1 INTRODUCTION

Progress in understanding the nature of high and ultra-high-energy cosmic-ray sources located in both our Galaxy and the Metagalaxy follows three directions:

(1) the study of secondary gamma-ray photons produced by cosmic rays (CRs) in the immediate neighborhood of the source where CRs are accelerated;

(2) the detection of high-energy neutrinos produced during the interactions of high-energy particles in sources;

(3) an accurate determination of the CR energy spectrum, mass composition, and anisotropy through the detection of extensive air showers (EASs).

The TAIGA astrophysical facility [1, 2] allows the nature of CR sources to be investigated within both the first and the third approaches. A unique feature of the facility consists in combining arrays with detectors of different types into a unified system to detect all EAS components. This will allow one to search for PeVatrons, i.e., galactic objects in which protons are accelerated to energies ~1015–1017 eV, to find the energy limits for particle acceleration in supernova remnants and pulsar nebulae, and to carry out a search for correlations with neutrino events recorded by the IceCube [3] and Baikal-GVD [4] neutrino observatories. In this paper we will discuss only the results obtained within the third of the listed approaches when measuring the energy spectrum and mass composition through the detection of EASs.

Studying the energy spectrum and mass composition of primary CRs in the energy range 1014–1018 eV is of crucial importance for understanding the origin and propagation of CRs in the Galaxy. The increasing predominance of heavy nuclei from the “knee” energy to 1017 eV points to the energy limit for CR acceleration in the standard model of particle acceleration in supernova remnants (for a review, see [5]). At higher energies the mass composition again becomes lighter at 2 × 1018 eV. This may point to the transition to an extragalactic origin of CRs or the dominance of new galactic sources of an unknown nature (for a discussion, see [6, 7]).

The paper has the following structure. The main arrays of the astrophysical facility, including the arrays that are already nonoperational at present, are briefly described in Section 2. The EAS parameter reconstruction methods are presented in Section 3. Our main results are presented in Section 4.

2 DEVELOPMENT OF THE ASTROPHYSICAL FACILITY IN THE TUNKA VALLEY

2.1 The First Experiments (1993–2005)

The experiments to detect EASs by their Cherenkov radiation in the Tunka valley 50 km to the west from Lake Baikal (51.49 N, 103.04 E) were begun in 1993. The first experimental array Tunka-4 consisted of four optical detectors [8]. Hybrid QUASAR-370 photomultiplier tubes (PMTs) with a photocathode diameter of 37 cm designed for the Baikal neutrino telescope NT200 [9] were used in the optical detectors of the array.

The next array Tunka-13 consisted of 13 detectors based on PMTs of the same type and operated from 1996 to 1999. In 2000 the number of detectors was increased to 25. This array called Tunka-25 [10] operated until 2005. The detectors of the array were located at the grid points of a square grid with a 85-m spacing on an area ~0.1 km2 (Fig. 1).

Fig. 1.
figure 1

The arrangement of detectors in the Tunka-25 array. The black squares indicate the detectors involved in triggering the array.

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Below we briefly describe the arrays that have operated and are operating in the astrophysical facility since 2005. At present, the following arrays operate in the facility: Tunka-133, Tunka-Grande, TAIGA-HiSCORE, TAIGA-MUON, and TAIGA-IACT. All these arrays of the facility are synchronized with one another with a 10-ns precision [11].

2.2 Tunka-133

The Tunka-133 array consists of 175 Cherenkov detectors [12, 13] arranged over an area of 3 km2 (Fig. 2). The detectors are grouped into 25 clusters of seven detectors each — six detectors at the vertices of a regular hexagon and one at the center. The detector spacing in one cluster is 85 m. Nineteen clusters are arranged in the form of a dense central part of the array with a radius ~500 m. These 133 detectors gave the name Tunka-133 to the array. The remaining 6 outer clusters are located at a distance ~1 km around the center. Each cluster is an independent array with its own local trigger. Each detector contains a PMT with a photocathode diameter of 20 cm the signal from which is sent to the cluster center, where it is digitized by a 12-bit ADC with a sampling frequency of 200 MHz. The main task of the Tunka-133 array is to investigate the energy spectrum and mass composition of CRs with energies above 1016 eV. The EAS parameter reconstruction methods and the main results will be presented in Sections 3 and 4.

