# Probing the origin of ultra-high-energy cosmic rays with neutrinos in the
EeV energy range using the Pierre Auger Observatory

The Pierre Auger Collaboration

###### Abstract

Neutrinos with energies above $10^{17}$ eV are detectable with the Surface
Detector Array of the Pierre Auger Observatory. The identification is
efficiently performed for neutrinos of all flavors interacting in the
atmosphere at large zenith angles, as well as for Earth-skimming $\tau$
neutrinos with nearly tangential trajectories relative to the Earth. No
neutrino candidates were found in $\sim\,14.7$ years of data taken up to 31
August 2018. This leads to restrictive upper bounds on their flux. The $90\%$
C.L. single-flavor limit to the diffuse flux of ultra-high-energy neutrinos
with an $E_{\nu}^{-2}$ spectrum in the energy range $1.0\times
10^{17}\leavevmode\nobreak\ {\rm eV}-2.5\times 10^{19}\leavevmode\nobreak\
{\rm eV}$ is $E^{2}{\rm d}N_{\nu}/{\rm d}E_{\nu}<4.4\times
10^{-9}\leavevmode\nobreak\ {\rm GeV\leavevmode\nobreak\
cm^{-2}\leavevmode\nobreak\ s^{-1}\leavevmode\nobreak\ sr^{-1}}$, placing
strong constraints on several models of neutrino production at EeV energies
and on the properties of the sources of ultra-high-energy cosmic rays.

##  1 Introduction

The origin, nature and production mechanisms of ultra-high-energy cosmic rays
(UHECRs) with energies above $10^{18}$ eV are some of the long-standing
questions in astroparticle physics [1, 2, 3]. The discovery of a large-scale
dipolar anisotropy in the arrival directions of UHECRs at energies $E>8\times
10^{18}$ eV indicates that above this threshold they have an extragalactic
origin [4]. The amplitude of the dipole increases steadily with energy above
$4\times 10^{18}$ eV [5]. Indications of anisotropy of UHECRs on intermediate
angular scales were also reported recently [6]. The pattern of UHECR arrival
directions for energies above $4\times 10^{19}$ eV is matched by a model in
which about $10\,\%$ of them arrive from directions that are clustered around
bright, nearby galaxies. An isotropic arrival direction distribution of UHECRs
was disfavored at 4.0 and 2.7 $\sigma$ by comparing the pattern of arrival
directions with flux models based on the local distribution of starburst
galaxies and $\gamma$-AGN, respectively.

Measurements of the UHECR spectrum have revealed a strong suppression of the
flux at energies above $\sim\,4\times 10^{19}$ eV with respect to that
extrapolated from lower energies [7, 8]. The position of the break is
compatible with that expected from the Greisen-Zatsepin-Kuzmin (GZK) effect
[9], namely the attenuation of the flux of UHECRs due to interactions with the
Cosmic-Microwave Background (CMB) radiation. While the position of the break
is compatible with both proton and heavier primaries, data collected with the
fluorescence technique at the Pierre Auger Observatory indicate that the
composition is getting heavier than protons as the energy increases beyond 2
EeV [10]. Models in which medium to heavy nuclei are photodisintegrated while
traveling through the Universe [11] cannot be excluded. It is also possible
that the suppression is due to a combination of the maximum energy to which
sources can accelerate particles and the interactions of the primaries with
the background fields [12]. The determination of the composition of the UHECRs
[10, 13] is key to distinguishing between these scenarios since they predict
different fractions of primaries heavier than protons as energy increases
[12].

All models of UHECR production predict UHE neutrinos as a result of the decay
of charged pions generated in interactions of UHECRs within the sources
themselves (“astrophysical” neutrinos), and/or in their propagation through
background radiation fields (“cosmogenic” neutrinos) [14]. However, the
predicted fluxes have large uncertainties, depending strongly on the spatial
distribution and redshift ($z$) evolution of the unknown UHECR sources, on the
transport model of UHECR assumed, as well as on the spectral features of
UHECRs at production [15, 16, 17]. Moreover, if UHECRs are heavy nuclei, the
UHE neutrino yield is expected to be strongly suppressed [18, 19]. In this
respect the (lack of) observation of UHE neutrinos would provide constraints
on the dominant scenario of UHECR production [20] as well as on the evolution
with $z$ of their sources, both of which can help in their identification [21,
15]. The fact that neutrinos travel unaffected by magnetic fields and
unattenuated implies that EeV neutrinos may be the only direct probe of the
sources of UHECRs at distances farther than $\sim\,100$ Mpc.

Astrophysical neutrinos have been observed by the IceCube experiment. Nearly a
hundred neutrino events in the energy range $\sim\,100$ TeV and a few PeV have
been detected representing a $5.6\leavevmode\nobreak\ \sigma$ excess with
respect to atmospheric backgrounds [22]. These include four neutrinos of
energies above $10^{15}$ eV [23, 22]. Most remarkable is the recent detection
of a $\sim\,3\times 10^{14}$ eV neutrino event (IceCube-170922A) in spatial
coincidence with the blazar TXS 0506+056 [24] and in temporal coincidence with
a flare of gamma rays detected by the Fermi satellite from the same source.
Also, an excess of neutrino events from the position of the blazar was found
in the TeV energy range prior to the IceCube event [25]. Unfortunately, and
despite the variety of techniques being used [26, 27], neutrinos have escaped
detection by existing experiments in the EeV energy range [28, 29, 30, 31],
i.e. about three orders of magnitude above the most energetic neutrinos
detected in IceCube.

In this work, we report on the search for UHE neutrinos in data taken with the
Surface Detector Array (SD) of the Pierre Auger Observatory [32]. A blind scan
of data from 1 January 2004 up to 31 August 2018 has yielded no neutrino
candidates. This corresponds to $\sim\,9.7$ equivalent years of operation of a
complete SD, or $\sim\,14.7$ years of lifetime (because the array was not
fully deployed until 2008) representing an increase of $5.2$ years of
lifetime, operated with a complete SD, over previous searches [28]. The non-
observation of neutrino candidates allows us to place stringent constraints on
the diffuse flux of UHE neutrinos with relevant implications for the origin of
the UHECRs. In this paper, we report upper limits to the diffuse flux of UHE
neutrinos. Corresponding limits to the neutrino flux from point-like sources
as a function of declination are the subject of a separate paper [33].

##  2 Searching for UHE neutrinos in Auger data

The Pierre Auger Observatory [32] is located in the province of Mendoza,
Argentina, at a mean altitude of 1400 m above sea level
($\sim\,880\leavevmode\nobreak\ {\rm g\leavevmode\nobreak\ cm^{-2}}$ of
vertical atmospheric column density). It was primarily designed to measure
extensive air showers (EAS) induced by UHECRs, and has been running and taking
data since its construction started in 2004. For that purpose a surface
detector (SD) [34, 32] samples the front of shower particles at the ground
level with an array of water-Cherenkov detectors (“stations”). Each SD station
contains 12 tonnes of water viewed by three nine-inch photomultiplier tubes.
The signals produced by the passage of shower particles through the SD
stations are recorded as time traces in 25 ns intervals. 1660 SD stations have
been deployed over an area of $\sim\,3000\leavevmode\nobreak\ {\rm km^{2}}$
arranged in a hexagonal pattern with 1.5 km spacing.

Although the primary goal of the SD of Auger is to detect UHECRs, it can also
identify ultra-high-energy (UHE) neutrinos. Neutrinos of all flavors can
interact in the atmosphere through charged (CC) or neutral current (NC)
interactions and induce a “downward-going” (DG) shower that can be detected
[35]. The probability of interacting per unit column density traversed is
essentially independent of the atmospheric depth. In addition, $\tau$
neutrinos ($\nu_{\tau}$) can undergo CC interactions and produce a $\tau$
lepton in the earth crust. The $\tau$ lepton leaves the earth and decays in
the atmosphere, inducing an “Earth-skimming” (ES) upward-going shower [36].
Tau neutrinos are not expected to be copiously produced at the astrophysical
sources, but as a result of neutrino oscillations over cosmological distances,
approximately equal fluxes for each neutrino flavour should reach the earth
[37, 38, 39].

Neutrino-induced showers must be identified in Auger data in the background of
showers initiated by UHECRs (protons and/or nuclei). The identification is
based on a simple idea: highly-inclined (zenith angle $\theta>60^{\circ}$), ES
and DG neutrino-induced showers initiated deep in the atmosphere near the
ground level have a significant electromagnetic component when they reach the
SD array, producing signals that are spread over time in several of the
triggered SD stations. In contrast, inclined showers initiated by standard
UHECRs are dominated by muons at the ground level, inducing signals in the SD
stations that have characteristic large peaks associated with individual muons
which are spread over smaller time intervals. Thanks to the fast sampling (25
ns) of the SD digital electronics, several observables that are sensitive to
the time structure of the signal can be used to discriminate between these two
types of showers. Auger data are searched for UHE neutrino candidates in both
the DG and ES analyses, in the zenith angle ranges
$60^{\circ}<\theta<90^{\circ}$ and $90^{\circ}<\theta<95^{\circ}$
respectively.

