Composition of UHECR and the Pierre Auger Observatory Spectrum
Katsushi Arisaka, Graciela B. Gelmini, Matthew Healy, Oleg Kalashev, Joong Lee
aa r X i v : . [ a s t r o - ph ] N ov CERN-PH-TH/2007-161
Composition of UHECR and the Pierre AugerObservatory Spectrum
Katsushi Arisaka a , Graciela B. Gelmini a,b , Matthew Healy a ,Oleg Kalashev c and Joong Lee a a Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547, USA b CERN, PH-TH, CH-1211 Gen`eve 23, Switzerland c INR RAS, 60th October Anniversary pr. 7a, 117312 Moscow, Russia
Abstract.
We fit the recently published Pierre Auger ultra-high energy cosmic rayspectrum assuming that either nucleons or nuclei are emitted at the sources. Weconsider the simplified cases of pure proton, or pure oxygen, or pure iron injection.We perform an exhaustive scan in the source evolution factor, the spectral index, themaximum energy of the source spectrum Z × E max , and the minimum distance tothe sources. We show that the Pierre Auger spectrum agrees with any of the sourcecompositions we assumed. For iron, in particular, there are two distinct solutions withhigh and low E max (e.g. 6 . × eV and 2 × eV) respectively which could bedistinguished by either a large fraction or the near absence of proton primaries at thehighest energies. We raise the possibility that an iron dominated injected flux may bein line with the latest composition measurement from the Pierre Auger Observatorywhere a hint of heavy element dominance is seen.PACS numbers: 98.70.Sa omposition of UHECR and the Pierre Auger Observatory Spectrum
1. Introduction
The Greisen-Zatsepin-Kuzmin (GZK) cutoff [1] at 4 × eV seems not to be presentin the data of the AGASA ground array [2] but it appears in the data of the HiResair fluorescence detector [3, 4]. This controversy can be addressed by the Pierre AugerObservatory [5], a hybrid combination of charged particle detectors and fluorescencetelescopes, as it continues to accumulate data. We study here the most recent spectrumpublished by the Pierre Auger Observatory [6].Using the surface array, the Pierre Auger Collaboration presented [6] an update totheir previous result [7] that includes two additional years of data and an integratedaperture (5165 km sr yr) nearly equivalent to that of the HiRes experiment. Theupdated spectrum begins at an energy of 2 . × eV, the energy at which the surfacearray becomes fully efficient within the zenith angle range 0-60 ◦ [8], and ends with ahighest observed energy of ∼ . × eV. Energies are determined in a simulationindependent way assuming constant intensity and calibrating the ground observableS(1000) against the fluorescence detector energy for the subset of showers (known asgolden hybrid showers) that contain reconstructions from both detectors. The methodleads to a statistical error of 8% and a systematic error of 22% [6, 9] on the energy.The origin of cosmic rays with energies beyond the GZK cutoff remain anoutstanding open question in astroparticle physics and cosmology [2, 3, 4, 10]. Nucleonscannot be significantly deflected by the magnetic fields of our galaxy for energies abovethe “ankle”, i.e. above 10 . eV. This and the absence of a correlation of arrivaldirections with the galactic plane indicate that, if nucleons are the primary particles ofthe ultra high energy cosmic rays (UHECR), these nucleons should be of extragalacticorigin. Moreover, nucleons as well as photons with energies above 5 × eV could notreach Earth from a distance beyond 50 to 100 Mpc [11, 12] thus sources should be foundwithin this distance. Nucleons scatter off the cosmic microwave background (CMB)photons with a resonant photoproduction of pions pγ → ∆ ∗ → N π , where the pioncarries away ∼
20% of the original nucleon energy. Photons with comparable energypair-produce electrons and positrons on the radio background.Intervening sheets of large scale intense extra galactic magnetic fields (EGMF),with intensities B ∼ . − × − G, could provide sufficient angular deflection forprotons to explain the lack of observed sources in the directions of arrival of UHECR.However, recent realistic simulations of the expected large scale EGMF show that strongdeflections could only occur when particles cross galaxy clusters. Except in the regionsclose to the Virgo, Perseus and Coma clusters the magnetic fields are not larger than3 × − G [13] and the deflections expected are not important (however see Ref. [14]).Heavy nuclei are an interesting possibility for UHECR primaries, since they couldbe produced at the sources with larger maximum energies and would more easilybe deflected by intervening magnetic fields. Both AGASA