# Neutrino Physics and Astrophysics

Francis<EMAIL_ADDRESS>Dept. of Physics and Wisconsin
IceCube Particle Physics Center
Univeristy of Wisconsin–Madison, Madison, WI 53706 USA  Ali
<EMAIL_ADDRESS>Dept. of Physics, Department of Astronomy
and Astrophysics, and Center for Multimessenger Astrophysics, Institute for
Gravitation and the Cosmos
The Pennsylvania State University, University Park, PA 16802, USA

###### Abstract

The IceCube experiment discovered PeV-energy neutrinos originating beyond our
Galaxy with an energy flux that is comparable to that of TeV-energy gamma rays
and EeV-energy cosmic rays. Neutrinos provide the only unobstructed view of
the cosmic accelerators that power the highest energy radiation reaching us
from the universe. We will review the rationale for building kilometer-scale
neutrino detectors that led to the IceCube project, which transformed a cubic
kilometer of deep transparent natural Antarctic ice into a neutrino telescope
of such a scale. We will summarize the results from the first decade of
operations: the status of the observations of cosmic neutrinos and of their
first identified source, the supermassive black hole TXS 0506+056.
Subsequently, we will introduce the phenomenology associated with cosmic
accelerators in some detail. Besides the search for the sources of Galactic
and extragalactic cosmic rays, the scientific missions of IceCube and similar
instruments under construction in the Mediterranean Sea and Lake Baikal
include the observation of Galactic supernova explosions, the search for dark
matter, and the study of neutrinos themselves. This review resulted from notes
created for summer school lectures and should be accessible to
nonexperts.333To be published in Neutrino Physics and Astrophysics, edited by
F. W. Stecker, in the Encyclopedia of Cosmology II, edited by G. G. Fazio,
World Scientific Publishing Company, Singapore, 2022.

##  Chapter 0 IceCube and High-Energy Cosmic Neutrinos

###### Contents

  1. 0 IceCube Neutrinos
    1. 1 Neutrino Astronomy: A Brief History
    2. 2 Rationale for the Construction of Kilometer-Scale Neutrino Detectors
      1. 1 Cosmic-Ray Accelerators
      2. 2 Neutrinos and Gamma Rays Associated with Cosmic Rays
    3. 3 IceCube
      1. 1 Detecting Very High Energy Neutrinos
      2. 2 The Neutrino Detector’s Telescope Area
      3. 3 Atmospheric Neutrinos: Calibration and Background
    4. 4 The Discovery of Cosmic Neutrinos
    5. 5 Multimessenger Astronomy: General Considerations
      1. 1 IceCube Neutrinos and Fermi Photons
      2. 2 IceCube Neutrinos and Ultra-High-Energy Cosmic Rays
      3. 3 Pinpointing the Astrophysical Sources of Cosmic Neutrinos: Relating the Point Source and Diffuse Fluxes
    6. 6 The First-Identified Cosmic-Ray Accelerator: the Rotating Supermassive Black Hole TXS 0506+056
    7. 7 Modeling Cosmic Beam Dumps
      1. 1 Active Galaxies: A Worked Example
      2. 2 Neutrino Flux Associated with the Plasma Blobs in the Jets of Active Galaxies
      3. 3 An Alternative Approach for the Modeling of Neutrino Production by Proton Beams on External Photons
      4. 4 Fireballs and GRB
      5. 5 The “Guaranteed” Cosmogenic Neutrinos
    8. 8 Galactic Sources
      1. 1 Galactic Neutrino-Producing Beam Dumps: Supernova Remnants
      2. 2 Identifying Galactic Sources of High-Energy Neutrinos
      3. 3 Diffuse Galactic Neutrino Emission
      4. 4 Closing In on Galactic Sources of High-Energy Neutrinos
    9. 9 Beyond Astronomy
      1. 1 IceCube, the Facility
      2. 2 From Discovery to Astronomy, and More…
    10. 10 Acknowledgements

###  1 Neutrino Astronomy: A Brief History

Soon after the 1956 observation of the neutrino [1], the idea emerged that it
represented the ideal astronomical messenger. Neutrinos travel from the edge
of the Universe without absorption and with no deflection by magnetic fields.
Having essentially no mass and no electric charge, the neutrino is similar to
the photon, except for one important attribute: its interactions with matter
are extremely feeble. So, high-energy neutrinos may reach us unscathed from
cosmic distances: from the inner neighborhood of black holes and from the
nuclear furnaces where cosmic rays are born. But, their weak interactions also
make cosmic neutrinos very difficult to detect. Immense particle detectors are
required to collect cosmic neutrinos in statistically significant numbers [2].
By the 1970s, it was clear that a cubic-kilometer detector was needed to
observe cosmic neutrinos produced in the interactions of cosmic rays with
background microwave photons [3]. Subsequent estimates for observing potential
cosmic accelerators such as Galactic supernova remnants and gamma-ray bursts
unfortunately pointed to the same exigent requirement [4, 5, 6]. Building a
neutrino telescope has been a daunting technical challenge.

Given the detector’s required size, early efforts concentrated on transforming
large volumes of natural water into Cherenkov detectors that collect the light
produced when neutrinos interact with nuclei in or near the detector [7].
After a two-decade-long effort, building the Deep Underwater Muon and Neutrino
Detector (DUMAND) in the sea off the main island of Hawaii unfortunately
failed [8]. However, DUMAND paved the way for later efforts by pioneering many
of the detector technologies in use today, and by inspiring the deployment of
a smaller instrument in Lake Baikal [9] as well as efforts to commission
neutrino telescopes in the Mediterranean [10, 11, 12]. These efforts in turn
have led towards the construction of KM3NeT. But the first telescope on the
scale envisaged by the DUMAND collaboration was realized instead by
transforming a large volume of transparent natural Antarctic ice into a
particle detector, the Antarctic Muon and Neutrino Detector Array (AMANDA). In
operation beginning in 2000, it represented a proof of concept for the
kilometer-scale neutrino observatory, IceCube [13, 14].

Neutrino astronomy has achieved spectacular successes in the past: neutrino
detectors have “seen” the Sun and detected a supernova in the Large Magellanic
Cloud in 1987. Both observations were of tremendous importance; the former
showed that neutrinos have mass, opening the first crack in the Standard Model
of particle physics, and the latter confirmed the basic nuclear physics of the
death of stars. Fig. 1 illustrates the neutrino energy spectrum covering an
enormous range, from microwave energies ($10^{-12}$ eV) to $10^{20}$ eV [15].
The figure is a mixture of observations and theoretical predictions. At low
energy, the neutrino sky is dominated by neutrinos produced in the Big Bang.
Nuclear fusion in the sun generates neutrinos with keV energy and a spectrum
that extends to MeV. At MeV energy, neutrinos are produced by supernova
explosions; the flux from the 1987 event is shown. At yet higher energies, the
figure displays the atmospheric-neutrino flux, up to energies of 100 TeV which
were measured at the Frejus underground laboratory [16] AMANDA experiment
[17]. Atmospheric neutrinos are a main player in our story, because they are a
dominant background for extraterrestrial searches. The flux of atmospheric
neutrinos falls with increasing energy; events above 100 TeV are rare, leaving
eventually a clear field of view for extraterrestrial sources at the highest
energies.

Figure 1: The cosmic-neutrino spectrum. Sources are the Big Bang (C$\nu$B),
the Sun, supernovae (SN), atmospheric neutrinos, active galactic nuclei (AGN)
galaxies, and GZK neutrinos.

The highest energy neutrinos in Fig. 1 are the decay products of pions
produced by the interactions of cosmic rays with microwave photons [18]. Above
a threshold of $\sim 4\times 10^{19}$ eV, cosmic rays interact with the
microwave background introducing an absorption feature in the cosmic-ray flux,
the Greisen-Zatsepin-Kuzmin (GZK) cutoff. As a consequence, the mean free path
of extragalactic cosmic rays propagating in the microwave background is
limited to roughly 75 megaparsecs, and, therefore, the secondary neutrinos are
the only probe of the still enigmatic sources at longer distances. What they
will reveal is a matter of speculation. The calculation of the neutrino flux
associated with the observed flux of extragalactic cosmic rays is
straightforward and yields one event per year in a kilometer-scale detector.
The flux, labeled GZK in Fig. 1, shares the high-energy neutrino sky with
neutrinos anticipated from gamma-ray bursts and active galactic nuclei (AGN)
[19, 4, 5, 6].

A population of extragalactic cosmic neutrinos with energies of 60 TeV–1 PeV
was revealed by the first two years of IceCube data. We will review the
present status of the observations, the identification of the sources by
multimessenger astronomy, as well as attempts to decipher the phenomenology of
the heavenly beam dumps producing neutrinos. Subsequently, we will describe
the status of the search for Galactic neutrino sources and conclude with brief
discussions of other uses of neutrino telescopes.

###  2 Rationale for the Construction of Kilometer-Scale Neutrino Detectors

The construction of kilometer-scale neutrino detectors was primarily motivated
by the prospect of detecting neutrinos associated with the sources of high-
energy cosmic rays. Cosmic accelerators produce particles with energies in
excess of $100$ EeV; we still do not know where or how [20]; see Fig. 2111We
will use energy units TeV, PeV and EeV, increasing by factors of 1000 from GeV
energy.. The bulk of the cosmic rays are Galactic in origin. Any association
with our Galaxy presumably disappears at EeV energy when the gyroradius of a
proton in the Galactic magnetic field exceeds its size. The cosmic-ray
spectrum exhibits a rich structure above an energy of a few PeV, the so-called
“knee” in the spectrum, but where exactly the transition to extragalactic
cosmic rays occurs is a matter of debate.

Figure 2: At the energies of interest here, the cosmic-ray spectrum follows a
sequence of three power laws. The first two are separated by the “knee,” the
second and third by the “ankle.” Cosmic rays beyond the ankle are a new
population of particles produced in extragalactic sources. Note that the
spectrum $F(E)(=dN/dE)$ has been multiplied by a power $E^{2.7}$ in order to
visually enhance the structure in the spectrum. Figure from Particle Data
Group [21].

####  1 Cosmic-Ray Accelerators

The detailed blueprint for a cosmic-ray accelerator must meet two challenges:
the highest-energy particles in the beam must reach energies beyond $10^{3}$
TeV ($10^{8}$ TeV) for Galactic (extragalactic) sources and their luminosities
must accommodate the observed flux. Both requirements represent severe
constraints that have guided theoretical speculations. Acceleration of protons
(or nuclei) to TeV energy and above requires massive bulk flows of
relativistic charged particles. Instead of accelerating protons, cosmic
accelerators can boost really large masses to relativistic velocities. The
radio emission reveals that the plasma in the jets of active galaxies flows
with velocities of $0.99\,c$. A fraction of a solar mass per year can be
accelerated to relativistic Lorentz factors of order 10 leading to
luminosities of $10^{46}$ erg/s close to the Eddington limit. In the collapse
of very massive stars $10^{51}\sim 10^{52}$ erg/s is released in a fireball
that expands with velocities of $0.999\,c$.

The blueprint of the accelerator can be copied from solar flares where
particles are accelerated to GeV energy by shocks and, possibly, magnetic
reconnection; see Fig. 3. Requiring that the gyroradius of the accelerated
particle be contained within the accelerating B-field region, $E/ZevB\leq R$,
leads to an upper limit on the energy of the particle, the Hillas [22] formula

$E\leq Ze\,v\,B\,R\,.$ (1)

Reaching energies much above 10 GeV in solar flares is dimensionally
impossible. In a solar flare, the extent $R$ of the accelerating region and
the magnitude of the magnetic fields $B$ are not large enough to accelerate
particles of charge $Ze$ to energies beyond GeV even if their velocity is
taken to be the speed of light, $c$. Another way to view the dimensional
argument is by estimating the energy of a particle from the Lorentz force

$E=-e\frac{d\Phi}{dt}=e\,\pi R^{2}\,\frac{dB}{dt}\simeq
1\rm{GeV}\big{(}\frac{R}{10^{4}\rm\,km}\big{)}^{2}\big{(}\frac{\Delta
B}{10^{3}\,\rm G})\big{(}\frac{10^{5}\rm\,s}{\Delta t}\big{)},$ (2)

where $\Phi$ is the magnetic flux set up in the loops of gyrating particles in
Fig. 3 that can reach several thousand Gauss in a time of order one day. The
result follows from the loops have radii of more than $10^{4}$ km. In the
spirit of dimensional analysis the two estimates above are the same by
identifying the velocity $v$ with $dR/dt$.

While it is not a challenge to find astronomical sources with larger B and R,
the other challenge is that the luminosity of the cosmic ray sources is large
as well, and here a central idea for accommodating the high luminosities of
the Galactic and extragalactic cosmic rays observed is that a fraction of the
gravitational energy released in a stellar collapse is converted into particle
acceleration, presumably by shocks.

Figure 3: Opportunities exist near intense charged particle flows, seen as
filaments in this X-ray picture of a solar flare, for solar particles to
accelerate to GeV energy. Figure 4: This X-ray picture of the supernova
remnant CasA reveals strong particle flows near its periphery. We believe they
are the site for accelerating Galactic cosmic rays to energies reaching the
“knee” in the spectrum. Figure 5: Colliding shocks in the simulation of a
gamma-ray burst (GRB) fireball may accelerate cosmic rays to the highest
energies observed. The filaments in the particle flow are directed along the
rotation axis of the black hole. Animated view at
http://www.nasa.gov/centers/goddard/news/topstory/2003/0618rosettaburst.html.

Baade and Zwicky [23] suggested as early as 1934 that supernova remnants could
be sources of the Galactic cosmic rays. It is assumed that, after the
collapse, $\sim\\!\\!10^{51}$ erg of energy is transformed into particle
acceleration by diffusive shocks associated with young ($\sim\\!\\!1000$ year
old) supernova remnants expanding into the interstellar medium. Like a
snowplow, the shock sweeps up the $\sim\\!\\!\,1\,\rm proton/cm^{3}$ density
of hydrogen in the Galactic plane. The accumulation of dense filaments of
particles in the outer reaches of the shock, clearly visible as sources of
intense X-ray emission, are the sites of high magnetic fields; see Fig. 4. It
is theorized that particles crossing these structures multiple times can be
accelerated to high energies following an approximate power-law spectrum
$dN/dE\sim\\!\\!E^{-2}$. The mechanism copies solar flares where filaments of
high magnetic fields, visible in Fig. 3, are the sites for accelerating
nuclear particles to tens of GeV. The higher energies reached in supernova
remnants are the consequence of particle flows of much larger intensity
powered by the gravitational energy released in the stellar collapse.

This idea has been widely accepted despite the fact that to date no source has
been conclusively identified, neither by cosmic rays nor by accompanying gamma
rays and neutrinos produced when the cosmic rays interact with Galactic
hydrogen. Galactic cosmic rays reach energies of at least several PeV, the
“knee” in the spectrum; therefore, their interactions should generate gamma
rays and neutrinos from the decay of secondary pions reaching hundreds of TeV.
Such sources, referred to as PeVatrons, have not been found; see, however,
reference 24. Nevertheless, Zwicky’s suggestion has become the stuff of
textbooks, and the reason is energetics: three Galactic supernova explosions
per century converting a reasonable fraction of a solar mass into particle
acceleration can accommodate the steady flux of cosmic rays in the Galaxy. It
is interesting to note that Zwicky originally assumed that the sources were
extragalactic since the most recent supernova in the Milky Way was in 1572.
After diffusion in the interstellar medium was understood, supernova
explosions in the Milky Way became the source of choice for the origin of
Galactic cosmic rays [25], although after more than 50 years the issue is
still debated [26].

