All-particle primary energy spectrum in the 3-200 PeV energy range

All-particle primary energy spectrum in the 3-200 PeV energy range

A.P. Garyaka R.M. Martirosov S.V. Ter-Antonyan samvel$_$ A.D. Erlykin N.M. Nikolskaya Y.A. Gallant L.W. Jones and J. Procureur Yerevan Physics Institute, 2 Alikhanyan Br. Str., 375036 Yerevan, Armenia Department of Physics, Southern University, Baton Rouge, USA P.N.Lebedev Physical Institute, Moscow, Russia LPTA, Universit Montpellier II, CNRS/IN2P3, Montpellier, France University of Michigan, Department of Physics, USA Centre d’Etudes Nucléaires de Bordeaux-Gradignan, Gradignan, France

We present all-particle primary cosmic-ray energy spectrum in the GeV energy range obtained by a multi-parametric event-by-event evaluation of the primary energy. The results are obtained on the basis of an expanded EAS data set detected at mountain level ( g/cm) by the GAMMA experiment. The energy evaluation method has been developed using the EAS simulation with the SIBYLL interaction model taking into account the response of GAMMA detectors and reconstruction uncertainties of EAS parameters. Nearly unbiased () energy estimations regardless of a primary nuclear mass with an accuracy of about in the GeV energy range respectively are attained.
An irregularity (‘bump’) in the spectrum is observed at primary energies of GeV. This bump exceeds a smooth power-law fit to the data by about 4 standard deviations. Not rejecting stochastic nature of the bump completely, we examined the systematic uncertainties of our methods and conclude that they cannot be responsible for the observed feature.

Cosmic rays, energy spectra, composition, extensive air showers
96.40.Pq, 96.40.De, 96.40.-z, 98.70.Sa

, , , Corresponding author: , , ,

1 Introduction

Study of the fine structure in the primary energy spectrum is one of the most important tasks in the very high energy cosmic ray experiments [1]. Commonly accepted values of the all-particle energy spectrum indexes of and before and after the knee are an average and may not reflect the real behavior of the spectrum particularly after the knee. It is necessary to pay special attention to the energy region of GeV, where experimental results have been very limited up to now. Irregularities of the energy spectrum in this region were observed a long time ago. They can be seen from energy spectrum obtained more than 20 years ago with AKENO experiment [2] as well as in later works of the GAMMA [3] and TUNKA [4] experiments. At the same time the large statistical errors did not allow to discuss the reasons of these irregularities.
On the other hand results of many experiments on the study of EAS charge particle spectra, the behavior of the age parameter and muon component characteristics point out that the primary mass composition at energies above the knee becomes significantly heavier. Based on these indications, additional investigations of the fine structure of the primary energy spectrum at GeV have an obvious interest.
There are two ways to obtain the primary energy spectra using detected extensive air showers (EAS). The first way is a statistical method, which unfolds the primary energy spectra from the corresponding integral equation set based on a detected EAS data set and the model of the EAS development in the atmosphere [5, 6, 7, 8, 9]. The second method is based on an event-by-event evaluation [2, 10, 11, 12, 13] of the primary energy of the detected EAS with parameters using parametric [2, 10, 11, 13] or non-parametric [12] energy estimator previously determined on the basis of shower simulations in the framework of a given model of EAS development.
Here, applying a new event-by-event parametric energy evaluation , the all-particle energy spectrum in the knee region is obtained on the basis of the data set obtained with the GAMMA EAS array [7, 8, 9, 11] and a simulated EAS database obtained using the SIBYLL [14] interaction model. Preliminary results have been presented in [10, 11].

2 GAMMA experiment

The GAMMA installation [7, 8, 9, 11] is a ground based array of 33 surface detection stations and 150 underground muon detectors, located on the south side of Mount Aragats in Armenia. The elevation of the GAMMA facility is 3200 m above sea level, which corresponds to 700 g/cm of atmospheric depth. A diagrammatic layout of the array is shown in Fig. 1.

Figure 1: Diagrammatic layout of the GAMMA facility.

