GZK Horizons and the Recent Pierre Auger Result on the Anisotropy of Highest-energy Cosmic Ray Sources
aa r X i v : . [ a s t r o - ph ] M a y GZK Horizons and the Recent Pierre Auger Result on theAnisotropy of Highest-energy Cosmic Ray Sources
Chia-Chun Lu
Institute of Physics, National Chiao-Tung University, Hsinchu 300, Taiwan.
Guey-Lin Lin
Institute of Physics, National Chiao-Tung University, Hsinchu 300, Taiwan andLeung Research Center for Cosmology and Particle Astrophysics,National Taiwan University, Taipei 106, Taiwan. (Dated: October 29, 2018)
Abstract
Motivated by recent Pierre Auger result on the correlation of the highest-energy cosmic rays withthe nearby active galactic nuclei, we explore possible ultrahigh energy cosmic ray (UHECR) sourcedistributions and their effects on GZK horizons. Effects on GZK horizons by local over-density ofUHECR sources are examined carefully with constraints on the degree of local over-density inferredfrom the measured UHECR spectrum. We include the energy calibration effect on the Pierre Augerdata in our studies. We propose possible local over-densities of UHECR sources which are testablein the future cosmic ray astronomy.
PACS numbers: 95.85.Ry, 96.50.sb, 96.50.Vg . INTRODUCTION Recently, Pierre Auger observatory published results on correlation of the highest-energycosmic rays with the positions of nearby active galactic nuclei (AGN) [1, 2]. Such a corre-lation is confirmed by the data of Yakutsk [3] while it is not found in the analysis by HiRes[4]. In the Auger result, the correlation is maximal for the threshold energy of cosmic raysat 5 . × eV, the maximal distance of AGN at 71 Mpc and the maximal angular sepa-ration of cosmic ray events at ψ = 3 . ◦ . With the same threshold energy, and the angularseparation ψ ≤ ◦ , the correlation remains strong for a range of maximal AGN distancebetween 50 Mpc and 100 Mpc. Due to increasing efforts on verifying the Auger result, it isworthwhile to examine the above correlation from a phenomenological point of view.Since the angular scale of the observed correlation is a few degrees, one expects that thesecosmic ray particles are predominantly light nuclei. The effect of GZK attenuations on thesecosmic ray particles [5, 6] can be described by a distance scale referred to as “GZK horizon”which is a function of the selected energy threshold for the arriving cosmic ray particles. Bydefinition, the GZK horizon associated with a threshold energy E th is the radius of a sphericalregion which is centered at the Earth and produce 90% of UHECR events arriving on Earthwith energies above E th . With continuous energy loss approximation, the GZK horizon forprotons with E th = 57 EeV is about 200 Mpc by assuming a uniformly distributed UHECRsources with identical cosmic ray luminosity and spectral index [7]. The calculations basedupon kinetic equation approach or stochastic energy loss also reach to similar conclusions[8, 9].The departure of theoretically calculated GZK horizon to the maximum valid distanceof the V-C catalog [10] employed in Pierre Auger’s analysis, which is around 100 Mpc, canbe attributed to several factors. As mentioned in [2], such a deviation may arise from non-uniformities of spatial distribution, intrinsic luminosity and spectral index of local AGN. Inaddition, the energy calibration also plays a crucial role since the GZK horizon is highlysensitive to the threshold energy E th . Energy values corresponding to the dip and the GZKcutoff of UHECR spectrum were used to calibrate energy scales of different cosmic ray ex-periments [11, 12]. It has been shown that all measured UHECR energy spectra can bebrought into good agreements by suitably adjusting the energy scale of each experiment[11]. Keeping the HiRes energy scale unchanged, the energy-adjustment factor λ is found to2e 1 .
2, 0 .
75, 0 .
83 and 0 .
