Cosmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV
aa r X i v : . [ a s t r o - ph . H E ] S e p Cosmogenic Neutrinos:parameter space and detectabilty from PeV to ZeV
K. Kotera , D. Allard and A. V. Olinto Department of Astronomy & Astrophysics, Enrico Fermi Institute, and KavliInstitute for Cosmological Physics, The University of Chicago, Chicago, Illinois60637, USA. Laboratoire Astroparticules et Cosmologie (APC), Universit´e Paris 7/CNRS, 10 rueA. Domon et L. Duquet, 75205 Paris Cedex 13, France.E-mail: [email protected]
Abstract.
While propagating from their source to the observer, ultrahigh energycosmic rays interact with cosmological photon backgrounds and generate to the so-called “cosmogenic neutrinos”. Here we study the parameter space of the cosmogenicneutrino flux given recent cosmic ray data and updates on plausible source evolutionmodels. The shape and normalization of the cosmogenic neutrino flux are verysensitive to some of the current unknowns of ultrahigh energy cosmic ray sourcesand composition. We investigate various chemical compositions and maximum protonacceleration energies E p, max which are allowed by current observations. We considerdifferent models of source evolution in redshift and three possible scenarios for theGalactic to extragalactic transition.We summarize the parameter space for cosmogenic neutrinos into three regions:an optimistic scenario that is currently being constrained by observations, a plausiblerange of models in which we base many of our rate estimates, and a pessimistic scenariothat will postpone detection for decades to come. We present the implications of thesethree scenarios for the detection of cosmogenic neutrinos from PeV to ZeV (10 − eV)with the existing and upcoming instruments. In the plausible range of parameters, thenarrow flux variability in the EeV energy region assures low but detectable rates forIceCube (0 . − . . − . osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV
1. Introduction
The idea that a “guaranteed” flux of detectable extragalactic neutrinos should beproduced by the propagation of ultrahigh energy cosmic rays as they interact withthe ambient photon backgrounds (Berezinsky and Zatsepin, 1969; Stecker, 1979) hasencouraged efforts to detect them for decades (see, e.g., Anchordoqui and Montaruli,2009). One important assumption, that cosmic rays are extragalactic at the highestenergy, has been verified by the detection of the Greisen-Zatsepin-Kuzmin (GZK)cutoff feature (Greisen, 1966; Zatsepin and Kuzmin, 1966) in the cosmic ray spectrum(Abbasi et al., 2009; Abraham et al., 2008 b ) and by the indication of anisotropies inthe cosmic ray sky distribution at the highest energies (Abraham et al., 2007, 2008 a ).These findings herald the possibility of detecting ultrahigh energy neutrinos in the nearfuture and a possible resolution to the mystery behind the origin of ultrahigh energycosmic rays (UHECRs).At present the sources of ultrahigh energy cosmic rays, their location, cosmologicalevolution, and maximum energy, as well as the injected composition, remain unknown.A multi-messenger approach with the detection of secondary neutrinos and photonscan lead to a resolution of this quest for sources. At the highest energies neutrinosare particularly useful because, unlike cosmic-rays and photons, they are not absorbedby the cosmic backgrounds while propagating through the Universe. These so-called“cosmogenic neutrinos” bear the information of ultrahigh energy cosmic ray sources upto high redshifts, and can help constrain their nature, injection spectrum, distribution,and evolution.A number of authors have estimated the cosmogenic flux with varying assumptions(e.g., Engel et al., 2001; Ave et al., 2005; Seckel and Stanev, 2005; Hooper et al., 2005;Berezinsky, 2006; Stanev et al., 2006; Allard et al., 2006; Takami et al., 2009, and seeAnchordoqui and Montaruli, 2009 for a recent review): in particular, Engel et al. (2001)and Ave et al. (2005) calculated analytically and semi-analytically the expected neutrinoflux around PeV energies, considering cosmic ray interactions with CMB photons anda homogeneous source distribution. They find that IceCube could detect ∼ . E ν & eV), EeV (= 10 eV), and ZeV (= 10 eV). ANTARES, osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV ∼
10 EeV. Experiments primarily dedicated to the detectionof cosmic rays like the Pierre Auger Observatory and the Telescope Array have theirbest neutrino sensitivities in the EeV energy range. The radio telescope ANITA andthe fluorescence telescope JEM-EUSO are most effective at the highest energy neutrinosaround 0.1 ZeV.We demonstrate in this paper that the detection of cosmogenic neutrinos ischallenging for current detectors, unless cosmic ray sources fall into the optimisticcategory. If neutrinos are observed, each energy range would enable one to explorethe influence of a specific set of cosmic ray source parameters. In section 2, we examinethe effects of the major unknown that concern ultrahigh energy cosmic ray sources:the injected chemical composition, the maximum proton acceleration energy, sourceevolution models, and outstanding scenarios for the Galactic to extragalactic transition.Section 3 discusses the implication of our calculations in terms of detection by currentand upcoming neutrino experiments. We focus alternately on the three main energyranges covered by these instruments and conclude in section 4.
