# Testing Lorentz Invariance with Neutrinos from Ultrahigh Energy Cosmic Ray
Interactions

Sean T. Scully<EMAIL_ADDRESS>Department of Physics and Astronomy, James
Madison University,Harrisonburg, VA 22807  Floyd W. Stecker
<EMAIL_ADDRESS>NASA/Goddard Space Flight Center, Greenbelt, MD 20771

###### Abstract

We have previously shown that a very small amount of Lorentz invariance
violation (LIV), which suppresses photomeson interactions of ultrahigh energy
cosmic rays (UHECRs) with cosmic background radiation (CBR) photons, can
produce a spectrum of cosmic rays that is consistent with that currently
observed by the Pierre Auger Observatory (PAO) and HiRes experiments. Here, we
calculate the corresponding flux of high energy neutrinos generated by the
propagation of UHECR protons through the CBR in the presence of LIV. We find
that LIV produces a reduction in the flux of the highest energy neutrinos and
a reduction in the energy of the peak of the neutrino energy flux spectrum,
both depending on the strength of the LIV. Thus, observations of the UHE
neutrino spectrum provide a clear test for the existence and amount of LIV at
the highest energies. We further discuss the ability of current and future
proposed detectors make such observations.

###### keywords:

cosmic rays; neutrinos; Lorentz invariance; quantum gravity

††journal: Astroparticle Physics

##  1 Introduction

Ultrahigh energy cosmic rays and neutrinos are of interest as possible probes
of new physics [1]. In particular, some quantum gravity models predict that
Lorentz invariance may be weakly broken at the very high energies, leading to
potentially observable consequences. The possibility of using ultrahigh energy
cosmic rays (UHECRs) to probe for a small violation of Lorentz invariance was
suggested over a decade ago [2]. Indeed, a detailed analysis of the effects of
LIV on the UHECR spectrum has yielded the tightest constraint on LIV to date
[3]. Shortly after the discovery of the CBR it was pointed out that photomeson
interactions of UHECRs with photons of the cosmic background radiation (CBR)
would result in a sharp steepening of their spectrum above E $\sim$ 50 EeV now
known as the ”GZK effect” [4, 5]. However, even a very small amount of LIV
will kinematically inhibit some of these interactions. It has been previously
shown that a possible signature of LIV in the UHECR spectrum would be a
recovery of the cosmic ray spectrum at energies greater than $\sim$ 200 EeV
([3, 6]).

Given the current state of the UHECR observational data as reported by HiRes
[7, 8] and Auger [9], it is possible to constrain LIV, but not to rule it out.
However, unlike cosmic-ray baryons and photons, ultrahigh energy neutrinos do
not suffer significant energy losses over cosmological distances. Studies of
UHE neutrinos with both ground-based and space-based detectors could provide a
new and less ambiguous test of LIV. This is because they are a guaranteed
byproduct of the photomeson interactions of UHECRs with the CBR followed by
subsequent pion decay [10, 11, 12, 13].

In this paper, we further consider the observational implications of the
effect of a very small amount of LIV, viz. the suppression of photomeson
production on the resulting UHE neutrino spectrum. In our previous work [3, 6]
we undertook a detailed calculation of the modification of the UHECR spectrum
caused by LIV using the formalism of reference [2] and the kinematical
approach originally developed in reference [14]. We employ the same techniques
used in references [3] and [6] to determine the resulting photomeson neutrino
spectrum. We again consider here the case where the primary UHECRs are
protons.

ANITA II [15] has placed an upper limit on the neutrino flux for energies
greater than $>10^{18}$ eV. IceCube is nearing completion and will be
sensitive to neutrinos of energies up to $\sim 10^{17}$eV. Proposed future
ground-based and space-based neutrino detectors could be capable of detecting
and studying photomeson neutrinos. We will discuss the ability of such
detectors to constrain LIV or observe its effect.

