A SIGNIFICANT PROBLEM WITH USING THE AMATI RELATION FOR COSMOLOGICAL PURPOSES

We start by using the compilation of data from Tables 1 and 2 of Amati (2006). These bursts all have redshifts and are from Beppo-SAX, Konus, HETE, BATSE, and Swift. Using the Amati relation, we calculate the S_bolo from the given data. We exclude bursts 050315, 050824, 050904, 981226, 000214, and 030723 because only limits to E_{peak, obs} are provided, and therefore are not useful to us. We also had to exclude burst 980329 because a redshift range is given, and we could therefore could not get an accurate measurement of (d_L) for converting E_iso into S_bolo. The results are in Figure 5.

This sample of GRBs was largely the same as used by Amati et al. (2002) to discover and calibrate the Amati relation, so it is no surprise that the bursts are spread out along the Amati limit line. The violator fraction is ξ = 34%, which is as expected given the usual scatter due to measurement errors. The sample has 〈log E^2.04_peak/S_bolo〉 = 8.90 which is close to the limit of log (1.13 × 10⁹) = 9.05 (in appropriate units). The rms scatter of log E^2.04_peak/S_bolo is 0.56, which is a measure of how tight the Amati relation is for the sample.

Next, we visit another compilation set, this time from Schaefer (2007). This data set also takes its burst sample from a variety of different detectors: Konus, BATSE, Beppo-SAX, HETE, and Swift bursts were included for this sample. While the paper studies 69 GRBs with known redshift, only 27 have the bolometric fluence reported and thus are the ones that we use (see Figure 6).

Of the 27 bursts, 11 fail the Nakar & Piran test (ξ = 41%). This is an expected failure rate for the Amati relation and is in agreement with previous analysis on these data (see Schaefer & Collazzi 2007). The average value for the log of 〈log E^2.04_peak/S_bolo〉 is 8.95 ± 0.57. So these data are similar to the previous data set (which is not surprising as they share some of the same bursts).

The two samples which contain exclusively bursts with known redshifts both agree well with the Amati relation, and this has long been the primary justification for accepting the Amati relation as a physical relation for GRBs. However, other samples (see below) do not agree with the Amati limit, and this suggests that bursts with redshifts might be a significantly different sample from those without redshift. This is the reason why we distinguish bursts with and without redshifts in Table 1 and in the figures.

Our data for BATSE are a part of the upcoming 5B catalog (A. Goldstein et al. 2011, in preparation). We present the values of E_peak and S_bolo for the most statistically preferred fitting model, CPL (cut-off power law) or Band. We also use bursts for which we have 40% relative errors or better. After applying these selection criteria, we are left with 1654 bursts, which are plotted in Figure 7. In the figure, we also introduce two new curved lines which represent illustrative thresholds for the trigger (dotted) and the ability to detect E_peak (dot-dashed). These are explained in detail in Section 4.

We find that the BATSE bursts fail at an extreme rate, with 93% violators. In addition, the BATSE bursts cover a large region of the disallowed zone, with very few bursts above the limit. We find that 〈log E^2.04_peak/S_bolo〉 to have a value of 10.18 ± 0.88. The failure rate is consistent with that observed in the past of BATSE bursts in previous works. The spread of BATSE bursts is so large, it hints that any future changes to the Amati relation (e.g., as more Swift bursts are detected with redshifts) will result in a high failure rate.

Our sample of HETE bursts comes from Sakamoto et al. (2005). The quoted values for S_bolo were covering the 2–400 keV range and had to be converted into bolometric fluences. This was done by using the given parameters for the spectral model (Band or CPL) to extrapolate a bolometric correction. This was generally a small correction, while even the large corrections are still small compared to scatter in Figure 8. The quoted error bars for both E_peak and S_bolo are given for the 90% level, so they had to be converted into standard 1σ confidence level error bars. The figure also has illustrative lines to represent trigger thresholds (dotted line) and the E_peak detection threshold (dot-dashed line). Again, these will be explained in detail in Section 4.

