CONSTRAINING THE HIGH-ENERGY EMISSION FROM GAMMA-RAY BURSTS WITH FERMI

We derive upper limits for the 288 GRBs that were detected by the GBM and fell in the LAT FOV from the LAT data using two methods. The first consists of the standard unbinned likelihood analysis using the software developed and provided by the LAT team, while the second method simply considers the total observed counts within an energy-dependent acceptance cone centered on the GBM burst location. The likelihood analysis will give more constraining upper limits, but since it uses the instrumental point-spread-function (PSF) information to model the spatial distribution of the observed photons, in cases where the burst location is inaccurate and burst photons are present, it can give less reliable constraints. The latter method will be less constraining in general, but it will also be less sensitive to errors in the burst location, as the analysis considers photons collected over a fixed aperture and does not otherwise use the burst or photon positions on the sky. We use both methods to obtain photon flux upper limits over a 0.1–10 GeV energy range.

For the unbinned likelihood analysis, we used the standard software package provided by the LAT team (ScienceTools version v9r15p6).⁵⁶ We selected "transient" class events in a 10° acceptance cone centered on the burst location, and we fit the data using the pyLikelihood module and the P6_V3_TRANSIENT response functions (Atwood et al. 2009). Each burst is modeled as a point source at the best available location, derived either from an instrument with good localization capabilities (e.g., Swift or LAT) or by the GBM alone. Of the 288 GRBs considered here, in the likelihood fitting, the expected distribution of counts is modeled using the energy-dependent LAT PSF and a power-law source spectrum. The photon index of the power law is fixed to either the β value found from the fit of the GBM data for that burst or, if the GBM data are not sufficiently constraining (i.e., δβ ⩽ 0.5), to β = −2.2, the mean value found for the population of BATSE-detected bursts (Preece et al. 2000; Kaneko et al. 2006). An isotropic background component is included in the model, and the spectral properties of this component are derived using an empirical background model (Abdo et al. 2009c) that is a function of the position of the source in the sky and the position and orientation of the spacecraft in orbit. This background model accounts for contributions from both residual charged particle backgrounds and the time-averaged celestial gamma-ray emission.

Since we are considering cases where the burst flux in the LAT band will be weak or zero, the maximum likelihood estimate of the source flux may actually be negative owing to downward statistical fluctuations in the background counts. Because the unbinned likelihood function is based on Poisson probabilities, a prior assumption is imposed that requires the source flux to be non-negative. This is necessary to avoid negative probability densities that may arise for measured counts that are found very close to the GRB point-source location because of the sharpness of the PSF. On average, this means that for half of the cases in the null hypothesis (i.e., zero burst flux), the "best-fit" value of the source flux is zero but does not correspond to a local maximum of the unconstrained likelihood function (Mattox et al. 1996).

Given the prior of the non-negative source flux, we treat the resulting likelihood function as the posterior distribution of the flux parameter. In this case, an upper limit may be obtained by finding the flux value at which the integral of the normalized likelihood corresponds to the chosen confidence level (Amsler et al. 2008). For a fully Bayesian treatment, one would integrate over the full posterior distribution, i.e., marginalize over the other free parameters in the model. However, in practice, we have found it sufficient to treat the profile likelihood function as a one-dimensional probability distribution function in the flux parameter. Again, in the limit of Gaussian statistics and a strong source, this method is equivalent to the use of the asymptotic standard error for defining confidence intervals. Hereafter, we will refer to this treatment as the "unbinned likelihood" method.

In the second set of upper limit calculations, we implement the method described by Helene (1983) and the interval calculation implemented in Kraft et al. (1991). Here, the upper limit is computed in terms of the number of counts and is based on the observed and estimated background counts within a prescribed extraction region. For the LAT data, the extraction region is an energy-dependent acceptance cone centered on the burst position. Since the burst locations from the GBM data have typical systematic uncertainties ∼32 (Connaughton et al. 2011), the size of the acceptance cone at a given energy is taken to be the sum in quadrature of the LAT 95% PSF containment angle and the total (statistical + systematic) uncertainty in the burst location. The counts upper limits are evaluated over a number of energy bands, converted to fluxes using the energy-dependent LAT exposure at the burst location, and then summed to obtain the final flux limit. Since this method relies on comparing counts without fitting any spectral shape parameters, we will refer to this as the "counting" method.