##  2 LIV and the Spectrum of UHECRs

We now extend the calculation of reference [3] to determine the photomeson
neutrino fluxes.111We use the usual convention $c=1$. A full description of
the LIV formalism we use is given in references [2] and [3]. We summarize the
salient points here. The free particle Lagrangian is modified by the inclusion
of a leading order perturbative, Lorentz violating term. This term leads to
the modified free particle dispersion relations

$E^{2}~{}=~{}\vec{p}\ ^{2}+m^{2}+2\delta\vec{p}\ ^{2}.$ (1)

These relations can be put in the standard form

$E^{2}~{}=~{}\vec{p\ }{{}^{2}}c_{MAV}^{2}+m^{2}c_{MAV}^{4},$ (2)

by shifting the renormalized mass by the small amount $m\rightarrow
m/(1+2\delta)$ and shifting the velocity by the amount

$c_{MAV}=\sqrt{(1+2\delta)}\simeq 1+\delta$ (3)

where $c_{MAV}$ is identified as the maximum attainable velocity of the free
particle in the reference frame of the CBR. Using this formalism, different
particles can have different maximum attainable velocities (MAVs) that can all
be different from $1$ as well as different from one another. Hereafter, we
denote the MAV of a particle of type $i$ by $c_{i}$ and the difference

$c_{i}-c_{j}~{}~{}\equiv~{}\delta_{ij}$ (4)

These modified dispersion relations are then applied to the kinematical
relations governing the dominant single meson photomeson interaction:

$p+\gamma\rightarrow N+\pi.$ (5)

From equations (1) and (4), a dispersion relation can be constructed for a
particle $a$

$E^{2}=p^{2}+2\delta_{a}p^{2}+{m_{a}}^{2}$ (6)

where $\delta_{a}$ is the difference between the MAV for the particle a and
the speed of light in the low momentum limit, $c=1$.

In order to modify the effect of photomeson production on the UHECR spectrum
above the GZK energy, $\delta_{\pi p}>0$ as shown in reference [2]. The
condition for photomeson interactions to take place is

$\delta_{\pi p}\leq 3.23\times 10^{-24}(\omega/\omega_{0})^{2}.$ (7)

where $\omega$ is the energy of the CBR photon and $\omega_{0}\equiv
kT_{CBR}=2.35\times 10^{-4}$ eV with $T_{CBR}=2.725\pm 0.02$ K [2, 3].

If LIV occurs and $\delta_{\pi p}>0$, photomeson production can only take
place for interactions of CBR photons with energies large enough to satisfy
equation (7). This condition implies that while photomeson interactions
leading to GZK suppression can occur for “lower energy” UHE protons
interacting with higher energy CBR photons on the Wien tail of the spectrum,
other interactions involving higher energy protons and photons with smaller
values of $\omega$ will be forbidden [3]. Thus, the observed UHECR spectrum
may exhibit the characteristics of GZK suppression near the normal GZK
threshold, but the UHECR spectrum can “recover” at higher energies owing to
the possibility that photomeson interactions at higher proton energies may be
forbidden [3, 6, 16, 17]. Even a small violation of Lorentz invariance changes
the inelasticity of the interaction, (i.e., the amount of energy transferred
from the incident proton to the created pion). This is the key to
understanding the effect of LIV on photomeson production. With an increase in
proton energy, the range of kinematically allowed angles of interaction
between it and the photon becomes more restricted, thus reducing the phase
space and, in turn, the total inelasticity. Figure 1, reproduced from
reference [3], shows the calculated inelasticity modified by LIV for a value
of $\delta_{\pi p}=3\times 10^{-23}$ as a function of both CBR photon energy
and incident proton energy. Other choices for $\delta_{\pi p}$ yield similar
plots but change the energy at which LIV effects become significant. The
inelasticity precipitously drops above a certain energy because the LIV term
in the pion rest energy from equation (6) becomes comparable to $m_{\pi}$.

Figure 1: The calculated proton inelasticity modified by LIV for $\delta_{\pi
p}=3\times 10^{-23}$ as a function of CBR photon energy and proton energy
(from reference [3]).

The proton energy loss rate by photomeson production is given by

${{1}\over{E}}{{dE}\over{dt}}=-{{\omega_{0}c}\over{2\pi^{2}\gamma^{2}}\hbar^{3}c^{3}}\int\limits_{\eta}^{\infty}d\epsilon~{}\epsilon~{}\sigma(\epsilon)K(\epsilon)\ln[1-e^{-\epsilon/2\gamma\omega_{0}}]$
(8)

where $\epsilon$ is the photon energy in the center of mass system,
$K(\epsilon)$ is the modified inelasticity calculated from the kinematics, and
$\sigma(\epsilon)$ is the total $\gamma$-p cross section. The lower limit of
the integration,$\eta$, is the photon threshold energy for the interaction in
the center of mass frame.

