# A relook at the GZK Neutrino – Photon Connection: Impact of Extra-galactic
Radio Background & UHECR properties

Sovan Chakraborty  Poonam Mehta  and Prantik Sarmah

###### Abstract

Ultra-high energy cosmic rays (UHECRs) beyond the Greisen-Zatsepin-Kuzmin
(GZK) cut-off provide us with a unique opportunity to understand the universe
at extreme energies. Secondary GZK photons and GZK neutrinos associated with
the same interaction are indeed interconnected and render access to multi-
messenger analysis of UHECRs. The GZK photon flux is heavily attenuated due to
the interaction with Cosmic Microwave Background (CMB) and the Extra-galactic
Radio Background (ERB). The present estimate of the ERB comprising of several
model uncertainties together with the ARCADE2 radio results in large
propagation uncertainties in the GZK photon flux. On the other hand, the
weakly interacting GZK neutrino flux is unaffected by these propagation
effects. In this work, we make an updated estimate of the GZK photon and GZK
neutrino fluxes considering a wide variation of both the production and
propagation properties of the UHECR like, the spectral index, the cut-off
energy of the primary spectrum, the distribution of sources and the
uncertainties in the ERB estimation. We explore the detection prospects of the
GZK fluxes with various present and upcoming UHECR and UHE neutrino detectors
such as Auger, TA, GRAND, ANITA, ARA, IceCube and IceCube-Gen2. The predicted
fluxes are found to be beyond the reach of the current detectors. In future,
proposed IceCube-Gen2, Auger upgrade and GRAND experiments will have the
sensitivity to the predicted GZK photon and GZK neutrino fluxes. Such
detection can put constraints on the UHECR source properties and the
propagation effects due to the ERB. We also propose an indirect limit on the
GZK photon flux using the neutrino-photon connection for any future detection
of GZK neutrinos by the IceCube-Gen2 detector. We find this limit to be
consistent with our GZK flux predictions.

##  1 Introduction

Cosmic rays (CRs), charged particles of astrophysical origin, first discovered
in 1912 by Hess [1], continue to intrigue us even today. Several questions
concerning their origin and propagation remain unanswered (see [1, 2, 3] for
recent reviews). The primary CR spectrum extends in energy from $\rm GeV$ to
around $10^{2}~{}\rm EeV$. The CR spectrum exhibits sharp steepening (the
spectral index increases from about $2.7$ to about $3.1$) at $\sim 1~{}\rm
PeV$ (known as the “knee”) and flattening at $\sim 3~{}\rm EeV$ (known as the
“ankle”). CRs below the ankle are believed to have their origin in galactic
sources while those above, referred to as the Ultra-high energy CRs (UHECRs),
are believed to originate from extra-galactic sources 111This is because the
gyration radius of protons in the galactic magnetic field is of the same order
as the size of our Galaxy and therefore containment is not possible and no
acceleration mechanism could be effective..

Soon after the discovery of the cosmic microwave background (CMB), Greisen
[4], Zatsepin and Kuzmin [5] (GZK) in 1966 suggested that the UHECR protons
can interact with CMB photons via $\Delta^{\star}$ resonance ($E_{th}\sim
4\times 10^{19}$ eV) and get attenuated if they originate in sources more
distant than $10$ Mpc.

The GZK interaction ($p+\gamma_{CMB}$) yields charged pions ($\pi^{\pm}$) and
neutral pions ($\pi^{0}$) that subsequently decay to produce high energy
neutrinos (GZK or cosmogenic neutrinos) and gamma rays (GZK photons) as
secondaries. However, the UHECR spectrum extending well beyond the GZK cut-off
(the trans-GZK UHECRs) demands rigorous understanding regarding the origin and
propagation of these UHECRs [6, 7, 8, 9]. One possible reason for this could
be attributed to the piling up of UHECR protons around the GZK cut-off due to
energy losses [see 10, for details]. In order to resolve this puzzle, one
needs to probe the sources and production mechanism of these UHECRs.
Concerning the production of these UHECRs, there are two broad classes of
models : bottom-up and top-down. In the bottom-up model, the initial UHECRs at
the source can be produced by accelerating low energy particles. Charged
particles, such as protons and electrons can be efficiently accelerated to
extreme energies (around $10^{20}$ eV) via Fermi’s diffusive shock
acceleration mechanism [1]. The acceleration depends on the size of the
acceleration region, forward shock speed and the magnetic field strength [1].
Thus, for production of UHE particles, one needs large sources with fast shock
and strong magnetic fields. Possible sources of UHECRs include Active Galactic
Nuclei (AGN) such as blazars or quasars, radio galaxies, pulsars and Gamma-ray
bursts (GRB). Recent detection of a PeV neutrino event by IceCube neutrino
observatory from the direction of the blazar TXS 0506+056 provides us with an
indirect evidence of particle acceleration to very high energies [11]. UHECRs
detected by various cosmic ray experiments such as Akeno Giant Air Shower
Array (AGASA) [8], High Resolution Fly’s Eye (HiRes) [7], Pierre Auger
Observatory (Auger) [6] and Telescope Array (TA) [12] support the idea that
UHECRs could emanate from these sources. Note that shock waves produced in
supernova explosion could also produce high energy protons upto a few
$\mathcal{O}(10~{}\rm PeV)$ [13, 14, 15, 16, 17]. In contrast to the bottom-up
model, the top-down model leads to the production of UHECRs via exotic sources
[18] such as decay of heavy dark matter particles [19, 2, 20, 21, 22, 23],
topological defects, cosmic strings etc [see Ref. 2, for details]. The generic
prediction of this class of models is that photons should dominate over
nucleons but this is not supported by Auger data [24]. Thus, the top-down
models are currently disfavoured. In what follows, we shall consider the
bottom-up model i.e., shock accelerated origin of UHECRs.

In addition to directly probing the UHECR primary, we can also probe the
secondaries such as cosmogenic or GZK neutrinos and GZK photons which are
expected to be produced in the GZK process [25, 26, 27, 28, 29, 30, 31]. With
the advancement in detection prospects of high energy neutrinos and gamma
rays, the possibility of probing the origin of UHECRs (trans-GZK) via multi-
messenger approach opens up.

The GZK neutrinos, being weakly interacting, are not impacted by the
propagation effects and can reveal useful information about the source
spectrum of the UHECR primary. On the other hand, the GZK photons are expected
to suffer attenuation due to pair production loss on the CMB and the extra-
galactic radio background (ERB). Apart from the pair production loss, other
propagation effects like inverse Comptonisation of these background photons
can also affect the GZK photon flux. The attenuation of GZK photon flux by the
CMB is well-estimated as the CMB spectrum is well-described by a thermal
distribution [32] However, the attenuation of GZK photons by the ERB is not
very well-estimated, primarily because models describing the ERB spectrum have
large uncertainties. [33, 34]. These model uncertainties can cascade to the
GZK photon flux prediction and can give rise to the substantial uncertainties
[35, 36, 37]. Apart from these uncertainties, the ARCADE2 experiment [38] has
detected an excess ERB at GHz frequencies and can influence the prediction of
the GZK photon flux.

Here, we exploit the multi-messenger approach to address the unresolved
questions in UHECRs. Since the GZK neutrinos and GZK photons are connected to
the same origin, detection of either of them allows us to constrain the other.
This multi-messenger connection proves to be helpful for understanding the
source characteristics of the UHECRs. Similarly, such muti-messenger detection
of UHECRs, GZK neutrinos and GZK photons will reveal the effect of propagation
and in turn allow us to constrain the ERB models.

In the present study, we estimate the fluxes of GZK protons, GZK photons and
GZK neutrinos produced by interaction of UHECR with CMB 222The UHECRs
propagate through the extra-galactic magnetic field (EGMF). Strong deflection
of the UHECRs in magnetic field could occur only in dense regions such as
UHECR sources and in Milky-way. The deflection due to the EGMF is taken to be
negligible [35, 36].. We consider both the source and propagation
uncertainties. In particular, it should be noted that the ERB uncertainties
are crucial for any multi-messenger estimate involving the GZK photons. We
take into account the different ERB models that account for uncertainties in
the estimation of ERB [33, 34]. We also include the rarely explored ARCADE2
radio results [35, 36]. We analyse the propagation of GZK photons with
different ERB models and uncertainties independently. Based on the
uncertainties of the source parameters and of the ERB, we estimate the viable
parameter space of GZK photon flux and GZK neutrino flux and discuss detection
prospects at various current and upcoming UHE photon detectors (Pierre Auger
Observatory (Auger) [6], Telescope Array (TA) [12], GRAND [39, 40]) as well as
UHE neutrino detectors (Auger, ANITA [41], ARA [42], IceCube [43], Square
Kilometer Array [44, 45, 46]).

