the simplest choice which can serve the purpose for GPR. There aremany other alternatives like the Matérn and Cauchy kernels. This covariance function canbe written in the form of a matrix, for a set of input points X as [ K ( X , X )] i,j = k ( x i , x j ) (5.4)Another requirement for this is the choice for the function µ ( x ) , which is the apriorimean of this quantity. A constant function is a good choice for this. Now this can be used toextrapolate and determine the mean and errors at other points in space. (cid:104) f ∗ (cid:105) = µ ∗ + K ( X ∗ , X )[ K ( X, X ) + C ] − ( y − µ ) ,cov ( f ∗ ) = K ( X ∗ , X ∗ ) − K ( X ∗ , X )[ K ( X, X ) + C ] − K ( X, X ∗ ) , (5.5)where X ∗ represents the points at which we want to predict the values for the quantity f ( x ) , (cid:104) f ∗ (cid:105) is the mean value predicted for the function f ( x ) at X ∗ , cov ( f ∗ ) is the error on thesevalues, y is the set of values of f ( x ) available from the data, and C is the covariance matrixfor the data set (for uncorrelated errors, this is simply diag ( σ i ) ). The set X represents thedata set. A much more detailed explanation of GPR can be found in [69]. We implementGPR using the publicly available code GaPP in Python [69].After a successful reconstruction of H ( z ) (and h ( z ) ) using GPR, we can estimate thevalue of any function of h ( z ) at any redshift along with its σ error bars. Using this re-constructed h ( z ) , the integral in Eq. 4.5, required to compute K ( z ) can be computed usingany standard numerical integration algorithm. In this work, we use the quad function in the scipy Python module.
We now discuss the analysis of the stacked spectral lag data. The first step in any modelcomparison involves parameter estimation. For this purpose, we write down the followinglikelihood ( L ) for the given model: L = N (cid:89) i =1 σ t √ π exp (cid:26) − [∆ t i − f (∆ E i , θ )] σ t (cid:27) , (6.1)where ∆ t i denote the spectral lag data corresponding to the energy intervals ∆ E i ; σ t denotesthe total uncertainty as discussed in Eq. 4.6; f (∆ E i , θ ) is the hypothesis used to fit the– 7 – arameter Prior Minimum Maximum α Uniform -1 1 τ Uniform -10 10 log ( E QG /GeV ) Uniform 6 19
Table 1 . List of priors used for the parameters of the three models (cf Eq. 4.1). Note that E QG isnot used for the null hypothesis of only intrinsic emission. data (cf. Eq. 4.1); and θ is the vector of parameters used to fit each hypothesis. For the nullhypothesis, ∆ t LIV would be equal to 0. The best-fit values for each of the models are obtainedby maximizing the posterior P ( θ | D, M ) ∝ P ( D | M, θ ) P ( θ ) [48], where P ( θ ) represents thepriors for each of the models. The priors used for each of the three models can be found inTable 1.Although our main goal is model comparison, we would like to get a sense of how goodthe best fits for each of the models are. For that, we use χ = − L , where L is definedin Eq. 6.1. For a good fit, χ /DOF ∼ , where DOF is the number of data points minustotal number of free parameters. Although this calculation of DOF assumes that the modelis linear as a function of the free parameters [70], this reduced χ provides a useful rule ofthumb to check if the fit is good. The χ /DOF are shown in Table 3. We can see that noneof the three models can adequately fit the data since χ /DOF ∼ for all the models. Thisis in accord with the results in [40], who also found that none of the three models provide agood fit to the spectral lag data for GRB 1606025B. However, [27] had found that all the LIVmodels provide a good fit to the spectral lag data of GRB 190114C. Therefore, our resultsshow when we stack the spectral lag data from different GRBs, no one model provides arobust description of the time lags.The corresponding 68% and 90% marginalized credible intervals for all the free param-eters can be found in Figs. 1, 2, and 3 for the null hypothesis, n = 1 LIV, and n = 2 LIVmodels, respectively. Unlike [7, 27, 39], we do not get closed contours for E QG (with E QG less than Planck scale) for the linear LIV model. Consequently, we cannot obtain bound 1 σ marginalized point estimates for E QG for the linear model. We get closed contours only forthe quadratic LIV model, with the marginalized central estimate for E QG = 7 . +0 . − . × GeV. The marginalized central estimates for all the other parameters can be found in Table 2.Therefore, for the linear LIV model, we can only calculate lower limit for E QG . Forthis purpose, we use the same method as in [5], which we briefly describe. We calculatethe marginal likelihood ( L marg ) over the nuisance parameters ( τ and α ), and obtain the 68%lower limit on E QG by solving the following equation E ∞ (cid:82) E QG L marg ( x ) dx E ∞ (cid:82) E L marg ( x ) dx = 0 . , (6.2)where similar to [5], E ∞ indicates the maximum value used for fixing the normalization andis chosen to be the Planck scale equal to GeV. E is the lower limit used in calculatingthe integral is equal to GeV, which is the lower limit of our prior on E QG (cf. Table 1).This lower limit on E QG for the linear LIV model can be found in Table 2.– 8 – ntrinsic (n=1) LIV (n=2) LIV α . ± .
02 0 . ± .