Energetics also guides speculations on the origin of extragalactic cosmic
rays. By integrating the cosmic-ray spectrum above the ankle at $\sim 4$ EeV,
it is possible to estimate [27] the energy density in extragalactic cosmic
rays as $\sim 3\times 10^{-19}\rm\,erg\ cm^{-3}$. This value is rather
uncertain because of our ignorance of the energy where the transition from
Galactic to extragalactic sources occurs. The power required for a population
of sources to generate this energy density over the Hubble time of $10^{10}$
years is $2\times 10^{37}\,\rm\,erg\ s^{-1}$ per Mpc3. Long-duration gamma-ray
bursts have been associated with the collapse of massive stars to black holes,
and not to neutron stars, as is the case in a collapse powering a supernova
remnant. A gamma-ray-burst fireball converts a fraction of a solar mass into
the acceleration of electrons, seen as synchrotron photons. The observed
energy in extragalactic cosmic rays can be accommodated with the reasonable
assumption that shocks in the expanding gamma-ray burst (GRB) fireball convert
roughly equal energy into the acceleration of electrons and cosmic rays [28];
see Fig. 5. It so happens that $2\times 10^{51}$ erg per GRB will yield the
observed energy density in cosmic rays after $10^{10}$ years, given that their
rate is on the order of 300 per $\textrm{Gpc}^{3}$ per year. Hundreds of
bursts per year over a Hubble time produce the observed cosmic-ray density,
just as three supernovae per century accommodate the steady flux in the
Galaxy.

Figure 6: Anticipated astrophysical neutrino fluxes compared with measured and
calculated fluxes of atmospheric neutrinos. Measurements of $\nu_{\mu}$ from
Super-K [29], Frejus [30], AMANDA, [31, 32] and IceCube [33, 34] are shown
along with the electron-neutrino spectrum at high energy from reference [35]
(green open triangles). Calculations of conventional $\nu_{e}$ (red line) and
$\nu_{\mu}$ (blue line) from Honda et al. [36], $\nu_{e}$ (red dotted line)
from Bartol, [37] and charm-induced neutrinos (magenta band) [38] are also
shown.

Problem solved? Not really: it turns out that the same result can be achieved
assuming that active galactic nuclei convert, on average, $2\times
10^{44}\,\rm\,erg\ s^{-1}$ each into particle acceleration [15]. This is an
amount that matches their output in electromagnetic radiation.

In contrast with our own Galaxy where the black hole is mostly dormant, in an
active galaxy the supermassive black hole is absorbing the matter in its host
galaxy at a very high rate. An active galactic nucleus (AGN) hosts a rotating
supermassive black hole. Fast spinning matter falling onto it swirls around
the black hole in an accretion disk, like the water approaching the drain of
your bath tub. When the accretion disk comes in contact with the rotating
black hole space-time drags on the magnetic field winding it into a tight cone
around the rotation axis into a jet of particles; see Fig. 7. Not just
particles but huge “blobs” of plasma from the accretion disk are flung out
along these field lines. It is not clear whether it is the rotation energy of
the black hole or the magnetic energy in the rotating plasma that powers the
accelerator. When this jet runs into a target material, for instance the
ubiquitous 10 eV ultraviolet photons in some galaxies, neutrinos can be
produced.

Active galaxies are actually complex systems with many possible sites for
accelerating cosmic rays and for targets to produce neutrinos. Acceleration of
particles may occur at the spectacular termination shocks of the jets in
intergalactic space [39] at distances of hundreds of Mpc from the center where
there would be little target material available and few neutrinos produced. In
contrast, production of neutrinos near the black hole [40], or in collisions
with interstellar matter of the accelerated particles diffusing in the
magnetic field of the galaxy hosting the black hole [41], could yield fluxes
at the level observed. We will work through these examples further on.

Figure 7: The accretion disk meets the spinning black holes that winds up the
disk’s magnetic field lines. Credit:
http://www.physics.ucc.ie/gabuzda/Nadia.html.

####  2 Neutrinos and Gamma Rays Associated with Cosmic Rays

Neutrinos must be produced at some level in association with the cosmic-ray
beam. Cosmic rays accelerated in regions of high magnetic fields near black
holes or neutron stars inevitably interact with the matter or radiation
surrounding them. Thus, cosmic-ray accelerators are also part of a “beam
dump”, like the ones producing neutrino beams in accelerator laboratories: the
beam is dumped is a target where it produces pions and kaons that decay into
neutrinos. Dense targets absorb all secondary particles except for the
neutrinos. This is typically not the case for a cosmic beam dump where
neutrinos are expected to be accompanied by other stable particles: protons,
neutrons and photons. For example, cosmic rays accelerated in supernova shocks
interact with gas in the Galactic disk, producing equal numbers of pions of
all three charges that decay into pionic photons and neutrinos. A larger
source of secondaries is likely to be produced by the interaction of
accelerated particles with the gas near the sources, for example cosmic rays
interacting with high-density molecular clouds that are ubiquitous in the
star-forming regions where supernovae are more likely to explode. For
extragalactic sources, the neutrino-producing target may be electromagnetic,
for instance photons radiated by the accretion disk of an AGN, or synchrotron
photons that coexist with protons in the expanding fireball producing a GRB.

How many neutrinos and, inevitably, gamma rays are produced in association
with the cosmic-ray beam? A Galactic supernova shock is an example of a
hadronic beam dump. Cosmic rays interact with the hydrogen in the Galactic
disk, producing equal numbers of pions of all three charges in hadronic
collisions $p+p\rightarrow n_{\pi}\,[\,\pi^{0}+\pi^{+}+\pi^{-}]+X$; $n_{\pi}$
is the pion multiplicity. In the case of a photon target, neutral and charged
pion secondaries are produced by the photoproduction processes

$p+\gamma\rightarrow\Delta^{+}\rightarrow\pi^{0}+p\mbox{ \ and \
}p+\gamma\rightarrow\Delta^{+}\rightarrow\pi^{+}+n.$ (3)

Only the neutrinos and neutrons will escape the source. While secondary
protons may remain trapped and loose energy in the high magnetic fields of the
accelerator, neutrons will decay and the decay products escape the dump with
high energy. The energy escaping the source is therefore distributed among
cosmic rays, gamma rays and neutrinos, particles produced by the decay of
neutrons, neutral pions and charged pions, respectively. Photoproduction
produces charged and neutral pions according to Eq. 3, with probabilities of
2/3 and 1/3, respectively. Subsequently, the pions decay into gamma rays and
neutrinos that carry, on average, 1/2 and 1/4 of the energy of the parent
pion. It is a good approximation to assume that, on average, the four leptons
in the decay
$\pi^{+}\rightarrow\nu_{\mu}+\mu^{+}\rightarrow\nu_{\mu}+\left(e^{+}+{\nu}_{e}+\bar{\nu}_{\mu}\right)$
equally share the charged pion’s energy. The energy of the pionic leptons
relative to the proton is:

$x_{\nu}=\frac{E_{\nu}}{E_{p}}=\frac{1}{4}x_{\pi}\;\simeq\,\frac{1}{20}$ (4)

and

$x_{\gamma}=\frac{E_{\gamma}}{E_{p}}=\frac{1}{2}x_{\pi}\;\simeq\,\frac{1}{10}.$
(5)

Here,

$x_{\pi}\,=\,\langle\frac{E_{\pi}}{E_{p}}\rangle\;\simeq\,0.2$ (6)

is the average energy transferred from the proton to the pion that produces
the neutrino [42].

Interestingly, these relations are approximately valid whether the pions are
produced in $pp$ or $p\gamma$ interactions. Pion production is usually
described in terms of their average multiplicity $n_{\pi}$ and a pion’s
average energy $E_{\pi}$ in the final state. After interaction, the initial
state energy is distributed between the proton and the production of pions.
The fraction of energy going into the production of pions is referred to as
the inelasticity $\kappa$ which is $0.5(0.2)$ for $pp(p\gamma)$ interactions.
The fraction of energy going into a single pion $x_{\pi}$ is

$x_{\pi}\,=\,\frac{\kappa}{n_{\pi}}\simeq 0.2.$ (7)

It turns out that $x_{\pi}\simeq 0.2$ yields the pion energy for both $pp$ and
$p\gamma$ interactions, despite the very different particle processes. For
photoproduction, the charged pion takes 0.2 of the initial energy and the pion
multiplicity is 1.

While both gamma-ray and neutrino fluxes can be calculated knowing the
luminosity of the accelerated protons and the density of the target material,
their relative flux is independent of the details of the production mechanism.
Their production rates ${\rm d}N/{\rm d}E{\rm d}t$ of neutrinos and gamma rays
are related by known particle physics. The above discussion can be summarized
as:

$\frac{1}{3}\sum_{\nu_{\alpha}}E_{\nu}\frac{{\rm d}N_{\nu}}{{\rm d}E_{\nu}{\rm
d}t}(E_{\nu})\simeq\frac{K_{\pi}}{2}E_{\gamma}\frac{{\rm d}N_{\gamma}}{{\rm
d}E_{\gamma}{\rm d}t}(E_{\gamma})\,.$ (8)

Here, $N$ and $E$ denote the number and energy of neutrinos and gamma rays and
$\nu_{\alpha}$ stands for the neutrino flavor. Note that this relation is
solid and depends only on the charged-to-neutral secondary pion ratio, with
$K_{\pi}=1(2)$ for $p\gamma$($pp$) neutrino-producing interactions. In
deriving the relative number of neutrinos and gamma rays, one must be aware of
the fact that the neutrino flux represents the sum of the neutrinos and
antineutrinos, which cannot be separated by current experiments: in short, a
$\pi^{0}$ produces two $\gamma$ rays for every charged pion producing a
$\nu_{\mu}+\bar{\nu}_{\mu}$ pair. A more detailed discussion of this relation
will follow in the context of multimessenger astronomy; see Section 7.

The production rate of gamma rays at their origin described by Eq. 8 is not
necessarily the emission rate observed. For instance, in cosmic accelerators
that efficiently produce neutrinos via $p\gamma$ interactions, the target
photon field can also efficiently reduce the energy of the pionic gamma rays
produced via pair production. Gamma rays with energies above the threshold for
pair production will lose energy in the source. Their maximum energy, in the
comoving frame, is determined by the energy of the target photons:

$E_{\gamma}^{\prime\rm max}E^{\prime}_{\rm ph}<m^{2}_{e}.$ (9)

This is a calorimetric process that will, however, conserve the total energy
of hadronic gamma rays. The production of photons in association with cosmic
neutrinos is inevitable but unlike neutrinos, photons may reach Earth with
reduced energy after losing energy in the target and after propagation in the
universal microwave and infrared photon backgrounds. However, their cascaded
energy must appear in some electromagnetic wave band because of energy
conservation. Furthermore, one must be aware of the fact that inverse-Compton
scattering and synchrotron emission by accelerated electrons in magnetic
fields in the source have the potential to produce gamma rays; not every high-
energy gamma ray is pionic.

A couple of decades of modeling potential neutrino sources yielded generic
predictions shown in Fig. 6, estimates of astrophysical neutrino fluxes are
compared with measurements of atmospheric neutrinos. While the models varied,
the result were typically dictated by the relation between the gamma ray and
neutrino fluxes discussed above. The shaded band indicates the level of model-
dependent expectations for high-energy neutrinos of astrophysical origin. The
estimates that we will discuss in more detail further on predicted a neutrino
flux at a level of

$E_{\nu}^{2}\,{\rm d}N_{\nu}/{\rm d}E_{\nu}\,{\simeq}\,10^{-8}\,{\rm
GeV\,cm}^{-2}{\rm s}^{-1}{\rm sr}^{-1}$ (10)

per flavor. The figure illustrates the rationale for building a kilometer-
scale detector because it yields about 100 neutrino events per year in a cubic
kilometer detector. We now know that this is indeed the magnitude of the
cosmic component of the neutrino spectrum above 100 TeV revealed by IceCube’s
data although we have found no evidence for the specific sources shown in the
picture! The ongoing search by IceCube for neutrinos in coincidence with and
in the direction of GRB alerts issued by astronomical telescopes has limited
the GRB neutrino flux to less that 1% of the diffuse cosmic neutrino flux
actually observed by the experiment [43, 44]. However, this may not
conclusively rule out GRBs as a source of cosmic rays; the events that produce
the spectacular photon displays catalogued by astronomers as GRBs may not be
the stellar collapses that are sources of high-energy neutrinos. We will
return to this point further on when we discuss acceleration of cosmic rays in
GRB fireballs.

The failure of IceCube to observe neutrinos from GRBs has lately promoted AGNs
as the best-bet source of the cosmic neutrinos observed. Here again Gaisser
[45, 46] has emphasized the relation between IceCube limits and the
electromagnetic energy of the sources; see Fig. 8 [47]. In this context, we
introduce Fig. 8 [47] showing IceCube upper limits [48] on the neutrino flux
from nearby AGNs as a function of their distance. The sources at red shifts
between 0.03 and 0.2 are Northern Hemisphere blazars for which distances and
intensities are listed in TeVCat [49] and for which IceCube also has upper
limits. In several cases, the muon-neutrino limits have reached the level of
the TeV photon flux. One can sum the sources shown in the figure into a
diffuse flux. The result, after accounting for the distances and luminosities,
is $3\times 10^{-9}\,\rm GeV\,cm^{-2}\,s^{-1}\,sr^{-1}$, or approximately
$10^{-8}\,\rm GeV\,cm^{-2}\,s^{-1}\,sr^{-1}$ for all neutrino flavors. This is
at the level of the generic astrophysical neutrino flux of Eq. 10. At this
intensity, neutrinos from theorized cosmic-ray accelerators will cross the
steeply falling atmospheric neutrino flux above an energy of $\sim 300$ TeV;
see Fig. 6. The level of events observed in a cubic-kilometer neutrino
detector is $10\sim 100$ $\nu_{\mu}$-induced events per year. Such estimates
reinforce the logic for building a cubic kilometer neutrino detector [50].

Figure 8: Limits on the neutrino flux from selected active galaxies derived
from IceCube data taken during construction, when the instrument was operating
with 40 and 59 strings of the total 86 instrumented strings of DOMs [48].
These are compared with the TeV photon flux for nearby AGNs. Note that energy
units are in ergs, not TeV.

###  3 IceCube

####  1 Detecting Very High Energy Neutrinos

Cosmic rays have been studied for more than a century. They reach energies in
excess of $10^{8}$ TeV, populating the extreme universe that is opaque to
photons because they interact with the background radiation fields, mostly
microwave photons, before reaching Earth. We don’t yet know where or how
cosmic rays are accelerated to these extreme energies, and with the recent
observation of a the rotating supermassive black hole TXS 0506+056 in
coincidence with the direction and time of a very high energy muon neutrino,
neutrino astronomy might have taken a first step in solving this puzzle [51,
52]. The rationale is however simple: near neutron stars and black holes,
gravitational energy released in the accretion of matter or binary mergers can
power the acceleration of protons or heavier nuclei that subsequently interact
with gas (“$pp$”) or ambient radiation (“$p\gamma$”). Neutrinos are produced
by cosmic-ray interactions at various epochs: in their sources during their
acceleration, in the source environment after their release, and while
propagating through universal radiation backgrounds from the source to Earth.