The time intervals over which the upper limits are calculated are important for their interpretation. For both upper limit methods, we consider three time intervals: two fixed intervals of 30 and 100 s post-trigger, and a "T100" interval that is determined through the use of the Bayesian Blocks algorithm (Jackson et al. 2005) to estimate the duration of burst activity in the NaI detector that has the largest signal above background. For the T100 interval, an estimate of the time-varying background count rate is obtained by fitting a third-degree polynomial to the binned data in time intervals outside of the prompt burst phase. Nominally, we take T₀ − dt to T₀ − 100 s and T₀ + 150 s to T₀ + dt, where T₀ is the GBM trigger time and dt = 200 s, although we increased the separation of these intervals in some cases to accommodate longer bursts. The counts per bin is then subtracted by the resulting background model throughout the T₀ − dt to T₀ + dt interval, and the binned reconstruction mode of the Bayesian Blocks algorithm is applied. The T100 interval is then defined by the first and last change points in the Bayesian Blocks reconstruction.

The two fixed time intervals have been introduced so as to not bias our results through assumptions regarding the durations of the high-energy components. The brighter LAT-detected GRBs have exhibited both delayed and extended high-energy emission on timescales that exceed the durations traditionally defined by observations in the keV–MeV energy range (Abdo et al. 2011). Hence, we search for and place limits on emission over intervals that may, in some cases, exceed the burst duration. We will discuss the implications of the limits found for the various time intervals in Section 5.1.

For the 92 bursts in the bright BGO subsample, we performed spectral fits to the NaI and BGO data and estimated the flux expected to be seen by the LAT between 0.1–10 GeV using the GBM-fitted Band function (Band et al. 1993) parameters. The selection of background and source intervals for all bursts were performed manually through the use of the RMFIT (version 3.3) spectral analysis software package.⁵⁷ Because the number of counts in the highest BGO energy bins is often in the Poisson regime, we use the Castor modification (J. Castor 1995, private communication) to the Cash statistic (Cash 1976), commonly referred to as C-Stat,⁵⁸ since the standard χ² statistic is not reliable for low counts. The variable GBM background for each burst is determined for all detectors individually by fitting an energy-dependent, second-order polynomial to the data several hundred seconds before and after the prompt GRB emission. The standard 128 energy bin CSPEC data (Meegan et al. 2009) from the triggered NaI and BGO detectors were then fit from 8 keV to 1 MeV and from 200 keV to 40 MeV, respectively, for each burst.

As we noted above, only 30 bursts in the bright BGO subsample have sufficient signal to noise to constrain the high-energy power-law index β of the Band function to within ±0.5. Although we considered a variety of models in our spectral analysis, we found that the Band function was sufficient to describe the spectral shape for all of these bursts.

Of the 288 GRBs in our sample, we were able to obtain upper limits, at 95% confidence level (CL), for 270 bursts using the unbinned likelihood method and 95% CL upper limits for 250 bursts using the counting method for the T100 intervals derived from the GBM data. The GRBs for which upper limits could not be calculated were bursts that occurred either during spacecraft passages through the South Atlantic Anomaly or at angles with respect to Earth's zenith that were ≳ 100°, thereby resulting in diffuse emission at the burst locations that was dominated by γ-rays from Earth's limb produced by interactions of cosmic rays with Earth's atmosphere. These cases where the burst occurred at a high angle with respect to the zenith primarily affect the counting method, because it requires a reliable estimate of the background during the burst, and our method to estimate the background does not account for Earth limb emission. The likelihood method can fit for an Earth limb as a diffuse component, but it may give weaker limits since the background level is not as tightly constrained in this case compared to when the empirical background estimate can be used to model all of the non-burst emission. The photon flux upper limits found for the likelihood method for all three time intervals are presented in the last three columns of Table 1.