The HETE bursts have ξ = 33% and 〈log E^2.04_peak/S_bolo〉 similar to that of the original Amati sample, so it appears that these bursts are consistent with the Amati relation. Nevertheless, the scatter apparent in Figure 8 is so large that there is little utility in applying the Amati relation to these bursts. The difference is insignificant for 〈log E^2.04_peak/S_bolo〉 between bursts with and without redshifts. Of all the single-satellite data sets that we consider, the HETE data are the only ones that apparently obey the Amati relation, although their large scatter limits their usefulness.

For the Swift data, we use the catalog in Butler et al. (2007). We use their E_{peak, obs} as derived from frequentist statistics as it is the most common approach to finding E_peak. (Their Bayesian values were made with unreasonable priors that significantly skew the results.) The Swift burst detector only goes up to 150 keV, so the reported values of E_{peak, obs} are almost all lower than 200 keV. We have adopted their bolometric fluences and we have converted their non-standard 90% error bars into standard 1σ error bars. Figure 9 plots the results; bursts without known redshifts are represented as empty circles and bursts with known redshift are represented by a filled diamond. This will be true for all future plots. This is the last of the three plots in which we have plotted illustrative lines representing trigger (dashed line) and E_peak detection thresholds (dot-dashed line), which again, are detailed in Section 4.

The Swift bursts violate the Amati limit at a rate of 76%–82%. That is, the Amati relation does not work for Swift. This result is not the result of small number statistics, and we can see from the distribution that the disagreement is highly significant. This is another version of the same conclusion first reported by Butler et al. (2007).

When first confronted with the discrepancy that bursts with redshifts agreed with the Amati relation (see Figures 5 and 6), while bursts without redshifts disagreed with the Amati relation (Band & Preece 2005), our initial thought was that the bursts with redshifts might be somehow selected from a separate population for which the Amati relation applied. However, with this large sample of well-measured bursts from Swift, the distributions of bursts with and without redshifts are essentially identical. Thus, we have a proof that the success or failure of the Amati relation does not depend on some selection effect that correlates with the measuring of spectroscopic redshifts. Another thing to remember is that since Swift bursts are the bursts that account for a majority of the bursts with known redshifts, there is a built-in selection effect that will eventually develop that will bias future iterations of the Amati limit toward the area Swift bursts cover.

Suzaku data for long GRBs are available through the GCN circulars (http://gcn.gsfc.nasa.gov). Typically, the reported fluence covers the 100 keV–1 MeV range, so we apply a bolometric correction based on the reported spectral fit. A typical bolometric correction value is a factor of ∼1.7. The E_{peak, obs} and fluence values reported in the circulars are preliminary and made soon after the burst, yet any likely changes to get to the final best fits are greatly smaller than the scatter shown in Figure 10 and are thus not important. The Suzaku bursts all have E_{peak, obs} > 200 keV, a result of the relatively high energy range of sensitivity of the detector.

Most of the Suzaku bursts violate the Amati limit, usually by a large factor (ξ ∼ 94%) and the small fraction that are not violators are very close to the limit (Figure 10). Therefore, the Amati relation does not work for Suzaku bursts. This is true for both bursts with redshifts and without.

Krimm et al. (2009) presented a catalog of bursts for which there was both Swift and Suzaku data. The expanded energy range gave better fits to the spectra, with Swift covering the lower energies and Suzaku covering the higher energies. With the joint spectral fits over a very wide range of photon energies, the sample has a wide range of E_{peak, obs} values from 30 keV to 2000 keV. The catalog lists the best fit for the three major spectral models (power law, power law with an exponential cutoff, and Band model) for the majority of the listed bursts, while we use the E_{peak, obs} values from just the Band function.

The joint sample largely stretches between the Amati limit line and the Ghirlanda limit line (Figure 11). The fraction of violators is 86% for the 28 bursts with spectroscopic redshifts and 74% for the 38 bursts without redshifts. Again, the Amati relation fails, and there is no significant difference related to whether the burst has a spectroscopic redshift or not. No burst significantly violates the Ghirlanda limit.

The GRB detectors on the Wind satellite (Aptekar et al., 1995) are long-running instruments with a stable background that have measured many bursts, with the fluence and E_{peak, obs} values promptly reported in the GCN circulars. These reported values are preliminary, with no final analysis having been published, but any plausible errors due to the preliminary nature of the report are greatly smaller than the observed scatter in the S_bolo–E_{peak, obs} diagram (see Figure 12). We find a total of 97 bursts, 33 of which have associated spectroscopic redshifts, with reported fluences and E_{peak, obs} for the entire burst interval. We applied a bolometric correction factor, based on the given spectral fits. This factor was typically very small, due to the large range the reported Konus fluences usually cover.