###  2.1 General search strategy

The search strategy consists in selecting showers that arrive at the SD array
in the inclined directions and identifying those that exhibit a broad time
structure in the signals induced in the SD stations. Such signals are
indicative of an early stage of development of the shower, a signature of the
shower developing close to the ground. To define the selection algorithms and
optimize the numerical values of the cuts needed to identify neutrino-induced
showers we follow a blind analysis procedure. A fraction of $\sim\,15\,\%$ of
the whole data sample (from 1 January 2004 up to 31 August 2018), along with
Monte Carlo simulations of UHE neutrinos, is dedicated to define the selection
algorithms, the most efficient observables for neutrino identification, and
the value of the cuts. This “training” data set is assumed to be constituted
of background UHECR-induced showers. The remaining fraction of data (“search
data”) is “unblinded” to search for neutrino candidates but only after the
selection procedure is established.

The selection of inclined showers is tailored to the different zenith angle
ranges where the search is performed, namely DG and ES. Since standard angular
reconstruction techniques have larger uncertainties for nearly horizontal
events [40, 41], another strategy to select inclined events has been followed.
For purely geometrical reasons, in inclined events the pattern of the
triggered SD stations typically exhibits an elliptical shape on the ground
with the major axis of the ellipse along the azimuthal arrival direction.
These patterns can be characterized by a length $L$ (major axis) and a width
$W$ (minor axis). Inclined events typically exhibit large values of $L/W$, and
an appropriate cut in $L/W$ is therefore an efficient selector of inclined
events [42, 43, 44]. Another indication of the arrival direction of the event
is given by the average (apparent) speed $\langle V\rangle$ of the trigger
time from one station to another. This is calculated from the projected
distance between pairs of stations along the major axis of the ellipse and the
trigger times, and it is averaged over all pairs of stations in the event. In
vertical showers $\langle V\rangle$ exceeds the speed of light since all
triggers occur at roughly the same time, while in very inclined events
$\langle V\rangle$ is close to the speed of light. In addition, the Root-Mean-
Square (RMS($V$)) of the apparent speed (as obtained from the values of $V$
using different pairs of stations) is typically below $\sim\,25\%$ of $\langle
V\rangle$ [42, 43, 44].

For the purpose of identifying those inclined events that interact deep in the
atmosphere, several observables that contain information on the spread in time
in the SD stations can be extracted from the time traces. Among them the Area-
over-Peak111The Area-over-Peak is defined as the ratio of the integral of the
time trace to its peak value normalized to the average signal produced by a
single vertical muon. (AoP) has been shown to serve as an efficient observable
to discriminate broad from narrow shower fronts [28]. Inclined background
showers of hadronic origin, in which the muons arrive at the SD stations in a
short time interval, exhibit AoP values close to one by definition
corresponding to the average AoP of single vertical muons used for
calibration. In contrast in neutrino-induced showers the values of AoP are
typically larger. This can be seen in Fig. 1 where we show examples of traces
of stations belonging to a vertical and an inclined event detected with the SD
of Auger, as well as the trace of a station in a neutrino-induced simulated
event. The optimization of the algorithms based on AoP and related observables
to identify deeply-initiated showers is done separately in each angular range
as described briefly in the following and in more detail in [28].

Figure 1:  FADC traces of stations at a distance of approximately 1 km to the
shower core. From top to bottom panel the station belongs to a vertical event,
to an inclined event, and to a neutrino-simulated event. The reconstructed
energy ($E$) and zenith angle ($\theta$) for the events, as well as the
simulated $E$ and $\theta$ of the neutrino-induced shower, are indicated in
each panel. The value of the Area-over-Peak (AoP) of each trace is also given.

###  2.2 Earth-skimming (ES) neutrinos

The algorithms to reconstruct the zenith angle of arrival of the primary
particles in Auger [40] have been designed for showers that have an impact
point within the array. However, a $\nu_{\tau}$-induced ES shower travels
nearly parallel to the earth’s surface, in a slightly upward direction, and
its core does not intersect the ground. The shower particles deviate laterally
from the axis and can reach the ground triggering the SD stations. For this
reason, the selection of inclined ES showers is just based on the properties
of the signal pattern obtained with the triggered SD stations and the apparent
propagation speed of the signal at ground level. Above the energy threshold of
the SD (around $10^{17}$ eV), $\tau$ leptons are efficiently produced only at
zenith angles between $\theta=90^{\circ}$ and $\theta=95^{\circ}$ [45, 46].
Moreover, from Monte Carlo simulations of $\tau$-lepton-induced showers we
have determined that the trigger efficiency of the SD decreases rapidly above
$\theta\sim 95^{\circ}$. For these reasons, we place restrictive cuts to
select quasi-horizontal showers with highly elongated signal patterns, namely,
$L/W>5$ and $\langle V\rangle\in[0.29,0.31]\leavevmode\nobreak\ {\rm
m\,ns^{-1}}$ with RMS($V$)$\,<0.08\leavevmode\nobreak\ {\rm m\,ns^{-1}}$ (see
also Table I in [28]). In the ES analysis the search is performed in all
events with at least 3 triggered stations $N_{\rm stat}\geq 3$.

For data prior to 31 May 2010, the neutrino identification variables included
the fraction of stations with Time-over-Threshold (ToT) [47] trigger222This is
a type of trigger designed to select spread-in-time sequences of small signals
in the time traces [47]. and having AoP$\,>1.4$ [42, 43]. This fraction is
required to be above $60\,\%$ of the triggered stations in the event. For data
beyond 1 June 2010, an improved selection is adopted using the average value
of AoP ($\langle{\rm AoP}\rangle$) over all the triggered stations in the
event as the only observable to discriminate between hadronic showers and ES
neutrinos. The value of the cut on $\langle$AoP$\rangle$ is fixed using the
tail of the distribution of $\langle$AoP$\rangle$ in real data, which is
consistent with an exponential function. This tail is fitted and extrapolated
to find the value of $\langle$AoP$\rangle$ corresponding to less than 1
expected background event per 50 yr on the full SD array (see [28] for full
details).

Applying these criteria, a search for ES neutrino-induced showers is performed
in the Observatory data from 1 January 2004, when data taking started, up to
31 August 2018. No neutrino candidates are identified. In Fig. 2 we show the
distribution of $\langle$AoP$\rangle$ for the whole data period compared to
that expected in Monte Carlo simulations of $\nu_{\tau}$-induced ES showers,
along with the optimized value of the cut ($\langle$AoP$\rangle=1.83$) above
which an event would be regarded a neutrino candidate. After the inclined
selection and the neutrino identification criteria, $\sim\,95\%$ of the
simulated neutrinos that induce triggers are kept. This proves that the Pierre
Auger Observatory is highly efficient as a neutrino detector, with its
sensitivity mostly governed by its lifetime and the available target matter
for neutrino interactions along the earth’s chord. The neutrino search is not
limited by the background due to UHECR-induced showers since this can be very
efficiently reduced as shown in Fig. 2.

Figure 2:  Distribution of $\langle{\rm AoP}\rangle$ after the Earth-skimming
inclined selection. Black histogram: full data set up to 31 August 2018
containing 25904 events. Red-shaded histogram: Monte Carlo simulated ES
$\nu_{\tau}$ events.

###  2.3 Downward-going (DG) neutrinos

For optimization purposes, the DG category of events is further subdivided
into two sets for Low (DGL) and High (DGH) zenith angles, between
$60^{\circ}<\theta<75^{\circ}$ [28, 48] and $75^{\circ}<\theta<90^{\circ}$
[44] respectively.

Since the core of DG showers always hits the ground, standard angular
reconstruction techniques [40] can be used to obtain an estimate of the zenith
angle of the shower. However, these techniques have larger uncertainties for
nearly horizontal events [40, 41]. For this reason the primary observables for
inclined selection in the DGH case are the ratio $L/W$ of the signal pattern
of the shower at ground as well as the apparent average velocity of the signal
$\langle V\rangle$, in addition to a simple estimate of the zenith angle
$\theta_{\rm rec}$ [28, 41]. In the case of DGH showers the cuts on the
properties of the signal pattern are $L/W>3$, $\langle
V\rangle<0.313\leavevmode\nobreak\ {\rm m\,ns^{-1}}$ and RMS($V$)/$\langle
V\rangle<0.08$, along with a further requirement on the estimated shower
zenith angle $\theta_{\rm rec}>75^{\circ}$ (see Table I in [28]). In contrast,
in the DGL case, corresponding to $60^{\circ}<\theta<75^{\circ}$, restrictions
on the signal patterns have been found to be less efficient in selecting
inclined events than $\theta_{\rm rec}$ [48], and only an angular cut
$58.5^{\circ}<\theta_{\rm rec}\leq 76.5^{\circ}$ is applied, including some
allowance to account for the resolution in the angular reconstruction of the
simulated neutrino events [48]. In both the DGH and DGL cases, at least 4
stations $(N_{\rm stat}\geq 4)$ are required in the event.

For DG showers, we use a multivariate analysis (Fisher method [49]) combining
several observables that carry information on the time spread of the signals
in the SD stations. These observables are constructed from the AoP values of
individual stations. This analysis is optimized with Monte Carlo simulations
of UHE neutrinos and the training sample of real data. Based on the
information obtained from simulations the selection is further improved when
dividing the DGH category of events into three sets depending on the number of
triggered stations $N_{\rm stat}\leq 6$, $7\leq N_{\rm stat}\leq 11$ and
$N_{\rm stat}\geq 12$. Correspondingly for the DGL category the division is in
five sets depending on the reconstructed zenith angle
$58.5^{\circ}<\theta_{\rm rec}\leq 61.5^{\circ},\leavevmode\nobreak\
61.5^{\circ}<\theta_{\rm rec}\leq 64.5^{\circ},\leavevmode\nobreak\
64.5^{\circ}<\theta_{\rm rec}\leq 67.5^{\circ},\leavevmode\nobreak\
67.5^{\circ}<\theta_{\rm rec}\leq 70.5^{\circ}$, and $70.5^{\circ}<\theta_{\rm
rec}\leq 76.5^{\circ}$. A different Fisher discrimination variable is
constructed in each subcategory and the cut value independently optimized.

