PHYSICAL REVIEW D 72, 081301(R) (2005)

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On the shape of the ultrahigh energy cosmic ray spectrum

Daniel De Marco and Todor StanevBartol Research Institute, University of Delaware, Newark, Delaware 19716, USA

(Received 21 June 2005; revised manuscript received 26 July 2005; published 13 October 2005)

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We fit the ultrahigh energy cosmic ray spectra above 1019 eV with different injection spectra at cosmicray sources that are uniformly and homogeneously distributed in the Universe. We conclude that thecurrent ultrahigh energy spectra are consistent with power laws of index � between 2.4 and 2.7. There is aslow dependence of these indices on the cosmological evolution of the cosmic ray sources, which in thismodel determines the end of the galactic cosmic rays spectrum.

DOI: 10.1103/PhysRevD.72.081301 PACS numbers: 98.70.Sa, 13.85.Tp, 98.80.Es

I. INTRODUCTION

The current results on the energy spectrum of the highestenergy cosmic rays are not fully consistent. The two highstatistics experiments, AGASA [1] and HiRes [2], do notagree on the normalization of the ultrahigh energy cosmicray (UHECR) spectrum. In addition, HiRes results areconsistent with a Greisen-Zatsepin-Kuzmin (GZK) [3]suppression from proton photo-pion interactions on themicrowave background, and AGASA claims a spectrumextended to higher energy. With the current statistics thedifferences are not very significant—the number of eventsabove 1020 eV differs by less than 3� [4]. The normal-izations of the spectra are quite different, but a renormal-ization of the energy assignment by 15%–20%, which iswithin the reported systematic uncertainty, of both data setsleads to a good agreement [4,5]. The high-energy extensionand exact normalization of the UHECR spectrum are thusnot well known, but after the renormalization both experi-ments show the same spectral shape between 1018:5 and1020 eV.

There have been recently several attempts [6–13] toexplain this shape with different injection spectra ofextra-galactic protons after propagation to the observerfrom isotropically and homogeneously distributed sources.The assumed injection spectra and to a certain extent thecosmological evolution of the sources determine the shapeof the extra-galactic cosmic ray spectrum at Earth. Asubtraction from the observed cosmic ray spectrum in thesemodels determines also the end of the galactic cosmic rayspectrum. There are two types of solutions. Flat injectionspectra, dN=dE � AE�2, are suggested in Refs. [7,8]. Inthis case the galactic cosmic rays spectrum extends above1019 eV. The other popular solution is to use much steeperinjection spectra with spectral indices � � 2:6–2:7. Suchsolutions set the end of the galactic cosmic ray spectrum atlower energy. The assumption that extra-galactic cosmicrays have a large fraction of heavy nuclei [14] modifies thepropagation process and can potentially initiate a new classof models, in which the composition at injection introducesmore model parameters.

Cosmic ray data is not yet good enough to prove that anyof these models are correct. Fitting the spectra with several

05=72(8)=081301(5)$23.00 081301

different parameters and the uncertain knowledge of thecosmic ray composition at 1019 eV makes all modelsplausible.

We describe an attempt to fit the renormalized AGASAand HiRes spectra with injection spectra and cosmologicalevolution parameters covering practically the whole phasespace and to use the quality of the fits as a measure of theplausibility of the models. This involves several assump-tions that, although used in many previous publications,may not be correct. The main one is that extra-galacticcosmic rays are protons and that the renormalized experi-mental spectra represent correctly the shape of the UHECRspectrum. One of the fit parameters is the required extra-galactic cosmic ray luminosity at present time. The uncer-tainty of the luminosity depends on the arbitrary renormal-ization of the experimental data, as well as on the crosscorrelation with other parameters and assumptions.

This paper is organized as follows: Sec. II describes thefitting procedures that we use, Sec. III gives the mainresults of the fits, and Sec. IV discusses the results, com-pares to other fits, and derives the main conclusions fromthis research.

II. FITTING THE UHECR SPECTRUM

It was shown in Refs. [4,5] that a renormalization ofabout 15% of the energy assignment of the AGASA andHiRes events would bring the two spectra in very goodagreement in the energy region below 1020 eV in bothnormalization and shape. In Ref. [4] it was also shownthat the statistics of events above 1020 eV is too small toachieve a conclusive result about the end of the UHECRspectrum. In this paper we study how the best fit to theinjection spectrum depends on the source parameters: in-jection spectrum, luminosity, and luminosity evolutionwith redshift. To do so, we fit these injection parametersto the energy-shifted spectra in the energy range1019–1020 eV. We chose as a lower bound 1019 eV becausewe expect at this energy the particles to be mostly extra-galactic and 1020 eV as a higher bound because of thesparseness of data above this threshold.

