Where is the End of the Cosmic-Ray Electron Spectrum?

Abstract

Detecting the end of the cosmic-ray (CR) electron spectrum would provide important new insights. While we know that Milky Way sources can accelerate electrons up to at least ∼\sim1~PeV, the observed CR electron spectrum at Earth extends only up to 5~TeV (possibly 20~TeV), a large discrepancy. The question of the end of the CR electron spectrum has received relatively little attention, despite its importance. We take a comprehensive approach, showing that there are multiple steps at which the observed CR electron spectrum could be cut off. At the highest energies, the accelerators may not have sufficient luminosity, or the sources may not allow sufficient escape, or propagation to Earth may not be sufficiently effective, or present detectors may not have sufficient sensitivity. For each step, we calculate a rough range of possibilities. Although all of the inputs are uncertain, a clear vista of exciting opportunities emerges. We outline strategies for progress based on CR electron observations and auxiliary multi-messenger observations. In addition to advancing our understanding of CRs in the Milky Way, progress will also sharpen sensitivity to dark matter annihilation or decay.

Full text

Where is the End of the Cosmic-Ray Electron Spectrum?
Takahiro Sudoh 1, 2, 3, 4, ∗and John F. Beacom 1, 2, 3, †
1Center for Cosmology and AstroParticle Physics (CCAPP), Ohio State University, Columbus, OH 43210, USA
2Department of Physics, Ohio State University, Columbus, OH 43210, USA
3Department of Astronomy, Ohio State University, Columbus, OH 43210, USA
4Graduate School of Artificial Intelligence and Science, Rikkyo University,
Nishi-Ikebukuro 3-34-1, Toshima-ku, Tokyo 171-8501, Japan
(Dated: August 29, 2023)
Detecting the end of the cosmic-ray (CR) electron spectrum would provide important new insights. While we
know that Milky Way sources can accelerate electrons up to at least ∼1 PeV, the observed CR electron spec-
trum at Earth extends only up to 5 TeV (possibly 20 TeV), a large discrepancy. The question of the end of the
CR electron spectrum has received relatively little attention, despite its importance. We take a comprehensive
approach, showing that there are multiple steps at which the observed CR electron spectrum could be cut off. At
the highest energies, the accelerators may not have sufficient luminosity, or the sources may not allow sufficient
escape, or propagation to Earth may not be sufficiently effective, or present detectors may not have sufficient
sensitivity. For each step, we calculate a rough range of possibilities. Although all of the inputs are uncertain,
a clear vista of exciting opportunities emerges. We outline strategies for progress based on CR electron obser-
vations and auxiliary multi-messenger observations. In addition to advancing our understanding of CRs in the
Milky Way, progress will also sharpen sensitivity to dark matter annihilation or decay.
I. INTRODUCTION
While the Milky Way is known to host powerful acceler-
ators of cosmic-ray (CR) hadrons (protons and nuclei), the
nature of these sources is uncertain because CR directions
are smeared by propagation through magnetic fields [1–4].
Even so, we gain important clues about possible sources from
other observables, including the CR spectrum and composi-
tion. One of the most important clues is the spectral break at
∼3 PeV (the “knee”) in the CR spectrum [5–8], which is com-
monly interpreted as the maximum proton energy that can be
reached in typical accelerators. This sets a scale that should be
predicted by theory; the fact that this energy is so high greatly
restricts the properties of possible sources [9–11].
Similarly, while the Milky Way is also known to host pow-
erful accelerators of CR leptons (electrons and positrons), the
nature of these sources is also uncertain [1–4]. (Hereafter, we
use electrons to mean the sum of electrons and positrons un-
less otherwise specified.) CR electrons have a much lower
flux and a steeper spectrum than CR hadrons [12–19], which
is partially a consequence of CR electrons having greater en-
ergy losses [20]. While this makes CR electron observa-
tions more challenging, it means that the possible sources are
closer, which will allow more detailed observations. Many as-
pects of CR electrons have received attention in the theoretical
literature, e.g., see Refs. [21–65].
In this paper, we address a central question about CR elec-
trons — where does their spectrum end? — that has received
less attention, likely because a clear end has not yet been ob-
served. In direct CR observations, electrons have been ro-
bustly detected up to 5 TeV and possibly 20 TeV [14–19].