2. Neutrino fluxes
We calculate the fluxes of neutrinos generated by the propagation of ultrahigh energycosmic rays over cosmological distances. For this purpose we use a complete numericalMonte Carlo method that takes into account the interactions of nuclei with cosmicbackground radiations (see Allard et al., 2005 and Kotera et al., 2009 for more details).Our modeling of the background radiations includes redshift evolutions of the cosmicmicrowave background (CMB) and of the diffuse infrared, optical, and ultravioletbackgrounds modeled according to Stecker et al. (2006), hereafter referred to as theIR/UV background. The IR/UV background and its cosmological evolution are not aswell known as the CMB, though recent measurements (see discussion in Stecker et al.,2006 and Allard et al., 2006) have lead to better constraints. One can notice thatKneiske et al. (2004) find higher photon densities in the far infrared bump at largeredshift compared to Stecker et al. (2006), and lower densities in the optical and UVrange. The latter difference implies neutrino fluxes a factor of two higher in our studycompared to Takami et al. (2009) around 3 × eV, and the agreement becomesvery good above 10 eV. The dispersion of neutrino fluxes due to the astrophysicalhypotheses we consider in the following is far larger than the one introduced by differentmodelings of the IR/UV backgrounds. The latter do not have a large impact on thediscussion of the detectability of cosmogenic neutrinos in the PeV region.Particles are injected with energy between E min = 10 eV and Z × E p, max , followinga power-law spectrum: d N/ d E ∝ E − α , with an exponential cut-off above Z × E p, max .Apart from section 2.3, we use a fiducial maximum proton acceleration energy of osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV Table 1.
Spectral indices corresponding to models presented in section 2. Abreviationsfor source evolution models are defined in section 2.1. Galactic to extragalactictransition models are described in section 2.2 and chemical compositions in section 2.4.composition transition source evolution α pure protons dip uniform 2.6” ” SFR1 2.5” ” SFR2 2.5” ” GRB1 2.4” ” GRB2 2.4” ” FRII 2.3” WW SFR1 2.1Galactic mix Galactic mix SFR1 2.1pure iron Galactic mix SFR1 2.0iron rich, low E p , max Galactic mix SFR1 1.2 E p, max = 10 . eV. Cosmic rays are followed down to energy E cr = 10 eV where theinteraction probability becomes negligible. The redshifts of their sources are distributedbetween z = 10 − and z = 8 and are weighted according to source evolution models.In the following, we explore different source evolution and transition models,chemical compositions and maximum acceleration energies. For each scenario, thespectral index and the overall normalization of the cosmic ray flux is chosen to best fit theAuger data. These combined parameters lead to a particular shape and normalizationof the produced cosmogenic neutrino fluxes. Table 1 presents the spectral indicescorresponding to each model.Let us note that our cosmogenic neutrino fluxes all present the same basic shape thatcan be understood from Fig. 1. Neutrinos are produced via two principal channels: bypion decay or by neutron decay. The contribution of the latter channel is plotted in reddashed line in Fig. 1. Pions are produced through the interaction of neutrons, protons,and heavier nuclei with the CMB and the IR/UV radiation. Those two backgrounds areresponsible for the presence of the two bumps around a few PeV and a few EeV in theneutrino flux. We will see in the following that the height of the low energy bump mainlydepends on the value of the injection spectral index, α , corresponding to the variousmodels. The high energy bump remains fairly independent of α and of transition models,depending mostly on the composition and on the maximum acceleration energy. Several observation-based estimates of the evolving star formation rate have been madein recent years, mostly by measuring the evolution of the galaxy luminosity functionsover a broad range of wavelengths. Results from SDSS, GALEX, COMBO17, andSpitzer now allow a tighter constraint on the cosmic star formation history up toredshifts of z ∼ z ∼
1, the results of thedifferent studies disagree on the shape of the SFR, though a tendency towards a plateau osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV Figure 1.
Contribution of the different processes to the neutrino flux, considering allflavors. The case of a pure proton composition, assuming a star formation rate typeevolution for the source emissivity (Hopkins and Beacom, 2006) and a dip transitionmodel (Berezinsky et al., 2006) is presented. The black solid line indicates the totalflux. The green solid line represents the neutrino emission due to the interaction ofcosmic rays with CMB photons and the blue dotted line with UV, optical, and IRphotons. The red dashed line is the contribution of the neutron decay (neutrons areproduced through photo-hadronic interactions). in the range of z ∼ − z & z ∼ −
2. The cosmic ray meanfree path of interaction with the IR/UV background will consequently evolve with theredshift. The CMB photon density also increases with redshift in (1 + z ) , implyingthat the high energy bump will also be affected by the source emissivity evolution. Notethat the IR/UV background evolves less than the CMB because unlike the latter it iscontinuously produced during the cosmic history. The decrease of this background withredshift is thus slower than the one of the CMB. The effect of the evolution is actuallysmaller in the IR/UV region than in the CMB region. Nevertheless, the difference inthe steepness of the injected spectral indices required to adjust the propagated cosmicray spectrum induces large variations between the fluxes at low energy.Not many astrophysical objects fulfill the stringent energetic requirements to bepotential sources of ultrahigh energy cosmic rays. The main candidate sources are thefollowing: transient sources such as gamma ray bursts (GRB) or young magnetars, and osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV Figure 2.
