Near-future discovery of the diuse ux of ultra-high-energy cosmic neutrinos V ctor B. Valera and Mauricio Bustamante

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Near-future discovery of the diffuse flux of ultra-high-energy cosmic neutrinos
V´ıctor B. Valera ∗and Mauricio Bustamante †
Niels Bohr International Academy, Niels Bohr Institute,
University of Copenhagen, DK-2100 Copenhagen, Denmark
Christian Glaser ‡
Department of Physics and Astronomy, Uppsala University, Uppsala, SE-752 37, Sweden
(Dated: March 2, 2023)
Ultra-high-energy (UHE) neutrinos, with EeV-scale energies, carry with them unique insight into
fundamental open questions in astrophysics and particle physics. For fifty years, they have evaded
discovery, but maybe not for much longer, thanks to new UHE neutrino telescopes, presently under
development. We capitalize on this upcoming opportunity by producing state-of-the-art forecasts of
the discovery of a diffuse flux of UHE neutrinos in the next 10–20 years. By design, our forecasts are
anchored in often-overlooked nuance from theory and experiment; we gear them to the radio array
of the planned IceCube-Gen2 detector. We find encouraging prospects: even under conservative
analysis choices, most benchmark UHE neutrino flux models from the literature may be discovered
within 10 years of detector exposure—many sooner—and may be distinguished from each other.
Our results validate the transformative potential of next-generation UHE neutrino telescopes.
I. INTRODUCTION
Ultra-high-energy (UHE) neutrinos, with energies in
the EeV scale (1 EeV ≡1018 eV), were first predicted in
the late 1960s [1], as a natural consequence [2,3] of the in-
teraction of UHE cosmic rays (UHECRs), with compara-
ble energies, and cosmological photon fields, like the cos-
mic microwave background. They are the most energetic
neutrinos expected to be produced from standard particle
processes, at least 10–100 times more energetic than the
TeV–PeV neutrinos discovered by the IceCube neutrino
telescope [4–10] (there may be higher-energy neutrinos
made in exotic processes [11–14], but we do not consider
them). UHE neutrinos provide unique insight into long-
standing open problems in astrophysics—what are the
most energetic astrophysical sources in the Universe—
and particle physics—how do neutrinos, in particular,
and fundamental physics, in general, behave at the high-
est energies [15–24]. Yet, despite efforts, they remain
undiscovered; however, maybe not for much longer.
Over the last fifty years, UHE neutrinos have re-
ceived considerable attention from experiment and the-
ory. Progress, while steady, has been challenging: past
and present experiments have placed upper limits on
their flux [25–29], but even the tightest present-day lim-
its [25,28] leave vast swathes of the space of UHE neu-
trino flux models unconstrained; see Fig. 2. On the ex-
perimental front, the main challenge is that the flux of
UHE neutrinos is possibly tiny [30,31]. This motivates
the need to build larger neutrino telescopes and explore
more suitable detection strategies [23]. On the theory
front, the main challenge is that predictions of the UHE
neutrino flux are uncertain because they depend on prop-
erties of UHECRs and their sources, which are known
only uncertainly, such as the evolution with redshift of
the source number density, the UHECR mass compo-
sition, the UHECR acceleration mechanism, including
the maximum cosmic-ray energies achievable, and the
neutrino production mechanism, among others; for de-
tails, see, e.g., Refs. [32–36]. This motivates the need to
consider a host of competing, representative flux predic-
tions [36–43]. We tackle both challenges.
Upcoming UHE neutrino telescopes, presently in dif-
ferent stages of planning, design, and prototyping, and
built around different detection strategies, will have a
real chance of discovering UHE neutrinos in the next 10–
20 years, even if their flux is low [21,23,24]. We carve
out this opportunity by providing the most detailed fore-
casts, to our knowledge, of the prospects of discovering
a diffuse flux of UHE neutrinos. Our results, even under
conservative analysis choices, are encouraging.