and HiRes data favor adominance of light hadrons, consistent with being all protons, in the composition ofUHECR above 10 eV [15]. These data are consistent with models in which all UHECR omposition of UHECR and the Pierre Auger Observatory Spectrum eV are due to extragalactic protons [16]. The Pierre Auger Observatory haspresented an elongation rate that is better represented by a fit containing a break pointin the slope at 2 × eV. Below the break point the spectrum is consistent witha progressively lighter composition but above the break the composition is consistentwith a constant and mixed composition up to the highest energies [17]. This raises thepossibility of a significant fraction of heavier elements in the range of the GZK cutoff.Whether particles can be emitted with the necessary energies by astrophysicalaccelerators, such as active galactic nuclei, jets or extended lobes of radio galaxies,or even extended objects such as colliding galaxies and clusters of galaxies, is still anopen question. The size and possible magnetic and electric fields of these astrophysicalsites make it plausible for them to accelerate protons and nuclei to a maximum energyof Z × eV, where Z is the number of protons in each nucleus. Larger emissionenergies would require a reconsideration of possible acceleration models or sites.A galactic component of the UHECR flux, which could be important up to energies10 eV, should consist of heavy nuclei, given the lack of correlation with the galacticplane of events at this energy (outside the galactic plane galactic protons would bedeflected by a maximum of 15-20 o at this energy [18]).In this paper we fit the Pierre Auger UHECR spectrum above the energy E cut =1 × eV (and for comparison we also use two other values of E cut , 2 . × eVand 4 × eV) assuming that either protons or nuclei are emitted at the sources.The UHECR spectrum predicted depends on the slope and maximum energy of thenucleon or nucleus spectrum emitted at the source, the distribution of sources, and theintervening backgrounds. We take a phenomenological approach in choosing the range ofthe several relevant parameters which determine the cosmic ray flux, namely we take foreach of them a range of values mentioned in the literature, without attempting to assignthem to particular sources or acceleration mechanisms. We consider the simplified casein which either only protons, or only oxygen nuclei, or only iron nuclei would be emittedby the sources. Although these are not realistic models for the injected composition, weexpect to gain some understanding of how well a heavy or intermediate or light elementsdominated composition in the injected spectrum can account for the observed spectrum.The ankle in the UHECR spectrum at energies 10 eV - 10 eV can be explainedeither by e ± pair production by extragalactic protons interacting with the CMB [16] orby a change from one component of the UHECR spectrum to another. We take intoaccount the first possibility by fitting the Pierre Auger spectrum above 2 . × eVwith a flux of protons emitted at the sources. This possibility can still be consistentwith the proton-dominated composition observed by HiRes.The second explanation of the ankle, in which the extragalactic componentdominates at energies above the ankle, assumes the existence of a low energy component(LEC) when necessary to fit the UHECR spectrum at energies lower than 1 to 4 × eV.This LEC can be dominated by galactic Fe or by a different population of lower energyextragalactic nucleons. Here we do not address the issue of what the LEC is. We onlyassume that, if it exists, it becomes negligible at energies above the energy E cut at which omposition of UHECR and the Pierre Auger Observatory Spectrum × eV or 4 × eV. In this case we study both thecase of protons as well as that of nuclei (Fe or O) emitted at the sources.Our calculations do not take into account deflections. Since we assume typicalextragalactic magnetic fields not larger than 3 × − G [13] outside large clusters, thedeflections of iron nuclei become important for energies below 1 × eV. Thereforewe only consider nucleons below this energy.When E cut > . × eV, besides fitting the spectrum above E cut , we requirethat the spectrum we predict is never above the measured spectrum at energies between2 . × eV and E cut .The plan of the paper is the following. In Section II, we explain how we model thesources and the propagation of particles. In Section III, we show the goodness of fit ofthe many models we consider. In Section IV we show the average composition and thespectra of some of the models. We conclude in Section V.