While accessing the GZK neutrino flux is beyond the reach of current UHE
neutrino detectors (Auger, ANITA, ARA, IceCube), the upcoming efforts such as,
IceCube-Gen2 and GRAND will be able to detect GZK neutrinos up to $10^{20}$
eV. These detectors might be sensitive to the UHECR spectral index for pure
proton primary. However, presence of heavy element primary such as iron in the
UHECR spectrum would produce lower fluxes of GZK secondaries leading to lower
detection sensitivity. On the other hand, cut-off energies of the UHECR
spectrum, only smaller than $10^{20}$ eV will be accessible to these
detectors. On the other hand, the GZK photon flux is found to be consistent
with the present upper limits of Auger and TA. The upcoming Auger upgrade and
the GRAND detector will be able to detect the GZK photon flux. Interestingly,
the detection of GZK neutrinos in IceCube-Gen2 will allow us to put a lower
bound on the GZK photon flux.

The paper is organised as follows. We discuss the UHECR source spectrum
modeling, production of GZK neutrinos and GZK photons in GZK process, followed
by their propagation through the inter-galactic space in Sec. 2. In Sec. A, we
describe the method to compute the fluxes of these GZK particles. The
dependence of the GZK secondaries on the source parameters as well as the
propagation effects due to ERB have been analysed in details in Sec. 3. We
present a broad discussion on the detection prospects of the GZK neutrinos and
GZK photons together with possible multi-messenger phenomenology in Sec. 5.
Finally, we conclude the paper with a brief summary in Sec. 6.

##  2  GZK flux: Production and Propagation effects

UHECRs upto $10^{20}$ eV have been detected by CR observatories [6, 7, 8, 12].
These UHECRs are believed to be produced in high energy astrophysical
environments through the mechanism of diffusive shock acceleration [1]. The
diffusive shock acceleration results in a power law distribution of CRs with a
cut-off at higher energies related to the maximum CR energy [2]. This power
law behaviour is also well-motivated from the CR observations [6, 7, 8, 12].
Thus, we can model the primary UHECR (assumed to be proton) spectrum to be a
power law with an exponential cut-off [47],

$J_{\rm p}(E_{\rm p})=\frac{\mathrm{d}N_{\rm p}(E_{\rm p})}{\mathrm{d}E_{\rm
p}}\propto\begin{cases}\left(\frac{E_{\rm p}}{E_{\rm
p,min}}\right)^{-\alpha},&E_{\rm p}\leq E_{\rm cut}\\\ \left(\frac{E_{\rm
p}}{E_{\rm p,min}}\right)^{-\alpha}\exp{\left(1-\frac{E_{\rm p}}{E_{\rm
cut}}\right)},&E_{\rm p}>E_{\rm cut}\ ,\end{cases}$ (2.1)

where, $E_{\rm p,min}$ is the minimum energy of the UHECR protons and $\alpha$
is the power-law index. The spectrum falls rapidly for energies larger than
the cut-off energy, $E_{\rm cut}$.

The standard diffusive shock acceleration theory predicts $\alpha$ to be
around $2$ [1, 48]. However, complex nature of shock acceleration, magnetic
field amplification and effect of cosmic ray pressure can lead to deviations
from $\alpha=2$ [49, 50]. Both $\alpha<2$ and $\alpha>2$ have been indicated
via observation of secondary high energy particles believed to be originating
from parent CRs. Some examples are, Galactic PeV gamma-ray observations
($\alpha<2$) [51, 52, 14], the diffuse neutrino background at IceCube
($\alpha>2$) [53], and the diffuse CR spectra ($\alpha\neq 2$) [54, 6]. The
other source parameter, $E_{\rm cut}$ also depends on the source and the
confinement duration of CRs in the source environment. These CRs escaping the
confinement zone due to dynamical and/or advective and/or diffusion losses can
render acceleration at very high energies inefficient [2, 1]. Thus, the CR
spectrum is expected to fall rapidly at very high energies resulting in a cut-
off energy [35, 2]. Note that, the Eq. 2.1 is not normalised. The
normalisation is obtained from the cosmic ray data above $10^{17}$ eV [6, 7,
8], thus fixing the minimum proton energy, $E_{\rm p,min}=10^{17}$ eV.

Based on this modelling of the primary UHECR spectra, one may estimate the CR
and secondary fluxes and compare with observational limits. Secondary
particles such as photons and neutrinos are produced as a result of
interaction of UHECR with the CMB photons above a certain threshold energy. We
next describe the GZK process and the production of secondaries. We also
discuss the reach of upcoming UHE neutrino and photon detectors.

###  2.1 The GZK effect and secondary production

Greisen, Zatsepin and Kuzmin [4, 5] proposed that the UHECR spectrum beyond
$E_{th}\simeq 4\times 10^{19}$ eV should be suppressed due to the interaction
of CR particles (protons or nuclei) with the low energy photons of the CMB
primarily due to the $\Delta^{+}$ resonance. For protons, the pion
photoproduction produces neutral or charged pions,

$\displaystyle p+\gamma_{\rm CMB}$ $\displaystyle\to$
$\displaystyle\Delta^{+}{~{}\rm}\to N+\pi^{\pm}(\pi^{0})\quad$ (2.2)

The pions produced in the GZK process (Eq. 2.2) decay to give rise to
secondaries [55] which include electrons, neutrinos and photons. The charged
secondaries (electrons, positrons) from GZK process can not reach Earth due to
interaction with inter-galactic magnetic field and ambient matter [35]. The
neutral pions decay to produce “GZK photons” and charged pions decay to
produce “GZK or cosmogenic neutrinos”,

$\displaystyle\pi^{0}$ $\displaystyle\to$ $\displaystyle\gamma+\gamma\quad$
$\displaystyle\pi^{\pm}$ $\displaystyle\to$
$\displaystyle\mu^{\pm}+\nu_{\mu}(\bar{\nu}_{\mu})$ (2.3)
$\displaystyle~{}\hookrightarrow
e^{\pm}+\nu_{e}(\bar{\nu}_{e})+\bar{\nu}_{\mu}(\nu_{\mu})~{}.\quad$

The CMB spectrum follows a blackbody distribution with a mean temperature of
$2.7$ K and peaks at around $4\times 10^{-4}$ eV. Using this, we can estimate
the cut-off energy of the protons to be $\sim 4\times 10^{20}$ eV. This is
referred to as the GZK cut-off. The UHECR protons observed above the GZK cut-
off are termed as “GZK protons”. While the charged secondaries get deflected
and can not reach Earth, the neutral ones, GZK photons and GZK neutrinos have
promising detection prospects.

The GZK photons and GZK neutrinos propagate through the intergalactic space
filled with low energy photon backgrounds, i.e., CMB and ERB. These
backgrounds are crucial for the propagation of GZK photons as they can take
part in interactions such as pair production and inverse Compton. In order to
estimate the impact of CMB and ERB on the propagation of GZK photons, one
needs a proper estimate of the background photons (ERB and CMB). While the CMB
is well-described by a blackbody spectrum [56] as mentioned above,
understanding of ERB is not at the same level. The models describing the ERB
spectrum, in general, have large uncertainities and therefore the attenuation
of GZK photons by the ERB is not very well-estimated. In the following, we
shall adopt a model of ERB to demonstrate the impact of ERB on GZK photon
propagation [33]. It should be noted that the GZK neutrinos do not suffer
propagation losses, owing to their weakly interacting nature.

