Robust constraint on Lorentz violation using Fermi-LAT gamma-ray burst data

Models of quantum gravity suggest that the vacuum should be regarded as a medium with quantum structure that may have nontrivial effects on photon propagation, including the violation of Lorentz invariance. Fermi Large Area Telescope (LAT) observations of gamma-ray bursts (GRBs) are sensitive probes of Lorentz invariance, via studies of energy-dependent timing shifts in their rapidly varying photon emissions. We analyze the Fermi-LAT measurements of high-energy gamma rays from GRBs with known redshifts, allowing for the possibility of energy-dependent variations in emission times at the sources as well as a possible nontrivial refractive index in vacuo for photons. We use statistical estimators based on the irregularity, kurtosis, and skewness of bursts that are relatively bright in the 100 MeV to multi-GeV energy band to constrain possible dispersion effects during propagation. We find that the energy scale characterizing a linear energy dependence of the refractive index should exceed a $\text{few}\times {10}^{17}\mathrm{GeV}$, and we estimate the sensitivity attainable with additional future sources to be detected by Fermi-LAT.

Figure 1

Time evolution of the profile of a Gaussian wave packet injected into a dispersive medium characterized by (8), convoluted at two different points in time $t_2>t_1>0$ with a power-law energy spectrum (17). The (green) solid line represents the profile of the wave envelope at the injection time ($t=0$). The (orange) dashed and (magenta) dashed-dotted lines represent the profiles modified by the propagation effects at times $t_1$ and $t_2$, respectively. The energy range spans 2 orders of magnitude in dimensionless units, so as to reproduce a typical spectral range of the high-energy GRB emissions measured by Fermi-LAT used in the analysis.

Figure 2

Energies vs arrival times of Fermi-LAT photons that passed the transient off-line selection (solid green squares), as outlined in Sec. 3, which are consistent with the direction of (left panel) GRB090510A and (right panel) GRB080916C. The earliest arrival time is set to zero in each case. The detector-frame delays indicated in the left panel by the open triangles, stars, and circles are calculated assuming a dependence $F(E)=E$, with the irregularity estimator, kurtosis estimator, and skewness estimators, respectively (see the text for details), for compensation parameters with $\tau =-0.081\mathrm{s}/\mathrm{GeV}$, $\tau =-0.099\mathrm{s}/\mathrm{GeV}$ and $\tau =-0.104\mathrm{s}/\mathrm{GeV}$. The same is shown in the right panel for $\tau =0.930\mathrm{s}/\mathrm{GeV}$, $\tau =0.935\mathrm{s}/\mathrm{GeV}$, and $\tau =0.801\mathrm{s}/\mathrm{GeV}$.

Figure 3

Values of $D(\tau )$ for a selected discrete set of trial values ${\tau }^j$ of the compensation parameter, where $j$ runs over a grid of several hundred values. The plots show the $D(\tau )$ dependences for the signal from (left panel) GRB090510A and (right panel) GRB080916C. The positions of the maxima are marked by the vertical (red) solid lines, (left panel) $\tau =-0.103\mathrm{s}/\mathrm{GeV}$ and (right panel) $\tau =0.90\mathrm{s}/\mathrm{GeV}$, as estimated by calculating the weighted average of the ${\tau }_{\gamma }$ values for which $D({\tau }_{\gamma })$ exceeds 95% of its peak value. The (green) dashed lines mark the positions of the maxima in the absence of any propagation effect.

Figure 4

Curves of the kurtosis $\mathcal{K}(\tau )$ as functions of the compensation parameter $\tau$, as calculated for a set of discrete values of the compensation parameter ${\tau }^j$, with $j$ running over a grid of several hundred values. The calculations are for (left panel) GRB090510A and (right panel) GRB080916C. The positions of the maxima are marked by the vertical (red) solid lines, with the values (left panel) $\tau =-0.098\mathrm{s}/\mathrm{GeV}$ and (right panel) $\tau =0.80\mathrm{s}/\mathrm{GeV}$. The (green) dashed lines mark the positions of the maxima in the absence of any propagation effect.

Figure 5

The skewness $\mathcal{S}(\tau )$ calculated for a set of discrete values of the compensation parameter ${\tau }^j$ values of the compensation parameter $\tau$ with $j$ running over a grid of several hundred points, for (left panel) GRB090510A and (right panel) GRB080916C. The position of the maximum is marked by the vertical (red) solid line: (left panel) $\tau =-0.073\mathrm{s}/\mathrm{GeV}$ and (right panel) $\tau =0.93\mathrm{s}/\mathrm{GeV}$. The (green) dashed lines mark the positions of the maxima in the absence of any propagation effect.