The distributions of the 95% CL photon flux upper limits obtained via the likelihood and counting methods for the 30 s, 100 s, and T100 time intervals are shown in upper-left, upper-right, and lower-left panels of Figure 2, respectively. As expected, the likelihood limits are systematically deeper than those found using the counting method over the same time interval. For either method, the upper limits for the 100 s integrations are roughly half an order of magnitude deeper than for the 30 s integrations. In the photon-limited case, this is expected since the flux limit at a specified confidence level should be inversely proportional to the exposure. The doubly peaked upper limit distribution that appears in the upper-left panel of Figure 2 for the T100 duration reflects the bimodal duration distribution for the short and long GRB populations. The median of the T100 upper limit distribution for the likelihood method is ${{\tilde{F}_{{\rm UL,{T100}}}}} = 1.20\times 10^{-4}$ photons cm⁻² s⁻¹ with a standard deviation of σ_T100 = 1.57 × 10⁻³; whereas the counting method distribution has a median of ${{\tilde{F}_{{\rm UL,{T100}}}}} = 1.27\times 10^{-4}$ photons cm⁻² s⁻¹ and σ_T100 = 1.52 × 10⁻³. The median of the 30 s upper limit distribution for the likelihood method is ${{\tilde{F}_{{\rm UL,{30\rm s}}}}} = 4.76\times 10^{-5}$ photons cm⁻² s⁻¹ with a standard deviation of σ_30s = 3.20 × 10⁻⁴; whereas the counting method distribution has a median of ${{\tilde{F}_{{\rm UL,{30\rm s}}}}} = 5.46\times 10^{-5}$ photons cm⁻² s⁻¹ and σ_30s = 3.00 × 10⁻⁴. The median of the 100 s upper limit distribution for the likelihood method are ${{\tilde{F}_{{\rm UL,{100\rm s}}}}} = 1.74\times 10^{-5}$ photons cm⁻² s⁻¹ and σ_100s = 1.23 × 10⁻⁴ and ${{\tilde{F}_{{\rm UL,{100\rm s}}}}} = 2.59\times 10^{-5}$ photons cm⁻² s⁻¹ and σ_100s = 1.06 × 10⁻⁴ for the counting method.

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A comparison of the likelihood and counting methods for all three time intervals for is shown in the lower-right panel of Figure 2. The scatter in the upper limit distribution for both methods is largely due to the range of angles at which the GRBs occurred with respect to the LAT boresight, resulting in different effective areas and hence different exposures for each burst. The LAT exposure as a function of the off-axis angle drops steeply with increasing inclination, resulting in a shallowing of the LAT upper limits as a function of increasing off-axis angle, which can be seen in Figure 3. Overall, the two methods give consistent results for the bursts in our sample, and therefore we will hereafter focus primarily on the limits obtained with the likelihood method in our discussion of the implication of these results.

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Despite the dependence of the upper limit values on off-axis angle, the distribution of LAT photon flux upper limits is relatively narrow for angles <40°, allowing us to define an effective LAT sensitivity assuming a typical GRB spectrum (i.e., β ≈ −2.2). We can therefore set sensitivity thresholds for the corresponding median photon flux upper limit for each integration time of F_{lim, 30 s} = 4.7 × 10⁻⁵ photons cm⁻² s⁻¹ and F_{lim, 100 s} = 1.6 × 10⁻⁵ photons cm⁻² s⁻¹.

Finally, in Figure 4 we plot the location of each burst on the sky in Galactic coordinates, color-coded to represent the likelihood-determined photon flux upper limits. There is no evidence of a spatial dependence of the GBM detection rate nor of the magnitude of the LAT upper limit, as a function of Galactic latitude b.