However, we have indirect indications that electrons are accel-
erated to much higher energies, based on observations of the
emission they produce due to synchrotron losses and inverse-
Compton scattering [66–73]. For example, luminous sources
like the Crab accelerate electrons up to at least 1 PeV [66]
and less luminous (but more common) sources like Geminga
accelerate electrons up to at least 100 TeV [67]. However, it
is not known if such electrons significantly contribute to the
observed CR electron spectrum. Determining the end of the
CR electron spectrum would provide important new insights
about the sources and propagation of all CR particles. Near
the end of the spectrum, it may be that only one source con-
tributes, which would allow astronomy even without direc-
tionality. And beyond the end, the sensitivity to exotic sources
like dark matter would greatly improve.
Figure 1shows that the question of the end of the CR elec-
tron spectrum depends on four steps: (1) acceleration in the
sources, (2) escape from the sources, (3) propagation from
the sources to Earth, and (4) detection of the electrons. We
tackle these steps in an approximate but comprehensive way,
focusing on establishing a framework to be explored in more
detailed work. For each of the four steps — any of which
could cause the end of the CR electron spectrum — we show
that existing data and theory allow a wide range of possibil-
ities. We outline strategies for resolving these sub-questions
through multi-messenger observations and theory.
Figure 2quantifies the observational status and prospects
for the detection of CR electrons at the highest energies. We
show direct measurements of and limits on the CR electron
spectrum, along with a rough sense of how the experimental
sensitivity is limited by the background due to CR hadrons.
In addition, we show relevant scales that may determine how
astrophysical foregrounds (e.g., due to gamma rays) limit the
sensitivity. The details of our calculations are given in Sec. IV,
where we also discuss the sensitivity of powerful new experi-
ments, especially the Large High Altitude Air Shower Obser-
vatory (LHAASO) [74].
In Sec. II, we discuss CR electron acceleration in and es-
cape from sources. In Sec. III, we discuss CR electron prop-
agation. In Sec. IV, we discuss their detection. Each of these
three sections starts with general considerations, followed by
estimates of the ranges of possibilities. In Sec. V, we discuss
multi-messenger strategies. In Sec. VI, we conclude.
arXiv:2308.13600v1 [astro-ph.HE] 25 Aug 2023

2
YES
End of spectrum at TeV
x
Do sources !
accelerate electrons!
to TeV ?
x
Do -TeV electrons!
escape the source?
x
Do -TeV electrons!
propagate to the!
Earth?
x
Is the -TeV flux !
detectable?
x
YES
YES
NO
NO
NO
End of spectrum at TeV
x
End of spectrum at TeV
x
Sources
(Sec. 2)
Propagation
(Sec. 3)
Detection
(Sec. 4)
NO
End of spectrum at TeV
x
YES
Spectrum observed!
above TeV
x
Acceleration Escape
FIG. 1. Flowchart to show the different steps (see the corresponding sections for details) that could each cause an end to the observed CR
electron spectrum. It is unknown where the spectrum ends and why.
101102103104105106
Energy [GeV]
10 5
10 4
10 3
10 2
E
3 [GeV2cm 2s1sr 1]
e±
e
e+
Isotropic
e
± and
exclusions
Hadronic × 10 6
× 10 5
× 10 4
× 10 3
secondary e+
(model)
LHAASO Disk
LHAASO Disk × 0.1
FIG. 2. Overview of CR electron and positron observations. For electrons (e−only; up to 1.4 TeV) and positrons (e+only; up to 1 TeV) we
show with bold dark-blue curves the fits to AMS data [14,15]. For e−+e+(denoted as e±), we show with black bars the data from Fermi
(up to 2 TeV [16]), CALET (up to 4 TeV [17]), and DAMPE (up to 4 TeV [18]); the data from VERITAS [19] and MAGIC are consistent but
are not shown. HESS results extend up to 20 TeV, but these data (grey open hexagons) are reported only in ICRC 2017 and still not yet in a
publication (we take the data from a theory paper [43]). At the highest energies, we show (gray shading) limits on the isotropic gamma-ray and
electron flux from HAWC [75] (open triangles) and KASCADE [76] (filled triangles; the limits are given relative to the cosmic-ray intensity,
for which we use the fit in Ref. [77]. See also Ref. [40], which used other experiments’ data, including preliminary data). We also show
relevant scales that may set a floor to the sensitivity of CR electron detection (details in Sec. IV). The CR electron spectrum is effectively
unprobed at the highest energies.

