Problem of the energy spectrum of ultrahigh-energy cosmic rays

ISSN 1063-7788, Physics of Atomic Nuclei, 2009, Vol. 72, No. 1, pp. 85–96. c© Pleiades Publishing, Ltd., 2009.Original Russian Text c© A.V. Glushkov, 2009, published in Yadernaya Fizika, 2009, Vol. 72, No. 1, pp. 91–102.

ELEMENTARY PARTICLES AND FIELDSTheory

Problem of the Energy Spectrum of Ultrahigh-Energy Cosmic Rays

A. V. Glushkov*

Institute of Cosmophysical Research and Aeronomy,Yakutsk Research Center, Siberian Division, Russian Academy of Sciences,

pr. Lenina 31, Yakutsk, 677980 RussiaReceived January 15, 2008; in final form, April 7, 2008

Abstract—A number of cosmic-ray energy spectra measured in the energy region E0 ≥ 1017 eV at theYakutsk array and at AGASA, Haverah Park, HiRes, Auger, and SUGAR within different periods of timewere considered. It was shown that, upon rescaling the energy of these spectra by factors of K = 0.75, 0.85,0.9, 1.02, 1.19, and 1.29, respectively, all of them agree with one another rather well in shape. The factorsK themselves exhibit a pronounced north–south dependence on the geographical latitude of the positionsof the above arrays.

PACS numbers: 95.30.-k, 96.40.Pq, 96.40.-z, 98.70.-fDOI: 10.1134/S1063778809010116

1. INTRODUCTION

The energy spectrum of ultrahigh-energy (E0 1017 eV) cosmic rays is a key link in the chain of in-volved problems of the origin of primary particles hav-ing such energies. Giant air showers (GAS) of limit-ing energy in the region E0 1019 eV have arousedgreat interest over almost 40 years—since the discov-ery of the first events recorded at the largest arraysworldwide such as Volcano Ranch (USA) [1], Hav-erah Park (England) [2], SUGAR (Australia) [3], andthe Yakutsk array [4]. The shape and the intensity ofthe spectrum are of paramount importance for reveal-ing the composition of primary cosmic rays (PCR)and sources of their formation. The experimental dataobtained at the different arrays are markedly differ-ent in absolute intensity but are similar in shape.This can clearly be seen in Fig. 1a. The open cir-cles there correspond to the differential spectrum ob-tained at SUGAR (Sydney University Giant Air-shower Recorder) and presented by the authors of [10]in an ultimate form when this facility had alreadybeen closed. The crosses and closed circles represent,respectively, the spectrum measured at the HaverahPark [7] and the spectrum obtained at the Yakutsk ar-ray [5]. The inclined straight line was drawn throughthe characteristic dips in the spectra. It correspondsto a change in the intensity by a factor of 3∆ log E0

in response to a change in the energy by a factorof ∆ log E0. In all spectra, there are events at anenergy in the region E0 1020 eV. Their presence

*E-mail: [email protected]

contradicts a steep decrease expected for the spec-trum in the region E0 6 × 1019 eV on the basis ofthe theoretical concepts developed by Greisen [11]and Zatsepin and Kuz’min [12] (GZK cutoff) be-cause of the intense interaction of giant air showerswith microwave background radiation. The problemof Greisen–Zatsepin–Kuzmin cutoff gave impetus tocreating the new arrays AGASA (Akeno Giant AirShower Array, Japan) and Fly’s Eye–HiRes (USA),which had a higher sensitivity. Hopes for ultimatelysolving the problem of the giant-air-shower originwere placed on these arrays, but these hopes provedto be illusory. The supergiant array Auger (its areais 3000 km2), commissioned recently in Argentina,brought about some sensational results, reported bythe Auger group in the summer of 2007 at the 30thCosmic Ray Conference in Mexico (for an overview,see [9, 13]), whereupon the situation became stillmore intriguing.

Here, we consider the problem of the energy spec-trum of ultrahigh-energy cosmic rays on the basis ofall statistically significant results without subjectingthem to any selection even if some of them may seemoutdated and not quite reliable. This yields a fullerpattern of the very problem of measuring the spectrumand opens some new possibilities for solving it.