Top: source emissivity evolution with redshift, normalized to unity at z = 0, for our six models described in the text. Bottom: effects of source evolution onneutrino fluxes for all flavors. We assume here a pure proton composition and a diptransition model. continuous sources like powerful active galactic nuclei (AGN). Among AGN, Faranoff-Riley type I (FRI) and II (FRII) galaxies are more specifically discussed, though FRIgalaxies are far from satisfying the energetic criteria to accelerate particles to the highestenergies (see Lemoine and Waxman, 2009). It might be worth mentioning as well thatno outstanding correlation has been observed between catalogues of FRII galaxies and osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV Figure 3.
Propagated cosmic ray spectra adjusted to the observational data by HiRes-I and HiRes-II (Abbasi et al., 2004) and the Auger Observatory (Abraham et al.,2008 b ). Top: for different source evolutions, assuming a pure proton composition anda dip transition model. Bottom: for different compositions as labeled in the legendand with the corresponding spectral indices listed in table 1, assuming a SFR1 typesource evolution. the most energetic events seen by Auger, which does not give strong credence to thesetypes of sources, unless the particle rigidity is unexpectedly low.To describe the redshift evolution of the emissivity of these candidate sources, we osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV Figure 4.
Contribution to the total neutrino flux of sources in different redshiftbins. Each panel represents the fluxes obtained for one of the source evolution modeldescribed in the text. Each line (for increasing thickness) corresponds to the sumof the fluxes produced by sources located in the following redshift bins: z < . . ≤ z < .
5, 1 . ≤ z < .
5, 2 . ≤ z < z > use the latest measurements and studies made on these objects. The star formation rate(SFR) being a general tracer of matter density in the Universe, and thus a (possiblybiased) tracer of ultrahigh energy cosmic ray sources, we also examine the effect of tworecently derived SFR trends. Star forming galaxies in particular might host transientobjects such as GRB and young magnetars.In this paper, we model the source evolution using six typical trends (see Fig. 2): • uniform: the source emissivity experiences no evolution. Beckmann et al. (2003)for example argue that this might be the case for FRI type galaxies. • SFR1: the source evolution follows the star formation rate derived inHopkins and Beacom (2006). In this model, the source emissivity increases as(1 + z ) . for z <
1, then (1 + z ) − . for 1 ≤ z < z ) − . for z ≥ • SFR2: the source evolution follows the star formation rate from Y¨uksel et al.(2008). In this model, the source emissivity increases as (1 + z ) . for z < z ) − . for 1 ≤ z < z ) − . for z ≥ • GRB1 and GRB2: the latest
Swift
GRB observations indicate that the GRB ratedeparts from the SFR at the highest redshifts (Daigne et al., 2006; Le and Dermer, osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV z ∼ z ) / [1 + ( z/ . ] for GRB1 and as (1 + 11 z ) / [1 + ( z/ . ] for GRB2, as derivedin Le and Dermer (2007). • FRII: recent measurements indicate that FRII type galaxies follow a steep evolutionfor z < ρ according to Wall et al. (2005) as log ˙ ρ = 2 . z + 1 . z +0 . z − . z . Hasinger et al. (2005) argue that FRI type galaxies might followa similar, though less steep trend.Figure 2 presents the cosmogenic neutrino fluxes for all flavors, obtained for thesesource evolutions, for a pure proton dip model case. We checked that the followingdiscussion remains identical for all other composition and transition models. Thenormalization of the fluxes was calculated by adjusting our propagated ultrahigh energycosmic ray spectra to the observational data, as shown in Fig. 3. The cosmic ray spectraobtained for the different source evolution models match well the data for the chosenspectral indices (see Table 1) and normalizations. Figure 3 shows how the UHECRspectra data alone cannot specify the source model.Figure 2 shows that the source evolution introduces an overall scaling of the neutrinofluxes. The fact that the neutrino fluxes obtained for SFR1, SFR2, GRB1 and GRB2models are very close demonstrates that the difference of evolution above z ∼ z .