Figure 6

The upper panels show distributions of the correct values of the compensation parameter estimated for 112 000 toy models of (left panel) GRB090510A and (right panel) GRB080916C. The (magenta) dotted lines show the results of applying of the irregularity estimator using simply the middle values of line segments at 95% of the heights of the KS difference curves. The positions of the maxima of the irregularity estimators found by averaging of the peaks of the KS difference curves are shown by (green) short-dashed three-dotted lines. The results of using the kernel density estimator (KDE) to estimate the positions of the KS difference curves of the irregularity estimator are shown by (orange) dashed lines. The (black) long-dashed dotted lines show the distribution of the results of the kurtosis estimator applied to the toy models. The results of the skewness estimators are shown by (blue) short-dashed dotted lines. Finally, the distributions called the overall distributions in the text are depicted by the (red) solid lines. The plots in the lower panels shows the results corrected for the systematic bias explained in the text.

Figure 7

Examples of one particular realization with injected quantum-gravity effects for the objects (left panel) GRB090510A and (right panel) GRB080916C. The solid squares represent the data recorded by Fermi-LAT, the stars represent the progenitor histogram obtained with the quantum-gravity compensation estimated using the kurtosis estimator, and the downward-pointing triangles represent a random simulation of the two-dimensional distribution constrained by the cell pattern of the progenitor histogram. The upward-pointing triangles are obtained by injecting ${\tau }^{\text{true}}=-0.2$ for GRB090510A and ${\tau }^{\text{true}}=0.4$ for GRB080916C into the simulated realization.

Figure 8

The results of studies of bias in the kustosis estimator for the objects (left panel) GRB090510A and (right panel) GRB080916C. The amounts of injected and reconstructed Lorentz-violating signals are shown on the horizontal and vertical axes, respectively. The injections have been made into 15 realizations of sources seeded by data with timings compensated for the value of the dispersion effect estimated by the kurtosis estimation procedure. The horizontal errors are $1\sigma$ uncertainties in the kurtosis estimation procedure indicated by the data.

Figure 9

Same as Fig. 8, but for the irregularity estimator with its maximum values calculated by averaging of the tops of the KS difference curves.

Figure 10

(Upper row) The overall distributions (left panel) of the corrected values of the compensated parameters for the eight sources studied (see Table 1), and their K-reduced normalized versions (right panel). Every individual distribution is obtained as an average of probability distributions obtained using the five estimation procedures described in Sec. 6. (Lower row) The same distributions as in the top row, corrected for the bias systematic as described in Sec. 6.

Figure 11

(Left panel) The normalized K-reduced overall PDFs of the values of the compensated parameters for all eight sources with standard deviations rescaled by a universal factor $\sqrt{{\chi }_{\mathrm{raw}}^2}$, together with the consolidated PDF shown as the (red) solid line. (Right panel) The normalized K-reduced overall PDFs with an additional contribution to the standard deviations together with their consolidated PDF, which is shown as the (red) solid line. The values of $\tau (z_0)$ to the right of the vertical dashed line are not compatible with zero at the 95% C.L.

Figure 12

Simulated statistical realizations of the pattern of mean value/variance measurements obtained from the K-reduced distributions. The pattern of the measured data is shown by the crosses, while the simulated measurements are marked with triangles. The left and right panels are for 8 and 16 simulated measurements, respectively.

Figure 13

(Left panel) Distribution of the 95% C.L. limits obtained in 100 000 realizations of eight measurements processed with the $\sqrt{{\chi }_{\mathrm{raw}}^2}$ standard deviation rescaling method. We find that 95% of entries do not exceed the value indicated by the vertical dotted-dashed line. (Right panel) The same as in the left panel, but for realizations processed by adding a universal stochastic spread.

Figure 14

(Left panels) Distribution of the 95% C.L. limits obtained in 100 000 realizations with (upper panels) 8 and (lower panels) 16 simulated measurements added to the current data, processed with the $\sqrt{{\chi }_{\mathrm{raw}}^2}$ rescaling method. We note that 95% of the entries do not exceed the value indicated by the vertical dotted-dashed line. (Right panels) The same as in the left panels, but processing the real and simulated data by adding a universal stochastic spread.

Figure 15

(Left panel) Distribution of the 95% C.L. limits obtained in 100 000 realizations of one simulated measurement added to the eight present sources, processed with the $\sqrt{{\chi }_{\mathrm{raw}}^2}$ rescaling method. (Right panel) The same as in the left panel, but with 28 simulated measurements added to the eight present sources.