Origin of the high energy neutrino flux at IceCube

TL;DR

This paper analyzes the origins of high-energy neutrinos detected by IceCube, proposing that a spectrum with a slight deviation from E^{-2} and a drop at PeV energies best explains the data, considering various astrophysical sources.

Contribution

It introduces a neutrino spectrum model with a specific spectral index and features, linking cosmic ray interactions and transport parameters to IceCube observations.

Findings

Diffuse galactic flux accounts for only 2 of 54 events.

Neutrino flux depends critically on cosmic ray transport parameter δ.

Proposed E^{-2.1} spectrum with a drop at PeV energies fits IceCube data.

Abstract

We discuss the spectrum of the different components in the astrophysical neutrino flux reaching the Earth and the possible contribution of each component to the high-energy IceCube data. We show that the diffuse flux from cosmic ray interactions with gas in our galaxy implies just 2 events among the 54 event sample. We argue that the neutrino flux from cosmic ray interactions in the intergalactic (intracluster) space depends critically on the transport parameter $\delta$ describing the energy dependence in the diffusion coefficient of galactic cosmic rays. Our analysis motivates a E^{-2.1} neutrino spectrum with a drop at PeV energies that fits well the data, including the non-observation of the Glashow resonance at 6.3 PeV. We also show that a cosmic ray flux described by an unbroken power law may produce a neutrino flux with interesting spectral features (bumps and breaks) related to changes in the cosmic ray composition.

Figures

Tables

Table 1. Table 1: Number of events at IceCube (1,347 days) implied by the atmospheric neutrino flux (conventional plus charm) and by the diffuse galactic neutrino flux. We have assumed a pure He composition in the CR flux at E > E knee 𝐸 subscript 𝐸 knee E>E_{\rm knee} . Events in parentheses correspond to atmospheric muons entering the detector from outside.

	Data	Atm	Gal	Data	Atm	Gal	Data	Atm	Gal
Tracks	2	0.6+0.2	0.0	1	0.1+0.1	0.0	0	0.0+0.0	0.0	UPGOING
Showers	6	0.2+1.0	0.2	1	0.0+0.5	0.1	0	0.0+0.1	0.0	()
Tracks	3	(3)+2.8+0.3	0.1	1	0.8+0.2	0.1	0	0.1+0.0	0.0	HORIZONTAL
Showers	8	(1)+1.1+1.4	0.3	2	0.2+0.8	0.2	1	0.0+0.2	0.1	()
Tracks	7	(6.4)+0.0+0.0	0.1	0	0.0+0.0	0.1	1	0.0+0.0	0.0	DOWNGOING
Showers	9	(2.2)+0.1+0.6	0.4	9	0.0+0.1	0.3	3	0.0+0.0	0.1	()
Total	35	(12.6)+4.9+3.4	1.1	14	1.2+1.7	0.9	5	0.1+0.3	0.3
	30–100 TeV			100–300 TeV			300–3000 TeV

Table 2. Table 2: Total number of events for 1,347 days at IceCube implied by the fluxes discussed in the text together with the excess deduced from Table  1 .

	30–100 TeV	0.1–0.3 PeV	0.3–3 PeV	3–10 PeV
Excess	13.0$\pm$8.6	10.2$\pm$1.1	4.3$\pm$0.1	0
$E^{-2.0}$	4.7	6.4	8.1	3.2
IG	7.6	9.3	5.2	0.4
AG	14.0	11.0	3.5	0.1

Equations

$${\Phi }_p=1.3{\left(\frac{E}{\mathrm{GeV}}\right)}^{-2.7}particles/(\mathrm{GeV}\mathrm{c}{\mathrm{m}}^2\mathrm{s}\mathrm{sr})$$

$${\Phi }_{\mathrm{He}}=0.54{\left(\frac{E}{\mathrm{GeV}}\right)}^{-2.6}particles/(\mathrm{GeV}\mathrm{c}{\mathrm{m}}^2\mathrm{s}\mathrm{sr}).$$

$$\Phi =330{\left(\frac{E}{\mathrm{GeV}}\right)}^{-3.0}particles/(\mathrm{GeV}\mathrm{c}{\mathrm{m}}^2\mathrm{s}\mathrm{sr}).$$

$${\bar{\Phi }}_{\nu }^{\mathrm{gal}}=3.7\times 10^{-6}{\left(\frac{E}{\mathrm{GeV}}\right)}^{-2.617}+0.9\times 10^{-6}{\left(\frac{E}{\mathrm{GeV}}\right)}^{-2.538},$$

$$({\nu }_e:{\nu }_{\mu }:{\nu }_{\tau }:{\bar{\nu }}_e:{\bar{\nu }}_{\mu }:{\bar{\nu }}_{\tau })=(1.13:1.07:0.99:0.91:0.99:0.91),$$

