Interpretation of neutrino flux limits from neutrino telescopes on the Hillas plot

Walter Winter

Institut fur Theoretische Physik und Astrophysik, Universitat Wurzburg, 97074 Wurzburg, Germany(Received 16 November 2011; published 27 January 2012)

We discuss the interplay between spectral shape and detector response beyond a simple E�2 neutrino

flux at neutrino telescopes, using the example of time-integrated point source searches using IceCube-40

data. We use a self-consistent model for the neutrino production, in which protons interact with

synchrotron photons from coaccelerated electrons, and we fully take into account the relevant pion and

kaon production modes, the flavor composition at the source, flavor mixing, and magnetic field effects on

the secondaries (pions, muon, and kaons). Since some of the model parameters can be related to the Hillas

parameters R (size of the acceleration region) and B (magnetic field), we relate the detector response to the

Hillas plane. In order to compare the response to different spectral shapes, we use the energy flux density

as a measure for the pion production efficiency times luminosity of the source. We demonstrate that

IceCube has a very good reach in this quantity for active galactic nuclei and jets for all source declinations,

while the spectra of sources with strong magnetic fields are found outside the optimal reach. We also

demonstrate where neutrinos from kaon decays and muon tracks from � decays can be relevant for the

detector response. Finally, we point out the complementarity between IceCube and other experiments

sensitive to high-energy neutrinos, using the example of 2004–2008 Earth-skimming neutrino data from

Auger. We illustrate that Auger, in principle, is more sensitive to the parameter region in the Hillas plane

from which the highest-energetic cosmic rays may be expected in this model.

DOI: 10.1103/PhysRevD.85.023013 PACS numbers: 95.85.Ry, 14.60.Pq, 25.20.�x

I. INTRODUCTION

Neutrino telescopes, such as IceCube [1] or ANTARES[2], are designed to detect neutrinos from astrophysicalsources. If protons are accelerated in these astrophysicalobjects, as we expect from the observation of the highest-energetic cosmic rays, the collision with target photons orprotons will lead to charged pion production, and thereforeto an extraterrestrial neutrino flux. Therefore, neutrinotelescopes are an indirect method to search for the originof the cosmic rays. There are numerous source candidates,the most prominent extragalactic ones being active galacticnuclei (AGNs) [3–6] and gamma-ray bursts (GRBs) [7];see Ref. [8] for a review and Ref. [9] for the general theoryof the astrophysical neutrino sources. In generic estimatesor bounds, such as in Refs. [10,11], the cosmic-ray flux isrelated to the potentially expected neutrino flux. Veryinterestingly, IceCube-40, referring to the 40 string con-figuration of IceCube, is currently starting to touch thesegeneric estimates for particular source candidates; seeRef. [12] for GRBs. In addition, time-integrated [13] andtime-dependent [14] point source searches have beenperformed, so far, without success. On the other hand,there may be sources for which the optical counterpartis absorbed, so-called ‘‘hidden sources’’; see, e.g.,Refs. [15–18]. This immediately raises questions abouthow to interpret the data (see also Ref. [19] for AGNmodels); in particular, what does it mean that IceCubehas not seen anything? What parts of the parameter space

is IceCube actually most sensitive to? In this study, weaddress these questions in terms of the interplay betweenspectral shape expected from the sources and the detectorresponse. In order to quantify the detector response, we usethe time-integrated point source analysis in Ref. [13] fordifferent source declinations. In addition, we comparethe parameter space coverage to other experiments, usingthe example of 2004–2008 Earth-skimming neutrino datafrom Auger [20].A convenient description of the parameter space of

interest is the Hillas plot [21]. In order to confine a particlein a magnetic field at the source, the Larmor radius has tobe smaller than the extension of the acceleration region R.This can be translated into the Hillas condition for themaximal particle energy

Emax½GeV� ’ 0:03 � � � Z � R½km� � B½G�: (1)

Here Z is the charge (number of unit charges) of theaccelerated particle, B is the magnetic field in Gauss, and� can be interpreted as an efficiency factor or linked to thecharacteristic velocity of the scattering centers. Potentialcosmic-ray sources are then often shown in a plot as afunction of R and B, as it is illustrated in Fig. 1 by thenumbered disks (see the end of the figure caption forpossible source correspondences). Assuming that a sourceproduces the highest-energetic cosmic rays with E ’1020 eV, one can interpret Eq. (1) as a necessary condition,excluding the region below the dashed line in Fig. 1 (forprotons with � ¼ 0:1). However, this method does not takeinto account energy loss mechanisms, which may lead to aqualitatively different picture—as we shall see later. In the*winter@physik.uni-wuerzburg.de

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following, we will study the complete parameter spacecovered by Fig. 1 without any prejudice. Since the locationof the sources in Fig. 1 cannot be taken for granted, we willrefer to the individual sources in Fig. 1 as ‘‘test points’’(TP) in most cases, and leave the actual interpretation tothe reader.

Our main focus is the particle physics perspective; i.e.,we start off with the minimal ingredients of neutrinoproduction and the main impact factors on the spectralshape. First of all, note that most neutrino telescope analy-ses (and even some models) use an E�2 neutrino spectrumas an initial assumption (see, e.g., Ref. [13]), which is oftenbelieved to be consistent with a proton injection spectrum(/E�2) coming from Fermi shock acceleration. However,in p� interactions, the neutrino spectrum follows the pionspectrum, which depends on the proton and photon spec-tral shape. Therefore, the E�2 assumption for the neutrinosonly holds for an E�1 target photon density (for the E�2

proton injection spectrum), as it is often assumed for theprompt emission from GRBs up to the photon break. In

self-consistent models, where the target photons originate,for example, from synchrotron emission of coacceleratedelectrons, such a hard spectrum is very difficult to obtain,which results in a different neutrino spectrum; see, e.g.,Ref. [5] for an example. In this work, we study the detectorresponse to different shapes of the neutrino spectra. Forthat purpose, we require a toy model which can predict theneutrino spectral shape for wide regions of the parameterspace without any bias from a multimessenger observation,since, after all, neutrino sources may not necessarily beseen as photon sources. The model needs to take intoaccount the dominant particle physics processes especiallyaffecting the spectral shape, which are multipion processesin the pion production (see, e.g., Refs. [22–24]), neutronand kaon production (see, e.g., Refs. [25–28]), magneticfield effects on the secondary muons, pions, and kaons(see, e.g., Refs. [23,25,28–31]), and the appropriate maxi-mal energies. Note that for the neutrino spectral shape, themagnetic field is an especially important control parame-ter; see, e.g., Refs. [25,32] for quantitative discussions.First of all, the maximal proton energy, which is recoveredin the neutrino spectrum, can be limited by proton syn-chrotron emission. Second, for high enough magneticfields, both the maximal energy of the neutrino spectrumand the spectral shape will be determined by the synchro-tron cooling of the secondaries. Therefore, in order topredict the neutrino spectral shape, it is, especially forlarge B, more important to accurately model the particlephysics of the secondaries instead of the cooling andescape processes of the primaries (protons, electrons, pho-tons). Especially if the target photons come from synchro-tron emission, there is little sensitivity to the spectral shapeof the parents (only the square root of the parent spectralindex enters the photon spectrum).We use the model in Ref. [25] for the prediction of the

spectral shapes, where one of the starting points was theminimal set of assumptions required to describe the neu-trino flavor ratios and spectral shapes as accurately aspossible, given the above boundary conditions. In thismodel, charged pions are produced from photohadronic(p�) interactions between protons and the synchrotronphotons from coaccelerated electrons (positrons). The pho-tohadronic interactions are computed using the methoddescribed in Ref. [24], based on the physics of SOPHIA[33], including higher resonances, t-channel charged pionproduction, and multipion production. The helicity-dependent muon decays are taken into account as de-scribed in Ref. [23]. The toy model relies on relativelyfew astrophysical parameters, the most important onesbeing the size of the acceleration region (R), the magneticfield strength at the source (B), and the injection index (�)which is assumed to be universal for protons and electrons/positrons. Naturally, the parameters R and B can bedirectly related to the Hillas plot. The leading kaon pro-duction mode and the energy losses of all secondaries

