All-particle primary energy spectrum in the 3-200 PeV energy range
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.
keywords:Cosmic rays, energy spectra, composition, extensive air showers
Pacs:96.40.Pq, 96.40.De, 96.40.-z, 98.70.Sa
, , , , , ,
Study of the fine structure in the primary energy spectrum is one
most important tasks in the very high energy cosmic ray experiments
Commonly accepted values of the all-particle energy spectrum indexes of
and before and after the knee are an average and may not
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  as well as in later
the GAMMA  and TUNKA  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  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  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.
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
3 plastic scintillation detectors with the dimensions
Each of the central 9 stations contains an additional (the 4th)
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  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 :
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
is consisted of.
In this case, the expression (1) is unfolded numerically
and the slope of detected energy spectrum can differ from primary
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
and at .
It is also accepted that the mass spectra of primary nuclei
can be divided into separate nuclear groups and below,
as in ,
just 4 nuclear species (, , -like and -like)
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  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 
with the SIBYLL  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  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 ), 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  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 . 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.
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
for equilibrium accompanying charged particles obtained from preliminary
simulations with the FLUKA code  in the TeV muon
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.
Parametric representation for the energy estimator we obtained by minimizing
with respect to
for different empirical
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
the number of degrees of freedom
The expected errors of
corresponding parameters were negligibly small ()
due to very high values of .
The corresponding average biases versus energies
( and )
for the primary proton (), iron () nucleus and uniformly mixed
composition are presented in Fig. 3 (symbols).
lines correspond to approximations
and for the upper and lower limits
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.
The results for the uniformly mixed composition
with the shower core selection of m  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).
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
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.
Using the aforementioned unbiased () event-by-event method
of primary energy evaluation (10), we obtained the all-particle energy
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
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
It follows from our preliminary data [10, 11], that
the all-particle energy spectrum derived by event-by-event analysis
estimator (Section 3) depends only slightly on the interaction
model ( QGSJET01
 or SIBYLL2.1  ) 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  for the EAS selection criteria m and .
The obtained energy spectrum agrees within errors with the KASCADE
AKENO  and Tibet-III  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 ()
() 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  (Fig. 9), TUNKA  and Tibet-III  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 .
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
However, the observed
bump in Fig. 9 at energies GeV both directly points out the
of additional component in the primary nuclei flux and displays
a very flat
() energy spectrum before a cut-off
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:
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
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
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  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.
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
evaluation method (Section 4, Fig.9) are shown in Fig.12 (symbols) as
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 .
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.
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  and Fly’s Eye  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 .
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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