2. EXPERIMENTAL RESULTS

Let us return once again to Fig. 1a, in which onecan see that all three of the aforementioned spec-tra agree fairly well with one another in shape. Itshould be recalled that, at the SUGAR array, the

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Fig. 1. Differential energy spectrum (a) as measured at (closed circles) Yakutsk array (for showers whose zenith angles fallwithin the range θ 60) [5], (open triangles) AGASA (θ 60) [6], (crosses) Haverah Park (θ 60) [7], (closed diamonds)HiRes [8], (closed stars) Auger (θ 70) [9], and (open circles) SUGAR (θ 60) [10] and (b) upon rescaling their energiesby factors of K = 0.75, 0.85, 0.9, 1.02, 1.19, and 1.29, respectively.

energy of primary particles was found from the totalnumber of vertically going muons at the thresholdof Eµ = 0.75 sec θ [GeV]. At the Haverah Park array,the energy was determined by means of rescaling thequantity ρ600, which is defined as the charged-particledensity measured by ground-based water Cherenkovtanks 1.2 m thick at a distance of 600 m from theextensive-air-shower axis. At the Yakutsk array, the

respective procedure consisted in rescaling S600 toE0, where S600 is the charged-particle density mea-sured by ground-based scintillation detectors 5 cm inthickness and 2 m2 in area at a distance of 600 m fromthe extensive-air-shower axis.

It can be assumed, as was done by Berezinskyand his coauthors in [14], that the observed differ-ence in the intensity of the spectra is caused by

PHYSICS OF ATOMIC NUCLEI Vol. 72 No. 1 2009

PROBLEM OF THE ENERGY SPECTRUM OF ULTRAHIGH-ENERGY COSMIC RAYS 87

1.0

–90 –30

K

Latitude, deg

0.530 90

1.5

Fig. 2. Normalization coefficients versus the latitude of the positions of the arrays used (the notation for the points is identicalto that in Fig. 1).

some methodological features of one experiment oranother—for example, by an inaccuracy in deter-mining E0. We will now try to render the spectrain Fig. 1a as close as possible to one another. Byconvention, we require that, at E0 = 3.15 × 1018 and1.4 × 1019 eV, the intensities of J(E0) ∼ E−γ

0 power-law approximations in the energy ranges (1–8) ×1018 and (8–50) × 1018 eV be equal to JE3

0 = 2 ×1024 m−2 s−1 sr−1 eV2 and that the exponents inthese ranges be γ = 3.2 and 2.68, respectively. In or-der to accomplish this shift, it is necessary to changethe energies at the Yakutsk array, Haverah Park, andSUGAR by factors of K = 0.75 ± 0.04, 0.9 ± 0.07,and 1.29 ± 0.1, respectively. The factors K and theerrors in them were found by means of the χ2 test.The results of this procedure are illustrated in Fig. 1b,where the errors are not shown in order to avoidencumbering the figure.

We can see rather good agreement in shape be-tween all three spectra over the entire energy rangeunder study. This situation around the energy spec-trum was prevalent just before the appearance of da-ta from the new arrays AGASA [6], HiRes [8], andAuger [9]. Those data are also displayed in Fig. 1a.It is noteworthy that scintillation detectors similarto those of the Yakutsk array and the procedure ofestimating E0 on the basis of S600 were employed atAGASA. In all probability, this was reflected in thefact that the spectra at the two arrays were rather

close both in shape and in intensity. The only excep-tion here is that there are eleven giant air showersin the energy region E0 1020 eV at AGASA, butthere is only one such event at the Yakutsk array.At the HiRes and Auger arrays, the energy of pri-mary particles was found from the fluorescence of theatmosphere because of the development of extensiveair showers. However, the energy of the overwhelm-ing majority of giant air showers was determined atAuger from ρ1000, which is the charged-particle den-sity measured by water Cherenkov tanks 1.2 m thickat a distance of 1000 m from the extensive-air-showeraxis.