$$({\nu }_e:{\nu }_{\mu }:{\nu }_{\tau }:{\bar{\nu }}_e:{\bar{\nu }}_{\mu }:{\bar{\nu }}_{\tau })=(1.04:1.06:0.97:1.00:1.00:0.93).$$

$$\bar{\Phi}_{\nu}^{\rm gal}=\left\{\begin{array}[]{ll}4.4\times 10^{-4}\,\left({E\over{\rm GeV}}\right)^{-2.918}&{\rm\;(100\%\;proton)}\,,\\ 1.2\times 10^{-4}\,\left({E\over{\rm GeV}}\right)^{-2.938}&{\rm\;(100\%\;helium)}\,,\\ 1.3\times 10^{-5}\,\left({E\over{\rm GeV}}\right)^{-2.974}&{\rm\;(100\%\;iron).}\end{array}\right.$$

$${\Phi }_p$$

$${\Phi }_{\mathrm{He}}$$

$$I_p$$

$$I_{\mathrm{He}}$$

$${\tau }_G\approx 10^7\mathrm{years}\times {\left(\frac{E}{Z\times 10\mathrm{GeV}}\right)}^{-0.5}.$$

$$I_A=\frac{254}{\sqrt{Z}}C{E}^{-2.5},$$

$${\bar{\Phi }}_{\nu }(E)=R{\bar{\rho }}_{\mathrm{IG}}\sum _A\frac{F_A}{m_p}{\int }_0^1\mathrm{d}x{\sigma }_{Ap}(E/x){\bar{\Phi }}_A^{\mathrm{IG}}(E/x)x^{-1}{f}_A^{\nu }(x,E/x).$$

$${\bar{\Phi }}_{\nu }^{\mathrm{IG}}(E)=R{\bar{\rho }}_{\mathrm{IG}}\sum _A\frac{F_A{\sigma }_{Ap}^0{n}_A}{m_p}{Z}_A^{\nu }{E}^{-({\alpha }_A-{\beta }_A)},$$

$$Z_A^{\nu }={\int }_0^1\mathrm{d}x{x}^{{\alpha }_A-{\beta }_A-1}{f}_A^{\nu }(x).$$

$${\bar{\Phi }}_{\nu }^{\mathrm{IG}}=2.8N{E}^{-2.12}+1.0N{E}^{-2.04},$$

$$\bar{\Phi}^{\rm IG}_{\nu}=\left\{\begin{array}[]{ll}290\,N\,E^{-2.42}&{\rm\;(100\%\;proton)}\,,\\ 106\,N\,E^{-2.44}&{\rm\;(100\%\;helium)}\,,\\ 11\,N\,E^{-2.47}&{\rm\;(100\%\;iron)}\,.\end{array}\right.\,$$

$${\bar{\Phi }}_{\nu }^{\mathrm{IG}}=aN{E}^{-2.44}$$

$${\bar{\nu }}_{e}e\to W^{-}\to q\bar{q},\ell {\bar{\nu }}_{\ell }$$

$${\bar{\Phi }}_{\nu }^{\pi /K}(100\mathrm{TeV})$$

$${\bar{\Phi }}_{\nu }^{\mathrm{charm}}(100\mathrm{TeV})$$

$${\bar{\Phi }}_{\nu }^{\mathrm{gal}}(100\mathrm{TeV})$$

$${\bar{\Phi }}_{\nu }^{(2.0)}(100\mathrm{TeV})$$

$${\bar{\Phi }}_{\nu }^{(2.58)}(100\mathrm{TeV})$$

$${\bar{\Phi }}_{\nu }^{\mathrm{IG}}(100\mathrm{TeV})$$

$${\bar{\Phi }}_{\nu }^{\mathrm{AG}}(100\mathrm{TeV})$$

Full text

**Origin of the high-energy neutrino flux

at IceCube**

J.M. Carcellera, J.I. Illanaa, M. Masipa, D. Melonib

*a**CAFPE and Departamento de Física Teórica y del Cosmos

Universidad de Granada, E-18071 Granada, Spain

*b**Dipartimento di Matematica e Fisica, Università di Roma Tre

Via della Vasca Navale 84, 00146 Rome, Italy

jmcarcell@correo.ugr.es, jillana@ugr.es, masip@ugr.es, meloni@fis.uniroma3.it

1 Introduction

The IceCube observatory has recently discovered a flux of TeV–PeV neutrinos whose origin is not atmospheric [1, 2, 3]. In particular, four years of data include a total of 54 events of energy above 28 TeV. The spectrum, the track to shower ratio and the angular distribution of these events are consistent with a diffuse (isotropic) flux of astrophysical neutrinos harder than the atmospheric one. Where do these neutrinos come from?

