An unbinned test for Quantum Gravity effects in highenergy lightcurves
Abstract
Some models of quantum gravity can predict observable effects on the propagation of light: most notably an energy dependent dispersion, where the speed of light is seen to vary with the energy of the photon. As quantum gravity effects should appear at the Planck scale they will be very small and so require very high energy photons to travel large distances before even becoming noticeable. Precisely because this effect is greater for the most energetic photons (dt 10 s/TeV/Gpc), groundbased gammaray measurements of large AGN flares are the ideal resource for performing such tests. The modest photon flux combined with the fact that these experiments are capable of recording the photon times with great resolution suggests the use of unbinned algorithms as an optimal solution for testing models of quantum gravity. In this paper we discuss the application of a nonparametric test to such datasets, analysing its limitations and exploring the potential benefits.
Lorentz invariance violation, statistical methods, quantumgravity
1 Introduction
The study of phenomena in which both quantum mechanical and general relativistic effects are important, motivated theoretical efforts to construct a theory capable of describing gravitation at the subatomic level: the socalled Quantum Gravity (QG) theories. One of the most fundamental results, common to several competing approaches to QG, is the quantization of the spacetime continuum, which appears in the form of a spacetime uncertainty relation (e.g., [8]).
A consequence of this discreteness of spacetime is that the vacuum
will interact with energetic photons on the Planck scale, acting
analogously to a medium that absorbs and reemits radiation by
excitation of its internal degrees of freedom [11].
Lorentz invariance violation (LIV) arises in this context due to a
modified dispersion relation for the photon, resulting from a
nontrivial, spectral dependent refractive index for the vacuum of
the form
For photons of energies , the perturbed dispersion relation can be approximated by a series expansion of the form [5]:
(1) 
Despite the vanishingly small velocity corrections, of the order of for a 1 TeV photon, the observation of extragalactic gammaray sources such as gammaray bursts (GRBs) and active galactic nuclei (AGNs) is a promising laboratory to test this prediction of QG theories. This is because the variations on the speed of light, integrated over the large propagation distances of the photons, result in sizeable delays that could be directly measured by highaccuracy timing experiments [5], which including the cosmological effects of propagation on an expanding universe [7]:
(2) 
where and are respectively the Hubble constant and its associated dimensionless parameter [13], and is the redshift of the source. In the analysis of broad spectral band lightcurves, this delay will manifest as a timelag between the arrival times of the lowest and the highest energy photons of s Gpc TeV.
Traditionally, highenergy experiments have drawn from this principle and, by splitting the lightcurves in two or more energy bins, have looked for significant shifts in the times of bursts or sharp features between them, deriving upper limits to the magnitude and energyscale of the QG effects. In the following section we will briefly review the current status and results of these searches, before proceeding to the presentation of our method.
2 Current Results of Timelag Measurements
In recent years, several highenergy experiments have begun to perform timing analysis in order to identify energy dependent lags in the light curves of distant sources such as GRBs and AGNs. In principle, the former would be the preferred targets for the study because of their large distances and extremely short burst features, reaching down to subsecond and even millisecond timescales ([12] and [6]). The advent of FERMI brings great prospects to the search for LIV signatures, due to a significant increase in sensitivity. The most constraining GRB results to date come from recent FERMI observations of GRB080916C, and give a robust lower limit of to the energy scales of QG [10].
Nevertheless, AGN observations with groundbased gammaray telescopes carry the advantage of observations at much higher energies, increasing the magnitude of the QGinduced lags one is seeking to tens of seconds. Recent results by HESS [1] and MAGIC [3] provide lower limits for the onset of QG effects of GeV and GeV respectively, in agreement with the newest GRB results. For specialized reviews of these latest results see [9] and [17].