625 respectively for Auger, AGASA, Akeno and Yakutsk. Further-more, it has been shown that a different shower energy reconstruction method infers a 30%higher UHECR energy than that determined by Auger’s fluorescence detector-based showerreconstruction [13].In this paper, we investigate the consistency between Auger’s UHECR correlation studyand its spectrum measurement. As just stated, the V-C catalog used by Pierre Auger forthe correlation study is complete only up to 100 Mpc while the GZK horizon for E th = 57EeV is generally of the order 200 Mpc. We first consider the local over-density of UHECRsources as a possible resolution to the above discrepancy. It is motivated by the existence ofLocal Supercluster (LS) which has a diameter of the order 60 Mpc. In LS, the over-densityof galaxies has been estimated to be ∼ · − ) eVregion. Hence fittings to the measured UHECR spectrum [20] can provide information onthe degree of local over-density. Subsequently, the magnitude of GZK horizon can be betterestimated.We next study the energy calibration effect on the estimation of GZK horizon and thespectrum of UHECR. Certainly a 20% −
30% upward shift on UHECR energies reduces thedeparture of theoretically calculated GZK horizon to the maximum valid distance of V-Ccatalog [2]. The further implications of this shift will be studied in fittings to the shiftedAuger spectrum.We fit the UHECR spectrum for events with energies above 10 eV. This is the energyrange where the GZK attenuation exhibits its effect. It is also the energy range where thelocal over-density of UHECR sources shows significant effects. In our analysis, we take theUHECR as protons, which is hinted in the Auger events with energies ≥
57 EeV althoughthe composition study by the same group suggests a heavier composition for E ≤
40 EeV[21]. The HiRes experiment measures the composition up to 50 EeV [22] and obtains acomposition lighter than that of Auger. For
E >
50 EeV, the event number is still toosmall for the composition study. To fit the UHECR spectrum at the highest energy, it ismore appropriate to treat the cosmic ray energy loss as a stochastic process [23]. Thereare numerical packages available for treating stochastic energy loss of cosmic ray particles324, 25]. We employ the latter package for our calculations. Although UHECR loses itsenergy mostly by scattering off CMB photons, it also loses some amount of energy byscattering off infrared background photons [26, 27, 28, 29, 30]. Thus we include the infraredphoton contribution to the UHECR energy attenuation. Source evolution n ( z ) = n (1 + z ) is adopted in the calculation of GZK horizon and spectrum, where n is the source numberdensity at the present epoch. It is from the generally-accepted soft evolution model whichtraces the star formation history and has been adopted in previous works [31].We discuss about GZK horizons in Sec. II. We calculate the accumulated event probabil-ities of UHECR for E th = 57 EeV, 70 EeV, 80 EeV and 90 EeV respectively. GZK horizonscorresponding to different E th are tabulated. We also calculate GZK horizons with local over-density of UHECR sources taken into account. In Sec. III, we fit the measured UHECRspectrum with various local over-densities of UHECR sources and obtain information on thedegree of local over-density. To study the energy calibration effect, we also perform fittingsto the shifted UHECR spectrum. Sec. IV contains discussions and conclusions. II. THE ACCUMULATIVE EVENT PROBABILITIES OF UHECR
For single UHECR source, the cosmic-ray energy attenuation is governed by the equation ∂φ N ( E, t ) ∂t = ∂∂E (cid:20)(cid:18) − dEdt (cid:19) φ N ( E, t ) (cid:21) , (1)in the continuous energy loss approximation. This equation results from the number con-servation of cosmic-ray particles in the energy attenuation process. The cosmic-ray energyloss per unit time − dE/dt is due to the cosmic expansion and its scattering with cosmicmicrowave background photons through photo-pion production process P γ → N π and pairproduction process
P γ → P e + e − . The above attenuation equation is well known [32]. In thecurrent context, the solution of Eq. (1) can be expressed in terms of the red-shift variable[17] φ N ( E, z ) = φ N ( ¯ E, z s ) × exp (cid:20)Z z s z dz ′ (cid:18) (1 + z ′ ) H ( z ′ ) × ∂b ((1 + z ′ ) ¯ E ) ∂ ¯ E + 11 + z ′ (cid:19)(cid:21) , (2)where z s is the red-shift of the UHECR source and the function b is related to the rate ofcosmic-ray energy loss at the present epoch by − dEdt ( z = 0) = b ( E ) + H E, (3)4here H is the present value of Hubble constant. The UHECR has an energy ¯ E at thesource with red-shift z s and its energy is downgraded to E at the red-shift z . The energy ¯ E is a function of E and z so that ¯ E ( E, z s ) = E and d ¯ Edz = − b (cid:0) (1 + z ) ¯ E (cid:1) H ( z ) (1 + z ) − ¯ E z . (4)Due to the non-trivial form of b , one resorts to numerical methods for computing thefunction ¯ E and the flux φ N ( E, z ).We have mentioned that the stochastic nature of UHECR energy loss can not be over-looked for shorter propagation distances [23]. One then treats the energy attenuation byphoto-pion production as a stochastic process while treating other attenuations as continu-ous processes.