Figure 1: The figure shows the intensity ($I_{\nu}$) of ERB photons as a
function of frequency ($\nu$) for the Protheroe and Biermann model [33]

###  2.2 Extra-galactic radio background and secondary propagation

The ERB comprises of radio photons of KHz-GHz frequencies emitted by radio
galaxies and star forming galaxies [33, 34, 57]. These radio photons are
produced through free-free emission of thermal electrons and synchrotron
radiation of high energy electrons [34]. At high frequencies (GHz), the ERB is
dominated by the radio galaxies, whereas at low frequencies (below MHz), the
contributions from star forming galaxies are more pronounced. Both radio and
star forming galaxies contribute almost equally to the ERB in the intermediate
frequencies, i.e., MHz-GHz. The intensity of this ERB can be estimated by
using observed luminosity functions of star forming and radio galaxies [33,
34]. This estimation also requires radio emission of individual galaxy which
can be obtained by modeling the thermal and non-thermal electron densities of
the galaxy. Two models of ERB are available in the literature, namely the
Protheroe and Biermann (PB96) and Nitu et. al. (Nitu21) [33, 34]. These models
shall be discussed in detail in Sec. 4. Fig. 1 shows the lower estimate of ERB
PB96 [33]. This shows that the ERB peaks at around 100 KHz frequency and then
falls rapidly. Note that various sources of uncertainty such as luminosity
function and electron density impact the ERB estimate. Also, the predictions
of the two ERB models [33, 34] have substantial difference. To demonstrate the
role of ERB on GZK photon propagation, we choose the conservative estimate of
ERB from the literature as shown in Fig. 1 [33]. In Sec. 4, we provide a
detailed discussion on the different ERB models, their associated
uncertainties and the effect of these on the GZK photon flux propagation.

Estimating the GZK photon flux at Earth includes various propagation effects
like pair production, subsequent cascading and inverse Compton scattering on
different photon backgrounds like CMB, ERB and Extra-galactic Background Light
(EBL) [35]. It should be noted that the effects of EBL dominate at energies
around $100-1000$ TeV [58]. However as we are interested in energies above
$10^{18}$ eV, the effects are expected to be negligible and we shall not
consider these effects henceforth. To calculate the GZK fluxes, we utilize the
publicly available CRPropa 3.2 code [47, 59]. CRPropa 3.2 is primarily written
in C++ interfaced to Python. CRPropa 3.2 contains different modules for the
aforementioned propagation effects. Note that the distribution UHECR sources
evolves in redshift due to the expansion of the Universe. CRPropa 3.2 has
inbuilt source evolution as a function of redshift [47]. In addition, it also
takes into account the redshift evolution of photon fields such as CMB.
Further details of computational analysis with CRPropa 3.2 are outlined in
Appendix A.2.

The GZK fluxes are dependent on various source characteristics ($\alpha$,
$E_{\rm cut}$ and composition) as well as the minimum distance to these
sources from Earth. In the following section, we discuss the significance and
selection of specific ranges of these parameters.

##  3 GZK flux: Dependence on source properties

The properties of the primary UHECRs at the source impact the estimation of
the secondary GZK photon and GZK neutrino fluxes at Earth. These properties
depend on the acceleration mechanism as well as CR interactions within source.
The primary UHECR proton spectrum (Eq. 2.1) depends on two parameters, the
spectral index, $\alpha$ and the cut-off energy, $E_{\rm cut}$. These
parameters depend on the underlying acceleration mechanism at the source. The
GZK fluxes produced from the UHECR also depend on the propagation length.
Thus, the distribution of these sources over distances (redshift) becomes
crucial for the flux estimate. We model the UHECR sources to be distributed
over the space in the region $d_{\rm min}\leq d\leq d_{\rm max}$, where
$d_{\rm min}$ and $d_{\rm max}$ denote the minimum and maximum distance to the
source. Since, the likely sources of UHECRs are extra-galactic, we choose to
vary $d_{\rm min}$ in the range $[0.1,100]$ Mpc. On the other hand, we fix
$d_{\rm max}$ to be $10$ Gpc which corresponds to a redshift of $1.4$. In the
following, we discuss the role played by these parameters in the computation
of the GZK fluxes.

The spectral index ($\alpha$): The spectral index ($\alpha$) at source is a
crucial parameter for UHECR flux at the highest energies. Several indirect
galactic and extra-galactic observations [51, 60, 2, 6, 61] suggest
substantial deviation from the conventional diffusive shock acceleration
theory predicted spectral index of $\alpha=2.0$ [49, 50, 54]. Note that the
observed spectral index ($\alpha^{\prime}$) at Earth might be significantly
different from the spectral index at the acceleration site. This is because
the shock accelerated CRs can undergo different interactions within the source
environment [2]. Thus, the spectral index not only depends on the
acceleration, but may also depend on the nature of CR interactions within the
source. The observed UHECR spectrum (above $10^{18}$ eV) at Earth mostly
follows a $E^{-3}$ ($\alpha^{\prime}\sim 3$) like behaviour and shows breaks
at different energies indicating different spectral indices [see 61, for
details]. In addition, recent studies by Auger collaboration [see e.g., 62,
63] have shown that the mass composition of UHECRs may also influence the
spectral index. In fact, these studies suggest that CR spectrum much harder
than that predicted by diffusive shock acceleration theory is plausible due to
substantial heavy element contribution at highest energies. For our analysis,
we primarily focus on the impact of ERB background on the GZK fluxes and do
not intend to differentiate the various spectral parameter scenarios at
source. In this regard, we consider a conservative scenario of softer primary
spectra at source, i.e., $\alpha$ in the range $[2.2,2.7]$ with $100\%$ proton
composition. This choice allows us to demonstrate how the impact of the ERB
uncertainties on the GZK fluxes may get affected due to the spectral index
variation at source.

The spectral index variation impacts the GZK secondary fluxes substantially
(see Appendix B for more details). For instance, smaller $\alpha$ yields
harder GZK secondary fluxes. The difference due to variation of $\alpha$ is
found to be more prominent at higher energies.

The composition of UHECR: So far, the UHECR primary is considered to be proton
only. However, the recent data by Auger [64, 65, 62, 66, 63, 67, 6] suggests
that the UHECR could be composed of heavy elements but the fractional
component of these heavy elements is still debatable [36, 68, 69]. The
presence of heavy elements along with protons might lead to lowering of the
fluxes of GZK neutrinos and GZK photons [70, 71, 72]. For example, we note
that GZK neutrinos and GZK photon fluxes for pure iron primary are an order of
magnitude smaller than the corresponding flux for pure proton case (see
Appendix B). In turn, this would impact the detection prospects of these GZK
neutrinos and GZK photons. We would like to remark that in all our
computations, we consider UHECRs to be composed of protons only. As a result,
our flux estimates represent the upper limits of the fluxes of the secondaries
produced via the GZK interaction. For reference, we also discuss the extreme
scenario of $100\%$ iron primary to evaluate the most conservative estimate.

The cut-off energy ($E_{\rm cut}$): The behaviour of the UHECR spectrum at the
extreme high energies is governed by the $E_{\rm cut}$ (see Eq. 2.1). The
UHECR spectrum at these energies may also depend on the nature of the cut-off
function at source [73, 10, 74]. The cut-off energy influences the fluxes of
GZK protons, GZK photons and GZK neutrinos. The present observation of UHECRs
suggests $E_{\rm cut}\sim 10^{20}$ eV. The conventional diffusive shock
acceleration theory also allows acceleration of CRs upto energies larger than
$10^{20}$ eV that can go up to $10^{22}$ eV [2]. Thus, we vary the $E_{\rm
cut}\in[5\times 10^{20}-10^{22}]$ eV in order to incorporate the acceleration
mechanism at the highest energies. The variation of $E_{\rm cut}$ can produce
large differences between the GZK secondary fluxes of different $E_{\rm cut}$
at the highest energies (see Appendix B for details). For lower values of
$E_{\rm cut}$ for instance, $E_{\rm cut}=$ $10^{20}$ eV, both GZK photons and
GZK neutrinos produce softer spectra at higher energy and fluxes fall rapidly
at energies larger than $10^{20}$ eV. Thus, the GZK photon and GZK neutrino
fluxes at higher energies will be sensitive to $E_{\rm cut}$.