Cosmic rays (CRs) are the key to understand any (atmospheric, galactic or extragalactic) neutrino fluxes, and it seems clear that IceCube’s discovery will have profound implications in CR physics. Moreover, the remarkable simplicity observed in the CR spectrum suggests that the high-energy neutrino flux may admit an equally simple description. Let us be more specific.

We observe that at energies below $E_{\rm knee}=10^{6.5}$ GeV CRs reaching the Earth are dominated by hydrogen and 4He nuclei, having both species slightly different spectral index [4, 5]. In particular, we estimate

[TABLE]

and

[TABLE]

These expressions imply an all-nucleon flux $\Phi_{N}\approx 1.8\,({E/{\rm GeV}}\,)^{-2.7}\;{\rm(GeV\,cm^{2}\,s\,sr)^{-1}}$ and a similar number of protons and He nuclei at $E\approx 10$ TeV. Beyond $E_{\rm knee}$ up to $E_{\rm ankle}=10^{9.5}\;{\rm GeV}$ the CR composition is uncertain, while the total flux becomes

[TABLE]

These spectral features, together with the almost perfect isotropy observed in the flux and the primary to secondary CR composition (the B/C ratio, the frequency of antimatter or of radioactive nuclei in CRs) can be accommodated within the following general scheme.

Galactic CRs are accelerated according to a power law $E^{-\alpha_{0}}$. The spectral index $\alpha$ that we see would then result after including propagation effects: CRs diffuse from the sources and stay trapped by galactic magnetic fields [6] during a time proportional to $E^{-\delta}$. As a consequence $\alpha=\alpha_{0}+\delta$, expressing that higher energy CRs are less frequent both because they are produced at a lower rate and because they propagate with a larger diffusion coefficient and leave our galaxy faster. The transport parameter $\delta$ is universal, in the sense that it is identical for CRs with the same rigidity $R=E/(Ze)$, and its value would be determined by the magnetic fields in the interstellar (IS) medium. For example, a Kraichnan or a Kolmogorov spectrum of magnetic turbulences imply $\delta=1/2$ or $1/3$, respectively [7]. The value of $\alpha_{0}$, in turn, may include some dependence with the CR composition; notice, in particular, that the difference in the proton and He spectral indices observed at $E<E_{\rm knee}$ requires that $\alpha_{0}^{p}\approx\alpha_{0}^{\rm He}+0.1$. As for the spectral break observed at $E_{\rm knee}$, it is thought to be associated to the sources rather than the transport: up to subleading effects [8] $\delta$ could be constant at all energies between 10 GeV (where the effects of the heliosphere become important) and a critical energy near $E_{\rm ankle}$ where the Larmor radius of CRs equals the maximum scale of the magnetic turbulences in the IS medium.

Although the basic parameters $\alpha_{0}$ and $\delta$ can be fit using the observables mentioned above, there is some degree of degeneracy that allows for different possibilities [9, 10]. Take the CR flux below $E_{\rm knee}$ in Eqs. (1,2). The spectral index $\alpha=2.6$–$2.7$ in the proton and He fluxes may result from $\alpha_{0}=2.1$–$2.2$, which is expected from diffusive acceleration at supernova remnants, with $\delta=0.5$. The data, however, could also be fit with a smaller diffusion parameter $\delta=0.3$–$0.4$ if $\alpha_{0}\approx 2.3$, which could be explained in models with significant CR reacceleration, or even with $\delta\approx 0.8$ in scenarios with strong convective winds if $\alpha_{0}\approx 1.9$.

Here we will argue that the analysis of the high-energy IceCube signal gives us not only an indication of its origin, but also it may provide a hint of what the value of $\delta$ is. Our objective is to discuss the spectrum of the different components in the diffuse neutrino flux reaching the Earth and propose one of these components as the main source of the IceCube neutrinos. In particular, we will show that the simplest case with $\delta=0.5$ implies a consistent picture where the bulk of the signal comes from neutrinos produced in the extragalactic (intracluster) medium.

First we will review the expected contribution to the IceCube data set from neutrinos produced in CR interactions inside our own galaxy. This will let us estimate the excess of events relative to the atmospheric plus galactic background. Then we will discuss the spectrum of two extragalactic components in the diffuse $\nu$ flux, and we will calculate their possible contribution to the IceCube signal. Finally, we will consider the possibility that the flux discovered by IceCube is not a power law: we will show that the interactions of a CR flux described by a single power law may introduce bumps and breaks in the secondary neutrino spectrum associated to sudden changes in the CR composition at different energies.

2 Diffuse flux of galactic neutrinos

Astrophysical neutrinos of $E>1$ GeV are non thermal, they appear always as secondary particles produced in the collisions of high-energy CRs with matter. Let us start considering the ones produced inside our own galaxy. CR collisions may take place mainly in two different environments: the interstellar (IS) medium where CRs are trapped for a long time (diffuse flux) and the dense regions at or near the acceleration sites (pointlike sources). The diffuse $\nu$ flux comes predominantly from directions along the galactic disk, whereas the local sources include pulsars and supernova remnants.