3
II. SOURCES
In this section, we discuss possible sources of TeV–PeV
electrons, focusing on acceleration and escape. While we
show results for varying propagation models in Sec. III, here
we fix this to focus on source properties. We use isotropic
diffusion with Diso = 3 ×1029 cm2s−1(E/TeV)1/3, a mag-
netic field strength of 3 µG, and the radiation fields from the
cosmic microwave background (2.7 K, 0.26 eV cm−3) and the
local infrared background (20 K, 0.2 eV cm−3). We take into
account the Klein-Nishina effect following Ref. [78].
Here and in subsequent sections, our calculations are ap-
proximate. We seek to elucidate the range of possibilities,
showing where more detailed calculations — especially ex-
tending to the PeV range, which is not typical — would be
important to reducing uncertainties.
A. General Considerations
In the GeV range, supernova remnants (SNRs) are believed
to be the dominant sources of the CR electron flux. SNRs pro-
duce electron (e−) primaries by directly accelerating them as
well as positron and electron secondaries through the pp inter-
actions of accelerated CR hadrons. The global fit to the GeV-
range CR e−flux allows us to determine the total energy per
source emitted in nonthermal electrons (Ee,SNR ≃1048 erg)
and the typical spectral index (γe,SNR ≃2.6) [55,63]. Present
data suggest that the injected e−spectrum from SNRs may
have a cutoff near 10 TeV, though this is uncertain [63].
In the TeV range, young and middle-aged pulsars, which
produce both electron and positron primaries, may start to be
important. The total GeV flux from pulsars must be much
smaller than that from SNRs because pulsars inject approxi-
mately equal numbers of e−and e+and the observed positron
flux is about an order of magnitude smaller than the electron
(e−) flux. Nevertheless, at high energies, the pulsar contribu-
tion can be important, due the long lifetimes of pulsars and the
hard spectra they produce. It is well established that the max-
imum energies in such sources reach extreme energies. An
iconic young (≃1 kyr) pulsar, the Crab, produces PeV gamma
rays [66], which indicates that electrons of even higher ener-
gies are present. Another well-studied pulsar, Geminga, pro-
duces electrons above about 100 TeV [67], despite its rela-
tively old age (≃300 kyr). Together with a variety of X-ray
and gamma-ray observations, these observations indicate that
pulsars can produce >100 TeV electrons over a time as long
as 300 kyr (and possibly even longer).
The total energy released by a pulsar over its lifetime is
Epul ∼1049(P0/50 ms)−2erg (P0is the initial pulsar spin
period), with electrons carrying ∼10% of this [79]. As the
pulsar power decreases over time as ∝(1 + t/τsd)−2, most
of the power is released before the spindown timescale, which
is τsd ∼4(P0/50 ms)2kyr. The distribution of P0is un-
certain, with possible average values ranging between ∼50–
300 ms [80–85]. The spectral shape of electrons from pulsars
is uncertain; above ∼1 TeV, the spectral index is usually con-
strained to be in the range γe,pul ≃2.0–2.5, though γe,pul ≃
1.5has also been commonly assumed for Geminga [86–92].
What is more uncertain and more important is the spectrum
of escaped electrons. This has gained renewed attention since
the discovery of “TeV halos” around pulsars [67,79,93–98],
which indicate that electrons do escape from the shocked re-
gions in the pulsar wind nebulae but remain efficiently con-
fined in the source vicinity. The degree of confinement by
TeV halos is under debate. Geminga observations are com-
monly interpreted as indicating that the diffusion coefficient
in the source vicinity is suppressed from its typical galac-
tic value by a factor of ∼100–1000, in which case electrons
above ∼10 TeV would lose their energy in the halo (for a
size of 50 pc). However, alternative models with no sup-
pression have been proposed [99], in which case particles of
>1 PeV could escape. Even if the strong suppression is true
for Geminga, it is a separate and unresolved question if this is
ubiquitous [100].