To include realistic experimental nuance, we gear
our forecasts to the radio array of IceCube-Gen2 [46]
(“IceCube-Gen2 Radio” in our figures), the planned high-
energy upgrade of IceCube, whose target UHE neutrino
flux sensitivity is among the best [23]. The array will
instrument Antarctic ice with radio antennas that look
for radio signals emitted by showers induced by UHE
neutrinos [47–49], a technique tested by ARA [29] and
ARIANNA [27] (and by ANITA [26], from the air).
RNO-G [50], currently under deployment, will serve as
a pathfinder for the radio array of IceCube-Gen2. To
make our forecasts comprehensive, we consider a large
number of benchmark UHE neutrino flux models that
span the full allowed space of models, in size and shape,
from optimistic to pessimistic [9,10,36–43].
To produce our forecasts, we adopt the same flow of
calculations as Ref. [51]. For each UHE neutrino flux
model, we propagate it through the Earth, computing
neutrino interactions with matter along the way, and
model its detection in the radio array of IceCube-Gen2.
We use the same state-of-the-art ingredients at every
stage of the calculation as Ref. [51]: in the choice of
diffuse UHE neutrino flux models (Section II), the UHE
neutrino-nucleon cross section, the propagation of neutri-
nos through the Earth (Section III), the neutrino detec-
arXiv:2210.03756v2 [astro-ph.HE] 1 Mar 2023
2
0.1 1 10
IceCube-Gen2 Radio exposure time, T[yr]
0.1
1
101
102
Mean UHE νflux discovery Bayes factor, hlog10 Bi
IceCube HESE (7.5 yr) extrapolated
IceCube νµ(9.5 yr) extrapolated
Heinze et al., fit to Auger UHECRs
Bergman & van Vliet, fit to TA UHECRs
Rodrigues et al., all AGN
Rodrigues et al., all AGN
Rodrigues et al., HL BL Lacs
Fang & Murase, CR reservoirs
Fang et al., newborn pulsars
Padovani et al., BL Lacs
Muzio et al., max. extra pcomp.
Muzio et al., fit to Auger & IceCube
IceCube νextrapolated
Cosmogenic ν
Source ν
Cosmogenic + source ν
Decisive
Very strong
Strong
Substantial
Negligible
1
2
3
4
5
6
7
8
910
11
12
1
2
3
4
5
6
7
8
9
10
11
12
FIG. 1. Discovery potential of benchmark diffuse ultra-high-
energy (UHE) neutrino flux models 1–12 [9,10,36–43] (Fig. 2)
in the radio array of IceCube-Gen2. The background to dis-
covery consists of atmospheric muons [44,45], for all models,
plus the tentative UHE tail of the IceCube 9.5-year through-
going νµflux [10], for models 3–12; see Section IV E. All anal-
ysis choices are baseline and conservative; see Table II and
Section VB1.Decisive discovery may be achievable for most
flux models after only a handful of years. See the main text,
especially Sections V A and V B 2, for details.
tion, including the emission, propagation, and detection
of radio signals in ice, and the neutrino and non-neutrino
backgrounds (Section IV). See Section II of Ref. [51] for
an overview. Further, our forecasts account for random
statistical fluctuations in the predicted event rates (Sec-
tions Vand VI). Below, we expand on all of the above.
Figure 1shows our main results: the discovery
prospects of the benchmark UHE neutrino flux models,
computed under our baseline analysis choices, chosen to
be largely conservative. Because our statistical analysis
is Bayesian, we report the flux discovery potential—and,
later, the potential to tell apart different flux models—via
Bayes factors. Figure 1reveals encouraging prospects:
conservatively, most benchmark UHE neutrino flux mod-
els may be discovered after only a handful of years. Later,
we show that less conservative analysis choices, still well-
motivated, lead to even better prospects.
The overarching goal of our detailed forecasts is to help
map the potential science reach that upcoming UHE neu-
trino telescopes will usher in in the next 10–20 years.