The diffuse flux of galactic neutrinos has been estimated by a number of authors [11, 12, 13, 14, 15, 16]. It basically depends on three factors: (i) The CR density at each point in our galaxy, (ii) the gas density in the disk and the halo, and (iii) the neutrino yield in the collisions of the CRs with the IS gas (hydrogen and helium in a proportion near 3 to 1 in mass). We will take the approximate analytical expressions for the diffuse flux obtained in [17], where we can find a detailed account of these three factors. At $10^{3}$–$10^{5.5}\,{\rm GeV}$ this neutrino flux is [in ${\rm(GeV\,cm^{2}\,sr\,s)^{-1}}$]

[TABLE]

where the two terms correspond to the contributions from the protons and the He nuclei in the CR flux, respectively, and the uncertainty is estimated at the $20\%$. Eq. (4) provides the total neutrino plus antineutrino flux averaged over all directions. The angular dependence ($\Phi_{\nu}$ is 100 times stronger from the galactic disk than from high latitudes) can also be found in [17]. After oscillations the relative frequency of each flavor reads

[TABLE]

for the component in the flux coming from protons and

[TABLE]

for the neutrinos from He (or from any nucleus with a similar number of protons and neutrons).

At energies $E\geq 10^{5.5}\,{\rm GeV}$ the expression in Eq. (4) is no longer valid, and at $E>10^{6.5}\,{\rm GeV}$ the flux $\bar{\Phi}_{\nu}$ is determined by the the CR composition beyond $E_{\rm knee}$ [17]:

[TABLE]

We will use these expressions and will interpolate with a power law in the $10^{5.5}$–$10^{6.5}$ GeV energy interval.

If compared with the atmospheric $\nu$ flux, the galactic flux in Eqs. (4,7) is small but not negligible. The atmospheric flux has two main components: the so called conventional neutrinos from light-meson decays (a detailed calculation can be found in [18]) and neutrinos from the prompt decay of forward charm [19]. In Fig. 1 we plot these fluxes and their dependence with the zenith angle $\theta$ at IceCube, where the declination is just $\theta-\pi/2$. At these energies the conventional flux contains muon and electron neutrinos in an approximate 30 to 1 proportion, whereas the $\nu$ flux from charm decays has a similar frequency of both flavors and a $2\%$ of $\nu_{\tau}$. Conventional and charm neutrinos are described by different spectral indices (see Fig. 1), and although the first ones dominate the atmospheric flux up to $E\approx 250$ TeV, charm hadrons are the main source of electron neutrinos already at 10 TeV [19]. The atmospheric $\nu$ flux also has a strong dependence on the CR composition at $E>E_{\rm knee}$ (it is proportional to $1/A$ [20, 21]). In the plot we see that the diffuse galactic flux is below the conventional one at $E<1$ PeV, and it is just one fourth of the flux from charm decays at all IceCube energies.

We can readily estimate the number of events that these fluxes imply at IceCube and compare it with the data. In Table 1 we have defined three energy bins, three direction bins, and we have separated shower from track events [22].

Our estimate includes the attenuation by the Earth of the neutrino flux reaching IceCube from different zenith angles, the (energy and flavor-dependent) effective volume of the detector [1], and the veto due to the accompanying muon in downgoing atmospheric events [23]. Events in parentheses are not genuine neutrino interactions but atmospheric muons entering the detector from outside. The uncertainty in the atmospheric background estimated by IceCube [3] for these three energy bins is around 8.6, 1.1 and 0.1 events, respectively.

The comparison with the data reveals that there is an excess that is (i) more significant at higher energies and (ii) stronger for showers than for tracks and for down-going and near horizontal directions than for upgoing events (see [24] for a fit to the anisotropy between the North and the South skies). For example, there are a total of 14 events with a deposited energy between 100 and 300 TeV but just $2.9\pm 1.1$ expected from atmospheric neutrinos. The table also reflects that galactic neutrinos only provide around 2 events among the 54 data sample. Our calculation of the galactic signal seems robust, as we reproduce within a 10% the signal estimated by IceCube for different astrophysical neutrino fluxes (their Fig. 3, page 49 in [3]). Although our result is significantly larger than the one obtained by other authors***This discrepancy is mostly due to the calculation of IceCube events implied by a given neutrino flux, i.e., to the cuts and the effective volume of the detector at different energies and event topologies (shower or track). (e.g., only 0.1 galactic events among IceCube’s three year data in [15]), it is still unable to explain the high-energy IceCube excess. It is also consistent with a recent likelihood analysis [25] indicating that the galactic contribution to total IceCube signal is subleading (around $7\%$) or with the total estimate (including point-like sources) in [26] (4–8%).