To calculate the expected spectrum from a isotropichomogeneous distribution of proton sources we use the

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DANIEL DE MARCO AND TODOR STANEV PHYSICAL REVIEW D 72, 081301 (2005)

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analytical approach presented in Refs. [6,15]. In this ap-proach all proton energy losses are included (redshiftlosses, pair and pion production losses) and they are alltreated as continuous. This is the correct treatment forredshift losses and it is well suited for pair productionand for pion production at large propagation distances.At small propagation distance, however, the large inelas-ticity of the pion production process produces large fluc-tuations in the expected fluxes that cannot be reproduced inthe continuous energy loss approximation. A Monte Carlosimulation is better suited in this case. In the present paper,since pion production affects mostly the highest part of theenergy spectrum, and we are only interested in the spectrabelow 1020 eV, we can safely use the continuous energyloss approximation for all energy loss processes.

We assume the sources inject protons with a power-lawspectrum, E��, with a sharp cutoff at Emax � 1021:5 eV.Changing the value of Emax does not appreciably affect theresults in the energy region we are interested in. Weassume the source emissivity to evolve as L�z� � L0�1�z�m, where L0 is the present cosmic ray emissivity of thesources and m � 0 corresponds to absence of evolution.There is no cutoff to the evolution of the luminositybecause we are only interested in the region above1018:5 eV and in this region the contributions to the ob-served flux can come only from sources with z & 0:5. Tocalculate the energy losses we use the loss lengths ofRef. [6], plotted in Fig. 1, that were shown to be indistin-guishable from the ones of Ref. [16] in the low-energyregion and within 15% at high energy [6]. We assume a�CDM universe with �� � 0:7, �m � 0:3, and H0 �75 km s�1 Mpc�1.

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FIG. 1. Solid line: loss length for photo-pion and photo-pairproduction for protons as calculated in Ref. [6]. The dashed linesreport the separate contribution of the two processes. The dottedline shows the loss length for redshift losses. The squaresrepresent the loss length of Ref. [16].

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We explore the parameter region in � from 2.05 to 3.00in steps of 0.05 and in m from 0 to 4 in steps of 0.25. Foreach ��;m� pair we calculate the expected flux and then thebest fit emissivity, L0, minimizing the �2 indicator. Toemulate the experimental energy resolution we include30% Gaussian error distribution in the spectra after propa-gation. The effect of the energy resolution is to smooth thefeatures produced by the propagation on the photon back-ground and to lessen the GZK suppression [4].

For AGASA and Akeno we use the data from Ref. [1],whereas for HiRes 1&2 we use the data from their website[17] which is very close to the published results [2,12].Following the suggestion of Ref. [4] we shift the AGASAand HiRes energies, respectively, by �15% and �15%,while we leave the Akeno energies unchanged [18]. To dothe shift we proceed in the following way: we calculate

dNdlog10�E�

�E� and we assign this flux, calculated in E, to the

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FIG. 2 (color online). Upper panel: Best fits to the AGASA�15% data set in the 1019–1020 eV energy range. Squares witherror bars: HiRes data set. Circles with error bars:AGASA&Akeno data set. Filled symbols: points used in thefit. Smaller data points: data from HiRes2 and from Akeno (notshifted) to show the low-energy shape of the spectrum. Solidlines: best fits for different values of m. Dotted lines: galacticcomponent needed in order to fit the spectrum at lower energy.The numbers attached to the lines indicate the value of m. Lowerpanel: best fit for the HiRes �15% data set.

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ON THE SHAPE OF THE ULTRAHIGH ENERGY COSMIC . . . PHYSICAL REVIEW D 72, 081301 (2005)

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energy kE. This means that �dNdE �kE��new � 1

k �dNdE �E��

old or�k3E3 dN

dE �kE��new � k2�E3 dN

dE �E��old, where k � 1:15 for

HiRes 1&2 and k � 0:85 for AGASA. These new, shifteddata sets agree quite well almost over the whole energyrange as shown in Fig. 2.

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FIG. 3 (color online). Best fits to the HiRes �15% data set inthe 1018:5–1020 eV energy range. Symbols are the same as inFig. 2.

III. RESULTS FROM THE FITS

We applied the method presented in the previous para-graph to the AGASA �15% data fitting the points in theenergy range 1019–1020 eV. The results are shown in theupper panel of Fig. 2 where we plot the best fits for m �0; 3; 4 (solid lines). We only show results for m � 3 and 4because these values bracket the cosmological evolutionderived from star forming regions and from gamma raybursts. The corresponding slopes are, respectively, � �2:55; 2:45; 2:45 with 1� errors of about 0.20. As it is clearfrom the plot all three curves fit well the data in the regionconsidered, with different degrees of goodness at lowenergy. The dotted lines represent the needed galacticcomponent to fit the spectrum. The best fit with m � 4does not allow for a galactic component above 1017 eV.