As in reference [3], we assume that the source spectrum of UHE protons can be
approximated over a limited energy range by a power-law that fits the UHECR
data below 60 EeV. This spectrum is then of the form $A(z)E_{i}^{-\Gamma}$
where $E_{i}$ is the energy of the proton. The UHECRs suffer energy losses
from pair and pion production through interactions with the CBR and also
cosmological redshifting. The energy losses from pion production are
determined according to equation (8). The pair-production loss rate comes from
[18]. In order to determine redshift loses, a flat $\Lambda$CDM universe with
a Hubble constant of H0 = 70 km s-1 Mpc-1 is assumed, taking
$\Omega_{\Lambda}$ = 0.7 and $\Omega_{m}$ = 0.3. The source evolution is
additionally assumed $\propto(1+z)^{\zeta}$ with $\zeta$ = 3.6, out to a
maximum redshift of 2.5 which tracks the star formation rate. This value is a
mean between the fast evolution and baseline models used in reference [19].
(See also references [20, 21].) The spectrum of UHECRs on Earth can then be
determined from

$\displaystyle J(E)={{3cA(0)}\over{8\pi
H_{0}}}E^{-\Gamma}\int_{0}^{z_{max}}{{(1+z)^{(\zeta-1)}}\over{\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}}}\left({E_{i}\over{E}}\right)^{-\Gamma}{{dE_{i}}\over{dE}}dz.$
(9)

where A(0) is determined by fitting our final calculated spectrum to the
observational UHECR data assuming $\Gamma=2.55$, which is consistent with the
spectrum derived by the Pierre Auger Observatory (PAO) collaboration below 60
EeV.222We have chosen a maximum UHECR energy of $3\times 10^{21}$ eV. However,
our results are insensitive to this value because they are determined by the
LIV kinematics. The results of this calculation for various choices of the
parameter $\delta_{\pi p}$ are shown in Figure 2 plotted along with the most
recent results from PAO.

Figure 2: Comparison of the PAO data [9] with calculated spectra for various
values of $\delta_{\pi p}.$ From top to bottom, the curves give the predicted
spectra for $\delta_{\pi p}=1\times 10^{-22},6\times 10^{-23},3\times
10^{-23},1\times 10^{-23},0$ (no Lorentz violation).

It can be seen from Figure 2 that a small amount of LIV can still preserve the
GZK suppression effect, but produces a ”recovery” of the UHECR spectrum at
higher energies. Since this recovery is due to the virtual elimination of
photomeson interactions at higher UHECR energies, it will also suppress the
production of higher energy photomeson neutrinos.

##  3 The Photomeson Neutrino Spectrum

We now turn our attention to calculating the photomeson neutrino spectrum that
would result from the UHECR calculation detailed in the previous section. We
use the data on the cross section for pion production compiled in reference
[22] and summarized in reference [23]. Near threshold the the total photomeson
cross section is dominated by the emission of single pions. The most
significant channel to consider involves the intermediate production of the
$\Delta$ resonance [24]:

$p+\gamma\rightarrow\Delta\rightarrow N+\pi$ (10)

This channel strongly dominates the photomeson production process near
threshold. Since the UHECR flux falls steeply with energy, it follows that the
bulk of the pions leading to the production of neutrinos will be produced
close to the threshold.

For a proton interacting with the CBR, a pion and a nucleon are produced. The
outgoing nucleon has probability of 2/3 to be a proton and 1/3 probability to
be a neutron from isospin considerations. Should the resulting nucleon be a
neutron, then the resulting pion is a $\pi^{+}$. Thus approximately twice the
number of neutral pions are produced relative to charged pions from resonant
pion production. However direct pion production, which accounts for about 20%
of the total cross section, produces charged pions almost exclusively meaning
that all told, approximately equal numbers of neutral and charged pions are
produced around threshold. Neutral pions decay into photons so we need only
consider the charged pions for neutrino production. Three neutrinos of roughly
equal energy result from the decay chain of the
$\pi^{+}\rightarrow\mu^{+}\nu_{\mu}\rightarrow
e^{+}\bar{\nu{{}_{\mu}}}\nu_{e}$.

