About the Energy Interval beyond the Ankle Where the Cosmic Radiation Consists Only of Ultraheavy Nuclei from Zinc to the Actinides

According to recent measurements the tendency of the chemical composition above the ankle is characterized by increasing fractions of intermediate and heavy nuclei and a dominance of light nuclei around the ankle. Calculation of the chemical composition in the range 3.5 × 10 5 × 10 eV according to new principles explains both the rising tendency of the heavy component. The calculation is prolonged to the adjacent interval 5 × 102.4 × 10 eV using the same theoretical background and some features of the observed cosmic-ray spectrum. It results that above the energy of 6.7 × 10 eV, where the flux is estimated to be 1.8 × 10 particles/m s sr GeV, the cosmic radiation consists only of nuclei heavier than Zinc. Measurements of the spectrum of present and past experiments are compared with the calculations.


Introduction
In the years 2005-2007 the HiRes Collaboration reported unequivocal evidence for a break of the cosmicray spectrum close and above to 5×10 19 eV [1]. This depression was confirmed by the Auger Collaboration at a significant lower energy [2] in the range (2−3)×10 19 eV. Presently this fundamental feature of the spectrum can be further investigated by the Telescope Array (hereafter T A) and Auger Collaborations which operate the two largest instruments gathering data above 10 19 eV. The T A experiment located in Utah, North America, has a collecting area of about 700 Km 2 and is deployed in the historical site of the Fly 's Eye experiment which first recorded the florescence light of air showers in order to measure the energy of primary cosmic nuclei. The technique improves the energy resolution achievable otherwise. The Auger experiment located in Argentina, South America, has a collecting area of about 3000 Km 2 . The difference in the collecting areas of the two instruments is reflected in the error bars of the measurements. In fact both instruments use the same hybrid technique which jointly exploits the florescence light yield and the muon density at ground of giant air cascades. This study necessitates the outcomes of both TA and Auger experiments in spite of the different precisions in the energy spectrum and chemical composition of two observatories.
The calculation of the cosmic-ray spectrum in the range 10 19 -2.4×10 21 eV presented in this paper is based on two empirical and one theoretical inputs designated by A, B and C. The limited size of this paper impedes a critical examination of the empirical inputs A and B. Consequently they are concisely summarized by two statements: (A) the chemical composition of the cosmic radiation evolves from light to heavy in the range 5 × 10 18 -10 20 eV [3,4,5,6] ; (B) there is a suppression in the energy spectrum above 2.6 × 10 19 eV [1, 2] with respect to a power-law extrapolation from the vast, adjacent, lower energy band, for instance the energy decade 2.6×10 18 -2.6×10 19 eV [7]. Let us anticipate that the energy scales of the instruments are of critical importance for the validation of the energy spectrum computed in this paper above 2.6 × 10 19 eV because imperfections in the calibration of the energy scales affect also the observed fluxes. The Auger apparatus in the hybrid mode of operation at 10 19 eV has an energy resolution of about 15 per cent and 10 per cent at 10 20 eV. The TA instrument has a comparable energy resolution. As a matter of fact, if the energies reported in the published measurements are regarded as real and not preliminary, there was (2007) and there is (2015) an evident mismatch in the energy scales of the HiRes, Auger and TA instruments. Our sentiment is that the interexperiment calibration is not resolvable by a rigid shift in the range  10 19 -10 20 eV but involves non linear adjustments of the energy scales.
2. The power law of the energy spectrum of the cosmic radiation with a single index A brief description of the third input C follows. Two decades of research via numerical simulation of the properties of Galactic cosmic rays have led to the explanation of the knee and ankle of the energy spectrum of the cosmic radiation [8]. The knee and the ankle are effects caused by the particle transport in the Galaxy (propagation effects) and not by the acceleration mechanism. This explanation is achievable only by introducing a notable assumption called Principle o f Constant Indices [9] : the physical process accelerating cosmic rays in the Galaxy releases particles with an energy spectrum featured by a power law and an index of 2.67 ± 0.05 in the energy range 10 9 -5 × 10 19 eV. Cosmic-ray propagation in the Galaxy has a negligible effect (see Chapter 17 of ref. 10) on the index inherent the acceleration process. Why this assumption is called principle is justified elsewhere [9]. The cosmic-ray spectrum shown in figure 1 in the limited range 10 9 -2.6×10 19 eV is comprised between two rails, the blue lines, which differ by a factor 38 marked in fig. 1. The theoretical framework which explains the knee and ankle features also provides the characteristic gap expressed by the factor 38 [8].