The minimum distance to source ($d_{\rm min}$): The minimum distance to the
source, $d_{\rm min}$ is also an important parameter and can affect the UHECR
spectrum. As mentioned above, there are no known sources in the Milky way or
any nearby galaxies that are capable of producing CRs upto energies as large
as $10^{18}$ eV [2] . In fact, such CR accelerators might be located very far
away from Earth. Thus, this is encoded in $d_{\rm min}$. The nearest galaxy
(M31) to Milky way is about $1$ Mpc away, this can be treated as a reasonable
choice for $d_{\rm min}$. To study the effect of $d_{\rm min}$, we take the
lowest value of $d_{\rm min}$ to be $0.1$ Mpc. The highest value of the
$d_{\rm min}$ is taken to be $100$ Mpc [35]. We note that the variation of
$d_{\rm min}$ does not play any role in case of diffuse flux of GZK neutrinos
pertaining to their weakly interacting nature. In contrast, the diffuse flux
of GZK photons depends on $d_{\rm min}$. The larger the value of $d_{\rm
min}$, the larger are the losses. Note that the suppression is more pronounced
at higher energies and this is due to the fact that ERB is more sensitive to
GZK photons at higher energies. This causes more attenuation for larger values
of $d_{\rm min}$ (see Appendix B).

Figure 2: The figure shows the intensity ($I_{\nu}$) as a function of
frequency ($\nu$) of different ERB models. The red curve and the red band
together show the ERB model of Nitu et al., (ERB Nitu21) [34]. The red curve
represents the best fit estimates, whereas the red band shows the associated
uncertainty. The ERB model of Protheroe and Biermann (ERB PB96), [33] is shown
in blue colour. The blue curve corresponds to the mid estimate of ERB PB96 and
blue band shows the associated uncertainty. The ERB measured by the ARCADE2
experiment has been shown by the brown curve (ERB Fixsen) [38].

##  4 Dependence of the GZK photon flux on ERB

The ERB is distributed over seven decades in frequency and impacts
significantly the GZK photon flux. Any reasonable prediction of the GZK photon
flux requires a good understanding and estimation of the ERB. There are two
models in literature, PB96 [33] and Nitu21 [34]. The observational constraints
on ERB is only available in certain ranges of the frequencies. The ERB in the
frequency range of $0.1-1$ GHz is constrained by radio source counts [34].
However, uncertainties due to unresolved radio sources cannot be ignored [75].
Both the ERB models have adopted a similar approach for the estimation of ERB.
The estimation is based on the modelling of radio emission of individual
galaxies. Based on this individual galaxy emission model the ERB from all
possible sources (radio and star forming galaxies) is obtained by using
luminosity functions [33, 34]. Below 100 MHz, there is significant uncertainty
due to the uncertainty in the evolution of luminosity functions of radio and
normal galaxies [33]. On the other hand, the galactic foreground emission
severely distorting the ERB produces extremely large uncertainties at KHz
frequencies [34]. Note that, these uncertainties at low frequencies might also
get significantly impacted by different competing radiative processes such as
free-free emission, synchrotron self-absorption [76, 77].

In addition to these uncertainties, ARCADE2 experiment has reported an excess
in the ERB [38]. This excess was realised through the measurement of absolute
sky temperature at different frequencies ($3,8,10,30$ and $90$ GHz). This
analysis estimated the excess temperature, $T=24.1\pm 2.1~{}({\rm
K})~{}(\nu/\nu_{0})^{-2.599\pm 0.036}$ ($\nu_{0}=310~{}\rm MHz$) in the
frequency range of $22$ MHz$-10$ GHz in addition to the CMB temperature of
$2.725\pm 0.001$ K. The origin of this excess is not well understood. A
possible explanation could be attributed to the unresolved radio sources [75].
The authors in [78] adopted a simple geometry (plane parallel slab) to model
the Galactic emission and estimated the extragalactic contribution. The
residual ERB component is found to exceed the integrated contribution of the
known population of extragalactic radio sources by factors of five or more
[38, 79]. However, it was shown that [80], a more realistic modeling led to
estimates for the uniform extragalactic brightness that were consistent with
expectations from known extragalactic radio source populations. The excess ERB
has motivated dark matter interpretations for its origin [81] and could be a
consequence of gravitational wave modeling for dark energy [82]. Note that the
CMB component at GHz frequencies dominates over the ERB component. However,
excess measured by ARCADE2 experiment is still larger than the CMB. Note that
observation of radio sources across a wide range of frequencies with
telescopes like Low frequency Array (LOFAR) [83] and the upcoming Square
Kilometer Array (SKA) [84, 85] might help to resolve this issue.

Fig. 2 shows two different ERB models, PB96 [33] and Nitu21 [34] with their
associated uncertainties. The red curve shows the best-fit ERB spectra
corresponding to ERB Nitu21 model along with the associated uncertainties
shown as red band. Similarly, the blue curve and the blue band depict the
typical estimate and the uncertainties of ERB PB96 model, respectively. The
excess detected by ARCADE2 has also been depicted by the brown curve (ERB
Fixsen). This shows that ERB has large uncertainties at both low and high
frequencies which impact the prediction of the GZK photon flux (see Sec. 2.1
for details). Therefore, in what follows, we analyse the effect of the ERB
uncertainties and the ARCADE2 results on the GZK photon flux.

The GZK neutrinos do not interact with ERB and are insensitive to the ERB. The
propagation of GZK protons is impacted by CMB only and effect of ERB is
negligible in the energy range our interest. Fig. A.2 shows that the ERB plays
a crucial role in the propagation of high energy (above $10^{18}$ eV) GZK
photons and the uncertainty in the ERB leads to variation in the estimated GZK
photon flux.

Figure 3: Effect of different ERB models on GZK photon propagation. The model
parameters are chosen as $\alpha=2.0$, $E_{\rm cut}=10^{22}$ eV and $d_{\rm
min}=1$ Mpc. The left panel shows the GZK photon flux with different ERB
models: PB96 (blue), Nitu21 (red) and Fixsen (brown). The blue and red curves
represent the GZK photon flux with the medium estimate of PB96 and the best
estimate of Nitu21 respectively. Lower ERB yields larger GZK photon flux and
vice-versa. The effect of ARCADE2 ERB results (ERB Fixsen) on GZK photon flux
in addition to the effect of ERB Nitu21 is also shown by the brown curve. The
right panel shows uncertainty in the GZK photon flux due to the model
uncertainty. The blue band shows the uncertainty in the ERB model of PB96
whereas the red band shows the uncertainty in the model of Nitu21. This shows
the uncertainty in the ERB models gives rise to large uncertainties in the GZK
photon flux. Note that we also plot the UHECR flux with the CR data for
reference.

The effect of the above ERB models (PB96 and Nitu21) and the ARCADE2 results
on the GZK photon flux are shown in the left panel of Fig. 3. For this, we
have adopted the GZK photon flux corresponding to a primary UHECR spectra with
$\alpha=2.0$, $E_{\rm cut}=10^{22}$ eV and $d_{\rm min}=0.1$ Mpc. The GZK
proton flux at Earth, normalized to Auger and HiRes data, is also shown on the
plot. The solid blue curve shows the GZK photon flux with the medium estimate
of PB96 (see Fig. 2), whereas the solid red curve depicts the GZK photon flux
for the best-fit ERB model of Nitu21. The ERB measured by ARCADE2 at high
frequencies can create additional GZK photon absorption. Thus, the effect of
ARCADE2 ERB (in addition to ERB Nitu21) is shown by the brown curve. Note that
the ERB models have large effect at energies above $10^{18}$ eV.

The ERB models of PB96 and Nitu21 have associated uncertainties as shown in
Fig. 2 that can result in uncertainty in GZK photon flux as well. The right
panel of Fig. 3 shows the uncertainty in the GZK photon flux due to the ERB
models. The blue band shows the uncertainty due to ERB model of PB96. The
lower edge in this band comes from the upper estimate of PB96, whereas the
upper edge is due to the lower estimate of PB96. Similarly, the uncertainty in
the GZK photon flux due to ERB Nitu21 is shown by the red band. Clearly, the
uncertainty due to ERB Nitu21 is smaller than that of PB96. This is because
the ERB above frequency $10^{5}$ Hz mostly contributes to this uncertainty and
at those frequencies, the uncertainty in Nitu21 model is smaller than that of
PB96. The ARCADE2 results along with the upper limit of ERB Nitu21 gives rise
to a lower GZK photon flux as shown by the brown curve. All these ERB model
estimates together with the model uncertainties yield a large (about an order
of magnitude) uncertainty for the GZK photon flux. A proper understanding of
these uncertainties is crucial for the detection of the GZK photon flux.