One can see that new spectra rendered the generalpattern even more intricate. First, a strong differencein the intensity between the spectra measured atHiRes and Auger was quite bizarre, since the sameprocedure of estimating the energy of primary par-ticles from the fluorescence of extensive air showerswas used at those two arrays. Second, the distinc-tion between the spectra at Haverah Park and Augerproved to be even more surprising, since similar waterCherenkov tanks were used as the main detectors atthose two arrays. Watson, who headed investigationsat those arrays, has made no comments on this point.Third, the spectra at Auger and SUGAR proved tobe astoundingly similar to each other. A very lowintensity of the latter spectrum aroused the mistrusttoward it and was the reason why it unfortunatelysank into oblivion for a long time, but, as we now


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Fig. 3. Deviations of the observed number N1 of showers from the expected average value 〈N〉 in units of nσ = (N1 −〈N〉)/

√〈N〉 on the developed celestial sphere in terms of the galactic coordinates for E0 8 × 1018 eV giant air showers.

The SG curve represents the Supergalaxy plane, and N and S stand for, respectively, the North and South Poles of the Earth.

see, a decent successor to the SUGAR spectrum hasappeared. This wide variety of the spectra furnishes asufficient motivation for seeking some general reasonbehind their disagreement.

It can readily be proven that all spectra in Fig. 1aare rather close to one another in shape. For this, it isonly necessary to change the energy of the AGASA,HiRes, and Auger spectra by factors of K = 0.85 ±0.06, 1.02 ± 0.07, and 1.19 ± 0.11, respectively. Thecorresponding results are shown in Fig. 1b. This rel-ative normalization of the spectra was proposed byBerezinsky et al. [14]. Those authors assume thatthe absolute flux of ultrahigh-energy cosmic rays assuch is isotropic and that the problem of distinctionsbetween the spectra lies entirely in the errors in esti-mating the energy at one array or another. To someextent, this is a rational explanation.

However, the following intriguing circumstancesuggested by the data in Fig. 1a and illustrated inFig. 2 is worthy of note. The normalization factorsK exhibit a specific dependence on the geographicallatitude of the positions of the arrays considered here:the spectrum measured by an array positioned fartherto the south goes lower. This dependence is hardlyan accidental one caused by methodological errorsin determining E0. In all probability, the reason hereis different. It is the opinion of the present authorthat the reason is predominantly of an astrophysicalcharacter.

3. DISCUSSIONThe astrophysical nature of the north–south de-

pendence of the normalization factors on the geo-graphical latitude of the position of arrays is sug-gested by the anisotropy of giant air showers [15]. InFig. 3, the map of the distribution of the density ofevents characterized by energies in the region E0 8× 1018 eV and by zenith angles in the range θ 60is shown on the developed celestial sphere in termsof the galactic coordinates. This distribution includesdata coming from the Yakutsk array (519 showers)and from SUGAR (522 showers [16]) and appearingin the spectra in Fig. 1. The analysis here was per-formed in terms of the deviation of the observed num-ber N1 of events from the expected average number〈N〉 = N2(Ω1/Ω2) in units of the standard deviationσ =

√〈N〉; that is,

nσ = (N1 − 〈N〉)/σ. (1)

Here, N1 and N2 are the numbers of showers with-in the solid angles Ω1 = 2π(1 − cos θ1) and Ω2 =2π(1 − cos θ2) (θ1 = 20 and θ2 = 60), respectively.The values of the expression in (1) were found by suc-cessively moving the center of a 1 × 1 area over theentire sphere. The limits of nσ are shown in the lowerpart of Fig. 3 as a toned scale. The darkest regionscorrespond to the giant-air-shower flux exceeding theaverage level by not less than three standard devia-tions, nσ 3σ. The SG curve represents the plane ofthe Supergalaxy (local supercluster of galaxies).



270

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90Galaxy

plane

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Fig. 4. Density of galaxy distributions in a local volume of radius about 100 Mpc in the Supergalaxy plane, the North Pole ofthe Supergalaxy facing the observer [23]. The supergalactic longitude is reckoned from the “0” axis in the counterclockwisedirection. The symbols NG and SG stand for, respectively, the North and the South Pole of the Milky Way Galaxy, whoseplane is orthogonal to the figure and is situated along the 90–270 line. The GA arrow is directed toward the Great Attractor.The closed circle indicates the position of the Yakutsk array. The figures are the numbers of superclusters, their names beingindicated in the table.