Because of their weak interactions, high-energy neutrinos will reach our
detectors without deflection or absorption. They essentially act like photons;
their small mass is negligible relative to the TeV to EeV energies targeted by
neutrino telescopes. They do however oscillate over cosmic distances. For
instance, for an initial neutrino flavor ratio of
$\nu_{e}:\nu_{\mu}:\nu_{\tau}\simeq 1:2:0$ from the decay of pions and muons,
the oscillation-averaged composition arriving at the detector is approximately
an equal mix of electron, muon, and tau neutrino flavors,
$\nu_{e}:\nu_{\mu}:\nu_{\tau}\simeq 1:1:1$ [53].

Figure 9: The concept of a neutrino telescope from the point of view of
IceCube located at the South Pole. Muon neutrinos interact with the nuclei of
atoms in a transparent medium like water or ice and produce a secondary muon
track that is detected by the wake of Cherenkov photons it leaves inside the
detector. The background of high-energy muons (solid blue arrows) produced in
the atmosphere can be reduced by placing the detector underground. The
surviving fraction of muons is further reduced by looking for upgoing muon
tracks that originate from muon neutrinos that have traversed the Earth
(dashed blue arrows) interacting close to the detector. This still leaves the
contribution of muons generated by atmospheric muon neutrino interactions.
This contribution can be separated from the diffuse cosmic neutrino emission
by an analysis of the combined neutrino spectrum.

High-energy neutrinos interact predominantly with matter via deep inelastic
scattering off nucleons: the neutrino scatters off quarks in the target
nucleus by the exchange of a $Z$ or $W$ weak boson, referred to as neutral
current (NC) and charged current (CC) interactions, respectively. Whereas the
NC interaction leaves the neutrino state intact, in a CC interaction a charged
lepton is produced that shares the initial neutrino flavor. The average
relative energy fraction transferred from the neutrino to the lepton is at the
level of $80$% at high energies. The inelastic CC cross section on protons is
at the level of $10^{-33}~{}{\rm cm}^{2}$ at a neutrino energy of $10^{3}$ TeV
and grows with neutrino energy as $\sigma_{\rm tot}\propto E_{\nu}^{0.36}$
[54, 55]. The struck nucleus does not remain intact and its high-energy
fragments typically initiate hadronic showers in the target medium.

Immense particle detectors are required to collect cosmic neutrinos in
statistically significant numbers. Already by the 1970s, it had been
understood [3] that a kilometer-scale detector was needed to observe the
cosmogenic neutrinos produced in the interactions of cosmic rays with
background microwave photons [56, 57]. A variety of methods are used to detect
the high-energy secondary particles created in CC and NC neutrino
interactions. One particularly effective method observes the radiation of
optical Cherenkov light radiated by secondary charged particles produced in CC
and NC interactions that travel faster than the speed of light in the medium.

The detection concept is that of a conventional Cherenkov detector, a
transparent medium is instrumented with photomultipliers that transform the
Cherenkov light into electrical signals by the photoelectric effect; see Figs.
9 and 10. IceCube consists of 80 strings, each instrumented with 60 10-inch
photomultipliers spaced by 17 m over a total length of 1 kilometer. The
deepest module is located at a depth of 2.450 km so that the instrument is
shielded from the large background of cosmic rays at the surface by
approximately 1.5 km of ice. Strings are arranged at apexes of equilateral
triangles that are 125 m on a side. The instrumented detector volume is a
cubic kilometer of dark, highly transparent and sterile Antarctic ice. The
radioactive background in the detector is dominated by the instrumentation
deployed into this natural ice.

Each optical sensor consists of a glass sphere containing the photomultiplier
and the electronics board that captures and digitizes the signals locally
using an on-board computer. The digitized signals are given a global time
stamp with residuals accurate to less than 3 ns and are subsequently
transmitted to the surface. Processors at the surface continuously collect the
time-stamped signals from the optical modules, each of which functions
independently. The digital messages are sent to a string processor and a
global event trigger. They are subsequently sorted into the Cherenkov patterns
emitted by secondary muon tracks, or particle showers for electron and tau
neutrinos, that reveal the flavor, energy and direction of the incident
neutrino [58].

Figure 10: Sketch of the IceCube observatory (left) and the digital optical
module (right).

There are two principle classes of Cherenkov events that must be separated by
the detector, “tracks” and “cascades”. The two basic topologies are
illustrated in Fig. 11: tracks initiated by $\nu_{\mu}$ neutrinos and cascades
from $\nu_{e}$, $\nu_{\tau}$ neutrinos as well as the neutral current
interactions from all flavors. On the scale of IceCube, PeV cascades, with a
length of less than 10 m, are therefore essentially point sources of Cherenkov
light in a detector of kilometer size.

Figure 11: Contrasting Cherenkov light patterns produced by muons (left) and
by showers initiated by electron and tau neutrinos (right) and by neutral
current interactions. The patterns are referred to as tracks and cascades (or
showers). Cascades are produced by the radiation of particle showers, whose
dimensions are less than 10 m, i.e., essentially a point source of light
relative to the dimensions of the detector.

The term “tracks” refers to the Cherenkov emission of long-lived muons
produced in CC interactions of muon neutrinos interacting inside or in the
vicinity of the detector. Energetic electrons and taus produced in CC
interactions of electron and tau neutrino interactions, respectively, will not
produce elongated tracks: electrons will initiate electromagnetic showers in
the ice and, because of the relatively short lifetime, taus will decay and
also produce a shower in the ice. Because of the large background of muons
produced by cosmic ray interactions in the atmosphere, the observation of muon
neutrinos is typically limited to upgoing muon tracks that are produced in
interactions inside or close to the detector by neutrinos that have passed
through the Earth as illustrated in Fig. 9. The remaining background consists
of atmospheric neutrinos, which are indistinguishable from cosmic neutrinos on
an event-by-event basis. However, the steeply falling spectrum ($\propto
E^{-3.7}$) of atmospheric neutrinos allows identifying diffuse astrophysical
neutrino by a spectral analysis as we will highlight in the following
sections. Above $\sim 300$ TeV atmospheric neutrinos are relatively rare, even
in a cubic kilometer detector, and every neutrino exceeding this energy is
likely to be of cosmic origin. The atmospheric background is further reduced
when looking in the specific direction of a point source, in particular
transient neutrino sources like GRB.

The hadronic particle shower generated by the target struck by a neutrino also
radiates Cherenkov photons. Because of the large multiplicity of secondary
particles at these energies and the repeated scattering of the Cherenkov
photons in the ice, the light pattern is essential spherical tens of meters
from its point of origin. The light patterns produced by the particle showers
initiated by the electron or tau produced in CC interactions of electron or
tau neutrinos, respectively, will be superimposed on the hadronic cascade. The
two are not typically not separated. The direction of the initial neutrino can
only be reconstructed from the Cherenkov emission of secondary particles
produced close to the neutrino interaction point, and the angular resolution
is inferior to that for track events.

In contrast, the energy resolution of the neutrino is superior for cascades
than for tracks. For both, the observable energy of the secondaries can be
estimated from the total number of Cherenkov photons after accounting for
kinematic effects and detection efficiencies. The Cherenkov light observed in
cascades is proportional to the energy transferred to the cascade and is often
fully contained in the instrumented volume. It is actually sufficient that the
cascade is partially contained, as long as one can reconstruct its actuals
size. In contrast, muons produced by CC muon neutrino interactions lose energy
gradually by ionization, bremsstrahlung, pair production, and photo-nuclear
interactions while passing through the detector. The secondary charged
particles produced in each energy-loss interaction radiate Cherenkov photons
that allow for a measurement of the total energy lost by the muon in the
detector. The energy deposited by the muon in the detector represents a lower
limit on the initial neutrino energy.

Catastrophically losing energy by the process mentioned above, muon tracks
range out, over kilometers at TeV energy to tens of kilometers at EeV energy.
Because the energy of the muon thus degrades along its track, the energy of
the secondary showers decreases, which reduces the distance from the track
over which the associated Cherenkov light can trigger a PMT. The geometry of
the light pool surrounding the muon track is therefore a kilometer-long cone
with a gradually decreasing radius. On average, in its first kilometer, a
high-energy muon loses energy in a couple of showers with one-tenth of the
muon’s initial energy. So the initial radius of the cone is the radius of a
shower with 10% of the muon energy. At lower energies of hundreds of GeV and
less, the muon becomes minimum-ionizing.

Because of the stochastic nature of the muon’s energy loss, the relationship
between the observed energy loss inside the detector and the muon energy
varies from muon to muon. Additionally, only the muon energy lost in the
detector can be determined; we do not know the energy lost before entering the
instrumented volume, nor how much energy it carries out upon exiting. An
unfolding process is required to derive the neutrino energy from the energy
lost inside the detector; fortunately, it is described by well-understood
Standard Model physics. One derives a probability distribution for the energy
of the initial neutrino that determines its most probable value. The neutrino
energy may only be determined within a factor of 2 or thereabout, depending on
the energy, but the uncertainties drop out when measuring a neutrino spectrum
involving multiple events. In contrast, for $\nu_{e}$ and $\nu_{\tau}$, the
detector is a total energy calorimeter capturing all or most of the Cherenkov
light produced, and the determination of their energy is superior.

The different topologies each have advantages and disadvantages. For
$\nu_{\mu}$ charged-current interactions, the long lever arm of muon tracks
allows for a measurement of the muon direction with an angular resolution of
better than $0.4^{\circ}$. Superior angular resolution can be reached for
selected high-energy events. At the highest energies the neutrino is aligned
with the muon within the angular resolution and the sensitivity to point
sources thus maximized. The disadvantages are a large background of
atmospheric neutrinos below 100 TeV, and of cosmic-ray muons at all energies,
and the indirect determination of the neutrino energy that must be inferred
from sampling the energy loss of the muon when it transits the detector.

Observation of $\nu_{e}$ and $\nu_{\tau}$ flavors represents significant
advantages. They are detected from both Northern and Southern Hemispheres.
(This is also true for $\nu_{\mu}$ with energy in excess of several hundred
TeV, where the background from the steeply falling atmospheric spectrum
becomes negligible.) At TeV energies and above, the background of atmospheric
$\nu_{e}$ is lower by over an order of magnitude because long-lived pions, the
source of atmospheric $\nu_{e}$, no longer decay, and relatively rare K-decays
become the dominant source of background $\nu_{e}$. Atmospheric $\nu_{\tau}$,
produced by oscillations, are rare above an energy of $100$ GeV. High-energy
$\nu_{\tau}$ are therefore of cosmic origin; one such event with an energy of
$100$ TeV has been identified and represents an independent discovery of
cosmic neutrinos [59]. Furthermore, one can establish the cosmic origin of a
single single cascade event by demonstrating that the energy cannot be reached
by muons and neutrinos of atmospheric origin.

Finally, $\nu_{\tau}$ are not absorbed by the Earth [60]: $\nu_{\tau}$
interacting in the Earth produce a secondary $\nu_{\tau}$ of lower energy,
either directly in a neutral current interaction or via the decay of a
secondary tau lepton produced in a charged-current interaction. High-energy
$\nu_{\tau}$ will thus cascade down to energies of hundred of TeV where the
Earth becomes transparent. In other words, they are detected with a reduced
energy, but not absorbed. By this mechanism GZK $\nu_{\tau}$ with EeV energies
produce a signal at PeV energy [61].

Although cascades are nearly point-like sources of Cherenkov light and, in
practice, spatially isotropic, the pattern of arrival times of the photons at
individual optical modules reveals the direction of the secondary lepton.
While a fraction of cascade events may be reconstructed to within a degree
[62], the precision is inferior to that reached for $\nu_{\mu}$ events,
typically $8^{\circ}$ using the present techniques. A campaign is underway to
better characterize the optical properties of the ice and the calibration of
the detector in order to improve this resolution.

At energies above about $100$ PeV, electromagnetic showers begin to elongate
because of the Landau-Pomeranchuk-Migdal effect [63]. An extended length
scale, associated with the abundant radiation of soft photons, results in the
interaction of the secondary shower particles with two atoms. Negative
interference of this process relative to interactions with a single atom
results in a reduction of the energy loss.

####  2 The Neutrino Detector’s Telescope Area

Cosmic neutrinos must be separated from the large backgrounds of atmospheric
neutrinos and atmospheric cosmic-ray muons. Two principal methods have been
developed: isolating neutrinos that interact inside the instrumented volume
(“starting events”), and specializing to events where a muon enters the
detector from below, created by a neutrino that has traversed the Earth
(“throughgoing events”), thus pointing back to its origin. In the latter case,
the Earth is used as a filter for cosmic-ray muons.

For starting events, neutrinos are detected provided they interact within the
detector volume, i.e., within the instrumented volume of one cubic kilometer.
That probability is

$P(E_{\nu})=1-\exp[-l/\lambda_{\nu}(E_{\nu})]\simeq
l/\lambda_{\nu}(E_{\nu})\,,$ (11)

where $l$($\theta$) is the path length traversed by a neutrino with zenith
angle $\theta$ within the detector volume and
$\lambda_{\nu}(E_{\nu})=[\rho_{\rm ice}\ N_{A}\ \sigma_{\nu N}(E_{\nu})]^{-1}$
is the mean free path in ice for a neutrino of energy $E_{\nu}$. Here,
$\rho_{\rm ice}=0.9~{}{\rm g}{\rm cm}^{-3}$ is the density of the ice,
$N_{A}=6.022\times 10^{23}$ is Avogadro’s number, and $\sigma_{\nu
N}(E_{\nu})$ is the neutrino-nucleon cross section. The path length is
determined by the detector’s geometry and is typically much shorter than the
neutrino mean free path $\lambda_{\nu}$.

A neutrino flux $dN/dE_{\nu}$ (with typical units neutrinos per GeV or erg per
cm2 per second) crossing a detector with energy threshold $E_{\nu}^{\rm th}$
and cross sectional area A($E_{\nu}$) facing the incident neutrino beam will
produce

$N_{ev}=T\,\int_{E_{\nu}^{\rm
th}}A(E_{\nu})\,P(E_{\nu})\,\frac{dN}{dE_{\nu}}\,\,dE_{\nu}$ (12)

events after a time $T$. The “effective” detector area A($E_{\nu}$) is a
function of neutrino energy and of the zenith angle $\theta$. It isn’t
strictly equal to the geometric cross section of the instrumented volume
facing the incoming neutrino, because even neutrinos interacting outside the
instrumented volume may produce enough light inside the detector to be
detected. In practice, A($E_{\nu}$) is determined as a function of the
incident neutrino direction and zenith angle by a full-detector simulation,
including the trigger and the cuts that are used to isolate “starting” events.

In contrast, throughgoing muon neutrinos will be detected provided the
secondary muon reaches the detector with sufficient energy to trigger it.
Because the muon travels kilometers at TeV energy and tens of kilometers at
PeV energy, neutrinos are detected outside the instrumented volume with a
probability

$P=\lambda_{\mu}/\lambda_{\nu}$ (13)

obtained by the substitution

$l\rightarrow\lambda_{\mu}\,,$ (14)

in Eq. 11. Here, $\lambda_{\mu}$ is the range of the muon determined by its
energy losses. Values for the neutrino nucleon cross section and the range of
the muon can be found in reference 64.