We make our forecasts realistic by factoring in nuance,
from experiment and theory, that is often considered only
partially, or not at all. We present our methods in con-
siderable detail so that they can be readily adapted to
produce forecasts for other upcoming UHE neutrino tele-
scopes. We hope that they help to assess and compare
the complementary capabilities of competing designs.
This paper is organized as follows. Section II presents
the benchmark diffuse UHE neutrino flux models that
we use in our forecasts. Section III sketches the effects of
neutrino propagation inside Earth on them. Section IV
introduces the method that we use to compute neutrino-
induced event rates and the backgrounds that we con-
sider. Section Vcontains forecasts of the discovery po-
tential of the benchmark flux models. Section VI contains
forecasts of the separation between different benchmark
flux models. Section VII outlines possible directions for
future work. Section VIII summarizes and concludes.
II. ULTRA-HIGH-ENERGY NEUTRINOS
Ultra-high-energy neutrinos [2], with energies above
100 PeV, are expected to be produced in the interaction
of UHECRs [1,3], with energies up to 1012 GeV, with
matter and radiation, inside the UHECR sources (source
neutrinos), outside them and en route to Earth (cosmo-
genic neutrinos), or both. See Ref. [23] for a review.
The interaction of UHECR protons on matter (pp) and
radiation (pγ) produces a short-lived ∆(1232) resonance
that decays into charged pions. Upon decaying, they pro-
duce high-energy neutrinos, via π+→µ++νµ, followed
by µ+→e++νe+ ¯νµ, and their charge-conjugated pro-
cesses. Each final-state neutrino carries, on average, 5%
of the energy of the parent proton. En route to Earth,
neutrino oscillations change the flavor composition of the
flux, i.e., the relative content of νe,νµ, and ντin it. (Our
benchmark UHE neutrino flux models below account for
this change; more on this later.)
In realistic neutrino production models, including in
some of our benchmark UHE neutrino flux models below,
different production channels become accessible or dom-
inant at different energies. In pγ interactions, neutrino
production occurs via resonances heavier than ∆(1232)
at intermediate energies, and via multi-pion production
at high energies [52–54]. In pp interactions, the pion
multiplicity changes with energy and affects the neutrino
yield [55]. The physical conditions inside the sources
may affect neutrino production, too. For instance, neu-
trino energies might be damped by strong magnetic fields
that cool intermediate charged particles—protons, pions,
muons—via synchrotron radiation [56–59], or by UHECR
interactions in dense source environments [60,61].
For UHE neutrinos produced in pp interactions, their
energy spectrum is a power law that follows the power-
law spectrum of the parent protons, and that may ex-
tend to low neutrino energies [39]. For UHE neutrinos
produced in pγ interactions, their energy spectrum is de-
termined by the spectra of the parent protons and pho-
tons. Because the photon spectrum is typically peaked
3
1061071081091010
Neutrino energy, Eν[GeV]
10−11
10−10
10−9
10−8
All-flavor neutrino flux, E2
νΦν+¯
ν[GeV cm−2s−1sr−1]
IceCube HESE (7.5 yr) extrapolated
IceCube νµ(9.5 yr) extrapolated
Heinze et al., fit to Auger UHECRs
Bergman & van Vliet, fit to TA UHECRs
Rodrigues et al., all AGN
Rodrigues et al., all AGN
Rodrigues et al., HL BL Lacs
Fang & Murase, cosmic-ray reservoirs
Fang et al., newborn pulsars
Padovani et al., BL Lacs
Muzio et al., maximum extra pcomponent
Muzio et al., fit to Auger & IceCube
IceCube νextrapolated
Cosmogenic ν
Source ν
Cosmogenic + source ν
IceCube-Gen2 Radio energy range
Auger u.l.
1
2
3
4
5
67
8
910
11
12
1
2
3
4
5
6
7
8
9
10
11
12
IC-Gen2 Radio sens. (10 yr)
IceCube u.l.