In Fig. 3, an excess flux of giant air showers isclearly seen in the region of intersection of the Galaxyand Supergalaxy planes at a galactic longitude ofabout 137. The wavelet analysis performed in [17]revealed that this local part of the sky manifests itselfmost distinctly (with a random-outcome probabilityof 0.007) in the arrival directions of primary cosmicrays characterized by E0 ≈ (1–2) × 1019 eV. In myopinion, the fact that this region is slightly elongatedalong the Supergalaxy plane is indicative of the extra-galactic origin of primary cosmic rays that have theabove energies. It is also noteworthy that this regionis situated in the vicinity of the North Pole of the Earth(N). Upon going over to south latitudes, the averageflux of giant air showers in Fig. 3 decreases, which isroughly in accord with the north–south dependenceof the intensity of the spectra in Fig. 1.

According to [18–20], quasars, which generateneutral particles, may be sources of primary cosmicrays that have such energies. On their passage to-

ward the Earth, these particles traverse the Super-galaxy and the Milky Way Galaxy. Probably, someof them take part in nuclear reactions with the gasof these structures. Along the ray of sight, this gashas the highest density precisely in the region ofapparent intersection of the Galaxy and Supergalaxydisks at a galactic longitude of about 137, and thisleads to the pattern observed in Fig. 3. However,the Auger group reported recently that active-coregalaxies (ACG) situated in the local volume of spacewith a radius 75 Mpc are main sources of primarycosmic rays whose energy lies in the region E0 5.7 × 1019 eV [21]. This conclusion was confirmed inpart in [22] for E0 4 × 1019 eV events recorded atthe Yakutsk array. In this connection, we will examinein more detail the structure of matter in the localvolume of radius about 100 Mpc. Figure 4 showsthe density of the galaxy distribution in this volumein the plane of the Supergalaxy, whose North Polefaces the observer [23]. The supergalactic longitude


90 GLUSHKOV

Largest superclusters of galaxies in a local spatial volume of radius about 100 Mpc

Number of Coordinates

super- Name of object supergalactic galactic

clusters lSG bSG lG bG

1 А 194 27.4 −6.0 144.5 −62.4

2 Pegasus 1 28.3 30.0 85.1 −47.6

3 Pisces 31.6 7.4 119.0 −56.8

4 Perseus 77.0 −13.3 150.4 −13.4

5 Camelopardalis 94.7 1.2 136.4 4.7

6 А 569 116.6 −27.2 168.3 22.0

7 А 779 149.9 −32.6 191.1 44.4

8 Coma 181.3 −1.3 244.3 85.5

9 А 1367 182.5 −13.4 236.0 73.4

10 Virgo 192.3 −6.5 279.7 74.4

11 7S 224 206.0 14.7 343.6 61.8

12 Hydra 231.4 −39.7 269.6 26.5

13 А 3581 237.2 6.5 323.2 32.9

14 Centaurus 247.1 −12.3 302.4 21.6

15 Pavo 294.9 12.3 332.2 −23.6

16 А 2870 340.8 −8.4 300.8 −70.2

17 А 2911 350.8 −12.3 271.3 −76.9

18 Fornax 1 353.5 −37.9 236.3 −54.6

19 Great Attractor 260.3 −9.7 307.0 9.0

is reckoned from the “0” axis in the counterclockwisedirection. The NG and SG axes indicate toward, re-spectively, the North and the South Pole of the MilkyWay Galaxy, whose plane is orthogonal to the plane ofthe figure and lies along the 90–270 line. The positionof the Yakutsk array is shown by a closed circle. TheGA (Great Attractor) arrow is directed toward thecenter of the Great Attractor, which will discussedbelow.

