The clustering of ultrahigh energy (>10^{20} eV) cosmic rays (UHECR) suggests
that they might be emitted by compact sources. Statistical analysis (Dubovsky
et al., 2000) estimated the source density. We extend their analysis to give
also the confidence intervals (CI) for the source density using a.) no
assumptions on the relationship between clustered and unclustered events; b.)
nontrivial distributions for the source luminosities and energies; c.) the
energy dependence of the propagation. We also determine the probability that a
proton created at a distance r with energy E arrives at earth above a threshold
E_c. Using this function one can determine the observed spectrum just by one
numerical integration for any injection spectrum. The observed 14 UHECR events
above 10^{20} eV with one doublet gives for the source densities
180_{-165}^{+2730}\cdot 10^{-3} Mpc^{-3} (on the 68% confidence level).Comment: 4 pages, 1 figure, talk to be presented at the 27th International
  Cosmic Ray Conference, Hamburg, Germany, August 7-15, 200

Fodor, Z.

Katz, S. D.

English

arXiv

Fodor, Z

Katz, SD

ELTE Digital Institutional Repository (EDIT)

arXiv:hep-ph/0105347v1  31 May 2001Propagation of ultrahigh energy cosmic rays and compact sourcesZ. Fodor and S.D. KatzInstitute for Theoretical Physics, Eötvös University,Pázmány 1, H-1117 Budapest, HungaryOctober 26, 2018AbstractThe clustering of ultrahigh energy (> 1020 eV) cosmic rays(UHECR) suggests that they might be emitted by compactsources. Statistical analysis (Dubovsky et al., 2000) esti-mated the source density. We extend their analysis to givealso the confidence intervals (CI) for the source density us-ing a.) no assumptions on the relationship between clusteredand unclustered events; b.) nontrivial distributions for thesource luminosities and energies; c.) the energy dependenceof the propagation. We also determine the probability that aproton created at a distancer with energyE arrives at earthabove a thresholdEc. Using this function one can deter-mine the observed spectrum just by one numerical integra-tion for any injection spectrum. The observed 14 UHECRevents above1020 eV with one doublet gives for the sourcedensities180+2730−165 · 10−3 Mpc−3 (on the 68% confidencelevel).1 INTRODUCTIONThe interaction of protons with photons of the cosmic mi-crowave background predicts a sharp drop in the cosmic rayflux above the GZK cutoff around5 ·1019 eV (Greisen, 1966;Zatsepin and Kuzmin, 1966). The available data shows nosuch drop. About 20 events above1020 eV were observedby a number of experiments such as AGASA (Takeda et al.,1998), Fly’s Eye (Bird et al., 1993), Haverah Park (Lawrenceet al., 1991), Yakutsk (Efimov et al., 1991) and HiRes (Kiedaet al., 1999). Future experiments, particularly Pierre Auger(Boratav, 1996; Guerard, 1999; Bertou et al. 2000), will havea much higher statistics. Since above the GZK energy theattenuation length of particles is a few tens of megaparsecs(Yoshida and Teshima, 1993 ; Aharonian and Cronin, 1994;Protheroe and Johnson, 1996; Bhattacharjee and Sigl, 2000;Achterberg et al., 1999; T. Stanev et al., 2000) if an ultra-high energy cosmic ray (UHECR) is observed on earth it isusually assumed that it is produced in our vicinity.At these high energies the galactic and extragalactic mag-netic fields do not affect the orbit of the cosmic rays, thusthey should point back to their origin within a few degrees.In contrast to the low energy cosmic rays one can use UHE-CRs for point-source search astronomy. Though there aresome peculiar clustered events, which we discuss in detail,the overall distribution of UHECRs on the sky is practicallyisotropic. This observation is rather surprising since in pri -ciple only a few astrophysical sites are capable to acceleratsuch particles.There are several ways to look for the source inhomogene-ity from the energy spectrum and spatial directions of UHE-CRs. One possibility is to assume that the source density ofUHECRs is proportional to the galaxy densities (Waxman etal., 1997; Giller et al., 1980; Hill and Schramm, 1985); orone can analyze the clustering of the unknown sources bysome correlation length (Bahcall