In the DGH channel, the linear Fisher discriminants are constructed with ten
variables that exploit the fact that, due to the large inclination of the
shower, the electromagnetic component is less attenuated in the stations that
are first hit by a deep inclined shower than in those that are hit last [44,
41]. Using Monte Carlo simulations of $\nu-$induced showers with
$\theta>75^{\circ}$ we have found that in the first few stations hit in the
event the AoP values range between 3 and 5, while AoP is typically $\sim\,1$
in stations triggered later. Based on this we have found good discrimination
when constructing the Fisher discriminant with the AoP and (AoP)2 of the four
stations that trigger first in each event, the product of the four AoPs, and a
global parameter of the event that is sensitive to the asymmetry between the
average AoP of the early stations and those triggering last in the event (see
[28] for further details).

The selection of neutrino candidates in the zenith angle range DGL category is
more challenging because the electromagnetic component of UHECR-induced
showers at the ground increases as the zenith angle decreases, imposing a
larger background to the searches for neutrinos. Based on the information from
Monte Carlo simulations in this angular range, it has been determined that out
of all triggered stations of an event, the ones closest to the shower core
exhibit the highest discrimination power in terms of AoP. In fact, it has been
found that the first triggered stations can still contain some electromagnetic
component for background events and, for this reason, they are not used for
discrimination purposes. The last ones, even if they are triggered only by
muons from a background cosmic-ray shower, can exhibit large values of AoP
because they are far from the core where muons are known to arrive with a
larger spread in time. The variables used in the Fisher discriminant analysis
in the DGL channel are the individual AoP of the four (five) stations closest
to the core for events with $\theta_{\rm rec}\leq 67.5^{\circ}$ ($\theta_{\rm
rec}>67.5^{\circ}$) and their product [28, 48]. Finally, in the DGL analysis
it is also required that at least $75\,\%$ of the triggered stations closest
to the core have a ToT local trigger [28, 48].

Once the Fisher discriminant $\cal F$ is defined, an optimized value for the
cut is selected to efficiently separate neutrino candidates from regular
hadronic showers for both the DGH and DGL channels. The value is chosen
performing an exponential fit to the Fisher distribution of the training data,
extrapolating it, and requiring less than 1 event per 50 yr on the full SD
array for each multiplicity sub-sample in the DGH channel, and 1 event per 20
yr in the DGL channel [28, 48].

Applying these criteria, a search for DG neutrino-induced showers is performed
in the Observatory data since 1 January 2004 up to 31 August 2018. No neutrino
candidates are identified in any of the three subcategories of DGH.

In Fig. 3, we show the distribution of the Fisher variable in the DGH
subsample with $N_{\rm stat}\in[7,\leavevmode\nobreak\ 11]$ for the whole data
period compared to that expected in Monte Carlo simulations of $\nu$-induced
DGH showers, along with the optimized value of the cut above which an event
would be regarded a neutrino candidate. Of all the simulated inclined $\nu$
events that trigger the Observatory, a fraction between $\sim\,81\,\%$ and
$\sim 85\,\%$ (depending on $N_{\rm stat}$) are kept after the cuts on the
Fisher variable in the DGH analysis.

Figure 3:  Distribution of the Fisher variable after the downward (DGH)
inclined event selection for events with number of triggered stations between
7 and 11. Black histogram: full data set up to 31 August 2018 containing 33885
events. Blue-shaded histogram: Monte Carlo simulated $\nu$ DGH events.

No candidates are found in any of the five DGL subcategories during the same
period. As an example, we show in Fig. 4 the distribution of the Fisher
variable in the DGL subset with $\theta_{\rm
rec}\in(64.5^{\circ},\leavevmode\nobreak\ 67.5^{\circ}]$ and that obtained in
Monte Carlo simulations of $\nu$-induced DGL showers in the same zenith angle
range. In this case, a fraction between $\sim\,51\,\%$ and $\sim\,77\,\%$ of
the simulated inclined $\nu$ events that trigger are kept after applying the
cuts on the Fisher variable, with the fraction depending on the $\theta_{\rm
rec}$ range considered, the lowest fraction achieved at the smallest values of
$\theta_{\rm rec}$. The lower identification efficiency on average in the DGL
selection when compared to that in DGH, is due to the more stringent criteria
in the angular bin $\theta\in(58.5^{\circ},76.5^{\circ}]$ needed to reject the
larger contamination from cosmic-ray induced showers.

Figure 4:  Distribution of the Fisher variable after the downward (DGL)
inclined event selection for events with reconstructed zenith angle
$64.5^{\circ}<\theta\leq{}67.5^{\circ}$. Black histogram: full data set up to
31 August 2018 containing 3948 events. Green-shaded histogram: Monte Carlo
simulated $\nu$ DGL events.

##  3 Exposure

The non-observation of neutrino candidates can be converted into an upper
limit to the diffuse flux of UHE neutrinos. For this purpose, the exposure of
the SD of Auger needs to be calculated for the period of data taking. This is
done with Monte Carlo simulations of neutrino-induced showers. The same
selection and identification criteria applied to the data were also applied to
the results of these simulations. The identification efficiencies for each
channel were obtained as the fraction of simulated events that trigger the
Observatory and pass the selection procedure and identification cuts [44]. The
results presented here can be obtained after solid angle integration of the
directional exposure for point-like sources addressed in [33].

Figure 5:  Exposure of the SD of the Pierre Auger Observatory (1 January 2004
- 31 August 2018) to UHE neutrinos as a function of neutrino energy for each
neutrino flavor and for the sum of all flavors assuming a flavor mixture of
$\nu_{e}:\nu_{\mu}:\nu_{\tau}=1:1:1$. Also shown are the exposures to upward-
going Earth-skimming $\nu_{\tau}$ only and to the Downward-Going neutrinos of
all flavors including CC and NC interactions.

For downward-going neutrinos, the detection efficiency, $\varepsilon_{i,c}$,
depends on neutrino flavor $i=\nu_{e},\nu_{\mu},\nu_{\tau}$, the type of
interaction ($c=$CC, NC), neutrino energy $E_{\nu}$, zenith $\theta$ and
azimuth $\varphi$ angles, the point of impact of the shower core on the
ground, and the depth in the atmosphere $X$ measured along the shower axis at
which the neutrino is forced to interact in the simulations [28, 44]. The
detection efficiency also has some dependence on time because the SD array
grew steadily from 2004 up to 2008 when it was completed, and because the
fraction of working stations - typically above $95\%$ \- is changing
continuously with time. For downward-going neutrinos the calculation of the
exposure involves folding the detection efficiencies with the area of the SD
projected onto the direction perpendicular to the arrival direction of the
neutrino, and with the $\nu$ interaction probability for a neutrino energy
$E_{\nu}$ at a depth $X$ which also depends on the type of interaction (CC or
NC). Integrating $\varepsilon$ over the parameter space
($\theta,\leavevmode\nobreak\ \varphi,\leavevmode\nobreak\ X$), detector area,
and time over the search period, yields the exposure for flavor $i$ and
channel $c$ [28, 44],

${\cal E}_{i,c,\rm DG}(E_{\nu})=\leavevmode\nobreak\
\int_{A}\leavevmode\nobreak\ \int_{\theta}\leavevmode\nobreak\
\int_{\varphi}\leavevmode\nobreak\ \int_{X}\leavevmode\nobreak\
\int_{t}\leavevmode\nobreak\ \cos\theta\sin\theta\leavevmode\nobreak\
\varepsilon_{i,c}\leavevmode\nobreak\ \sigma^{c}_{\nu}\leavevmode\nobreak\
m_{\rm p}^{-1}\leavevmode\nobreak\ {\rm d}A\leavevmode\nobreak\ {\rm
d}\theta\leavevmode\nobreak\ {\rm d}\varphi\leavevmode\nobreak\ {\rm
d}X\leavevmode\nobreak\ {\rm d}t.$ (3.1)

The term $\sigma^{c}_{\nu}\leavevmode\nobreak\ m_{p}^{-1}\leavevmode\nobreak\
{\rm d}X$ , with $m_{p}$ the mass of a proton, and $\sigma^{c}_{\nu}$ the
neutrino-nucleon cross-section [50], represents the probability of neutrino-
nucleon interactions along a depth ${\rm d}X$ (in g cm-2). An effective
exposure can be obtained by summing over channels (CC and NC) and neutrino
flavors, weighting each with the relative flavor ratios (for instance weight 1
for each flavor corresponding to a 1:1:1 flavor ratio at earth).