3 Unbinned Tests for the Detection of Photon Dispersion
Given the discrete nature of the highenergy data, tests that exploit the full information content of the lightcurves by looking at individual photons are a natural choice to exploit the maximum sensitivity of the experiments. The new method we propose for detecting spectral timelags in the lightcurves of highenergy sources has its fundamental idea drawn from the original approaches of [4] or [16]. It consists of using the linear approximation to the energydependent delay given in equation (2) to apply a systematic correction to the arrival times of each individual recorded photon of the form:
(3) 
so as to cancel any putative QG effects on photon propagation. Since the applied correction is to be the exact inverse of the original dispersion, the optimal correction is a direct estimate of the QG energy scale and dispersion magnitude .
The QG signature is asymmetric, always introducing a dispersion to the original burst profile. Therefore the correction assumed to most closely cancel the timeofflight delays is expected also to return a burst profile which is maximally sharp, according to an appropriately chosen measure, so that the problem is then reduced to that of the maximization of a cost function. In [16], the two proposed cost functions are the Shannon Information (or an alternative informationentropy measure) and the average intrapulse photoninterval, whereas in [4] the authors sought to maximize the total power of the burst around its maximum.
In any approach of this sort there are two basic assumptions in play, which represent limitations to the method:

all energydependent dispersion corrected in the cancelation algorithm is supposed extrinsic to the source and due to QG, since we cannot account for effects intrinsic to the emission process;

the maximally sharp burst retrieved by the linear correction is assumed to be an accurate representation of the original burst profile.
Whilst (i) is an unavoidable condition in (preferential)
nonparametric approaches
4 Kolmogorov distance
Given two random variables and in , the simplest measure of the difference between their probability distributions (pdfs) is the Kolmogorov distance, introduced by Kolmogorov in 1933 as a metric for random variables in probability space (see [14] and [18]). For prob and prob, cumulative distribution functions (cdfs) of and respectively, the Kolmogorov distance is defined as
(4) 
the maximum vertical distance between the two cdfs.
Given a broad spectral range lightcurve with sufficient photon statistics, we can meaningfully bin the data in low and high energy bands, creating two lightcurves that should in principle superpose, provided that the high and low energy photons were produced simultaneously at the source, without any intrinsic net delay (same as condition 1 from last paragraph). After propagation, if QG effects are present, the profiles of any given burst in the light curve will differ in the two energy bands, due to the different amount of dispersion of the photons, so that the high energy ones will be more strongly shifted towards later arrival times.
Following [16], we represent the bursts by a normalised
probability distribution. We construct a photon cell at the place of each photon , where is the
waiting time of each photon and is then indicative of the
photon density at each time. We then transform these densities into
normalised probability measures by defining for every cell
A natural way to quantify the relative dispersion suffered by the low and high energy components of the burst is to calculate the Kolmogorov distance (Kdist) between the two constructed pdfs. By doing so, we are using the less affected (and usually better sampled) low energy burst profile as a reference for the process of finding the best cancelation parameter to the dispersion of the highenergy lightcurve, which is more sensitive to the dispersion.
By applying a simultaneous correction as in equation (5) to the arrival times of each photon in the low and high energy bins, we want to find the optimum correction which minimizes the Kdist between the two bursts:
(5) 
corresponding to the QGinduced delay. It is important to note that
the cdf is an ideal (and simple) representation to be used for this
purpose of comparing two distributions, which acts like a fitting
measure of the two profiles as the temporal dispersion is
canceled.
5 Performance of the Method
Following [5], we define a sensitivity factor
(6) 
where represents the relative delay that two photons of different energies acquired on their travel from the source as in equation (2), and the width of the burst under study. This parameter measures the power of the method in relation to the size of the lightcurve features, which is the most important factor in detecting the delays, and by definition will always be calculated relatively to the delay suffered by a photon of energy equal to the average energy of all events in the burst.
Typical values for the delays are so small ( 110 s/TeV) that in order to detect their effect we have to either rely on the observation of extremely short bursts, for which [6], or on analysis methods sensitive to small deformations of the burst profile. Apart from GRBs for which is frequent, for AGNs detected in the TeV range, the best case to date is from the large flare of PKS 2155304 in 2006 [2]: its shortestduration burst of min and average photon energy of TeV, imply a sensitivity factor .