2. Modeling of the sources and particle propagation
We use a numerical code originally described in Ref. [19] to compute the flux of GZKphotons produced by a uniform distribution of sources emitting originally only protonsor nuclei. The code uses the kinematic equation approach and calculates the propagationof nuclei, nucleons, stable leptons and photons using the standard dominant processes.This is the same numerical code as in Ref. [20], where the latest version of the code isdescribed in detail.UHE particles lose their energy in interactions with the electromagneticbackground, which consists of CMB, radio, infra-red and optical (IRO) components, aswell as EGMF. Protons are sensitive essentially to the CMB only, while for UHE photonsand nuclei the radio and IRO components are respectively important, besides the CMB.Secondary photons are always subdominant and thus do not contribute significantlyto the fits. Therefore the radio background assumed is not important. For the IRObackground component we used the model of Ref. [21]. This background is importantfor the photodisintegration of nuclei and to transport the energy of secondary photonsin the cascade process from the 0.1 - 100 TeV energy range to the 0.1-100 GeV energyrange observed by EGRET, and the resulting flux in this energy range is not sensitiveto details of the IRO background models. The possible deflection due to extragalacticmagnetic fields is not included in the calculations. These deflections could considerablyextend the path of heavy nuclei below 1 × eV, but we do not consider the propagationof nuclei at these energies.Notice that if neutrons are produced at the sources, the results at high energies arevery close to those obtained with protons. The interactions of neutrons and protons withthe intervening backgrounds are almost identical and when a neutron decays practicallyall of its energy goes to the final proton (while the electron and neutrino are producedwith energies 10 eV or lower).As is usual, we take the spectrum of an individual UHECR source to be of the omposition of UHECR and the Pierre Auger Observatory Spectrum F ( E ) = f E − α Θ( ZE max − E ) , (1)where f provides the flux normalization, α is the spectral index and E max ( ZE max ) isthe maximum energy to which protons (or nuclei with charge Z ) can be accelerated atthe source.We are implicitly assuming that the sources are astrophysical, since these are theonly ones which could produce solely protons (or neutrons) and nuclei as UHECRprimaries. Astrophysical acceleration mechanisms often result in α > ∼ α < ∼ . eV or higher [25]. Here, we consider the power law indexto be in the range 1 ≤ α ≤ .
7. An injected proton spectrum with α ≥ . E < eV [26]. For α ≤ E < × eV. Here we will consider values of E max up to 10 eV.We assume a standard cosmological model with a Hubble constant H =70 km s − Mpc − , a dark energy density (in units of the critical density) Ω Λ = 0 . m = 0 .
3. The total source density in this model can bedefined by n ( z ) = n (1 + z ) m Θ( z max − z )Θ( z − z min ) , (2)where m parameterizes the source density evolution, in such a way that m = 0corresponds to non-evolving sources with constant density per comoving volume, and z min and z max are respectively the redshifts of the closest and most distant sources.The energy of the background photons increases linearly with ( z + 1) thus the GZKenergy, about 3 × eV at z = 0, decreases as 1 / ( z + 1) at redshift z . Moreover, theparticles produced with that energy at redshift z will arrive to us with energy redshiftedas 1 / ( z +1), namely with characteristic energy E = 3 × eV / ( z +1) . This means thatfor z > E < (3 / × eV, and for z > E < (3 / × eV. We conclude thatsources with z > × eVand those with z > × eV. Thus any value of z max ≥ m , i.e. m = 4 , , , − m = 4 and m = 3respectively at z < z = 1 up to z > m = 3 below z = 2 (reaches amaximum at a z between 2 and 3 and then decreases-see Fig. 6 of [28]). Smaller positivevalues of m up to m = 0, correspond to an older star population evolution and is takenhere as a lower limit to the value of m at low redshifts for protons. Negative values of m have been mentioned in the literature only for very massive clusters, which only formed omposition of UHECR and the Pierre Auger Observatory Spectrum z min is connected to the density of sources. Quite often in the literaturethe minimal distance to the sources is assumed to be negligible (i.e. comparable to theinteraction length). We also consider non-zero minimum distances of up to 50 Mpc( z min = 0 . × − Mpc − , which makes plausible the existence of one source within 25 Mpcof us. However, the HiRes negative result on clustering requires a larger density ofsources and, as a result, a smaller distance to the nearest one of them. Larger values ofthe EGMF (as found in Ref. [14]), and/or some fraction of iron in the UHECR, havethe effect of reducing the required number of sources and, consequently, increasing theexpected distance to the nearest one.Most of the energy in GZK photons cascades down to below the pair productionthreshold for photons on the CMB and infrared backgrounds. In general, for α <
3. Goodness of fit of different source models