At energies above tens of TeV, one has to account for the fact that the
neutrinos may be absorbed in the Earth before reaching the detector. The
reduced flux of $\nu_{\mu}$-induced muons reaching the detector is given by
[4, 5, 6]:

$\phi_{\mu}(E_{\mu}^{\rm min},\theta)=\int_{E_{\mu}^{\rm
min}}\,P(E_{\nu},E_{\mu}^{\rm min})\,\exp[-\sigma_{\rm
tot}(E_{\nu})\,N_{A}\,X(\theta)]\,\phi(E_{\nu},\theta)dE_{\nu}.$ (15)

The additional exponential factor accounts for the absorption of neutrinos
along a chord through the Earth of length $X(\theta)$ at zenith angle
$\theta$.

For back-of-the-envelope calculations, the $P$-function describing the
probability that a neutrino is detected can be approximated by

$\displaystyle P\simeq 1.3\times 10^{-6}E^{2.2}$ $\displaystyle\mbox{for
$E=10^{-3}$--1 TeV}\,,$ (16) $\displaystyle\simeq 1.3\times 10^{-6}E^{0.8}$
$\displaystyle\mbox{for $E=1$--$10^{3}$ TeV}\,.$ (17)

At EeV energy, the increase is reduced to only $E^{0.4}$. The parametrization
describes how neutrinos of higher energy are more likely to be detected
because of the increase with energy of both the cross section and muon range.
But at neutrino energies of tens of TeV and above, this gain is partially
mitigated by absorption in the Earth. The highest energy neutrinos reach the
detector from zenith angles near the horizon.

Tau neutrinos interacting outside the detector can be observed provided the
secondary tau lepton reaches the instrumented volume within its lifetime. In
Eq. 11, $l$ is replaced by

$l\rightarrow\Gamma c\tau=\frac{E}{m}c\tau\,,$ (18)

where $m$, $\tau$, and $E$ are the mass, lifetime, and energy of the tau,
respectively. The tau’s decay length $\lambda_{\tau}=\Gamma c\tau\approx
50~{}{\rm m}\times(E_{\tau}/\rm{PeV})$ grows linearly with energy and exceeds
the range of the muon near 1 EeV. At yet higher energies, the tau eventually
ranges out by catastrophic interactions, just like the muon, despite the
reduction of the leading energy-loss cross sections by a factor of
$(m_{\mu}/m_{\tau})^{2}$.

At sub-PeV energies, tracks and showers produced by tau neutrinos are
difficult to distinguish from those initiated by muon and electron neutrinos,
respectively. Only at PeV energies it is possible to detect both the initial
neutrino interaction and the subsequent tau decay that are separated by tens
of meters. Additionally, both must be contained within the detector volume;
for a cubic-kilometer detector, this can be realized for neutrinos with
energies from a few hundreds of TeV to a few tens of PeV [65].

For an in-depth discussion of neutrino detection, energy measurement, and
flavor separation, and for detailed references, see the IceCube Preliminary
Design Document [13] and reference 2.

####  3 Atmospheric Neutrinos: Calibration and Background

The 3-kHz trigger rate of the IceCube detector is dominated by atmospheric
muons from the decay of pions and kaons produced in the atmosphere above the
detector. Their distribution peaks near the zenith and decreases with
increasing angle because the muon energy required to reach the deep detector
increases. Most atmospheric muons are identified as tracks entering the
detector from above and are rejected because the Earth shields the detector
from atmospheric muons in the Northern Hemisphere. Even after the removal of
cosmic ray muons, the neutrinos from decay of mesons produced by cosmic-ray
interactions in the atmosphere are a residual background in the search for
neutrinos of extraterrestrial origin. Because of the large ratio of
atmospheric muons to neutrinos misreconstructed atmospheric muons remain an
important source of background for most searches.

Measurement of the relatively well-established spectrum of atmospheric
neutrinos is an important benchmark and a useful calibration tool for a
neutrino telescope. IceCube detects an atmospheric neutrino every few minutes,
more than one hundred thousand per year. The spectrum of atmospheric
$\nu_{\mu}$ has been measured by unfolding the measured rate and energy
deposition of neutrino-induced muons entering the detector from below the
horizon [33], as shown in Fig. 6. More challenging is the measurement of the
flux of atmospheric electron neutrinos. This has been achieved by making use
of DeepCore, the more densely instrumented subarray in the deep center of
IceCube, to identify contained shower events. The measured spectrum of
$\nu_{\mu}$ is used to calculate the contribution of neutral current
interactions to the observed rate of showers. Subtracting the neutral current
contribution leads to the measurement of the spectrum of atmospheric electron
neutrinos from 100 GeV to 10 TeV [35]; see Fig. 6.

In general, atmospheric neutrinos are indistinguishable from astrophysical
neutrinos. An important exception is when muon neutrinos reaching the detector
from above can be tagged as atmospheric by detecting the muon produced in the
same decay as the neutrino. The neutrino energy must be sufficiently high and
the zenith angle sufficiently small that this muon to reach the detector [66].
Monte Carlo simulation are performed to evaluate the rejection rate or,
equivalently, the atmospheric neutrino passing rate. Also high-energy muons
other than the muon associated directly with the neutrino and produced in the
same cosmic-ray shower as the neutrino are included in the veto. In this way,
the method can be extended to electron neutrinos. In practice, the passing
rate is significantly reduced for $E_{\nu}>100$ TeV and zenith angles
$\theta<70^{\circ}$ where the ice overburden is not too large.

The spectrum of atmospheric neutrinos becomes one power steeper than the
spectrum of primary nucleons at high energy because the competition between
interaction in the atmosphere and the decay of pions and kaons increasingly
suppresses their decay. For the kaon channel, dominant at high energies, the
characteristic energy for the steepening is $E_{\nu}\sim 1~{}{\rm
TeV}/\cos\theta$. A further steepening occurs above $100$ TeV as a consequence
of the knee in the primary cosmic ray spectrum. In contrast, astrophysical
neutrinos should reflect the cosmic-ray spectrum of the cosmic accelerator
expected to be significantly harder spectrum than atmospheric neutrinos.
Establishing an astrophysical signal above the steep atmospheric background
requires an understanding of the atmospheric neutrino spectrum at energies of
$100$ TeV and above.

Although there is some uncertainty associated with the composition through the
knee region [67], the major uncertainty in the spectrum of atmospheric
neutrinos at high energy is the level of charm production. The short-lived
charmed hadrons preferentially decay, up to a characteristic energy of
$10^{7}$ GeV, producing “prompt” muons and neutrinos with the same spectrum as
their parent cosmic rays. This prompt flux of leptons has yet to be measured.
Existing limits [68, 69] allow a factor of two or three around the level
predicted by model calculations [38]. For reasonable assumptions, the charm
contribution is expected to dominate the conventional spectrum above $\sim 10$
TeV for $\nu_{e}$, above $\sim 100$ TeV for $\nu_{\mu}$, and above $\sim 1$
PeV for muons [70].

The expected hardening in the spectrum of atmospheric neutrinos due to prompt
neutrinos, is partially degenerate with a hard astrophysical component.
However, the spectrum of astrophysical neutrinos should reflect the spectrum
of cosmic rays at their sources, which is expected to be harder than the
spectrum of cosmic rays at Earth. It should eventually be possible with
IceCube to measure the charm contribution by requiring a consistent
interpretation of neutrino flavors and cosmic-ray muons for which there is no
astrophysical component. An additional signature of atmospheric charm is the
absence of seasonal variations for this component [71].

As we will discuss further on, with a good understanding of the energy and
zenith angle dependence of the atmospheric neutrino spectrum, supplemented by
the veto technique and the shielding of the muons by the Earth, IceCube has
been successful in separating a diffuse flux of cosmic neutrinos from the
atmospheric backgrounds. The next step is to find the origin of this flux by
identifying individual sources.

The strategy of searching for neutrino sources is to look for spatial
clustering in the arrival direction of neutrinos to find any excess over the
expected isotropic distribution of background. The technique used by IceCube
to search for point sources is described in reference 72. In this method, an
unbinned maximum likelihood is constructed to search for spatial clustering of
the events. Significances are estimated by repeating each hypothesis test on
data sets that are randomized in right ascension and dominated by background.
This provides robust p-values that are largely independent of detector
systematic uncertainties.

The unbinned maximum likelihood ratio method used to look for a localized,
statistically significant excess of events above the background allows full
use of spatial and spectral information from the data. The data are
hypothesized to be a mixture of events from signal and background.

For an event with reconstructed direction
$\vec{x}_{i}=(\alpha_{i},\delta_{i})$, the probability of originating from the
source at $x_{s}$ is modeled as a circular two-dimensional Gaussian. The
signal probability distribution function (PDF) $\mathcal{S}_{i}$ incorporates
directional information for each individual event and its angular uncertainty,
$\sigma_{i}$, and the angular difference between the reconstructed direction
of the event and the source:

$\mathcal{S}_{i}={S}_{i}(|\vec{x}_{i}-\vec{x}_{s}|,\sigma_{i})\,\mathcal{E}_{i}(\delta_{i},E_{i},\gamma),$
(19)

where the spatial distribution is modeled as a two-dimensional Gaussian

$S_{i}(|\vec{x}_{i}-\vec{x}_{s}|,\sigma_{i})=\frac{1}{2\pi\sigma_{i}^{2}}\exp\Big{(}{-\frac{|\vec{x}_{i}-\vec{x}_{s}|^{2}}{2\sigma_{i}^{2}}}\Big{)}$
(20)

The background PDF, $\mathcal{B}_{i}$, contains similar terms that describe
the angular and energy distributions of background events. The likelihood
function for a point source is defined as

$\mathcal{L}(n_{s},x_{s},\gamma)=\prod_{i}^{events}\bigg{(}\frac{n_{s}}{N}\mathcal{S}_{i}(|x_{i}-x_{s}|,\sigma_{i},E_{i},\gamma)+\frac{N-n_{s}}{N}\mathcal{B}_{i}(\delta_{i},E_{i})\bigg{)}$
(21)

The likelihood ratio test statistic (TS) is used to perform statistical tests.
The results of these tests can be clearly defined in the context of testing
between two hypotheses: the null hypothesis $H_{0}$ and the alternative
hypothesis $H_{1}$ that signal events exceed the background. $H_{0}$
represents the case of background only ($n_{s}=0$).

After maximizing and determining the best fit number of signal events
$\hat{n}_{s}$ and their spectral index $\hat{\gamma}$, the test statistic (TS)
is defined as the log likelihood ratio between the null and signal hypothesis.
In this case, the null hypothesis is that all events are generated from the
isotropic background distribution, i.e., $n_{s}=0.$ The alternative hypothesis
is that $n_{s}$ neutrinos originate from the source. As is the case for the
diffuse flux, a harder spectrum can indicate a signal. The TS is calculated
as:

${\rm
TS}=2\log\big{[}\frac{\mathcal{L}(\hat{n}_{s},\hat{\gamma})}{\mathcal{L}(n_{s}=0)}\big{]}$
(22)

The significance of an observation is determined by comparing the observed TS
to the TS distribution from data sets randomized in right ascension. The TS
distribution for randomized data sets represents the probability that a given
observation could occur by random chance within the data set. For large sample
sizes, this distribution approximately follows a chi-squared distribution,
where the number of degrees of freedom corresponds to the difference in the
number of free parameters between the null hypothesis and the alternate
hypothesis.

In addition to triggered and untriggered searches for neutrino sources,
stacking a collection of candidate sources could be an effective way to
enhance the discovery potential. in stacking searches, the correlation to a
catalog of sources is tested instead of searching in the direction of a single
source. The stacking likelihood is defined as

$\mathcal{L}(n_{s},\gamma)=\prod_{i}^{events}\sum_{j}^{M}\frac{n_{s}}{N}\frac{w_{j}}{M}\mathcal{S}_{i}^{j}(|x_{i}-x_{j}|,\sigma_{i},E_{i},\gamma)+\frac{N-n_{s}}{N}\mathcal{B}_{i}(\delta_{i},E_{i})\,,$
(23)

where $\mathcal{B}_{i}$ represents the isotropic background PDF, and
$\mathcal{S}_{i}$ the signal PDF, for each event. $M$ is the number of sources
in the catalog and $w_{j}$ the normalized theoretical weight for each source.
This weight could be associated with properties of the individual sources in
the catalog such as distance or the magnitude of their flux at some
wavelength.

Either the point source search or the stacking search can be modified in order
to search for the extended sources of neutrinos. For this purpose, the angular
uncertainty in the spatial PDF has to be modified using the extension of the
source. In this case the effective angular uncertainty for an extended source
is given by

$\sigma_{\rm eff}=\sqrt{\sigma_{i}^{2}+\sigma_{\rm ext}^{2}}$ (24)

The technique described above can be extended to a time-dependent search for
transient neutrino sources [73]. A time component is added to the signal
function in Eq. (19). This component will take into account the temporal
structure of the neutrino emission. This temporal dependency can be modeled as
a Heaviside function or a Gaussian, for example.

Time-dependent searches are generally more sensitive than time-integrated
searches because they accumulate lower background rates. These searches,
therefore, offer an alternative opportunity to pinpoint the origin of cosmic
neutrinos.

###  4 The Discovery of Cosmic Neutrinos

For neutrino astronomy, the first challenge is to select a pure sample of
neutrinos, more than 100,000 per year above a threshold of 0.1 TeV for
IceCube, in a background of ten billion cosmic-ray muons, while the second is
to identify the small fraction of these neutrinos that is astrophysical in
origin, expected to be at the level of ten to hundred events per year
according to the expectations of Fig. 6. Atmospheric neutrinos are a
background for cosmic neutrinos, at least at neutrino energies below $\sim
300$ TeV. Above this energy, the atmospheric neutrino flux reduces to less
than one event per year, even in a kilometer-scale detector, and thus events
in that energy range are predominantly cosmic in origin.

Searching for high-energy neutrinos of cosmic origin, IceCube continuously
monitors the whole sky collecting very high statistics data sets of
atmospheric neutrinos. Neutrino energies cover more than six orders of
magnitude, from 5 GeV in the highly instrumented inner core (DeepCore) to
beyond 10 PeV. Soon after the completion of the detector, with two years of
data, IceCube discovered an extragalactic flux of cosmic neutrinos with an
energy flux in the local universe that is, surprisingly, similar to that in
gamma rays; see references 74 and 75.

Two principal methods are used to identify cosmic neutrinos. The first method
reconstructs upgoing muon tracks initiated by muon neutrinos and the second
identifies neutrinos of all flavors interacting inside the instrumented volume
of the detector. For illustration, the Cherenkov patterns initiated by an
electron (or tau) neutrino of 1 PeV energy and a neutrino-induced muon losing
2.6 PeV energy while traversing the detector are contrasted in Fig. 12.

Figure 12: Left Panel: Light pool produced in IceCube by a shower initiated by
an electron or tau neutrino of $1.14$ PeV, which represents a lower limit on
the energy of the neutrino that initiated the shower. White dots represent
sensors with no signal. For the colored dots, the color indicates arrival
time, from red (early) to purple (late) following the rainbow, and size
reflects the number of photons detected. Right Panel: A muon track coming up
through the Earth, traverses the detector at an angle of $11^{\circ}$ below
the horizon. The deposited energy, i.e., the energy equivalent of the total
Cherenkov light of all charged secondary particles inside the detector, is 2.6
PeV.