2. The contribution of sources at redshifts z & Around energies E ∼ . − eV, the cosmic ray spectrum hardens, creating a featurecommonly referred to as the ‘ankle’. In the standard picture, the ankle is associated withthe transition between the Galactic and the extragalactic components. Early versionsof transition models proposed that the extragalactic cosmic rays emerge at very highenergy ( E > eV) and be composed of 100% protons, as the measurements seemedto indicate at that time (see e.g. Waxman, 1995). Wibig and Wolfendale (2004) followthis idea and fit the shape of the ankle by injecting particles with spectral index 2 . − . E ∼ × eV. The injected composition is highly enriched in protons. osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV Figure 5.
Effects of various transition models on neutrino fluxes for all flavors.We present the case of a source evolution following the star formation rate fromHopkins and Beacom (2006). Black solid line: the pure proton ‘dip model’ with aninjection spectrum of 2.5, pink dotted: transition slightly below the ankle for a Galacticmixed composition with an injection spectrum of 2.1, blue dashed line: pure proton‘WW model’ with a transition at energy > eV with a 2.1 injection spectrum (seetext for description of models). Throughout this paper, we will refer to this ankle transition model, most recentlydeveloped by Wibig and Wolfendale (2004) as the ‘WW model’.For the mixed chemical composition model (for which the extragalactic cosmicray composition at the source is assumed to be similar to that of low energy Galacticcosmic rays), Allard et al. (2005) demonstrated that the shape of the spectrum can bewell reproduced, assuming an injection spectrum α of order 2 . − .
3. In this model,the transition between Galactic and extragalactic components happens at lower energy( E ∼ EeV) and ends at the ankle.Berezinsky et al. (2006) proposed that this transition occurs at even lower energy,around E ∼ . − . eV, where the cosmic ray spectrum may steepen, creating the so-called ‘second knee’. The combination of the second knee and the ankle is viewed in thismodel as a dip due to pair production energy losses during the intergalactic propagation.This scenario eases the issue of particle acceleration up to high energy inside the Galaxy,that is raised by the other models. It requires however a relatively steep injectionspectrum (2 . − . osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV −
10 PeV is directlyrelated to the number of extragalactic cosmic rays present at low energy. At the highestenergy end of the cosmic ray spectrum, the effect of the harder spectral injection neededfor the WW model is counterbalanced by the fact that the extragalactic component doesnot account for the whole cosmic ray flux at 10 EeV, energy at which we normalize ourcalculated flux. As a result the flux is similar to what is found for the dip model. In themixed composition model, although the cosmic ray flux is purely extragalactic above theankle, the gain of the harder spectral index (compared to the dip model), is compensatedby the presence of nuclei (that produce less neutrinos than protons for a 2.1 spectralindex) in the source composition. For these reasons, the neutrino fluxes expected at thehighest energies are almost identical for the three transition models although the WWand mixed composition models require harder spectral indices.The weak dependence of neutrino fluxes on transition models at EeV energieswas highlighted by Takami et al. (2009). We see in the following that the maximumacceleration energy and the composition can affect the neutrino flux in this energyrange significantly.
The maximal energy at which particles can be accelerated depends on the sourceenergetics and on numerous physical parameters of the acceleration site. For a givensource model, the maximum energy can in principle be estimated by comparing thecosmic ray acceleration time to the escape time, the source lifetime, and energyloss times due to various processes like interaction and dynamical expansion (see,e.g., Norman et al., 1995; Allard and Protheroe, 2009). Since acceleration sites arenot yet known the physical processes that may reach the highest energies are farfrom clear. Moreover, one expects that E p, max varies among the sources (see, e.g.,Kachelrieß and Semikoz, 2006).We present in Fig. 6 the effects of three different maximum acceleration energyon the shape of the neutrino flux. We present the case of a pure proton dip transitionmodel, assuming a SFR1 type source evolution. This figure demonstrates the robustnessof the high energy neutrino peak around E ν ∼ − . eV, with respect to themaximum acceleration energy. The main difference in flux occurs at E ν & eV,which corresponds to the energy range covered by ANITA and JEM-EUSO. In the next osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV Figure 6.
Effects of various maximum acceleration energy for protons E p, max onneutrino fluxes for all flavors. We present here the case of a pure proton dip transitionmodel (see section 2.2 for description of the model), assuming a SFR1 type sourceevolution. E p, max = 10 , . and 10 eV for respectively: pink dotted, black solidand blue dashed lines. section, we discuss however that a too low E p, max , that would fall below the protonphoto-pion production threshold, can lead to a drastic suppression of the neutrino flux,especially around ∼ EeV energies.