Measurements of the energy spectra of 12 individual cosmic nuclei in the preknee region 10 11 -10 15 eV indicate that the spectral indices are compatible with a common value of 2.67 ± 0.05 [11]. Above the ankle energy the all-particle spectrum measured by Haverah Park, Yakutsk, Fly ' s Eye, HiRes, AGASA, Auger and TA experiments is also compatible with the common index of 2.67 ± 0.05 observed in the preknee energy region (see for example fig Presently (2015) the physical process in the Galaxy accelerating cosmic rays in the range 10 9 -2.6 × 10 19 eV is unknown, nevertheless it has some identified features [9,10] : (1) it is distributed in space; (2) the accelerated particles obey a power law compatible with a single index of 2.67 ± 0.05 [11]; (3) it operates in the range 10 9 -2.0 × 10 20 eV. For conciseness in this paper the ensemble of these features is designated by Galactic Accelerator. Thus the Principle o f Constant Indices [9] has been rephrased here by the properties of the Galactic Accelerator.
Other parameters of the Galactic Accelerator, conceivable a priori, could be the maximum energy of operation E max and the efficiency of the acceleration cycle versus energy denoted here F. The efficiency may be a function of some variables such as the energy, the time interval elapsed between the birth and the extinction of the cosmic ray, the nuclei abundances at the injection stage of the acceleration cycle and others. The acceleration cycle denotes a sequence of subprocesses that convert nuclei of the quiescent matter at the injection stage, up to the highest observed energies of 3 × 10 20 eV [12] and eventually beyond this empirical limit. The Galactic Accelerator is expected to attain E max above 2.6 × 10 19 eV since below this energy the spectrum conforms perfectly to a power law with a parameter of 2.67 (see fig. 36 ref. 10). The tentative energy for E max resulting from this paper is likely to be 2.4 × 10 21 eV regarded as a lower limit to the Galactic Accelerator.

The failure of particle injection to the Galactic Accelerator
Data useful for the following inference are the flux and the chemical composition in the range 10 19 -3 × 10 20 eV. A scheme of the logical paths of the inference is summarized in fig.2. Simplicity is an ingredient of the reasoning and intervenes in more than one link of the logical chain. The intrinsic acceleration process is also assumed not to alter the chemical composition at the injection, which is a common assumption recurrent in the literature.
A priori any incipient deviation from a power-law spectrum with a single parameter could be a depression or an enhancement. The data clearly manifest a depression above the energy 2.6×10 19 eV marked by a vertical green segment in fig. 1. This particular energy is designated by E LI (H) and shortly by E LI where H is for Hydrogen and LI for Lack of particle In jection to the Galactic Accelerator. These wordings are justified below.
According to the Auger Collaboration the flux depression is halved with respect to a power-law extrapolation with a single parameter at the energy of 4.01 ± 0.21 ×10 19 eV [13] which is quite consistent with E LI = 2.6 × 10 19 eV being this value the lowest energy where the deviation manifests itself (see fig. 36 ref. 10). Since the hypothetical extragalactic component would yield an enhancement relative to the extrapolation, the observed spectrum rules out the extragalactic component Figure 3: Illustration of the mechanism giving rise to a staircase pattern in the energy spectrum. The mechanism is designated in this paper by Lack o f particle In jection to the Galactic Accelerator and the lowest energy where it manifests itself is E LI = 2.6 × 10 19 eV. As far as the energy increases above E LI quiescent atoms at the injection are suppressed initiating with light nuclei (H, He, etc.) and terminating with the heaviest one, Uranium at the energy E LI = 92×E LI (H)= 2.4 × 10 21 eV.
(level 1 of fig. 2). It follows that the Galactic Accelerator is still operating at this energy but a subprocess of the entire acceleration cycle initiates to fail, or equivalently, that the Galactic Accelerator is loosing its global efficiency exhibited and demonstrated below 2.6 × 10 19 eV via the constant index of 2.67 ± 0.05 . Since the break in the spectrum at the energy E LI does exist [1,2], either the intrinsic acceleration process is becoming inefficient or nuclear abundances are changing with the energy. The constraint of the simplicity dictates that the chemical composition and the acceleration efficiency do not vary simultaneously in the same energy range.