5 10 15 20

5

0

5

10

15

Log R (km)

Log

B(G

auss

)

No acceleration

NeuCosmA 2011

UHECR

1

2

3

4

5

6

7

89 1011

12

105 GeV

107 GeV

109 GeV

1011 GeV

FIG. 1 (color online). Maximal proton energy as a function ofR and B in our model (contours) for an acceleration efficiency� ¼ 0:1 on the Hillas plot; data taken from Fig. 2 in Ref. [25].The dashed line indicates the Hillas condition in Eq. (1) for1020 eV protons. The dotted line separates the regions wheresynchrotron (above line) and adiabatic (below line) energy lossesdominate the maximal proton energy in the model; i.e., themaximal proton energy follows the Hillas condition below thedotted line. The region UHECR indicates where 1020 eVcosmic-ray protons are expected to be produced in the model.The ‘‘no acceleration’’ region refers to either inefficient accel-eration or inefficient pion production. The different ‘‘test points’’(numbered disks) correspond to (1) neutron stars, (2) whitedwarfs, (3) active galaxies: nuclei, (4) active galaxies: jets,(5) active galaxies: hot spots, (6) active galaxies: lobes, (7) col-liding galaxies, (8) clusters, (9) galactic disk, (10) galactic halo,(11) supernova remnants, and (12) additional test points. Testpoints are taken from Ref. [56].

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(muons, pions, kaons) are taken into account. Our keyargument goes as follows: for each set of R, B, and �,we can predict the spectral flux shape using this model.Using the exposures from IceCube or Auger, we can thencompute the sensitivity limit (normalization of the flux) forexactly this parameter set. In order to compare sensitivitiesto different spectra, the (integrated) neutrino energy fluxdensity at the detector (erg cm�2 s�1) is used. We illustrate,at the end of Sec. III, that this quantity is typically a directmeasure for the luminosity� pion production efficiency ofthe source. Thus, ‘‘good sensitivity’’ in this work meansthat the source will be found for relatively small values ofthis product. Note that we do not interpret the data in termsof the deeper astrophysics involved, and we do not predictthe absolute levels of the neutrino fluxes. In addition, notethat our conclusions will naturally be somewhat modeldependent. However, as we will illustrate, some of thequalitative results should be rather robust.

This study is organized as follows: In Sec. II we sum-marize the key ingredients of the model, where a criticaldiscussion of the limitations of the model can be found atthe end of this section. Then, in Sec. III we describe ourmethod to limit individual fluxes using the neutrino effec-tive areas from IceCube-40. Furthermore, in Sec. IV weshow the result for the whole Hillas plane and differentsource declinations; we illustrate the impact of the injec-tion index �, neutrinos from kaon decays, and ��-inducedmuon tracks. In Sec. V we emphasize the complementarityof different experiments and data sets, using Auger as anadditional example. Finally, we summarize in Sec. VI.

II. REVIEW OF THE SOURCE MODEL

The model in Ref. [25] describes neutrino production viaphotohadronic (p�) processes for transparent sources(optically thin to neutrons) and includes magnetic fieldeffects on the secondary particles (pions, muons, kaons).It can be used to generate neutrino fluxes as a function of afew astrophysical parameters. Below we outline the keyingredients of the model relevant for this study; for detailssee Refs. [24,25], or Ref. [34] for a shorter summary. All ofthe following quantities refer to the frame where the targetphoton field is isotropic, such as the shock rest frame (SRF).

The protons and electrons/positrons are injected withspectra / E��, where � is one of the main model parame-ters. The maximal energies of these spectra are determinedby balancing the energy loss and acceleration time scalegiven by

t�1acc ¼ �

c2eB

E; (2)

with � an acceleration efficiency depending on the accel-eration mechanism, where we typically choose � ¼ 0:1. Ifsynchrotron losses dominate, the maximal energy is there-fore given by

Emax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9��0�

B

sm2c7=2

e3=2: (3)

It scales / m2, which means that the protons are acceler-

ated to much higher energies, and / 1=ffiffiffiffiB

p, which means

that strong magnetic fields limit the maximal energies. Ifadiabatic energy losses dominate, t�1

ad ’ c=R, the maximal

energy is (for protons) given by Eq. (1), i.e., the Hillascondition. Since the neutrino energy follows the protonenergy, the maximal energy of the protons determines (fornot too strong magnetic field effects) the peak of theneutrino spectrum in E2dN=dE. We show the maximalproton energy in Fig. 1 as a function of R and B for thismodel, where the dotted line separates the regions wheresynchrotron (above line) and adiabatic (below line) energylosses dominate the maximal proton energy (here� ¼ 0:1).1 One can easily see that in the upper part (abovethe dotted line) the Hillas condition, Eq. (1), does notapply, since the synchrotron losses dominate, whereasbelow the dotted line, the maximal proton energy followsEq. (1). One can also take an educated guess for the bestIceCube spectral shape sensitivity already, since its differ-ential limits are minimal at around 104 to 106 GeV (seeSec. III)—which should be a very robust prediction. In theabsence of magnetic field effects, one can estimate that theoptimal detector response is roughly obtained when thesedifferential limit minima coincide with a certain fraction ofthe maximal proton energy in Fig. 1 (the neutrinos takeabout 0.05 to 0.1 of the proton energy in photohadronicinteractions). Note that, in this model, the potential sourcesof the highest-energetic cosmic-ray protons are to be foundin the lower right corner in the region labeled ‘‘UHECR’’within the 1011 GeV contour (unless there is a strongLorentz boost of the source). In this case, the neutrinoenergies extend up to about 1010 GeV. However, recentresults on the cosmic-ray composition, such as from Auger[37], indicate that, at least above about 3� 109 GeV,heavier nuclei may dominate the cosmic-ray composition.Since these heavier nuclei may reach the same energies forlower magnetic fields according to Eq. (1), they wouldoccupy a slightly different region in the Hillas plane.Therefore, one should keep in mind that the mappinginto the Hillas plane only exactly applies to the protoncontribution at ultrahigh energies.

1Note that these assumptions are consistent with Ref. [35](efficient acceleration case) and Ref. [36], who have emphasizedthese effects earlier. In particular, the dotted line approximatelycorresponds to the middle solid curve in Fig. 6 of Ref. [36],which limits the region where the Hillas condition for themaximal proton energy applies. That, however, does not meanthat the region on the right-hand side of the dotted line isexcluded; it only means that the maximal energy is limitedotherwise (by synchrotron losses, and, in the very lower rightcorner of our plot, also by interactions with the CMB).Compared to Ref. [36], we show in Fig. 1 the maximal energydirectly.

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For each particle species, the injection and energylosses/escapes are balanced by the steady state equation

QðEÞ ¼ @

@EðbðEÞNðEÞÞ þ NðEÞ

tesc; (4)

with tescðEÞ the characteristic escape time, bðEÞ ¼ �Et�1loss

with t�1lossðEÞ ¼ �1=EdE=dt the rate characterizing energy

losses, QðEÞ the particle injection rate ½ðGeV s cm3Þ�1�,and NðEÞ the steady particle spectrum ½ðGeV cm3Þ�1�. Forall charged particles, synchrotron energy losses and adia-batic cooling are taken into account. In addition, unstablesecondaries, i.e., pions, muons, and kaons, may escape viadecay. As a consequence, for pions, muons, and kaons,neglecting the adiabatic cooling, the (steady state) spec-trum is loss-steepened above the energy

Ec ¼ffiffiffiffiffiffiffiffiffiffiffi9��0�0

sm5=2c7=2

e2B; (5)

where synchrotron cooling and decay rates are equal. Fromthis formula one can see that the different secondaries,which have different masses m and rest frame lifetimes

�0, will exhibit different break energies Ec /ffiffiffiffiffiffiffiffiffiffiffiffiffiffim5=�0

q,

which solely depend on particle physics properties andthe value of B. These different break energies will lead toa spectral split of the neutrino spectra, which is an imprintof the magnetic field.