and Waxman 2000).Clearly, the arrival directions of the UHECRs measuredby experiments show some peculiar clustering: some eventsare grouped within∼ 3o, the typical angular resolution ofan experiment. Above4 · 1019 eV 92 cosmic ray eventswere detected, including 7 doublets and 2 triplets. Above1020 eV one doublet out of 14 events were found (Uchihoriet al., 2000). The chance probability of such a clusteringfrom uniform distribution is rather small (Uchihori et al.,2000; Hayashida et al., 1996; Tinyakov and Tkachev 2001a,Tinyakov and Tkachev 2001b,).The clustered features of the events initiated an interest-ing statistical analysis assuming compact UHECR sources(Dubovsky et al., 2000). The authors found a large number,∼ 400 for the number of sources inside a GZK sphere of25 Mpc. We extend their analysis to give also the CIs for thesource density using a.) no assumptions on the relationshipbetween clustered and unclustered events; b.) nontrivial dis-tributions for the source luminosities and energies; c.) theenergy dependence of the propagation. We also determinethe probability that a proton created at a distancer with en-ergyE arrives at earth above a thresholdEc.As we show the most probable value for the source densityis really large; however, the statistical significance of this re-sult is rather weak. At present the small number of UHECRevents allows a 95% CI for the source density which spreadsover four orders of magnitude. Since future experiments, par-ticularly Pierre Auger, will have a much higher significanceon clustering (the expected number of events of1020 eV andabove is 60 per year, we present give results also for largernumber of UHECRs above1020 eV.In order to avoid the assumptions of (Dubovsky et al.,2000) a combined analytical and Monte-Carlo technique willbe presented adopting the conventional picture of protons as1the ultrahigh energy cosmic rays. Our analytical approachof Section 2 gives the event clustering probabilities for anyspace, luminosity and energy distribution by using a singleadditional functionP (r, E;Ec), the probability that a protoncreated at a distancer with energyE arrives at earth abovethe threshold energyEc (Bahcall and Waxman, 2000). Withour Monte-Carlo technique of Section 3 we determine theprobability functionP (r, E;Ec) for a wide range of parame-ters. Our results for the present and future UHECR statisticsare presented in Section 4. We summarize in Section 5.2 ANALYTICAL APPROACHThe number of UHECRs emitted by a source ofλ lumi-nosity during a periodT follows the Poisson distribution.However, not all emitted UHECRs will be detected. Theymight loose their energy during propagation or can simplygo to the wrong direction. For UHECRs the energy lossis dominated by the pion production in interaction with thecosmic microwave background radiation. In a recent anal-ysis (Bahcall and Waxman, 2000) the probability functionP (r, E,Ec) was presented for three specific threshold ener-gies. This function gives the probability that a proton createdat a given distance from earth (r) with some energy (E) isdetected at earth above some energy threshold (Ec).The features of the Poisson distribution enforce us to takeinto account the fact that the sky is not isotropically observed.The probability of detectingk events from a source at dis-tancer with energyE can be obtained by simply includingthe factorP (r, E,Ec)Aη/(4πr2) in the Poisson distribution:pk(x, E, j) =exp[−P (r, E,Ec)ηj/r2]k!×[P (r, E,Ec)ηj/r2]k, (1)where we introducedj = λTA/(4π) andAη/(4πr2) is theprobability that an emitted UHECR points to a detector ofareaA. The factorη represents the visibility of the source,which was determined by spherical astronomy. We denotethe space, energy and luminosity distributions of the sourcesby ρ(x), c(E) andh(j), respectively. The probability of de-tectingk events above the thresholdEc from a single sourcerandomly positioned within a sphere of radiusR isPk =∫SRdV ρ(x)∫∞EcdE c(E)∫∞0dj h(j)×exp[−P (r, E,Ec)ηj/r2]k![P (r, E,Ec)ηj/r2]k. (2)Denote the total number of sources within the sphere ofsufficiently large radius (e.g. several times the GZK radius)by N and the number of sources that gavek detected eventsby Nk. Clearly,N =∑∞0 Ni and the total number of de-tected events isNe =∑∞0 iNi. The probability that forNsources the number of different detected multiplets areNkis:P (N, {Nk}) = N !