Notice the following important circumstance about the chemical composition of the cosmic radiation around and above to 10 19 eV. The fractions of cosmic nuclei above 10 19 eV cannot change by nuclear interactions as they do at lower energies. In fact the matter density in the interstellar medium is about 1 g/cm 3 . A nucleus crossing the Milky Way Galaxy at very high energy accumulates an average grammage between 6 to 8 millig/cm 2 . Since interaction cross sections are known, it follows that the nuclear abundances cannot change by significant amounts. Due to this circumstance the fractions of nuclei at 10 19 eV cannot be very different from those measured in the preknee energy region. Detailed calculations of the chemical composition in the range 10 12 -10 19 eV confirm this crude estimate (see fig. 7 of T he new horizon disclosed by the measurements o f the chemical composition o f the cosmic radiation above   fig. 2). Moreover they change in a very selective manner: the fraction of heavy ions augment with energy, or equivalently, the fraction of light ions decrease with energy (level 4 in fig. 2). The trend of the chemical composition with energy is certain but the absolute fractions of individual nuclei composing the cosmic radiation are still not known. The simple conclusion to be drawn is that : (1) the acceleration mechanism performs with the same efficiency above E LI = 2.6 × 10 19 eV and (2) the relative abundances of cosmic nuclei at the sources, before acceleration, change with the energy.
Major alternatives excluded by data as structured by the scheme of fig. 2 in the energy interval E LI -E max are : (first alternative) (1) the efficiency of the acceleration process changes with the energy and (2) the relative abundances of the cosmic nuclei are energy independent ; (second alternative) (1) the efficiency of the acceleration process does not change with the energy and (2) the relative abundances of cosmic nuclei are energy independent. According to the conclusion given in fig. 2 the cosmic-ray flux is expected to abruptly fall as the energy increases and then to stabilize in plateaux (staircase pattern) as shown in figure 3. The gap between adjacent plateaux are directly proportional to the abundances of the nuclear species in the interval, 1 ≤ Z ≤ 92 . Thus the observed flux above E LI can deviate from the lower rail in fig. 1 by discrete amounts; amounts related to the fractions of nuclei composing the cosmic radiation. After the flux fall of a given nucleus, the spectrum regains the same slope of 2.67. Virtually each nucleus gives an intensity step. As the energy increases the lighter nuclear species disappear and the total flux grows thinner and thinner (see fig. 3).

The predicted energy spectrum in the range 10 19 -2.4 × 10 21
The determination of the energy spectrum above E LI ≡ E LI (H) = 2.6 × 10 19 eV requires two additional parameters: (I) the abundances of quiescent nuclei around  Figure 6: The predicted spectrum (green squares) is compared with the recent measurements of Auger (blue dots) [18] and TA experiments (red squares) [19]. Notice that the last two blue dots are upper limits. The horizontal blue line is the extrapolated spectrum based on the Auger data [7] in the energy interval 10 18 -10 19 eV.
.10, F-Ca, 0.012, Sc-Ni .089 and Zc-Ge 7.4×10 −4 based on a variety of measurements [14,15,16]. They are expressed in total energy per particle and not in energy per nucleon. This unit is appropriate for very high energies. Nuclei are bunched in small groups with adjacent atomic numbers to simplify the calculations.
The relative amounts of nuclei at the injection do not suffice to calculate the energy spectrum because the particular energy of a nucleus Z for which the injection process is hampered, is not assigned. The suppression mechanism delineated in fig. 3 only implies the disappearance of lighter nuclei before the heavier ones, as the energy increases. The simplest rule reflecting the fact that the chemical composition is becoming heavier above 10 19 eV is to admit that nuclei of atomic number Z are depleted above the threshold energies E LI (Z) , that is, E LI (Z) = Z E LI (H) . The implication is that above the energy E LI (Z) the element Z is not available as cosmicray source matter for any reason whatsoever. According to this linear relationship between E and Z, Helium is expected to be depleted above the energy, Z E LI (H) = 5.2 × 10 19 eV , Nitrogen above 1.75 × 10 20 eV, Silicon above 3.5 × 10 20 eV, Fe nuclei above 6.7 × 10 20 eV and so on. If the rule were a linear relationship between E  The predicted spectrum in the range 10 19 -2.3 × 10 20 eV is given in fig. 4 (green squares) along with the positions of the threshold energies E LI (Z) of the abundant elements (H, He, C, N and O). The observed spectrum E LI is multiplied by E 2.67 and the blue line represents the flux extrapolation. The blue line is normalized to the Auger data at the intensity of 798 part/m 2 s sr GeV 1.67 [7] which corresponds to 1.159 × 10 −25 part/m 2 s sr GeV in the flux unit reported in figure 1. Note that the normalization to the Auger data, and not to those of the T A experiment, is arbitrary.