It is straightforward to determine the neutrinos produced and their energies
from the UHECRs. We follow closely the calculation of the neutrino flux as
described in reference [12] and references therein. The key is to determine
the amount of energy that is carried away by the pion. This follows directly
from the inelasticity and the incident proton energy calculated using equation
(8). We assume that all of the sources have the same primary injection
spectrum and distribution as detailed in section 2. We calculate the total
neutrino flux by integrating over proton energy, photon energy, and redshift,
assuming the standard $\Lambda$CDM cosmology. We find that the shape of the
neutrino spectrum we obtain when assuming Lorentz invariance is very similar
to that obtained from the more detailed Monte Carlo calculations that include
all the relevant baryonic resonances and possible meson and multi-pion
production channels (See, e.g., reference [13]).

The effect of LIV on the photomeson neutrino production is again manifested
through the modification of the inelasticity of the interaction, since this
determines the amount of energy that is carried away by the pion and therefore
the resultant neutrino energy. The biggest impact of including LIV is to
suppress the production of the higher energy photopions and therefore the
resulting higher energy neutrinos.

Figure 3 shows the corresponding total neutrino flux (all species) for the
same choices of $\delta_{\pi p}$ as the UHECR spectra presented in figure 2.
As expected, increasing $\delta_{\pi p}$ leads to a decreased flux of higher
energy photomeson neutrinos as the interactions involving higher energy UHECRs
are suppressed [3]. It is also evident that the peak energy of the neutrino
energy flux spectrum (EFS), $E\Phi(E)$, shifts to lower energies with
increasing $\delta_{\pi p}$.

Figure 3: Neutrino fluxes (of all species) corresponding to the UHECR models
considered in Figure 2. From left to right, the curves give the predicted
fluxes for $\delta_{\pi p}=1\times 10^{-22},6\times 10^{-23},3\times
10^{-23},1\times 10^{-23},0$.

We note that a similar effect can be produced on the photomeson neutrino
spectrum from another possible mechanism involving LIV, namely ”neutrino
splitting” [25]. With LIV, the decay of one neutrino into three neutrinos can
be kinematically allowed. This effect may also produce a decreased flux at the
high energy end of the neutrino spectrum. The feature that distinguishes
between the two possible LIV effects is that neutrino splitting results in an
increase in the flux of lower energy photomeson neutrinos.

##  4 Considerations of UHECR Composition

Throughout this paper, we have made the assumption that the highest energy
cosmic rays, i.e., those above 100 EeV, are protons. The composition of these
primary particles is presently unknown. The highest energy events for which
composition measurements have been attempted are in the range between 40 and
50 EeV, and the composition of these events is uncertain [26],[27],[28].

We note that in the case where the UHECRs with total energy above $\sim$100
EeV are not protons, both the photomeson threshold and the LIV effects are
moved to higher energies because (i) the threshold is dependent on
$\gamma\propto E/A$, where $A$ is the atomic weight of the UHECR [29], and
(ii) it follows from equation (1) that the LIV effect depends on the
individual nucleon momentum. We also note that the neutrino spectrum at the
high energy end is the same for the mixed composition case as in the pure
proton case [30, 31, 32].