100 200 300 400 500D H Mpc L P H D,E th L E th = H Mpc L P H D,E th L E th = H Mpc L P H D,E th L E th = H Mpc L P H D,E th L E th = FIG. 1: The accumulative event probability P ( D, E th ) as a function of D for E th = 57 EeV, 70 EeV,80 EeV and 90 EeV respectively. The horizontal dash line in each panel denotes P ( D, E th ) = 0 . n ( l < /n = 1 , , , and 10 respectively. The intrinsic spectrum index γ = 2 .
4, energy cut E cut = 1000 EeV and the source evolution model n ( z ) = n (1 + z ) are used for calculations. To facilitate our discussions, we define the accumulative event probabilities of UHECR5s P ( D, E th ) = R D dl · N ( l, E th ) R ∞ dl · N ( l, E th ) , (5)where N ( l, E th ) · dl is the number of cosmic ray events which are originated from sourcesat distances between l and l + dl from the Earth and arrive at the detector with energiesabove E th . We calculate P ( D, E th ) for various local over-densities of UHECR sources. Thesource distribution over the red-shift is taken as n ( z ) = n (1 + z ) and the energy spectrumof each source is taken to be the form, φ N ( E ) ≡ dN/dE = AE − γ , with the maximal energy E cut = 1000 EeV. We choose γ = 2 .
4, 2 . . γ = 2 . P ( D, E th ) for E th = 57 EeV, 70 EeV, 80 EeV and 90 EeV are shown inFig. 1 for γ = 2 .
4. Results for γ = 2 . γ = 2 . γ = 2 .
4. In each panel, the red, green, blue, and black curves represent local over-density n ( l < /n = 1 , , , and 10 respectively. The local over-density n ( l < /n = k is defined explicitly as n ( l < /n = k (1 + z ) ,n ( l ≥ /n = (1 + z ) . (6)The horizontal dash line in each panel denotes P ( D, E th ) = 0 .
9. The intersection of this linewith each color curve gives the GZK horizon corresponding to a specific local over-densitycharacterized by the ratio n ( l < /n . TABLE I: GZK horizons of UHECR calculated with the local over-density n ( l < /n =1 , , , and 10, and arrival threshold energy E th = 57 EeV, 70 EeV, 80 EeV and 90 EeV respec-tively. The listed numbers are in units of Mpc. n ( l < /n E th = 57 EeV E th = 70 EeV E th = 80 EeV E th = 90 EeV1 220 150 115 902 210 140 105 754 195 120 85 6010 155 85 50 30 GZK horizons corresponding to different local over-densities and E th are summarized inTable I. It is seen that local over-densities up to n ( l < /n = 4 do not alter GZK6orizons significantly for a given E th . One could consider possibilities for higher local over-densities. However, there are no evidences for such over-densities either from astronomicalobservations [14] or from fittings to the measured UHECR spectrum. We note that GZKhorizons are rather sensitive to E th . Table I shows that GZK horizons are ∼