In this section we estimate the flux predicted by the models by fitting the Pierre AugerUHECR spectrum. We proceed using the method explained in Ref. [36].We fit the Pierre Auger UHECR data assuming many different injected spectra.We assume an injected spectrum given by Eq. 1, a uniform distribution of sources witha density as in Eq. 2 with z max = 3 and, z min = 0 or 0.005 or 0.01 and m = 4 or2 or 0 or -2. We consider then many different spectra resulting from changing theslope α and the maximum energy E max in Eq. 1 within the ranges 1 ≤ α ≤ . eV ≤ E max ≤ . × eV in steps α n = 1 + 0 . n , with n = 0 to 17 and E ℓ = 1 × eV × ℓ , with ℓ = 0 to 7. For each one of the models so obtained wecompute the predicted UHECR spectrum arriving to us from all sources.In order to compare the predicted flux with the data, we also take into accountthe experimental error in the energy determination as proposed in Ref. [37]. We takea lognormal distribution for the error in the energy reconstructed by the experimentwith respect to the true value of energy of the UHECR coming into the atmosphere.To find the expected flux we convolute the spectrum predicted by each model with omposition of UHECR and the Pierre Auger Observatory Spectrum E/E = 8% [9] (the parameter σ in Eq. (5) of Ref. [37], the standard deviation oflog E , is σ = (∆ E/E ) / ln(8) ≃ (∆ E/E ) / . ZE max . Somewhat arbitrarily we consider the energy beyondwhich no event is predicted to be (1 + 10∆ E/E ) ZE max . Moreover, we take into accountthat there is about a factor of 2 between the energy of a photon event and the energymeasured if the event is reconstructed assuming it is a proton [38]. Thus we divide theenergy of the predicted GZK photon energy by 2 before comparing it with the observedPierre Auger spectrum. However, the GZK photons are always subdominant in theflux of UHECR [20, 36] thus they do not affect the goodness of the fits (and at presentthe GZK photon fractions are not constrained by Auger upper bounds- see Fig. 18 ofRef. [39]).With each predicted spectrum we fit the UHECR data from E cut up to a binpast the last published bin of the spectrum (which is the 10 . eV bin of the PierreAuger Observatory). The extra bin extends from the maximum experimental pointof the observed spectrum, 10 . eV [6] (which is also empty) to (1 + 10∆ E/E ) ZE max (where ZE max is the maximum energy assumed for the injected spectrum in Eq. 1).We do this because the assumed injection spectrum could produce an event in thisbin even though the experiment did not observe one. If the maximum possible energy,(1 + 10∆ E/E ) ZE max , is less than the maximum bin of the published spectrum theadditional bin is not needed and therefore not added. In certain assumed injectionspectra the maximum possible energy is less than the energy of the most energeticevent observed. In this case the assumption is not valid on the face of it and thereforeimmediately disqualified. Situations like this can be seen in Fig. 1 and Fig. 2 as theempty regions.We also change E cut and fit the UHECR data from 2 . × eV (only with injectedprotons) or 4 × eV and compare the results with those of 1 × eV. We showhow this affects the goodness of the fit in Fig. 5 using proton sources.The expected number of events in each bin between E cut and the maximum energybin is computed using the exposure of the Pierre Auger Observatory, 5165 km sr yr [6].The aperture remains constant with increasing energy.Fitting the UHECR data with a predicted spectrum follows a procedure similar tothat of Ref. [40] applied to the bins just mentioned. We compare the observed numberof events in each bin with the number of events predicted by the models and choosethe value of the parameter f in Eq. 1, i.e. the amplitude of the injected spectrum, bymaximizing the Poisson likelihood function. This is equivalent to minimizing − λ ,(i.e. the negative of the log likelihood ratio) [41]. This procedure amounts to choosingthe value of f so that the mean total number of events predicted (i.e. the sum of theaverage predicted number of events in all fitted bins) is equal to the total number ofevents observed. We then compute, using a Monte Carlo technique, the goodness of omposition of UHECR and the Pierre Auger Observatory Spectrum p -value of the distribution, defined as the mean fraction of hypotheticalexperiments (observed spectra) with the same fixed total number of events whichwould result in a worse, i.e. smaller, Poisson likelihood than the one obtained (inthe maximization procedure that fixed f ). These hypothetical experiments are chosenat random according to the multinomial distribution of the model (with f fixed asdescribed). We have checked that this procedure when applied to bins with a largenumber of events gives the same result as a Pearson’s χ fit, both for the value of thenormalization parameter f and for the goodness of fit. A higher p -value corresponds toa better fit since a greater number of hypothetical experimental results would yield a fitworse than the one we obtained.We make one additional requirement on the fit to insure the predicted flux doesnot exceed the observed flux at energies below E cut and above 2 . × eV, the lowestenergy of the published Auger spectrum. When E cut > . × eV, for each assumedspectrum (with f fixed as described above) we calculate the χ for the data at energiesbelow E cut using only the data points in which the predicted flux is above the observedflux (i.e. we take as zero the contribution to the χ of each data point for which thepredicted flux is below the observed flux). We then require the p -value of the χ soobtained to be larger than 0.05. This constraint eliminates many combinations of α and E max values. The regions eliminated by this requirement are the cross hatched regionsin Fig. 1, 2, 4 and 5 . This low energy constraint would, however, be too restrictive ifsomehow the extragalactic cosmic rays below some threshold energy between 2 . × eV and E cut do not reach Earth — for example, due to magnetic confinement at thesource. In this case, the deficit of extragalactic flux below the threshold energy shouldbe made up by a (possibly galactic) LEC.Fig. 1 and 2 show in a logarithmic scale the color coded p -value of the maximumPoisson likelihood value obtained for each model as a function of E max and α , for m = 4and m = 0, respectively. The top, middle and lower panels correspond to proton,oxygen, and iron emitted by the sources, respectively, while the columns from left toright correspond to z min = 0, 0.005, 0.01, respectively. Overall, the cross hatched region(in which the flux predicted at energies 2 . × eV < E < E cut exceeds the observedone) includes many regions of E max , α which would otherwise provide good fits (red andorange regions where p -value ≥ . p -value ≥ .