##  5 Observational Prospects

Several experiments currently place upper limits on the photomeson neutrino
flux. The ANITA long duration balloon experiment launched in December of 2008
searched for electromagnetic cascades initiated by UHE neutrinos within the
Antarctic ice shelf via the Askaryan effect. Their analysis yielded a model-
independent 90% CL limit on neutrino fluxes in the range of 1018 – 1023 eV
with a sensitivity capable of excluding several optimistic photomeson neutrino
flux models [15]. However, ANITA does not have sufficient sensitivity or
energy range to distinguish a possible LIV effect on the neutrino spectrum.
ANITA has an effective threshold energy $\sim 10^{18}$ eV and its sensitivity
about an order of magnitude too weak. IceCube is capable of detecting
neutrinos $<10^{17}$ eV, but its sensitivity is two orders of magnitude too
weak to detect photomeson neutrinos [33]. Proposed future space-borne missions
such as the Extreme-Universe Space Observatory (EUSO) [34], super EUSO [35],
and Orbiting Wide-angle Light Detectors (OWL) [36] would have much larger
effective aperatures than presently available detectors. They would be capable
of making accurate determinations of the energy, arrival direction, and
composition of the cosmic-rays and the associated photomeson neutrinos using a
target volume far greater than is possible from ground-based experiments
presently in operation. While such experiments could potentially provide the
statistics necessary to observe the recovery of the UHECR spectrum at high
energies, both EUSO and OWL as proposed would not achieve either the
sensitivity or energies necessary to distinguish LIV effects in the neutrino
spectrum.

However, the proposed full Antarctic Ross Ice shelf ANtenna Neutrino Array
(ARIANNA) would be capable of detecting photomeson neutrinos with a
sensitivity an order of magnitude better than ANITA and other existing
detectors. Like ANITA, ARIANNA exploits the Askaryan effect, i.e. the
detection of coherent Čerenkov emission at radio wavelengths produced in the
ice shelf by neutrino induced cascades. Because the power of coherent radio
emission grows as the square of the shower energy and therefore neutrino
energy, ARIANNA is capable of detecting lower energy neutrinos than ANITA
since the distances to balloon-borne detectors can be quite large while
ARIANNA utilizes detectors situated directly on the ice shelf surface. ARIANNA
is expected to observe $\sim$40 events per 6 months in the energy range of
photomeson neutrinos with energies in excess of $\sim 10^{17}$ eV [37]. This
lower energy threshold is crucial for searching for LIV effects. As such, we
shall restrict ourselves to the discussing the potential of ARIANNA for
distinguishing the effect of LIV on the photomeson neutrino spectrum. We note
that another proposed detector called IceRay would also make use of the
Askaryan effect. IceRay would be placed at the location of IceCube and would
also be capable of detecting photomeson neutrinos [38].

ARIANNA’s high event rate, combined with its low energy threshold can
distinguish LIV effects if $\delta_{\pi p}\leq 3.0\times 10^{-23}$ with a 5
year exposure. We note that this limit is close to the upper limit indicated
by the current PAO data [3]. Figure 4 shows our calculated fluxes along with
the sensitivity of ARIANNA for exposure times of 6 months [37] and 5 years.
Here we have plotted $E\Phi(E)$ for all neutrino flavors and $\nu$-$\bar{\nu}$
combinations compared with the proposed ARIANNA sensitivities to more clearly
illustrate the threshold effect. We note that ARIANNA is sensitive to all
neutrino flavors since it would primarily detect the hadronic shower and not
the outgoing lepton. In Figure 4, we present our neutrino fluxes and
experimental sensitivities as the total of all flavors and $\nu$-$\bar{\nu}$
combinations.

Figure 4: All-flavor neutrino flux spectra (EFS) that correspond to the UHECR
models considered in Figure 2. The six month sensitivity from the proposed
ARIANNA array is shown as a dashed curve. The dot-dashed curve shows the
sensitivity scaled to a 5 year exposure. From bottom to top, the solid curves
give the predicted spectra for $\delta_{\pi p}=1\times 10^{-22},6\times
10^{-23},3\times 10^{-23},1\times 10^{-23},0$.

It is clear from Figure 4 that at the lower energy threshold of $\sim 10^{17}$
eV for ARIANNA is very close to the peak energy in $E\Phi(E)$ for $\delta_{\pi
p}=3\times 10^{-23}$ and would have sufficient sensitivity at $10^{17}$ eV to
detect the expected neutrino flux provided it runs long enough to produce the
desired sensitivity. ARIANNA is therefore very promising in terms of its
ability to distinguish the effect of an amount of LIV that is compatible with
the current PAO results.