The chemical composition of ultrahigh energy cosmic rays remains an open question.Measurements prior to the Pierre Auger Observatory indicated an increasingly lightercomposition above E ∼ eV (Bird et al., 1993; Shinozaki and et al., 2005;Abu-Zayyad et al., 2000; Abbasi et al., 2005, 2010 a ). The latest results of the PierreAuger Observatory suggest a mixed composition at all energies, that gets heavier atthe highest end (Abraham et al., 2010). Furthermore, there is no reliable theoreticalprediction of the expected composition at the source, mainly because very little is knownabout the physical parameters that govern the acceleration and survival of nuclei in thosepowerful objects.We thus consider in this study four typical compositions that have been shown to fitthe shape of the observed ultrahigh energy spectrum: (i) a pure proton composition inthe dip model case, (ii) a proton dominated mixed composition based on Galactic cosmicray abundances as in Allard et al. (2006), (iii) a pure iron composition and (iv) a mixedcomposition that was proposed by Allard et al. (2008), that contains 30% of iron. For osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV E p, max = 10 eV and a hard injectionspectrum of 1.2 in the SFR evolution case is needed in order to correctly adjust theobserved cosmic ray spectral shape (see Fig. 3). In this scenario, the propagated cosmicray composition would be heavy at the highest energy, as favored by Abraham et al.(2010). For the first three models, we choose the maximum proton injection energy of E p, max = 10 . eV. In all cases, we assume E Z, max = Z × E p, max for a nucleus of chargenumber Z . We take an exponential cut-off for the spectrum.Figure 7 presents the neutrino fluxes obtained in these four scenarios. It appearsthat the high energy peak is only mildly dependent on all composition models, exceptfor the case of the iron rich low E p, max scenario. Indeed, as illustrated in Fig. 8,neutrinos are mainly produced via photopion production of secondary nucleons thatare themselves produced by photo-disintegration processes of the primary nuclei. Forheavy nuclei, the high energy neutrino flux thus depends on the rate of particles thathave an energy per nucleon higher than the pion production threshold of protons onthe CMB ( E A /A > E pγ CMB , where A is the atomic number of a nucleus of energy E A ).When comparing with fluxes expected in the pure proton case at a similar maximumenergy per charge, the difference will ultimately depend on the spectral index neededto fit experimental data (see Allard et al., 2006 for more details). In the SFR sourceevolution case the spectral indices for proton and iron sources are respectively 2.5 and2.0, resulting in a factor of 3 different in the high energy neutrino peak. In the FRIIcase, the spectral indexes required are respectively 2.3 and 1.6, and the two compositionhypotheses result in similar neutrino fluxes at the highest energies. In the iron rich case(iv), the pion production threshold is not reached for most of the particles because ofthe low E p , max . The direct photopion production of primary nuclei is also suppressedbecause of the too low E Z, max . Let us note here that in our calculation we assume thatsources are standard candles and all have the same maximum energy. In particular inthe E p , max all the sources have an exponential cut-off above 10 eV (note that then, E Z =26 , max = 2 . × eV). This assumption could be somewhat relaxed to allow somefraction of the sources to reach higher maximum energy for protons without changing theglobal phenomenological feature of the low E p , max model. The expectations for neutrinofluxes at high energies would be higher in this case but would obviously remain belowthese obtained for standard proton dominated models, except if the maximum energyreached by the sources is strongly correlated with the redshift. Indeed, in the lattercase, if the sources at large redshift had a larger proton maximum energy, one couldobtain large neutrino fluxes together with a heavy composition at the highest energy,the protons accelerated up to the highest energies being produced outside the energylosses horizon.The strong variations in amplitude at lower energy stem from the injected spectralindices that change the number of particles available for producing ∼ −
10 PeVenergy neutrinos. Thus the low energy peak is strongly affected by changes in injectedcomposition. osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV Figure 7.
Effects of various compositions on neutrino fluxes for all flavors. We presentthe cases of (i) a pure proton injection assuming a dip transition model (black solid),(ii) a proton dominated Galactic type mixed composition (pink dotted), (iii) pure ironcomposition (blue dashed) and (iv) the iron rich low E p , max model (red dash-dotted).
3. Implications for the existing and upcoming detectors
Figure 9 summarizes our results and compares our fluxes to the existing, upcoming,and possible future neutrino detector sensitivities. Our estimates for neutrino fluxesare divided into three possible regions: an optimistic scenario (pink dot-dashed line),a plausible range of models in which we base many of our rate estimates (grey shadedarea), and a more pessimistic scenario (blue lines). The optimistic scenario correspondsto the FRII strong source evolution case with a pure proton composition, dip transitionmodel and E p, max = 10 . eV. The most pessimistic scenario is given by a pure ironinjection and the iron rich composition with low E p, max , assuming in both cases a uniformevolution of sources. The shaded area brackets a wide range of parameters: all discussedtransition models, all source evolutions except for uniform and FRII, and varying cosmicray injection composition from pure protons to a mixed Galactic type model, with E p, max ≥ eV. The black long-dashed line indicates the minimum neutrino flux onecould obtain in the case of a uniform source evolution, when the composition and themaximum acceleration energy are chosen among reasonable values. Namely, this linerepresents the case of a Galactic mixed composition with E p, max = 10 eV for a uniformsource evolution.From the discussion elaborated at the beginning of section 2.1, it stands out that auniform UHECR source evolution should be deemed rather extreme. Indeed, under the osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV Figure 8.