While being accelerated, the electrons lose energy intosynchrotron photons, which serve as the target photonfield. Charged meson production then occurs via

pþ � ! �þ p0; (6)

pþ � ! Kþ þ�=�; (7)

with these synchrotron photons, where the leading kaonproduction mode is included and p0 is a proton or neutron.In addition, two-pion and multipion production processesare included (not listed here); see Ref. [24] for details. Theinjection of the charged mesons is computed from thesteady state proton NpðEpÞ and photon N�ð"Þ spectra

with Ref. [24],

QbðEbÞ ¼Z 1

Eb

dEp

Ep

NpðEpÞZ 1

0d"N�ð"ÞRbðx; yÞ: (8)

Here x ¼ Eb=Ep is the fraction of energy going into the

secondary, y � ðEp"Þ=mp is directly related to the center of

mass energy, andRbðx; yÞ is a ‘‘response function’’ [24]. Theweak decays of the secondary mesons, such as

�þ ! �þ þ ��; (9)

�þ ! eþ þ �e þ ���; (10)

are described in Ref. [23], including the helicity dependenceof the muon decays. Thesewill finally lead to neutrino fluxesfrom pion, muon, kaon, and neutron decays.

In order to compute the � ( ¼ e,�, �) neutrino flux at

the detector, i.e., including flavor mixing, we sum over allthese initial neutrino fluxes of flavor �� weighted by theusual flavor mixing,

P� ¼ X3i¼1

jU�ij2jUij2; (11)

where U�i are the entries of the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix. We use sin212 ¼0:318, sin223 ¼ 0:5, and sin213 ¼ 0 for the sake of sim-plicity (see, e.g., Ref. [38]), leading to flavor equipartitionbetween �� and �� at the detector. We sum over neutrinos

and antineutrinos; i.e., if we refer to ‘‘��,’’ we mean the

sum of the �� and ��� fluxes.

Concerning the limitations of the model, this certainlydoes not apply exactly to all types of sources. For example,in supernova remnants, pp (proton-proton) or pA (proton-nucleus) interactions may dominate the neutrino produc-tion, which would require additional parameters todescribe the target protons or nucleons. In addition, atultrahigh energies, heavier nuclei may be accelerated(see discussion above). The spirit of this model is different:It is developed as the simplest (minimal) possibility in-cluding nontrivial magnetic field and flavor effects. Therelevant point for this study is that the instrument responsedepends on the neutrino energies and spectral features. Thespectral features, as a peculiarity of neutrinos, are, in casesof strong magnetic fields, dominated by the cooling anddecay of the secondaries (pions, muons, kaons). The en-ergy of the secondaries follows the proton energy, to a firstapproximation, Eb � xEp with x ¼ Oð0:1Þ to Oð1Þ. Thesetwo observations apply to any of these mentioned inter-actions, no matter if p�, pp, or pA, if the energy of theaccelerated particle is much larger than the one of theinteraction partner (Feynman scaling). Therefore, we ex-pect qualitatively similar results for these cases.Another variable is the target photon density, which is

assumed to come from synchrotron emission of coaccel-erated electrons here. In more realistic models, typically acombination of different radiation processes is at work. Forexample, there may be thermal photons radiated off anaccretion disk, proton synchrotron photons, inverseCompton up-scattered photons, photons from pair annihi-lation, and photons from �0 decays cascaded down tolower energies. It is therefore a frequently used approachfor a particular type of source to compute the neutrinospectrum from a target photon spectrum which corre-sponds to the observation without describing the originof these photons; see, e.g., Ref. [12] for gamma-ray bursts,where a band function parametrization of the observedgamma-ray fluxes is used. Qualitatively, the target photonscontrol the interaction rate of the primaries in p� inter-actions (see, e.g., [24]), which enters the normalization andis not used here. In addition, the spectral shape depends on

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the target photon spectrum in the energy range relevant forthe interactions. In many examples with strong magneticfields, a spectral break in the photon spectrum is lessimportant than the cooling and decay of the secondaries,which depend on particle physics only. For example, in ourmodel, different hypotheses for the cooling of the elec-trons, leading to the target photons, have been tested, andin most cases the relevant part of the spectral shape hardlychanges. Thus, while it is unlikely that the model appliesexactly to a particular source, it may be used as a goodstarting hypothesis. In addition, note that from Eq. (8), onlythe product of the proton and photon (and therefore elec-tron) density normalizations enters the final result. Thefinal normalization of the neutrino spectrum will dependon the source luminosity, the interaction volume, a possibleLorentz boost of the acceleration region �, and the redshiftz of the source. Since we only need the spectral shape, wedo not compute the normalization explicitly. Note, how-ever, that also the final neutrino energies depend on � andz. For the sake of simplicity, we neglect these effects,but one should keep in mind that the actually observedenergy spectrum could be significantly shifted in energy asE / �=ð1þ zÞ.

The main parameters of the model are R, affecting theshape and maximal energy (via t�1

ad ) of the primaries; B,affecting the maximal energy (via t�1

synchr) of the primaries

and the break (via Ec) of the secondaries; and �, affectingthe spectral slope of the primaries. The parameters Rand B can be directly related to the Hillas parameters;see Fig. 1.

III. METHOD AND IMPACT OF THESPECTRAL SHAPE

Here we describe our method, i.e., how we constrainindividual fluxes from IceCube data and quantify the re-sponse of the instrument. The simplest possible approach isbased on total event rates, which means that no informationon the reconstructed neutrino or muon energy is usedexplicitly. It can be described with the exposureExpðE; �Þ � Aeff

� ðE; �Þtexp, where Aeff� is the neutrino ef-

fective area and texp is the observation time. Here Aeff� ðE; �Þ

is a function of the flavor or interaction type (which we donot show explicitly), the incident neutrino energy E, andthe declination of the source �. The neutrino effective areaalready includes Earth attenuation effects and event selec-tion cuts to reduce the backgrounds, which depend on thetype of source considered, the declination, and the assump-tions for the input neutrino flux, such as the spectral shape.Normally, the cuts are optimized for an E�2 flux, whichmeans that for specific fluxes with different shapes, thefollowing analysis may slightly improve by an optimiza-tion of the detector response. On the other hand, onehas to understand that the experiments cannot optimizetheir event selection for any possible input spectrum.The total event rate of a neutrino telescope can be

obtained by folding the input neutrino flux with the expo-sure as

N ¼Z

dEExpðE; �ÞdNðEÞdE

¼Z

dEAeff� ðE; �Þtexp dNðEÞ

dE: (12)