Due to CR electron cooling during propagation (discussed
in Sec. III), only sources within approximately 700 pc can be
relevant for CR electron observations above 10 TeV. Based on
Refs. [22,101], the following sources are often considered as
promising: G114.3+00.3 (700 pc, 7.7 kyr), Vela Jr. (750 pc,
1.7–4.3 kyr; we use 3 kyr), Cygnus Loop (440 pc, 20 kyr),
Monogem (300 pc, 86 kyr, associated with PSR B0656+14),
Vela YZ (300 pc, 11 kyr, associated with PSR B0833-45),
Loop I (170 pc, 200 kyr), and Geminga (250 pc, 330 kyr, as-
sociated with PSR J0633+1746). (See also Ref. [102], which
discusses the possible importance of PSR B1055-52.) We
quote the distances and ages from Refs. [22,101], except for
Geminga, for which we use Ref. [103].
Sources besides those listed above can also be important.
Based on the ATNF pulsar catalog [104], there are 17 pul-
sars within 700 pc that are younger than 1 Myr. These are
only a minor fraction of the total population; due to the the
pulsar emission being beamed, only ∼25% of pulsars are vis-
ible [105], so there could be ∼50 more that remain undiscov-
ered. Their parent SNRs may be difficult to detect if their age
significantly exceeds 100 kyr, which is when SNRs start to
blend into the interstellar medium (for example, the SNR for
Geminga is not observed). In most previous studies, under the
assumption of isotropic diffusion, such “unnamed” sources
are thought to be unimportant compared to the above-listed,
bright, and well-studied objects. (For an exception, see, e.g.,
Ref. [45].) However, diffusion may be strongly anisotropic,
with particles preferentially propagating along magnetic field
lines (see Sec. III). In such scenarios, a random source that
happens to be close to a local magnetic field line can be signif-
icantly more important than sources that are more luminous.
Other class sources may also contribute to the CR elec-
tron and positron fluxes. In particular, millisecond pulsars
could be important [106], because various observations point
to the production of nonthermal particles by millisecond pul-
sars [107–110]. They are individually dim compared to the
young pulsar sources discussed above, but are more numer-
ous, and hence possibly make a sizable contribution to the
observed flux. Sources like Galactic black holes [111–113],
white-dwarf pulsars [114], or exotic sources like dark mat-
ter [115–121] could also be relevant.

4
102103104105106
Electron Energy
Ee
[GeV]
10 5
10 4
10 3
10 2
10 1
E
3
ee
[GeV2cm 2s1sr 1]
Free
Escape
Initial
Loss
Moderate
Loss
Strong
Loss
e±
e
e+
Isotropic
e
± and
exclusions
FIG. 3. For a single pulsar, variations in the CR electron spectrum at Earth in various escape scenarios. Where the predictions for a single
source are above the data, such scenarios would be ruled out. Where the predictions for a single source are below the data, additional studies
would be needed to assess if the scenario can reproduce observations as the sum of multiple sources. As the preliminary HESS results (grey
open hexagons, same as in Fig. 2) could have a huge impact on constraining models, it is critical to test those results.
B. Range of Specific Possibilities
Here, we illustrate that models for particle escape from
pulsars are crucial to where and how the electron spectrum
ends. To do so, we solve the one-dimensional diffusion-
loss equation with the method of Ref. [63] (which builds on
Ref. [21]) to obtain the density of electrons near Earth, ne.
The observed electron intensity (flux per solid angle) is then
Φe=cne/(4π).
While this method is standard, two points need attention.
First, we should propagate particles up to the actual age, Tage,
rather than the observed age, tage,obs, because what appears
in the diffusion-loss equation as tis the physical time. As
the observed age is estimated based on electromagnetic ob-
servations, we expect Tage =tage,obs +R/c, where Ris the
distance to the source. Second, the ordinary diffusion equa-
tion has a “superluminal propagation” problem. This can be
an issue in particular for the case of continuous injection: no
particles that are injected between tage,obs and Tage should
arrive at the Earth, while some do in the naive diffusion ap-
proximation. While phenomenological solutions exist in the
literature [122], this is not yet extended to include the case
of anisotropic diffusion, which we discuss below. Here, we
avoid the superluminal propagation problem by simply setting
the injection term to zero between t=tage,obs and t=Tage.
Though this treatment does not entirely remove the problem,
it does remove particles that are a priori too young and should
not arrive at Earth, while keeping other particles unaffected.