We repeated the same exercise for the HiRes�15% dataset (lower panel of Fig. 2) and the results were similar. Theslopes are somewhat steeper, by about 0.05–0.1, wellwithin the similarly large uncertainties. The main differ-ence is at energies much lower than the fitting range, wherethe fits of the HiRes data set do not allow for a galacticcomponent in cases with cosmological evolution. The fluxof extra-galactic cosmic rays below 1019 eV has to beslightly decreased by some additional process in ordernot to exceed the Akeno and HiRes measurements. Theshape of the spectra in the considered region is, however,the same for both experiments.

It has to be noted that the inclusion of the error distri-bution in the fit affects the spectral shape—the pile-upapproaching 1020 eV is visibly smoother. Since the pointsimmediately above 1019 eV have the lowest error bars, andthus affect the fit the most, the slope of the spectrum isincreased by at most 0.05, much smaller than the uncer-tainties from the fits.

The best fitting parameters depend slowly on the fittedenergy range. Figure 3 shows the fit of the HiRes �15%data set for the energy range between the 1018:5–1020 eVrange. The best fit with m � 4 again does not leave spacefor a galactic component below 1018 eV. The best spectralindices are 2.6, 2.5, and 2.5, respectively, for m � 0, 3, and4, slightly smaller than those for the higher fitting thresholdand almost identical to the AGASA �15% set shown inFig. 2. The 1� error bars decrease to about�0:1. The effectof the wider fitting range on the AGASA �15% spectrumis similarly small, although for this data set it increases thespectral slopes for all m values by 0.05–0.1. The 1� fiterrors decrease to slightly less than �0:1.

As a consistency check we also fitted the unmodifiedAGASA and HiRes spectra. The results we obtain are

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much like the ones presented above. The best fit parametersdiffer by about 0.05–0.1, with the same �0:2 error bars.

We performed several other fits varying the fittingthreshold between 1018:5 and 1019:2 eV and convincedourselves that all fits returned consistent results withinthe 1� errors of the presented fits as shown in Fig. 4.

In this figure we plot in the top panel log10�2 as a

function of ��;m� for the AGASA �15% fit above1019 eV, i.e. for 6 degrees of freedom. The analogousfitting of the HiRes �15% set is shown in the bottompanel. The white contours are the confidence bands for1�, 2�, and 3�. All m values can provide a good fit to thedata as the best fit� value slowly decreases with increasingm. This correlation is easily understood; with a smallervalue of �, less low-energy particles are injected and tocompensate for that one needs a stronger evolution of thesources to increase the number of low-energy particlesreaching the observer. For the HiRes �15% data set thebest fit parameters are in the strip connecting �� �2:6; m � 0� and �� � 2:4; m � 4�. As it is clear from theconfidence bands in the plot, the present data sets do notrestrict very much the values of the parameters, � beingdetermined with an uncertainty of �0:2 for a given m andm being almost free for a given �. It is obvious, though,that fits with a flat injection spectrum do not give good �2

values even with a strong cosmological evolution. Injectionspectrum with � � 2:0 would be in the 3� range only inthe case of m � 4. Flat injection spectrum models requirethat the galactic cosmic ray spectrum extends to 1019:5 eV.

If the shape of the cosmic ray spectrum is the same as theone derived from the existing experimental statistics, evenmuch higher future statistics from the Auger observatory[19] would not help to solve it. We performed a fit with thecurrent spectral shape and increased statistics that corre-sponds to the one expected from Auger. The 1� errors on�for the fits above 1019 eV became only slightly smaller�0:15. The measurement of the cosmic ray chemicalcomposition, or, a measurement of the flux of cosmogenic

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FIG. 5 (color online). Required emissivity as a function of��;m�. Lines: best fit emissivity (above 1019 eV) as a functionof �. The numbers attached to the lines give the value of m.Every line is highlighted with different thicknesses correspond-ing to the confidence bands for 1�, 2�, and 3�. The blacksquares are the best fits obtained in other works. W&B corre-sponds to Refs. [7,21]; BGG corresponds to Ref. [6].

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DANIEL DE MARCO AND TODOR STANEV PHYSICAL REVIEW D 72, 081301 (2005)

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neutrinos generated by UHECR in propagation to us [20],is needed to disentangle the two parameters.

In Fig. 5 we plot the best fit present day emissivitiesabove 1019 eV as a function of � for different values of m.In this plot we also show the values obtained inRefs. [6,7,21]. The differences with the results of Ref. [6]are likely due to the slightly different data set, to thedifferent range of data used for the fits, and to the inclusionin our calculation of the experimental energy resolution.There is also a factor because the fits were performed witha �10% shift of the AGASA data set instead of the �15%shift used here. It is interesting to note that the requiredvalues of the emissivity above 1019 eV cover a narrowrange between 2 and 6 � 1044 erg Mpc�3 yr�1 and thatthe required luminosity increases with the flattening ofthe injection spectrum. This is a consequence of the factthat we only present the luminosity required above1019 eV. If we were to extend the energy spectrum to lowerenergy, say to 1017 eV, we would observe exactly theopposite trend—steeper injection spectra would requiremuch higher luminosity than flatter ones.