Here dNðEÞ=dE is, for point sources, given in units ofGeV�1 cm�2 s�1. If backgrounds are negligible, the 90%(Feldman-Cousins) sensitivity limit K90 for an arbitrarilynormalized input flux used in Eq. (12) can be estimated asK90 � 2:44=N [39]. This implies that a predicted flux at thelevel of the sensitivity limit, irrespective of the spectralshape, would lead to the same number (2.44) of events. The90% confidence level differential limit E2dN=dE can bedefined as 2:3E=ExpðE; �Þ; see, e.g., Ref. [20].2 We willcomment on the impact of the backgrounds below.In the following, we use the neutrino effective areas for

time-integrated point source searches in IceCube-40 [13],based on 2008–2009 data with texp ¼ 375:5 days. We show

these neutrino effective areas for muon tracks and for sixdeclination bands, as in the original reference, in Fig. 2. Forthe sake of readability, we use ‘‘downgoing,’’ ‘‘upgoing,’’or ‘‘(quasi)horizontal’’ as additional descriptions of thedeclination bands. Note that for IceCube located at theSouth Pole, a particular source will always appear undera specific declination, which means that the position in thesky will have some impact on the sensitivity, and thedeclination bands correspond to different source classesor data sets. Since upgoing muon neutrinos become ab-sorbed in Earth matter above about 30 PeVand strong cutshave to be applied for the downgoing events to reduce thecosmic muon background, quasihorizontal events implythe optimal performance.The effective areas shown in Fig. 2 are for muon tracks

only. However, note that there are two contributions: Notonly do �� produce muon tracks, but also ��, since the tau

lepton decays into a muon with a branching ratio of about17%.We show the individual contributions in Fig. 2, wherethin solid curves represent the �� effective areas, the

dashed curves the �� effective areas, and the thick solidcurves the total effective areas for either �� or �� if flavor

equipartition ��: �� ¼ 1:1 is assumed. Note that the total

effective areas for the individual �� or �� fluxes are given

by the maximum of the individual effective areas, not thesum. Assuming flavor equipartition, one can read off from

2This is the factor weighting E2dN=dE in the integrand ofEq. (12) if integration over LogðEÞ is used. If the differentiallimit and flux are smooth enough on a logarithmic energy scale,the flux limit typically is below the differential limit sinceenough contribution to the integral is obtained. However, ifspikes or sudden flux or differential limit changes are present,the flux limit may also exceed the differential limit locally. Anexample is the Glashow resonance process, which is sometimestaken out of the analysis for this reason. Note that in this study,LogðEÞ ¼ Log10ðEÞ everywhere.

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Fig. 2 that quasihorizontal muon tracks are dominated by�� events. The downgoing muon tracks also come mostly

from ��, but there is a small region at high energies where

�� dominates. In this case, the muon tracks are recovered atlower energies. The upgoing muon tracks above 30 PeV,where the Earth becomes opaque to ��, are dominated by

�� events. Therefore, it is a priori not clear if and when themuon tracks from �� can be neglected. In general, ouranalysis will be based on �� interactions, but we will

come back to this point in Sec. IVD. Note that beyondthe limits shown in Fig. 2, we use linear extrapolation,which turns out to be a good approximation because it onlyaffects the regions of poor sensitivity.

The backgrounds for the point source analysis in

Ref. [13] mostly come from atmospheric neutrinos forupgoing events, and from cosmic muons for downgoing

events. For our analysis, we assume that backgrounds can

be suppressed to a sufficient level, which, for point sources,can be achieved by the angular resolution of about 1� for

upgoing events. For example, if the photon counterpart isused to preselect potential sources (‘‘a priori source can-

didate search’’), the number of background events is rela-tively low, see Table 3 in Ref. [13]. For downgoing events,

the suppression of the cosmic-ray muon background is

more difficult, which means that here the assumptionof low backgrounds has to be interpreted with care and

2 3 4 5 6 7 8 910 3

10 2

10 1

100

101

102

103

104

effe

ctiv

ear

ea(m

2)

90°, 60° downgoing

2 3 4 5 6 7 8 910 3

10 2

10 1

100

101

102

103

104

Log E (GeV)Log E (GeV)

Log E (GeV)Log E (GeV)

effe

ctiv

ear

ea(m

2)

60°, 30° downgoing

2 3 4 5 6 7 8 910 3

10 2

10 1

100

101

102

103

104

effe

ctiv

ear

ea(m

2)

30°,0° horizontal

2 3 4 5 6 7 8 910 3

10 2

10 1

100

101

102

103

104

effe

ctiv

ear

ea(m

2)

0°,30° horizontal

2 3 4 5 6 7 8 910 3

10 2

10 1

100

101

102

103

104

Log E (GeV)

effe

ctiv

ear

ea(m

2)

30°,60° upgoing

2 3 4 5 6 7 8 910 3

10 2

10 1

100

101

102

103

104

Log E (GeV)

effe

ctiv

ear

ea(m

2)

60°,90° upgoing

FIG. 2 (color online). Neutrino effective area as a function of the neutrino energy for different declinations in IceCube-40. The thinsolid curves correspond to muon tracks from ��, and the dashed curves to muon tracks from �� after � decay. The thick solid curves

show the neutrino effective area for either �� or �� if flavor equipartition between �� and �� is assumed. Data are taken from Fig. 8 in

Ref. [13] and rearranged.

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somewhat higher backgrounds are expected.3 In eithercase, we can reproduce the sensitivity for an E�2 flux inFig. 19 of Ref. [13] very well. We have also checked theflux of atmospheric neutrinos to be expected within 1�angular resolution; see, e.g., Ref. [40]. It turns out that,

especially in the energy range between 100 GeV and

10 TeV, where Earth attenuation effects are relatively

small, the IceCube-40 point source sensitivities for up-

going events are already touching the atmospheric neutrino

background in the most conservative (systematics) case for

the atmospheric neutrino flux. This means that in this

energy range, we expect that the IceCube sensitivity will

be background-limited very soon, whereas for the present

analysis, neglecting the atmospheric neutrino background

is a good approximation.We apply the above mechanism to obtain a limit for

arbitrary fluxes to our model. We show in Fig. 3 thelimits for selected �� spectra (including flavor mixing)

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FIG. 3 (color online). Limits for selected �� spectra (including flavor mixing) for different declination bands inIceCube-40 (90% CL). The numbers of the individual curves correspond to the test points in Fig. 1. The thick lines showthe limits for an E�2 flux (in the dominant energy range), and the thick curves the differential limits. Here the injectionindex � ¼ 2 is chosen.

3In fact, the cuts for the downgoing events seem to be chosensuch that the backgrounds are roughly a factor of 2 higher for thedowngoing events than the upgoing events; see examples inTable 3 in Ref. [13]. However, the absolute number of expectedbackground events is nevertheless only a few, which means thatthe impact on the final sensitivity limit using the Feldman-Cousins approach [39] is relatively small on a logarithmic scale.

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for different declination bands in IceCube-40. We alsoshow as thick lines the limits for an E�2 flux (in thedominant energy range4), and as thick curves the differen-tial limits. The �� spectra correspond to the test points as

marked in the figure, which correspond to the parameters Rand B in Fig. 1 and � ¼ 2. From the differential limits, wecan read off that the optimum sensitivity for downgoingevents is found at higher energies�PeV, whereas upgoingevents are best constrained at about 10 TeV [13]. Thequasihorizontal events exhibit a relatively broad differen-tial limit. As far as the absolute sensitivity is concerned(see, e.g., thick solid lines for integrated limits), horizontaland upgoing events have similar sensitivities, but thedowngoing events face lower statistics because of thenecessary cuts to reduce the cosmic-ray muon background.As far as the limits for the individual spectra are concerned,we make a few observations. First of all, the best sensitiv-ity, i.e., lowest normalization, for a particular spectrum istypically obtained if the spectral peak in E2dN=dE coin-cides with the minimum of the differential limit. Theoptimum sensitivity is then, to a first approximation, givenby the maximum proton energy, which roughly determinesthe position of the peak and can be read off from Fig. 1. Forexample, TP 11 is best constrained by upgoing events.However, especially if strong magnetic field effects arepresent and therefore the spectral shape becomes morecomplicated, such as for TP 2 for downgoing events, thisrule does not necessarily apply anymore. In this case, therecan be quite some impact of neutrinos from kaon decays, aswe will discuss in Sec. IVC. Note that, for a particularsource, the declination is, of course, predetermined. Theway to interpret Fig. 3 therefore goes as follows: if aparticular spectral shape for a source of declination � isdescribed by a TP of this model, the sensitivity can be readoff from the corresponding panel of Fig. 3.