One can see that galaxies form giant superclus-ters, between which there are enormous voids. Thenames of the superclusters and their coordinates aregiven in the table. The centers of these superclus-ters are shown in Fig. 5 (crosses accompanied byfigures) on the developed celestial sphere in terms of(Fig. 5a) galactic and (Fig. 5b) supergalactic coor-dinates. From Fig. 5a, it can be seen that, in thevolume being considered, the overwhelming major-ity of the superclusters are situated with respect toeach other in the form of a disk—specifically, in thesupergalactic-latitude band |bG| 30. The open cir-

cles in Fig. 5 show the arrival directions of 27 E0 5.7 × 1019 eV giant air showers that underlay theconclusion drawn in [21] that active-core galaxiesare main sources of such events. The closed circlesrepresent the arrival directions of 28 giant air showersrecorded by SUGAR [16] that have the same energyand which appear in the spectrum presented in [10].Finally, the closed triangles stand for the data thatwere recorded by the Yakutsk array (49 events) andwhich are characterized by energies in the regionE0 3.3 × 1019 eV. In view of the difference in thenormalization factors for the spectra in Fig. 2, thelower boundary of the energy in the Yakutsk data wasreduced by a factor of 1.72 (=1.29/0.75).

From Fig. 5a, one can see that some of the giantair showers recorded by Auger come from the regionof the А 3581 (13) and Centaurus (14) superclusters.The Great Attractor (GA) lies in the same region,which is not far from the locus of intersection ofthe Galaxy and Supergalaxy planes (in Fig. 5b at agalactic latitude of lG ≈ 317). Some of the SUGAR



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Fig. 5. Centers of superclusters from Fig. 4 in terms of (a) supergalactic and (b) galactic coordinates (crosses accompaniedby figures): (open circles) arrival directions of 27 E0 5.7 × 1019 eV giant air showers according to Auger data from [21],(closed circles) arrival directions of 28 E0 5.7 × 1019 eV giant air showers according to SUGAR data from [16], and (closedtriangles) 49 E0 3.3 × 1019 eV events recorded at the Yakutsk array. The SG curve represents the Supergalaxy plane; N andS are, respectively, the North and the South Pole of the Earth; and GA stands for the Great Attractor.

events are adjacent to the region of the Hydra su-percluster (12). Some part of the showers recordedby this array refers to the region of the Pavo super-cluster (15). About half of all events recorded by thetwo arrays seem to stem from the voids clearly seen inFig. 4. Despite the fact that the Auger and SUGARarrays scan the same part of the southern sky, the ar-rival directions of giant air showers recorded by thesearrays almost do not intersect. As for the Yakutskarray, some part of its events are seen to be con-centrated in the vicinity of the enormous superclusterPerseus (4); some events are also concentrated in

the regions of the Camelopardalis (5) and А 569 (6)superclusters. By and large, giant air showers areconcentrated around the direction to the apparentintersection of the Galaxy and Supergalaxy planes (inFig. 5b at a galactic latitude of lG ≈ 137), where alarge region of an enhanced density of such events canbe seen in Fig. 3. In view of all this, the conclusiondrawn in [21] that active-core galaxies of the localvolume about 4.2 × 105 Mpc3 in dimension are mainsources of such events becomes questionable, themore so as, according to [18–20], quasars generating


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X

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Fig. 6. Radial velocity vectors of individual elliptic galaxies (open circles) and their groups featuring not less than threemembers (closed circles) in a cone of apex angle 45 around the direction specified by the galactic coordinates lG = 307

and bG = 9 (cross at the center of the figure) [24]. The clean horizontal band around Y = 0 is due to light absorption in thedisk of the Milky Way Galaxy.

neutral particles may be sources of primary cosmicrays having such energies.

In order to obtain deeper insight into the problemat hand, we will additionally consider the structureof space on a larger scale. There are some pieces ofevidence (see, for example, [24–27]) that galaxies ofthe local volume about 3 × 106 Mpc3 in dimensionexecute a collective motion. One can clearly see itin Fig. 6, which shows peculiar velocities of the 400brightest elliptic galaxies [24]. All of these galaxies(open circles) and their clusters featuring three ormore galaxies (closed circles) move toward the GreatAttractor, which occurs at the point (cross at thecenter of the panel) whose galactic coordinates arelG ≈ 307 and bG ≈ 9. The simulation of this gravi-tating mass in [24] yielded the estimate MGA ∼ 5 ×1016MS ≈ 1047 kg (where MS is the Sun’s mass).Its center lies at a distance of RGA = 58 ± 5 Mpcfrom us. The result presented in Fig. 6 complies wellwith the direction of the collective motion of 2395spiral galaxies characterized by redshift values in the

range z 0.03 [27]. In [27], the peculiar velocities ofgalaxies were investigated over the entire sky, and itwas found there that the galaxies move in the direc-tion specified by the coordinates lG = 299 ± 10 andbG = −5 ± 8.

