In Vacuo Dispersion-Like Spectral Lags in Gamma-Ray Bursts

Abstract

Some recent studies exposed preliminary but rather intriguing statistical evidence of in vacuo dispersion-like spectral lags for gamma-ray bursts (GRBs), a linear correlation between time of observation and energy of GRB particles, which is expected in some models of quantum geometry. Those results focused on testing in vacuo dispersion for the most energetic GRB particles, and in particular only included photons with energy at emission greater than 40 GeV. We here extend the window of the statistical analysis down to 5 GeV and find results that are consistent with what had been previously noticed at higher energies.

1. Introduction

Over the last 15 years, there has been considerable interest (see e.g., Refs. [1,2,3,4,5,6,7,8,9,10] and references therein) in descriptions of spacetime based on quantum geometry that produce in vacuo dispersion, the possibility that spacetime itself might behave essentially like a dispersive medium for particle propagation: there might be an energy dependence of the travel times of ultrarelativistic particles from a given source to a given detector. This is the most studied scenario within a broader research program [1] concerning the possibility that relativistic symmetries might be affected by quantum-gravity effects (A partly related research program is known as the SME (Standard Model Extension), a framework considering all possible (with or without “quantum-gravity motivation”) Lorentz-violating modifications of the Standard Model of particle physics [11]. The SME phenomenology mainly focuses Ref. [12] on dimension-four operators added to the Standard-Model lagrangian, whereas the effects we here consider would be described in field-theory in terms of dimension-five operators. However, describing photons within the SME the effects of dispersion would be combined with birefringence [13,14,15,16,17], which we here assume to be absent consistently with results obtained in frameworks alternative to the SME, such as those studied in Refs. [1,3,7] and references therein).

It is well established [1,2,3,4] that the analysis of GRBs could allow us to test the in vacuo dispersion hypothesis. Some of us were involved in the first studies using IceCube data for searching for GRB-neutrino in vacuo dispersion candidates [9,18,19,20]. Analogous investigations were performed in a series of studies [21,22,23] (also see [24]) focusing on the highest-energy GRB photons observed by the Fermi telescope. As summarized in Figure 1 these studies provided rather intriguing statistical evidence of in vacuo dispersion-like spectral lags. The values of $\Delta t$ used in Figure 1 reflect the difference between the time of observation of the relevant particle and the time of observation of the first low-energy peak in the GRB to which the particle might be associated. The values of $E^{*}$ used in Figure 1 are obtained from the formula

$$E^{*}\equiv E\frac{D\left(z\right)}{D\left(1\right)}$$

(1)

where z is the redshift of the relevant GRB and

$$D\left(z\right)={\int }_0^{z}d\zeta \frac{(1+\zeta )}{H_0\sqrt{{\Omega }_{\Lambda }+{(1+\zeta )}^3{\Omega }_m}}.$$

(2)

denoting, as usual, with ${\Omega }_{\Lambda }$, $H_0$ and ${\Omega }_m$ respectively the cosmological constant, the Hubble parameter and the matter fraction, for which we take the values given in Ref. [25].

The most studied [1,2,3,4,5,6,7,8,9,10] model of quantum-gravity-induced in vacuo dispersion is (This formula for $\Delta t$ is expected to arise from Planck-scale modifications of the dispersion relation [4,5,6] and such modifications of the dispersion relation would affect many aspects of physics, with however effects that are irrelevantly small [1] in nearly all experimental context, these astrophysical time-of-travel studies being one rare exception in which, thanks to the huge amplification provided by the travel times, we might have a chance to test the effects.)

$$\Delta t={\eta }_X\frac{E}{M_P}D\left(z\right)\pm {\delta }_X\frac{E}{M_P}D\left(z\right),$$

(3)

which in terms of $E^{*}$ takes the form

$$\Delta t={\eta }_{X}D\left(1\right)\frac{E^{*}}{M_P}\pm {\delta }_{X}D\left(1\right)\frac{E^{*}}{M_P}.$$

(4)

$M_P$ denotes the Planck scale ($\simeq 1.2·{10}^{28} \mathrm{eV}$) and the values of the parameters ${\eta }_X$ and ${\delta }_X$ in (3) are to be determined experimentally. In (3) the notation “$\pm {\delta }_X$" reflects the fact that ${\delta }_X$ parametrizes the size of quantum-uncertainty (fuzziness) effects. Instead the parameter ${\eta }_X$ characterizes systematic effects: for example in our conventions for positive ${\eta }_X$ and ${\delta }_X=0$ a high-energy particle is detected systematically after a low-energy particle (if the two particles are emitted simultaneously). The label X for ${\delta }_X$ and ${\eta }_X$ intends to allow for a possible dependence [1,10] of these parameters on the type of particles (so that for example for neutrinos and photons one would have ${\eta }_{\nu }$, ${\delta }_{\nu }$, ${\eta }_{\gamma }$, ${\delta }_{\gamma }$) and in principle also on spin/helicity (so that for example for neutrinos one would have ${\eta }_{\nu +}$, ${\delta }_{\nu +}$, ${\eta }_{\nu -}$, ${\delta }_{\nu -}$).

The black points in Figure 1 are “GRB-neutrino candidates" in the sense of Ref. [18], while the blue points correspond to GRB photons with energy at emission greater than 40 GeV. The linear correlation between $\Delta t$ and $E^{*}$ visible in Figure 1 is just of the type expected for quantum-gravity-induced in vacuo dispersion. It might of course be accidental, but it has been estimated [18] that for the relevant GRB-neutrino candidates such a high level of correlation would occur accidentally only in less than $1\%$ of cases, while GRB photons could produce such high correlation (in absence of in vacuo dispersion) only in less than $0.1\%$ of cases [20]. The “statistical evidence" summarized in Figure 1 is evidently intriguing enough to motivate us to explore whether or not the in vacuo dispersion-like spectral lags persist at lower energies.