100 Mpc orless for E th ≥
80 EeV.
III. FITTINGS TO THE UHECR SPECTRUM MEASURED BY PIERRE AUGER
As mentioned earlier, the local over-density of UHECR sources affects the cosmic-rayspectrum at the highest energy, especially at energies higher than 5 · eV. Hence thedegree of local over-density can be examined through fittings to the measured UHECRspectrum as will be shown momentarily.Fittings to the Auger spectrum have been performed in [33, 34, 35, 36].. In our work,we take into account the over-density of UHECR sources in the distance scale l ≤ TABLE II: The values of total χ from fittings to the Auger measured UHECR spectrum. Numbersin the parenthesis are χ values from fittings to the 8 data points in the energy range 19 . ≤ log ( E/ eV) ≤ .
75. The last 4 data points record events with energy greater than 71 EeV. n ( l < /n γ = 2 . . .
34) 14 . .
93) 17 . .
50) 28 . . γ = 2 . . .
28) 15 . .
90) 16 . .
83) 20 . . fittings to the Auger measured UHECR spectrum with γ = 2 . . n ( z ) = n (1 + z ) m with m = 3.We have fitted 12 Auger data points beginning at the energy 10 eV. We make a fluxnormalization at 10 eV while varying the power index γ and the the degree of local over-density, n ( l < /n . Part of χ values from our fittings are summarized in Table II. Wefound that γ = 2 . , n ( l < /n = 1 gives the smallest χ value with χ / d . o . f . = 1 . n ( l < /n = 10 is ruled outat the significance level α = 0 . γ = 2 . n ( l < /n = 10 is ruled out at thesignificance level α = 0 .
02. 7 @ Energy H eV LD Log @ E J H eV m - s - sr - LD FIG. 2: Fittings to the Auger measured UHECR spectrum where the red, green, blue and blackcurves denote the model with the local over-density n ( l < /n = 1 , , , and 10 respectively.Solid curves correspond to γ = 2 . γ = 2 .
5. We take the sourceevolution parameter m = 3 throughout the calculations. We note that, for both γ = 2 . γ = 2 .
6, the GZK horizon with n ( l < /n = 10, E th = 57 EeV, m = 3 and E cut = 1000 EeV is about 155 Mpc. Since n ( l < /n = 10is clearly disfavored by the spectrum fitting, one expects a GZK horizon significantly largerthan 155 Mpc for E th = 57 EeV.We next perform fittings to the shifted Auger spectrum. The results are shown in Fig. 3where the cosmic ray energy is shifted upward by 30%. Part of χ values are summarized inTable III. The smallest χ value occurs approximately at γ = 2 . n ( l < /n = 2 with χ / d . o . f = 0 .
82. For γ = 2 . χ / d . o . f = 1 .
31, 0 .
96 and 0 .
87 for n ( l < /n = 1, 2 and4 respectively. It is seen that χ values from current fittings are considerably smaller thanthose from fittings to the unshifted spectrum. Given a significance level α = 0 .
1, it is seenthat every local over-density listed in Table III except n ( l < /n = 10 is consistentwith the measured UHECR spectrum. It is intriguing to test such local over-densities aswill be discussed in the next section.We note that, with a 30% upward shift of energies, the cosmic ray events analyzed inAuger’s correlation study would have energies higher than 74 EeV instead of 57 EeV. TheGZK horizon corresponding to E th = 74 EeV is 120 Mpc for n ( l < /n = 2 and 105Mpc for n ( l < /n = 4.We have so far confined our discussions at m = 3. In the literature, m has been taken as8 @ Energy H eV LD Log @ E J H eV m - s - sr - LD FIG. 3: Fittings to the Auger measured UHECR spectrum with a 30% upward shift on UHECRenergies where the red, green, blue and black curves denote the model with the local over-density n ( l < /n = 1 , , , and 10 respectively. Solid curves correspond to γ = 2 . γ = 2 .