The results obtained by studying the structureof matter over a still larger volume of radius about380 Mpc are presented in [28]. The spatial distributionof 535 rich galaxy clusters from the catalog of Abell etal. [29] that are characterized by redshift values in therange z 0.1 was examined in that study. Figure 7displays the deviations

nz = (〈z〉 − 〈z〉0)/∆z (2)

of the average redshifts 〈z〉 in a circle of radius 45from the average redshift 〈z〉0 = 0.056 ± 0.001 overthe entire sample, the center of this circle being suc-cessively moved over all areas of the sphere 1 × 1in dimension. The limits of changes in nz are shownat the bottom of the panel as a toned scale with astep of ∆z = 0.004. Curve 1 represents the equa-tor of the Earth. Figure 7 demonstrates clearly the



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Fig. 7. Distribution of the deviations (2) of the mean redshifts of 535 z 0.1 rich galaxy clusters presented in the catalog ofAbell et al. [29] from 〈z〉0 = 0.056 for the entire sample over the developed celestial sphere in terms of the galactic coordinates.The toned scale indicates the limits of changes in nz with a step of ∆z = 0.004. Curve 1, N, S, GA, and curve SG stand for,respectively, the equator, the North Pole of the Earth, the South Pole of the Earth, the Great Attractor, and the Supergalaxyplane.

anisotropy of redshifts. The redshifts have an averagevalue of 〈z〉N ≈ 0.064 in the Northern Hemisphere ofthe Earth and an average value of 〈z〉S ≈ 0.043 in theSouthern Hemisphere; that is, they differ by a factor ofabout 1.5. The north–south character of this asym-metry may have a common origin with the north–south dependence of the normalization coefficients inFig. 2.

Of particular interest is the black hole in the di-rection to the Great Attractor (GA). In [28], a similarblack hole was found in the spatial distribution ofz 6 Seyfert galaxies and quasars. In the dispositionof all of the aforementioned objects, there is a globalanisotropy caused by the shift of the observer’s posi-tion by about 50 Mpc from the center of their symme-try (presumably the Metagalaxy center) toward thevector specified by the galactic coordinates lG ≈ 123and bG ≈ 7. In the direction opposite to this, thereis a large region where redshifts decrease graduallyto a minimum value (in the vicinity of the directionspecified by the galactic coordinates lG ≈ 303 andbG ≈ −7).

In my opinion, this topological inhomogeneity isone of many indications that a Metagalaxy, a gravita-tionally coupled aggregate of matter in the Universewhere we live and beyond which our sight cannotpenetrate in principle, does in fact exist. The idea ofa Metagalaxy as such is not new; from time to time, itis developed by researchers (see, for example, [18, 28,

30–36]). In [32, 33], it was proposed to consider theUniverse as an object that consists of many isolatedand closed “miniuniverses.” Possibly, it is one of suchminiuniverses that is our Metagalaxy.

Anyway, this idea makes it possible, in my opinion,to offer some explanation of the results presentedabove. In Fig. 7, the South Pole of the Earth roughlyfaces the Metagalaxy center, while the North Polespecifies the opposite direction. Under such condi-tions, the gravitational potential of the Metagalaxyand its central spherical directivity—this potentialdecelerates primary particles going from the south-ern direction (from the Metagalaxy center) and, onthe contrary, additionally accelerates primary parti-cles going from the northern direction—play an im-portant role in energy transfer to cosmic rays (alongwith the sources of their formation). This would resultin the shift of the energy spectrum to the right alongthe energy scale in the Northern Hemisphere of theEarth and to the left in the Southern Hemisphere,presumably leading to the pattern observed in Fig. 1a.

Let us now estimate this change in energy. Eventhe first physicists involved in studying quasars (see,for example, [37]) noticed that the gravitational po-tential of the Metagalaxy may contribute to the red-shift of these objects. The redshift is defined as theshift

z = (λ2 − λ1)/λ1 (3)


94 GLUSHKOV

of the wavelength λ1 measured for electromagneticradiation from a remote object at the radiation-emission instant by an observer residing near thatobject at point 1 with respect to the wavelength λ2

of the same radiation that arrived at point 2, wherewe occur, together with a receiver of radiation, ourstandard of wavelength λ1 being assumed to beidentical to that at point 1.