Fig. 2.
figure 2

The arrangement of detectors in the central part of the Tunka-133 array.

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2.3 Tunka-Grande

The Tunka-Grande array [14] is a network of scintillation detectors combined into 19 stations, each of which consists of the ground and underground parts. The ground part contains 12 counters with a total area ~8 m2 that detects charged EAS particles. The underground part located under a 1.5-m-thick layer of soil consists of 8 counters with a total area ~5 m2 and is designed to detect the EAS muon component. The scintillation station electronics, which is analogous to the Tunka-133 electronics, provides local station trigger generation, signal shape digitization with a 5-ns step, and data transmission to the unified data acquisition center.

2.4 TAIGA–HiSCORE

The TAIGA-HiSCORE (High Sensitivity COsmic Rays and gamma Explorer) array [15, 16] is a network of wide-field optical stations to detect the Cherenkov radiation from EASs. At present, the array consists of 120 stations arranged on an area of 1.1 km2, with the station spacing being 106 m (Fig. 3). The stations are grouped into 4 clusters with independent data acquisition centers. Each station is connected with the cluster data acquisition center by a fiber-optic cable for data transmission and synchronization [11]. One optical station contains four PMTs with a diameter of 20 cm (ET 9352 and Hamamatsu R5912). The light collection area of each PMT was increased by a factor of 4 using a Winston cone with a diameter of 0.4 m and a viewing angle of 30° (a solid angle of 0.6 sr). The signals from the PMT anode and intermediate dynode are digitized with a 0.5-ns step. The effective energy threshold of the array when four or more stations are triggered is ~80 TeV for EASs from charged CR particles and ~40 TeV for EASs from gamma-ray photons. The angular resolution of the array changes from 0.4–0.5 deg. near the array threshold to 0.1 deg. when more than 10 stations are triggered [17]. The TAIGA-HiSCORE array was created not only to investigate CRs, but also to search for local high-energy gamma-ray sources [18]. The large angular aperture of the array and its good angular resolution have made it possible to begin a search for astrophysical nanosecond optical transients [19].

Fig. 3.
figure 3

Left: the arrangement of optical stations of the TAIGA-HiSCORE array (squares) and three ACTs of the TAIGA-IACT array (circles). Right: the TAIGA-HiSCORE optical station and the first two ACTs.

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2.5 Tunka-REX

The detection of radio emission is yet another method that can be used to reconstruct the EAS parameters. The radio emission arises mainly from the current produced by the separation of relativistic electrons and positrons of a shower in the geomagnetic field. The Tunka-REX (Radio Extension) array [20] is a system of antennas that operated for 7 years (2012–2019) in the astrophysical facility [21] and was used in experiments to detect the radio emission from EASs. The array consisted of 63 radio antennas arranged on an area of 3 km2. The antennas were connected to FADC boards in the data acquisition electronics of the Tunka-133 and Tunka-Grande arrays. The signals from the antennas were read out when Tunka-133 and Tunka-Grande local triggers were generated.

2.6 TAIGA–Muon

A new array, TAIGA-Muon [22], is being created to investigate the mass composition of CRs with energies above 1016 eV and to suppress the hadron background when detecting high-energy gamma-ray photons. New scintillation counters with an area of 1 m2 were designed for this array [23]. The design of the counters allows them to be buried in soil without any additional protection. The first three clusters of the TAIGA-Muon array were deployed in 2019. Each cluster has 8 ground and as many underground detectors. The counters are arranged in pairs; the ground ones are located strictly above the underground ones. All 8 pairs are arranged along the perimeter of a square with a 5-m side. The distance between the neighboring pairs is 1 m. The area of the muon detectors in the TAIGA-Muon array is planned to be increased to 150 m2 by 2024.