The distribution of the Konus bursts in the S_bolo–E_{peak, obs} diagram has a flat lower cutoff, likely due the trigger threshold (although this cutoff is higher than the reported trigger threshold in Aptekar et al. 1995 of ∼5 × 10⁻⁷ erg cm⁻²). The distribution also shows a fairly high upper limit on E_{peak, obs} due to the sensitivity of the detectors to high photon energies. From 22% to 27% of the bursts are above the Amati limit line, while all the bursts are above the Ghirlanda limit line. The Amati relation fails for the Konus bursts. Again, the bursts with redshifts are distributed identically to those without, so there is no apparent selection effect based on spectroscopic redshifts.

Guidorzi et al. (2011) provide a large catalog of both E_peak and S from the Beppo-SAX GRBM. In the catalog, data for the brightest 185 bursts are given; for which we are able to use 129. Of the useable 129 bursts, we have only ten bursts with spectroscopic redshifts. The provided S_bolo were over the 40–700 keV range. We apply the same type of bolometric correction as before, using the provided spectral indices for the CPL used. This correction is typically small, with a typical correction value of 1.5.

While it is impossible to make a strong statement about the shape of the distribution of Beppo-SAX bursts with only the bright bursts, there is still an important result from the data. The data are plotted in Figure 13. We find that even among the brightest bursts, 90% of bursts with redshifts and 85% of bursts without redshifts are violators. What is particularly provocative about this result is that these are the brightest bursts, and thus the most likely to not be violators. It is unlikely that there are a significant number of "missing" bursts that would be non-violators. Any such burst would have to be both bright and have a low E_peak while still being sim enough to be missed in the bright burst catalog. Finally, we provide information as to the average energy ratio of these bursts, but we once again stress that these are only the brightest bursts, and therefore these values should be taken with caution. We therefore feel confident in saying that the Amati relation fails for Beppo-SAX bursts, although this statement is not as strong as it is for the other detectors because of the sample used.

Previously, Butler et al. (2007) had pointed out that the normalization constant for the Amati relation was slightly different depending on whether Swift or pre-Swift bursts were used, and this is like noting that 〈log E^2.04_peak/S_bolo〉 has changed. Previously, Band & Preece (2005) and Goldstein et al. (2010) pointed out that >80% of BATSE bursts violate the Amati limit. In this section, we have generalized these analyses, both to looking at many GRB detector instruments and to looking at the two-dimensional distribution in the S_bolo–E_{peak, obs} diagram.

All of these data sets give consistent conclusions. (1) the distribution of bursts in the S_bolo–E_{peak, obs} diagram varies significantly and greatly from satellite-to-satellite. (2) The only data sets to pass the generalized Nakar & Piran test for the Amati relation are the early heterogeneous sample of bursts with measured spectroscopic redshifts. (3) The bursts detected by BATSE, Swift, Suzaku, and Konus all have a high fraction (ξ > 70%) of bursts which violate the Amati limit, with the violations being highly significant and by large factors. That is, the Amati relation fails for bursts from these four satellites. (4) The Amati limit is satisfied for the HETE bursts, to the extent that the violator fraction is consistent the Amati relation plus normal observational scatter, however, the scatter in the S_bolo–E_{peak, obs} diagram is so large that we conclude that the Amati relation does not satisfactorily apply to the HETE data. (5) We find that no bursts, from any satellite, significantly violate the Ghirlanda limit. (6) These conclusions are true whether we examine only bursts with spectroscopic redshifts or without redshifts.

The normalization factor for the Amati relation will scale closely with 〈log E^2.04_peak/S_bolo〉. These values are listed in Table 1 and can be taken as an intercept with some dispersion. That is, these values can possibly be seen as a means of describing a sort of Amati relation to each detector. However, this is not a useful interpretation as the dispersion is often as big as the population itself, rendering such an interpretation nearly meaningless. If anything, the results from Table 1 demonstrate that the Amati relation must vary from detector-to-detector by over an order of magnitude. With the bursts seen in the sky not depending on the satellite, the large variations in the Amati relation from detector to detector imply that there must be some selection effect which biases the visible bursts, with these biases being instrument specific.