05 to be acceptable).When m = 4, good models with pure proton injection have α = 2 . E max = 10 . − . eV if z min = 0 .
000 or E max = 10 . − . eV if z min = 0 . z min = 0 . E ≤ eV. As m decreases there are relatively more sources near by, thus the initial energiesare less redshifted and the sources contribute less to the spectrum at lower energies.This change is compensated in the models providing good fits by an increase in α (a omposition of UHECR and the Pierre Auger Observatory Spectrum Alpha1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 Figure 1.
Color coded p -value plots as function of E max and α for E cut = 1 × eVand m = 4, for p, O, or Fe emitted at the sources (top to bottom) and z min = 0, 0.005,0.01 (left to right). White regions for p are eliminated because of energetic reasons.Cross hatched regions eliminated by the requirement at 2 . × eV < E < E cut (seetext). Only orange and red regions have p > . steeper initial spectrum). However this leads to too large a flux at energies below E cut and the models are rejected by the low energy constraint. For m = 0, for example, wesee in Fig. 2 that a good fit region allowed by the low energy constraint exists only for α = 2 . E max = 10 . eV and z min = 0. For large E max the proton accumulationbelow the GZK energy increases, thus the normalization of the predicted flux need tobe lower to provide a good fit, this also lowers the flux predicted at low energies and themodel is accepted by the low energy constraint. For lower values of α the p -values ofthe pure proton injection models are low because the predicted flux becomes too low atlow energies, above but close to E cut = 1 × eV. This conclusion could be avoided ifthere was a non-negligible contribution from the LEC still contributing to the spectrumat energies above 1 × eV.No model with pure oxygen injection provides a good fit (outside the cross hatchedregion) with m = 4 because for those models allowed by the low energy constraint, theprotons resulting from the spallation of the O nuclei produce a too large bump in thepredicted spectrum at low energies above but close to E cut . However, for m = 0 a high E max region of good fits is present for pure O injected, with α = 2 . E max > . eVif z min = 0 or 0.05, which disappears for larger distance to the sources. omposition of UHECR and the Pierre Auger Observatory Spectrum Alpha1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 Figure 2.
Same as Fig. 1 but for m = 0. Notice that the best fit regions have shiftedto higher values of α . log10(Energy (eV))18.5 19 19.5 20 20.5 ) e V - s r - s - ( m E · F l u x ICRC07 =2.0, m=0 a Fe, Emax=2E19, =2.2, m=0 a Fe, Emax=6.4E20, =2.2, m=4 a P, Emax=6.4E20, =2.2, m=0 a Fe, Emax=8E19, vs log(E)) Energy Spectrum (dF/dE*E
Figure 3.
Examples of predicted UHECR spectra compared to the Auger data forseveral models providing good fits for p (blue), low E max Fe (red) and high E max Feinjected (green) and a bad fit (the intermediate E max = 8 × eV, α = 2 . m = 0Fe case, in teal). The respective p -values of these models are: 0.119, 0.816, 0.744 and0.0025. Recall that the maximum energy is ZE max . omposition of UHECR and the Pierre Auger Observatory Spectrum Alpha1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6m -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 log10(Energy (eV))18.5 19 19.5 20 20.5 ) e V - s r - s - ( m E · F l u x ICRC07 =1.2 a Fe, Emax=2E19, m=4, =1.6 a Fe, Emax=2E19, m=2, =2.0 a Fe, Emax=2E19, m=0, =2.2 a Fe, Emax=2E19, m=-2, vs log(E)) Energy Spectrum (dF/dE*E
Figure 4.
Degeneracy in m and α for p injection with E max = 6 . × eV (topleft), Fe injection with maximum energy ZE max = 26 × × eV (top right) and26 × . × eV (bottom left). Predicted spectra for best fit cases as function of m for Fe injection with E max = 2 × eV (bottom right). The p -values of the modelslisted are 0.458 ( m = 4), 0.855 ( m = 2), 0.816 ( m = 0) and 0.828 ( m = − Again for m = 4, pure iron injection only provides acceptable models if the sourcesare close by, i.e. z min = 0 if α = 1 . − .
6, and E max = 10 . − . eV. In contrast, m = 0 leads to a larger region of satisfactory fits for iron injection which for z min = 0runs across all the range of E max with α from 1.7 to 2.2. As the minimum distance to thesources increases to 25 Mpc ( z min = 0 .