02 0 . ± . τ (sec) . ± . . ± . . ± . E qg / GeV > . × (68% c.l.) . +0 . − . × Table 2 . Best-fit values of the models for the three hypotheses considered. The intrinsic hypothesisis given by Eq. 4.3 and the two LIV models are obtained by plugging n = 1 , in the relevant equationin Eq. 4.1. For the linear LIV model, we only report 68% c.l. lower limits on E QG . No LIV (n=1) LIV (n=2) LIV χ / DOF
Table 3 . Bayesian statistical significance of Lorentz invariance violation (LIV) for the two models(linear and quadratic LIV) as compared to the null hypothesis of only intrinsic emission. We alsoprovide the χ /DOF for all the three models. We can see that none of the three models provide agood fit to the spectral lag data since χ /DOF >> . The Bayes factor show negligible evidence forthe linear LIV model and strong evidence for the quadratic LIV model. To calculate the Bayesian evidence, we again use the same priors described in Table 1.The evidence was computed using the
Dynesty [71] package in Python, which is based on theNested sampling algorithm [72, 73]. As discussed in Sec. 2, the Bayes factor is the ratio ofthe Bayesian Evidence. These Bayes factors are shown in Table 3. We can see that the Bayesfactor for the n = 1 is close to one. According to Jeffrey’s scale [48], this corresponds toinconclusive evidence for any of the linear LIV model. For the quadratic LIV model, we geta Bayes factor of about 25. According to the Jeffreys scale, this only corresponds to strongevidence and not decisive evidence. Therefore, we conclude that when we stack the spectrallag data for GRB 160625B, GRB 190114C, and the 35 GRB sample analyzed in [5], Bayesianmodel comparison does not show decisive evidence for either the linear or quadratic LIV overpure astrophysical emission. A large number of works have analyzed the spectral lag data of individual GRBs, to look foran energy dependent speed of light, characteristic of a signature of LIV. The spectral timelags have been modelled as a sum of intrinsic time lags due to astrophysical emission alongwith an energy dependent speed of light, characteristic of Lorentz violation. These workshave found found evidence for LIV with varying levels of significance [5, 7, 27, 39, 40]. Ifthese spectral time lags can be self-consistently described by a unified model, which includesLIV, one would expect the significance to be enhanced, when one stacks the spectral lag datafrom different GRBs.Therefore, to test this ansatz , we stack the spectral lag data from GRB 190114C (19lags) [27], GRB 1606025B (37 lags) [39] and 35 additional lags from 35 different GRBs (withone lag per GRB) [5]. Therefore, in all we have a total of 91 spectral lag measurements from37 GRBs. We then analyze the stacked data using the unified model described in Eq. 4.1. Themodel for the intrinsic astrophysical contribution to the spectral lag is described in Eq. 4.2).– 9 – = 0.536 +0.1240.097 . . . . . . = 0.157 +0.0180.018 Figure 1 . The marginalized 68% and 90% credible regions for the parameters of the null hypothesisof only intrinsic astrophysical (cf. Eq. 4.2). The marginalized best-fit estimates for τ and α are shownin the figure. The contribution due to the energy dependent speed of light can be found in Eq. 4.4. One alsoneeds to model the cosmic expansion history in order to evaluate Eq. 4.4. For this purpose,(similar to [7]), we have reconstructed this is a non-parametric method using GPR withindividual H ( z ) measurements from cosmic chronometers [55]. We then carry out a Bayesianmodel comparison to check if the stacked data show evidence for an energy dependent speedof light caused by LIV, over purely intrinsic astrophysical emission. The priors on the theparameters for these models used for regression and model comparison can be found in Table 1.The marginalized credible intervals for the parameters of the intrinsic only, combinedintrinsic and linear LIV model, combined intrinsic and quadratic LIV model can be foundin Figs 1, 2, 3, respectively. For the linear LIV model, we do not obtain closed σ bounds.Therefore, we obtain 68% lower limits on E QG . This lower limit along with the marginalizedestimates for the other parameters in all the three models can be found in Table 2.As a sanity check on the viability of each of these models using the stacked data, we– 10 – igure 2 . The marginalized 68% and 90% credible regions for the linear LIV model, corresponding to n = 1 in Eq. 4.4. We do not get a closed contour for E QG . Therefore, marginalized central estimatesfor E QG cannot be defined. The marginalized best-fit estimates for τ and α can be found in the figure. calculate the reduced χ (cf. Table 3). We find that the reduced χ for all the three modelsare O (10) , indicating that none of the three models can adequately fit the stacked data. Thisis contrast to the results in [27], who had found that the spectral lag data of GRB 190114Cshow reduced χ of around one for all the LIV models.Our results for Bayesian model comparison between the two LIV models and the nullhypothesis (of only intrinsic astrophysical emission) can be found in Table 3. We find that theBayes factor for the linear LIV model is close to one, indicating that there is no preference forit. The Bayes factor for the quadratic LIV model is about 25, indicating strong evidence forthe quadratic model. However, the Bayes factor does not cross 100, needed to claim decisiveevidence. Therefore, we find that with the stacked data, the Bayes factor for the LIV models– 11 – og( E QG ) = 7.168 +0.0740.055 . . . = 0.319 +0.0830.060 . . . log( E QG ) . . . . . . . = 0.209 +0.0220.023 Figure 3 . The marginalized 68% and 90% credible regions for the quadratic LIV parameters, corre-sponding n = 2 defined in Eq. 4.4. gets degraded compared to the decisive evidence found for GRB 190114C.Therefore, we conclude that the stacked GRB spectral lag data cannot be uniformlyexplained with the same intrinsic emission model and an energy-dependent speed of light.It is possible that one needs more complicated GRB-specific model for the intrinsic spectrallag, which depends on additional parameters besides the energy. However, with more GRBdata with spectral measurements ranging from KeV to TeV ranges, one could get betterstatistics with more samples having a spectral turnover, which could enable us to disentanglethe astrophysics from possible LIV signatures. This should soon be possible with the adventof CTA, LHASSO, and other gamma-ray observatories References [1] J. D. Tasson,
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