Note that, the large ERB uncertainties at low frequencies (Fig. 2) may not
yield significant effect on the GZK photon flux. This is because the threshold
energy of GZK photons for pair production on such low energy ERB photons is
above $10^{21}$ eV. The primary cosmic ray flux above these energies ($E_{\rm
cut}\sim\mathcal{O}(10^{22}~{}\rm eV)$) being extremely small, the low energy
ERB impact is much below the sensitivity limitations of various UHE photon and
UHE neutrino detectors.

##  5 Detection prospects: multi-messenger approach with GZK photons and GZK
neutrinos

As pointed out before, along with the flux of CRs there is a guaranteed flux
of neutrinos and photons. Naturally measuring the flux of these different
messengers at UHE would allow us to probe the nature of these UHECRs and their
sources much better. As far as the detection of UHECR and GZK photons is
concerned, we have the currently operational UHECR detectors such as the
Pierre Auger Observatory (Auger) [24, 86, 6, 67], Telescope Array (TA) [12] as
well as proposed detectors such as Giant Radio Array for Neutrino Detection
(GRAND) 333Auger and GRAND can also detect UHE neutrinos along with the
UHECRs. [39, 40]. In addition, for the GZK neutrino detection, we also have
Antarctic Impulsive Transient Antenna (ANITA) [41], the IceCube neutrino
observatory (IceCube and upcoming IceCube-Gen2) [53, 43], the Askaryan Radio
Array (ARA) [42], the Radio Ice Cherenkov Experiment Observatory (RICE) [87].
These neutrino experiments rely on the principle of Askaryan effect [41, 42,
40, 43]. Using lunar Cherenkov technique, SKA will also be able to detect the
GZK neutrinos. For a comparison of sensitivities of different detectors, see
[86, 43].

Figure 4: Detection prospects of GZK photon and neutrino (single flavour)
fluxes. The upper panel shows the typical estimate of GZK photon flux (red
curve) with associated uncertainty (red band). The lower panel depicts the
typical estimate of GZK neutrino flux (blue curve) with associated uncertainty
(blue band). The typical estimates are obtained by obtained by choosing the
parameters as listed in Table 1. For the upper limit of the GZK photon flux we
consider minimum estimate of ERB PB96, whereas for the lower limit we take ERB
Fixsen and the maximum estimate of ERB Nitu. The sensitivities of the UHE
photon and neutrino detectors have been also plotted. The flux predictions are
consistent with the experimental limits (Auger, TA) and theoretical limit i.e,
Waxman-Bahcall for neutrino. Most of the present detectors are not sensitive
to the fluxes. However, future proposals will have sensitivity for detection
like Auger 2023 (proposed upgrade) for photon, GRAND for photon and neutrino,
IceCube-Gen2 for neutrino. We also show the GZK photon (red dashed) and GZK
neutrino (blue dashed) fluxes for $100\%$ iron primary with spectral
properties, $\alpha=2.7$ and $E_{\rm cut}=5\times 10^{20}$ eV.

In Fig. 4, we show the flux of GZK photons (top panel) and GZK neutrinos
(bottom panel) as a function of energy for the benchmark values of parameters
given in Table 1, considering UHECR primary to be protons only. The units of
the fluxes in each plot are appropriately chosen so as to draw a comparison
with the existing sensitivities of different experiments. The GZK photon flux
has been plotted in units of $\rm km^{-2}~{}y^{-1}~{}sr^{-1}$, whereas the GZK
neutrino flux is shown in units of $\rm eV~{}cm^{-2}~{}sr^{-1}$. In the top
(bottom) panel, the red (blue) solid curve corresponds to the benchmark values
whereas the red (blue) band corresponds to the range of variation of
parameters given in Table 1 for gamma-rays (neutrinos). The photon band also
incorporates the ERB uncertainty. For the upper limit of the red band, we
consider the minimum estimate of PB96 ERB, whereas the lower limit is obtained
by considering the maximum estimate of ERB Nitu21 together with the ARCADE2
results (ERB Fixsen). The sensitivities of existing detectors are depicted on
the plots which allow for a direct comparison with our prediction. Note that
the GZK neutrino band at energies between $(10^{18}-10^{20})$ eV is narrower
compared to the GZK photon band. This is due to the fact that the effect of
UHECR source parameters ($\alpha$ and $E_{\rm cut}$) is more prominent at the
higher energies (see Sec. 3) and there is no propagation effect on the GZK
neutrinos. It is evident from the observations [64, 65, 62, 66, 63, 67, 6]
that there are heavy elements in the UHECR spectrum. The heavy elements in the
UHECRs can result in lowering the GZK photon and GZK neutrino fluxes. Hence,
in addition to the pure proton primary case, we also show the GZK photon (red
dashed) and GZK neutrino (blue dashed) fluxes due to heavy composition by
considering hypothetically extreme $100\%$ iron for the primary. For this
estimation, we consider the parameter values as $\alpha=2.7$, $E_{\rm
cut}=5\times 10^{20}$ eV and $d_{\rm min}=100$ Mpc. These fluxes show that the
uncertainty in the GZK photon and GZK neutrino fluxes could be an order of
magnitude lower (red dashed and blue dashed) than the lower limit of the pure
proton case. For any mixed UHECR composition, the corresponding GZK photon
(neutrino) flux should lie in between the upper edge of the red (blue) band
and the red (blue) dashed curve.

Parameters | Benchmark value | Range
---|---|---
$\alpha$ | $2.4$ | $2.2-2.7$
$E_{\rm cut}$ (eV) | $10^{21}$ | $5\times 10^{20}-10^{22}$
$d_{\rm min}$ (Mpc) | $1$ | $0.1-100$
Table 1: The choice of benchmark parameter values and the range considered in
our analysis for Fig. 4, see Sec. 3 for justification.

From top panel of Fig. 4, it is evident that the existing detectors do not
have the sensitivity to probe the predicted GZK photon flux. However, Auger SD
2025 (brown line) [67] and GRAND 200k 3 years (gray line) [39] may be able to
access the GZK photon flux. Likewise, from the bottom panel it is clear that
existing neutrino detectors are insensitive to the predicted GZK neutrino
flux. Only IceCube-Gen2 [43] and GRAND200k [39, 40] could provide us with
opportunities of detection of these fluxes in the future. Note that the
IceCube-Gen2 sensitivity is a band rather than a line due to the radio array
design uncertainties of the detector [43].

###  5.1 What can we learn from non-observation GZK photon and GZK neutrino
flux ?

Current upper limits on the UHE photon flux by different detectors (Auger and
TA) have been shown by the downward arrows in the top panel of Fig. 4. The
sensitivities of the Auger 2025 upgrade (brown line) [67] and the future
proposal, GRAND200k 3 years (gray line) [39] have also been plotted. The upper
limit on the GZK photon flux is well below the sensitivity of the Auger Hybrid
[24] and TA [12] detector. However, the sensitivity of the Auger SD 1500 m
[24] detector is found to be close to the benchmark flux in the energy range
$(1-4)\times 10^{19}$ eV. The photon fluxes (in the red band) in this energy
range above the typical flux mostly come from the smaller power law index
($\alpha<2.4$) and large cut-off energy ($E_{\rm cut}=10^{22}$ eV) of the
primary UHECR spectra. The upper limit at these photon energies ($<10^{20}$
eV) also corresponds to a smaller ERB flux at the medium and higher
frequencies. The non-observation of the GZK fluxes at Auger and the
sensitivity of Auger SD 1500 m slightly overlapping with these GZK photon
fluxes suggests either $\alpha\geq 2$ for pure proton primary or $\alpha\sim
2$ for heavy element primary. However, one needs to detect the GZK photon flux
over a wide energy range to constrain these parameters. The upcoming Auger
upgrade and the GRAND detector will be able to put such constraints more
firmly.