Contribution of different neutrino production channels to the totalcosmogenic neutrino flux for all flavors (black solid line). Case of pure iron, withinjection spectral index 2.0 and assuming a SFR1 type source evolution. Bluedashed line: neutrinos produced by secondary nucleons, red dotted line: neutrinosproduced directly by photo-pion process of the primary nuclei, green dash-dotted line:contribution of the decay of secondary neutrons. assumption that UHECRs are produced in astrophysical sources, the majority of theirplausible progenitors should follow – with a possible bias – the star formation history.Though Beckmann et al. (2003) suggest that FRI-type galaxies might have experienceda quasi-uniform emissivity evolution throughout time, one should be aware that theseradio-galaxies are, as already discussed, very bad candidates for UHECR productiondue to their poor energetics. As for the FRII source evolution, we chose to consider itas an extreme scenario as well, and not to include it in our ‘reasonable’ grey shadedarea. This optimistic scenario from the neutrino flux point of view is not favored byUHECR observations due to the lack of correlations between the FRII galaxies and thehighest energy events seen by Auger.Figure 9 reveals that the cosmogenic neutrino flux can vary of many orders ofmagnitude throughout the whole energy range. In the PeV energy region, the full setsof models imply a three order of magnitude variation while one has an uncertainty offour orders of magnitude in the EeV region. The ZeV region is unbound from below,making a detection in this range the most speculative. If one focuses on the ‘reasonable’domain (the grey shaded area in Figure 9), it appears that the spread around EeVenergies becomes fairly limited in regard to the various parameters. The one exceptionto this robust behavior is a very low E p, max . In the PeV region, our ‘reasonable’ models osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV Figure 9.
Cosmogenic neutrino fluxes for all flavors, for different parameters comparedto various instrument sensitivities. The pink dot-dashed line corresponds to the FRIIstrong source evolution case with a pure proton composition, dip transition model and E p, max = 10 . eV. Blue lines are our extreme pessimistic cases: the blue dotted linerepresents the iron rich, low E p, max composition, and the blue dashed line the pureiron injection case, with E p, max = 10 eV; both lines assume a uniform evolution ofsources. The shaded area brackets a wide range a parameters: all transition models andall source evolutions except uniform and FRII, for pure protons and a mixed ‘Galactic’composition are considered. Including the uniform source evolution would broaden theshaded area down to the black long-dashed line. Instrument sensitivities: differentiallimits for super Auger North multiplied by 3 (green dashed, see text), IceCube 80lines averaged over the three flavors (blue dash-dotted, acceptance from S. Yoshida,private communication, see also Karle, 2010 and Abbasi et al., 2010 b ), and JEM-EUSOmultiplied by 3 (purple solid, acceptance from Medina-Tanco et al., 2009, see text). Inred solid line: differential limit and integral flux limit on a pure E − spectrum (straightline), both multiplied by 3 (see text) for Auger South, using the optimistic acceptancefrom Abraham et al. (2009). In black solid line: ANITA-II differential limit at 90%CL, for 27.1 day livetime, for all flavors, the straight line indicates the integral fluxlimit on a pure E − spectrum (Gorham et al., 2010). introduce a wider span in flux, which can be helpful to distinguish among the variousmodels, if neutrinos are found in both energy ranges. osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV Figure 10.
Cumulative number of neutrinos per year above a given energy expectedfor Auger South, in the ‘reasonable’ parameter range represented by the grey shadedarea in Fig. 9: thick lines for the upper bound and thin lines for the lower bound. Thenumbers are calculated using the optimistic exposures given in Abraham et al. (2009).
In terms of detectability, one may first note that the optimistic scenario is currentlybeing constrained by observations. For more plausible sets of parameters (grey region),the EeV energy range is close to the sensitivity of the Auger Observatory and of IceCube-80 (see also the numbers in Figs. 10 and 11). At PeV energies, the situation is moreuncertain due to the high variation of the flux according to the parameters. It should benoted however that IceCube-80 will start to detect neutrinos or at least give interestingconstraints in this region very soon. ZeV neutrino observatories, such as ANITA andJEM-EUSO will constrain the maximum acceleration energy in the most optimistic case.The sensitivity in the ZeV energy range will however have to be greatly improved inorder to explore this parameter space.The green line in Fig. 9 labeled ‘Super Auger North’ represents the sensitivity ofan EeV neutrino detector covering the area proposed for Auger North (20,000 km ,Olinto et al., 2009; Bl¨umer and the Pierre Auger Collaboration, 2010). If Auger Northhad 100% detection efficiency for neutrino showers around an EeV (as assumed inthis plot), for example, through a denser array than currently proposed, then itssensitivity would be vastly superior to currently planned observatories in this energyrange. An alternative to a denser array of Auger surface detectors is the possibility ofnew atmospheric air-shower detection techniques such as radio or microwave.Figures 10 and 11 give the cumulative numbers of neutrinos that are expected forthe different instruments, in the case of a pure proton composition, dip transition model, osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV Figure 11.