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We compare the LAT upper limits calculated over the burst duration to the expected 0.1–10 GeV photon fluxes found through extrapolations of spectral fits presented in the first GBM spectral catalog (Goldstein et al. 2012). We focus this analysis on bursts for which a Band spectral model was a preferred fit compared to models with fewer degrees of freedom, since alternative models such as Comptonized spectra suffer sharp drops in expected flux at high energy and are not expected to result in LAT detections without the presence of additional spectral components. Of the 487 GRBs presented in that catalog, a Band model fit was preferred over simpler models for 161 bursts, 75 of which appeared in the LAT FOV. For this comparison, the LAT upper limits were recalculated for a duration that matched the interval used in the GBM spectral catalog (see Goldstein et al. 2012 for a detailed discussion of their interval selection). We next performed a simulation in which we varied the expected LAT photon flux fitted values using the associated errors for each burst in order to determine the median number of bursts over all realizations that would fall above the LAT upper limit. In a total of 10⁵ realizations, we find that 50% of the GRBs in the GBM spectral catalog, which prefer a Band model fit, have expected 0.1–10 GeV photon fluxes that exceeds the LAT upper limit.

We investigate the differences between the GBM-based extrapolations and the LAT upper limits further by performing detailed spectral fits to our spectroscopic subsample. The spectral parameters obtained from the fits to the GBM data only for the 30 GRBs in this spectroscopic subsample are listed in Table 2. The median values of the low- and high-energy power-law indices and the peak of the νF_ν spectra are α = −0.83, β = −2.26, and E_pk = 164 keV, with standard deviations of σ_α = 0.44, σ_β = 0.25, and $\sigma _{{E_{\rm pk}}} = 177$ keV, respectively. The distributions of spectral parameters for these bursts are consistent with similar distributions found for BATSE-detected GRBs (Preece et al. 2000; Kaneko et al. 2006). The time durations used in the spectral fits and the time-averaged photon flux values in the 0.02–20 MeV energy range for these GRBs are given in Table 3. In the third column, we list the expected flux in the 0.1–10 GeV energy range assuming a power-law extrapolation of the Band function fit to the GBM data; and in the fourth column, we give the measured LAT photon flux upper limit found for the same time interval. The errors on the expected LAT photon fluxes were determined using the covariance matrices obtained from the GBM spectral fits.

A comparison of the LAT photon flux upper limits versus the expected 0.1–10 GeV photon fluxes for each burst in our spectroscopic subsample is shown as blue data points in Figure 5. The downward arrows on the expected flux values indicate values that are consistent with zero within the 1σ errors shown. The dashed line represents the line of equality between the expected LAT photon flux and the LAT photon flux upper limits when calculated for the durations presented in Figure 5. In a total of 10⁵ realizations, we find that 53% of GRBs in our spectroscopic subsample have expected 0.1–10 GeV photon fluxes that exceed their associated 95% CL LAT upper limit. As with the flux comparison, roughly 50% in our sample also have expected fluence values that exceed the 95% CL LAT fluence upper limit. Figure 6 shows that the degree to which the expected flux in the LAT energy range from these bursts exceeds our estimated LAT upper limits correlates strongly with the measured high-energy spectral index, with particularly hard bursts exceeding the estimated LAT sensitivity by as much as a factor of 100. Again, the spectral fits to the bright bursts detected by the BGO clearly shows that a simple extrapolation from the GBM band to the LAT band systematically overpredicts the observed flux.

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Including the LAT data in the spectral fits drastically alters the best-fit Band model parameters and the resulting expected photon flux in the LAT energy range. The best-fit parameters of the joint spectral fits for the spectroscopic subsample can be found in Table 4. The high-energy spectral indices are typically steeper (softer) than found from fits to the GBM data alone.