The surface stations of the EAS array are arranged in 5 concentric circles of 20, 28, 50, 70 and 100 m radii, and each station contains 3 plastic scintillation detectors with the dimensions of m. Each of the central 9 stations contains an additional (the 4th) small scintillator with dimensions of m (Fig. 1) for high particle density ( particles/m) measurements.
A photomultiplier tube is placed on the top of the aluminum casing covering each scintillator. One of three detectors of each station is viewed by two photomultipliers, one of which is designed for fast timing measurements.
150 underground muon detectors (’muon carpet’) are compactly arranged in the underground hall under 2.3 kg/cm of concrete and rock. The scintillator dimensions, casings and photomultipliers are the same as in the EAS surface detectors.
The shower size thresholds of the shower detection efficiency are equal to and at the EAS core location within m and m respectively [7, 8, 11].
The time delay is estimated by the pair-delay method [15] to give the time resolution of about ns. The EAS detection efficiency () and corresponding shower parameter reconstruction errors are equal to: , , , , and m. The reconstruction errors of the truncated muon shower sizes for m from the shower core are equal to at respectively [8, 9, 11].

3 Event-by-event energy estimation

3.1 Key assumptions

Suppose that is an estimator of energy of unknown primary nuclei which induced showers with the detected parameter . Then the expected all-particle energy spectrum is defined by


where are the energy spectrum of primary nuclei and are the corresponding () transformation probability density function.
If and are the log-normal distributions with and parameters, the expression (1) has the analytic solution for the expected spectrum of the energy estimator [16]:


It is seen that evaluation of energy spectrum from (2) is possible to perform only at a priori known , and parameters and spectral slope () of detected energy spectra coincides with spectral slope of primary energy spectra . The values of and may depend on the primary energy () and mass of primary nuclei () from which the all-particle energy spectrum is consisted of. In this case, the expression (1) is unfolded numerically and the slope of detected energy spectrum can differ from primary energy spectrum.
For example, the dependence at leads to the numerical solutions which can be approximated by the expression (2) replacing with . The corresponding approximation errors is about in the energy range of and .
However, the evaluation of energy spectra can be simplified provided


are satisfied for given energy range of . Then, the all-particle energy spectrum can be evaluated from


The corresponding error of evaluation (6) with approaches (3-5) is determined by a sum of the statistical errors and systematic errors due to approaches being used:

where the systematic relative errors is


The values of , and parameters from expressions (3-7) and corresponding uncertainties , and essential for the reconstruction of primary energy spectrum using the GAMMA facility EAS data and approach (6) are considered in Sections 3.2-3.4 below.

3.2 Uncertainty of spectral slope

The results of different experiments [9, 17, 18, 19] and theoretical predictions [20, 21, 22] indicate that the elemental energy spectra can be presented in the power law form


where at and at . It is also accepted that the mass spectra of primary nuclei can be divided into separate nuclear groups and below, as in [9], just 4 nuclear species (, , -like and -like) are considered. Dependence of the knee energy on the primary nuclei type assumed to be either rigidity dependent, [9, 20, 21, 22] or -dependent [9, 23], , where and are the charge and mass of primary nuclei correspondingly.
As a result, the all-particle energy spectrum slowly changes its slope and can be roughly approximated by a power law spectrum with power index at GeV, at GeV and at GeV. This presentation of the all-particle spectrum agrees with world data [23] in the range of uncertainty and energy interval GeV.
The values of and parameters are presented in Section 3.4 and depend on efficiency of energy estimator .
Notice that it follows from the expression (7) that for and the contribution of in the systematic errors (7) is negligible and the difference of all-particle spectra evaluated by expression (6) for and is less than at .