2. Analysis

One challenge for this is that evidently we cannot simply apply to lower-energy photons the reasoning which led to Figure 1: as stressed above the $\Delta t$ in Figure 1 is the difference between the time of observation of the relevant particle and the time of observation of the first low-energy peak in the GRB, so it is a $\Delta t$ which makes sense for in vacuo dispersion studies only for photons which one might think were emitted in (near) coincidence with the first peak of the GRB. This assumption is (challengeable [26] but) plausible [23] for the few highest-energy GRB photons relevant for Figure 1, with energy at emission greater than 40 GeV, but of course it cannot apply to all photons in a GRB. Conceptually the main aspect of novelty of our analysis concerns a strategy for handling this challenge.

We consider the same GRBs relevant for the analysis summarized in Figure 1, but now including all photons from those GRBs with energy at the source greater than 5 GeV, thereby lowering the cutoff by nearly an order of magnitude. Only 11 photons took part in the previous analyses whose findings were summarized in our Figure 1, whereas the analysis we are here reporting involves a total of 148 photons. For the reasons discussed above, we do not consider the $\Delta t$ (with reference to the first peak of the GRB), but rather we consider a $\Delta t_{pair}$, which gives for each pair of photons in our sample their difference of time of observation. Essentially each pair of photons (from the same GRB) in our sample is taken to give us an estimated value of ${\eta }_{\gamma }$, by simply computing

$${\eta }_{\gamma }^{\left[pair\right]}\equiv \frac{M_P\Delta t_{pair}}{D\left(1\right)E_{pair}^{*}},$$

(5)

where $E_{pair}^{*}$ is the difference in values of $E^{*}$ for the two photons in the pair. Of course the $\Delta t_{pair}$ for many pairs of photons in our sample could not possibly have anything to do with in vacuo dispersion: if the two photons were produced from different phases of the GRB (different peaks) their $\Delta t_{pair}$ will be dominated by the intrinsic time-of-emission difference. Those values of ${\eta }_{\gamma }^{\left[pair\right]}$ will be spurious, they will be “noise" for our analysis. However we also of course expect that some pairs of photons in our sample were emitted nearly simultaneously, and for those pairs the $\Delta t_{pair}$ could truly estimate ${\eta }_{\gamma }$. Since estimating ${\eta }_{\gamma }$ from the photons in Figure 1 one gets ${\eta }_{\gamma }=30\pm 6$, the preliminary evidence here summarized in Fig.1 would find additional support if this sort of analysis showed that values of ${\eta }_{\gamma }^{\left[pair\right]}$ of about 30 are surprisingly frequent, more frequent than expected without a relationship between arrival times and energy of the type produced by in vacuo dispersion.

This is just what we find, as shown perhaps most vividly by the content of Figure 2, which was obtained including in the analysis all possible pairs of photons belonging to the same GRB. The main point to be noticed in Figure 2 is that we find in our sample a frequency of occurrence of values of ${\eta }_{\gamma }^{\left[pair\right]}$ between 25 and 35 which is tangibly higher than one would have expected in absence of a correlation between $\Delta t_{pair}$ and $E_{pair}^{*}$. Following a standard strategy of analysis (see, e.g., Refs. [24,27]) we estimate how frequently $25\le {\eta }_{\gamma }^{\left[pair\right]}\le 35$ should occur in absence of correlation between $\Delta t_{pair}$ and $E_{pair}^{*}$ by producing ${10}^5$ sets of simulated data, each obtained by reshuffling randomly the times of observation of the photons in our sample. More details on this and other aspects of our analysis are given in Appendix A. Also in Appendix A we show that our findings are strikingly robust with respect to restricting the analysis to only part of our data set: values of ${\eta }_{\gamma }^{\left[pair\right]}$ between 25 and 35 occur at a rate higher than expected for all meaningful portions of our data set. Most notably, values of ${\eta }_{\gamma }^{\left[pair\right]}$ between 25 and 35 occur at a rate higher than expected even if we exclude from the analysis the photons whose energy at emission is greater than 40 GeV (the photons that were taken into account in the analyses leading to the content of our Figure 1).

It is also noteworthy that we find (see Appendix A) that an excess of results for ${\eta }_{\gamma }^{\left[pair\right]}$ between 25 and 35 as big as shown by our data should occur accidentally (in absence of in vacuo dispersion) in less than 0.5% of cases.

Also intriguing is the content of our Figure 3, which offers an intuitive characterization of the consistency that emerged from our analysis between what had been found in previous studies of GRB photons with energy at emission greater than 40 GeV, and what we now find for GRB photons with energy between 5 and 40 GeV.

3. Closing Remarks

We used data that were already available at the time of the studies that led to Figure 1 (which in particular focused on photons with energy at emission greater than 40 GeV) but nobody had looked before at those data for photons with energy at emission between 5 and 40 GeV, from the perspective of Figure 1. Our findings should therefore be viewed as providing further motivation for investigating this scenario. Of course, if this feature is confirmed by future studies the next task would be to establish whether it is truly connected with quantum-gravity-induced in vacuo dispersion, rather than being related to some intrinsic property of GRB signals. Within our analysis the imprint of in vacuo dispersion is coded in the $D\left(z\right)$ for the distance dependence and, while that does give a good match to the data, one should keep in mind that only a few redshifts (a few GRBs) were relevant for our analysis. If we are actually seeing some form of in vacuo dispersion it would most likely be of statistical (“fuzzy”) nature since other studies have provided evidence strongly disfavoring the possibility that this type of in vacuo dispersion effects would affect systematically all photons [28].