The top panel of figure 2 shows the simulated performance of the Kdist method in function of for a source with spectral index 2.5 at a distance of 500 Mpc. The sensitivity factor is calculated assuming . Each simulated burst is a MC realisation of an inhomogeneous poisson process with 1000 events distributed according to a gaussian rate function. The three adjacent points for each value of represent respectively 0, 10 or 20% error in the energy resolution of the observations, which is introduced in the simulation at the moment of the correction for the lag. We can see that the method is only little affected by it, and that the energy error introduces a small systematic underestimation on the size of the lag, but always compatible with the true lag within one standard deviation.
The method as it stands is capable of detecting delays within 3 for (), corresponding to a burst of 30 s duration for a source at 500 Mpc. Taking a feature equivalent to the shortest burst in the PKS 2155304 large flare, we can probe the presence of QG effects in a scale up to GeV, a factor of 10 above the Planck energy, and of the order of the most accurate upper limits to date on the QG scale.
Finally, we have tested how the performance of the method depends on the source spectral index and how it varies in the presence of multiple overlayed bursts; the results are presented in figure 3. To test the effect of burst superposition, we generated 4 sets of 10000 MC realisations of a lightcurve each, consisting of two identical gaussian bursts separated by 1, 2, 3 and 5 times the individual profile widths, respectively. We can see from the upper plot of figure 3 that the superposition affects the performance of the method as it broadens the effective width of the feature in which the lag must be detected. The situation is progressively worsened as the distance between the individual bursts increase, further spreading the combined profile, until both are completely separated (around 3 sigma separation), and the curve reaches its assymtoptic limit. At this point, in practice, the individual bursts should be treated individually for better results, which should not represent a problem as long as their separation is comparable or larger than the average photon lag to be tested; otherwise, there could have occurred significant ”leakage” of photons between the bursts, which might reflect on the effectiveness of the cancelation method.
The bottom panel of figure 3 shows the sensitivity curve in function of the photon index of the burst, which presents an optimal minimum around an index of 2.5, for a fixed minimum energy threshold. All bursts generated for testing this effect had the same number of photons (1000) and the energy boundaries of the low and highenergy profiles are chosen so that the average energy difference between the two profiles and the number of photons in each of them are maximised. The global minimum of the curve at 2.5 results from the fact that for very steep or very hard photon indexes both these factors cannot be ideally optimised.
6 Conclusions and Possible Variations on the Method
We have exposed here the general accords of an alternative method to test for energy dependent lags in HE lightcurves. The method draws inspiration from the unbinned dispersioncancelation algorithms independently derived by [16] and [3], but it evolves from a maximum sharpness costfunction approach to the minimization of an appropriate distance metric between low and highenergy components of the burst. By doing so we aim to avoid problems such burst asymmetry that can weaken the assumption of maximum sharpness. In this regard, the metric minimization approach has the role of a dynamic fit between the two components of the profile. We are currently testing ways for further increasing the sensitivity of the method. The search for new, more appropriate, distance measures are also under way and are encouraged to be tested. Applications of this method to VHE and GRB data is underway.
Acknowledgements
U.Barres de Almeida acknowledges a PhD scholarship from CAPES Foundation, Ministry of Education of Brazil.
Footnotes
 Notice that here, unlike an ordinary medium, the vacuum refractive index increases for smaller wavelengths in QG models.
 See [15] for an alternative nonbinned approach that try to surpass this limitation by introducing a modeldependent cost function, incorporating properties of the source emission process.
 We actually construct the s from the to reduce the influence of extremely highdensity cells that might arise from statistical fluctuations.
 Notice also that we are not relying on confidence intervals and pvalues from the KolmogorovSmirnov test to compare the distributions, but rather using the Kdist as a metric to differentiate them in probability space.
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