Similarly, sensitivities of different UHE neutrino detectors are plotted in
the bottom panel of Fig. 4. The neutrino flux corresponding to our benchmark
parameter values is found to be consistent with the Waxman-Bahcall limit [88]
(Black dashed line). The neutrino benchmark flux is also below the sensitivity
of IceCube (magenta curve) [43] and Auger (purple curve) [86] and is
consistent with the GZK non-observation at both the detectors. The combined
sensitivity of ANITA $\rm I+II+III$ (green curve) [89] is found to be above
the upper limit on GZK neutrinos. The sensitivities of the future detectors
ARA37 (orange curve) [42] and SKA low (cyan curve) [86, 43] are also found to
be above the GZK neutrino flux upper limit. Thus, none of these detectors are
capable of probing the benchmark neutrino flux. However, the proposed
sensitivity (brown band) of the next generation of IceCube experiment
(IceCube-Gen2) will be able to probe the GZK neutrino flux upto $10^{20}$ eV.
It should be noted that these neutrino detectors can not probe the effect of
the parameter $E_{\rm cut}$. Further, the neutrino flux is nearly independent
of the parameter $d_{\rm min}$, hence $d_{\rm min}$ can only be probed through
the detection of GZK protons and photons. In addition, the GZK neutrino flux
at lower energies between $10^{18}$ eV and $10^{19}$ eV is almost independent
of the parameters $\alpha$, $E_{\rm cut}$ and $d_{\rm min}$. The variation of
the neutrino flux at these energies is largely due to composition effects.
Hence, detection of GZK neutrinos in IceCube-Gen2 and GRAND will contribute to
our understanding of the UHECR composition. Note that the GZK neutrino flux at
lower energies i.e, $10^{18}$ eV can vary substantially due to the variation
of the spectral properties [see e.g., 90].

Figure 5: Constraints on GZK photon flux from detection/non-detection of GZK
neutrinos at IceCube-Gen2. The blue and red bands show our estimated GZK
neutrino ($\nu$, single flavour) and GZK photon ($\gamma$) fluxes
respectively. The brown band shows the sensitivity of IceCube-Gen2 to GZK
neutrinos. Following the neutrino sensitivity (brown band), the corresponding
constraints on the GZK photon flux has been estimated, shown by the black
band. This band also takes into account of the ERB uncertainties. Detection of
GZK neutrinos by IceCube-Gen2 would mean that the GZK photon flux is above (or
within due to ERB uncertainties) this black band.

###  5.2 What can IceCube-Gen2 limit say about GZK photon flux ?

Having discussed the flux prediction and sensitivity limits of the existing
detectors for the GZK photons and GZK neutrinos separately, we now address if
observation of any one of the component can have implications for detection of
the other component, thereby utilizing the power of multi-messenger approach.
In particular, we discuss the implications of GZK neutrino flux measured by
IceCube-Gen2 on GZK photon flux. As we have seen from Fig. 4, IceCube-Gen2
detector is sensitive to the GZK neutrinos upto $10^{20}$ eV. Using IceCube-
Gen2 neutrino sensitivity, we compute the corresponding GZK photon flux limit.
We compute this limit for the scenario of UHECR primary to be protons only.

For this estimation, we use the IceCube-Gen2 limit to obtain the normalization
and the spectral index of the primary UHECR proton flux. However, this primary
UHECR proton flux may produce different GZK photon flux due to different
propagation effects. In this regard, we estimate the GZK photon flux
corresponding to the neutrino flux sensitivity of IceCube-Gen2 considering
propagation effects of both the CMB and ERB. Our results are shown in Fig. 5.
The constraints from IceCube-Gen2 together with the GZK neutrino (blue) and
photon (red) fluxes (see Fig. 4) from our analysis are plotted in Fig. 5. The
brown band shows the GZK neutrino flux sensitivity of IceCube-Gen2 whereas the
black band represents the corresponding constraints on the GZK photon flux.
The GZK photon band is wider in comparison to the neutrino sensitivity band
primarily due to the effect of ERB uncertainties. Indeed, our limit of the GZK
photon flux is well below the model prediction in the corresponding energies.
Thus pointing at the following crucial multi-messenger implications:

  * •

If IceCube-Gen2 detects GZK neutrinos - the GZK photon flux is guaranteed from
the GZK interaction and expected to lie above or within the black band (Fig.
5).

  * •

The IceCube-Gen2 sensitivity being almost one order of magnitude lower than
the GZK neutrino flux model estimate would create a tension between the
observed UHECR data and our understanding of the GZK process in case no GZK
neutrinos are detected by IceCube-Gen2 in future. This can also be understood
from Fig. 4. Even the hypothetically extreme and most conservative composition
scenario of $100\%$ iron primary, the GZK neutrino flux is at the level of
IceCube-Gen2 sensitivity. It may be noted that the actual GZK neutrino flux
would lie in between these two extreme scenarios.

This photon flux corresponding to the IceCube-Gen2 neutrino sensitivity can be
interpreted as multi-messenger limit on the GZK photon flux. However, it
shouldn’t be confused with the lowest GZK photon flux that nature can produce.
The present limit is for $100$ percent protons, even different composition of
UHECR primary will lead to a very similar multi-messenger limit as the overall
normalization is fixed by the IceCube-Gen2 limit. There would be a little
variation in the spectral shape of this limit due to the choice of the primary
UHECR spectral index. Nevertheless, this difference is found to be negligible
in comparison to the uncertainties due to the ERB.

However, a similar limit for the GZK neutrinos from the detection of GZK
photons is not possible as the GZK photons suffer from propagation
uncertainties. Different UHECR parameters (source and propagation) will lead
to the same GZK photon flux as shown in Fig. 4. This arbitrariness makes the
reverse correspondence from GZK photons to GZK neutrinos inconclusive.

##  6 Summary and Conclusion

With impressive progress in gamma-ray astronomy, neutrino astronomy as well as
gravitational wave astronomy, we are ushering into an exciting phase in high
energy astroparticle physics where we can utilize multi-messenger signals to
understand the origin of UHECRs and their nature. UHECRs (at energies above
$0.1$ EeV) have been observed by different CR observatories such as AGASA,
HiRes and Auger. Assuming the UHECR primary to be proton, these will suffer
attenuation due to the CMB and a cut-off in the UHECR spectrum is expected at
$\sim 4\times 10^{19}$ eV (as predicted by GZK [4, 5]). The GZK interaction
also produces secondaries i.e., GZK photons and GZK neutrinos. In recent
times, due to the advancement of high energy gamma-ray experiments (e.g.
Fermi-LAT, Large High Altitude Air Shower Observatory (LHAASO)) [11, 91, 92,
93] and high energy neutrino (e.g. IceCube) experiments [11, 94, 95],
exploring the multi-messenger connections has emerged as a powerful tool in
understanding the complex phenomena at the highest energies. These detectors
have made significant discoveries of point sources (galactic and extra-
galactic) as well as diffuse fluxes, mainly at energies in the PeV range [11,
91, 92, 93, 94, 95]. Thus, with the advancement in the detection techniques in
both gamma-ray astronomy and neutrino astronomy, probing the origin and nature
of UHECR with the multi-messenger signals would become a reality in the coming
years.

In the present work, we numerically estimate the diffuse flux of GZK
secondaries (neutrinos and photons) and discuss possibilities of their
detection with current and upcoming UHECR and UHE neutrino experiments. For
estimation of the fluxes, we use the code CRPropa 3.2 assuming that UHECR
primary has a power-law distribution with an exponential cut-off. We have
analysed the dependence on various parameters such as the spectral index
($\alpha$), the cut-off energy ($E_{\rm cut}$) and the minimum distance to the
source ($d_{\rm min}$). In order to obtain the expected flux of the GZK
photons, it is important to consider the attenuation of GZK photons by the CMB
as well as ERB. We have analysed the impact of both the CMB and ERB (including
the impact of associated uncertainties) on the propagation of GZK photons.
Finally, we also discuss possible multi-messenger phenomenological
implications of GZK neutrinos (especially with IceCube-Gen2) on the flux of
GZK photons.

The predicted GZK photon flux is found to be below the sensitivities of the
telescopes, Auger and TA. However, in future, Auger 2025 and GRAND200k will be
able to probe the GZK photon flux parameter space. The sensitivity of
GRAND200k is expected to be one order smaller than our predicted upper limit
of the GZK photon flux and will be the best option for GZK photon detection.
In addition, we have found that the uncertainties at the high frequency part
of the ERB spectra can have significant impact on the low energy GZK photon
flux. This low energy flux detectable at the upcoming UHE photon detectors,
Auger SD 2025 and GRAND200k can provide us better understanding of the high
energy ERB spectra uncertainties (like, ARCADE2 radio excess). The IceCube-
Gen2 neutrino detector being sensitive in these low energies can also provide
multi-messenger constraints.