The kilometer-long muon range makes it possible to identify neutrinos that
interact outside the detector and to separate them from the the background of
atmospheric muons using Earth as a filter. Using this method, IceCube has
measured the background atmospheric neutrino flux over more than five orders
of magnitude in energy with a result that is consistent with theoretical
calculations. However, with eight years of data, IceCube has observed an
excess of neutrino events at energies beyond 100 TeV [76, 77, 78] that cannot
be accounted for by the atmospheric flux; see Fig. 13. Although the detector
only records the energy of the secondary muon inside the detector, from
Standard Model physics we can infer the energy spectrum of the parent
neutrino. The high-energy cosmic muon neutrino flux is well described by a
power law with a spectral index of $2.28\pm 0.10$ and a normalization at 100
TeV neutrino energy of $(1.44^{+0.25}_{-0.24})\,\times 10^{-18}\,\rm
GeV^{-1}\rm cm^{-2}\rm sr^{-1}$ [79].

Figure 13: Distribution of muon tracks arriving from the Northern Hemisphere,
i.e., with declination greater than -5∘, as a function of median neutrino
energy. The black crosses correspond to observed data. The blue band
represents the expected conventional atmospheric neutrino. The red lines shows
the high-energy cosmic neutrino flux with the best fit spectral index of 2.13.
The black dashed line demonstrates the probability density function for the
highest energy neutrino event when the best fit spectrum is assumed.

The arrival directions of the muon tracks have been analyzed by a range of
statistical methods, yielding a first surprise: there is no evidence for any
correlation to sources in the Galactic plane. IceCube is recording a diffuse
flux of extragalactic sources. Only after analyzing 10 years of data [80] has
evidence emerged at the $3\sigma$ level that the neutrino sky map might not be
isotropic. The anisotropy results from four sources–TXS 0506+056 among them
(you will hear more about that source later on), that emerge as point sources
at the $4\sigma$ level (pretrial); see Fig. 14. The strongest of these sources
is the nearby active galaxy NGC 1068, also known as Messier 77.

NGC 1068 is one of the best-studied Seyfert 2 galaxies that motivated the AGN
unification model. Its measured infrared luminosity implies a high level of
starburst activity. In addition, NGC 1068 has a heavily obscured nucleus and
the column density of the gas surrounding it reaches to $\simeq
10^{25}\rm\,cm^{-2}$ [81]. Therefore, the high-energy electromagnetic emission
is absorbed in the Compton thick molecular gas, which even makes measuring its
intrinsic X-ray emission challenging [82, 81]. As such, the optically thick
environment at the core of NGC 1068 provides a favorable environment for the
efficient production of high-energy neutrinos. The $\sim$51 signal neutrinos
identified in the direction of NGC 1068 [80] implies a neutrino flux that
considerably exceeds the gamma-ray emission measured by the Fermi satellite,
implying a very efficient neutrino emission concurrent with suppression of the
very high energy gamma rays.

Modeling of the high-energy neutrino emission from the corona of AGN [83, 84]
can accommodate these conditions. In Seyfert galaxies and quasars, a
magnetized corona is formed above the accretion disk on the central black
hole; see Miller & Stone (2000) [85] for details. The hot, turbulent, and
highly magnetized corona of AGN facilitates particle acceleration. Combined
with the high density and abundance of target gas and radiation in the
vicinity of the AGN core, a sizable neutrino flux can be expected [86]. The
accompanying pionic photons will cascade down due to the high opacity of the
target. The efficient production of neutrinos in the vicinity of supermassive
black holes by this mechanism can accommodate the measured flux of high-energy
neutrinos. We will discuss this further in Sec. 6 where we highlight
implications of astrophysical beam dumps for neutrino emission, especially in
the presence of major accretion activity onto a supermassive black hole.

Figure 14: Upper limits on the flux from candidate point sources of neutrinos
in 10 years of IceCube data assuming two spectral indices of the flux. Four
sources exceed the $4\sigma$ level (pretrial) and collectively result into a
$3\sigma$ anisotropy of the sky map.

The second method for separating cosmic from atmospheric neutrinos exclusively
identifies high-energy neutrinos interacting inside the detector, so-called
high-energy starting events. It divides the instrumented volume of ice into an
outer veto shield and a $\sim 500$-megaton inner fiducial volume. The
advantage of focusing on neutrinos interacting inside the instrumented volume
of ice is that the detector functions as a total absorption calorimeter [87]
allowing for a good energy measurement that separates cosmic from lower-energy
atmospheric neutrinos. With this method, neutrinos from all directions in the
sky and of all flavors can be identified, which includes both muon tracks as
well as secondary showers produced by charged-current interactions of electron
and tau neutrinos and neutral current interactions of neutrinos of all
flavors. A sample event with a light pool of roughly one hundred thousand
photoelectrons extending over more than 500 meters is shown in the left panel
of Fig. 12. The starting event sample revealed the first evidence for
neutrinos of cosmic origin [88, 89]. Events with PeV energies, and no trace of
accompanying muons from an atmospheric shower, are highly unlikely to be of
atmospheric origin. The present seven-year data set contains a total of 60
neutrino events with deposited energies ranging from 60 TeV to 10 PeV that are
likely to be of cosmic origin.

The deposited energy and zenith dependence of the high-energy starting events
[78] is compared to the atmospheric background in Fig. 15. The expected number
of events for the best-fit astrophysical neutrino spectrum following a two-
component power-law fit is shown as dashed lines in the two panels. The
corresponding neutrino spectrum is also shown in Fig. 16. It is, above an
energy of $200$ TeV, consistent with a power-law flux of muon neutrinos
penetrating the Earth inferred by the data shown in Fig. 13. A purely
atmospheric explanation of the observation is excluded at $8\sigma$.

Figure 15: Left Panel: Deposited energies, by neutrinos interacting inside
IceCube, observed in six years of data [78]. The grey region shows
uncertainties on the sum of all backgrounds. The atmospheric muon flux (blue)
and its uncertainty is computed from simulation to overcome statistical
limitations in our background measurement and scaled to match the total
measured background rate. The atmospheric neutrino flux is derived from
previous measurements of both the $\pi,K$, and charm components of the
atmospheric spectrum [90]. Also shown are two fits to the spectrum, assuming a
simple power-law (solid gray) and a broken power-law (dashed gray). Right
Panel: The same data and models, but now showing the distribution of events
with deposited energy above 60 TeV in declination. At the South Pole, the
declination angle $\delta$ is equivalent to the distribution in zenith angle
$\theta$ related by the identity, $\delta=\theta-\pi/2$. It is clearly visible
that the data is flat in the Southern Hemisphere, as expected from the
contribution of an isotropic astrophysical flux. Figure 16: Unfolded spectrum
for six years of HESE neutrino events starting inside the detector. The yellow
and red bands show the $1\sigma$ uncertainties on the result of a two-power-
law fit. Superimposed is the best fit to eight years of the upgoing muon
neutrino data (pink). Note the consistency of the red and pink bands. Figure
from reference 78.

Both measurements of the cosmic neutrino flux using cascades [91] and
throughgoing muons yield consistent determinations of the cosmic neutrino
flux; see Fig. 17. The data are consistent with an astrophysical component
with a spectrum close to $E^{-2.2}$ above an energy of $\sim 100$ TeV [78].

Figure 17: The flux of cosmic muon neutrinos [78] inferred from the eight-year
upgoing-muon track analysis (red solid line) with $1\sigma$ uncertainty range
(shaded range; from fit shown in upper-right inset) is compared with the flux
of showers initiated by electron and tau neutrinos [91]. The measurements are
consistent assuming that each flavor contributes the same flux to the diffuse
spectrum. Figure 18: The flux of high-energy neutrinos (per flavor) above 1
TeV. At low energies, IceCube’s measured atmospheric muon and electron
neutrinos [92, 93] are shown in light and dark grey, respectively. The modeled
prompt flux [38] from atmospheric charm production in cosmic ray showers is
shown in black. The red line shows the fitted spectra for the HESE analysis
covering 7.5 years of data and the shaded region shows the uncertainties [94].
The dark red points show segmented flux data points in the HESE 7.5-year
analysis. The results for the 10-year muon neutrino flux measurement is shown
as the blue shaded area [79]. At higher energies: the predicted flux of
cosmogenic neutrinos [95] shown in green, as well as the upper limit on the
extremely high energy flux of neutrinos from IceCube [96] and Pierre Auger
Observatory [97].

In summary, IceCube has observed cosmic neutrinos using both methods for
rejecting background. Based on different methods for reconstruction and energy
measurement, their results agree, pointing at extragalactic sources whose flux
has equilibrated in the three flavors after propagation over cosmic distances
[98] with $\nu_{e}:\nu_{\mu}:\nu_{\tau}\sim 1:1:1$.

Figure 18 summarizes the measurements of the cosmic neutrino flux using
starting events and throughgoing muons from the northern sky. An extrapolation
of this high-energy flux to lower energy may suggest an excess of events in
the $10-100$ TeV energy range over and above a single power-law fit. This
conclusion is however not statistically compelling [99].

In Fig. 19 we show the arrival directions of the most energetic events in the
eight-year upgoing $\nu_{\mu}+\bar{\nu}_{\mu}$ analysis ($\odot$) and the six-
year HESE data sets. The HESE data are separated into tracks ($\otimes$) and
cascades ($\oplus$). The median angular resolution of the cascade events is
indicated by thin circles around the best-fit position. The most energetic
muons with energy $E_{\nu}>200$ TeV in the upgoing $\nu_{\mu}+\bar{\nu}_{\mu}$
data set accumulate near the horizon in the Northern Hemisphere. Elsewhere,
muon neutrinos are increasingly absorbed in the Earth before reaching the
vicinity of the detector because of their relatively large high-energy cross
sections. This causes the apparent anisotropy of the events in the Northern
Hemisphere. Also HESE events with deposited energy of $E_{\rm dep}>100$ TeV
suffer from absorption in the Earth and are therefore mostly detected when
originating in the Southern Hemisphere. After correcting for absorption by the
Earth, the arrival directions of cosmic neutrinos are isotropic, suggesting
extragalactic sources. In fact, no correlation of the arrival directions of
the highest energy events, shown in Fig. 19, with potential sources or source
classes has reached the level of $3\sigma$ [99].

Figure 19: Mollweide projection in Galactic coordinates of the arrival
direction of neutrino events. We show the results of the eight-year upgoing
track analysis [78] with reconstructed muon energy $E_{\mu}\gtrsim 200$ TeV
($\odot$). The events of the six-year high-energy starting event (HESE)
analysis with deposited energy larger than 100 TeV (tracks $\otimes$ and
cascades $\oplus$) are also shown [100, 101, 78]. The thin circles indicate
the median angular resolution of the cascade events ($\oplus$). The blue-
shaded region indicates the zenith-dependent range where Earth absorption of
100 TeV neutrinos becomes important, reaching more than 90% close to the
nadir. The dashed line indicates the horizon and the star ($\star$) the
Galactic Center. We highlight the four most energetic events in both analyses
by their deposited energy (magenta numbers) and reconstructed muon energy (red
number).

We should comment at this point that there is yet another method to
conclusively identify cosmic neutrinos: the observation of very high energy
tau neutrinos. Below an energy of 100 GeV, tau neutrinos are abundantly
produced in the atmosphere by the oscillation of muon into tau neutrinos.
Above that energy, they must be of cosmic origin, produced in cosmic
accelerators whose neutrino flux has approximately equilibrated between the
three flavors after propagating over cosmic distances. Tau neutrinos produce
two spatially separated showers in the detector, one from the interaction of
the tau neutrino and the second one from its decay; the mean tau lepton decay
length is $\lambda_{\tau}=\Gamma c\tau\approx 50~{}{\rm
m}\times(E_{\tau}/\rm{PeV})$, where $m$, $\tau$, and $E$ are the mass,
lifetime, and energy of the tau, respectively. Two such candidate events have
been identified [102]. An event with a decay length of 17 m and a probability
of 98% of being produced by a tau neutrino is shown in Fig. 20.

Figure 20: Event view of a tau neutrino. The Cherenkov photons associated with
the production and subsequent decay of the tau neutrino are identified by the
double-peaked photon count as a function of time for the bright DOMs, for
instance, the one shown in the top-right corner. The best fit (solid line)
corresponding to a 17 m decay length and is far superior to fits assuming a
single electromagnetic or hadronic shower (dashed lines).

Yet another independent confirmation of the observation of neutrinos of cosmic
origin appeared in the form of a Glashow resonance event shown in Fig. 21. The
event was identified in a search for partially contained events in a 4.5 year
data set: an antielectron neutrino interacting with an atomic electron
produced an event compatible with an incident neutrino energy of 6.3 PeV,
characteristic of the resonant production of a weak intermediate $W^{-}$ with
a mass of $M_{W}=80.38\,\rm GeV$ [103].

Given its energy and direction, the event is classified as an astrophysical
neutrino at the $5\sigma$ level. Furthermore, data collected by the sensors
closest to the interaction point, as well as the measured energy, are
consistent with the hadronic decay of a $W^{-}$ produced on the Glashow
resonance. In the observer frame, where the electron mass ($m_{e}=0.511\,\rm
MeV$) is at rest, the resonance energy is given by
$E_{R}=M_{W}^{2}/2m_{e}=6.32\,\rm PeV$ for $M_{W}=80.38\,\rm GeV$. The
measured energy of $6.05\pm 0.72$ PeV translates into a neutrino energy of 6.3
PeV after correcting the visible energy produced by the hadronic decay of the
W for shower particles that do not radiate. Taking into account the detector’s
energy resolution, the probability that the event is produced off resonance by
deep inelastic scattering is only 0.01 assuming a spectrum with a spectral
index of $\gamma=-2.5$. Its observation extends the measured astrophysical
flux to 6.3 PeV. Assuming the Standard Model resonant cross section, we expect
1.55 events in the sample assuming an antineutrino:neutrino ratio of 1:1
characteristic of a cosmic beam dump producing an equal number of pions of all
three electric charges.

The observation of a Glashow resonant event heralds the presence of electron
antineutrinos in the cosmic neutrino flux. Its unique signature illustrates a
method to disentangle neutrinos from antineutrinos, thus opening a path to
distinguish astronomical accelerators that produce neutrinos via hadronuclear
or photohadronic interactions with or without strong magnetic fields [103]. As
such, knowledge of both the flavor and charge of the incident neutrino will
add a new tool for doing neutrino astronomy.

Figure 21: Particle shower created at the Glashow resonance. The energy,
measured via Cherenkov radiation in the Antarctic ice sheet, is reconstructed
at the resonant energy for the production of a weak intermediate boson $W^{-}$
in the interaction of an antielectron neutrino with an atomic electron in the
ice. Features consistent with the production of secondary muons in the
particle shower indicate the hadronic decay of a resonant $W^{-}$ boson.

###  5 Multimessenger Astronomy: General Considerations

The most important message emerging from the IceCube measurements may not be
apparent yet: the prominent and surprisingly important role of neutrinos
relative to photons in the nonthermal universe. To illustrate this point, we
show in Fig. 22 the observed energy flux of neutrinos, $E^{2}\phi$. One can
see that the cosmic energy density of high-energy neutrinos is comparable to
that of gamma-rays observed with the Fermi satellite [74] and to that of the
ultra-high-energy (UHE) cosmic rays above $10^{9}$ GeV, observed, by the Auger
observatory [104]. This may indicate a common origin and, in any case,
provides an excellent opportunity for multi-messenger studies.

The pionic gamma rays accompanying the neutrino flux is shown as the solid
blue. It has been derived from Eq. 10 but, for extragalactic neutrinos, this
gamma-ray emission is not directly observed because of strong absorption of
photons by $e^{+}e^{-}$ pair production in the extragalactic background light
(EBL) and CMB. The high-energy photons initiate electromagnetic showers by
repeated pair production and inverse-Compton scattering, mostly on CMB
photons, that eventually yield the lower energy photons observed by Fermi in
the GeV-TeV range.