As an example electron source, we consider a pulsar at
440 pc and 20 kyr (similar to the Cygnus Loop), and fix its
total electron energy (for particles above 1 GeV) to 1048 erg
and assume a spindown timescale of 10 kyr. For the spec-
trum of electrons that escape into the interstellar medium, we
estimate results for the following scenarios:
•Free Escape: Particles easily escape from the source,
and the escaped spectrum of electrons is similar to the
accelerated spectrum. We assume γe,pul = 2.3and set
a cutoff energy Ee,cut = 1 PeV. Note that the spectral
index might be even harder.
•Initial Loss: Particles can escape the source only after
the initial evolution of the pulsar wind nebula. This may
occur if the magnetic field is strong and ordered when
the source is young (like the Crab), effectively confin-
ing electrons and leading to strong synchrotron losses.
To estimate this effect, we assume that no particles pro-
duced in the initial 10 kyr of the pulsar’s lifetime can
escape the source region. After 10 kyr, the particle spec-
trum is set to be the same as for the Free Escape case.
•Moderate Loss: Particles lose some energy before es-
caping the source region. The impact of energy loss
at the source has been extensively studied using multi-
zone models (e.g., Ref. [123]); generally, the escaping
spectrum is softer than the accelerated spectrum. We
assume a slightly softer index (γe,pul = 2.5) than above
and a smaller cutoff (Ee,cut = 100 TeV). Note that the
spectral index might be even softer.
•Strong Loss: Particles are efficiently confined by the
TeV halo, losing much of the energy. We assume

5
γe,pul = 2.5and even smaller cutoff Ee,cut = 10 TeV.
Figure 3shows that the end of the electron spectrum is very
sensitive to the details of the particle-escape model. Even for
a single pulsar, the “Free Escape” scenario could be strongly
in tension with the preliminary HESS results above 5 TeV and
mildly in tension with the HAWC limits near 100 TeV. For the
“Initial Loss” scenario, the tension with HESS is removed but
that with HAWC remains, also even for a single pulsar. For
both the “Free Escape” and the “Initial Loss” scenarios, fu-
ture LHAASO measurements at very high energies will be of
crucial importance. Note that the spectral index for these two
cases might be even harder, in which case the tension would
be more severe. For the “Moderate Loss” scenario, the con-
tribution from this single pulsar would fall below all measure-
ments, and the CR electron spectrum would end smoothly.
However, it needs to be asked if the contributions from multi-
ple sources would make make the overall spectrum too large;
in the next section we show that above ∼10 TeV, typically
only one or at most a few pulsar contributes, suggesting that
the “Moderate Loss” scenario is likely allowed even if mul-
tiple pulsars are considered. Finally, for the “Strong Loss”
scenario, the contribution from this pulsar would fade below
∼10 TeV, making pulsars of little importance near the end of
the electron spectrum. In this case, SNRs would likely be
more important sources than pulsars all the way from GeV
energies to the end of the electron spectrum.
As a next step, more detailed theoretical efforts will be im-
portant to investigate the impact of particle escape. Such stud-
ies have been carried out in, e.g., Ref. [30], though their atten-
tion is typically at lower energies than we discuss here. Focus-
ing on the end of electron spectrum would be fruitful because
it is very sensitive to the source model, as we show above.
III. PROPAGATION
In this section, we discuss the propagation of TeV–PeV
electrons for fixed source models. For SNRs, we assume that
electrons are injected into the interstellar medium with a spec-
trum dNe/dEe∝E−2.6
eexp(−Ee/Ee,cut), normalizing it to
1048 erg above 1 GeV and using Ee,cut = 30 TeV. For pul-
sars, we use “Moderate Loss” model above, again fixing the
integrated electron energy to 1048 erg. Again, our focus is on
approximate calculations that show the range of possibilities.
A. General Considerations
Once accelerated by and escaped from sources, charged
particles propagate in the galaxy’s ∼µG interstellar magnetic
field, where they scatter with turbulence, leading to diffu-
sive CR propagation. The nominal Larmor radius is rL≃
1pc (E/PeV)(B/µG)−1and the coherence length of the tur-
bulence is lcoh ∼1–30 pc [124–127], depending on the loca-
tion in the Milky Way. The physical scales are obtained with
phenomenological transport models fit to various primary
and secondary CR data; the obtained diffusion coefficient is
Diso ≃3×1029 cm2s−1(E/TeV)1/3[12,128–131], which
corresponds to a mean free path of lmfp ∼10 pc (E/TeV)1/3.