2.7 TAIGA–IACT

Five atmospheric Cherenkov telescopes (ACTs) with the reconstruction of the angular distribution (image) of Cherenkov light from EASs are planned to be deployed in the astrophysical facility. Such telescopes are the main instruments of ground-based high-energy gamma-ray astronomy that allow the events from gamma-ray photons to be separated from the events from charged CR particles. The first two telescopes already operate; the third telescope will begin to operate in the winter of 2022. The fourth and fifth telescopes will begin to operate by 2023. Each ACT of the TAIGA-IACT (Imaging Atmospheric Cherenkov Telescope) array has composite Davis–Cotton mirrors with an area ~10 m2 and a focal length of 4.75 m [24]. Imaging cameras consisting of 600 PMTs with a photocathode diameter of 2 cm each (XP1911) are installed at the mirror focus. The diameter of the cameras is ~110 cm. The camera viewing angle is 9.6°, the viewing angle per pixel is 0.36°, and the point spread function (PSF) of the telescope is ~0.07° [24]. The imaging camera and the data acquisition system are described in detail in [25]. Our telescopes correspond in their characteristics to the Small Size Telescopes (SSTs) of the CTA project [26]. The viewing angle of such telescopes is more than twice the viewing angle of previous-generation telescopes. Such a viewing angle allows EASs with an axis position up to 500 m from the telescope to be detected. The telescope energy threshold is 2–3 TeV, depending on the zenith angle at which the gamma-ray source is seen. At energies above 10 TeV it becomes possible to use a stereoscopic approach — EASs from gamma-ray photons are detected by two or more telescopes [27]. At energies above 40 TeV a new “hybrid” approach to the detection of gamma-ray photons becomes possible — the detection of EASs both by the telescopes and by the TAIGA-HiSCORE array. The main advantage of the joint operation of the ACTs and the network of wide-field Cherenkov stations is a more efficient selection of events from gamma-ray photons from the EAS background from charged CRs [28].

2.8 Small Cherenkov Telescopes

One of the shortcomings of the joint operation of the TAIGA-HiSCORE array and the Cherenkov telescopes is a significant difference in apertures. The ACT aperture is a factor of 20 smaller than that of the TAIGA-HiSCORE array and, accordingly, only 5% of the events recorded by the TAIGA-HiSCORE array fall into the ACT field of view. Small Cherenkov telescopes with cameras with a field-of-view diameter of 25–30 deg and an effective detection area ~1 m2 are planned to be used to investigate the range of energies above 50 TeV. During the operation of such telescopes, the percentage of joint (with the TAIGA-HiSCORE array) events will increase by almost a factor of 10 and a high efficiency of the selection of events from gamma-ray photons will be retained for joint events. At present, the prototype of such a telescope, the Small Imaging Telescope (SIT) that uses silicon PMTs as imaging detectors, operates in the facility [29].

3 EAS PARAMETER RECONSTRUCTION METHODS

The EAS parameters reflecting the energy and mass A of the primary nucleus are measured to investigate the CR energy spectrum and mass composition by the EAS detection method. Since ~80% of the energy of the primary nucleus during the development of a cascade in the atmosphere passes into the electromagnetic component (electrons and photons), the total Cherenkov light flux from an EAS is a good measure of the primary energy, being independent of the mass number. However, it is impossible to measure the Cherenkov light flux at most arrays and, therefore, the flux density at a certain distance from the axis is chosen as a measure of the total flux. The depth of EAS maximum Xmax is a parameter sensitive to the mass number. The depth of maximum for individual EASs fluctuates greatly (σ(Xmax)) for protons ~60–80 g cm–2, depending on the energy; the mean value of Xmax (〈Xmax〉) for the EAS from a nucleus with a mass number A is well described by a simple fit [30]:

$$\left\langle {{{X}_{{\max }}}} \right\rangle = A + B\log E\;[{\text{TeV}}]{\text{/}}A,$$
(1)

with constants A and B (A = 370 g cm–2 and B = 65 g cm–2).

In principle, the values of these parameters depend on the energy range under study, but, to a first approximation, this dependence may be neglected. As the energy changes by a factor of 10, 〈Xmax〉 increases approximately by 65 g cm–2. The change in 〈Xmax〉 when passing from a proton to an iron nucleus is 110–120 g cm–2 in the range 1016–1018 eV. The simulation results presented in Section 4 qualitatively coincide with this simple fit.