In the $\nu_{\tau}$ Earth-skimming channel, the calculation of the exposure is
more involved. The efficiency $\varepsilon_{\rm ES}$ depends on the energy of
the emerging $\tau$ leptons $E_{\tau}$, the position of the signal pattern on
the ground, on the zenith $\theta$ and azimuth $\varphi$ angles, and on the
altitude $h_{\rm dec}$ of the decay point of the $\tau$ above ground [43].
$\varepsilon_{\rm ES}$ is averaged over the decay channels of the $\tau$ with
their corresponding branching ratios, that determine the amount of energy fed
into the shower. The detection efficiency $\varepsilon_{\rm ES}$ is folded
with the projected area of the SD, with the differential probability $p_{\rm
exit}(E_{\nu},\theta,E_{\tau})={\rm d}p_{\rm exit}/{\rm d}E_{\tau}$ of a
$\tau$ emerging from the earth with energy $E_{\tau}$ (given a neutrino with
energy $E_{\nu}$ crossing an amount of earth determined by the zenith angle
$\theta$), as well as with the differential probability $p_{\rm
dec}(E_{\tau},\theta,h_{\rm dec})={\rm d}p_{\rm dec}/{\rm d}h_{\rm dec}$ that
the $\tau$ decays at an altitude $h_{\rm dec}$ [43]. An integration over the
whole parameter space ($E_{\tau},\leavevmode\nobreak\
\theta,\leavevmode\nobreak\ \varphi,\leavevmode\nobreak\ h_{\rm dec}$), area,
and time yields the exposure [28, 43]:

${\cal E}_{\rm ES}(E_{\nu})=\int_{A}\leavevmode\nobreak\
\int_{\theta}\leavevmode\nobreak\ \int_{\varphi}\leavevmode\nobreak\
\int_{E_{\tau}}\leavevmode\nobreak\ \int_{h_{\rm dec}}\leavevmode\nobreak\
\int_{t}\leavevmode\nobreak\ |\cos\theta|\leavevmode\nobreak\
\sin\theta\leavevmode\nobreak\ p_{\rm exit}\leavevmode\nobreak\ p_{\rm
dec}\leavevmode\nobreak\ \varepsilon_{\rm ES}\leavevmode\nobreak\ {\rm
d}A\leavevmode\nobreak\ {\rm d}\theta\leavevmode\nobreak\ {\rm
d}\varphi\leavevmode\nobreak\ {\rm d}E_{\tau}\leavevmode\nobreak\ {\rm
d}h_{\rm dec}\leavevmode\nobreak\ {\rm d}t.$ (3.2)

The probability $p_{\rm exit}(E_{\nu},E_{\tau},\theta)$ is calculated using a
dedicated Monte Carlo simulation of $\nu_{\tau}$ propagation inside the earth
[45] that includes $\nu_{\tau}$ interaction through CC (NC) channels and
sampling of the resulting $\tau$ ($\nu_{\tau}$) energy, $\tau$ energy loss,
and $\tau$ decay inside the earth before emerging with $\nu_{\tau}$
regeneration following the decay [45, 46]. The probability of $\tau$ decay
after traveling a distance $l=h_{\rm dec}/|\cos\theta|$ can be obtained
analytically $p_{\rm dec}(E_{\tau},\theta,h_{\rm dec})=(\lambda_{\rm
dec}|\cos\theta|)^{-1}\leavevmode\nobreak\ \exp{(-l/{\lambda_{\rm dec}})}$
where $\lambda_{\rm dec}=\gamma_{\tau}c\tau_{0}=E_{\tau}c\tau_{0}/m_{\tau}$ is
the average $\tau$ decay length with $m_{\tau}\simeq 1.777$ GeV the mass of
the $\tau$ lepton, and $c\tau_{0}\simeq 86.93\times 10^{-6}$ m the lifetime of
the $\tau$ multiplied by the speed of light $c$.

The total exposure, ${\cal E}_{\rm tot}$, obtained assuming a flavor mixture
of $\nu_{e}:\nu_{\mu}:\nu_{\tau}=1:1:1$ is plotted in Fig. 5. Also shown are
the exposures for each neutrino flavor ${\cal E}_{i}$ with
$i=\nu_{e},\nu_{\mu},\nu_{\tau}$, as well as the separate contribution to the
exposure due to Earth-skimming $\nu_{\tau}$ alone and the sum of all the
exposures due to the three flavors and two channels (CC & NC) contributing to
the downward-going neutrino category.

When comparing the different neutrino flavors, the $\nu_{\tau}$ dominates the
contribution to the total exposure mainly because of the enhanced sensitivity
of the Earth-skimming channel that stems from the denser target provided by
the earth’s crust for the ES $\nu_{\tau}$ to interact, and the large range of
$\tau$ leptons at EeV energies ($\sim\,10$ km at 1 EeV in rock [51]). The
contribution to the $\nu_{\tau}$-flavor exposure ${\cal E}_{\nu_{\tau}}$ due
to downward-going $\nu_{\tau}$ is typically small, but increases as the energy
rises reaching a fraction of $\sim\,30\,\%$ of ${\cal E}_{\rm ES}$ at
$E_{\nu}=10^{20}$ eV. This can be seen in Fig. 5 by comparing ${\cal
E}_{\nu_{\tau}}$ and ${\cal E}_{\rm ES}$. Next in importance is the exposure
due to electron neutrinos ${\cal E}_{\nu_{e}}$ (CC and NC interactions
together) dominating over that due to muon neutrinos ${\cal E}_{\nu_{\mu}}$.
This is partly due to the larger fraction of neutrino energy going into the
shower in $\nu_{e}$ CC interactions as compared to $\nu_{\mu}$ CC
interactions, and partly due to the larger electromagnetic content of the
shower induced by the electron produced in the $\nu_{e}$ CC interaction. When
comparing the exposure for ES $\nu_{\tau}$ with the sum of all flavors and
channels contributing to the downward-going exposure, it can be seen in Fig. 5
that the latter dominates only above $\sim\,3-4\times 10^{19}$ eV. Above this
energy the exposure to ES $\nu_{\tau}$ starts to decrease due to the growing
decay length of the emerging $\tau$ leptons in air at these energies,
favouring decays at high altitude above the detector and reducing the
probability that the $\tau$-induced shower triggers the SD array.

##  4 Limits to diffuse fluxes

The total exposure ${\cal E}_{\rm tot}$ folded with a single-flavor flux of
UHE neutrinos per unit energy, area $A$, solid angle $\Omega$ and time,
$\phi(E_{\nu})={\rm d}^{6}N_{\nu}/({\rm d}E_{\nu}\leavevmode\nobreak\ {\rm
d}\Omega\leavevmode\nobreak\ {\rm d}A\leavevmode\nobreak\ {\rm d}t)$ and
integrated in energy gives the expected number of events for that flux:

$N_{\rm evt}=\leavevmode\nobreak\ \int_{E_{\nu}}\leavevmode\nobreak\ {\cal
E}_{\rm tot}(E_{\nu})\leavevmode\nobreak\ \phi(E_{\nu})\leavevmode\nobreak\
{\rm d}E_{\nu}.$ (4.1)

Assuming a differential neutrino flux $\phi=k\cdot E_{\nu}^{-2}$, an upper
limit to the value of $k$ at $90\%$ C.L. is obtained as

$k_{90}=\frac{2.39}{\int_{E_{\nu}}\leavevmode\nobreak\
E_{\nu}^{-2}\leavevmode\nobreak\ {\cal E}_{\rm
tot}(E_{\nu})\leavevmode\nobreak\ {\rm d}E_{\nu}},$ (4.2)

where $2.39$ is the Feldman-Cousins factor [52] for non-observation of events
in the absence of expected background accounting for systematic uncertainties
[53, 28]. The integrated limit represents the value of the normalization of a
$E_{\nu}^{-2}$ differential neutrino flux needed to predict $\sim\,2.39$
expected events.

Several sources of systematic uncertainty have been considered in the
calculation of the exposure and limit. The uncertainty due to simulations
includes the effects of using several neutrino interaction generators, shower
simulations, hadronic interaction models and thinning level. These would
modify the event rate for a $\phi(E_{\nu})\propto E_{\nu}^{-2}$ flux in Eq.
(4.1) between $-3\%$ and $4\%$ with respect to the reference calculation of
the exposure shown in Fig. 5. The uncertainty due to different models of
$\nu_{\tau}$ cross-section and $\tau$ energy-loss affects mainly the ES
channel with a corresponding range of variation of the event rate between
$-28\%$ and $34\%$. The topography around the Observatory is not accounted for
explicitly in the calculation of the exposure and is instead taken as a
systematic uncertainty that would increase the event rate by an estimated
$\sim\,20\,\%$ [42, 43]. The total uncertainty, obtained by adding these bands
in quadrature, ranges from $-28\%$ to $39\%$, and is incorporated in the value
of the limit itself through a semi-Bayesian extension [53] of the Feldman-
Cousins approach [52].

The single-flavor $90\%$ C.L. integrated limit is:

$k_{90}\leavevmode\nobreak\ <\leavevmode\nobreak\ 4.4\leavevmode\nobreak\
\times\leavevmode\nobreak\ 10^{-9}\leavevmode\nobreak\ {\rm
GeV\leavevmode\nobreak\ cm^{-2}\leavevmode\nobreak\ s^{-1}\leavevmode\nobreak\
sr^{-1}},$ (4.3)

or equivalently $1.4\leavevmode\nobreak\ {\rm EeV\leavevmode\nobreak\
km^{-2}\leavevmode\nobreak\ yr^{-1}\leavevmode\nobreak\ sr^{-1}}$. It mostly
applies in the energy interval $10^{17}\leavevmode\nobreak\ {\rm eV}-2.5\times
10^{19}\leavevmode\nobreak\ {\rm eV}$ for which $\sim\,90\,\%$ of the total
event rate is expected in the case of a $E_{\nu}^{-2}$ spectral flux. The
relative contributions to the expected rate of events for a $E_{\nu}^{-2}$
flux due to the three neutrino flavors and to the ES and DG channels are
displayed in Table 1. For such a spectral shape $\tau$ neutrinos contribute to
$\sim\,86\,\%$ of the total event rate, and in particular ES neutrinos
dominate the rate of $\nu_{\tau}$ events over the downward-going $\nu_{\tau}$.
The contribution of $\nu_{e}$ and $\nu_{\mu}$ together is smaller than
$15\,\%$ in this case.