∞∏k=01Nk!PNkk. (3)For a given set of unclustered and clustered events (N1 andN2, N3,...) inverting theP (N, {Nk}) distribution functiongives the most probable value for the number of sources andalso the CI for it.Note, thatPk and thenP (N, {Nk}) are easily determinedby a well behaving four-dimensional numerical integrationfor anyc(E), h(j) andρ(r) distribution functions. In orderto illustrate the uncertainties and sensitivities of the resultswe used a few different choices for these distribution func-tions.Forc(E) we studied three possibilities. The most straight-forward choice is the extrapolation of the ‘conventional highenergy component’∝ E−2. Another possibility is to use astronger fall-off of the spectrum at energies just below theGZK cutoff, e.g. ∝ E−3. The third possibility is to as-sume that UHECRs are some decay products of metastablesuperheavy particles (Berezinsky et al., 1997; Kuzmin andRubakov, 1998; Birkel and Sarkar, 1998; Sarkar 2000; Fodorand Katz 2001b). According to (Birkel and Sarkar, 1998)the superheavy particles decay into quarks and gluons whichinitiate multi-hadron cascades through gluon bremstrahlung.In the recent analysis (Dubovsky et al., 2000) the authorshave shown that for a fixed set of multiplets the minimal den-sity of sources can be obtained by assuming a delta-functiondistribution for h(j). We studied both this limiting lumi-nosity: h(j) = δ(j − j∗) and a more realistic one withSchechter’s luminosity function (Schechter 1976), which canbe given as:h(j)dj = h · (j/j∗)−1.25 exp(−j/j∗)d(j/j∗).The space distribution of sources can be given based onsome particular survey of the distribution of nearby galax-ies or on a correlation lengthr0 characterizing the clusteringfeatures of sources. For simplicity the present analysis dealswith a homogeneous distribution of sources.3 MONTE-CARLO STUDY OF THEPROPAGATIONIn our Monte-Carlo approach we determined the propaga-tion of UHECR protons on an event by event basis. The in-elasticity of Bethe-Heitler pair production is small (≈ 10−3),thus we used a continuous energy loss approximation for thisprocess. The inelasticity of pion-photoproduction is larger(≈ 0.2 − 0.5) in the energy range of interest, thus there areonly a few tens of such interactions during the propagation.Due to the Poisson statistics and the spread of the inelastic-ity, we will see a spread in the energy spectrum even if theinjected spectrum is mono-energetic.In our simulation protons are propagated in small steps(10 kpc), and after each step the energy losses due to pair pro-duction, pion production and the adiabatic expansion are cal-culated. During the simulation we keep track of the current2energy of the proton and its total displacement. For the pro-ton interaction lengths and inelasticities we used the valuesof (Bhattacharjee and Sigl, 2000; Achterberg et al., 1999).The deflection due to magnetic field is not taken into account(for a recent Monte-Carlo on it see eg. Stanev et al., 2000).To cover a broad energy range we used the parametrizationP (r, E,Ec) = exp[−a · (r/1 Mpc)b]. (4)Fig. 1 shows the functionsa(E/Ec) andb(E/Ec) for a rangeof three orders of magnitude and for five different thresholdenergies. Just using the functions ofa(E/Ec) andb(E/Ec),thus a parametrization ofP (r, E,Ec) one can obtain the ob-served energy spectrum for any injection spectrum withoutadditional Monte-Carlo simulation.4 RESULTSIn order to determine the CIs for the source densities we usedthe frequentist method (Groom et al., 2000). We wish to setlimits on S, the source density. Using our Monte-Carlo basedP (r, E,Ec) functions and our analytical technique we deter-minedp(N1, N2, N3, ...;S; j∗), which gives the probabilityof observingN1 singlet,N2 doublet,N3 triplet etc. events ifthe true value of the density isS and the central value of lu-minosity isj∗. For a given set of{Ni, i = 1, 2, ...