Every burst detector has a substantially different distribution of bursts in Figures 7–12. Since the population of bursts that appear in the skies above the Earth does not change with the satellite, the large changes from detector to detector can only be due to some selection effect where bursts in various regions of the S_bolo–E_peak diagram are not selected. The next section will investigate and identify these selection effects that create the Amati relation.

The best-known selection effect is the detector trigger threshold. For example, a burst would trigger BATSE only if it was produced a peak flux (in a 0.064, 0.256, or 1.024 s time bin) brighter than 5.5σ above background in at least two detectors over the 50–300 keV energy range. Other satellites have more complex trigger algorithms (for example, the Gamma-ray Burst Monitor (GBM) has overlapping triggers), but they all come down to the same essentials. The trigger threshold depends on the E_{peak, obs}, the spectral energy range of the trigger, the background flux, and the effective area of the detector. The triggers operate off the peak flux (P_max), so the limiting fluence will depend on the effective duration (S_bolo/P_max), which can vary widely from burst to burst. Thus, the limit due to trigger thresholds will be "fuzzy," with no sharp edge but rather a gradient as S_bolo is reduced. Approximately, the trigger threshold will produce a horizontal cutoff at the bottom of the S_bolo–E_{peak, obs} diagram.

In principle, the exact trigger thresholds can be calculated for every detector and burst. In practice, the conditions (E_{peak, obs}, background flux, incidence angle, burst light curve) vary greatly from burst to burst, creating substantial scatter in the thresholds. For this paper, we do not need an accurate distribution for the S_bolo threshold, so instead we calculate the typical S_bolo threshold as a function of E_{peak, obs} for average conditions. In particular, we adopt an average spectral shape as the Band function (Band et al. 1993) with a low-energy power-law index of −1.0 and a high-energy power-law index of −2.0. We also take the effective duration of the peak (S_bolo/P_max) such that it fits the observed distribution of the detectors. For each detector, we take its trigger energy range, face-on effective area, and the average background flux. The formalism and many of the input parameters were taken from Band (2003). The result is a lower limit in the S_bolo–E_{peak, obs} diagram, as displayed in Figure 7 for BATSE, Figure 8 for HETE, and Figure 9 for Swift. We do not have enough information to calculate trigger thresholds for some satellites, but the threshold is usually fairly obvious (e.g., Figure 12 has a nearly flat and moderately sharp lower limit to S_bolo). These thresholds are not sharp, so bursts can easily appear somewhat below the threshold. Indeed, by varying the input conditions somewhat, the trigger threshold lines can be translated up and down substantially.

A second detector selection effect is that the burst must have enough photons recorded for the analyst to be able to determine the E_{peak, obs} value. This will depend on both S_bolo and E_{peaks, obs} as well as the detector properties. For example, a burst just above the trigger threshold will have just enough photons to be detected but not enough photons to allow any constraints on the E_{peak, obs} value, so this burst will not be included in a sample for plotting on the S_bolo–E_{peak, obs} diagram. For another example, consider a burst with E_{peak, obs} at the upper edge of the measured spectral range for a detector, such that a very bright burst will have a well-measured turnover that accurately defines the fitted E_{peak, obs} value, whereas a fainter burst will have poor photon statistics near the turnover in the spectrum and the E_{peak, obs} value will remain unmeasured and the burst will not be included in any of our samples.

In general, for a given E_{peak, obs}, there will be some lower limit on S_bolo, below which there will be too few photons to measure E_{peak, obs}. As E_{peak, obs} moves to higher energies, the limit on S_bolo will sharply increase. The result will roughly be a diagonal line across the S_bolo–E_{peak, obs} diagram, from lower left to upper right, with any burst below that line not having a measured E_{peak, obs} and not appearing in any sample of bursts in Section 3.