The difference in the β values for the joint fits with respect to the fits to the GBM data alone can be found in Column 8 of Table 4. The resulting β distributions are shown in Figure 7. The GBM-only β distribution (red histogram) peaks at β = −2.2, matching the β distribution found for the population of BATSE-detected bursts presented in Preece et al. (2000). In contrast, the β distribution found from the joint fits (blue histogram) indicates spectra that are considerably softer, with a median value of β = −2.5. While the GBM-only β distribution includes five GRBs with β > −2.0, no bursts had β values this hard from the joint fits. The low-energy power-law index α and the peak of the νF_ν spectra, E_pk distribution remain relatively unchanged. In Figure 5, we compare the LAT photon flux upper limits calculated over the burst duration presented in Table 4 versus the expected 0.1–10 GeV photon fluxes for each burst, now using a power-law extrapolation of the Band function that was fit to both the GBM and LAT data. The softer β values obtained through the joint fits yield expected LAT photon flux values that are more consistent with the LAT non-detections, with only 23% of the bursts in our spectroscopic subsample with expected flux values that exceed the 95% CL LAT flux upper limit given 10⁵ realizations of the data about their errors. We find that a similar ratio of bursts have expected fluence values that exceed their associated 95% CL LAT fluence upper limit.

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Although the discrepancy between the predicted 0.1–10 GeV fluxes from the GBM-only fits and the LAT upper limits can be explained by the softer β values in the joint fits, intrinsic spectral breaks at energies ≳ 40 MeV can also reconcile the conflicting GBM and LAT results. Determining whether softer β values or spectral breaks are present has at least two important implications: if the spectral breaks or cutoffs arise from intrinsic pair production (γγ → e⁺e⁻) in the source, then the break or cutoff energy would provide a direct estimate of the bulk Lorentz factor of the emitting region within the outflow. On the other hand, an intrinsically softer distribution of β values would mean that theoretical inferences based on the β distributions found by fitting BATSE or GBM data alone may need to be revised. Evidence for either spectral breaks or softer β values could also provide support for multi-component models that have been used to describe novel spectral features detected by the GBM and LAT (e.g., Guiriec et al. 2011).

For the joint fitting of the GBM and LAT data, deciding between the two possibilities for any single burst can be cast as a standard model selection problem. Under the null hypothesis, we model the GRB spectrum using a simple Band function, as we have done in Section 5.3. As an alternative hypothesis, we could extend the Band model to account for the presence of a spectral break. This may be done via an additional break energy above the Band E_pk, effectively using a doubly broken power law in the fit; or it could be accomplished by adding an exponential cutoff to the Band model with cutoff energy E_c > E_pk. In either case, the null and alternative hypotheses are "nested" such that the former is a special case of the latter for some values of the extra model parameters that are introduced. Assuming there are n_alt additional free parameters under the alternative model, then whether the alternative model is statistically preferred would be given by the ΔC-Stat value assuming it follows a χ² distribution for n_alt degrees of freedom.

For the purposes of this analysis, we have adopted an alternative model consisting of a Band function plus a step function fixed at 50 MeV. The step function is not intended to be a physical model; instead its use is simply designed to test consistency between the GBM and LAT data. By using a step function we are explicitly avoiding making any assumptions as to the physical mechanism producing the emission, which allows us instead to focus on simply comparing the LAT upper limits to the extrapolation of the best fit to the GBM data. The additional degree of freedom introduced by the step function represents the normalization of the Band function's high-energy component above 50 MeV, which is left to vary, leading to the normalization of the power law above 50 MeV being adjusted such that it is always consistent with the LAT upper limits. For this analysis, the index of the power law above the break is fixed to match the Band function's high-energy power-law index, which is allowed to vary as a free parameter. Since this introduces a single extra degree of freedom, a value of ΔC-Stat >9 would represent a >3σ improvement in the fit. We adopt this criterion as the threshold for a statistical preference for a break in the high-energy spectrum of an individual GRB.

An example of such a fit can be seen in Figure 8, where the three panels show (clockwise) a Band model fit to GBM data alone, a Band model fit to both the GBM and LAT data, and a Band model plus a step function fit to the GBM and LAT data. The difference between the first two panels demonstrates the degree to which the high-energy spectral index can steepen to accommodate the LAT data, despite being outside of the range allowed by the statistical uncertainty in the β determination made through the GBM fit alone. The third panel shows the effect of introducing a step function between the two instruments, in which the requirement for a softer β value is alleviated. For the fit shown in Figure 8, the β value determined through the Band model plus a step function fit is consistent with the value found by fitting a Band model to the GBM data alone.