Let us recast the definition in (3) into an alternativeform

1 + z = λ2/λ1 = (ν1/ν2)(c1/c2), (4)

where ν1 = c1/λ1 and ν2 = c2/λ2 are the corre-sponding radiation frequencies and c1 and c2 are thelocal speeds of light at, respectively, points 1 and 2of the Metagalaxy. In the argument presented below,we consider that the local speed of light at a distanceR from the Metagalaxy center is determined by theEinstein formula [38]

c(R) = c0(1 − |Φ(R)|), (5)

where Φ(R) = ϕ(R)/(c0)2 is the dimensionless grav-itational potential of the Metagalaxy and c0 is thespeed of light far from the Metagalaxy, where itsgravitational potential is zero. Within regions off theMetagalaxy (for R RMG), this gravitational poten-tial is given by the well-known Newton formula

ϕ(R) = −GMMG/R, (6)

where G = 6.67 × 10−11 kg−1 m3 s−2 is the gravita-tional constant, RMG ≈ 1026 m ≈ 4000 Mpc is theradius of the outer boundary of the Metagalaxy, andMMG is its mass. The Metagalaxy mass is approxi-mately given by

MMG ≈ (4π/3)(RMG)3ρ ≈ 4 × 1052 kg, (7)

where ρ ≈ 10−26 kg/m3 is the average density ofmatter in the Metagalaxy [39]. The question ofhow the gravitational potential is distributed withinthe Metagalaxy is quite involved. Within a uniformsphere, this potential has the form [39]

ϕ(R) = −GMMG(3 − (R/RMG)2)/2. (8)

At R = RMG, expressions (7) and (8) match eachother, yielding the same value: ϕMG = ϕ(RMG) ≈3 × 1016 m2/s2. At the Metagalaxy center, this po-tential is 1.5 times greater than that, the gravitationalfield itself being zero. The gravitational potential isthe work to be performed in order to remove a unitmass from a given gravitational field. For a bodyof mass m to be able to escape from the gravita-tional field of the Metagalaxy, it must perform thework A = −mϕMG; for this, it must possess the ki-netic energy K = mv2/2 = −A and, hence, the speed

v =√−2ϕMG ≈ 2.5 × 108 m/s, which is close to the

speed of light.The following intriguing fact is worthy of spe-

cial note. The square of the speed of light, c2 = 9 ×1016 m2/s2, agrees with the potential of the Meta-galaxy (within the errors in estimating ϕMG). Accord-ing to the Einstein formula, the total energy of a bodyis E = mc2. It turns out that this energy is approxi-mately equal to the gravitational potential created bythe entire mass of the Metagalaxy. Some astonishingconsequences of this circumstance were consideredin [40], but they will not be revisited here.

Taking into account relation (5), we will recast (4)into the form

1 + z = (ν1/ν2)((1 − |Φ1|)/(1 − |Φ2|)), (9)

which is more convenient for the subsequent analysis.In general, the change from the frequency ν1 of a

moving source to the observed frequency ν2 owing tothe Doppler effect follows the law [39]

ν1/ν2 = (1 − β1 cos θ)/√

1 − β21 , (10)

where θ is the angle between the source velocityvector v1 (β1 = v1/c1) and the direction of wave prop-agation. The source moves away from the source(redshift) for θ > π/2 and approaches it (violet shift)for θ < π/2. The denominator on the right-hand sideof (10) characterizes the relativistic slowing down ofthe course of time in a moving source.