The predicted GZK neutrino flux is beyond the reach of the UHE neutrino
detectors like Auger, IceCube, ANITA, ARA37 and SKA low. The GZK neutrino flux
estimate is found to be in excellent agreement with the Waxman-Bahcall limit
[88]. The reach of proposed IceCube-Gen2 will be better and hence, it is
expected that IceCube-Gen2 would be able to detect GZK neutrinos. It should be
noted that uncertainties in source parameters have little impact on the GZK
neutrino flux at  energies between $(10^{18}-10^{19})$ eV  and can be used to
calibrate the GZK photon flux for any multi-messenger study.

Apart from discussing the flux prediction and sensitivity limits of the
existing detectors for the GZK photons and GZK neutrinos, we have also
addressed the impact of detection of any one of the component on the other
component utilizing the power of multi-messenger approach. Using IceCube-Gen2
neutrino sensitivity, we have computed the corresponding GZK photon flux
limit. This limit is consistent with our model predictions and yields the
following crucial conclusions. Any future detection of GZK neutrinos in
IceCube-Gen2 will imply a guaranteed GZK photon flux and lie above this limit.
However, non-detection of GZK neutrinos at IceCube-Gen2 will create a tension
between the observed UHECR flux and the present understanding of the GZK
process. Moreover, detection of the GZK secondaries could also indirectly lead
to an improved understanding of the ERB.

Note that our prediction of GZK neutrinos and GZK photon fluxes is based on
the assumption of pure proton composition of UHECR primary. However, the
presence of heavy elements in the UHECR primary might result in lower fluxes
depending on the fraction of different heavy elements. Though this is expected
to be a minor effect, in principle, the fluxes of GZK neutrinos and GZK
photons presented in this work are representing the upper limits, in regards
to the presence of heavier elements in the UHECR fluxes.

## Acknowledgments

P.S. & S.C. thank Lopamudra Mukherjee, Aparajitha Karthikeyan for the initial
discussion on the project. P.S. & S.C. also thank Irene Tamborra for valuable
comments on the paper. P.S. thanks Günter Sigl and Simone Rossoni for useful
discussion and hosting his stay at DESY, Hamburg. P.M. would like to thank
Ravi Subrahmanyan and Subir Sarkar for useful discussions. The use of HPC
cluster at SPS, JNU funded by DST-FIST is acknowledged. S.C. acknowledges the
support of the Max Planck India Mobility Grant from the Max-Planck Society,
supporting the visit and stay at MPP during the project. S.C has also received
funding from DST/SERB projects CRG/2021/002961 and MTR/2021/000540. P.M.
acknowledges the warm hospitality from ICTS, Bengaluru during the final stages
of writing the manuscript. The work of P.M. is partially supported by the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skodowska-Curie grant agreement No 690575 and 674896.

##  Appendix A Numerical estimates of the GZK fluxes

###  A.1 Primary UHECR spectrum:

The UHECR source spectrum described by Eq. 2.1 is governed by the parameters,
$\alpha$ and $E_{\rm cut}$. Here, we show the dependence of the UHECR source
spectrum on these parameters by varying in the range, $\alpha\in[2.0-2.7]$ and
$E_{\rm cut}\in[10^{20},10^{22}]$ eV. Fig. A.1 depicts the dependence of the
primary UHECR spectrum (in arbitrary units (a.u.)) of a point source on the
different model parameters. The left panel shows CR spectra for three values
of the spectral index, $\alpha=2.0,~{}2.4,~{}2.7$ with a fixed cut-off energy,
$E_{\rm cut}=10^{22}$ eV, while the right panel displays CR spectra for
different cut-off energies, $E_{\rm cut}=10^{20},~{}10^{21},~{}10^{22}$ ${\rm
eV}$ with $\alpha=2.0$. It is clear that smaller spectral index produces
harder spectra. As discussed above, the cut-off energy plays an important role
in determining the spectral shape at very high energies, evident in our model
spectra. The spectra in the right panel shows that the CR spectrum falls
rapidly for energies $E_{\rm p}>E_{\rm cut}$.

###  A.2 GZK flux: estimation

In order to demonstrate the GZK proton and secondary production, we compute
the fluxes of GZK protons, photons and neutrinos for a point source at 50 Mpc
using 1D simulation in CRPropa 3.2. The point source fluxes are plotted in
arbitrary units in the left panel of Fig. A.2. For this demonstration purpose,
we consider a typical example with $\alpha=2.0$ and $E_{\rm cut}=10^{22}$ eV (
Eq. 2.1) for the source spectrum. This spectrum is shown as dashed black line
in the left panel of Fig. A.2. The interaction of these protons with CMB
photons creates the resultant GZK protons (black solid curve). The sudden
attenuation of the GZK proton flux above $\sim 10^{20}$ eV is due to the GZK
effect. Among the GZK secondaries, the GZK photons will suffer attenuation due
to their interaction with the CMB and ERB. The secondary GZK photon flux
without these propagation losses is shown by the red dotted curve in the left
panel of Fig. A.2. To understand the propagation effect on the GZK photons, we
first study the attenuation on the CMB and then add the ERB effect. The red
dashed curve in the left panel of Fig. A.2 shows the sole effects of CMB on
the GZK photon flux. The relevant interactions for CMB are pair production and
Inverse Compton. The ERB also impacts the GZK photon flux and is dominated by
pair production loss on ERB at energies above $10^{19}$ eV [35]. The resultant
GZK photon flux including attenuation on ERB is shown by the red solid curve
in the left panel of Fig. A.2. We consider the PB96 ERB model for
demonstrating the attenuation of the GZK photon flux. On the other hand, the
GZK neutrino flux remains unaffected by these backgrounds and only undergo
flavour equalisation ($1:1:1$) during propagation [86]. The GZK neutrino flux
(single flavour) is shown by the blue curve in the left panel of Fig. A.2.

The diffuse flux of GZK protons, photons and neutrinos for the benchmark
parameter values as $E_{\rm cut}=10^{22}$ eV and $\alpha=2.0$ are shown in the
right panel of Fig. A.2. The source spectra appear as a horizontal flat line
(black dashed) as the spectral index is $\alpha=2.0$. These protons undergo
attenuation due to the GZK effect. The resultant GZK protons at Earth are
shown by the black curve. The GZK proton flux is normalized to the observed
cosmic ray data at energy  $4\times 10^{19}$ eV [7, 6, 8]. The magenta data
points show the Auger data [6], whereas the gray data points represent the
HiRes data [7]. The corresponding secondaries have also been scaled with this
normalization to obtain the GZK photons (red curve) and GZK neutrinos (blue
curve) at Earth.

The diffuse flux of GZK photons without any propagation effects is shown by
the dotted red curve in the right panel of Fig. A.2. The attenuation of the
GZK photon flux due to CMB is shown by the red dashed curve and the resulting
diffuse GZK photon flux after interaction with ERB is shown by the red solid
curve in the right panel of Fig. A.2. Clearly, the effect of ERB becomes
important above energies $10^{19}$ eV as in the point source case.

Figure A.1: UHECR proton spectra at the source for different values of power
law index, $\alpha$ and cut-off energy, $E_{\rm cut}$.

Figure A.2: Left: Fluxes of GZK protons (black solid), GZK photons (red) and
GZK neutrinos (blue, single flavour) for a point source at 50 Mpc. The primary
UHECR spectrum (black dashed) parameters are, $\alpha=2.0$, $E_{\rm
cut}=10^{22}$ eV. The red dotted and blue curves show the gamma-rays and
neutrinos (single flavour) without propagation effects respectively. The
gamma-rays after interaction with CMB is shown by the red dashed curves,
whereas the red solid curve shows the effects of both CMB and ERB. The
neutrinos do not suffer any losses. Right: Diffuse fluxes of GZK protons, GZK
photons and GZK neutrinos for similar cases as in left panel. The diffuse GZK
proton flux at Earth (black curve) is normalised to the observed cosmic ray
data by Auger (magenta data points) [6] and HiRes (gray data points) [7]. Both
the panels show that the effect of CMB on GZK photons is more pronounced at
lower energies, whereas the effect of ERB is crucial at higher energies.