05) two separate regions of good fits remain for Feinjection: one at high E max and α = 1 . − . E max with α = 1 . − . z min = 0 .
1) the low E max region of good fits disappears and the high E max region shrinks to a single combinationof parameters, E max = 10 . eV, α = 2 . ZE max .Closer sources, z min = 0, always provide better fits, irrespective of the m value, thusin the following figures (Fig. 3 to 6) z min is set to zero. A few examples of the predictedspectra of models which provide good fits (i.e. having p ≥ .
05) or bad fits are shownFig. 3. omposition of UHECR and the Pierre Auger Observatory Spectrum m and α is shown for three of the models providinggood fits, p injected with E max = 6 . × eV (top left), Fe injected with maximumenergy ZE max either 26 × × eV (top right) or 26 × . × eV (bottom left).The figure shows clearly that decreasing m while increasing α yields the same results.The bottom right panel shows the predicted spectra for best fit cases as function of m from − E max = 2 × eV.As we clearly see, requiring that the predicted flux does not exceed the observedflux below E cut and above 2 . × eV (the hatched regions in Figs. 1 to 5) excludesa large number of otherwise good fits. Thus, the caveat we mentioned earlier againstthis constraint is relevant: the constraint would not hold if the extragalactic cosmic rayswith energy below some threshold energy between 2 . × eV and E cut somehow donot reach Earth (are not emitted at the sources).The best fits for proton injection happen for larger values of α (steeper spectrum) incomparison to the best fits for iron injection. As mentioned above, the steeper spectrumfor p injection results in excess flux at low energies, whereas the harder spectrum forFe injection tends to give a deficit of flux at low energies. As m decreases (there arerelatively more sources nearby), in order to get a good fit α must increase to compensatehaving less particles at low energy (close but above E cut ) which means excess flux atenergies E < E cut . This means that only large values of m give acceptable solutions forproton.The best fits for iron and oxygen, on the other hand, are forbidden by the lowenergy constraint due to an excess of flux at E < E cut = 1 × eV due to a bumpconsisting of protons produced by photodisintegration (see the red Fe spectrum examplefor m = 4 in the bottom right panel of Fig. 4). For larger m values, good fits requiresmaller α values which result in a larger flux at the higher energies which also meansmore photodisintegrated protons.So far we have fitted the data above E cut = 1 × eV. In Fig. 5 we explore thechanges in the fits due to different choices of E cut , namely 2 . × eV and 4 × eVbesides 1 × eV for the case of proton sources with m = 4. As mentioned earlier,each E cut is appropriate for different hypotheses for the energy at which the transitionto extra-galactic sources occurs. The effect of E cut on the goodness of fit is shown inFig. 5: the regions with acceptable p -values increase progressively with increasing E cut .This is easily understood, since there are more events per bin at low energies, thus theerror bars are smaller and fewer models provide a good fit for lower E cut .For E cut = 2 . × eV, the point α = 2 . E max = 10 . eV provides the best fitalthough with p < .
05. If the first data bin, at the 10 . eV, is eliminated from the fit,the p -value becomes larger than 0.05. This is because the models with non-negligible p -value for this low E cut have a deficit of flux at the 10 . eV bin, the bin which hasthe smallest error bar. So presumably, if an LEC is added to match the flux exactly atthat first bin, their low goodness of fit could be improved.Fitting the spectrum only above 4 × eV, on the other hand, is easier and modelswith a wide range of α and E max values provide good fits, especially for small values omposition of UHECR and the Pierre Auger Observatory Spectrum Alpha1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 E m ax ( l og ( e V )) -6 -5 -4 -3 -2 -1 log10(Energy (eV))18.5 19 19.5 20 20.5 ) e V - s r - s - ( m E · F l u x ICRC07 =1 a P, Emax=6.4E20, m=4, vs log(E)) Energy Spectrum (dF/dE*E
Figure 5.
Color coded p -value plots for z min = 0 and m = 4 and only protons injectedfor different E cut values: 2 . × eV (top left), 1 × eV (top right), and 4 × eV E cut (bottom left). Example of the predicted flux for E cut = 4 × eV with α = 1, E max = 6 . × eV (bottom right), model with p -value 0.809. of α . Obviously these models require an LEC that makes up for the deficit in the fluxbelow 4 × eV. Even models with α = 1, a very flat spectrum, provide good fits.With such a hard injection spectrum the flux below 4 × eV is well under that ofthe Pierre Auger Observatory, as shown in the bottom right plot of Fig. 5. We can seefrom Fig. 5 that if the LEC is assumed to extend all the way to 4 × eV, almost anycombination of parameters is satisfactory.Although Fig. 5 only shows the case of proton injection with m = 4, the samegeneral consideration apply to nuclei and other values of m as well (although, as wementioned earlier, we cannot extend the fit all the way down to 2 . × eV fornuclei since we do not take magnetic deflections into account). In the following we use E cut = 1 × eV.