Below, we consider sources much more distant than lcoh and
lmfp, so that diffusive approximation is valid.
The horizon distance an electron can travel strongly de-
pends on the propagation model. The underlying reason is
that the diffusion and cooling of electrons limit the distance
they can travel (the “horizon distance”) to lcool ∼√2Dtcool,
which is only less than 700 pc above 10 TeV (see below) and
decreases with energy due to tcool ∼60(E/10 TeV)−1kyr.
To become relevant, a source needs to be located within a
volume of V(iso.)
cool ∼πl3
cool (for isotropic diffusion). As the
particle energy increases, the volume shrinks, decreasing the
number of sources inside it. Though we use the global dif-
fusion coefficient from CR hadron data, the local diffusion
coefficient near Earth — needed for CR electrons — could be
different.
While isotropic diffusion is often assumed, anisotropic dif-
fusion may be more appropriate. Isotropic diffusion is appro-
priate if particles experience a number of random field con-
figurations, averaging out the CR directions. However, the
Milky Way’s large-scale, ordered magnetic field may have a
significant impact on CR propagation. The role of anisotropic
diffusion has been intensively studied for CR hadrons [132–
138]. It should be even more critical for CR electrons because
their propagation distances are smaller and the effects can be
large [31,32,59].
Anisotropic diffusion occurs when particle transport is
102103104105106
Electron Energy
Ee
[GeV]
101
102
103
Horizon Distance [pc]
Isotropic
Anisotropic
(Perpendicular)
FIG. 4. Horizon distances for CR electrons in the isotropic and
anisotropic diffusion cases, as labeled, with the bands showing the
effect of varying the magnetic field strength from 1 µG (upper) to
6µG (lower). For the anisotropic case, we take D∥=Diso and
D⊥=D∥/100.The horizon distance for observable CR electron
sources is small, especially at very high energies.

6
more efficient parallel to the magnetic field lines than perpen-
dicular [139–141], as CRs will travel in helical trajectories
along the field lines as opposed to isotropic random walks.
The ratio D⊥/D∥could be as small as ∼1/1000–1/100 and
might be energy-dependent [142–148]. Correspondingly, the
horizon distance perpendicular to the field, l⊥, is smaller than
the parallel one, l∥, by a factor of pD∥/D⊥. Anisotropic
diffusion thus reduces the CR electron horizon volume to
V(aniso.)
cool ∼πl∥l2
⊥, where these separate cooling lengths are
defined analogously to that in the isotropic case. To obtain the
same CR flux at Earth, a smaller number of sources must then
contribute brighter fluxes.
Figure 4shows how the horizon distance can depend on
the propagation model (isotropic or anisotropic) as well as the
uncertain magnetic field strength. In this figure and below, we
assume D∥=Diso (required to be consistent with hadronic
CR data) and D⊥=D∥/100. For the isotropic case, we can
see that the sources must be nearby; above 10 TeV, the horizon
distance is below 700 pc. For the anisotropic case, the source
locations are even more strongly constrained; above 10 TeV,
the horizon distance perpendicular to the magnetic field is less
than 100 pc, meaning that the source has to be very close to a
magnetic field line that passes near Earth.
The number of sources that can contribute to the observed
CR electron flux is determined by the horizon volume. We
estatime the expected source count, n, as
n∼Vcool/Vsrc ,(1)
where Vsrc is the typical volume in which one source is con-
tained. To contribute, a source need not only be within Vco ol;
it must also inject CR electrons within a time tcool before
the present. Therefore, the source age and electron cooling
time must be accounted for in estimating Vsrc. For SNRs, this
means that the ages need to be younger than tcool, so then the
source volume can be estimated as Vsrc ∼VMW/Γsnr tcool,
where VMW is the volume of the Milky Way, for which we
assume 2π(15 kpc)2(500 pc), and Γsnr, for which we assume
0.03 yr−1, the Galactic core-collapse supernova rate [149].
For pulsars, the volume is estimated using the pulsar birth
rate Γpul, which we assume to be the same as Γsnr , and the
lifetime of pulsars as electron sources (i.e., how long a pul-
sar can inject electrons after their birth), tinj. We thus obtain
Vsrc ∼VMW/Γpul (tcool +tinj ). Note that tinj could be energy
dependent (i.e., pulsars might produce PeV electrons only
when they are young, while producing lower-energy electrons
over a much longer time.)