3.3 The simulated EAS database

To obtain the parametric representation for unbiased () energy estimator of the primary energy we simulated showers database using the CORSIKA(NKG) EAS simulation code [24] with the SIBYLL [14] interaction model for , , and primary nuclei.
Preliminary, the showers simulated with NKG mode of CORSIKA code for each of the primary nuclei were compared with the corresponding simulations using EGS mode of CORSIKA [24] taking into account the detector response, contribution of EAS -quanta and shower parameter reconstruction uncertainties. Simulated statistics were equal to 200 events for each of primary nuclei with log-uniform primary energy distribution in the range of GeV. Using the threshold energy of shower electrons (positrons) for NKG mode at observation level as a free parameter (the same as it was performed in [9]), the biases and were minimized for all simulated primary nuclei ().
Applied method of calibration of the NKG mode of CORSIKA for the GAMMA EAS array differed from [9] only by the expanded range of selected shower core coordinates () and zenith angles . The obtained biases of shower size and age parameter in the range of statistical errors () agreed with data [9]. The values of were used further for correction of the shower size obtained by NKG simulation mode.
The simulated primary energies () for shower database were distributed according to a power law spectrum with total number of detected (, m) and reconstructed showers for each primary nucleus. The energy thresholds of primary nuclei were set as GeV and GeV. The simulated showers had core coordinates distributed uniformly within the radius of m and zenith angles .
The reconstruction errors of shower size are presented in Fig. 2 for different primary nuclei and different zenith angles. The right and left ends of diagonals of the rectangular in Fig. 2 show the average primary energies (in units of GeV) responsible for corresponding shower sizes for the primary proton and Iron nuclei respectively.

Figure 2: Shower size reconstruction errors for different primary nuclei () and zenith angles ( and ). The right and left ends of diagonals of the rectangular show the average primary energies () and corresponding shower sizes computed for the primary proton and Iron nuclei respectively.

All EAS muons with energies of GeV at GAMMA observation level have passed through the 2.3 kg/cm of rock to the muon scintillation carpet (the underground muon hall, Fig. 1). The muon ionization losses and electron (positron) accompaniment due to muon electromagnetic and photonuclear interactions in the rock are taken into account using the approximation for equilibrium accompanying charged particles obtained from preliminary simulations with the FLUKA code [26] in the TeV muon energy range. The resulting charged particle accompaniment per EAS muon in the underground hall is equal to () and () at muon energies TeV and TeV respectively.
Due to absence of saturation in the muon scintillation carpet, the reconstruction errors () of truncated muon size are continuously decreasing with increasing muon truncated sizes in the range . Corresponding magnitudes of reconstruction errors for primary protons and Iron nuclei were equal to and for EAS muon truncated sizes respectively.
Fluctuations of the shower size for given primary energies GeV and were equal to and respectively.
Corresponding fluctuations of muon truncated size were equal to and . For zenith angles of primary nuclei , the fluctuations are increased about times due to the aging of detected showers.
The EAS simulated events with reconstructed , m, and shower parameters for the and parameters of primary nuclei made up the simulated EAS database.

3.4 Energy estimator

The event-by-event reconstruction of primary all-particle energy spectrum using the GAMMA facility is mainly based on high correlation of primary energy and shower size (). The shower age parameter () zenith angle () and muon truncated shower size () have to decrease the unavoidable biases of energy evaluations due to abundance of different primary nuclei. In Table 1 the correlation coefficients and between shower parameters , , and primary energy () and mass of primary nuclei () are presented.

Table 1: Correlation coefficients and between shower parameter and primary energy () and nuclei mass for two zenith angular intervals.

Parametric representation for the energy estimator we obtained by minimizing


with respect to for different empirical functions with a different number () of unknown parameters. The values of , and corresponding reconstructed shower parameters , , and for estimation of were taken from simulated EAS database (Section 3.3).
The best energy estimations as a result of the minimization (9) were achieved for the 7-parametric () fit:


where , , , is the shower age and energy is in GeV. The values of parameters are shown in Table 2 and were derived at and , where the number of degrees of freedom . The expected errors of corresponding parameters were negligibly small () due to very high values of .

Table 2: Approximation parameters of primary energy evaluation (10) obtained from -minimization (9) for the SIBYLL interaction model, and .