4. Composition and predicted spectra
The two extreme best fit cases for iron with maximum energy ZE max = 26 × × eVand ZE max = 26 × . × eV have interesting implications for the composition of theUHECR. As explained above, if m = 4 only the high E max case provides good fits to omposition of UHECR and the Pierre Auger Observatory Spectrum log10(Energy (eV))18.5 19 19.5 20 20.5 ) e V - s r - s - ( m E · F l u x p+nA 2-4A 5-9A 10-19A 20-29A 30-39A 40-49A 50-55Fe ICRC07 =2.0 a Fe, Emax=2E19, m=0, vs log(E)) Energy Spectrum (dF/dE*E log10(Energy (eV))18.5 19 19.5 20 20.5 ) e V - s r - s - ( m E · F l u x p+nA 2-4A 5-9A 10-19A 20-29A 30-39A 40-49A 50-55Fe ICRC07 =2.2 a Fe, Emax=6.4E20, m=0, vs log(E)) Energy Spectrum (dF/dE*E log10(E)18.6 18.8 19 19.2 19.4 19.6 19.8 20 20.2 20.4 A -3 -2 -1 Fractional Flux (A/All) log10(E)18.6 18.8 19 19.2 19.4 19.6 19.8 20 20.2 20.4 A -3 -2 -1 Fractional Flux (A/All)
Energy (log(eV))18.6 18.8 19 19.2 19.4 19.6 19.8 20 20.2 20.4 æ A Æ vs Energy æ A Æ vs Energy æ A Æ Energy (log(eV))18.6 18.8 19 19.2 19.4 19.6 19.8 20 20.2 20.4 æ A Æ vs Energy æ A Æ vs Energy æ A Æ Figure 6.
Detailed breakdown of the UHECR composition from photodisintegrationof pure Fe injected for ZE max = 26 × × eV (left column) and ZE max =26 × . × eV (right column) with m = 0 ( α = 2.0 and 2.2 respectively).Comparison of the flux of different nuclei groups with the Pierre Auger spectrum(top panel) — the black line is the flux of photodisintegrated protons. Fractional fluxfor different nucleus species, divided into 3 groups according to their atomic number A:from 1 to 10, from 11 to 30, and from 30 to 56, color coded according to the fractionalflux (middle panels). The flux is dominated by moderately heavy to heavy nuclei asindicated by the red regions, but for ZE max = 26 × . × eV it still contains asignificant proton flux (middle right). The light mass flux (A = 1–10) is almost allprotons and constitutes about 40% of the total. The average mass as function of energyshown in the bottom panels. omposition of UHECR and the Pierre Auger Observatory Spectrum E max case is dominated by heavy elements with almost a total absence ofprimaries lighter than Boron (atomic number five) throughout the entire energy range.The proton flux from photodisintegration, the endpoint of which is 1/56 of ZE max , endsbelow 10 eV. The high E max case does contain a significant proton fraction even upto the highest energies. The average mass of the composition for the low E max case isupwards of 40 amu (see bottom left in Fig. 6) nearing a flux of pure iron above an energyof 1 × eV. The high E max case has an average mass that varies with a minimumof 30 amu (see bottom right in Fig. 6). The detailed breakdown of the compositionalmakeup of the UHECR flux is the key to distinguishing the two cases (see the twomiddle panels of Fig. 6 where the fraction of light, intermediate and heavy nuclei isrepresented in a color coded logarithmic scale). Indeed, the composition is dominatedby the moderately heavy to heavy nuclei in both cases, but the fraction of protons forthe high E max case is significant and can be upwards of 40% (notice the orange line atA = 1 to 10 for all energies in the middle right plot).The latest data seem to indicate that the average X max , which is characteristic of aflux dominated by light elements below 2 × eV, transitions to a value that reflectsheavier primaries especially above ∼ × eV [17]. If this turns out to be true, itwill be important to identify proton shower candidates in the high energy regime.The two types of Fe injection solutions providing a good fit to the Pierre Augerspectrum were also found, with a different statistical analysis and modeling of predictedspectrum, in a very recent paper (see Ref. [42]) in which also the X max data of Augerare used (they seem to fix m = 0 and z min = 0).Very likely the UHECR sources will accelerate a mixed composition rather than alliron or oxygen or protons. We can think of a scenario, then, where the endpoint of eachnucleus species is Z × eV at the source where Z is the charge of the nucleus. Then,the proton injection would end at a low energy, at 1/26 the maximum iron energy,and the maximum energy for heavy nuclei would not be so high as to result in toomany protons from photodisintegration of heavy nuclei. In this scenario, unless thefraction of Fe (and other heavy elements) injected is very small, we can have a mixedcomposition spectrum that is dominated by heavy elements at the highest energies. If,on the contrary, the endpoint of each nucleus species is high, say Z × eV. Thenpresumably, protons would be the dominant component up to energies close to 10 eV,since hydrogen is the most abundant element. The addition of the significant nucleonfraction from photodisintegrated heavy elements would only serve to strengthen theproton dominance. A very recent paper (Ref. [42]) addressed this issue using injectedmixtures of iron nuclei and protons for an E max close to 4 × eV. They found thatfor these energies a small component of a few percent iron still dominated the spectrumand composition at high energies. omposition of UHECR and the Pierre Auger Observatory Spectrum