For the example of SNR sources, we find that the num-
bers of contributing sources are quite different for the cases
of isotropic and anisotropic diffusion:
n(iso.)∼100 TeV
E2
,(2)
and
n(aniso.)∼1TeV
E2
.(3)
The energy dependence is due to Vcool ∝E−1and Vsrc ∝
t−1
cool ∝E. For pulsars, we expect that the energy dependence
might be weaker because tcool ≪tinj, so that Vsrc may not
increase as rapidly as ∝E. For more careful estimates, we
would need to take the geometry of the Milky Way disk into
account, but doing so would not change our general points.
The above calculation reveals three energy ranges for how
many sources contribute to the observed CR electron flux:
• Continuum (n≫10): A large number of sources con-
tribute, leading to a smooth spectral shape. The relevant
energy range is E≪3 TeV for isotropic diffusion and
E≪0.3 TeV for anisotropic diffusion.
• Transition (n∼1–10): At higher energies, the spectrum
may show fluctuations at the tens-of-percent level due
to individual source contributions not averaging out.
• Individual (n∼1): Statistically, we expect only a single
source. The relevant energy range is E∼10 TeV for
isotropic diffusion and E∼1 TeV for anisotropic dif-
fusion. In this regime, statistical fluctuations are domi-
nant, and the expected source may be present or not.
These estimates take into account only propagation effects.
If only a fraction of sources in a given class can accelerate
electrons that then escape, then the numbers would be lower.
B. Range of Specific Possibilities
Here, we show that models for particle propagation are cru-
cial to where and how the CR electron spectrum ends. We
calculate results for the following two scenarios:
•Standard Isotropic: We apply the same methods and
parameters as in Sec. II to calculate the fluxes from in-
dividual sources, focusing on the seven specific sources
listed there. We assume that each SNR has an asso-
ciated pulsar, even if there is none reported. On top
of the individual fluxes, we add a continuum of various
sources with a cutoff at 3 TeV, normalizing the intensity
and spectrum from the observed GeV CR e−data.
•Strong Anisotropic: We estimate the flux as in the
“Standard Isotropic” case, but increases it by hand by a
factor of D∥/D⊥, which we assume to be 100; this ap-
propriately takes the impact of anisotropic diffusion as
long as D∥/D⊥is energy independent and r⊥/l⊥≪1,
where r⊥is the distance perpendicular to the magnetic
field. In the presence of strong anisotropic diffusion,
sources that appear promising may make little contri-
bution, while sources that are otherwise unremarkable
could be important. Here, for illustrative purposes,
we assume a single system of a SNR + pulsar of age
100 kyr and distance 800 pc. We add a continuum com-
ponent, as above, but with a lower cutoff of 0.3 TeV.
Figure 5(top panels) shows our calculations for the CR
electron spectra at Earth for the cases of isotropic and
anisotropic diffusion. For the isotropic case, the spectrum is
smooth up to energies >1 TeV. In the regime where the source
count is n∼1–10, fluctuations are present, though they are

7
102103104105106
10 4
10 3
10 2
10 1
E
3
ee
[GeV2cm 2s1sr 1]
n
10
n
1
Isotropic Diffusion
102103104105106
10 1
Anisotropic Diffusion
102103104105106
Electron Energy
Ee
[GeV]
10 3
10 2
10 1
100
Anisotropy
102103104105106
Electron Energy
Ee
[GeV]
Gray:
Limits from Fermi Black:
LHAASO Sensitivity
FIG. 5. How the composite CR electron spectrum (upper panels) and source dipole anisotropies (lower panels) depend on the propagation
model (isotropic diffusion in the left panels, anisotropic diffusion in the right panels). The dash-dotted lines show contributions from individual
SNRs, while dashed lines show those from individual pulsars. Shaded areas show the approximate energy ranges where a single source could
dominate. A key test for the diffusion models is the energy range where only a single source contributes and then the spectrum plummets.
small because each source has a broad spectrum. Note that
while our results broadly agree with the existing data, they do
not reproduce some spectral features, including the possible
drop near 1 TeV; physical mechanisms not included in this
calculation (e.g., electrons from SNRs may have a small cut-
off energy) would be needed to explain such features. At the
highest energies, Vela dominates the end of the spectrum. For
the anisotropic case, the dominance of a single source starts
at ∼1 TeV. This means that the observed electron flux is al-
ready reflecting the end of the electron spectrum and that the
spectral shape is determined by this single source.