The corresponding average biases versus energies ( and ) for the primary proton (), iron () nucleus and uniformly mixed composition are presented in Fig. 3 (symbols). The boundary lines correspond to approximations , where and for the upper and lower limits respectively. The shaded area corresponds to and and were used to estimate the errors according to (7) for the reconstruction of all-particle energy spectrum (Section 4).
The dependence of standard deviations of systematic errors of energy evaluations (10) on primary energy is presented in Fig. 4 for 4 primary nuclei and uniformly mixed composition with equal fractions of p, He, O and Fe nuclei.

Figure 3: Mean biases versus energy and for the primary proton () and iron () nucleus and the uniformly mixed compositions (All).
Figure 4: Errors of the energy estimator (6) versus primary energy for 4 primary nuclei and uniformly mixed (All) composition. The empty rhombic symbols are taken from our previous data [10] computed for the mixed composition and shower core selection criteria m.
Figure 5: (in units of GeV) scatter plots for four () primary nuclei. The white lines show the corresponding dependence.
Figure 6: Distribution functions of errors for different primary nuclei (symbols) and uniformly mixed composition (symbols and solid line).

The results for the uniformly mixed composition with the shower core selection of m [10] are presented in Fig. 4, as well. It is seen that the value of responsible for (expression (9)) with uncertainty (expression (5)) encloses the data presented in Fig. 4.
Such high accuracies of the energy evaluation regardless of primary nuclei is a consequence of the high mountain location of the GAMMA facility (700 g/cm), where the correlation of primary energy with detected EAS size is about (Table 1).
The scatter plot of simulated primary energy and estimated energy according to expression (10) and Table 2 are shown in Fig. 5. The corresponding distributions of energy errors or the kernel function of integral equation (1) for different primary nuclei and uniformly mixed composition are presented in Fig. 6 (symbols). The average values and standard deviations of these distributions depending on energy of primary nucleus () are presented in Figs. 3,4. The dashed line is an example of log-normal distribution with and parameters corresponding to the uniformly mixed composition.
It is seen, that the errors can be described by the log-normal distributions and the key assumptions (3-5) are approximately valid.

The test of applied approaches (expressions (3-6), Section 3.2) for the reconstruction of all-particle primary energy spectrum was carried out by the direct folding of the power law energy spectrum (expression (8)) with the log-normal kernel function according to expression (1) for primary proton and Iron nucleus. The values of and were derived from the log-parabolic interpolations of corresponding dependencies presented in Figs. 3,4. The event-by-event reconstructed energy spectrum (the left hand side of expression (1)) was obtained from expression (6) using approaches (3-5) with and . The boundary lines of shaded area in Fig. 3 were used as estimations of uncertainties of condition (4). In Fig. 7 the values of are presented (symbols) for primary proton and Iron nuclei and different ”unknown” spectral indices of primary energy spectra (8) with the rigidity dependent knee at GeV. The shades area is the expected errors computed according to expression (7).

Figure 7: Discrepancies of initial () and reconstructed () energy spectra (symbols) for the different primary nuclei and spectral indices of initial spectrum. The shaded area shows the expected errors according to expression (7). The star symbols are the spectral discrepancies for a pulsar component (Section 5).

It is seen, that all spectral discrepancies are practically covered by the expected errors according to expression (7). The star symbols in Fig. 7 represent the discrepancies of singular spectra with knee at energy GeV described in Section 5.

4 All-particle primary energy spectrum

The EAS data set analyzed in this paper has been obtained for sec of live run time of the GAMMA facility, from 2004 to 2006. Showers to be analyzed were selected with the following criteria: , m, , , and (where is the number of scintillators with non-zero signal), yielding a total data set of selected showers. The selected measurement range provided the EAS detection efficiency and similar conditions for the reconstruction of showers produced by primary nuclei with energies GeV. The upper energy limit is determined from Fig. 4, where the saturation of surface scintillators in the shower core region begins to be significant.
The independent test of energy estimates can be done by the detected zenith angle distributions which have to be isotropic for different energy thresholds. In Fig. 8 the corresponding detected distributions (symbols) are compared with statistically equivalent simulated isotropic distributions (lines). The agreement of detected and simulated distributions at GeV gives an additional support to the consistency of energy estimates in the whole measurement range. The anisotropic spectral behavior at low energies ( GeV) is explained by the lack of heavy nuclei at larger zenith angles in the detected flux due to the applied shower selection criteria.