Flavor | Relative contribution
---|---
$\nu_{e}$ | 0.10
$\nu_{\mu}$ | 0.04
$\nu_{\tau}$ | 0.86
Channel | Relative contribution
Earth-skimming $\nu_{\tau}$ | 0.79
Downward-going $\nu_{e}+\nu_{\mu}+\nu_{\tau}$ | 0.21
Table 1:  Top of table: Relative contribution of the three neutrino flavors to
the event rate in Auger due to a neutrino flux $\phi_{\nu}\propto
E_{\nu}^{-2}$. Bottom: Relative contribution to the rate in the Earth-skimming
(ES) and Downward-going (DG) channels.

The denominator of Eq. (4.2) can also be integrated in bins of neutrino energy
of width $\Delta E_{\nu}$, and a limit $\hat{k}_{90}$ can be obtained in each
energy bin. This is displayed in Fig. 6 for logarithmic energy intervals
$\Delta\log_{10}E_{\nu}=0.5$. The differential limit is an effective way of
characterizing the energy dependence of the sensitivity of a neutrino
experiment. For the case of Auger it can be seen that the best sensitivity is
achieved for energies around 1 EeV.

Figure 6:  Pierre Auger Observatory upper limit (90$\%$ C.L.) to the
normalization $k$ of the diffuse flux of UHE neutrinos
$\phi_{\nu}=k\leavevmode\nobreak\ E_{\nu}^{-2}$ as given in Eqs. (4.2) and
(4.3) (solid straight red line). Also plotted are the upper limits to the
normalization of the diffuse flux (differential limits) when integrating the
denominator of Eq. (4.2) in bins of width 0.5 in $\log_{10}E_{\nu}$ (solid red
line - Auger all channels and flavours; dashed red line - Auger Earth-skimming
$\nu_{\tau}$ only). The differential limits obtained by IceCube [30] (solid
green) and ANITA I+II+III [29] (solid dark magenta) are also shown. The
expected neutrino fluxes for several cosmogenic [15, 54, 55, 56] and
astrophysical models of neutrino production, as well as the Waxman-Bahcall
bound [57] are also plotted. All limits and fluxes are converted to single
flavor.

##  5 Constraints on the origin of UHECR

With the upper limit obtained with the Observatory, we can constrain several
classes of models of neutrino production in interactions of UHECR with the
Cosmic-Microwave Background and Extragalactic-Background Light (EBL), often
referred to as cosmogenic neutrino models. The expected event rate in the
Auger Observatory due to cosmogenic neutrinos depends strongly on the redshift
evolution of the UHECR sources, on the nature of the primaries, namely whether
they are protons or heavier nuclei, on the maximal redshift at which UHECR are
accelerated, $z_{\rm max}$, and on the maximum energy acquired in the
acceleration process, $E_{\rm max}$. Commonly, cosmogenic neutrino models
assume that the observed UHECR flux-suppression [7, 8] is based solely on the
GZK effect, i.e. on energy losses of protons or nuclei in the CMB. The highest
fluxes of cosmogenic neutrinos are then expected for injection of protons,
while those expected for injection of iron nuclei are down typically by about
an order of magnitude [18, 19, 15] (c.f. Fig. 6). We note, however, that the
possibility of pure proton (or iron) primaries in the energy range of interest
is disfavored by recent results on the composition of UHECR [10, 59, 60, 61].
Instead, a gradually increasing fraction of heavier primaries is observed with
increasing energy up to at least $E\sim 5\times 10^{19}$ eV [59]. In addition
to this, adopting a simple astrophysical model fitting the energy spectrum and
the mass composition suggests that the observed flux suppression is primarily
an effect of the maximum rigidity of the sources of UHECR rather than only the
effect of energy losses in the CMB and EBL [66, 67]. In consequence,
cosmogenic neutrino fluxes would be reduced much further and may escape
detection for the foreseeable future [16, 17, 68]. Thus, fluxes of cosmogenic
neutrinos provide an independent probe of source properties and of the origin
of the UHECR flux suppression at the highest energies.

In Table 2, we show the expected number of events in the present lifetime of
the Observatory for several cosmogenic neutrino models and the associated
Poisson probability of observing no events. Scenarios assuming sources that
accelerate only protons and that have a strong evolution with $z$, similar to
Fanaroff-Riley type II (FRII) radio-loud AGN [55], are strongly constrained by
the Auger results at more than $90\,\%$ C.L. The increased exposure reported
here allows us to constrain a larger parameter space of cosmogenic neutrino
models than was possible previously [28].

If instead of protons, the primaries are heavier nuclei, photodisintegration
is more likely than pion production, and the flux of cosmogenic neutrinos is
suppressed [18, 19, 15]. For models assuming mixed (Galactic) composition of
the UHECR such as those in [15], the constraints placed by the Observatory
limits have improved with respect to previous publications [28] but they are
still below the $90\,\%$ C.L. threshold of exclusion as can be seen in Table
2. An about threefold increase in the current exposure will be needed to
constrain at $90\,\%$ C.L. the lower edge of the gray band in Fig. 6 for a
mixed composition. The constraints are weaker if pure iron would be produced
at the sources. Ruling out at $90\,\%$ C.L. the most optimistic predictions of
cosmogenic neutrino flux at $10^{18}-10^{19}$ eV in such a scenario [55],
would require at least a sixfold increase in exposure. This is out of the
range of sensitivity of the current configuration of Auger.

Another class of cosmogenic neutrino models [54] are constructed in such a way
that the associated UHE photons produced along with the neutrinos, after
cascading in the Universe and being converted into GeV photons, saturate
measurements of the diffuse extragalactic $\gamma$-ray background performed by
Fermi-LAT [69]. Such approaches often neglect contributions from known
populations of unresolved gamma-ray sources [70, 71], and are also now in
tension with the direct limits on cosmogenic neutrinos obtained with the Auger
Observatory.

Also shown in Table 2 are some astrophysical models that are also excluded at
$90\,\%$ C.L. such as those assuming UHECR acceleration to trans-GZK energies
in radio-loud AGN [62].

The IceCube Neutrino Observatory detected a flux of astrophysical neutrinos
including a $\sim\,6.3$ PeV event compatible with being produced by the
Glashow resonance [64, 65]. Extrapolating the detected flux normalized at the
energy of the Glashow event to Auger energies, using a simple power-law in
energy $dN_{\nu}/dE_{\nu}\propto E^{-2.3}$, would yield $\sim\,0.4$ events in
Auger, with an uncertainty in the normalization of the flux of a factor of
$\sim\,3$. This suggests that if the measured flux actually extends into the
EeV energy range, it could be within reach of Auger in the next decade in an
optimistic scenario.

Neutrino Model | Expected number |  Probability of
---|---|---
(Diffuse flux) | of $\nu$ events |  observing $0$
Cosmogenic |  |
(Kampert et al. [55]) |  |
proton, FRII | $\sim\,5.9$ |  $\sim\,2.7\times 10^{-3}$
proton, SFR | $\sim\,1.4$ |  $\sim\,0.25$
iron, FRII | $\sim\,0.4$ |  $\sim\,0.67$
(Aloisio et al. [56]) |  |
proton, SFR | $\sim\,2.3$ |  $\sim\,0.10$
(Ahlers et al. [54]) |  |
proton, $E_{\rm min}=10^{19}$ eV |  $\sim\,4.6$ | $\sim\,1.0\times 10^{-2}$
proton, $E_{\rm min}=10^{17.5}$ eV |  $\sim\,2.4$ | $\sim\,9.0\times 10^{-2}$
(Kotera et al. [15]) |  |
p or mixed, SFR & GRB | $\sim\,0.8\leavevmode\nobreak\ -\leavevmode\nobreak\ 2.0$ |  $\sim\,0.45\leavevmode\nobreak\ -\leavevmode\nobreak\ 0.13$
Astrophysical |  |
(Murase et al. [62]) |  |
Radio-loud AGN | $\sim\,2.9$ | $\sim\,5.5\times 10^{-2}$
(Fang et al. [63]) |  |
Pulsars - SFR | $\sim\,1.5$ | $\sim\,0.22$
Table 2:  Number of expected neutrino events $N_{\rm evt}$ in the period 1 Jan
2004 - 31 August 2018 for several models of UHE neutrino production (see Fig.
6), given the exposure of the Surface Detector Array of the Pierre Auger
Observatory shown in Fig. 5. The last column gives the Poisson probability
$\exp({-N_{\rm evt}})$ of observing 0 events when the number of expected
events is $N_{\rm evt}$.