##  Appendix B Dependence of the GZK fluxes on the various parameters

In Sec. 4, we discussed the significance of the various parameters such as
$\alpha$, the UHECR composition, $E_{\rm cut}$ and $d_{\rm min}$. In the
present Appendix, we provide a numerical estimate of the impact of these
parameters on the GZK fluxes.

###  B.1 Dependence on the spectral index ($\alpha$)

To explore the spectral index dependence of the GZK fluxes, we vary $\alpha$
in the range $[2.0,2.7]$ for the primary proton spectrum. Fig. B.1 shows the
fluxes of GZK protons (left panel), GZK photons (red curves in right panel)
and GZK neutrinos (blue curves in right panel) for three different values of
$\alpha$. The dotted curves correspond to $\alpha=2.7$, the solid curves
represent $\alpha=2.0$ and the dashed curves show the fluxes for $\alpha=2.4$.
The other parameters are kept fixed, $E_{\rm cut}=10^{22}$ eV and $d_{\rm
min}=0.1$ Mpc. We also depict the observed CR data by HiRes and Auger [7, 6]
for comparison. Clearly, smaller spectral index produces harder spectra. For
$\alpha=2.4$ and $\alpha=2.7$, the GZK proton spectrum shows minimum deviation
from the HiRes and Auger below $10^{20}$ eV data [96, 62, 97, 98]. However,
$\alpha=2.0$ scenario shows larger deviation at these energies. This shows
that different values of $\alpha$ are possible for the primary UHECR proton
spectrum. The variation of $\alpha$ also results in significantly different
secondary fluxes. The dependence of GZK photon and GZK neutrino flux on
$\alpha$ is shown in the right panel of Fig. B.1. As expected, the fluxes of
GZK neutrinos are much larger than GZK photons. For both the scenarios, the
difference due to the variation of $\alpha$ is more prominent at higher
energies.

Figure B.1: Diffuse fluxes of GZK protons, GZK photons and GZK neutrinos
(single flavour) for different power law indices, i.e., $\alpha=2.0$ (solid),
$\alpha=2.4$ (dashed) and $\alpha=2.7$ (dotted). The remaining parameters are
kept fixed as $E_{\rm cut}=10^{22}$ eV and $d_{\rm min}=0.1$ Mpc. The left
panel shows the GZK protons fluxes for different $\alpha$ normalised to Auger
and HiRes data. The corresponding normalised GZK photon flux (red curves) and
the GZK neutrino flux (blue curves) are shown in the right panel. These plots
show that smaller $\alpha$ produces harder fluxes at high energies. Figure
B.2: Effect of heavy elements in the primay UHECR spectrum on the GZK protons,
GZK photons and GZK neutrinos (single flavour). The left panel shows the UHECR
flux at Earth for $100\%$ proton (black continuous) and $100\%$ Iron (black
dashed). The corresponding GZK neutrinos (blue) and GZK photons (red) for each
case are shown in the right panel.

###  B.2 Dependence on the composition of UHECR

The Auger data [64, 65, 62, 66, 63, 67, 6] suggests the presence of heavy
elements in the UHECR spectrum i.e., the composition effect. Here, we discuss
the composition effect on the GZK secondaries. [36, 68, 69]. The presence of
heavy elements along with protons would lead to lowering of the fluxes of GZK
neutrinos and GZK photons [70, 71, 72]. In order to illustrate this effect, we
consider a hypothetical scenario where the UHECR is composed of $100~{}\%$
iron nuclei. Fig. B.2 depicts the GZK proton flux, GZK photon flux and GZK
neutrino flux at Earth after incorporating the effect of heavy nuclei primary.
In the left panel of Fig. B.2, we show a comparison of the two constituents
(protons and iron). The pure proton UHECR spectra is shown as black solid line
and pure iron UHECR spectra is shown as black dashed line in the left panel.
In the right panel, we show the secondary fluxes of GZK neutrinos (blue solid
line) and GZK photons (red solid line). We note that GZK neutrino and GZK
photon fluxes are an order of magnitude smaller than the corresponding GZK
fluxes for the pure proton case and would impact their detection prospects.

Figure B.3: Diffuse fluxes of GZK protons (left panel), photons and single
flavour neutrinos (right panel) for different cut-off energies, i.e., $E_{\rm
cut}=10^{20}$ eV (dashed), $E_{\rm cut}=10^{21}$ eV (solid) and $E_{\rm
cut}=10^{22}$ eV (dotted). The remaining parameters are kept fixed as
$\alpha=2.0$ eV and $d_{\rm min}=0.1$ Mpc.

###  B.3 Dependence on the cut-off energy ($E_{\rm cut}$)

The spectrum of primary UHECRs is expected to fall-off rapidly at higher
energies due to the maximum energy of the CRs [35]. This effect is implemented
to the UHECR spectrum via the cut-off energy, $E_{\rm cut}$ as defined in Eq.
2.1. To demonstrate the effect of $E_{\rm cut}$ on the GZK fluxes, we vary the
$E_{\rm cut}$ in the range $[10^{20},10^{22}]$ eV. The left panel of Fig. B.3
shows the GZK proton fluxes for three values of $E_{\rm cut}$ i.e., $10^{20}$
eV (dashed), $10^{10^{21}}$ eV (solid) and $10^{22}$ eV (dotted). The other
parameters are chosen as $\alpha=2.0$ and $d_{\rm min}=0.1$ Mpc for all
$E_{\rm cut}$ values. The fluxes of GZK photons and GZK neutrinos
corresponding to these $E_{\rm cut}$ are shown in the right panel of Fig. B.3.
The variation of $E_{\rm cut}$ produces large differences between the fluxes
of different $E_{\rm cut}$, especially at higher energies. When considering
lower values of the cut-off energy ($E_{\rm cut}$), such as $E_{\rm cut}=$
$10^{20}$ eV, both GZK photons and GZK neutrinos fluxes decrease rapidly at
energies less than $10^{20}$ eV. The higher $E_{\rm cut}$ values result in
harder spectra at higher energies.

Figure B.4: Diffuse fluxes of GZK protons (left), photons and single flavour
neutrinos (right) for different minimum distances, i.e., $d_{\rm min}=0.1$ Mpc
(solid), $d_{\rm min}=50$ Mpc (dashed) and $d_{\rm min}=100$ Mpc (dotted). The
remaining parameters are kept fixed as $\alpha=2.0$ eV and$E_{\rm
cut}=10^{22}$ eV.

###  B.4 Dependence on minimum distance to source ($d_{\rm min}$)

The explore the minimum distance to the source ($d_{\rm min}$) we vary the
parameter in between $0.1$ Mpc to $100$ Mpc (see Sec. 4). The left panel of
Fig. B.4 shows GZK proton fluxes for three different $d_{\rm min}$ i.e., $0.1$
Mpc (solid curve), $50$ Mpc (dashed curve) and $100$ Mpc (dotted curve). The
flux corresponding to $d_{\rm min}=0.1$ Mpc has the maximum flux, i.e.,
minimum loss above the GZK cut-off due to the close proximity of sources. On
the other hand, the flux for $d_{\rm min}=100$ Mpc has the minimum flux
estimate as the propagation loss is larger for the far away sources. Thus, we
see that the GZK suppression is more pronounced (less pronounced) for larger
values (smaller values) of $d_{\rm min}$, and the smaller $d_{\rm min}$ have
lower attenuation.

The effect of $d_{\rm min}$ on the diffuse flux of GZK photons (red) and GZK
neutrinos (blue) is shown in the right panel of Fig. B.4 for the three
different values of $d_{\rm min}$, i.e., $0.1$ Mpc (solid), $50$ Mpc (dashed)
and $100$ Mpc (dotted). We note from Fig. B.4 (right panel) that the variation
of $d_{\rm min}$ does not play any role in case of diffuse flux of GZK
neutrinos. This is because neutrinos are weakly interacting in nature.
However, $d_{\rm min}$ value does impact the GZK photon flux. As $d_{\rm min}$
increases, so do the losses to the GZK photon flux. It’s worth noting that
this reduction in flux becomes more pronounced at higher energies. This occurs
as the losses on ERB are the largest at higher energies, resulting in more
pronounced attenuation for larger values of $d_{\rm min}$.

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