} the aboveprobability distribution as a function ofS andj∗ determinesthe 68% and 95% confidence level regions in theS−j∗ plane.For different choices ofc(E) andh(j) see Table 1 for the cal-culated densities. Our results for the Dirac-delta luminositydistribution are in agreement with the result of (Dubovsky etal., 2000) within the error bars. Neverthless, there is a veryimportant message. The CIs are are so large that on the 95%confidence level two orders of magnitude smaller densitiesthan suggested as a lower bound by (Dubovsky et al., 2000)are also possible.It is of particular interest to study higher statistics too,anddetermine CIs for these cases. We performed a detailed anal-ysis on this question (Fodor and Katz 2001a). An interestingfeature of the results is that ”doubling” the present statisticswith the same clustering features (in the case studied by thetable this means one new doublet out of 10 new events) re-duces the CIs by an order of magnitude. The study of evenhigher statistics leads to the conclusion that experimentsinthe near future with approximately 200 UHECR events cantell at least the order of magnitude of the source density.5 SUMMARYWe presented a technique in order to statistically analyze theclustering features of UHECR. The technique can be appliedfor any model of UHECR assuming small deflection. Thekey role of the analysis is played by thePk functions definedby eqn. (2), which is the probability of detectingk eventsabove the threshold from a single source. Using a combina-torial expression of eqn. (3) the probability distributionforFigure 1: The functionsa(E/Ec) –left panel– andb(E/Ec) –right panel– for the probability distributionfunction P (r, E,Ec) using the parametrizationexp[−a ·(r/1 Mpc)b] for five different threshold energies (5 ·1019 eV,1020 eV, 2 · 1020 eV, 5 · 1020 eV and1021 eV).3c(E) h(j) 14 events 1 doublet∝ E−2 ∝ δ 2.77+96.1(916)−2.53(2.70)∝ E−2 ∝ SLF 36.6+844(4268)−34.3(35.9)∝ E−3 ∝ δ 5.37+80.2(624)−4.98(5.25)∝ E−3 ∝ SLF 180+2730(8817)−165(174)∝ decay ∝ δ 3.61+116(1060)−3.30(3.51)∝ decay ∝ SLF 40.9+856(4345)−38.3(40.1)Table 1: The most probable values for the source densitiesand their error bars given by the 68% and 95% confidencelevel regions (the latter in parenthesis). The numbers are inunits of 10−3 Mpc−3 The three possible energy spectrumsare given by a distribution proportional toE−2, E−3, or bythe decay of a1012 GeV particle (denoted by “decay”). Theluminosity distribution can be proportional to a Dirac-deltaor to Schechter’s luminosity function (denoted by “SLF”).any set of multiplets can be given as a function of the sourcedensity.We discussed several types of energy and luminosity dis-tributions for the sources and gave the most probable sourcedensities with their CIs for present and future experiments.The probabilityP (r, E,Ec) that a proton created at a dis-tancer with energyE arrives above the thresholdEc (Bah-call and Waxman, 2000) is determined and parametrized fora wide range of threshold energies. This result can be usedto obtain the observed energy spectrum of the UHECR forarbitrary injection spectrum.In ref. (Dubovsky et al., 2000) the authors analyzed thestatistical features of clustering of UHECR, which providedconstraints on astrophysical models of UHECR if the num-ber of clusters is small, by giving a bound from below. Inour paper we have shown that there is some constraint, butit is far from being tight. At present statistics the 95% CIsusually span 4 orders of magnitude. Two orders of magni-tude smaller numbers than the prediction of (Dubovsky etal., 2000) (their eqn. (13) suggests for the density of sources∼ 6 · 10−3 Mpc−3) can also be obtained. Adding 10 newevents with an additional doublet the CI can be reduced to 3orders of magnitude and the increase of the UHECR eventsto 200 can tell at least the order of magnitude of the sourcedensity.More details of the present analysis can be found in (Fodorand Katz, 2001a). Note, that a similar study based on the Z-burst scenario (Fargion et al., 1999; Weiler, 1999) can be car-ried out which gives the mass of the heaviest neutrino (Fodoret al., 2001).6 ACKNOWLEDGEMENTSWe thank K. Petrovay for clarifying some issues in sphericalastronomy. 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