3.1 Reconstructing the Primary Particle Energy from Cherenkov Light

For arrays with a detector spacing ~100 m (Tunka-133 and TAIGA-HiSCORE) the primary particle energy is reconstructed from the Cherenkov light flux density at a distance of 200 m from the EAS axis (Q200). The lateral distribution function (LDF) obtained from Monte Carlo simulations is used for the recalculation to a distance of 200 m. As simulations show [10] (Fig. 4), at such a distance from the axis the light flux density for a fixed energy depends weakly on the depth of EAS maximum Xmax and EAS zenith angle.

Fig. 4.
figure 4

The lateral distribution function of light from an EAS with an energy of 5 × 1015 eV computed with the CORSIKA code for various distances from the shower maximum to the array, Hmax = (1) 3.2, (2) 4.1, and (3) 5 km.

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The relation between the primary particle energy and Q200 derived from Monte Carlo simulations (CORSIKA, QGSJET-II-04) can be expressed by the following formula [13]:

$${{E}_{0}} = CQ_{{200}}^{g},$$
(2)

where g = 0.94 for the range 1016–1017 eV and g = 0.95 for the range 1017–1018 eV. The results of our computation are presented in Fig. 5.

Fig. 5.
figure 5

Computation (CORSIKA, QGSJET-II-04) of the correlation between the primary particle energy (E0) and the light flux density at a distance of 200 m from the axis (Q200). The black dots indicate the mean values of Q200 for primary protons; the red dots indicate the mean values of Q200 for primary iron nuclei.

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The constant C is determined by normalizing the derived experimental integral energy spectrum to the known total flux of CRs with an energy above 3 × 1015 eV measured in the QUEST experiment [31].

The relative error of the reconstructed shower parameters can be estimated by the well-known chessboard method [32]. For energies E0 ≥ 1016 eV the EAS axis position reconstruction error is less than 6 m for the central part of the Tunka-133 array (R ≤ 450 m) and less than 10 m for 450 m ≤ R ≤ 800 m and E0 ≥ 5 × 1016 eV. Given the uncertainty in the mass composition and the EAS axis position determination accuracy, the energy reconstruction error is ~10%. The presence of the Tunka-Rex array recording the EAS radio emission in the TAIGA astrophysical facility and of the LOPES array in the KASCADE-GRANDE array has allowed the EAS energy reconstruction accuracies between Tunka-133 and KASCADE-GRANDE to be compared [33]. The idea of the method is to compare the amplitudes of the signals from the Tunka-Rex and LOPES radio antennas, depending on the EAS energies reconstructed using the techniques developed in the Tunka-133 and KASCADE-GRANDE experiments. The systematic difference in measuring the energy turned out to be 5%.

The main EAS parameters for the TAIGA-HiSCORE array are reconstructed using the same algorithms as those for the Tunka-133 array. At the same time, for the TAIGA-HiSCORE array it is possible to reconstruct the EAS Cherenkov light flux density at a distance of 200 m from the axis by interpolating the amplitudes measured with its optical stations located at distances greater or smaller than 200 m for 100% of the events only at EAS energies above 1015 eV, whereas the energy threshold of the TAIGA-HiSCORE array, when its four or more optical stations are triggered, is ~80 TeV for EASs from charged CR particles. Therefore, a different EAS parameter reconstruction algorithm was developed for lower energies. The axis position is defined as the centroid of the pulse amplitudes at 4–5 stations. The light flux density near the axis is calculated as a mean for two stations closest to the axis. The computation with the CORSIKA code showed that because of large light flux fluctuations, the energy measurement error near the EAS axis is considerably larger than that from the light flux density at a distance of 200 m from the axis, but, on average, the light flux can be transformed to the EAS energy using the formula [34]

$$\log ({{E}_{0}}) = C + 0.87{{Q}_{{65}}},$$
(3)

where Q65 is the photon flux density at a distance of 65 m from the axis.