We have made calculations of this threshold curve for BATSE, HETE, and Swift. To do this, in a Monte Carlo sense, we constructed many simulated bursts over each detector's spectral range for many values of E_{peak, obs} where the normalization and error bars of the spectra were determined by the burst fluence. These spectra were also for each spectrum, we then fitted both a power law times an exponential model (with the calculated E_{peak, obs} value) and a simple power-law model. If the chi-square values for the two fits differed by more than 15.0 (so that the model with the peak was a sufficiently good improvement on power-law model given the extra degree of freedom), then we took the S_bolo for the burst to be above the threshold. By varying the E_{peak, obs}, we were able to determine the threshold for measurement as a curve in the S_bolo–E_{peak, obs} diagram. As these lines are merely for illustration, we do not use the DRM for these simulations. For BATSE, HETE, and Swift, our calculated thresholds are presented as curves in Figures 7–9.

Among bursts appearing in the skies, the E_{peak, obs} distribution is not flat, but rather bursts appear with a roughly log-normal distribution of E_{peak, obs}. For bright bursts, the mean value is 335 keV, with the FWHM stretching from roughly 150–700 keV (Mallozzi et al. 1995). This mean value shifts significantly as the bursts get dimmer, being 175 keV just above the BATSE trigger threshold (Mallozzi et al. 1995). The so-called "X-ray Flashes" are simply bursts in the low-energy tail of the distribution (Kippen et al. 2004; Sakamoto et al. 2005; Pélangeon et al. 2008). The existence of this single peak in the E_{peak, obs} histogram is highly significant and not from any instrumental or selection effect (Brainerd et al. 2000). In all, most bursts are between 100 and 700 keV, and bursts <30 keV or >1000 keV are rare. This will directly translate to unpopulated regions of the S_bolo–E_{peak, obs} diagram. A direct simulation of this distribution is given in Figure 4.

The E_{peak, obs} distribution will cause definite but gradiated cutoffs in the S_bolo–E_{peak, obs} diagram. These cutoffs will be nearly vertical. The drop in the average E_{peak, obs} will make the cutoff on the right have a slope down to the lower left.

Unsurprisingly, bright bursts are rare, while faint bursts are more frequent. The distribution of burst fluences is traditionally represented by the log N(> P)–log P curve, for which the best observations come from the BATSE catalog (Fishman et al. 1994; Paciesas et al. 1999). For bright bursts, the slope of the curve is nearly the ideal −3/2. The slope flattens out for faint bursts, approaching −0.7. In the S_bolo–E_{peak, obs} diagram, the density of bursts falls off drastically from bottom to top (see Figure 4).

The intrinsic distribution of bursts in the S_bolo–E_{peak, obs} diagram is determined by the E_{peak, obs} log-normal distribution that changes with S_bolo (Mallozzi et al. 1995) and by the log N(> P)–log P distribution (Fishman et al. 1994). With these two effects, the burst density across the diagram is displayed in Figure 4. Together, the two effects produce contour lines of burst density in the diagram, and two such curves are displayed in Figure 14. The combined effects make for the upper left corner of the S_bolo–E_{peak, obs} diagram, simply because bursts up there are doubly rare (both low in E_{peak, obs} and very bright). This means that there are few bursts significantly above (and to the left) of the Amati limit line. That is, there is a natural cutoff on the top side of the Amati relation.

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The detector selection effects then operate on the natural distribution. The well-known trigger threshold is actually below the threshold for measuring an E_{peak, obs} value, so only the later selection effect is really operating. This selection effect cuts on a sort of diagonal from lower left to upper right, and its position depends greatly on the detector sensitivity and energy band for the trigger. For a relatively poor sensitivity and a trigger energy band that effectively does not get much above a few hundred keV, the threshold will be quite high. Indeed, for many detectors, the threshold will be just below the Amati limit line (Figure 14), so there will be few bursts significantly below the Amati limit line. That is, there is a selection effect from the intrinsic distribution of bursts such that there is a natural cutoff below the Amati limit.

For some detectors, we can see that the Amati relation is a natural and expected consequence of the intrinsic burst distribution combined with normal detector selection effects. This is illustrated in Figure 14, where the allowed region is confined to an area along the Amati limit line. From Figure 3, we know that bursts along the Amati limit in the S_bolo–E_{peak, obs} diagram will then imply a relation close to the Amati relation. Thus, the natural distribution of bursts makes for bursts above the Amati limit (i.e., the very-bright low-E_{peak, obs} bursts) to be rare, while the detector selection effects make for bursts below the Amati limit (i.e., the faint high-E_{peak, obs} to be too faint to have a measured E_{peak, obs}. With the only bursts remaining being close to the Amati limit line, a relation like the Amati relation would be apparent. Thus, we conclude that the Amati relation is simply a result of selection effects and there is no physical basis.