From Fig. 3, one can easily see that it is not trivial how tocompare two different spectra with different spectralshapes. Consider, for example, the fluxes 2 and 11 in theupper left panel, both leading to the same event rate bydefinition. Which of the two neutrino sources leading tothese fluxes can be better constrained by the downgoingevents? In order to quantify this aspect, it is useful to assigna single number to each spectrum which measures howmuch energy in neutrinos can be tested for a specificspectrum and event type. We choose the energy flux density

� ¼Z

EdNðEÞdE

dE (13)

as this quantity, which we show in units of erg cm�2 s�1 forpoint sources in order to distinguish it from E2dN=dE inunits of GeV cm�2 s�1 (1 erg ’ 624 GeV). This quantity

measures the total energy flux in neutrinos, and it is usefulas a performance indicator measuring the efficiency ofneutrino production in the source.In order to see that, consider the transformation of the

injection spectrum of the neutrinos Q0� (in units of

GeV�1 cm�3 s�1) from a single source into a point sourceflux (in units of GeV�1 cm�2 s�1) at the detector dN=dE,which is (before flavor mixing and neglecting a possibleLorentz boost or beaming) given by (see, e.g., Ref. [32])

dNðEÞdE

¼ Vð1þ zÞ24�d2L

Q0�; E ¼ 1

1þ zE0: (14)

Here, V is the volume of the interaction region and dLðzÞ isthe luminosity distance. For the energy flux density, onehas

� ¼ L�

4�d2L; where L� ¼ V

ZE0Q0

�dE0 (15)

is the ‘‘neutrino luminosity.’’ Since the neutrinos originatemostly from pion decays and take a certain fraction of thepion energy (about 1=4 per produced neutrino for eachcharged pion), the neutrino luminosity is directly propor-tional to the (internal) luminosity of protons Lint (or theproton energy dissipated within a certain time frame �T)and the fraction of the proton energy going into pionproduction, commonly denoted by f�. This quantity is ameasure of the efficiency of pion production.5 Since apossibly emitted photon flux can be linked to Lint by energyequipartition arguments, one has � / f� � Lint /f� � L�, and � is a measure for the pion production

efficiency times luminosity of the source (if no photoncounterpart is observed), or even the pion production effi-ciency itself (if a photon counterpart is observed). Forexample, for GRBs, this can be nicely seen inAppendix A of Ref. [41], from which it is also clear thatredshift and � cancel if the neutrino flux is related to aphoton observation (the neutrinos and photons will, to afirst approximation, experience the same Lorentz boosts,beamings, and redshifts). Therefore, the sensitivity to� fordifferent spectra yielding the same number of events (as thefluxes do at the sensitivity limit) can be regarded as theprime performance indicator. Consider, for instance, twovery similar sources producing neutrino spectra withslightly different shapes. Then the sensitivity to �, andtherefore to the pion production efficiency, may be verydifferent, and it is fair to say that one source can be betterconstrained than the other. Consider, on the other hand, twosources producing very different neutrino spectra. In thatcase, the sensitivity to � could be even similar, whichmeans a similar f� � Lint is required for detection. Our

4The choice of this energy range is somewhat arbitrary, as longas the main contribution to the integral, i.e., the minimum of thedifferent limit, is well contained in the integration range. Oftenthe range where 90% of the events are expected is shown.

5It can be roughly approximated as f� ’ 0:2R= p� for the �resonance, where p� ¼ 1=ðn��Þ is the mean free path of theproton, n� is the photon number density, and � is the interactioncross section.

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sensitivities to � in the following sections have to beinterpreted in that way. Finally, note again that our methoddoes not predict the expected level of a neutrino flux,which means that in specific cases f� � Lint might actuallybe much higher than in other ones for astrophysical rea-sons. Here, however, we discuss only the interplay betweenspectral shape and detector response.

IV. CONSTRAINTS FROM ICECUBE

Here we present our main result, the sensitivity to thespectral shape of sources as a function of the Hillas pa-rameters, and we discuss the dependence on the injectionindex �, the impact of kaon decays, and �� detection viamuon tracks.

A. Sensitivity on Hillas plane

For a comprehensive parameter space scan, we computefor each set ðR; BÞ in Fig. 1 the �� spectrum including

magnetic field and flavor effects. Then we normalize itwith Eq. (12) for each event type, as it is illustrated in Fig. 3for several examples, and we compute the energy fluxdensity according to Eq. (13). The resulting sensitivity to� is shown in Fig. 4 as a function of R and B for � ¼ 2 andthe different declination bands in the different panels. Thedarkest regions mean the highest sensitivities, as it isshown in the plot legends. For downgoing events, sensitiv-ities as low as 10�8 erg cm�2 s�1 can be achieved, and forupgoing events, sensitivities as low as 10�10 erg cm�2 s�1.The best parameter space coverage is actually obtained fornear-horizontal upgoing events [� ¼ ð0�; 30�Þ], whichmeans that sources with relatively low f� � Lint can bedetected there.

As one result, the detector responds very well to theusual suspects, such as AGN cores and jets (TP 3 and 4),and to sources on galactic scales, such as TP 9 and 10. Forinstance, in the upper right panel (downgoing events),these are just in the optimal region. However, the bestabsolute sensitivities are obtained for TP 2 and 11 ratherthan TP 3 and 4 (see upgoing events in the lower row). Thereason is the relatively low optimal energy for upgoingevents (see Fig. 2). In addition, TP 5, 7, and 8 are not withinthe optimal sensitivity ranges, since these spectra peak atrelatively high energies (see, e.g., Fig. 3 for TP 8). In fact,in our model, exactly these spectra stem from very high-energy protons; i.e., these test points may be the bestcandidates to produce the highest-energetic cosmic-rayprotons (see regions marked UHECR in Fig. 4). This justshows an intrinsic feature of IceCube, which should berather model independent: the differential limits in Fig. 2peak at relatively low energies, while the neutrino energiesfor interactions of 1021 eV protons in the source may beexpected at 1019 to 1020 eV (1010 to 1011 GeV). Therefore,IceCube may not be the best instrument to test the nature ofthe highest-energetic cosmic-ray sources—although it maycompensate for that by its size. Note that this argument

does not depend on the composition of the UHECR, sinceit depends on the energy of the particles, not on the regionof the Hillas parameter space. Moreover, there is a part ofthe parameter space which is very difficult to test: sourceswith B> 105 Gauss from downgoing events may be espe-cially difficult to find in IceCube, which means that galac-tic sources with strong magnetic fields are per se difficultto access, unless they are very luminous.6 Here DeepCoremay improve the sensitivity substantially, since it has alower threshold.