Figure 8: Detected zenith angular distributions for different energy thresholds (symbols). The lines are corresponding simulated isotropic distributions with the same statistics.

Using the aforementioned unbiased () event-by-event method of primary energy evaluation (10), we obtained the all-particle energy spectrum. Results are presented in Fig. 9 (filled circle symbols, GAMMA07) in comparison with the same spectra obtained by the EAS inverse approach (line with shaded area, GAMMA06) from [6, 9] and our preliminary results (point-circle symbols, GAMMA05) obtained using the 7-parametric event-by-event method with the shower core selection criteria m and [10].

Figure 9: All-particle energy spectrum in comparison with the results of EAS inverse approach [6, 9] and our preliminary data [10]. The AKENO, Tibet-III, Fly’s Eye Stereo, Hires/MIA and Hires-2 data were taken from [2, 27, 28, 29, 30] respectively.

It follows from our preliminary data [10, 11], that the all-particle energy spectrum derived by event-by-event analysis with the multi-parametric energy estimator (Section 3) depends only slightly on the interaction model ( QGSJET01 [25] or SIBYLL2.1 [14] ) and thereby, the errors of obtained spectra are mainly determined by the sum of statistical and systematic errors (7) presented in Fig. 9 by the dark shaded area.
Shower size detection threshold effects distort the all-particle spectrum in the range of GeV depending on the interaction model and determine the lower limit GeV of the energy spectrum in Fig. 9 whereas the upper limit of the spectrum GeV is determined by the saturation of our shower detectors which begins to be significant at GeV and GeV (see Fig. 4) for primary proton and nuclei. The range of minimal systematic errors and biases is GeV, where about and errors were attained (Fig. 3,4) for primary and nuclei respectively.
In Table 3 the numerical values of the obtained all-particle energy spectrum are presented along with statistical, total upper and lower errors according to (7) and corresponding number of detected events. The energy spectra for low energy region (the first four lines) were taken from our data [10] for the EAS selection criteria m and .

- -
[10] - -
[10] - -
[10] - -
Table 3: All-particle energy spectrum () in units of and corresponding statistical (), total upper, () and total lower () errors and number of events (). The first four lines represent our data [10] obtained for selection criteria m and .

The obtained energy spectrum agrees within errors with the KASCADE [6], AKENO [2] and Tibet-III [27] data both in the slope and in the absolute intensity practically in the whole measurement range. Looking at the experimental points we can unambiguously point out at the existence of an irregularity in the spectrum at the energy of GeV. As it is seen from Figs 3 and 4, the energy estimator (10) has minimal biases () and errors () at this energy. With these errors the obtained bump has an apparently real nature. If we fit all our other points in the GeV energy range by a smooth power-law spectrum, the bin at GeV exceeds this smooth spectrum by standard deviations. The exact value for this significance of the bump depends somewhat on the energy range chosen to adjust the reference straight line in Fig. 9, but it lies in the range .
We conservatively included the systematic errors in this estimate, although they are not independent in the nearby points but correlated: the possible overestimation of the energy in one point cannot be followed by an underestimation in the neighboring point if their energies are relatively close to each other. Systematic errors can change slightly the general slope of the spectrum but cannot imitate the fine structure and the existence of the bump.
The results from Fig. 7 show that in the range of ”bump” energy ( GeV) the systematic errors can not increase significantly the flux. To test this hypothesis more precisely we tested the reconstruction procedure for singular energy spectra with power indices and below and above the knee energy GeV in the GeV energy range. Results are presented in Fig. 7 (star symbols) and show that there are no significant discrepancies of reconstructed spectra observed, which stems from high accuracy of energy reconstruction.
The detected shower sample in the bump energy region did not reveal any discrepancies with the showers from adjacent energy bins within statistical errors neither with respect to reconstructed shower core coordinates, zenith angular and distributions, nor with respect to distribution. The only difference is that the average age of showers is increased from at GeV up to at GeV, instead of the monotonic shower age decrease with energy increment, which is observed at a low energy region ( GeV).
It is necessary to note that some indications of the observed bump are also seen in KASCADE-Grande [6] (Fig. 9), TUNKA [4] and Tibet-III [27] data but with larger statistical uncertainties at the level of 1.5-2 standard deviations. Moreover, the locations of the bump in different experiments agree well with each other and with an expected knee energy for -like primary nuclei according to the rigidity-dependent knee hypothesis [8, 9]. However, the observed width (% in energy) and height of the bump at the energy of GeV, which exceeds by a factor of ( 4 standard deviations) the best fit straight line fitting all points above GeV in Fig. 9, are difficult to describe in the framework of the conventional model of cosmic ray origin [21].
As it will be shown below (Section 5, Fig. 10,11) the detected EAS charged particle () and muon size () spectra [8, 9] independently indicate the existence of this bump just for the obtained energies and as it follows from the behavior of shower age parameter versus shower size [8, 9], the bump at energy GeV is likely formed completely from nuclei.