As demonstrated in Fig. 6, the presently achieved upper bounds of cosmogenic
neutrinos start to constrain astrophysical models that aim at describing the
UHECR flux suppression above $4\times 10^{19}$ eV by energy losses of protons
in the CMB. To further investigate the excluded parameter space of such proton
models in more detail, we performed a comprehensive scanning of the evolution
function of the sources with redshift, $\Psi(z)\propto(1+z)^{m}$ up to $z_{\rm
max}$. In all these scenarios, the spatial distribution of sources is assumed
to be homogeneous, and all the sources are assumed to have the same UHECR
luminosity. The simulations predicting the associated cosmogenic neutrino
fluxes were performed with the CRPropa package [72] for a fixed spectral index
of the UHECR proton spectrum $E^{-\alpha}$ at the sources (with $\alpha=2.5$),
and a maximum energy of the protons $E_{\rm max}=6\times 10^{20}$ eV. In the
energy range $\sim\,3\times 10^{17}$ to $\sim 3\times 10^{18}$ eV, where the
sensitivity of Auger peaks as shown in Fig. 6, the cosmogenic neutrino flux is
barely affected by the choice of $\alpha$ and $E_{\rm max}$ as long as the
latter is not below $10^{20}$ eV (see Fig. 1 in [73]). These calculations
include single- as well as multi-pion production in the proton-$\gamma$
interactions, and interactions with both the CMB and the EBL [72, 73]. The
proton flux at Earth is normalized to the Auger spectrum at $E=7\times
10^{18}$ eV. For each pair of $m$ and $z_{\rm max}$, the cosmogenic flux was
obtained, and the expected number of neutrino events in Auger was calculated.
Those models predicting more than 2.39 neutrinos are disfavored at $>90\,\%$
C.L. The resultant exclusion plot is shown in the top panel of Fig. 7 with the
contours representing the $68\,\%$ and $90\,\%$ C.L. exclusion limits. The
non-observation of neutrino candidates in the Observatory data allows us to
disfavor a significant region of the parameter space $(m,z_{\rm max})$. This
improves earlier constraints by the Pierre Auger Observatory [74] based on a
simplified analytical approach [75], also used by the IceCube neutrino
observatory [20, 30].

Finally, we explore the possibility of a proton component above $E\gtrsim 50$
EeV [58], i.e. in the energy range presently not covered by the direct
$X_{\mathrm{max}}$ measurements of Auger [59]. Such a subdominant proton
component would have a limited effect on the observed spectrum and
composition, but would strongly alter the expected cosmogenic neutrino flux
[73]. This is because protons produce significantly more neutrinos when
propagating through the universe than heavier nuclei of the same total energy,
particularly if the latter were not accelerated significantly beyond the GZK
energy threshold, such as indicated by a global fit to Auger data in [66]. To
be conservative, it is assumed that the cosmogenic neutrinos result only from
the proton component. The rejection power achieved on such a model then
depends on the relative contribution of protons normalized to the fixed all-
particle flux denoted as $f_{p}$ [73]. The corresponding exclusion plot is
depicted in the bottom panel of Fig. 7. For instance, proton flux
normalizations down to $f_{p}\simeq 0.2$ can be ruled out with the present
data for sources following a cosmological evolution of $m\simeq 3.8$ up to
redshift $z_{\rm max}=5$.

The ongoing upgrade of the Surface Detector of the Pierre Auger Observatory,
AugerPrime, will add a plastic scintillation detector [76] and a radio antenna
[77] to each Water-Cherenkov Detector. Additional measurements of composition-
sensitive observables will be possible with the data collected simultaneously
with the Water-Cherenkov and the scintillation detectors for a high statistics
sample of ultra-high energy events, not affected by the low duty cycle of the
Fluorescence Detector. This will enhance the potential to identify the primary
masses to the highest energies and will further add discrimination power for
the primary mass on an event-by-event basis. It has been calculated that the
upgrade will enable us to directly identify a possible proton component at the
highest energies as small as $f_{p}=0.1$ [76, 78]. This will be crucial for
determining the role of cosmic-ray observations in UHE astronomy, and for
establishing the potential of present and future detectors to the detection of
the cosmogenic neutrino flux [27].

---
Figure 7:  Constraints on UHECR source evolution models parameterized as
$\psi(z)\propto(1+z)^{m}$ for sources distributed homogeneously up to a
maximum redshift $z_{\rm max}$ and emitting protons following a power-law
${\rm d}N/{\rm d}E\propto E^{-2.5}$ up to $E=6\times 10^{20}$ eV. A proton-
only flux is matched to the Auger spectrum at $7\times 10^{18}$ eV (benchmark
calculation for $f_{p}=1$, see text). The cosmogenic neutrino fluxes for each
combination of $m$ and $z_{\rm max}$ were obtained with the Monte Carlo (MC)
propagation code CRPropa [72]. Top panel: Exclusion plot for source evolution
parameter $m$ and $z_{\rm max}$ with $f_{p}=1$. The colored areas represent
different levels of C.L. exclusion. In particular the solid and dashed lines
represent the contours of $68\%$ and $90\%$ C.L. exclusion, respectively. The
dashed-dotted blue line represents the $90\%$ CL contour exclusion for
cosmogenic neutrino models obtained with the analytical calculation in [75].
Bottom panel: Exclusion plot for source evolution model parameter $m$ and
variable $f_{p}\leq 1$. The regions above the colored lines corresponding to
several values of $z_{\rm max}$ are excluded at 90 % C.L. from the lack of
neutrino candidates in Auger data.

## Acknowledgments

The successful installation, commissioning, and operation of the Pierre Auger
Observatory would not have been possible without the strong commitment and
effort from the technical and administrative staff in Malargüe. We are very
grateful to the following agencies and organizations for financial support:

Argentina – Comisión Nacional de Energía Atómica; Agencia Nacional de
Promoción Científica y Tecnológica (ANPCyT); Consejo Nacional de
Investigaciones Científicas y Técnicas (CONICET); Gobierno de la Provincia de
Mendoza; Municipalidad de Malargüe; NDM Holdings and Valle Las Leñas; in
gratitude for their continuing cooperation over land access; Australia – the
Australian Research Council; Brazil – Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq); Financiadora de Estudos e Projetos (FINEP);
Fundação de Amparo à Pesquisa do Estado de Rio de Janeiro (FAPERJ); São Paulo
Research Foundation (FAPESP) Grants No. 2010/07359-6 and No. 1999/05404-3;
Ministério da Ciência, Tecnologia, Inovações e Comunicações (MCTIC); Czech
Republic – Grant No. MSMT CR LTT18004, LO1305, LM2015038 and
CZ.02.1.01/0.0/0.0/16_013/0001402; France – Centre de Calcul IN2P3/CNRS;
Centre National de la Recherche Scientifique (CNRS); Conseil Régional Ile-de-
France; Département Physique Nucléaire et Corpusculaire (PNC-IN2P3/CNRS);
Département Sciences de l’Univers (SDU-INSU/CNRS); Institut Lagrange de Paris
(ILP) Grant No. LABEX ANR-10-LABX-63 within the Investissements d’Avenir
Programme Grant No. ANR-11-IDEX-0004-02; Germany – Bundesministerium für
Bildung und Forschung (BMBF); Deutsche Forschungsgemeinschaft (DFG);
Finanzministerium Baden-Württemberg; Helmholtz Alliance for Astroparticle
Physics (HAP); Helmholtz-Gemeinschaft Deutscher Forschungszentren (HGF);
Ministerium für Innovation, Wissenschaft und Forschung des Landes Nordrhein-
Westfalen; Ministerium für Wissenschaft, Forschung und Kunst des Landes Baden-
Württemberg; Italy – Istituto Nazionale di Fisica Nucleare (INFN); Istituto
Nazionale di Astrofisica (INAF); Ministero dell’Istruzione, dell’Universitá e
della Ricerca (MIUR); CETEMPS Center of Excellence; Ministero degli Affari
Esteri (MAE); México – Consejo Nacional de Ciencia y Tecnología (CONACYT) No.
167733; Universidad Nacional Autónoma de México (UNAM); PAPIIT DGAPA-UNAM; The
Netherlands – Ministry of Education, Culture and Science; Netherlands
Organisation for Scientific Research (NWO); Dutch national e-infrastructure
with the support of SURF Cooperative; Poland -Ministry of Science and Higher
Education, grant No. DIR/WK/2018/11; National Science Centre, Grants No.
2013/08/M/ST9/00322, No. 2016/23/B/ST9/01635 and No. HARMONIA
5–2013/10/M/ST9/00062, UMO-2016/22/M/ST9/00198; Portugal – Portuguese national
funds and FEDER funds within Programa Operacional Factores de Competitividade
through Fundação para a Ciência e a Tecnologia (COMPETE); Romania – Romanian
Ministry of Research and Innovation CNCS/CCCDI-UESFISCDI, projects PN-
III-P1-1.2-PCCDI-2017-0839/19PCCDI/2018 and PN18090102 within PNCDI III;
Slovenia – Slovenian Research Agency, grants P1-0031, P1-0385, I0-0033,
N1-0111; Spain – Ministerio de Economía, Industria y Competitividad
(FPA2017-85114-P and FPA2017-85197-P), Xunta de Galicia (ED431C 2017/07),
Junta de Andalucía (SOMM17/6104/UGR), Feder Funds, RENATA Red Nacional
Temática de Astropartículas (FPA2015-68783-REDT) and María de Maeztu Unit of
Excellence (MDM-2016-0692); USA – Department of Energy, Contracts No. DE-
AC02-07CH11359, No. DE-FR02-04ER41300, No. DE-FG02-99ER41107 and No. DE-
SC0011689; National Science Foundation, Grant No. 0450696; The Grainger
Foundation; Marie Curie-IRSES/EPLANET; European Particle Physics Latin
American Network; and UNESCO.