The extragalactic gamma-ray background observed by Fermi [74] has
contributions from identified point-like sources on top of an isotropic gamma-
ray background (IGRB) shown in Fig. 22. This IGRB is expected to consist
mostly of emission from the same class of gamma-ray sources that are
individually below Fermi’s point-source detection threshold (see, e.g.,
reference 105). The significant contribution of gamma-rays associated with
IceCube’s neutrino observation has the somewhat surprising implication that
indeed many extragalactic gamma-ray sources are also neutrino emitters, while
none has been detected so far.

Another intriguing observation is that the high-energy neutrinos observed at
IceCube could originate in the sources of the highest energy cosmic rays.
These could, for instance, be embedded in environments that act as “storage
rooms” for cosmic rays with energies below the “ankle” ($E_{\rm CR}\ll 1$EeV).
This energy-dependent trapping can be achieved via cosmic ray diffusion in
magnetic fields. While these cosmic rays are trapped, they can produce gamma-
rays and neutrinos via collisions with gas. If the conditions are right, this
mechanism can be so efficient that the total energy stored in low-energy
cosmic rays is converted to that of gamma rays and neutrinos. These
“calorimetric” conditions can be achieved in starburst galaxies [106] or
galaxy clusters [107]. We will discuss these multimessenger relations in more
detail next.

Figure 22: The spectral flux ($\phi$) of neutrinos inferred from the eight-
year upgoing track analysis (red fit) and the six-year HESE analysis (magenta
fit) compared to the flux of unresolved extragalactic gamma-ray sources [74]
(blue data) and ultra-high-energy cosmic rays [104] (green data). The neutrino
spectra are indicated by the best-fit power-law (solid line) and $1\sigma$
uncertainty range (shaded range). We highlight the various multimessenger
interfaces: A: The joined production of charged pions ($\pi^{\pm}$) and
neutral pions ($\pi^{0}$) in cosmic-ray interactions leads to the emission of
neutrinos (dashed blue) and gamma-rays (solid blue), respectively. B: Cosmic
ray emission models (solid green) of the most energetic cosmic rays imply a
maximal flux (calorimetric limit) of neutrinos from the same sources (green
dashed). C: The same cosmic ray model predicts the emission of cosmogenic
neutrinos from the collision with cosmic background photons (GZK mechanism).

####  1 IceCube Neutrinos and Fermi Photons

Recall that photons are produced in association with neutrinos when
accelerated cosmic rays produce neutral and charged pions in interactions with
photons or nuclei. Targets include strong radiation fields that may be
associated with the accelerator as well as concentrations of matter or
molecular clouds in their vicinity. Additionally, pions can be produced in the
interaction of cosmic rays with the EBL when propagating through the
interstellar or intergalactic background medium from their source to Earth. A
high-energy flux of neutrinos is produced in the subsequent decay of charged
pions via $\pi^{+}\to\mu^{+}+\nu_{\mu}$ followed by $\mu^{+}\to
e^{+}+\nu_{e}+\bar{\nu}_{\mu}$ and the charge–conjugate processes. High-energy
gamma rays result from the decay of neutral pions, $\pi^{0}\to\gamma+\gamma$.
Pionic gamma rays and neutrinos carry, on average, 1/2 and 1/4 of the energy
of the parent pion, respectively. With these approximations, the neutrino
production rate $Q_{\nu_{\alpha}}$ (units of ${\rm GeV}^{-1}{\rm s}^{-1}$) can
be related to the one for charged pions as

$\sum_{\alpha}E_{\nu}Q_{\nu_{\alpha}}(E_{\nu})\simeq
3\left[E_{\pi}Q_{\pi^{\pm}}(E_{\pi})\right]_{E_{\pi}\simeq 4E_{\nu}}\,.$ (25)

Similarly, the production rate of pionic gamma-rays is related to the one for
neutral pions as

$E_{\gamma}Q_{\gamma}(E_{\gamma})\simeq
2\left[E_{\pi}Q_{\pi^{0}}(E_{\pi})\right]_{E_{\pi}\simeq 2E_{\gamma}}\,.$ (26)

Note, that the relative production rate of pionic gamma rays and neutrinos
only depends on the ratio of charged-to-neutral pions produced in cosmic-ray
interactions, denoted by $K_{\pi}=N_{\pi^{\pm}}/N_{\pi^{0}}$. Pion production
of cosmic rays in interactions with photons can proceed resonantly in the
processes $p+\gamma\rightarrow\Delta^{+}\rightarrow\pi^{0}+p$ and
$p+\gamma\rightarrow\Delta^{+}\rightarrow\pi^{+}+n$. These channels produce
charged and neutral pions with probabilities 2/3 and 1/3, respectively.
However, the additional contribution of nonresonant pion production changes
this ratio to approximately 1/2 and 1/2.

In contrast, cosmic rays interacting with matter, for instance with hydrogen
in the Galactic disk, produce equal numbers of pions of all three charges:
$p+p\rightarrow n_{\pi}\,[\,\pi^{0}+\pi^{+}+\pi^{-}]+X$, where $n_{\pi}$ is
the pion multiplicity. From above arguments we have $K_{\pi}\simeq 2$ for
cosmic ray interactions with gas ($pp$) and $K_{\pi}\simeq 1$ for interactions
with photons ($p\gamma$).

With this approximation we can combine Eqs. (8) and (26) to derive a simple
relation between the pionic gamma-ray and neutrino production rates:

$\frac{1}{3}\sum_{\alpha}E^{2}_{\nu}Q_{\nu_{\alpha}}(E_{\nu})\simeq\frac{K_{\pi}}{4}\left[E^{2}_{\gamma}Q_{\gamma}(E_{\gamma})\right]_{E_{\gamma}=2E_{\nu}}\,.$
(27)

The prefactor $1/4$ accounts for the energy ratio $\langle
E_{\nu}\rangle/\langle E_{\gamma}\rangle\simeq 1/2$ and the two gamma rays
produced in the neutral pion decay. This powerful multimessenger relation
connects pionic neutrinos and gamma rays without any reference to the cosmic
ray beam; it simply reflects the fact that a $\pi^{0}$ produces two $\gamma$
rays for every charged pion producing a $\nu_{\mu}+\bar{\nu}_{\mu}$ pair,
which cannot be separated by current experiments.

Before applying this relation to a cosmic accelerator, we have to be aware of
the fact that, unlike neutrinos, gamma rays interact with photons of the
cosmic microwave background before reaching Earth. The resulting
electromagnetic shower subdivides the initial photon energy, resulting in
multiple photons in the GeV-TeV energy range by the time the photons reach
Earth. Calculating the cascaded gamma-ray flux accompanying IceCube neutrinos
is straightforward [108, 109].

An example of the relation between pionic gamma-ray (solid) and neutrino
(dashed) emission is illustrated in Fig. 22 by the blue lines. We assume that
the underlying $\pi^{0}$ / $\pi^{\pm}$ production follows from cosmic-ray
interactions with gas in the universe; $K_{\pi}\simeq 2$. In this way, the
initial emission spectrum of gamma-rays and neutrinos from pion decay is
almost identical to the spectrum of cosmic rays, after accounting for the
different normalizations and energy scales. The flux of neutrinos arriving at
Earth (blue dashed line) follows this initial CR emission spectrum, assumed to
be a power law, $E^{-\gamma}$. However, the observable flux of gamma-rays
(blue solid lines) is strongly attenuated above 100 GeV by interactions with
extragalactic background photons.

The overall normalization of the emission is chosen in a way that the model
does not exceed the isotropic gamma-ray background observed by the Fermi
satellite (blue data). This implies an upper limit on the neutrino flux shown
as the blue dashed line. Interestingly, the neutrino data shown in Fig. 22
saturates this limit above 100 TeV. Moreover, the HESE data that extends to
lower energies is only marginally consistent with the upper bound implied by
the model (blue dashed line). If the underlying assumptions hold, the neutrino
spectrum cannot be harder than $E^{-1.5}$, a result somewhat challenged by the
data. This example shows that multi-messenger studies of gamma-ray and
neutrino data are powerful tools to study the neutrino production mechanism
and to constrain neutrino source models [110].

Above exercise relating gamma rays and neutrinos comes with more warnings than
a TV commercial for drugs:

  * •

The target for producing the neutrinos may be photons. This changes the value
of $K_{\pi}$ and the shape of the flux. Yoshida and Murase recently worked
through this example in detail [111].

  * •

Electrons accelerated along with protons may contribute photons to the
observed flux on top of those of pionic origin.

  * •

The source itself may not be transparent to high-energy photons that will lose
energy in the source event before reaching the EBL. As a result, part of the
pionic flux will emerge below the threshold of the Fermi satellite, at MeV and
lower energies. This will be an important consideration when we discuss the
first identified source of high-energy neutrinos, the black hole TXS 0506+056.

While the details matter, the main message should not be lost: the matching
energy densities of the extragalactic gamma-ray flux detected by Fermi and the
high-energy neutrino flux measured by IceCube suggest that, rather than
detecting some exotic sources, it is more likely that IceCube to a large
extent observes the same universe astronomers do. Clearly, an extreme universe
modeled exclusively on the basis of electromagnetic processes is no longer
realistic. The finding implies that a large fraction, possibly most, of the
energy in the nonthermal universe originates in hadronic processes, indicating
a larger role than previously thought. The high intensity of the neutrino flux
below 100 TeV in comparison to the Fermi data might indicate that these
sources are even more efficient neutrino than gamma-ray sources [112, 113].

IceCube is developing methods that enable real-time multiwavelength
observations in cooperation with astronomical telescopes, to identify the
sources and build on the discovery of cosmic neutrinos to launch a new era in
astronomy [114, 115]. We will return to a coincident observation of a flaring
blazar on September 22, 2017, further on.

####  2 IceCube Neutrinos and Ultra-High-Energy Cosmic Rays

The charged pion production rate $Q_{\pi^{\pm}}$ is proportional to the
density of the cosmic-ray nucleons in the beam, $Q_{N}$, by a “bolometric”
proportionality factor $f_{\pi}$. For a target with nucleon density $n$ and
extension $\ell$, the efficiency factor for producing pions is $f_{\pi}\simeq
1-\exp(-\kappa\ell\sigma n)$, where $\kappa$ is the inelasticity, i.e., the
average relative energy loss of the leading nucleon going into the production
of pions, $(\sigma n)^{-1}$ its interaction length with $\sigma$ the cross
section for either $p\gamma$ or $pp$ interactions. The pion production
efficiency $f_{\pi}$ normalizes the conversion of cosmic-ray energy into pion
energy on the target as:

$E_{\pi}^{2}Q_{\pi^{\pm}}(E_{\pi})\simeq
f_{\pi}\,\frac{K_{\pi}}{1+K_{\pi}}\,\left[E^{2}_{N}Q_{N}(E_{N})\right]_{E_{N}=E_{\pi}/x_{\pi}}\,.$
(28)

We already introduced the pion ratio $K_{\pi}$ in the previous section, with
$K_{\pi}\simeq 2$ for $pp$ and $K_{\pi}\simeq 1$ for $p\gamma$ interactions.
The factor $x_{\pi}$ denotes, as before, the average inelasticity per pion
that depends on the average pion multiplicity $n_{\pi}$. For, both, $pp$ and
$p\gamma$ interactions this can be approximated as
$x_{\pi}=\kappa/n_{\pi}\simeq 0.2$. The average energy per pion is then
$\langle E_{\pi}\rangle=x_{\pi}E_{N}$ and the average energy of the pionic
leptons relative to the nucleon is $\langle E_{\nu}\rangle\simeq\langle
E_{\pi}\rangle/4=(x_{\pi}/4)E_{N}\simeq 0.05E_{N}$.

In general, the cosmic ray nucleon emission rate, $Q_{N}$, depends on the
composition of the highest energy cosmic rays and can be obtained by
integrating the measured spectra222Note that the integrated number of nucleons
is linear to mass number, $\int{\rm d}EQ_{N}(E)=\sum_{A}A\int{\rm
d}EQ_{A}(E)$. of nuclei with mass number $A$ as
$Q_{N}(E_{N})=\sum_{A}A^{2}Q_{A}(AE_{N})$. In the following we will derive a
upper limit on diffuse neutrino fluxes under the assumption that the highest
energy cosmic rays are mostly protons [116, 117]. The local emission rate
density, $\mathcal{Q}=\rho_{0}Q$, is at these energies insensitive to the
luminosity evolution of sources at high redshift and can be estimated to be at
the level of $\left[E_{p}^{2}\mathcal{Q}_{p}(E_{p})\right]_{10^{19.5}{\rm
eV}}\sim(0.5-2.0)\times 10^{44}{\rm erg}/{\rm Mpc}^{3}/{\rm yr}$ [95, 118,
119]. Note, that composition measurements indicate that the mass composition
above the ankle also requires a contribution of heavier nuclei. However, the
estimated local power density based on proton models is a good proxy for that
of cosmic ray models that include heavy nuclei, as long as the spectral index
is close to $\gamma\simeq 2$. For instance, a recent analysis of Auger [120]
provides a solution with spectral index $\gamma\simeq 2.04$ and a combined
nucleon density of $[E_{N}^{2}\mathcal{Q}_{N}(E_{N})]_{10^{19.5}{\rm eV}}\sim
2.2\times 10^{43}\,{\rm erg}/{\rm Mpc}^{3}/{\rm yr}$.

We construct the diffuse neutrino flux from the contribution of individual
sources. A neutrino point-source (PS) at redshift $z$ with spectral emission
rate $Q_{\nu_{\alpha}}$ contributes a neutrino flux summed over flavors (in
units ${\rm GeV}^{-1}{\rm cm}^{-2}{\rm s}^{-1}$)

$\phi^{\rm PS}_{\nu}(E_{\nu})=\frac{(1+z)^{2}}{4\pi
d^{2}_{L}(z)}\sum_{\alpha}Q_{\nu_{\alpha}}((1+z)E_{\nu})\,,$ (29)

where $d_{L}$ is the luminosity distance

$d_{L}(z)=(1+z)\int\limits_{0}^{z}\frac{{\rm d}z^{\prime}}{H(z^{\prime})}\,.$
(30)

Here, the Hubble parameter $H$ has a local value of $c/H_{0}\simeq 4.4$ Gpc
and scales with redshift as $H^{2}(z)=H^{2}_{0}[(1+z)^{3}\Omega_{\rm
m}+\Omega_{\Lambda}]$, with $\Omega_{\rm m}\simeq 0.3$ and
$\Omega_{\Lambda}\simeq 0.7$ assuming the standard $\Lambda$CDM cosmological
model [121]. Note that the extra factor $(1+z)^{2}$ in Eq. (29) follows from
the definition of the luminosity distance and accounts for the relation of the
energy flux to the differential neutrino flux $\phi$. The diffuse neutrino
flux from extragalactic sources is given by the integral over co-moving volume
${\rm d}V_{c}=4\pi(d_{L}/(1+z))^{2}{\rm d}z/H(z)$. Weighting each neutrino
source by its density per co-moving volume $\rho(z)$ gives [122]

$\phi_{\nu}(E_{\nu})=\frac{c}{4\pi}\int_{0}^{\infty}\frac{{\rm
d}z}{H(z)}\mathcal{\rho}(z)\sum_{\alpha}Q_{\nu_{\alpha}}((1+z)E_{\nu})\,,$
(31)

and, assuming that $Q_{\nu_{\alpha}}$ follows a power law $E^{-\gamma}$,

$\frac{1}{3}\sum_{\alpha}E_{\nu}^{2}\phi_{\nu_{\alpha}}(E_{\nu})=\frac{c}{4\pi}\frac{\xi_{z}}{H_{0}}\frac{1}{3}\sum_{\alpha}E_{\nu}^{2}\mathcal{Q}_{\nu_{\alpha}}(E_{\nu})\,,$
(32)

where $\mathcal{Q}_{\nu_{\alpha}}=\rho_{0}Q_{\nu_{\alpha}}$ is the neutrino
emission rate density and

$\xi_{z}=\int_{0}^{\infty}{\rm
d}z\frac{(1+z)^{-\Gamma}}{\sqrt{\Omega_{\Lambda}+(1+z)^{3}\Omega_{\rm
m}}}\frac{\rho(z)}{\rho_{0}}\,.$ (33)

A spectral index of $\gamma\simeq 2.0$ and no source evolution in the local
($z<2$) universe, $\rho(z)=\rho_{0}$, yields $\xi_{z}\simeq 0.5$. For sources
following the star-formation rate, $\rho(z)=(1+z)^{3}$ for $z<1.5$ and
$\rho(z)=(1+1.5)^{3}$ for $1.5<z<4$, with the same spectral index yields
$\xi_{z}\simeq 2.6$.