B. Dependence on injection index

So far, we have chosen � ¼ 2 as the universal injectionindex for protons and electrons, as it may be roughlyexpected from Fermi shock acceleration. This, however,is not the spectral index of the neutrino spectrum. In ourmodel, the target photons are produced by synchrotronradiation from the coaccelerated electrons, and the (syn-chrotron) energy losses of the electrons are taken intoaccount. As a result, the spectral index of the neutrinospectrum is approximately �=2 up to the maximal energycutoff or the critical energy in Eq. (5), where the secondaryspectra become loss-steepened by synchrotron losses. Thisspectral shape is similar, for instance, to neutrinos fromblazar jets; see, e.g., Ref. [42]. We show a correspondingexample for TP 4 in Fig. 5 for two different declinationbands. In this figure, the sensitivities and differential limitsare shown for several values of �. One can easily see thatthe spectral index of the neutrino spectrum between about1 GeVand 1 PeV is roughly�=2. Since energy losses of thesecondaries are small in this case, the spectral shape isrelatively simple. For the case of the downgoing tracks(left panel), the peak of the spectrum coincides with theminimum of the differential limit. The energy flux sensi-tivity decreases as � increases, since more energy can behidden in the neutrino spectrum at low energies. For theupgoing events (right panel), the peak of the spectrum isfound at higher energies than the minimum of the differ-ential limit. In this case, increasing � means that theneutrino spectrum can be better constrained and that thesensitivity, which is determined by the high energies,improves.It turns out that all test points in Fig. 1 can be separated

into two categories: TP 1, 2, 11, and 12 always follow thetrend in Fig. 5, left panel, since the neutrino energies arerelatively low. The other test points exhibit a behaviorsimilar to TP 2: the optimum sensitivity depends on thedeclination of the source. We show the energy flux sensi-tivity as a function of the injection index � for two ex-amples (TP 2 and TP 4) in Fig. 6. The different curvesrepresent the six different declination ranges used forIceCube-40, from downgoing [�¼ð�90�;�60�Þ] (dotted)

6Note that the energy losses of the secondaries decrease thefraction of energy going into neutrinos even further.

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to upgoing [� ¼ ð60�; 90�Þ] (solid) with decreasing dashgaps. In the right panel, we find the functionaldependence for TP 4, which we have qualitativelydescribed above for upgoing and downgoing events.Note that the actual dependence on � is relatively flat forthe downgoing and quasihorizontal events, at leastfor reasonable injection indices (gray-shaded range),whereas for upgoing events the sensitivity significantly

improves with �.7 For TP 2 (left panel), low injectionindices are always preferred, but the dependence on � ismoderate.

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FIG. 4 (color online). Energy flux density (erg cm�2 s�1) sensitivity as a function of R and B for different declination bands inIceCube-40 (90% CL). The different contours (colors) correspond to the regions where a specific energy flux sensitivity can beexceeded. Here the injection index � ¼ 2 is chosen. The dashed regions UHECR indicate where 1020 eV cosmic-ray protons areexpected to be produced in the model.

7At about � ¼ 4, a minimum is found in these cases. At thisvalue, the neutrino spectral index is roughly 2, which means thatthe slope in the E2dN=dE plot changes at around this value fromincreasing to decreasing.

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C. Impact of neutrinos from neutron and kaon decays

Here we discuss two neutrino fluxes often not taken intoaccount: For sources optically thin to neutrons, most of theneutrons which are produced in photohadronic interactions[see Eq. (6)] will decay into electron antineutrinos eitherinside or outside the source. The protons produced in thesedecays may contribute to the cosmic-ray flux. The neutri-nos carry only a very small fraction of the neutron energy,which means that this flux is normally present at very lowenergies. However, for sources with very strong magneticfields, the neutrino flux from neutrons may actually bedominant, since the parents are electrically neutral. Inaddition, kaons can be produced by interactions such asEq. (7). While the contribution of the neutrino flux fromkaon decays is usually small, the higher value of the criticalenergy Eq. (5) compared to muons and pions may lead to adominant neutrino flux from kaon decays at high energiesfor strong enough magnetic fields.

All these effects are fully taken into consideration in ourcalculations. Nevertheless, we show in Fig. 7 the impact ofthese two neutrino fluxes in two examples. In this figure, thelimits for several �� spectra for two different selected dec-

lination bands are shown. The solid curves include neutrinosfrom kaon and neutron decays; the dashed curves show thelimits if only neutrinos from the pion decay chain are con-sidered. In the left panel, one can see a clear enhancement ofthe flux at high energies in all cases, which comes from theadditional kaon decay component. For TP 2, the additionalhump coincides with the differential limit minimum, whichmeans that it contributes to the sensitivity (the solid curve isbelow the dashed curve for lower energies). The same appliesto TP 1 and 12, although not so clearly visible. In the rightpanel, the contribution for TP 2 is negligible, since the solidand dashed curves coincide. There is, however, some con-tribution in the other two cases. For TP 1 one can also seethe contribution from neutron decays dominating at low

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FIG. 5 (color online). Flux limit for several different injection indices � ¼ 1:5, 2, 2.5, and 3 for test point 4 (AGN jets) and twodifferent declination bands (90% CL). The upper thick curves show the differential limits.

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FIG. 6 (color online). Energy flux density sensitivity as a function of the injection index � for two selected test points (90% CL). Thedifferent curves represent the six different declination ranges used for IceCube-40, from downgoing [� ¼ ð�90�;�60�Þ] (dotted) toupgoing [� ¼ ð60�; 90�Þ] (solid) with decreasing dash gaps. The gray-shaded regions mark the � ranges which may be roughlyplausible for Fermi shock acceleration.

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energies. However, while kaon decays help in parts of theparameter space, especially for downgoing events (where thedifferential minimum is at higher energies), the impact ofneutron decays is, in general, small.

D. Impact of muon tracks from ��?

In Fig. 2 we have compared the neutrino effective areasfor muon tracks from �� and ��. Assuming flavor equi-

partition between �� and ��, are there parts of the parame-

ter space where the muon tracks from �� actually limit thesensitivity? In fact, we show in Fig. 8 several exampleswhere we find some impact. In this figure, the thick solidcurves show the differential limits including the �� events(corresponding to the thick solid curves in Fig. 2), andthe thick dashed curves the contributions from �� only.

The thin solid curves show selected spectra using theinclusive effective area, and the thin dashed curves showthe corresponding spectra for �� based events only. For the

downgoing events (left panel), there is especially some

impact at low energies, where the sensitivities can besignificantly improved. For the upgoing events, the im-proved effective area for high energies has hardly anyimpact, at least for � ¼ 2, for which the spectrum isparallel to the differential limit. Only for �< 2, someimprovement may be expected. Redrawing Fig. 4 includingthe �� events, the result mostly deviates a bit in the high-energy region in the lower right. The effect at low energiesis hardly visible, because the absolute sensitivities for TP 1and 12 are worse than the contours shown.

V. COMPLEMENTARITY TO OTHEREXPERIMENTS

Here we qualitatively point out the complementarityamong different data and different experiments, and wecomment on the potential of future experiments. There area number of high-energy neutrino data, such as fromAMANDA [43,44], IceCube-22 cascades [45], Auger[20], RICE [46], and ANITA [47], to name a few examples.

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FIG. 7 (color online). Impact of neutrinos from kaon decays: Limits for several �� spectra (including flavor mixing) for two differentselected declination bands in IceCube-40 (90% CL). The solid curves include neutrinos from kaon (and neutron) decays, and thedashed curves represent neutrinos from the pion decay chain only. The numbers of the individual curves correspond to the test points inFig. 1. The thick curves show the differential limits. Here the injection index � ¼ 2 is chosen.

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FIG. 8 (color online). Impact of �� detection: Limits for several �� spectra (including flavor mixing) for two different selecteddeclination bands in IceCube-40 (90% CL). The solid curves are based on the detection of muon tracks from both �� and �� assuming

flavor equipartition, and the dashed curves are based on the detection of muon tracks from �� only. The numbers of the individual

curves correspond to the test points in Fig. 1. The thick curves show the differential limits for both �� and �� (solid) and �� only

(dashed). Here the injection index � ¼ 2 is chosen.