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The Pierre Auger Collaboration

A. Aab75, P. Abreu67, M. Aglietta50,49, I.F.M. Albuquerque19, J.M. Albury12,
I. Allekotte1, A. Almela8,11, J. Alvarez Castillo63, J. Alvarez-Muñiz74, G.A.
Anastasi42,43, L. Anchordoqui82, B. Andrada8, S. Andringa67, C. Aramo47, H.
Asorey1,28, P. Assis67, G. Avila9,10, A.M. Badescu70, A. Bakalova30, A.
Balaceanu68, F. Barbato56,47, R.J. Barreira Luz67, S. Baur37, K.H. Becker35,
J.A. Bellido12, C. Berat34, M.E. Bertaina58,49, X. Bertou1, P.L. Biermannb, J.
Biteau32, A. Blanco67, J. Blazek30, C. Bleve52,45, M. Boháčová30, D.
Boncioli42,43, C. Bonifazi24, N. Borodai64, A.M. Botti8,37, J. Bracke, T.
Bretz39, A. Bridgeman36, F.L. Briechle39, P. Buchholz41, A. Bueno73, S.
Buitink14, M. Buscemi54,44, K.S. Caballero-Mora62, L. Caccianiga55, L.
Calcagni4, A. Cancio11,8, F. Canfora75,77, I. Caracas35, J.M. Carceller73, R.
Caruso54,44, A. Castellina50,49, F. Catalani17, G. Cataldi45, L. Cazon67, M.
Cerda9, J.A. Chinellato20, K. Choi13, J. Chudoba30, L. Chytka31, R.W. Clay12,
A.C. Cobos Cerutti7, R. Colalillo56,47, A. Coleman88, M.R. Coluccia52,45, R.
Conceição67, A. Condorelli42,43, G. Consolati46,51, F. Contreras9,10, F.
Convenga52,45, M.J. Cooper12, S. Coutu86, C.E. Covault80,h, B. Daniel20, S.
Dasso5,3, K. Daumiller37, B.R. Dawson12, J.A. Day12, R.M. de Almeida26, S.J.
de Jong75,77, G. De Mauro75,77, J.R.T. de Mello Neto24,25, I. De Mitri42,43,
J. de Oliveira26, V. de Souza18, J. Debatin36, M. del Río10, O. Deligny32, N.
Dhital64, A. Di Matteo49, M.L. Díaz Castro20, C. Dobrigkeit20, J.C. D’Olivo63,
Q. Dorosti41, R.C. dos Anjos23, M.T. Dova4, A. Dundovic40, J. Ebr30, R.
Engel36,37, M. Erdmann39, C.O. Escobarc, A. Etchegoyen8,11, H. Falcke75,78,77,
J. Farmer87, G. Farrar85, A.C. Fauth20, N. Fazzinic, F. Feldbusch38, F.
Fenu58,49, L.P. Ferreyro8, J.M. Figueira8, A. Filipčič72,71, M.M. Freire6, T.
Fujii87,f, A. Fuster8,11, B. García7, H. Gemmeke38, F. Gesualdi8, A. Gherghel-
Lascu68, P.L. Ghia32, U. Giaccari15, M. Giammarchi46, M. Giller65, D. Głas66,
J. Glombitza39, F. Gobbi9, G. Golup1, M. Gómez Berisso1, P.F. Gómez
Vitale9,10, J.P. Gongora9, N. González8, I. Goos1,37, D. Góra64, A.
Gorgi50,49, M. Gottowik35, T.D. Grubb12, F. Guarino56,47, G.P. Guedes21, E.
Guido49,58, S. Hahn37, R. Halliday80, M.R. Hampel8, P. Hansen4, D. Harari1,
T.A. Harrison12, V.M. Harvey12, A. Haungs37, T. Hebbeker39, D. Heck37, P.
Heimann41, G.C. Hill12, C. Hojvatc, E.M. Holt36,8, P. Homola64, J.R.
Hörandel75,77, P. Horvath31, M. Hrabovský31, T. Huege37,14, J. Hulsman8,37, A.
Insolia54,44, P.G. Isar69, J.A. Johnsen81, J. Jurysek30, A. Kääpä35, K.H.
Kampert35, B. Keilhauer37, N. Kemmerich19, J. Kemp39, H.O. Klages37, M.
Kleifges38, J. Kleinfeller9, D. Kuempel35, G. Kukec Mezek71, A. Kuotb Awad36,
B.L. Lago16, D. LaHurd80, R.G. Lang18, R. Legumina65, M.A. Leigui de
Oliveira22, V. Lenok37, A. Letessier-Selvon33, I. Lhenry-Yvon32, O.C.
Lippmann15, D. Lo Presti54,44, L. Lopes67, R. López59, A. López Casado74, R.
Lorek80, Q. Luce36, A. Lucero8, M. Malacari87, G. Mancarella52,45, D.
Mandat30, B.C. Manning12, J. Manshanden40, P. Mantschc, A.G. Mariazzi4, I.C.
Mariş13, G. Marsella52,45, D. Martello52,45, H. Martinez18, O. Martínez
Bravo59, M. Mastrodicasa53,43, H.J. Mathes37, S. Mathys35, J. Matthews83, G.
Matthiae57,48, E. Mayotte35, P.O. Mazurc, G. Medina-Tanco63, D. Melo8, A.
Menshikov38, K.-D. Merenda81, S. Michal31, M.I. Micheletti6, L.
Miramonti55,46, D. Mockler13, S. Mollerach1, F. Montanet34, C. Morello50,49,
G. Morlino42,43, M. Mostafá86, A.L. Müller8,37, M.A. Muller20,d, S. Müller36,
R. Mussa49, W.M. Namasaka35, L. Nellen63, M. Niculescu-Oglinzanu68, M.
Niechciol41, D. Nitz84,g, D. Nosek29, V. Novotny29, L. Nožka31, A Nucita52,45,
L.A. Núñez28, A. Olinto87, M. Palatka30, J. Pallotta2, M.P. Panetta52,45, P.
Papenbreer35, G. Parente74, A. Parra59, M. Pech30, F. Pedreira74, J. Pȩkala64,
R. Pelayo61, J. Peña-Rodriguez28, L.A.S. Pereira20, M. Perlin8, L.
Perrone52,45, C. Peters39, S. Petrera42,43, J. Phuntsok86, T. Pierog37, M.
Pimenta67, V. Pirronello54,44, M. Platino8, J. Poh87, B. Pont75, C.
Porowski64, M. Pothast77,75, R.R. Prado18, P. Privitera87, M. Prouza30, A.
Puyleart84, S. Querchfeld35, S. Quinn80, R. Ramos-Pollan28, J. Rautenberg35,
D. Ravignani8, M. Reininghaus37, J. Ridky30, F. Riehn67, M. Risse41, P.
Ristori2, V. Rizi53,43, W. Rodrigues de Carvalho19, J. Rodriguez Rojo9, M.J.
Roncoroni8, M. Roth37, E. Roulet1, A.C. Rovero5, P. Ruehl41, S.J. Saffi12, A.
Saftoiu68, F. Salamida53,43, H. Salazar59, G. Salina48, J.D. Sanabria Gomez28,
F. Sánchez8, E.M. Santos19, E. Santos30, F. Sarazin81, R. Sarmento67, C.
Sarmiento-Cano8, R. Sato9, P. Savina52,45, M. Schauer35, V. Scherini45, H.
Schieler37, M. Schimassek36, M. Schimp35, F. Schlüter37, D. Schmidt36, O.
Scholten76,14, P. Schovánek30, F.G. Schröder88,37, S. Schröder35, J.
Schumacher39, S.J. Sciutto4, M. Scornavacche8, R.C. Shellard15, G. Sigl40, G.
Silli8,37, O. Sima68,h, R. Šmída87, G.R. Snow89, P. Sommers86, J.F. Soriano82,
J. Souchard34, R. Squartini9, M. Stadelmaier37, D. Stanca68, S. Stanič71, J.
Stasielak64, P. Stassi34, M. Stolpovskiy34, A. Streich36, M. Suárez-Durán28,
T. Sudholz12, T. Suomijärvi32, A.D. Supanitsky8, J. Šupík31, Z. Szadkowski66,
A. Taboada36, O.A. Taborda1, A. Tapia27, C. Timmermans77,75, P. Tobiska30,
C.J. Todero Peixoto17, B. Tomé67, G. Torralba Elipe74, A. Travaini9, P.
Travnicek30, M. Trini71, M. Tueros4, R. Ulrich37, M. Unger37, M. Urban39, J.F.
Valdés Galicia63, I. Valiño42,43, L. Valore56,47, P. van Bodegom12, A.M. van
den Berg76, A. van Vliet75, E. Varela59, B. Vargas Cárdenas63, A. Vásquez-
Ramírez28, D. Veberič37, C. Ventura25, I.D. Vergara Quispe4, V. Verzi48, J.
Vicha30, L. Villaseñor59, J. Vink79, S. Vorobiov71, H. Wahlberg4, A.A.
Watsona, M. Weber38, A. Weindl37, M. Wiedeński66, L. Wiencke81, H.
Wilczyński64, T. Winchen14, M. Wirtz39, D. Wittkowski35, B. Wundheiler8, L.
Yang71, A. Yushkov30, E. Zas74, D. Zavrtanik71,72, M. Zavrtanik72,71, L.
Zehrer71, A. Zepeda60, B. Zimmermann37, M. Ziolkowski41, F. Zuccarello54,44