We can now derive the relation between the diffuse neutrino flux from the
cosmic ray rate density by combining Eqs. (26) and (28):

$\frac{1}{3}\sum_{\alpha}E_{\nu}^{2}\phi_{\nu_{\alpha}}(E_{\nu})\simeq 3\times
10^{-8}f_{\pi}\left(\frac{\xi_{z}}{2.6}\right)\left(\frac{[E^{2}_{p}\mathcal{Q}_{p}(E_{p})]_{E_{p}=10^{19.5}{\rm
eV}}}{10^{44}\,{\rm erg}/{\rm Mpc}^{3}/{\rm yr}}\right)\frac{\rm GeV}{\rm
cm^{2}\,s\,sr}\,.$ (34)

Here, we have assumed $pp$ interactions with $K_{\pi}=2$. The calorimetric
limit, previously discussed corresponds to the assumption that $f_{\pi}\to 1$,
and is also referred to as the Waxman-Bahcall “bound” [116, 117]. Sources with
$f_{\pi}>1$ are referred to as “dark” sources; their thick target efficiently
converts nucleons to neutrinos and rendering them opaque to high-energy
photons. Accelerator-based beam dumps are the ultimate dark sources of
neutrinos.

It is intriguing that the observed intensity of diffuse neutrinos indicates
that $f_{\pi}\simeq 1$. The correspondence is illustrated by the green lines
in Fig. 22 showing a parametrization that accounts for the most energetic
cosmic rays (green data). Note, that the cosmic ray data below $10^{10}$ GeV
are not described by this fit and must be supplied by additional sources,
e.g., in our own Galaxy. Assuming that the cosmic ray energy density is
transformed in neutrinos we derive the maximal neutrino emission (green dashed
line). Interestingly, the observed neutrino flux saturates this calorimetric
limit. It is therefore possible that the highest energy cosmic rays and
neutrinos have a common origin. If this is the case, the neutrino spectrum
beyond 200 TeV should reflect the energy-dependent release of cosmic rays from
the calorimeters. Future studies of the neutrino spectrum beyond 1 PeV can
provide supporting evidence for this and, in particular, the transition to a
thin environment ($f_{\pi}\ll 1$), that is a necessary condition for the
emission of the highest energy cosmic rays by the source, implies a break or
cutoff in the neutrino spectrum.

Note that the proton model in Fig. 22 also contributes to the flux of EeV
neutrinos shown as a dotted green line. Ultra-high-energy cosmic rays are
strongly attenuated by resonant interactions with background photons, as first
pointed out by Greisen, Zatsepin and Kuzmin [123, 124] (GZK). This GZK
mechanism is responsible for the suppression of the proton flux beyond
$5\times 10^{10}$ GeV (“GZK cutoff”) in Fig. 22 (green solid line) and
predicts a detectable flux of cosmogenic neutrinos [56] (green dotted line) in
a kilometer-scale detector. In case a significant fraction of the highest
energy cosmic rays are heavy nuclei as indicated by the Auger data, the GZK
flux is reduced accordingly because the individual nucleons fall below the
threshold for producing neutrinos. This may explain the failure of IceCube to
observe GZK neutrinos at the level predicted in Fig. 22.

Figure 23: The effective local density and (maximal) neutrino luminosity of
various neutrino source candidates from reference 125. The green solid (green
dotted) line shows the local density and luminosity of the population of
sources responsible for the diffuse neutrino flux of $E^{2}\phi\simeq
10^{-8}~{}{\rm GeV}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}~{}{\rm sr}^{-1}$ observed
with IceCube, assuming source evolution following the star-formation rate
($\xi_{z}\simeq 2.6$) or no source evolution ($\xi_{z}\simeq 0.5$),
respectively. The grey-shaded area indicates source populations that are
excluded by the non-observation of point sources in the Northern Hemisphere
($f_{\rm sky}\simeq 0.5$) with discovery potential $E^{2}\phi^{\rm PS}\simeq
2\times 10^{-12}~{}{\rm TeV}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$ [126].

####  3 Pinpointing the Astrophysical Sources of Cosmic Neutrinos: Relating
the Point Source and Diffuse Fluxes

The diffuse flux of neutrinos measured by IceCube provides a constraint on the
flux from the individual sources that it is composed of. In the sense of
Olbers’ paradox, one can investigate under what circumstances IceCube can
detect the neutrino emission from individual, presumably nearby, point sources
that contribute to the quasi-diffuse emission.

One can relate the average luminosity of individual neutrino sources to the
diffuse flux

$\frac{1}{3}\sum_{\alpha}E_{\nu}^{2}\phi_{\nu_{\alpha}}(E_{\nu})\simeq\frac{c}{4\pi}\frac{\xi_{z}}{H_{0}}\rho_{0}\frac{1}{3}\sum_{\alpha}E_{\nu}^{2}Q_{\nu_{\alpha}}(E_{\nu})\,,$
(35)

where the $\xi_{z}$ summarizes the evolution of the sources, $H_{0}$ is the
Hubble constant, and $\rho_{0}$ is the effective source density.

Equation (35) relates the average luminosity of individual neutrino sources to
the diffuse flux on the left hand side, which is measured by the experiment to
be at the level of $E^{2}\phi_{\nu}\simeq 10^{-8}~{}{\rm GeV}{\rm cm}^{-2}{\rm
s}^{-1}{\rm sr}^{-1}$ for energies in excess of $\simeq 100$ TeV; see Fig. 18.
From the measurement, we can infer the average emission from a single source
[122] as a function their local density,

$\frac{1}{3}\sum_{\alpha}E_{\nu}^{2}Q_{\nu_{\alpha}}(E_{\nu})\simeq 1.7\times
10^{45}\left(\frac{\xi_{z}}{2.6}\right)^{-1}\left(\frac{\rho_{0}}{10^{-10}{\rm
Mpc}^{-3}}\right)^{-1}{\rm erg}\,{\rm s}^{-1}\,.$ (36)

This relation builds in the very strong assumption that all sources
corresponding to some astronomical classification are identical.

For a homogenous distribution of sources, we expect, within the partial field
of view of the full sky, $f_{\rm sky}$, of the telescope, one source within a
distance $d_{1}$ determined by

$f_{\rm sky}4\pi d_{1}^{3}/3\,\rho_{0}=1\,.$ (37)

In other words, $d_{1}$ defines the volume containing one nearby source for a
homogeneous source density $\rho_{0}$. Defining $\phi_{1}$ as the flux of a
source at distance $d_{1}$, given by Eq. (29) with $z\simeq 0$ and
$d_{L}=d_{1}$, we can write the probability distribution $p(\phi)$ of finding
the closest source of the population with a flux $\phi$ as

$p(\phi)=\frac{3}{2}\frac{1}{\phi}\left(\frac{\phi_{1}}{\phi}\right)^{\frac{3}{2}}e^{-\left(\frac{\phi_{1}}{\phi}\right)^{\frac{3}{2}}}\,.$
(38)

(This result takes into account fluctuation in the source distribution; for
details see appendix of reference 122). The average flux from the closest
source is then $\langle\phi\rangle\simeq 2.7\,\phi_{1}$, with median
$\phi_{\rm med}\simeq 1.3\,\phi_{1}$. Applying this to the closest source
introduced above, we obtain its neutrino flux:

$\frac{1}{3}\sum_{\alpha}E_{\nu}^{2}\phi_{\nu_{\alpha}}\simeq 2\times
10^{-12}\left(\frac{f_{\rm
sky}}{0.5}\right)^{\frac{2}{3}}\left(\frac{\xi_{z}}{2.6}\right)^{-1}\left(\frac{\rho_{0}}{10^{-10}{\rm
Mpc}^{-3}}\right)^{-\frac{1}{3}}{\rm TeV}\,{\rm cm}^{-2}\,{\rm s}^{-1}\,.$
(39)

Interestingly, this value is not far from IceCube’s average point-source
discovery potential at the level of $2\times 10^{-12}\,{\rm TeV}\,{\rm
cm}^{-2}\,{\rm s}^{-1}$ in the Northern Hemisphere [80].

In Fig. 23, we show the local density and luminosity of theorized neutrino
sources [125]. The grey-shaded region is excluded by the failure to observe
these sources as individual point sources. The green lines show the
combination of density and luminosity for which sources produce a diffuse
neutrino flux at the level observed by IceCube. We show the result for a
source densities following the star formation rate (solid line) or no
evolution (dotted line). We conclude that IceCube is presently sensitive to
source populations with local source densities smaller than, conservatively,
$10^{-8}\,{\rm Mpc}^{-3}$. Much lower local densities, like FSRQs, are
challenged by the nonobservation of individual sources. Some source classes,
like Fanaroff-Riley (FR) radio galaxies, have an estimated neutrino luminosity
that is likely too low for the observed flux. This simple estimate can be
refined by considering not only the closest source of the population, but the
combined emission of known local sources; see, e.g., reference 122.

The luminosity-density parameter space discussed above is often exploited to
study sources identified in astronomical catalogs as potential contributors to
the cosmic neutrino flux. However, it is critical to note that astronomical
classifications are meant to explain the collective behavior of sources with
regard to common spectral features and power. Thus, they may lack the
necessary condition for a neutrino source: provide a beam dump with sufficient
target density to produce neutrinos. Spectral features, deviating from a
simplified assumption of the single power law, together with the variations of
the power and target densities, modifies the allowed parameter space. In
addition, note that the source luminosity distribution is neglected by
assuming an average luminosity for all sources. Finally, the evolution of the
sources plays a key role in the allowed parameter space. As such, the built-in
assumption of a specific cosmological evolution that is common to all sources
is problematic.

In addition to the strong dependence of these bands on the spectrum of the
sources and their evolution, we should note that the all-sky untriggered
searches for the sources of high-energy neutrinos allocate a large number of
trial factors that penalizes the significance for any potential signal excess.
Therefore, the presence of outlying sources in the neutrino data could result
in the violation of such bands when additional information provided by other
messengers distinguishes a specific source or direction in the sky.

Is it possible that the sources of the extragalactic cosmic rays are
themselves neutrino sources? We previously derived the emission rate density
of nucleons from the measured cosmic-ray spectrum [95, 118]

$\mathcal{L}_{p}=\rho_{0}E^{2}_{p}Q_{p}(E_{p})\simeq{(1-2)\times 10^{44}\,{\rm
erg}\,{\rm Mpc}^{-3}\,{\rm yr}^{-1}}\,.$ (40)

Combining this with Eq.28 we can derive the diffuse neutrino flux:

$\frac{1}{3}\sum_{\alpha}E_{\nu}^{2}\phi_{\nu_{\alpha}}(E_{\nu})\simeq{f_{\pi}}{\frac{\xi_{z}K_{\pi}}{1+K_{\pi}}}(2-4)\times
10^{-8}\,{\rm GeV}\,{\rm cm}^{-2}\,{\rm s}^{-1}\,{\rm sr}\,.$ (41)

The equation has been rewritten and some notation adjusted in order to
accommodate both $pp$ and $\gamma p$ interactions. Counting particles we
derive that

$\frac{1}{3}\sum_{\alpha}E_{\nu}Q_{\nu_{\alpha}}(E_{\nu})=E_{\pi}Q_{\pi}\,,$
(42)

from which the relation for energy follows by multiplying both sides with
$E_{\nu}$; an additional factor of 1/4 multiplies the right-hand side
accounting for the ratio of neutrino to pion energy within the approximations
routinely used throughout.

The requirement $f_{\pi}\leq 1$ may limit the neutrino production by the
actual sources of the cosmic rays as pointed out by the seminal work by Waxman
and Bahcall [116]. For optically thin sources, $f_{\pi}\ll 1$, neutrino
production is only a small by-product of the acceleration process. The energy
loss associated with pion production must not limit the sources’ ability to
accelerate the cosmic rays. On the other hand, optically thick sources,
$f_{\pi}\simeq 1$, are more efficient neutrino emitters. Realistic sources of
this type need different zones, one zone for the acceleration process
($f_{\pi}\ll 1$) and a second zone for the efficient conversion of cosmic rays
to neutrinos ($f_{\pi}\simeq 1$). An example for this scenario are sources
embedded in starburst galaxies, where cosmic rays can be stored over
sufficiently long timescales to yield significant neutrino production. A more
relevant example is the multimessenger source TXS 0506+056 which we will
discuss in detail further on.

For $\xi_{z}\simeq 2.4$ and $K_{\pi}\simeq 1-2$, the upper bound resulting
from Eq. (41) and $f_{\pi}=1$ is at the level of the neutrino flux observed by
IceCube. Therefore, it is possible that the observed extragalactic cosmic rays
and neutrinos have the same origin. A plausible scenario is a “calorimeter” in
which only cosmic rays with energy below a few $10$ PeV interact efficiently.
An energy dependence of the calorimetric environment can be introduced by
energy–dependent diffusion. If $D(E)$ is the diffusion coefficient, then the
timescale of escape from the calorimeter is given by the solution to
$6D(E)t=d^{2}$, where $d$ is the effective size of the region. Typically, we
have $D(E)\propto E^{\delta}$ with $\delta\simeq 0.3-0.6$. In the following,
we again consider the case of protons. Taking $\sigma_{pp}\simeq 8\times
10^{-26}\,{\rm cm}^{2}$ at $E_{p}=100$ PeV and the diffusion coefficient of
$D(E_{p})\simeq D_{\rm GeV}(E_{p}/1{\rm GeV})^{1/3}$, the $pp$ thickness can
be expressed as $\tau_{pp}\simeq ctn_{\rm gas}\sigma_{pp}$ or

$\tau_{pp}\simeq 0.18\left(\frac{d}{100\,{\rm
pc}}\right)^{2}\left(\frac{D_{\rm GeV}}{10^{26}\,{\rm cm}^{2}/{\rm
s}}\right)^{-1}\left(\frac{E_{p}}{10~{}{\rm
PeV}}\right)^{-1/3}\left(\frac{n}{100\,{\rm cm}^{-3}}\right)\,.$ (43)

Here, we have used feasible parameters of starburst galaxies [106, 110].
Therefore, depending on the calorimetric environment, it is possible that the
flux below a few PeV is efficiently converted to neutrinos and contributes to
the TeV–PeV diffuse emission observed by IceCube.