3.2 Reconstructing the Primary Particle Energy from Charged EAS Particles

For the Tunka-Grande array the primary particle energy is reconstructed from the EAS particle flux density at a distance of 200 m from the axis [14]:

$$\log (E{\text{/}}1\;{\text{eV}}) = {{C}_{a}}\log ({{\rho }_{{200}}}) + {{C}_{b}}.$$
(4)

The coefficients Ca and Cb in this expression are determined by substituting the EAS energy reconstructed from the Tunka-133 data into the left-hand side of the equation for the events recorded by both Tunka-Grande and Tunka-133. The experimental dependence of E on ρ200 is shown in Fig. 6.

Fig. 6.
figure 6

Correlation between the EAS energy reconstructed from the Tunka-133 data and the charged particle density at a distance of 200 m from the axis from the Tunka-Grande data.

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3.3 Reconstructing the Depth of EAS Maximum

The parameter P that characterizes the steepness of the LDF of Cherenkov light is used to determine the depth of EAS maximum (Xmax). P is defined as the ratio of the light flux at a distance of 80 m from the EAS axis, Q80, to the flux at a distance of 200 m, Q200:

$$P = {{Q}_{{80}}}{\text{/}}{{Q}_{{200}}}.$$
(5)

Our Monte Carlo simulations confirmed that the parameter P is determined only by the thickness of the atmosphere between the array and the depth of maximum (ΔXmax = 965/cos θ – Xmax [g cm–2]) and does not depend on the energy, shower zenith angle, and type of the primary nucleus (see Fig. 7). Here, 965 g cm–2 is the depth of the atmosphere at the location of the arrays [35]. The calculated relation between ΔXmax and parameter P is nearly linear for the P range from 2.5 to 9. The standard deviation of the points from the fit for this range is ~15 g cm–2. The computed points for heavier nuclei or zenith angles greater than 30° lie largely at values of the parameter P less than 2.5, i.e., refer to the events with LDPs that are virtually flat near the EAS axis for which the described method is inapplicable.

Fig. 7.
figure 7

Relationship between the LDF steepness P = Q80/Q200 and the thickness of the atmosphere between the array and the depth of EAS maximum ΔXmax [35] from the bank of events obtained by the Monte Carlo method.

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4 MAIN RESULTS

4.1 Energy Spectrum in the Range 1014–1018 eV

The Tunka-133 and TAIGA-HiSCORE data obtained in 2175 and 327 h of operation, respectively, were used to construct the CR energy spectrum from the data of the TAIGA astrophysical facility. The joint CR energy spectrum [13, 34] is shown in Fig. 8 in comparison with the spectrum obtained previously from the Tunka-25 data [10]. The spectrum spans four orders of magnitude in the energy range from 3 × 1014 to 3 × 1018 eV. One of the main results that follows from the spectrum obtained is the proof of a more complex energy dependence of the CR intensity than was assumed previously. Apart from the “classical knee” at an energy of 3 × 1015 eV, two statistically significant features are observed in the spectrum. More specifically, at 2 × 1016 eV the slope of the energy spectrum decreases approximately by 0.2, while at 3 × 1017 eV the slope of the spectrum again increases approximately by 0.3. The energy spectrum at energies below 3 × 1015 eV can be described by a power law with an index of 2.70 ± 0.01. In the energy range 2 × 1016–3 × 1017 eV the spectrum is also described by a power law with an index of 2.99 ± 0.01. At energies above 3 × 1017 eV the slope of the spectrum increases to 3.29 ± 0.09. The steepening of the spectrum at 3 × 1017 eV can be interpreted as the “second knee” in the energy spectrum associated with the transition from galactic CRs to extragalactic ones. Figure 9 compares our data with the data of other experiments. There is good agreement both with the direct measurements at high-altitude balloons [36], satellite [37], and mountain [38] measurements at low energies and with the measurements at giant arrays at extremely high energies (the Pierre Auger Observatory (PAO) [41] and the TA array [42]). A decrease in the slope of the spectrum at 1.5–2.5 × 1016 eV by 0.2–0.3 is observed in all of the experimental data. At present, this effect has no astrophysical explanation.