3 Other components in the neutrino flux

Let us go back to the simple scheme for the galactic CR flux outlined in Section 1, with a spectral index $\alpha^{A}=\alpha^{A}_{0}+\delta$ for each species $A$. It is important to notice that, up to collisions and energy loss, the effects of the propagation will be identical for CRs with the same rigidity (same value of $E/Z$). A He nucleus ($Z=2$) of energy $E$ will describe exactly the same trajectory through the galaxy as a proton of energy $E/2$. This implies a relationship between the fluxes $\Phi_{p,{\rm He}}(E)$ that we see at the Earth and the production rate $I_{p,{\rm He}}(E)$ of each species at the sources:

[TABLE]

where the factor multiplying $I_{A}(E)$ includes the overall normalization $n$ and the propagation effects for $\delta=0.5$. Therefore, taking the fluxes in Eqs. (1,2) we obtain that at $E<E_{\rm knee}$ the relative production rate of CRs by our galaxy is

[TABLE]

Notice that within this scheme CRs would stay confined in the galaxy for a period of order

[TABLE]

We then assume a steady state, i.e., the number of CRs leaving the galaxy is similar to the number of CRs accelerated at the sources. This means that at energies below $E_{\rm knee}$ our galaxy is emitting protons and He nuclei at the rate given in Eq. (9). Once emitted, these CRs will stay inside the cluster and supercluster –generically, the intergalactic (IG) medium– containing our galaxy for a time that may be larger than the age of the universe [27, 28].

A similar argument applied to the CR flux at $E>E_{\rm knee}$ in Eq. (3) implies a galactic production/emission rate

[TABLE]

where $Z$ is the corresponding atomic number of the CRs dominating the galactic flux at these energies. Our basic statement in Eqs. (9,11) is that the spectral index and the relative composition of the CRs emitted by our galaxy into the IG medium are correlated with the ones we see reaching the Earth, and that this correlation depends on a single transport parameter $\delta$ that, in the simple scenario under consideration, takes the value $\delta=0.5$.

Let us now assume that galaxies (the supernova remnants and pulsars inside them) are the main source of CRs of energy up to $10^{8}$ GeV, and that ours is an average galaxy. CRs can then be found (i) inside the galaxies (including ours), where they exhibit the spectrum and composition in Eqs. (1–3), and (ii) in the IG space, where they appear with the spectrum and the composition in Eqs. (9,11). The interactions of these two types of CRs with the gas in the medium where they propagate will produce TeV–PeV neutrinos. Therefore, in the astrophysical neutrino flux discovered by IceCube we may consider the relative weight of the following three components:

• •

Neutrinos from CR interactions with the IS matter in our own galaxy. This component, $\Phi_{\nu}^{\rm gal}$, has been discussed in the previous section, and it is way too low to account for the number of events detected at IceCube. In addition, these neutrinos are concentrated near the galactic plane.

• •

Neutrinos from the same type of interactions but in other galaxies. As discussed above, if both the accelerators and the spectrum of magnetic turbulences are universal we may expect that the IS medium in other galaxies will confine CRs with the same spectrum and relative composition as in ours, given in Eqs. (1–3). Collisions with the gas there will then produce a neutrino flux from all galaxies, $\Phi_{\nu}^{\rm AG}$, proportional to the one in Eqs. (4,7). Such flux will be more isotropic than the one discussed in the previous section, but its $\approx E^{-2.6}$ spectrum seems too steep to account for a significant fraction of the IceCube events. In particular, it has been shown [29] (see also [11]) that the neutrino flux would come together with a 10–100 GeV diffuse gamma-ray flux inconsistent with Fermi-LAT data [30].

• •

Neutrinos from interactions of CRs with extragalactic gas [27, 28, 31, 32, 33]. As mentioned before, the CRs producing this IG neutrino flux $\Phi_{\nu}^{\rm IG}$ are steadily emitted by all the galaxies with the spectrum and composition in Eqs. (9,11). In the intracluster space these CRs will face a gas density typically $10^{-4}$ times smaller than the one inside the parent galaxy, but the time they spend there may be $10^{-5}$ times larger, resulting into a larger column density. In addition, while inside galaxies CRs are in a steady state, in the IG medium the total number of CRs grows with time.

In the next sections we will calculate $\Phi_{\nu}^{\rm IG}$ up to an overall normalization factor and will show that their spectrum may provide a good fit of the high-energy IceCube data, including the non-observation of the Glashow resonance at 6.3 PeV.