5 Possible origin of irregularities

Irregularities of all-particle energy spectrum in the knee region are observed practically in all measurements [2, 6, 8] and are explained by both the rigidity-dependent knee hypothesis and contribution of pulsars in the Galactic cosmic ray flux [20, 31, 32]. Two these approaches approximately describe the all-particle spectrum in GeV energy region. However, the observed bump in Fig. 9 at energies GeV both directly points out the presence of additional component in the primary nuclei flux and displays a very flat () energy spectrum before a cut-off energy GeV.
It is known [8, 9] that rigidity-dependent primary energy spectra can not describe quantitatively the phenomenon of ageing of EAS at energies GeV that was observed in the most mountain-altitude experiments [8, 15, 33]. It is reasonable to assume that an additional flux of heavy nuclei (-like) is responsible for the bump at these energies. Besides, the sharpness of the bump (Fig. 9) points out at the local origin of this flux from compact objects (pulsars) [31, 32].
The test of this hypothesis we carried out using parameterized inverse approach [7, 8, 9] on the basis of GAMMA facility EAS database and hypothesis of two-component origin of cosmic ray flux:

Figure 10: EAS size spectra detected by the GAMMA facility (empty symbols) and corresponding expected spectra (filled symbols) computed in the framework of the SIBYLL2.1 interaction model and 2-component parametrization of primary spectra (11). The lines correspond to expected size spectra computed for each of primary nuclei.
Figure 11: The same as Fig. 10 for truncated EAS muon size spectra.

where and

The first term in the right hand side of expression (11) (so called Galactic component) is the power law energy spectra with rigidity-dependent knees at energies and power indices and for and respectively, and the second term (so called ”pulsar component”) is an additional power law energy spectrum with cut-off energies and power indices and for and respectively.
The scale factors and along with particle rigidity , cut-off energy and power indices were estimated using combined approximation method [7, 8, 9] for two examined shower spectra showed in Figs. 10,11 (empty symbols): EAS size spectra, (Fig. 10) and EAS muon truncated size spectra, (Fig. 11) detected by the GAMMA facility using shower core selection criteria and m [8, 9]. We did not consider the , and pulsar components to avoid a large number of unknown parameters and corresponding mutually compensative pseudo solutions [34] for the Galactic components.
The folded (expected) shower spectra (filled symbols in Figs 10,11) were computed on the basis of parametrization (11) and CORSIKA EAS simulated data set [9, 8] for the and primary nuclei to evaluate the kernel functions of corresponding integral equations [8, 9]. The computation method, was completely the same as was performed in the combined approximation analysis [8, 9]. The initial values of spectral parameters for Galactic component were taken from [9, 8] as well. In Figs. 10,11 we also presented the derived expected elemental shower spectra (lines) for primary and nuclei respectively.
The parameters of two-component primary energy spectra (11) derived from the goodness-of-fit test of shower spectra and are presented in Table 4.