For the original bursts used to define the Amati relation (Amati et al. 2002), an additional selection effect is operating, in that the burst must also have a measured spectroscopic redshift for inclusion in the sample used for calibration. The selection effects for measuring a redshift are complex. There is certainly a selection based on redshift, with the cause being that more distant bursts are fainter, hence less likely to have a visible optical transient or host galaxy bright enough to get lines in a spectrum. An additional effect on redshift relates to the availability of spectral lines in the optical band. The efficiency of measuring redshift as a function of redshift has been quantified in Xiao & Schaefer (2011), with this effect being roughly an order of magnitude between z = 5 and nearby bursts. The efficiency of measuring redshifts also presumably depends on S_bolo which will roughly scale as the burst brightness in the optical band. The redshift measurement efficiency also is time dependent, as optical follow-up strategies and capabilities change within our community. Thus, Swift bursts started out with an average redshift of 2.8 in the first year after its launch (Jakobsson et al. 2006), while the average redshift has steadily declined to 2.1 over the last year (Jakobsson et al. 2008). The reason for this shift is unknown, but it must come from overall follow-up practices in our community. The bursts with redshift used in the original calibration of the Amati relation have an average redshift of 1.5, indicating that the effective threshold for this sample is quite high.

The distribution of bursts in the S_bolo–E_{peak, obs} diagram will depend greatly on the detector. The threshold for measuring E_{peak, obs} varies substantially detector to detector. For example, BATSE has a low threshold while Konus has a high threshold. The shape of the threshold (as a function of E_{peak, obs}) also varies, from a flat bottom for Konus due to its sensitivity to high energy, to the up-sloping threshold for Swift due to its lack of high-energy sensitivity and even more exaggerated in HETE with its small area. The ability to measure E_{peak, obs} depends critically on the energy range of the spectra. The Konus detectors have a very wide range of spectral energy resulting in a wide range of measured E_{peak, obs} values, the Swift detectors cutoff around a few hundred keV, while the Suzaku detectors can only record E_{peak, obs} ≳ 200 keV. The combination of these selection effects makes the distribution of bursts different for each detector and accounts for the wide range of distributions seen in Figures 5–12.

Still, the issue has been raised in recent tests (e.g., Ghirlanda 2011) that what we are seeing in these failures is just the scatter about a relation which is ever changing with new bursts every day. The primary argument is that the Nakar & Piran test limit should be formulated from the 3σ line about the model, instead of the model line itself. As a result, the Amati limit would be considerably higher. There are a variety of problems with this argument. The first of which being that there is already an allowance made for the Amati relation to have up to 40% violators and not be considered as failing for the data set. Therefore, the scatters are already being accounted for, and it is overkill to use such a generous limit to perform the test. If the test is done in this manner, allowances can no longer be made for any violators (or, more precisely, there needs to be less than 0.3% violators). Even by the groups own tests, there are violators on the order of a few percent, depending on the test. This is an unacceptable violator rate considering they are violating a limit from the 3σ deviation from the model. Finally, another question that arises is that the bursts we see all seem to be biased in one direction. If we were seeing the result of measurement scatter about the Amati relation, we should expect to see an equal fraction of bursts well above the limit line. Instead, we see that for almost all data sets, the bursts are systematically in the one direction from the limit.

The Amati relation will certainly see improvement in these tests in the future. With increasing number of Swift bursts with spectroscopic redshifts, it will undoubtedly eventually lie right in the middle of the Swift data set. Even then, the Amati relation will be failing for our best data sample, the BATSE data. Our argument is that there are undeniable systematic effects at play that are causing the Amati relation, and therefore even these "improvements" would be fairly meaningless as we would still see systematic differences in where bursts are observed in the diagrams. Therefore, the Amati relation is simply not good for making any kind of prediction, cosmological or otherwise.

Perhaps the simplest disproof of the Amati relation is simply that the violator fraction is much too high in most data sets. And perhaps the simplest proof that the Amati relation is caused by selection effects is the large differences between the various S_bolo–E_{peak, obs} diagrams for the many detectors.