4 Diffuse flux of intergalactic neutrinos

Our starting point is a CR number density in IG space with the spectrum and the relative composition given by Eqs. (9,11). Notice that an isotropic flux $\Phi_{A}$ of (relativistic) CRs type $A$ would be simply related to the number density $n_{A}$ by $\Phi_{A}=n_{A}\;c/(4\pi)$. Let us assume that the average gas density in the IG medium is $\bar{\rho}_{\rm IG}$, with a $(1\!:\!3)$ He to H ratio. The neutrino flux reaching the Earth from CR collisions with the gas along the line of sight is then [17]

[TABLE]

where $R$ is the maximum distance†††We neglect the redshift in the contribution to the neutrino flux from distant clusters. in our supercluster and beyond, $A$ runs over the different species in the CR flux, $f^{\nu}_{Ap}(x,E^{\prime})$ is the yield of neutrinos carrying a fraction $x$ of the incident energy produced in $A\,p$ collisions, and $F_{A}$ takes into account the mixed H/He composition of the IG gas ($F_{p}\approx 0.92$, $F_{\rm He}\approx 0.90$ and $F_{\rm Fe}\approx 0.86$ [17]). This expression gets simplified if one neglects the energy dependence of the yields and takes an unbroken power law both for the intergalactic CR flux [$\bar{\Phi}^{\rm IG}_{A}(E)=n\,I_{A}=n_{A}E^{-\alpha_{A}}$, with $I_{A}$ given in Eqs. (9,11)] and for the cross section [$\sigma_{Ap}(E)=\sigma^{0}_{Ap}E^{\beta_{A}}$]:

[TABLE]

being $Z^{\nu}_{A}$ the order-$(\alpha_{A}-\beta_{A}-1)$ moment of the yield,

[TABLE]

We see that the energy dependence of the cross sections will slightly change the spectral index of the IG neutrino flux from $\alpha_{A}$ to $\alpha_{A}-\beta_{A}$. In $p\,p$ collisions we have $\beta_{p}=0.082$ and $\sigma^{0}_{pp}=17.7$ mb, whereas in He$\,p$ and Fe$\,p$ collisions $\beta_{\rm He}=0.062$, $\sigma^{0}_{{\rm He}\,p}=60.5$ mb, $\beta_{\rm Fe}=0.026$ and $\sigma^{0}_{{\rm Fe}\,p}=551$ mb. Taking the yields from [17] and encapsulating the unknowns in a single normalization factor $N$, at $E<10^{5.5}$ GeV we obtain

[TABLE]

where the two terms come from the proton and the He contributions, respectively. At neutrino energies $E>10^{6.5}$ GeV we find

[TABLE]

The large uncertainty in $\Phi^{\rm IG}_{\nu}$ is related to the CR composition: its origin is the $Z$-moment of the neutrino yield, which is much smaller for heavy nuclei than for protons (see Fig. 2). If primary CRs above $E_{\rm knee}$ were mostly protons, then $\Phi^{\rm IG}_{\nu}$ would be 3.8 times larger than if they are pure helium, but this neutrino flux could also be a factor of 0.056 smaller if CRs were 100% iron.

In order to simplify our analysis, beyond $E_{\rm knee}$ we will consider the IG flux

[TABLE]

with $6<a<400$, and we will use a power law to interpolate between this flux and the one in Eq. (15) at $E<10^{5.5}$ GeV. Notice that the same value of $a$ may result from different CR compositions; for example, $a=106$ could correspond to 100% He or to $25\%$ proton plus $75\%$ iron. In the next section we will fit the high-energy IceCube data with the parameters $N$ and $a$ in this neutrino flux.

5 Fit of the high-energy IceCube data

Let us take the average IG flux obtained in the previous section to be isotropic.‡‡‡The flux could actually be modulated by the large-scale structure around our galaxy. In the first row of Table 2 we write the IceCube excess in each energy bin, i.e., the difference between the data and the sum of the atmospheric and the galactic events given in Table 1, including IceCube’s estimate of the background uncertainty. In the second row we give the number of events predicted by an unbroken power law with spectral indices $2.0$, which was the initial neutrino flux proposed by IceCube after three years of observations. In the third and fourth rows of Table 2 we give the number of events predicted by the IG flux $\Phi_{\nu}^{\rm IG}$ with $a=106$ and by the all-galaxies flux $\Phi_{\nu}^{\rm AG}$, which basically consists of the galactic flux in Eqs. (4,7) but isotropic and with an arbitrary normalization.

The normalization of each astrophysical flux has been fixed so that they reproduce the total IceCube excess at 100 TeV–3 PeV (i.e., the sum of the two high energy bins in Table 1). The flux $\Phi_{\nu}^{\rm AG}$ would imply a too strong gamma-ray signal [11]. In fact, as shown in [29], any neutrino flux steeper than $E^{-2.2}$ would appear with a gamma-ray flux that extrapolated to lower energies conflicts the Fermi-LAT data [30]. To be realistic those fluxes would require a mechanism that absorbs the gammas leaving the neutrino flux unaffected [34]. Another interesting possibility for the AG flux could be a radially dependent $\delta$ parameter [35] that would make the harder the CR flux near the galactic center, where most of the interactions occur.