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Most of these data (except from AMANDA) have beenapplied to diffuse flux limits, which makes a comparison tothe IceCube point source results, which are discussed inthis study, per se difficult. While AMANDA has, in prin-ciple, a lower threshold than IceCube, it is not clear fromthe present literature if the neutrino effective areas aresignificantly better at low energies than for IceCube-40,since the larger volume may compensate for that. TheRICE and ANITA experiments are based on radio detec-tion initiated by cascades in the Antarctic ice. In thesecases, as for cascades in IceCube, it is difficult to quantifythe performance for a particular flavor. For example, inRef. [45] (IceCube cascades), a flavor composition of 40%electron neutrinos, 45% tau neutrinos, and 15% muonneutrinos was given for an E�2 extragalactic test flux;see also the discussion in Appendix I in Ref. [46](RICE). The only exception in this list is Auger: Earth-skimming �� may produce tau leptons in the Earth, whichmay escape the Earth after energy losses, decay in theatmosphere, and produce an extensive quasihorizontal,slightly upgoing air shower detectable by the Auger sur-face detector. Since this signal is practically flavor-clean,we illustrate our main points with 2004–2008 data from theAuger experiment [20], which appears to be the simplestexample. However, note that the other experiments alsoimply complementary information.

A. Earth-skimming neutrinos in Auger

The sensitivity for arbitrary fluxes for Auger can be veryeasily obtained from Eq. (12). We use the ‘‘conservativesystematics’’ exposure from Table III of Ref. [20] for oursimulation. As the main differences to IceCube, diffuseflux limits are discussed, and the event sample is free ofbackgrounds from the atmospheric neutrino flux in thatenergy range. Since the viewing window constantlychanges, there is a strong declination dependence of thesensitivity. However, for the sake of comparison, one maycompute a ‘‘quasipoint source’’ flux by multiplying thediffuse flux with the viewing window solid angled�� 0:6.8 Since this number is of order unity, we usethe diffuse flux in the case of Auger directly. However, oneshould keep in mind this qualitative difference when onecompares the numbers, which cannot be easily avoided.9

We show in Fig. 9 the limits for selected �� spectra(including flavor mixing) for Auger 2004–2008 data(90% CL). The numbers of the individual curves corre-spond to the test points in Fig. 1. The thick lines showthe limit for an E�2 flux (in the dominant energy range),and the thick curves the differential limit. Comparing toFig. 3, one can easily see that the spectra peaking atvery high energies can especially be very well con-strained. Take, for instance, TP 8, and compare theresult to Fig. 3, lower left panel. It is obvious that thenormalization can in this case be better constrained byAuger, and, consequently, the energy flux density. Wehave also tested the impact of neutrinos from kaondecays here. Because the neutrinos from kaon decaysshow up at high energies and the Auger sensitivity (seedifferential limit) is dominant at about 109 GeV, it turnsout that the kaon component can be especially importantin that case; see also Ref. [27]. One example, TP 3, isshown in Fig. 9, where the rightmost hump is the addi-tional contribution from kaon decays which clearly lim-its the sensitivity.

B. Complementarity among different dataand experiments

In order to illustrate the complementarity of differentdata and experiments, we show in Fig. 10 the regionswhere the sensitivity exceeds 10�9 erg cm�2 s�1 ½sr�1�for �� detection (left panel) and �� detection (right

panel) for several selected data samples. In the case ofIceCube, only muon tracks from �� or �� have been

used; see neutrino effective areas in Fig. 2. In the case ofAuger, the diffuse flux limit for Earth-skimming tauneutrinos has been used. Note that (at least if the � trackcannot be separated) IceCube cannot distinguish theoriginal flavor of the neutrino leading to a muon track,

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FIG. 9 (color online). Limits for selected �� spectra (includingflavor mixing) for Auger 2004–2008 data (90% CL). The num-bers of the individual curves correspond to the test points inFig. 1. The thick line shows the limit for an E�2 flux (in thedominant energy range), and the thick curve the differentiallimit. Here the injection index � ¼ 2 is chosen.

8The solid angle d�� 2�Rd cos� 0:6, since is inte-

grated from �=2 to �=2þ �m with �m ¼ 0:1; see discussionafter Eq. 4 in Ref. [20].

9A point source analysis at Auger is presently being performed[48], using 2004–2010 data. For a more detailed analysis thanthe one presented here, including the strong declination depen-dence, the exposure as a function of energy and declinationwould be needed, which is currently not publicly available.However, from the comparison with Ref. [48], our sensitivitywith the 2004–2008 data roughly corresponds to a point sourcesensitivity at the declination 50� & j�j & 55� with the updated2004–2010 data set (replace units by GeV cm�2 s�1 on thevertical axis in Fig. 9).

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while Auger can. With this figure, we can qualitativelyillustrate a number of points:

1. Viewing window complementarity

Different data samples correspond to different viewingangles. In the case of IceCube, the upgoing events test thenorthern sky, and the downgoing events the southern sky.From the left panel, we clearly see the better sensitivityof, for instance, the quasihorizontal upgoing events com-pared to the downgoing ones. For almost vertical down-going events, the sensitivity is even poorer (cf., Fig. 4).New experiments built in the Northern hemisphere, suchas ANTARES or KM3NeT, will complement thissensitivity.

2. Energy range complementarity

Different experiments or data will test different energyranges. For example, from the right panel, IceCube cancover a wide region of the parameter space. However, forthe high B (low energy) region, which is almost empty,experiments with a lower threshold will be needed. TheDeepCore part of IceCube is especially expected to extendinto this region. In the lower right corner marked UHECR,however, where the highest-energetic cosmic-ray sourcesare suspected for protons, Auger in fact provides the bestsensitivity, given our assumptions. Therefore, Auger andthe radio-detection experiments may in fact be promisingtechniques to reveal the nature of these. The Auger Northproject [49], as well as the JEM-EUSO project [51], mayespecially achieve a substantially higher exposure in ex-actly that energy range [50].

3. Flavor complementarity

In Fig. 10 the left and right panels correspond to differ-

ent neutrino flavors at the detector. If, for some reason, the

equipartition between �� and �� is severely perturbed, it is,

first of all, interesting that IceCube can, at least for upgoing

or quasihorizontal upgoing events, still test most of the

parameter space relatively well (right panel). Of course, the

Auger limit applies to �� only, but in the case of flavor

equipartition it can be directly applied to ��. It is an

interesting theoretical question when one may expect a

strong deviation from the equipartition between �� and

�� at the detector, which is a consequence of nearly maxi-

mal atmospheric mixing.In the standard model including massive neutrinos,

flavor equipartition between these two fluxes relies ontwo necessary conditions: 13 ¼ 0 and 23 ¼ �=4, whichguarantee flavor equipartition regardless of the initialflavor composition at the source. Note that exact tribi-maximal mixings (sin212 ¼ 1=3) are not necessary. Wehave chosen these values in this study, which means thatall of our results are exactly ��-�� symmetric. However,

there may be deviations from these values. Therefore, wehave checked that for arbitrary initial flavor compositions(without �� contamination at the source) the ratiobetween �� and �� can, at most, vary between about

0.5 and 2 for the currently allowed 3� ranges of themixing parameters [38]. This means that within the stan-dard model, the �� and �� fluxes at the detector have to

be equal up to a factor of 2, and any limit on �� (��) can

be directly translated into a limit for �� (��) within a

5 10 15 20

5

0

5

10

15

Log R (km)

Log

B(G

auss

)

IC 40, 60°, 30° downgoing

IC 40, 0°,30° horizontal

1

2

3

4

5

6

7

89 1011

No acceleration

Sensitivity 10 9 erg cm 2 s 1

NeuCosmA 2011

UHECR

5 10 15 20

5

0

5

10

15

Log R (km)

Log

B(G

auss

)

IC 40, 60°,90° upgoing

IC 40, 0°,30° horizontal

Auger

1

2

3

4

5

6

7

89 1011

No acceleration

Sensitivity 10 9 erg cm 2 s 1 sr )( 1

NeuCosmA 2011

UHECR

FIG. 10 (color online). Regions where the sensitivity exceeds 10�9 erg cm�2 s�1 ½sr�1� for �� detection (left panel) and �� detection(right panel) for several selected data samples (90% CL, � ¼ 2). The dashed regions UHECR indicate where 1020 eV cosmic-rayprotons are expected to be produced in the model. See main text for details.