FIG. 2. Benchmark diffuse ultra-high-energy neutrino flux models [9,10,36–43] used here to assess the flux discovery
capabilities of the radio array of IceCube-Gen2 [46] (“IceCube-Gen2 Radio”). These flux models are representative of the
breadth of theoretical predictions in the literature. The upper limits on the flux are from IceCube [25] and the Pierre Auger
Observatory [28]. The shaded region indicates the approximate neutrino energy range to which the radio array of IceCube-Gen2
will be sensitive. In this figure, fluxes are all-flavor, i.e., summed over all neutrino flavors, but our analysis treats individually
the flux of each neutrino species, νe,νµ,ντ, ¯νe, ¯νµ, and ¯ντ. See Fig. 6 in Ref. [51] for a breakdown of the flux of each neutrino
species for each flux model. See Section II for details.
around a characteristic energy, the resulting neutrino en-
ergy spectrum is also peaked, at an energy set by the
energy requirements to produce a ∆ resonance.
Figure 2shows the energy spectra of the benchmark
UHE neutrino flux models 1–12 [9,10,36–43] that we
use in our forecasts below. They span predictions from
optimistic to pessimistic. The wide variety in their size
and shape is indicative of the present-day spread of the
flux predictions available in the literature, and reflects
large extant uncertainties in the properties of UHECRs
and of their sources [13,19]. The benchmark flux models
in Fig. 2are the same ones that Ref. [51] used to fore-
cast the measurement of the UHE neutrino-nucleon cross
section. Below, we only sketch the main features of the
models; we defer to Ref. [51] for a detailed overview, and
to the original Refs. [9,10,36–43] for full details.
Our benchmark UHE neutrino flux models are grouped
in four classes, depending on the origin of the flux:
(a) UHE extrapolation of the IceCube neutrino
flux (“IceCube νextrapolated”, •models 1
and 2): These are unbroken extrapolations to ultra-
high-energies of the power-law (∝E−γ
ν) neutrino flux
measured by IceCube in the TeV–PeV range.
Flux model 1 (“IceCube HESE (7.5 yr) extrap-
olated”) extrapolates the soft-spectrum flux (γ=
2.87) of the IceCube 7.5-year HESE analysis [9].
Flux model 2 (“IceCube νµ(9.5 yr) extrapolated”)
extrapolates the hard-spectrum flux (γ= 2.37) of the
IceCube 9.5-year through-going νµanalysis [10].
[In our forecasts below, we consider flux models 1
or 2, augmented with a high-energy cut-off (Sec-
tion IV E 2), as a background to the discovery of
the other flux models, 3–12; see Section V A. Sec-
tion VB9 forecasts the discovery of flux models 1
and 2 themselves.]
(b) Models of cosmogenic neutrinos (“Cosmo-
genic ν”, models 3–5, 7): These are models of
cosmogenic neutrinos made either by a population of
nondescript sources of UHECRs, or by known classes
of potential UHECRs sources.
Flux model 3 [36] (“Heinze al., fit to Auger UHE-
CRs”) considers UHECRs produced by nondescript
sources, and fits their flux and mass composition to
recent UHECR observations by the Pierre Auger Ob-
servatory [62,63]. (References [34,35] predict similar
fluxes using similar procedures and data.)
Flux model 4 [42]) (“Bergman & van Vliet, fit to
TA UHECRs”) is produced similarly to flux model
3, but using instead recent UHECR observations by
the Telescope Array (TA) [64,65]. (Reference [66]
predicts a similar flux.) Flux model 3 is significantly
4
smaller than flux model 4 because Auger observations
favor a heavier UHECR mass composition at the
highest energies, and because the fit of the UHECR
spectrum to Auger data favors a lower cosmic-ray
maximum rigidity [34–36] than the fit to TA data.
Flux model 5 [41] (“Rodrigues et al., all AGN”)
is the cosmogenic neutrino flux expected from the
full population of active galactic nuclei (AGN), which
are taken to be UHECR accelerators, including low-
and high-luminosity BL Lacs and flat-spectrum radio
quasars. The resulting UHECR flux is fit to Auger
data [62], and the associated cosmogenic neutrino
flux satisfies the IceCube upper limit on the UHE
neutrino flux [25]. We adopt the maximum allowed
predicted cosmogenic neutrino flux from the entire
AGN population (Fig. 2 in Ref. [41]).