Notice that in Table 2 we have added a fourth energy bin, 3–10 PeV, which provides an important piece of information in order to decide about the goodness of the fits. At these energies electron antineutrinos could reveal the Glashow resonance through collisions with electrons:

[TABLE]

In Fig. 3 we show that at $E=(6.3\pm 2.0)$ PeV the cross section for this process [36] goes well above $\sigma(\nu N)$ [37]. Since the IceCube target has 10 electrons per 18 nucleons and the $\bar{\nu}_{e}$ frequency in the IG neutrino flux is almost exactly 1:6 [see Eq. (6)], the Glashow resonance will clearly have an impact on the fit. Notice also that when the $W$ decays hadronically (with a 67.6% branching ratio) all the neutrino energy $E_{\nu}$ will be deposited in the ice, while in leptonic decays (32.4% of the times) the charged lepton will take an energy between 0 and $E_{\nu}$ with an average value of $0.33\,E_{\nu}$ [36].

We find that the $E^{-2.0}$ flux implies 3.2 events beyond 3 PeV, while all the other fluxes may fit the data while predicting less than one event in that bin. Therefore, the non-observation of the Glashow resonance after four years of data disfavors the harder $E^{-2.0}$ flux initially proposed by IceCube. The physically motivated flux $\Phi_{\nu}^{\rm IG}$ has a similar spectral index at lower energies [see Eq. (15)], however, a possible break caused by a change in the CR composition at $E_{\rm knee}$ may define an acceptable possibility. Indeed, the presence of the knee in the CR spectrum implies that we should not expect an unbroken power law in the neutrino flux at TeV–PeV energies. Although the amount of IceCube data is small, a recent analysis [38] that includes bounds from Fermi-LAT data excludes at the $3\sigma$ level an astrophysical neutrino flux described by a single power law, favoring a break in the spectrum at 200–500 TeV very similar to the one we obtain here.

In Fig. 4 we plot these astrophysical $\nu$ fluxes together with the total number of events that they imply at all IceCube energies.

For comparison, the normalization of the atmospheric and galactic neutrinos fluxes in Fig. 1 reads

[TABLE]

whereas the four fluxes in Fig. 4 have been normalized to

[TABLE]

A final comment concerns the possible North–South sky asymmetry of the IceCube signal in Table 2. Let us focus on the three bins of energy above 100 TeV, where the uncertainties are lower. The 14.5 excess in Table 2 is distributed as follows: 1.0 events from upgoing directions (Northern sky, with declinations $90^{\circ}>\delta>20^{\circ}$), 1.3 from near-horizontal directions ($20^{\circ}>\delta>-20^{\circ}$), and 12.2 from downgoing directions (Southern sky, $-20^{\circ}>\delta>-90^{\circ}$). In the same direction bins, the (isotropic) IG flux implies 2.1, 5.1, and 7.2 events, respectively, whereas the (also isotropic) AG flux would give 2.4, 5.9 and 6.2 events. Therefore, as emphasized in [24], although the sample is still small, the data seems to favor an anisotropic neutrino flux. For an IG origen this could simply reflect a larger total column density of intracluster gas along the directions in the Southern sky.

6 Dependence on the cosmic ray composition

The distribution of the IceCube data given in Fig. 4 shows a deviation from a power law at energies above 250 TeV. In particular, in the five bins between $10^{2.4}$ and $10^{3.4}$ TeV we find, respectively, 2, 1, 0, 2 and 1 events. This sequence suggests a flat event distribution following the steeper one observed at lower energies, or even a possible drop in the event rate at 0.25–1 PeV followed by a bump defined by the three events of highest energy. Obviously, the statistical significance of such deviations is limited (notice that in Tables 1 and 2 we have defined much wider bins in order to dilute them), but it is apparent that none of the neutrino fluxes that we have discussed so far would be able to accommodate such a spectral feature. Remarkably, the new IceCube analysis presented in ICRC2017 [39] corresponding to two more years of data taking does not include new events of $E>250$ TeV among a 28 event sample, which tends to steepen the neutrino flux and/or to increase the significance of the 0.25-1 PeV drop.

Therefore, it may be interesting to study what type of spectral irregularities may be expected from sudden changes in the CR composition, which could be associated, for example, to the maximum energy achieved by cosmic accelerators for CRs of a given charge. We will see that these spectral changes in the secondary neutrino flux would appear even when the primary CR flux exhibits an unbroken power law.