5. Conclusions
We have performed an exhaustive scan in the source evolution factor m , the spectralindex α and maximum energy ZE max of the source spectrum and the minimum distanceto the sources z min , for sources emitting only protons, or oxygen or iron nuclei andcompared the total predicted flux at Earth above E cut = 1 × eV with the latestPierre Auger spectrum. We have also imposed the predicted spectrum not to exceedthe observed one at energies below 1 × eV. For an evolution of sources with m = 4,consistent with evolution of AGN, the spectrum agrees with not only pure protoninjection (with α = 2 . E max = 10 . − . eV) but also iron injection with(with α = 1 . − . E max = 10 . − . ) if the sources are not further away than50 and 25 Mpc respectively.For smaller m , in particular m = 0, we find solutions with all injected compositions.The iron injection is particularly interesting in that it has two disparate regions ofhigh significance around ZE max = 26 × × eV (with α = 1 . − .
2) and ZE max = 26 × . × eV (with α = 2 . − .
3) with the intermediate E max casesmuch less favorable (only the high energy solution remains if the sources are very farway). Our results for m = 0 and z min seem to be in agreement with the results ofRef. [42], which appeared while we were finishing writing the present paper.We have also studied the effects of E cut and shown that the regions of parameterspace with good fits depends strongly on it. This is easily understood, since there aremore events per bin at low energies, thus the error bars are smaller and fewer modelsprovide a good fit for lower E cut . Each E cut is appropriate for different hypotheses for theenergy at which the transition to extra-galactic sources occurs. For E cut = 2 . × eVand pure proton injection, corresponding to the “dip model” of Ref. [16] only α = 2 . E max = 10 . eV providingthe best fit (although with p < . E cut have a deficit of flux in the first fitted bin, at the 10 . eV bin, which has the smallesterror bar. So presumably, if an LEC is added to match the flux exactly at that bin,their goodness of fit would improve. Also, if the first bin is eliminated from the fit,the best fit point just mentioned has p ≥ .
05. This disagreement of the fit of the “dipmodel” to the Auger spectrum using surface detector data due to the lowest energy bins,coincides with the recent findings of Berezinky [43] using a different statistical method.For E cut = 4 × eV, good models are found regardless of E max , but a suitable lowenergy component should become important up to energies close to 4 × eV.The spectrum favors a minimum distance to sources, z min , that is as small as possibleand the degeneracy between α and m was also demonstrated.The three models that have the highest probability to describe the observedspectrum paint very different pictures of cosmic ray composition. For the pure protoninjection at the source all UHECR primaries should be protons (possible with someGZK photons), while the two iron injection cases lead to a mixed composition separableby a distinctive abundance of UHECR proton primaries. The low E max Fe injection case omposition of UHECR and the Pierre Auger Observatory Spectrum ∼
90% of the primaries above an energy of 1 × eV are elements withan atomic weight greater than 30 amu, whereas the high E max case contains a fractionof protons only slightly smaller than the total flux of elements with an atomic weightgreater than 30 amu. In both cases the average atomic weight would be consideredheavy.If the hint of a transition from light element dominance to heavy element dominancein the composition of UHECR above 2 × eV seen in the latest results of the PierreAuger Observatory turns out to be true, then the highest energy cosmic rays are likelyto contain a large fraction of heavy elements. Both the low E max case and the high E max case present an intriguing scenario for a mixed composition. Pure iron injection at thesources is unlikely, so if cosmic rays are of mixed composition with maximum energy atthe source of each nucleus species equal to Z × eV, then the low E max case resultsin a composition that becomes heavier with energy until only iron primaries remain.In the high E max case the composition also becomes heavier with energy, but shouldmaintain a significant flux of protons well beyond the GZK energy, coming both fromthe proton injection itself and from the photodisintegration of the heavy elements. Thiscertainly presents an intriguing direction for composition and spectrum studies in thefuture. Acknowledgments
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