Of course the flux in fig. 4 continues to decrease with the power law, which is the characteristic of the Galactic Accelerator, but there is an additional decrement caused by particle injection failure generating a first break above 2.6 × 10 19 eV (proton depletion). In fig. 5 the computed spectrum is extended up to 2.4×10 21 eV where Uranium injection failure is predicted to occur. Above the Fe threshold, E LI (26) = 26 E LI (H) = 6.7 × 10 20 eV Iron disappears, giving rise to an almost vertical flux fall, due to the paucity of trans-iron nuclei. The trans-iron break is not out of reach of the planned JIM-EUSO experiment [17].   [21] is compared with the calculations (green squares). The blue line is normalized to the Auger data [7] while the black horizontal segment is an average value of the AGASA flux in the interval (1 − 5) × 10 19 eV.

Comparison between computed and observed spectra
The spectra measured by TA and Auger Collaborations are compared with the predicted spectrum (green squares) in fig. 6. In order to better focus on the minute aspects of the spectrum a linear scale of energy is used. Below 8 × 10 19 eV the computed spectrum lies between the TA and Auger data and above this energy is compatible with the TA data while those by Auger are deficient. Notice that the average gap between TA and Auger fluxes above 8 × 10 19 eV attains almost an order of magnitude signalling problems in the measurement procedures. Figures 7 and 8 show the comparison with the spectra measured by previous experiments having the largest exposures. An example of the spectrum measured by the Fly ' s Eye Collaboration, reported in the year 2000 [20], is shown in fig. 7 (black dots). In the half energy decade, between 5 × 10 19 eV and 10 20 eV there is a hint for the proton depletion. Above 10 20 eV some events are observed and the related flux is slightly above the prediction (green squares). The spectrum measured by the AGASA Collaboration [21] is shown in figure 8. With some imagination a small valley is visible in the interval 5 × 10 19 -10 20 eV, compared to the extrapolated AGASA spectrum rooted in the range (1−5)×10 19 eV and represented by the black horizontal segment. Data points above 10 20 eV in fig. 8 Figure 9: Energy spectrum of the cosmic radiation in a linear scale of energy up to 2.3 × 10 20 eV measured by the Auger Collaboration [13,18] . The horizontal blue line represents the extrapolation of the spectrum with a universal index γ u = 2.67. The normalization is slightly different from that of the previous figures. The green horizontal line is a visual guide which marks an arbitrary suppression factor of 12.5 relative to the upper blue line. The data suggest the staircase pattern of the spectrum consistent with the calculation reported in this paper.
The four quoted experiments together do not disprove the calculations described in this paper. If the comparison is limited to the data collected by the hybrid techniques of the TA and Auger instruments, the predicted flux is too high versus Auger data but not versus TA data. Should the comparison include the fluxes reported by AGASA and Fly ' s Eye experiments, the data are evenly scattered around the predicted spectrum and no firm conclusion emerges, neither for rejection nor for validation. The rejection of the predicted spectrum only on the basis of the Auger flux above 8 × 10 19 eV ( fig. 6) is premature until the large gap between TA and Auger spectra will not be clarified.
A second fact in favour of the computed spectrum reported in fig. 4 is that the Auger spectrum above 5×10 19 eV deviates from the extrapolation (blue line fig. 4) not by a power law with a single parameter but by steps, as the Auger data in fig. 9 demonstrate.
An alternative explanation of the observed cosmicray spectrum above 10 19 eV recurrent in the literature is that an hypothetical extragalactic component of the cosmic radiation would suffer a depletion by the impact with the ubiquitous photons of 6.7×10 −4 eV and density of 411 particles/cm 3 . The depletion would commence above 6.0 × 10 19 eV according to the original calculations made sixty years ago and is usually described by a power law with a single parameter, much softer than 2.67. This hypothetical phenomenon is known as GZK effect. Let us mention that HiRes, Auger and TA Collaborations have explicitly claimed evidence for the GZK effect [1,2] . The heavy chemical composition of the cosmic rays above 10 19 eV makes this alternative explanation quite unlikely as explained elsewhere (T he absence o f the GZK depression in the energy spectrum o f the cosmic radiation, ICRC 2013 by A. Codino ).