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The ΔC-Stat values obtained for the Band and Band+step function fits are listed in Column 9 of Table 4. For most of the bursts, a simple steepening of the high-energy power-law index was sufficient to explain the lack of a LAT detection. However, in six cases ΔC-Stat exceeded a value of 9, indicating a statistical preference for a break in the high-energy spectrum. Figure 9 shows the ratio of the expected LAT flux (based on GBM-only fits) to the LAT 95% CL upper limit plotted versus the ΔC-Stat values for the spectroscopic subsample. A weak correlation between the flux ratio and ΔC-Stat is apparent. In addition, Figure 10 shows an anti-correlation between the resulting ΔC-Stat values for this sample plotted versus the uncertainty in the high-energy spectral index found from fits to the GBM data alone. The bursts for which a spectral break is statistically preferred both have the most severe discrepancies between the GBM-only extrapolations and the LAT upper limits and also have the smallest uncertainties in their GBM-only β values.

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If we assume that the high-energy spectra in the six GRBs that prefer spectral cutoffs are a result of γγ attenuation, as opposed to a spectral turnover that is intrinsic to the GRB spectrum, then we can use the joint GBM and LAT spectral fits in conjunction with the LAT non-detections at 100 MeV to place limits on the maximum Lorentz factor. In this context, the high-energy γ-rays produced within the GRB jet may undergo γγ → e⁺e⁻ pair production and can be absorbed in situ. The interaction rate of this process and corresponding optical depth, τ_γγ, depend on the target photon density and can be significant when both the high-energy and target photons are produced in the same physical region. Highly relativistic bulk motion of such an emission region can reduce the implied γγ optical depth greatly by allowing for a larger emitting region radius and a smaller target photon density for a given observed flux and variability timescale. Observation of γ-ray emission up to an energy E_max ≫ m_ec² thus can be used to put a lower limit on the bulk Lorentz factor Γ of the emitting region (Lithwick & Sari 2001; Razzaque et al. 2004; Granot et al. 2008; Ackermann et al. 2010). This method is valid for Γ ⩽ E_max(1 + z)/m_ec², which follows from the threshold condition for e⁺e⁻ pair production, when both the incident and target photons are at the maximum observed energy.

If a high-energy γ-ray photon with energy E and the observed broadband photon emission originate from the same physical region, and if we assume the photons are quasi-isotropic in the comoving frame, then the γγ → e⁺e⁻ pair production optical depth can be written as

$$\begin{eqnarray} \tau _{\gamma \gamma }(E) &=& \frac{3}{4} \frac{\sigma _T d_L^2}{t_v\Gamma } \frac{m_e^4c^6}{E^2(1+z)^3} \int _{\frac{m_e^2c^4\Gamma }{E(1+z)}}^{\infty } \frac{d\epsilon ^\prime }{\epsilon ^{\prime 2}} n \nonumber\\ && \times \left( \frac{\epsilon ^\prime \Gamma }{1+z} \right) \varphi \left[ \frac{\epsilon ^\prime E(1+z)}{\Gamma } \right]. \end{eqnarray} \tag{ 1 }$$

Here, n() is the observed photon spectrum, is the target photon energy, ' is the target photon energy in the comoving frame of the emitting plasma, d_L is the luminosity distance, t_v is the γ-ray flux variability timescale, and σ_T is the Thomson cross-section. The function φ['E(1 + z)/Γ] is defined by Gould & Schréder (1967) and Brown et al. (1973). The value of Γ_{γγ, min} follows from the condition τ_γγ(E_max) = 1. This single-zone model, in which the spatial and temporal dependencies of τ_γγ have been averaged out, has been the technique used to measure the reported values of Γ_{γγ, min} for the LAT detections of GRBs 080916C, 090510, and 09092B in Abdo et al. (2009b), Ackermann et al. (2010), and Abdo et al. (2009a), respectively. It is important to note that these single-zone models may provide overestimated Lorentz factors compared to time-dependent multi-zone models that consider the possibility of multiple emitting regions and that take into account the time variability of τ_γγ. For a discussion of single and multi-zone models, see Zou et al. (2011).