###  5.1 A Caveat

We note that suggestions have been made to use the derivation of the
extragalactic $\gamma$-ray background flux up to 100 GeV using data from the
Fermi Gamma-ray Space Telescope [39] to place limits on the absolute value of
the photomeson neutrino flux, assuming that the 100 GeV $\gamma$\- rays are
produced by an electromagnetic cascade off the CBR initiated by UHE pion decay
$\gamma$-rays produced along with the neutrinos [40],[41]. Such a constraint
would lower the neutrino flux to almost an order of magnitude below the
expected value and would make it much more difficult to test Lorentz
invariance with photomeson neutrinos. These results follow earlier neutrino
constraints [42] obtained using various analyses of the EGRET-GRO results on
the extragalactic $\gamma$-ray background [43],[44]. However, the argument
here is contingent on the assumption that the extragalactic magnetic field is
so small that a UHE electromagnetic pair-production-Compton cascade leading to
the production of $\gamma$-rays in the GeV energy range will not be cut off by
synchrotron losses of the UHE electrons dominating over Compton losses [45,
11]. At this point in time, the strength of the extragalactic B-field is only
constrained to be within the range $\sim 3\times 10^{-16}$ – $\sim 3\times
10^{-9}$ G. [46],[47].

In addition, we note that the Fermi spectrum is not the result of direct
observation, but of analysis. It critically involves the subtraction of both
galactic foreground $\gamma$-rays and, in the case of Fermi instrumental
calibration, by Monte Carlo modeling [48]. These are non-negligible
uncertainties. As an example of the uncertainties involved, we note the
significant differences between the EGRET-GRO results [43],[44] and the Fermi
results [39] on the extragalactic $\gamma$-ray background.

We also note that since both the photomeson production cross section and the
CBR photon spectrum are very well determined, and since the GZK cutoff effect
is well documented [8],[9], a significant decrease in the predicted photomeson
neutrino flux could either imply an unexpectedly small production of UHECRs or
the existence of new physics.

##  6 Conclusion

With future improved data from PAO, tighter constraints can be placed on the
amount of LIV allowed by the UHECR spectrum. However even after a decade more
of operation it seems unlikely that PAO would be able to determine the UHECR
spectrum with adequate statistics at energies greater that 300 EeV, where the
effect of LIV would be manifested. While space-borne missions such as JEM-EUSO
and OWL could provide the statistics necessary to observe the effect, these
missions are currently only in the planning stage and are many years from
being realized.

We have shown here that additional information on LIV or its constraints can
be obtained from studying the spectrum of photomeson neutrinos. By calculating
the flux of high energy neutrinos generated by the propagation of UHECR
protons through the CBR in the presence of LIV, we find that LIV produces a
reduction in the flux of the highest energy photomeson neutrinos and a
reduction in the energy of the peak of the neutrino energy flux spectrum with
both effects increasing with the strength of the LIV. Thus, observations of
the UHE neutrino spectrum could provide a clear test for the existence and
amount of LIV that would be exhibited in the highest energy cosmic-ray
interactions.

ARIANNA would have a sufficiently low threshold energy and the sensitivity
necessary to determine the location of the energy peak in the photomeson
neutrino EFS to further test LIV in the range $\delta_{\pi p}\leq 3\times
10^{-23}$, consistent with the current limit indicated by the PAO data. The
amount of LIV or its nondetection has important consequences for Planck scale
physics and quantum gravity theories.

## Acknowledgments

We would like to thank Steve Barwick, Peter Gorham, and John Krizmanic for
enlightening discussions and comments about the various present and potential
future neutrino detectors.