IV. DISCUSSION AND CONCLUSIONS

After fitting the shifted AGASA and HiRes data sets interms of injection spectral index and cosmological evolu-tion of the cosmic ray sources for an isotropic and homo-geneous source distribution we obtained currentemissivities above 1019 eV that differ only by about afactor of 2. In this sense we confirm the statement ofWaxman [21] that approximately the same emissivity isrequired for a wide range on injection spectral indices. We

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disagree with the estimate of the central spectral index inRef. [21] and find a significantly steeper one.

Best fit spectral indices are, however, not well restrictedby current statistics. In Fig. 6 we show with a shaded areathe 1� errors of the best fit prediction from the AGASAdata above 1019 eV (top panel of Fig. 2) for m � 0. Thefigure emphasizes the perils of all fits of the extra-galacticcosmic ray component with the current statistics. Such fitsare most sensitive to, and attracted by, a small number ofexperimental points with the best statistics, in our case fourpoints between 1019 and 1019:4 eV. Fitting uncertainties donot affect much the higher energy spectra where the GZKsuppression prevails almost independently of the injectionspectral index but create a large uncertainty below1018:5 eV. This uncertainty makes estimates of the end ofthe galactic cosmic ray spectrum by subtraction of themodel predictions from the total observed flux unreliable.

The luminosities that we show in Fig. 5 apply only toenergies above 1019 eV. If one is interested in the totalcosmic ray luminosity of the cosmic ray sources oneshould continue the integration to much lower energies.This introduces several possible new astrophysical pa-rameters that come from the exact acceleration mechanismof the extra-galactic cosmic rays. One could integratedown to the proton mass and obtain the highest possibleemissivity. On the other hand, studies of particle accelera-tion at relativistic shocks [22] find a minimum accelerationenergy of m�2

shock which for �shock � 1000, as in gammaray bursts, could be 1015 eV and would decrease signifi-cantly the required emissivity. Modifications of the cosmicray spectrum on propagation, such as suggested inRefs. [10,11,23], because of magnetic horizon of thehigh-energy cosmic rays [16] would not change the re-quired emissivity. Such modifications would howeverchange very much the extra-galactic cosmic ray spectrum

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FIG. 6 (color online). One � error band (shaded) of theAGASA fit with m � 0 above 1019 eV.

ON THE SHAPE OF THE ULTRAHIGH ENERGY COSMIC . . . PHYSICAL REVIEW D 72, 081301 (2005)

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suppressing the flux at the lower energy end and by con-sequence changing the shape of the end of the galacticcosmic ray spectrum required in order to fit theobservations.

Fits of the monocular HiRes data have been performedby the HiRes group [24]. The best fit is obtained for � �2:38� 0:04 andm � 2:8� 0:3 for a total of 42 data pointsabove 1017 eV and, respectively, 39 degrees of freedom. Inaddition to the different energy range of the fit, HiResassumes a ‘‘toy’’ galactic cosmic ray model based on theircomposition measurement [25] that suggests dominationof the extra-galactic cosmic rays above 1018 eV.

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The HiRes fit is probably dominated by lower energycosmic rays (1017:5–18:5 eV) with much smaller error bars.The two fits give different central values for � and m butthey are qualitatively consistent in the conclusion that evenwith a strong cosmological evolution of the cosmic raysources the observed spectra do not support flat injectionspectra.

We have fitted the shape of the ultrahigh energy cosmicray spectrum above 1019 eV assuming that these cosmicrays are protons, and that the sources of these protons areuniformly and homogeneously distributed in the Universe.The fits of the scaled AGASA and HiRes data sets allow forpower-law injection spectra in the range AE��2:4–2:7� forcosmological evolution of the cosmic ray sources between�1� z�4 and �1� z�0. The cosmic ray emissivities above1019 eV required by different models are within about afactor of 2 in this range. The best fit spectral index de-creases for strong evolution models. Flatter injection spec-tra do not fit well the cosmic ray spectra above 1019 eV.This also means that the end of the galactic cosmic rayspectrum is at, or below, 1018 eV depending on the cos-mological evolution of the extra-galactic cosmic raysources. Consistent data on the cosmic ray compositionin the energy range above 1017 eV are required in order toreveal the end of the galactic cosmic ray spectrum and thushelp determine that of the extra-galactic sources.

ACKNOWLEDGMENTS

We thank D. Seckel for useful discussions. This researchis funded in part by NASA APT Grant No. NNG04GK86G.

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