Flux model 7 [41] (“Rodrigues et al., HL BL Lacs”)
isolates the contribution of high-luminosity (HL) BL
Lacs to the cosmogenic neutrino flux of model 5.
(c) Models of UHE neutrinos made inside as-
trophysical sources (“Source ν”,
models
6, 9, 10): These are models based on more de-
tailed descriptions of the physical properties of known
UHECR and neutrino source classes.
Flux model 6 [41] (“Rodrigues et al., all AGN”) is
the counterpart source neutrino flux to the cosmo-
genic flux model 5. We adopt the maximum allowed
predicted source neutrino flux from the entire AGN
population (Fig. 2 in Ref. [41]).
Flux model 9 [37] (“Fang et al., newborn pulsars”)
is the neutrino flux predicted from newborn, fast-
spinning pulsars with intense surface magnetic fields
that may accelerate UHECRs in the pulsar wind.
UHECR pp interactions on the surrounding super-
nova ejecta produce neutrinos. We adopt the flux
prediction from Ref. [37] for which the number den-
sity of pulsars evolves with redshift following the star
formation rate. (We include only the contribution of
neutrinos made inside the pulsar environment.)
Flux model 10 [38] (“Padovani et al., BL Lacs”) is
the neutrino flux produced by pγ interactions inside
the jets of BL Lacs, computed within the framework
of the simplified view of blazars. Following Ref. [67],
the ratio of the neutrino intensity to the gamma-ray
intensity, a key parameter of the model [38], is set to
Yνγ = 0.13 to satisfy the present IceCube upper limit
on the UHE neutrino flux [25].
(d) Models of joint cosmogenic and UHE source
neutrinos (“Cosmogenic + source ν”,
mod-
els 8, 11, 12): These are multi-messenger models
that aim to explain the joint production of UHECRs
and TeV–EeV neutrinos.
Flux model 8 [39] (“Fang & Murase, cosmic-ray
reservoirs”) is the flux of UHE neutrinos produced,
via pp and pγ interactions, by UHECRs accelerated
in the jets of radio-loud AGN embedded in galaxy
clusters that act as cosmic-ray reservoirs, within a
grand-unified multi-messenger model. The predicted
UHECR flux and mass composition are fit to Auger
data [68] and the predicted TeV–PeV neutrino flux,
to IceCube data [8,69].
Flux model 11 [40] (“Muzio et al., maximum extra
pcomponent”) is the neutrino flux produced in pγ in-
teractions within the UFA15 multi-messenger frame-
work [70], where the UHECR flux and mass com-
position are fit to Auger data. The model includes
a sub-dominant UHECR pure-proton component be-
yond 109GeV that enhances the UHE neutrino flux.
We adopt the maximum allowed neutrino flux from
the joint single-mass UFA15 plus pure-proton compo-
nents, computed using the Sybill 2.3c [71] hadronic
interaction model (Fig. 9 in Ref. [40]).
Flux model 12 [43] (“Muzio et al., fit to Auger &
IceCube”) is the neutrino flux produced in pγ interac-
tions within the UFA15 multi-messenger framework,
and in pp interactions of UHECRs in the source en-
vironment. The UHECR flux and mass composition
are fit to Auger data, and the neutrino flux is fit
to the IceCube TeV–PeV neutrino flux [72,73]. We
adopt the best-fit total neutrino flux, “UHECR ν”
plus “Non-UHECR ν”, from Fig. 1 in Ref. [43]).