5. We take the source evolution parameter m = 3 throughout thecalculations.TABLE III: The total χ values from fittings to the Auger measured UHECR spectrum with a30% upward shift on UHECR energies. Numbers in the parenthesis are χ values from fittings tothe 8 data points in the energy range 19 . ≤ log ( E/ eV) ≤ .
86. The last 4 data points recordevents with energy greater than 92 EeV. n ( l < /n γ = 2 . . .
30) 7 . .
67) 10 . .
35) 27 . . γ = 2 . . .
16) 8 . .
49) 7 . .
23) 16 . . any number between 0 and 5. It is demonstrated that the effect on UHECR spectrum causedby varying m can be compensated by suitably adjusting the power index γ [31]. Since GZKhorizons are not sensitive to γ and m , results from the above analysis also hold for other m ’s. 9 V. DISCUSSIONS AND CONCLUSIONS
We have investigated the consistency between Auger’s latest result on the correlationof UHECR sources with positions of nearby extra-galactic AGN and its measured UHECRspectrum. As stated before, this investigation is motivated by the fact that the V-C catalogused by Pierre Auger for the correlation study is reliable only up to 100 Mpc while theGZK horizon for E th = 57 EeV is generally of the order 200 Mpc. We have explored thepossibility for local over-density of UHECR sources, which is expected to shorten the GZKhorizon for a given threshold energy of arrival cosmic-ray particles. This is indeed the caseas can be seen from Table I. On the other hand, the effect is far from sufficient to shortenthe GZK horizon at E th = 57 EeV to ∼
100 Mpc for a local over-density of UHECR sourcesconsistent with the measured UHECR spectrum.We have performed a upward energy shift to the Auger measured UHECR spectrum. Assaid, a upward energy shift is motivated by simulations of shower energy reconstructions aswell as the requirement of reproducing the theoretically predicted GZK cutoff energy. Witha 30% energy shift, each cosmic ray event used by Auger for the correlation study wouldhave an energy above 74 EeV instead of 57 EeV. GZK horizons corresponding to E th = 74EeV then match well with the maximum valid distance of V-C catalog. Fittings to theshifted Auger spectrum indicate a possibility for the local over-density of UHECR sources.We point out that the local over-density of UHECR sources is testable in the futurecosmic ray astronomy where directions and distances of UHECR sources can be determined.Table IV shows percentages of cosmic ray events that come from sources within 30 Mpc fordifferent values of E th and n ( l < /n . We take γ = 2 . m = 3 and E cut = 1000EeV for calculating these percentages. We note that these percentages are not sensitive tothe above parameters. For E th = 57 EeV, only 17% of cosmic ray events come from sourcesless than 30 Mpc away for n ( l < /n = 1. For n ( l < /n = 2 and the samethreshold energy, 30% of cosmic ray events are originated from sources in the same region.It should be stressed that we have focused only on resolving the apparent discrepancybetween the GZK horizon at E th = 57 EeV and the maximum valid distance of V-C catalog.The statistics analysis for establishing the source correlation is an independent issue beyondthe scope of the current paper. We have found that the above discrepancy can not beresolved by merely introducing the local over-density of UHECR sources. On the other hand,10 ABLE IV: Percentages of cosmic ray events that come from sources within 30 Mpc for differentvalues of E th and local over-density n ( l < /n . n ( l < /n E th = 57 EeV E th = 70 EeV E th = 80 EeV E th = 90 EeV1 0 .
17 0 .
27 0 .
36 0 .
462 0 .
30 0 .
43 0 .
53 0 .
634 0 .
46 0 .
60 0 .
70 0 . .
68 0 .
79 0 .
85 0 . if Auger’s energy calibration indeed underestimates the UHECR energy, such a discrepancycan be reduced. More importantly, fittings to the shifted Auger spectrum indicate a possiblelocal over-density of UHECR sources, which is testable in the future cosmic ray astronomy. Acknowledgements
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