Param. G-component P-component
Table 4: Parameters of the primary energy spectra (11) derived from combined approximations of detected shower spectra. The scale factors have units of (m s sr TeV). The particle rigidity and cut-off energies are shown in PV and PeV units respectively.

Resulting expected energy spectra for the Galactic and nuclei (thin lines) along with the all-particle spectrum (bold line with shaded area) are presented in Fig. 12. The thick dash-dotted line corresponds to derived energy spectra of the additional component (the second term in the right hand side of expression (11)). The all-particle energy spectrum obtained on the basis of the GAMMA EAS data and event-by-event multi-parametric energy evaluation method (Section 4, Fig.9) are shown in Fig.12 (symbols) as well.

Figure 12: All-particle primary energy spectrum (symbols) and expected energy spectra (lines and shaded area) derived from EAS inverse problem solution for and primary nuclei using 2-component parametrization (11). The thin lines are the energy spectra of Galactic and components. The thick dash-dotted line is an additional component from compact objects.

It is seen, that the shape of 2-component all-particle spectrum (bold line with shaded area) calculated with parameters taken from the fit of EAS size spectra agrees within the errors with the results of event-by-event analysis (symbols) that points out at the consistency of applied spectral parametrization (11) with GAMMA data.
Notice that the flux of derived additional component turned out to be about of the total flux for primary energies GeV. This result agrees with expected flux of polar cap component [20].

Figure 13: Average logarithm primary nuclei mass number derived from rigidity-dependent primary energy spectra [8, 9] (dashed line) and 2-component model prediction (11) taking into account additional pulsar component (solid line).

The dependence of average nuclear mass number are presented in Fig. 13 for two primary nuclei flux composition models: one-component model, where the power law energy spectra of primary nuclei have rigidity-dependent knees at particle rigidity GeV/Z [8, 9] (so called Galactic component, dashed line) and two-component model (solid line), where additional pulsar (P) component was included according to parametrization (11) and data from Table 4 with very flat power index () before cut-off energy . The shaded area in Fig. 13 show the ranges of total (systematic and statistical) errors.

6 Conclusion

The multi-parametric event-by-event method (Sections 3,4) provides the high accuracy for the energy evaluation of primary cosmic ray nuclei regardless of the nuclei mass (biases) in the PeV energy region. Using this method the all-particle energy spectrum in the knee region and above has been obtained (Fig. 9, Table 3) using the EAS database from the GAMMA facility. The results are obtained for the SIBYLL2.1 interaction model.
The all-particle energy spectrum in the range of statistical and systematic errors agrees with the same spectra obtained using the EAS inverse approach [7, 6, 8] in the PeV energy range.
The high accuracy of energy evaluations and small statistical errors point out at the existence of an irregularity (‘bump’) in the PeV primary energy region.
The bump can be described by 2-component model (parametrization (11)), of primary cosmic ray origin, where additional (pulsar) component are included with very flat power law energy spectrum () before cut-off energy (Fig. 13, Table 4). At the same time, the EAS inverse problem solutions for energy spectra of pulsar component forces the solutions for the slopes of Galactic component above the knee to be steeper (Table 4), that creates a problem of underestimation of all-particle energy spectrum in the range of HiRes [29] and Fly’s Eye [28] data at PeV. From this viewpoint the underestimation () of the all-particle energy spectrum (bold solid line in Fig. 12) in the range of PeV can be compensated by the expected extragalactic component [35].
Though we cannot reject the stochastic nature of the bump completely, our examination of the systematic uncertainties of the applied method lets us believe that they cannot be responsible for the observed feature. The indications from other experiments mentioned in this paper provide the argument for the further study of this interesting energy region.


We are grateful to all our colleagues at the Moscow P.N.Lebedev Physical Institute and the Yerevan Physics Institute who took part in the development and exploitation of the GAMMA array. We thank Peter Biermann for fruitful discussions.
This work has been partly supported by research Grant No. 090 from the Armenian government, by the RFBR grant 07-02-00491 in Russia, and by the ”Hayastan” All-Armenian Fund and the ECO-NET project 12540UF.


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