In each UHE neutrino flux model above, we treat indi-
vidually the flux of each neutrino species, νe,νµ,ντ, ¯νe,
¯νµ, ¯ντ. To compute the flavor composition at Earth, after
oscillations, we follow the same detailed prescription as
in Ref. [51], based on recent values of the neutrino mixing
parameters from the NuFit 5.0 [74,75] global fit to neu-
trino oscillation data. See Section IV E 2 for a sketch of
our prescription (in the particular context of flux models
1 and 2 as background fluxes) and Ref. [51] for full details
of the flavor composition of each flux model. We main-
tain the individual treatment of the flux of each neutrino
species during their propagation through the Earth (Sec-
tion III) and when computing their contribution to the
predicted event rate (Section IV). However, we conser-
vatively assume no capability to distinguish events made
by different flavors in the radio array of IceCube-Gen2.
III. PROPAGATING NEUTRINOS THROUGH
EARTH
Once UHE neutrinos arrive at the surface of the Earth,
they propagate underground toward the detector, from
all directions. Because the neutrino-matter cross section
grows with energy (see below), for UHE neutrinos in-
teractions with matter underground are significant, and
attenuate the flux of neutrinos that reaches the detector.
The attenuation is energy- and direction-dependent: the
higher the energy and the longer the distance traveled
by a flux of neutrinos inside the Earth, the stronger it is
attenuated. The attenuation is also flavor-dependent: ντ
5
are relatively less affected compared to νeand νµ. In our
forecasts, we account in detail for the in-Earth propaga-
tion of UHE neutrinos from the surface of the Earth to
the radio array of IceCube-Gen2. Below, we elaborate.
At neutrino energies above a few GeV, the leading neu-
trino interaction channel is neutrino-nucleon (νN ) deep
inelastic scattering (DIS) [76–78]. In it, a neutrino scat-
ters off of one of the partons, i.e., a quark or a gluon,
inside a nucleon, N,i.e., a proton or a neutron. The
final-state parton promptly hadronizes into final-state
hadrons, X. A neutral-current (NC) DIS interaction,
mediated by a Zboson, produces in addition a final-
state neutrino, i.e.,να+N→να+X(α=e, µ, τ ). A
charged-current (CC) DIS interaction, mediated by a W
boson, produces in addition instead a final-state charged
lepton, i.e.,να+N→lα+X. The νN DIS cross sec-
tion has been measured at sub-TeV neutrino energies by
a variety of accelerator neutrino experiments [79–96]; in
the few-TeV range, by FASER [97] (and the upcoming
FASERν[98]), and in the TeV–PeV range, using Ice-
Cube data [99–101]. At higher energies, the cross section
is predicted [102–111] and may be measured in upcoming
UHE neutrino telescopes [51,112–114].
Computing the UHE νN DIS cross section requires
knowing the parton distribution functions in protons and
neutrons, which are measured in lepton-hadron collisions,
and extrapolating them beyond the regime where they
have been measured. (Concretely, it requires extrapolat-
ing them to values of Bjorken-x—the fraction of nucleon
momentum carried by the interacting parton—beyond
the measured ones.) At ultra-high energies, the NC and
CC νN cross sections, σNC
νN and σCC
νN , respectively, grow
roughly ∝E0.36
ν[103], are essentially equal for all flavors
of ναand ¯να, and σNC
νN ≈σCC
νN /3. Below, to produce
our forecasts, we adopt the state-of-the-art BGR18 cal-
culation of the νN DIS cross sections [111] in the prop-
agation and detection of neutrinos. The BGR18 is built
using recent experimental results and sophisticated next-
to-leading-order calculations, including the major correc-
tions described in Appendix B4 of Ref. [111]; for details,
see Ref. [111] andRef. [115], for a summary, see Ref. [51].
In a DIS interaction, the final-state hadrons receive
a fraction y—the inelasticity—of the neutrino energy,
and the final-state lepton receives the remaining frac-
tion (1 −y). In each interaction, the value of yis ran-
domly sampled from a probability density that is propor-
tional to the differential DIS cross sections, dσNC
νN /dy and
dσCC
νN /dy. At the energies relevant for our work, the aver-
age value of yis about 0.25 [102]. However, because the
distribution of values of yhas a large spread (see Fig. 4
in Ref. [51]), when propagating neutrinos through the
Earth below (and also when computing the event rates
that they induce, in Section IV D), we do it by using the
distributions of y, separately for NC and CC DIS, rather
than by using its average value.