## References

## References

  * [1] F. W. Stecker, High Energy Astrophysics Tests of Lorentz Invariance Violation, Intl. J. of Mod. Phys. A 20 (2005) 3139–3142.  arXiv:astro-ph/0409731.
  * [2] S. Coleman, S. L. Glashow, High-energy tests of Lorentz invariance, Phys. Rev. D 59 (1999) 116008.
  * [3] F. W. Stecker, S. T. Scully, Searching for new physics with ultrahigh energy cosmic rays, New J. Phys. 11 (2009) 085003.
  * [4] K. Greisen, End to the Cosmic Ray Spectrum?, Phys. Rev. Lett. 16 (1966) 748–750.
  * [5] G. T. Zatsepin, V. A. Kuzmin, Upper limit of the Spectrum of Cosmic Rays, JETP Lett. 4 (1966) 78–80.
  * [6] S. T. Scully, F. W. Stecker, Lorentz Invariance Violation and the Observed Spectrum of Ultrahigh Energy Cosmic Rays, Astropart. Phys. 31 (2009) 220–225.
  * [7] J. Belz, Overview of recent HiRes results, Nucl. Phys. B (Proc. Suppl.) 190 (2009) 5–11.
  * [8] P. Sokolsky, Observation of the GZK cutoff by the HiRes experiment, Nucl. Phys. B (Proc. Suppl.) 196 (2009) 67–73.
  * [9] J. Abraham, for the Pierre Auger Collaboration, Measurement of the energy spectrum of cosmic rays above 1018 eV using the Pierre Auger Observatory, Phys. Lett. B 685 (2010) 239–246.
  * [10] V. S. Berezinsky, G. T. Zatsepin, Cosmic neutrinos of superhigh energy, Yad. Fiz. 11 (1970) 200–205.
  * [11] F. W. Stecker, Ultrahigh energy protons electrons and neutrinos the microwave background and the universal cosmic ray hypothesis, Astrophys. and Space Sci. 20 (1973) 47–57.
  * [12] F. W. Stecker, Diffuse fluxes of cosmic high energy neutrinos, Astrophysical Journal 228 (1979) 919–927.
  * [13] R. Engel, D. Seckel, T. Stanev, Neutrinos from propagation of ultrahigh energy protons, Phys. Rev. D 64 (2001) 093010.
  * [14] J. Alfaro, G. Palma, Loop quantum gravity and ultrahigh energy cosmic rays, Phys. Rev. D 67 (2003) 083003.
  * [15] P. W. Gorham, et al., Observational constraints on the ultrahigh energy cosmic neutrino flux from the second flight of the ANITA experiment, Phys. Rev. D 82 (2010) 022004.
  * [16] L. Maccione, et al., Planck-scale Lorentz violation constrained by Ultra-High-Energy Cosmic Rays, J. Cosmology and Astropart. Phys. 04 (2009) 022\.
  * [17] X. Bi, et al., Testing Lorentz invariance with the ultrahigh energy cosmic ray spectrum, Phys. Rev. D 79 (2009) 083015.
  * [18] G. R. Blumenthal, Energy loss of high-energy cosmic rays in pair-producing collisions with ambient photons, Phys. Rev. D 1 (1970) 1596–1602.
  * [19] F. W. Stecker, M. A. Malkan, S. T. Scully, Intergalactic Photon Spectra from the Far-IR to the UV Lyman Limit for z between 0 and 6 and the Optical Depth of the Universe to High-Energy Gamma Rays, Astrophys. J. 648 (2006) 774–783.
  * [20] E. Le Floc’h, et al., Infrared Luminosity Functions from the Chandra Deep Field-South: The Spitzer View on the History of Dusty Star Formation at z between 0 and 1, Astrophys. J. 632 (2005) 169–190.
  * [21] A. M. Hopkins, J. F. Beacom, On the Normalization of the Cosmic Star Formation History, Astrophys. J. 651 (2006) 142–154.
  * [22] R. A. Arndt, W. J. Briscoe, I. I. Strakovsky, R. L. Workman, Analysis of pion photoproduction data, Phys. Rev. C 66 (2002) 055213.
  * [23] A. M. Bernstein, S. Stave, Energy dependence of the delta resonance: Chiral dynamics in action, Few-Body Systems 41.  arXiv:0710.3387.
  * [24] F. W. Stecker, Effect of Photomeson Production by the Universal Radiation Field on High-Energy Cosmic Rays, Phys. Rev. Letters 21 (1968) 1016–1018.