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factor of 2.10 On logarithmic scales, flavor equipartition istherefore guaranteed.

This picture may change if physics beyond the standardmodel is considered. First of all, strong perturbations of theflavor equipartition between �� and �� at the detector may

be unlikely if the physics beyond the standard model causeseffects at the production or propagation of the neutrinos,since maximal atmospheric mixing will lead to equilibra-tion of the two flavors again. Nevertheless, such exoticscenarios have been discussed in the literature, such as inthe context of CPT violation; see Ref. [52] for a discussion.The most plausible reason to perturb this equipartition maybe new physics contributing to the detection processes, or,for upgoing events, new physics modifying the interactionsin the Earth. Examples are nonstandard interactions at thedetection, such as �ud��, which are, however, limited to about

10% compared to the standard model [53]. Another possi-bility is a potential superluminal motion of muon neutrinos,which OPERA has recently claimed at the 6� confidencelevel [54], if a flavor-dependent effect is at work whichadvances or delays one flavor compared to the others (seealso Ref. [55]). In addition, it is not clear how the deepinelastic scattering cross sections evolve to high energies.Therefore, considering different flavors may still be con-sidered complementary.

VI. SUMMARYAND CONCLUSIONS

The main motivation of this study has been the dis-cussion of the interplay between spectral shape and de-tector response at neutrino telescopes beyond a simpleE�2 assumption for the neutrino flux, from a particlephysics perspective. Several effects often not taken intoaccount have been included, such as magnetic field ef-fects on the secondaries and flavor mixing. In particular,the impact of neutrinos from neutron and kaon decays hasbeen discussed, as well as the impact of � decays intomuon tracks in the detector. As data samples, we haveused 2008–2009 data from IceCube-40 for time-integrated point source searches, and 2004–2008 datafrom Auger. As our parameter space of interest, wehave used the Hillas plot, described by the parametersR (size of the acceleration region) and B (magnetic field).For the description of the neutrino spectra, we have useda self-consistent model for the neutrino production, inwhich protons interact with synchrotron photons fromcoaccelerated electrons (positrons) [25]. While this modelcertainly does not apply to all spectra on the Hillas plot, ithas been useful to illustrate some of the main qualitativepoints as a function of R and B.

In order to compare the detector response to differentspectral shapes, we have used the sensitivity to the energy

flux density. The energy flux density has been introducedas a measure for the source luminosity L times pionproduction efficiency f� (if no photon counterpart is ob-served), or even f� directly (if the neutrino spectrum iscompared to a photon observation). Here the pion produc-tion efficiency is the fraction of proton energy dumped intopion production. This means that the ‘‘sensitivity’’ in thiswork can be roughly interpreted as the sensitivity to f�.Conversely, a source in a region of high sensitivity will bemore easily detectable, i.e., for lower luminosities and f�,than a source in a region of low sensitivity. Note that we donot make any prediction for the expected level of theneutrino flux, as it can only come from astrophysicalarguments.Since a particular source will be seen with a specific

declination in IceCube, the final (shape) sensitivity will bedeclination dependent. As one result, we have confirmedthat IceCube has excellent sensitivity to spectral shapescorresponding to AGN cores or jets for the model consid-ered, which are among the usual suspects for significantneutrino production, for all declinations. For example, theyare found to be within the optimal region for downgoingtracks in IceCube. However, even better absolute sensitiv-ities are obtained for quasihorizontal and upgoing eventsfor the spectral shapes from potential source classes suchas white dwarfs, whereas the AGN blazar spectra may peakat energies a bit too high for these source declinations. Thestrong magnetic field region, for which the maximal protonenergy is synchrotron limited, cannot be easily testedby IceCube because of the relatively high threshold.Therefore, galactic sources with high magnetic fields andsignificant neutrino production are per se difficult to find.Data from DeepCore may significantly improve this pa-rameter region. Note that some optimization of the detectorresponse may be performed for specific spectral shapes,which we have not taken into account.The parameter space region from which the highest-

energetic cosmic-ray protons may be expected, i.e.,E� 1020 eV, is in fact somewhat better covered byAuger, at least in principle. This can be understood fromgeneric arguments, recovered in the model used: 1020 eVprotons colliding with much less energetic target photonswill lead to neutrinos of about 109 GeV to 1010 GeV,which is just around the optimum of the differential limitof Auger. The conclusion holds for neutrino spectra sig-nificantly harder than E�2, which is a common expectationif the target photons come from synchrotron emission.Therefore, the common picture of an E�2 neutrino fluxcan be clearly misleading, and the search for neutrinosfrom the sources of the highest-energetic cosmic raysmay greatly benefit from future experiments, such asAuger North or JEM-EUSO. This argument does notdepend on the composition of the highest-energeticcosmic rays, since it is related to their energy, not theirnature.

10The deviations from flavor equipartition between �� and ��

can, in fact, be largest if the initial flavor composition isdominated by �e.

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We have also tested the dependence on other modelparameters, such as the injection index �, which hassome impact, especially for spectra extending to highenergies for upgoing events in IceCube. Again, it is clearthat general conclusions on astrophysical sources cannotbe drawn independent of the neutrino spectral shape be-cause of the interplay with the detector response. We have,for instance, shown several examples where the neutrinospectrum peaks off the energy range IceCube is mostsensitive to, which means that rather large f� � L arerequired in order to be detectable. In turn, this means thata source may not be found in spite of a correct astrophys-ical prediction of the flux, often based on energy equipar-tition arguments, simply because the flux shape does notmatch the detector response.

As far as the particle physics is concerned, we haveshown that neutrinos from kaon decays improve the sensi-tivities in cases where magnetic fields lead to an additionalkaon decay hump in the spectrum, and this hump coincideswith the optimal sensitivity range. This effect can beespecially prominent in Auger, since the kaon decay partshows up at the high-energy end of the spectrum. InIceCube, the contribution from ��-induced muon tracks(via the leptonic � decay channel) improves the sensitiv-ities in parts of the parameter space if flavor equipartitionbetween �� and �� at the detector is assumed, which

means that it should not be a priori neglected.Finally, we have emphasized the complementarity

among different event samples and experiments, such as

with respect to the viewing window, accessible energyranges, and different measured flavors. For example, wehave demonstrated that even if the equipartition between�� and �� is strongly perturbed, IceCube can cover most of

the discussed parameter space at least for upgoing eventsalready from the ��-induced contribution to the muontracks. While such a perturbation can only be expectedwithin the (neutrino) standard model up to a factor of 2 forthe current mixing parameter uncertainties, new physicseffects may be in charge of larger deviations.We conclude that the interplay between spectral

shape and detector response is important for thedetection of astrophysical neutrino sources. While thecommon assumption of an E�2 neutrino flux is wellmotivated for GRBs, it is well known not to apply todifferent source classes, such as AGNs. Since the detectorresponse strongly depends on the shape of the neutrinospectrum, the amount of energy which is needed to bedumped into pion production at the source to guarantee aneutrino detection may have to be actually higher thanassumed in many astrophysical source models.

ACKNOWLEDGMENTS

I would like to thank Philipp Baerwald, Sandhya

Choubey, Raj Gandhi, Svenja Hummer, Michele Maltoni,

and Carlos Yaguna for useful discussions and comments.

This work has been supported by DFG Grants No. WI

2639/2-1, No. WI 2639/3-1, and No. WI 2639/4-1.

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