Inside the Earth, NC interactions shift the UHE neu-
trino flux to lower energies, by regenerating lower-energy
neutrinos, while CC interactions dampen the flux alto-
gether, by replacing neutrinos with charged leptons. The
one exception is the CC interaction of ντ: in them, the
final-state tauon may propagate for some distance inside
the Earth before decaying and generating a new, high-
energy ντ. As a result of this “ντregeneration,” the flux
of ντis less attenuated than that of νeand νµ.
The severity of the effects of in-Earth propagation on
the neutrino flux varies with neutrino energy, Eν, and
direction, expressed via the zenith angle, θz, measured
from the South Pole, where IceCube-Gen2 will be lo-
cated. Higher energies and directions corresponding to
longer path lengths inside the Earth yield more severe
effects. To illustrate this, we use a simplified calculation
of the number of neutrino-induced events in the detector,
Nsimp
ν, similar to the one in Ref. [100], i.e.,
Nsimp
ν(Eν, θz)∝Φν(Eν)σνN (Eν)e−L(θz)/LνN (Eν,θz),
(1)
where Φνis the neutrino flux at the surface of the Earth,
σνN is the νN cross section (for this simplified calcula-
tion, it is the sum of NC and CC cross sections), L(θz) =
qR2
⊕−2R⊕dcos2θz+ 2R⊕d−(R⊕−d) cos θzis the
distance traveled inside the Earth by a neutrino with in-
coming direction θz, where R⊕= 6371 km is the radius
of Earth, dis the detector depth, approximately 200 m
for the radio array of IceCube-Gen2, LνN ≡(σνN nN)−1
is the neutrino mean free path inside the Earth along this
direction, and nNis the average number density of nu-
cleons along this direction, based on knowledge of the in-
ternal matter density of Earth (more on this later). (We
use Eq. (1) only for illustration; later we describe the
detailed calculation with which we produce our results.)
Equation (1) accounts for flux attenuation during in-
Earth propagation, via the exponential dampening term,
but ignores the regeneration of lower-energy neutrinos.
Even so, it embodies essential features of the propaga-
tion and detection of high-energy and ultra-high-energy
neutrinos. Upgoing neutrinos (cos θz<0), i.e., neutrinos
that reach the detector from below after traveling under-
ground a distance of up to the diameter of the Earth, are
more strongly attenuated than downgoing (cos θz>0)
and horizontal neutrinos (cos θz≈0). For UHE neutri-
nos, the attenuation is so strong that virtually no upgoing
neutrinos reach the detector (see Fig. A2 in Ref. [100]),
unless the neutrino flux at the surface is extraordinarily
large; e.g., benchmark flux model 4 in Figs. 2,3, and 4.
This means that our forecasts below, which factor in the
contribution of neutrinos from all directions, are driven
primarily by downgoing and horizontal neutrinos.
Further, Eq. (1) shows that while flux attenuation is
∝e−σνN , the rate of neutrino interactions in the detec-
tor is ∝σνN . The interplay between these competing
effects is accentuated at high energies, where the cross
section is larger: a larger cross section makes the already
tiny flux of upgoing neutrinos vanish, which has little
marginal effect, but it appreciably increases the number
of downgoing and horizontal neutrinos detected.
Finally, Eq. (1) reveals important nuance in the rate
摘要：

Near-futurediscoveryofthedi useuxofultra-high-energycosmicneutrinosVctorB.ValeraandMauricioBustamanteyNielsBohrInternationalAcademy,NielsBohrInstitute,UniversityofCopenhagen,DK-2100Copenhagen,DenmarkChristianGlaserzDepartmentofPhysicsandAstronomy,UppsalaUniversity,Uppsala,SE-75237,Sweden(Dated:Ma...

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分类：图书资源 价格：10玖币 属性：43 页 大小：1.85MB 格式：PDF 时间：2025-05-02

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