To be definite, let us consider a CR flux proportional to $E^{-2.3}$ in the whole $10^{4}$–$10^{10}$ GeV energy interval. We will assume (see Fig. 5) that the dominant composition is proton up to $10^{6}$ GeV, then He up to $2\times 10^{6}$, and C up to $10^{7}$ GeV. At this energy the flux becomes proton dominated again up to $10^{7.5}$ GeV, then He up to $2\times 10^{7.5}$ GeV, C up to $10^{8.5}$ GeV, and Fe at higher energies. As expressed in Eq. (12), the secondary neutrino flux will then depend on the fraction of each species in this CR flux that interacts with IG matter (this fraction is proportional to the cross section) and on the neutrino yield in those interactions. In Fig. 5 we plot the neutrino flux up to an overall normalization factor. For comparison, we include in the plot the parent CR flux (the relative normalization between both fluxes is also arbitrary).

The key observation is that protons and nuclei of energy $E$ contribute to neutrinos of energy below $0.1E$ and $0.1A^{-1}E$, respectively. As a consequence, a change in the composition towards heavier nuclei tends to produce a drop (relative to a constant spectral index) in the neutrino flux, whereas a change in the opposite direction –heavy to light– may introduce a bump. In the plot we see that the change from proton to He and then to carbon at $E>1$ PeV translates into a neutrino drop in the energy region suggested by the data, whereas the change from C to p at $10$ PeV could induce a relative excess at energies around 1 PeV. The subsequent changes to heavier nuclei at $E>3$ PeV would be motivated by the absence in the IceCube data of the 6.3 PeV Glashow resonance. Although this CR flux is just a toy model, it shows that the CR composition at energies around $E_{\rm knee}$ and beyond may be a key factor to justify deviations from a power law in the high-energy neutrino flux detected by IceCube.

7 Summary and discussion

High energy neutrinos can only be produced in the collisions of charged CRs. It seems then clear that the discovery at IceCube of an astrophysical neutrino flux will have implications in our understanding of high energy CRs. In particular, a higher statistics should establish the spectral index of this flux at E\mathrel{\vbox{\hbox{<}\nointerlineskip\hbox{\sim}}}1 PeV and, most important, the presence or not of the Glashow resonance at $E\approx 6.3$ PeV. We have shown that these two observations will provide clear hints about the spectrum and the composition of the parent CRs, which in turn relate to the environment where the neutrinos have been produced.

Galactic CRs are described by a spectrum $\approx E^{-\alpha}$ that is steeper than the one they have at the sources: $\alpha_{0}=\alpha-\delta$ with $\delta=0.5$ in the simplest scenario. The neutrinos produced in their collisions will inherit the spectral index of the parent CR flux. If the main source of the IceCube neutrinos were the collisions of CRs inside galaxies, then their spectral index would be $2.6$ at $E<10^{5.5}$ GeV and around $2.9$ at higher energies. A few 1–2 PeV events at IceCube from such steep flux would then be correlated with a too large diffuse gamma-ray flux at 0.1–100 GeV [11, 29]. Once CRs leave into the IG space, however, their spectral index should be similar to the one they have at the sources. In Eq. (15) we provide a two-component (from proton and He collisions) IG neutrino flux with a spectral index near $2.1$. Such a hard spectrum, if unbroken, should have already revealed the Glashow resonance. We have shown, however, that if the CR knee brings a change in the composition towards heavier nuclei then the secondary neutrino flux may experience a sudden drop at $E>1$ PeV. Therefore, the observation (or not) of the Glashow resonance will provide important information about the CR composition at these energies.

As a viable possibility, we have studied the implications at IceCube of the IG neutrino flux that may appear if CRs above $E_{\rm knee}$ are dominated by He ($a=106$ in Eq. (17)). Our results are summarized in Table 2. We see that, normalizing the neutrino flux so that the total number of events in the two high energy bins matches the experimental excess, this single component $\Phi_{\nu}^{\rm IG}$ provides a good fit of all the data. We find that the (much steeper) galactic diffuse flux $\Phi_{\nu}^{\rm gal}$ contributes with just two events in the IceCube sample, a number that is significantly larger than previous estimates by other authors. Within our scheme, a pure proton composition above the CR knee is disfavored as $\Phi_{\nu}^{\rm IG}$ would imply around 2 events of $E>3$ PeV.

Our analysis depends basically on the transport parameter $\delta$. The value $\delta=0.5$ that we have considered is consistent with a Kraichnan spectrum of magnetic turbulences and diffusive shock acceleration at supernova remnants, although other possibilities could be accommodated. We have also discussed possible deviations from a power law in the neutrino flux caused by sudden changes in the primary CR composition. Therefore, we think that the astrophysical neutrino flux discovered by IceCube, once it is fully characterized, will provide very valuable information that will help to complete the CR puzzle.

Acknowledgments

We would like to thank Eduardo Battaner for discussions. This work has been supported by MICINN of Spain (FPA2013-47836, FPA2015-68783-REDT, FPA2016-78220 and Consolider-Ingenio MultiDark CSD2009-00064) and by Junta de Andalucía (FQM101). JMC acknowledges a Beca de Iniciación a la Investigación fellowship from the UGR.