Cumulative number of neutrinos (all flavors) per year above a givenenergy expected for Auger South (red solid, optimistic acceptance from Abraham et al.,2009), super Auger North (green dashed, see text), IceCube 80 lines for three flavors(blue dash-dotted, acceptance from S. Yoshida, private communication, see also Karle,2010), and JEM-EUSO (purple solid, acceptance from Medina-Tanco et al., 2009). Thenumbers are calculated for the ‘reasonable’ parameter range represented by the greyshaded area in Fig. 9: thick lines for the upper bound and thin lines for the lowerbound. with SFR1 type source evolution, which roughly corresponds to the upper limit of theshaded area in Fig. 9. The numbers are only represented in the instrument sensitivityrange. From these figures, one can infer that at EeV energies, one should detect 0 . − . . − .
06 with Auger South. Let us notehowever that current and planned experiments should be unable to detect cosmogenicneutrino fluxes predicted for the dip model if there is no evolution of the cosmic rayluminosity with redshift. In point of fact, the non observation of cosmogenic neutrinosin the next few years would certainly help us constrain UHECR source evolution models(see e.g. Berezinsky, 2009). On the other hand, it would not be possible to constrain thesource composition or the Galactic to extragalactic transition models without positivedetection over the next decades.Interestingly, it was pointed out by recent studies (Ahlers et al., 2010;Berezinsky et al., 2010) that the strongest source evolution models can be constrainedalso by the cascading of cosmogenic photons down to GeV energies. These authors haveshown that (at least for proton dominated compositions up to the highest energies) theFermi-LAT diffuse gamma-ray flux would be overshot by the products of cascading UHEphotons, for scenarios where the comoving source luminosity is very strong. Although osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV ν e : ν µ : ν τ = 1 : 2 : 0, subsequent neutrino oscillation during the propagation shouldlead to a proportion on Earth of 1 : 1 : 1 (Learned and Pakvasa, 1995). We will notconsider here other neutrino mixing behaviors than the canonical mixing described forexample in Pakvasa (2008).Figure 12 shows that these proportions are indeed reached in the case of a pureproton composition, in the range of energy that is of interest for our study. For theother chemical compositions, the contribution of the neutron decay at E ν < GeV(see Fig. 8 for the iron case) enhances strongly the proportion of electronic neutrinos.For the injection with a low E p, max , the production of secondary neutrons is lower dueto the low overall energy of nuclei and this effect is attenuated. In Fig. 9, we representthe sensitivity of IceCube averaged over the three neutrino flavors. The fact that theratio between ν µ and ν e departs from 2 for non pure proton compositions indicates thatat PeV energies, the detection of ν e can be favored against other flavors. The ratios areof ∼ / ν e : ∼ .
28 : 1 : 1, and the detection for a pure iron composition ishighly compromised in any case. Due to the slight difference between the sensitivities ofeach flavor (within a factor 2 −
3) for IceCube, we took into account the individual flavorfluxes and acceptances when calculating the number of expected neutrinos in Fig. 11.At EeV energies on the other hand, the ratio remains stable around 1 : 2 : 0. Forthis reason, we multiplied in Fig. 9 the sensitivities for τ neutrinos of Auger South andSuper Auger North by 3, assuming a final proportion at the Earth of 1 : 1 : 1. Weapplied the same multiplication to the JEM-EUSO sensitivity, though the assumptionof equal ratios may seem debatable. Indeed, as can be seen in Fig. 12, at the highestenergies ( E ν > GeV), the production of muonic neutrinos is enhanced due to thedecay of Kaons and other mesons that lead to different neutrino spectra than for muondecay. One can calculate however that the neutrino oscillation should lead in principleto flavor proportions of order 0 .
88 : 1 : 1 for flux ratios F ( ν µ ) /F ( ν e ) ∼
3, whichremains close to the standard 1 : 1 : 1 ratio. Our calculations do not consider the muonpolarization effects that should modify the ν µ to ν e flavor ratio at the highest energiesas was demonstrated by Lipari et al. (2007). This issue should again affect only slightlythe overall expected ν τ flux for JEM-EUSO. We checked that the ratios presented inFig. 12 are similar for our various source evolution models.Let us further note that in this study, we only take into account the cosmogenicneutrinos, i.e. the neutrinos produced by interactions with the radiative backgroundsduring the propagation of the UHECRs in the intergalactic medium. Therefore, the osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV Figure 12.