###  6 The First-Identified Cosmic-Ray Accelerator: the Rotating Supermassive
Black Hole TXS 0506+056

The qualitative matching of the energy densities of photons and neutrinos,
discussed in the previous section, may suggest that the unidentified neutrino
sources contributing to the diffuse flux might have already been observed as
strong gamma-ray emitters. Theoretical modeling [127, 128] and recent data
analyses [129, 130, 131] show that Fermi’s extragalactic gamma-ray flux is
dominated by blazars. However, a dedicated IceCube study [132] of Fermi-
observed blazars shows no evidence of neutrino emission from these source
candidates. The inferred limit on their quasi-diffuse flux leaves room for a
significant contribution to IceCube’s diffuse neutrino flux at the 10% level,
and increasing towards PeV. The multimessenger campaign launched by the
neutrino alert IC170922 not only identified the first source of cosmic
neutrinos, it will also shed light on this apparent contradiction.

Since 2016, the IceCube multimessenger program has grown from Galactic
supernova alerts and early attempts to match neutrino observations with early
LIGO/Virgo gravitational wave candidates to a steadily expanding set of
automatic filters that selects in real time rare very high energy events that
are potentially cosmic in origin. Within less than a minute of their detection
in the deep Antarctic ice, the astronomical coordinates are sent to the Gamma-
ray Coordinate Network for potential follow-up by astronomical telescopes.

On September 22, 2017, the tenth such alert, IceCube-170922A [133], reported a
well-reconstructed muon that deposited 180 TeV inside the detector, and with
an estimated neutrino energy of 290 TeV. Its arrival direction aligned with
the coordinates of a known blazar, TXS 0506+056, to within $0.06^{\circ}$. The
source was “flaring” with a gamma ray flux that had increased by a factor 7 in
recent months. A variety of estimates converge on the probability of order
$10^{-3}$ that the coincidence is accidental. The identification of the
neutrino with the source reaches the level of evidence, but no more. What
clinches the association is a series of subsequent observations, culminating
with the optical discovery of a switch of the source from an off to an on
state two hours after the emission of IC190922, conclusively associating the
neutrino with TXS 0506+056:

  * •

The redshift of the host galaxy was measured to be $z\simeq 0.34$ [134]. It is
important to realize that nearby blazars like the Markarian sources are at a
redshift that is ten times smaller, and therefore TXS 0506+056, with a similar
flux despite the greater distance, is one of the most luminous sources in the
Universe. It must belong to a special class of sources that accelerate proton
beams in dense environments, revealed by the neutrino. The source is somehow
special, and this eliminates any conflict between its observation and the lack
of correlation between the arrival directions of IceCube neutrinos and the
bulk of the blazars observed by Fermi [132].

  * •

Originally detected by NASA’s Fermi [135] and Swift[136] satellite telescopes,
the alert was followed up by the MAGIC and other air Cherenkov telescopes
[137]. MAGIC detected the emission of gamma rays with energies exceeding 100
GeV starting five days after the observation of the neutrino, indicating that
the source is a relatively rare TeV blazar given the large redshift of the
source. After correcting for the absorption on the microwave background, we
conclude that the source is a TeV blazar.

  * •

Given where to look, IceCube searched its archival neutrino data up to and
including October 2017 for evidence of neutrino emission at the location of
TXS0506+056. When searching the sky for point sources of neutrinos, two
analyses have routinely been performed: one that searches for steady emission
of neutrinos and one that searches for flares over a variety of timescales.
Evidence was found for 19 high-energy neutrino events on a background of less
than 6 in a burst lasting 110 days. This burst dominates the integrated flux
from the source over the last 9.5 years for which we have data, leaving the
2017 flare as a second subdominant feature. We note that this analysis applied
a published prescription to data; the chances that this observation is a
fluctuation are small.

  * •

Radio interferometry images of the source revealed a jet that loses its tight
collimation beyond 5 milliarcseconds running into target material that creates
the opportunity for producing the neutrinos. The origin of this target
material is still a matter of debate, and speculations include the merger with
another galaxy affecting the jet from the dominant supermassive black hole.
Alternatively, in a structured jet, the accelerated protons may collide with a
slower moving and denser region of jetted photons. The jet may interact with
the dense molecular clouds of a star-forming region, or simply with
supermassive stars in the central region of the host galaxy [138, 139]. The
VLBA observations not only identify a striking feature in the jetted material,
but also reveal that the neutrino burst occurs at the peak of enhanced radio
emission at 15 GHz, which started five years ago, see Fig. 24. The radio flare
may be a signature of a galaxy merger; correlations of radio bursts with the
process of merging supermassive black holes have been anticipated [140].

  * •

The robotic optical telescope MASTER network has been monitoring the source
since 2005 and found the strongest time variation of the source over a period
of two hours after the emission of IC170922, with a second variation following
the 2014-15 burst [141]. The blazar switches from the off to the on state two
hours after the emission of the neutrino. The observation conclusively
associates the source with the neutrino.

Figure 24: TXS 0506+056 radio light curve from Owen Valley Radio Observatory
(OVRO) at 15 GHz (red). THe dashed line illustrates the pattern of the radio
flux density. The 2014/15 110-day neutrino flare (yellow band) and the
IceCube-170922A episodes are shown. Radio data indicate that the neutrinos
arrive during periods of enhanced radio emission.

Additionally, it is important to note the striking fact that all high-energy
spectra, for both photons and neutrinos and for both the 2014 and 2017 bursts,
are consistent with a hard $E^{-2}$ spectrum, which is expected for a cosmic
accelerator. In fact, the gamma-ray spectrum hints at flattening beyond that
during the 110-day period of the 2014 burst [142, 143].

The problem with modeling TXS 05060+056 as a blazar is that it is not a blazar
at the times of neutrino emission as shown by the multimessenger observation
itemized above. It is evident that a source that is transparent to high-energy
gamma rays is unlikely to host the target material to produce neutrinos with
the opacity for $\gamma\gamma$ interactions two orders of magnitude larger
than that for $p\gamma$ interactions. We will work through these arguments
quantitatively further on.

In this context, an alert recorded by IceCube on July 30, 2019, provides
striking support for the idea that cosmic neutrinos are produced by
temporarily gamma-suppressed blazars. IC190730 and IC170922 are the two
highest energy alerts recorded so far. A well-reconstructed 300-TeV muon
neutrino is observed in spatial coincidence with the blazar PKS 1502+106
[144]. OVRO radio observations [145] show that the neutrino is coincident with
the highest flux density of a flare at 15 GHz that started five years ago
[146], matching the similar long-term radio outburst observed from TXS
0506+056. Even more intriguing is that the gamma ray flux observed by Fermi
shows a clear minimum at the time that the neutrino is emitted; see Fig. 25.
At the time that the target crosses the jet and produces a neutrino beam, the
high-energy gamma rays are absorbed and their energy cascades to energies
below the Fermi threshold, i.e., MeV or lower. For further discussion, see
reference 147.

Figure 25: Temporal variation of the $\gamma$-ray and radio brightness of PKS
1502+106. a: Fermi-LAT likelihood light curve integrated between 100 MeV and
300 GeV (marked by black dots with error bars). b: OVRO flux density curve of
PKS 1502+106 plotted with light blue dots, that is superimposed by the radio
flux density curve binned to the Fermi-LAT light curve (marked with dark blue
squares). The detection time of the neutrino IC-190730A is labeled by a
vertical purple line.

Other IceCube alerts have triggered interesting observations. Following up on
a July 31, 2016, neutrino alert, the AGILE collaboration, which operates an
orbiting X-ray and gamma-ray telescope, reported a day-long blazar flare in
the direction of the neutrino one day before the neutrino detection [148].
Even before IceCube issued automatic alerts, in April 2016, the TANAMI
collaboration argued for the association of the highest energy IceCube event
at the time, dubbed “Big Bird,” with the flaring blazar PKS B1424-418 [149].
The event was produced at a minimum of the Fermi flux [147], as was the case
for PKS 1502 +106. AMANDA, IceCube’s predecessor, observed three neutrinos in
coincidence with a rare flare of the blazar 1ES 1959+650, detected by the
Whipple telescope in 2002 [150]. However, none of these identifications reach
the significance of the observations triggered by IceCube-170922A.

Note that a subset of blazars, around $1\sim 10\,\%$ of all blazars, bursting
once in 10 years at the levels of TXS 0506+056, can accommodate the diffuse
cosmic neutrino flux observed by IceCube. The energy of the neutrino flux
generated by the flaring blazars is also at the same level as the flux in
extragalactic cosmic rays. In order to calculate the flux of the high-energy
neutrinos from a population of sources, we use the formalism developed in the
previous section [6] to relate the diffuse neutrino flux to the energy
injection rate of the cosmic rays and their efficiency to transfer this energy
from protons to neutrinos. For a population of sources with source density
$\rho$ and neutrino luminosity $L_{\nu}$, the diffuse neutrino flux is

$\displaystyle E^{2}\frac{dN}{dE}=\frac{1}{4\pi}\int d^{3}r\frac{L_{\nu}}{4\pi
r^{2}}\rho\,.$ (44)

which can be simplified into

$\displaystyle
E^{2}\frac{dN}{dE}=\frac{c}{4\pi}\,t_{H}\,\xi\,L_{\nu}\,\rho\,,$ (45)

where $\xi$ it the result of the integration over the redshift history of the
sources; see Eq. 68. We revise the equation to account for the duration of the
flares $\Delta t$, the total time of observation $T_{obs}$, and the fraction
of sources $\mathcal{F}$:

$\displaystyle
E^{2}\frac{dN}{dE}=\frac{c}{4\pi}\,t_{H}\,\xi\,L_{\nu}\,\rho\,\frac{\Delta
t}{T}\,\mathcal{F}\,,$ (46)

corresponding to

$\displaystyle 3\times
10^{-11}\,{\rm{TeVcm^{-2}s^{-1}sr^{-1}}}=\frac{\mathcal{F}}{4\pi}\bigg{(}\frac{R_{H}}{4.3\,\rm{Gpc}}\bigg{)}\bigg{(}\frac{\xi}{0.7\rm{}}\bigg{)}\bigg{(}\frac{L_{\nu}}{1.2\times
10^{47}\,\rm{erg/s}}\bigg{)}$
$\displaystyle\times\bigg{(}\frac{\rho}{10^{-8}\,\rm{Mpc^{-3}}}\bigg{)}\bigg{(}\frac{\Delta
t}{110\,{\rm d}}\frac{10{\,\rm yr}}{T_{\rm obs}}\bigg{)}$ (47)

which results in $\mathcal{F}=0.05$. In summary, a special class of BL Lac
blazars with a neutrino luminosity similar to TXS 05060+056, can accommodate
the diffuse flux of high-energy cosmic neutrinos observed by IceCube. These
high-energy neutrino flaring sources constitute 5% of the sources.

The energetics in neutrino production from these sources also matches the
energy flux of the highest energy cosmic rays, i.e.,

$\displaystyle
E^{2}\frac{dN}{dE}\simeq\frac{c}{4\pi}\,\bigg{(}\frac{1}{2}(1-e^{-f_{\pi}})\,\xi\,t_{H}\frac{dE}{dt}\bigg{)}\,.$
(48)

From the observed cosmic ray injection rate of $(1-2)\times 10^{44}\,\rm
erg\,Mpc^{-3}yr^{-1}$ we can determine the average pion efficiency of the
neutrino sources that makes the match of Eq. 6 possible:

$\displaystyle\bigg{(}\frac{L_{\nu}}{1.2\times
10^{47}\,\rm{erg/s}}\bigg{)}\bigg{(}\frac{\rho}{10^{-8}\,\rm{Mpc^{-3}}}\bigg{)}\bigg{(}\frac{\Delta
t}{110\,{\rm d}}\frac{10{\,\rm yr}}{T_{\rm
obs}}\bigg{)}\bigg{(}\frac{\mathcal{F}}{0.05}\bigg{)}$
$\displaystyle\simeq\frac{1}{2}(1-e^{-f_{\pi}})\,\frac{dE/dt}{(1-2)\times
10^{44}\,\rm erg\,Mpc^{-3}yr^{-1}}\,.$ (49)

We find that $f_{\pi}>0.8$.

In the section on blazar jets, we discussed how this can be achieved with a
proton beam with low boost factor interacting with the blue photons in the
active galaxy. The jet producing the neutrinos is not transparent to TeV
photons, only to photons with tens of GeV; this is indeed what is observed by
Fermi at the time of the 2014 flare.

That TXS 0506+056 belongs to a special class of sources is reinforced by the
fact that conventional blazar modeling of the multiwavelength spectrum has
been unsuccessful despite the opportunity of tuning 14 free parameters. Blazar
models at best meet the requirements for the 2017 burst by invoking Eddington
bias [151] for the single neutrino produced. The answer to this puzzle could
be a new subclass of high-energy gamma-ray sources, labeled as blazars by
astronomers, that is responsible for producing the high-energy cosmic
neutrinos and cosmic rays observed [152]. In fact, if every source labelled as
a blazar produced neutrinos at the level of TXS 0506+056, they would
overproduce the total diffuse flux observed by IceCube by close to two orders
of magnitude.

The observation that the energy content in neutrinos and very high energy
cosmic rays are similar, underscores the fact that the cosmic rays must be
highly efficient at producing neutrinos, requiring a large target density that
renders them opaque to high-energy gamma rays. A consistent picture emerges
when the source opacity exceeds a value of 0.4 [152], resulting in a gamma-ray
cascade where photons lose energy in the source before cascading to even lower
energies in the extragalactic background light. Most of their energy falls
below the Fermi threshold by the time they reach Earth. The 2014-15 burst
cannot be, and is not, accompanied by a large gamma-ray flare. This is
consistent with the premise that a special class of efficient sources is
responsible for producing the high-energy cosmic neutrino flux seen by
IceCube.

The nature of the special class of sources has not been settled. One
straightforward explanation could be a subclass of blazars selected by
redshift evolution: powerful proton accelerators producing neutrinos may have
been active in the past but are no longer today. This accommodates the large
redshift of TXS 0506+056 [153], which would be the closest among a set of
sources that only accelerated cosmic rays at early redshifts. It is a
requirement that the sources must have a sufficient target density in photons
or protons in order to produce neutrinos at the high flux level observed from
TXS 0506+056 [152]. If neutrino sources only represent a few percent of those
labelled by astronomers as blazars, attempts by IceCube and others to find
correlations between the directions of high-energy neutrinos and $\it{all}$
Fermi blazars must inevitably be unsuccessful.

Merger activity in blazars in not uncommon. In merging galaxies there is
plenty of material for accelerated cosmic rays to interact with the jet. This
fresh material provides optically thick environments, and allows for rapid
variation of the Lorentz factors. A cursory review of the literature on the
production of neutrinos in galaxy mergers is sufficient to conclude that it
can indeed accommodate the observations of both the individual sources
discussed above and the total flux of cosmic neutrinos; see, e.g., references
154, 155, 156.

There is another, possibly related, development associated with the search for
steady neutrino sources that recently delivered a new time-integrated neutrino
sky map covering ten years of IceCube data [157]. Evidence at the 3$\sigma$
level suggests that the neutrino sky map might no longer be isotropic. The
anisotropy results from four sources-TXS 0506+056 among them-that show
evidence for clustering at the 4$\sigma$ level. The strongest of these sources
is the nearby active Seyfert galaxy NGC 1068 (Messier 77) with starburst
activity. There is evidence for shocks near the core and for molecular clouds