1

Centro Atómico Bariloche and Instituto Balseiro (CNEA-UNCuyo-CONICET), San
Carlos de Bariloche, Argentina

2

Centro de Investigaciones en Láseres y Aplicaciones, CITEDEF and CONICET,
Villa Martelli, Argentina

3

Departamento de Física and Departamento de Ciencias de la Atmósfera y los
Océanos, FCEyN, Universidad de Buenos Aires and CONICET, Buenos Aires,
Argentina

4

IFLP, Universidad Nacional de La Plata and CONICET, La Plata, Argentina

5

Instituto de Astronomía y Física del Espacio (IAFE, CONICET-UBA), Buenos
Aires, Argentina

6

Instituto de Física de Rosario (IFIR) – CONICET/U.N.R. and Facultad de
Ciencias Bioquímicas y Farmacéuticas U.N.R., Rosario, Argentina

7

Instituto de Tecnologías en Detección y Astropartículas (CNEA, CONICET,
UNSAM), and Universidad Tecnológica Nacional – Facultad Regional Mendoza
(CONICET/CNEA), Mendoza, Argentina

8

Instituto de Tecnologías en Detección y Astropartículas (CNEA, CONICET,
UNSAM), Buenos Aires, Argentina

9

Observatorio Pierre Auger, Malargüe, Argentina

10

Observatorio Pierre Auger and Comisión Nacional de Energía Atómica, Malargüe,
Argentina

11

Universidad Tecnológica Nacional – Facultad Regional Buenos Aires, Buenos
Aires, Argentina

12

University of Adelaide, Adelaide, S.A., Australia

13

Université Libre de Bruxelles (ULB), Brussels, Belgium

14

Vrije Universiteit Brussels, Brussels, Belgium

15

Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, RJ, Brazil

16

Centro Federal de Educação Tecnológica Celso Suckow da Fonseca, Nova Friburgo,
Brazil

17

Universidade de São Paulo, Escola de Engenharia de Lorena, Lorena, SP, Brazil

18

Universidade de São Paulo, Instituto de Física de São Carlos, São Carlos, SP,
Brazil

19

Universidade de São Paulo, Instituto de Física, São Paulo, SP, Brazil

20

Universidade Estadual de Campinas, IFGW, Campinas, SP, Brazil

21

Universidade Estadual de Feira de Santana, Feira de Santana, Brazil

22

Universidade Federal do ABC, Santo André, SP, Brazil

23

Universidade Federal do Paraná, Setor Palotina, Palotina, Brazil

24

Universidade Federal do Rio de Janeiro, Instituto de Física, Rio de Janeiro,
RJ, Brazil

25

Universidade Federal do Rio de Janeiro (UFRJ), Observatório do Valongo, Rio de
Janeiro, RJ, Brazil

26

Universidade Federal Fluminense, EEIMVR, Volta Redonda, RJ, Brazil

27

Universidad de Medellín, Medellín, Colombia

28

Universidad Industrial de Santander, Bucaramanga, Colombia

29

Charles University, Faculty of Mathematics and Physics, Institute of Particle
and Nuclear Physics, Prague, Czech Republic

30

Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic

31

Palacky University, RCPTM, Olomouc, Czech Republic

32

Institut de Physique Nucléaire d’Orsay (IPNO), Université Paris-Sud, Univ.
Paris/Saclay, CNRS-IN2P3, Orsay, France

33

Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Universités
Paris 6 et Paris 7, CNRS-IN2P3, Paris, France

34

Univ. Grenoble Alpes, CNRS, Grenoble Institute of Engineering Univ. Grenoble
Alpes, LPSC-IN2P3, 38000 Grenoble, France, France

35

Bergische Universität Wuppertal, Department of Physics, Wuppertal, Germany

36

Karlsruhe Institute of Technology, Institute for Experimental Particle Physics
(ETP), Karlsruhe, Germany

37

Karlsruhe Institute of Technology, Institut für Kernphysik, Karlsruhe, Germany

38

Karlsruhe Institute of Technology, Institut für Prozessdatenverarbeitung und
Elektronik, Karlsruhe, Germany

39

RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany

40

Universität Hamburg, II. Institut für Theoretische Physik, Hamburg, Germany

41

Universität Siegen, Fachbereich 7 Physik – Experimentelle Teilchenphysik,
Siegen, Germany

42

Gran Sasso Science Institute, L’Aquila, Italy

43

INFN Laboratori Nazionali del Gran Sasso, Assergi (L’Aquila), Italy

44

INFN, Sezione di Catania, Catania, Italy

45

INFN, Sezione di Lecce, Lecce, Italy

46

INFN, Sezione di Milano, Milano, Italy

47

INFN, Sezione di Napoli, Napoli, Italy

48

INFN, Sezione di Roma “Tor Vergata”, Roma, Italy

49

INFN, Sezione di Torino, Torino, Italy

50

Osservatorio Astrofisico di Torino (INAF), Torino, Italy

51

Politecnico di Milano, Dipartimento di Scienze e Tecnologie Aerospaziali ,
Milano, Italy

52

Università del Salento, Dipartimento di Matematica e Fisica “E. De Giorgi”,
Lecce, Italy

53

Università dell’Aquila, Dipartimento di Scienze Fisiche e Chimiche, L’Aquila,
Italy

54

Università di Catania, Dipartimento di Fisica e Astronomia, Catania, Italy

55

Università di Milano, Dipartimento di Fisica, Milano, Italy

56

Università di Napoli “Federico II”, Dipartimento di Fisica “Ettore Pancini”,
Napoli, Italy

57

Università di Roma “Tor Vergata”, Dipartimento di Fisica, Roma, Italy

58

Università Torino, Dipartimento di Fisica, Torino, Italy

59

Benemérita Universidad Autónoma de Puebla, Puebla, México

60

Centro de Investigación y de Estudios Avanzados del IPN (CINVESTAV), México,
D.F., México

61

Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas
del Instituto Politécnico Nacional (UPIITA-IPN), México, D.F., México

62

Universidad Autónoma de Chiapas, Tuxtla Gutiérrez, Chiapas, México

63

Universidad Nacional Autónoma de México, México, D.F., México

64

Institute of Nuclear Physics PAN, Krakow, Poland

65

University of Łódź, Faculty of Astrophysics, Łódź, Poland

66

University of Łódź, Faculty of High-Energy Astrophysics,Łódź, Poland

67

Laboratório de Instrumentação e Física Experimental de Partículas – LIP and
Instituto Superior Técnico – IST, Universidade de Lisboa – UL, Lisboa,
Portugal

68

“Horia Hulubei” National Institute for Physics and Nuclear Engineering,
Bucharest-Magurele, Romania

69

Institute of Space Science, Bucharest-Magurele, Romania

70

University Politehnica of Bucharest, Bucharest, Romania

71

Center for Astrophysics and Cosmology (CAC), University of Nova Gorica, Nova
Gorica, Slovenia

72

Experimental Particle Physics Department, J. Stefan Institute, Ljubljana,
Slovenia

73

Universidad de Granada and C.A.F.P.E., Granada, Spain

74

Instituto Galego de Física de Altas Enerxías (I.G.F.A.E.), Universidad de
Santiago de Compostela, Santiago de Compostela, Spain

75

IMAPP, Radboud University Nijmegen, Nijmegen, The Netherlands

76

KVI – Center for Advanced Radiation Technology, University of Groningen,
Groningen, The Netherlands

77

Nationaal Instituut voor Kernfysica en Hoge Energie Fysica (NIKHEF), Science
Park, Amsterdam, The Netherlands

78

Stichting Astronomisch Onderzoek in Nederland (ASTRON), Dwingeloo, The
Netherlands

79

Universiteit van Amsterdam, Faculty of Science, Amsterdam, The Netherlands

80

Case Western Reserve University, Cleveland, OH, USA

81

Colorado School of Mines, Golden, CO, USA

82

Department of Physics and Astronomy, Lehman College, City University of New
York, Bronx, NY, USA

83

Louisiana State University, Baton Rouge, LA, USA

84

Michigan Technological University, Houghton, MI, USA

85

New York University, New York, NY, USA

86

Pennsylvania State University, University Park, PA, USA

87

University of Chicago, Enrico Fermi Institute, Chicago, IL, USA

88

University of Delaware, Department of Physics and Astronomy, Bartol Research
Institute, Newark, DE, USA

89

University of Nebraska, Lincoln, NE, USA

—–

a

School of Physics and Astronomy, University of Leeds, Leeds, United Kingdom

b

Max-Planck-Institut für Radioastronomie, Bonn, Germany

c

Fermi National Accelerator Laboratory, USA

d

also at Universidade Federal de Alfenas, Poços de Caldas, Brazil

e

Colorado State University, Fort Collins, CO, USA

f

now at Hakubi Center for Advanced Research and Graduate School of Science,
Kyoto University, Kyoto, Japan

g

also at Karlsruhe Institute of Technology, Karlsruhe, Germany

h

also at Radboud Universtiy Nijmegen, Nijmegen, The Netherlands