Ratio of fluxes of ν µ to ν e produced during the propagation of UHECRsfor the various injected chemical compositions described in section 2.4. We presentthe cases of (i) a pure proton injection assuming a dip transition model (black solid),(ii) a proton dominated Galactic type mixed composition (pink dotted), (iii) pure ironcomposition (blue dashed) and (iv) the iron rich low E p , max model (red dash-dotted). In our calculations, we neglected the effect of the extragalactic magnetic fields onthe neutrino fluxes. At the highest energies ( E cr > eV), their influence is indeednegligible unless their intensity becomes high enough to produce a sizable magnetichorizon effect at the energies where we normalize our neutrino flux to the cosmic raydata. This could happen if the mean magnetic field is h B i &
10 nG, which is not favoredby current observations (see e.g. Kotera and Lemoine, 2008 and Globus et al., 2008).It should be also remarked that the flux around PeV energies could be furtheramplified by more than one order of magnitude provided that most sources are locatedin strongly magnetized environments such as clusters of galaxies, if one stipulates thatthe radiation density in these environments is stronger than in the extragalactic medium.The amplitude of this effect depends on various assumptions on the source environmentas is discussed in detail in Kotera et al. (2009).Finally, because we normalized our neutrino fluxes to the observed ultrahigh energycosmic ray spectrum Abraham et al. (2008 b ), a possible shift in energy of the cosmic rayflux would consequently change the neutrino flux. The current systematic uncertainty on osmogenic Neutrinos: parameter space and detectabilty from PeV to ZeV ∼ inside the source (e.g. Chen and Hoffman, 2009; Allison et al.,2009; Gorham et al., 2010). Such a production is independent of the flux ofcosmogenic neutrinos that are produced outside the source during the propagation inthe extragalactic medium. Its detailed study requires assumptions on the opacity of theacceleration region and strongly depends on the shape of the injection spectrum as wellas on the phenomenological modelling of acceleration. In the same token, recently,Ahlers et al. (2009) claimed to obtain a ‘guaranteed’ neutrino flux (assumed to beproduced by each cosmic-ray proton escaping from the sources) as well as stringentconstraints on the proton fraction in the cosmic ray composition using AMANDA-IIand IceCube limits. One should be aware however that such estimates strongly dependon the physical modeling of particle acceleration and escape from the source. Especiallyif the source is optically thick, cosmic rays should not be accelerated up to the highestenergies as demonstrated by Allard and Protheroe (2009), and EeV energy neutrinoswould be sharply suppressed.In optimistic source scenarios, it was calculated that the cosmogenic neutrino fluxeswould be overwhelmed by the neutrinos from the acceleration site (see for exampleMannheim et al., 2001 for an AGN scenario and Murase and Nagataki, 2006 for aGRB scenario). It is interesting to note, as discussed by Takami et al. (2009) thatneutrinos from GRB would only overwhelm the PeV peak of the cosmogenic neutrinosand furthermore should be distinguishable, as they should be temporally and spatiallycorrelated with with the prompt GRB emission. Again, these estimates are optimisticcases and subject to high variability according to the parameters assumed for the source.
4. Conclusion
We calculated the flux of neutrinos generated by the propagation of ultrahigh energycosmic rays over cosmological distances, using a complete numerical Monte Carlomethod that takes into account the interactions of nuclei with cosmic backgroundradiations.We explored the influence of different cosmic ray source evolutions, transitionmodels, chemical compositions, and maximum acceleration energies on the cosmogenic
EFERENCES − eV) is quite robust in regard to these various cosmic rayparameters. We find that the normalization of the flux in this region ultimately dependson the source evolution up to redshift z ∼ E A /A > E pγ CMB , where A is the atomic number of a nucleus ofenergy E A ). Composition models and Galactic to extragalactic transition scenarios arenot uniquely determined by measurements above 10 . eV. These unknowns can onlybe constrained by complementary measurements of cosmogenic neutrinos in differentenergy ranges.In our ‘reasonable’ neutrino flux region, IceCube should observe 0 . − . . − .
06 neutrino per year in the EeVrange, and detection should happen in the next decade unless E p, max < × eV.If EeV neutrinos are detected, PeV information can help select between competingmodels of cosmic ray composition at the highest energy and the Galactic to extragalactictransition at ankle energies. With improved sensitivity, ZeV neutrino observatories,such as ANITA and JEM-EUSO could explore and place limits to the maximum protonacceleration energy. Acknowledgments
We thank Bruny Baret, Dariusz Gora, Aya Ishihara, Keiichi Mase, Teresa Montaruli,Kohta Murase, Markus Roth, V´eronique Van Elewyck, and Shigeru Yoshida forproviding us with the acceptances of IceCube and Auger and for very helpul discussions.KK and AVO are supported by the NSF grant PHY-0758017 and by Kavli Institute forCosmological Physics at the University of Chicago through grant NSF PHY-0551142and an endowment from the Kavli Foundation. AVO acknowledges the support fromAgence Nationale de Recherche in France and DA was supported by C.N.R.S. in Franceand by NSF grant PHY-0758017 and by Kavli Institute for Cosmological Physics at theUniversity of Chicago.
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