Let us recast relation (3) once again with al-lowance for (9) and (10) into the form

1 + z = λ2/λ1 (11)

= (λ2/λ)/(λ/λ1) = (1 + zv)(1 + zgr),

where the relation

(1 + zv) = ν1/ν2 = (1 + βv)/√

1 − β2v (12)

determines the redshift owing to the cosmologicalexpansion of the Metagalaxy with a relative radialvelocity βv and the coefficient

1 + zgr = (1 − |Φ1|)/(1 − |Φ2|) (13)

reflects the additional change in the observable in (3)because of the gravitational potential of the Meta-galaxy. At the observation point 2 in the vicinity ofthe Metagalaxy center, we have c2

2 = 9 × 1016 m2/s2

and Φ2 ≈ 1.5ϕMG/c22 ≈ 0.5. For cosmic-ray sources

(quasars) lying in the vicinity of its external bound-ary, c1 ≈ 1.5c2, ϕMG ≈ 3 × 1016 m2/s2, and Φ1 ≈ϕMG/с2

1 ≈ 0.148. From (13), we find the correction1 + zgr ≈ 1.7, which agrees with a general normaliza-tion factor of K = 1.29/0.75 ≈ 1.72 for matching theextreme Yakutsk and Australian spectra in Fig. 1.



Indeed, suppose that the energies of particles pro-duced at the boundary of the Metagalaxy and in thevicinity of its center are, respectively, E01 = m1c

21 =

(m0/γ1)c21 and E02 = m2c

22 = (m0/γ2)c2

2, where γ =√1 − β2. With allowance for (5), their ratio is then

given by

K = E01/E02 = (m1/m2)(c1/c2)2 (14)

= (γ2/γ1)((1 − |Φ1|)/(1 − |Φ2|))2.

According to Einstein’s special theory of relativ-ity [38], the Lorentz factor γ in a gravitational field isdetermined by the equation

γ = (1 − |Φ|). (15)

After the substitution into (14) with allowance for (13),it leads to the relation

K = 1 + zgr. (16)

4. CONCLUSIONS

We have examined six spectra obtained forultrahigh-energy cosmic rays from measurementsperformed in different periods of time at differentarrays for recording extensive air showers world-wide. These spectra have a rather high statisticalsignificance and arouse permanent interest. Basedon recording different components of extensive airshowers and on employing different data-treatmentmethods, all of these spectra proved to be different(see Fig. 1a). In recent years, attention has beengiven primarily to the results coming from the HiResand Auger arrays. Some researchers have been in-clined to discuss only those results. It is the opinionof the present author that no selectivity is of use here.A versatile approach that would make it possibleto cover and explain the entire body of availabledata seems more appropriate here. The method fornormalizing spectra that was developed in [13] reveals(see Fig. 1b) good agreement of all spectra in shape.The observed difference in the intensity between thespectra is likely to be due to scanning different regionsof the sky from the different arrays rather than tomaking errors in evaluating energies at these arrays.For the example of the Yakutsk array, it was alreadyshown in [20, 41] that the energy spectra of cosmicrays from different regions of the sky are markedlydifferent. The pronounced north–south dependenceof the normalization factors in Fig. 2 on the latitudeof the position of the arrays considered here andintended for recording extensive air showers suggeststhat they might have been of the astrophysical origin.The anisotropy of the giant air showers in Fig. 3 favorsthis assumption. Matter in the local volume of spacewith a radius of about 100 Mpc has a complicatedstructure (see Fig. 4), which correlates poorly with

arrival directions of giant air showers (see Fig. 5). Thecollective motion of galaxies that is shown in Fig. 6and the north–south anisotropy of redshifts of richgalaxy clusters in Fig. 7 demonstrate a strong effectof the Great Attractor on the dynamics and structureof matter on a cosmological scale. In [18, 28], it wasshown that a pronounced anisotropy is also observedin the spatial distribution of z 6 Seyfert galaxiesand quasars. This anisotropy is due to a shift of theobserver’s position by about 50 Mpc from their centerof symmetry (presumably the Metagalaxy center)toward the vector specified by the galactic coordinateslG ≈ 123 and bG ≈ 7. In the direction opposite tothis, there is a large region where redshifts decrease toa minimum value (in the vicinity of the vector specifiedby the galactic coordinates lG ≈ 303 and bG ≈ −7).The gravitational potential of the Metagalaxy (as wellas probably the gravitational potential of the GreatAttractor) may have an additional strong effect oncosmic rays and change their energy spectrum in thesame manner as is observed in Fig. 1. I am planningto continue the investigation of this problem.

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Translated by A. Isaakyan


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Problem of the energy spectrum of ultrahigh-energy cosmic rays