Fig. 8.
figure 8

CR energy spectrum based on data from the arrays of the TAIGA astrophysical facility: 1 — Tunka-25 [10], 2 — Tunka-133 [130, and 3 — TAIGA-HiSCORE [34].

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Fig. 9.
figure 9

Comparison of the CR energy spectra obtained in various experiments in a wide energy range: ATIC-2 [36], NUCLEON [37], HAWC [38], Tunka-25, TAIGA-HiSCORE, Runka-133, KASCADE-Grande [39], IceTop [40], PAO [41], and Ta+Tale [42].

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4.2 Mass Composition in the Range 1015–1018 eV

The mean depths of maximum derived at two arrays (Tunka-133 and TAIGA-HiSCORE) as a function of the primary particle energy are shown in Fig. 10. The data of both arrays, despite the difference in their geometry, agree well between themselves in the wide energy range from 1015 to 3 × 1017 eV. Events with a zenith angle ≥30° and an energy at which the EAS detection efficiency is ~100% are chosen to obtain undistorted estimates of Xmax. This energy is 1.25 × 1016 for Tunka-133 and 1.25 × 1015 eV for TAIGA-HiSCORE. In Fig. 10 our experimental data are compared with the Xmax measurements obtained by the detection of fluorescent light from EASs at TA [42] and PAO [43] and a mixture of fluorescent and Cherenkov light in the TALE experiment [44]. All of the experimental results are compared with the theoretical curves computed with the QGSJET-II-04 model [45] for primary protons and iron nuclei. At a fixed energy Xmax depends linearly on ln A (see Eq. (1)). Using this, the mean 〈ln A〉 is usually determined by interpolation between the computed curves for protons and iron nuclei presented in Fig. 10. The result is shown in Fig. 11.

Fig. 10.
figure 10

Energy dependence of the mean Xmax: Tunka-133 and TAIGA-HiSCORE [33], TA [42], TALE [43], and Auger [44].

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Fig. 11.
figure 11

Energy dependence of the mean logarithm of the atomic number 〈lnA〉: Tunka-133 and TAIGA-HiSCORE [33], TA [42], TALE [43], and Auger [44].

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The energy dependence of the mean value of ln A is well extrapolated to the Auger results at 3 × 1017 eV [44] and contradicts the results of the much more complex (in interpreting the data) TALE experiment [43].

5 FACILITY DEVELOPMENT PLAN FOR THE NEAR FUTURE. CONCLUSIONS

The development of the astrophysical facility in the Tunka valley over 25 years has led to the construction of a number of serious arrays to study ultra-high-energy CRs. The energy spectrum pf primary CRs in a range of four orders of magnitude was reconstructed from the results obtained with these arrays. A number of features that haven not yet got an astrophysical interpretation are observed in the energy spectrum. First of all, the spectrum near the “first knee” at (3–6) × 1015 eV cannot be described simply as an abrupt change in the slope. At present, the feature in the spectrum at 2 × 1016 eV that was firmly established in a number of experiments has not been explained either. Finally, the energy at which the “second knee” is located is still determined with a great uncertainty. According to our data, light nuclei, protons and helium, must dominate in the CR flux in the range 1015–1017 eV. This contradicts the break in the energy spectrum of the light CR component obtained at the ARGO-YBJ array at 0.7 × 1015 eV.

It should be noted that, according to the data on the mass composition in the region of the first knee obtained in various experiments, it is so far impossible to determine a consistent picture. It is possible that a detailed CR mass composition in this energy range will be ultimately understood only after the launch of a massive (several tons) detector [47] in a near-Earth orbit. In the coming 2–3 years the study of the mass composition will be continued at the astrophysical facility as one of the main themes. An attempt will be made to study the mass composition at energies below 1015 eV, possibly, based on the joint TAIGA-HiSCORE and TAIGA-IACT data. In the range 1016–1018 eV we hope to advance in studying the mass composition through an increase in the area of the muon detectors. We also associate the progress at energies above 1017 eV with the possible deployment of a new fluorescent detector in the TAIGA astrophysical facility.