A direct estimate of the bulk Lorentz factor Γ, as opposed to a minimum value, of the GRB jet can be made based on evidence of a cutoff in the spectral fits that are attributed to γγ attenuation, such as has been reported for GRB 090926A in Abdo et al. (2011).

In the case of the six GRBs that we consider here for which no direct evidence for a spectral cutoff is otherwise detected, we use our upper limits to calculate a maximum bulk Lorentz factor Γ_{γγ, max} from the condition τ_γγ(E_UL) = 1. To do so, we use the Band function fit to the GBM and LAT data and set E_UL = 100 MeV. We also assume a variability timescale of t_v = 0.1 s, which we believe represents a conservative estimate of t_v given the ubiquity of millisecond variability in BATSE-detected GRBs (Walker et al. 2000) as well as the short timescales observed in other LAT-detected GRBs (Ackermann et al. 2010).

We note that if the cutoff energy due to intrinsic pair opacity is small enough, E_cutoff < m_ec²Γ/(1 + z), then the Thomson optical depth of the pairs that are produced in the emitting region is $\tau _{T,e^\pm } > 1$ (Lithwick & Sari 2001; Abdo et al. 2009a). This should affect both the observed spectrum, thermalizing it for a large enough optical depth, and light curve, eliminating short timescale variability. For E_cutoff = 100 MeV, this condition is nearly violated at z ≲ 1.0, therefore a much lower cutoff energy would be hard to reconcile with an intrinsic pair opacity origin for GRBs at low redshift.

The resulting Γ_{γγ, min} and Γ_{γγ, max} values for previously reported LAT detections and from the upper limits presented here are shown in Figure 11. Since the Lorentz factor calculation depends on the redshift, which is unknown for the majority of GBM-detected bursts, we have plotted the Γ_{γγ, max} values as a function of the redshift (red lines). One GRB in our spectroscopic subsample, GRB 091127, has a measured redshift which allows us to constrain the burst's Γ_max value. Using a redshift of z = 0.490 (Cucchiara et al. 2009) and E_UL ∼ 100 MeV, we calculate a relatively small bulk Lorentz factor of Γ_max ∼ 155. Using the measurements of E_UL for these GRBs provides a relatively narrow distribution of Γ_max that range from 50 < Γ_max < 300 at z = 1 to 400 < Γ_{γγ, max} < 640 at z = 4. These values stand in stark contrast to the LAT-detected GRBs for which Γ_{γγ, min} was measured, all of which have Γ_{γγ, min} > 800. Our results are consistent with those presented by Beniamini et al. (2011) and Guetta et al. (2011), who used the non-detection of a smaller sample of GBM-detected bursts to also infer the Γ_{γγ, max} of the emitting region.

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The detection of spectral curvature by the LAT in the spectrum of GRB 090926 provides a case that appears to bridge the LAT detected and non-detected samples. The estimate of Γ of 200–700 presented in Abdo et al. (2011) reflects the systematic differences between Lorentz factors obtained through the use of time-dependent models by Granot et al. (2008) which yield systematic differences in τ_γγ and the inferred Γ when compared to the simple single-zone model used above. Granot et al. (2008), and more recently Hascoët et al. (2012), have shown that such time-dependent models, which include the temporal evolution of τ_γγ during the emission period, can yield inferred Γ estimates that are reduced by a factor of 2–3 compared to estimates made using single-zone models. In the context of these time-dependent model, the Γ_{γγ, min} and Γ_{γγ, max} presented in Figure 11 would all be systematically overestimated by a factor of 2–3, but the dichotomy between the LAT detected and LAT non-detected GRBs would persist since all Γ estimates would be effected by the same correction.

Note that the gray dashed line in Figure 11 demarcates the self-consistency line where the condition that Γ ⩽ E_max(1 + z)/m_ec² is violated, implying an incorrect determination of τ_γγ, for the bursts with no detected emission above E_max = 100 MeV. None of the bursts in our spectroscopic subsample violate this condition at any redshift for the choice of E_cutoff = 100 MeV.