  * [25] D. Mattingly, et al., Possible neutrino constraints on Planck scale Lorentz violation, J. Cosmology and Astropart. Phys. 02(2010)007.
  * [26] J. Belz, Composition Measurements near the Second Knee with the Telescope Array Low-Energy Extension (TALE), in: Intl. Cosmic Ray Conf., Vol. 5, 2008, pp. 1163–1166.
  * [27] M. Unger, for the Pierre Auger Collaboration, Composition Studies with the Pierre Auger Observatory, ArXiv: 0902.3787.
  * [28] R. Ulrich, for the Pierre Auger Collaboration, Investigation of Hadronic Interactions at Ultra-High Energies with the Pierre Auger Observatory, ArXiv: 0906.4691.
  * [29] F. W. Stecker, Photodisintegration of Ultrahigh-Energy Cosmic Rays by the Universal Radiation Field, Physical Review 180 (1969) 1264–1266.
  * [30] D. Hooper, A. Taylor, S. Sarkar, The impact of heavy nuclei on the cosmogenic neutrino flux, Astroparticle Physics 23 (2005) 11–17.
  * [31] M. Ave, et al., Cosmogenic neutrinos from ultra-high energy nuclei, Astroparticle Physics 23 (2005) 19–29.  arXiv:arXiv:astro-ph/0409316.
  * [32] Allard, et al., Cosmogenic neutrinos from the propagation of ultrahigh energy nuclei, J. Cosmology and Astropart. Phys. 9 (2006) 005.
  * [33] J. Ahrens, et al., Sensitivity of the IceCube detector to astrophysical sources of high energy muon neutrinos, Astropart. Phys. 20 (2004) 507–532.
  * [34] A. Santangelo, et al., The Science Case of the JEM-EUSO Mission-Unveiling the Universe at ultra-high energies, in: H. Susa and others (Ed.), AIP Conf. Ser., Vol. 1238, 2010, pp. 380–387.
  * [35] A. Petrolini, Ultra-High Energy Cosmic Particles studies from space: super-EUSO, a possible next-generation experiment, arXiv: 0909.5220.
  * [36] F. W. Stecker, et al., Observing the Ultrahigh Energy Universe with OWL Eyes, Nucl. Phys. B Proc. Suppl. 136 (2004) 433–438.  arXiv:astro-ph/0408162.
  * [37] S. W. Barwick, ARIANNA: A New Concept for UHE Neutrino Detection, in: Intl. Cosmic Ray Conf., Vol. 5, 2008, pp. 1601–1604.
  * [38] P. Allison, et al., IceRay: An IceCube-centered radio-Cherenkov GZK neutrino detector, Nucl. Instr. Meth. A 604 (2009) 64–69.  arXiv:0904.1309.
  * [39] A. A. Abdo, et al., Spectrum of the Isotropic Diffuse Gamma-Ray Emission Derived from First-Year Fermi Large Area Telescope Data, Phys. Rev. Letters 104 (2010) 101101.
  * [40] V. Berezinsky, et al., Fermi-LAT restrictions on UHECRs and cosmogenic neutrinos, arXiv: 1003.1496.
  * [41] M. Ahlers, et al., GZK neutrinos after the Fermi-LAT diffuse photon flux measurement, Astropart. Phys. 34 (2010) 106–115.
  * [42] D. V. Semikoz, G. Sigl, Ultra-high energy neutrino fluxes: new constraints and implications, J. Cosm. Astropart. Phys. 04(2004)003.
  * [43] P. Sreekumar, et al., EGRET Observations of the Extragalactic Gamma-Ray Emission, Astrophys. J. 494 (1998) 523–534.
  * [44] A. W. Strong, I. V. Moskalenko, O. Reimer, Diffuse Galactic Continuum Gamma Rays: A Model Compatible with EGRET Data and Cosmic-Ray Measurements, Astrophys. J. 613 (2004) 962–976.
  * [45] J. Wdowczyk, W. Tkaczyk, A. W. Wolfendale, Primary cosmic $\gamma$-rays above $10^{12}$ eV, J. Phys. A 5 (1972) 1419–1432.
  * [46] P. P. Kronberg, et al., A Global Probe of Cosmic Magnetic Fields to High Redshifts, Astrophys. J. 676 (2008) 70–79.
  * [47] A. Neronov, I. Vovk, Evidence for Strong Extragalactic Magnetic Fields from Fermi Observations of TeV Blazars, Science 328 (2010) 73–75.
  * [48] W. Atwood, et al., The Large Area Telescope on the Fermi Gamma-Ray Space Telescope Mission, Astrophys. J. 697 (2009) 1071–1102.