Figure 5(bottom panels) shows the expected source dipole
anisotropies. To obtain the total anisotropy, we would have to
add these with vectors, which would require the locations of
each source. At very high energies, where n∼1, the expected
anisotropy can be large. When the anisotropy approaches the
maximum value of unity, its calculated value may not be com-
pletely accurate due to the superluminal diffusion problem
noted above; a more careful treatment is needed.
As a next step, more detailed theoretical efforts will be im-
portant to deepen the knowledge on the role of anisotropic
propagation. This would not only be crucial to where the
CR electron spectrum ends, but also be important to un-
derstand the CR of all particles, including hadrons, as well
as diffuse emission of gamma rays and neutrinos (see, e.g.,
Refs. [150,151]).
IV. DETECTION
In this section, we assess some key factors for the de-
tectability of the CR electron spectrum at the highest en-
ergies, focusing on detector backgrounds and astrophysical
foregrounds. While detailed assessments of sensitivity must
be done by the experimental collaborations, our results show
that significant discovery space seems to be within reach.
A. General Considerations
In the TeV–PeV range, CR electrons must, due to their
small fluxes, be detected through the electromagnetic showers
they induce in Earth’s atmosphere. In the lower TeV range, the
best sensitivity has been through ground-based air-Cherenkov
detectors that register only the light from the showers. At
higher energies, the best sensitivity has been through ground-
based arrays that register the shower particles themselves.
Figure 6shows the uncertainties (dominantly systematics)

8
103104105106
Energy [GeV]
10 5
10 4
10 3
10 2
10 1
E
3 [GeV2cm 2s1sr 1]
equiv.
(IceCube)
LHAASO
Projected
Uncertainty
Range of "Gamma-Ray Floor"Range of "Gamma-Ray Floor"
e±
Isotropic
e
± and
exclusions
secondary e+
(model)
LHAASO Disk
LHAASO Disk × 0.1
FIG. 6. Detection prospects for CR electrons at the highest energies. Starting with a zoomed-in version of Fig. 2with some labels omitted,
we add the “Projected Uncertainty” band from LHAASO (which assumed that the spectrum follows the trend of the HESS preliminary
results) [74]. The yellow region shows our estimate of the range for the CR-electron detection “floor” due to the quasi-isotropic component of
the Milky Way’s gamma-ray emission. Note that the x-axis refers to either electron or gamma-ray energy. At the highest energies, probing CR
electrons requires doing the same for Milky Way gamma rays.
expected for a LHAASO measurement of the CR electron
spectrum [74]. In this estimate, they have assumed that the
CR electron spectrum continues as a power law that largely
follows the preliminary HESS results. To obtain this uncer-
tainty band, LHAASO had to take into account the projected
exposure of the detector as well as the expected detector back-
grounds. The results are quite encouraging for the detection
of the CR electron spectrum at very high energies, even if the
data does not follow the trend they assumed.
The most important background for CR electron detection
is caused by CR hadrons, which also produce showers in
Earth’s atmosphere, though these hadronic showers have dif-
ferent morphological features and have accompanying muons.
LHAASO can powerfully reject hadrons; they foresee hadron
survival fractions of ∼10−3and ∼10−5at 10 TeV and above
100 TeV, respectively, while retaining more than half of CR
electrons [74]. Despite such strong cuts, the level of back-
ground is still high, because the flux of CR hadrons is over-
whelmingly larger the flux of CR electrons. To illustrate this,
in Fig. 2and Fig. 6, we show the CR hadron flux, multi-
plied by the survival factors as marked. To achieve even better
hadron rejection power, LHAASO will also employ new tech-
niques, based on boosted decision trees, to statistically mea-
sure the CR electron fraction, even if individual events cannot
be classified with certainty.
We point out that it is also important to consider astro-
physical foregrounds, by which we mean other fluxes of CR
electrons or effectively identical particles. The most obvious
example is secondary electrons, which are produced by CR
hadrons interacting with gas and producing pions. In Figs. 2
and 6, we show the expected flux that we estimated with a
standard leaky-box model assuming that energy loss domi-
nates over escape for electrons [52]. Our estimates are sim-
ilar to those in Refs. [43,55,152], for which some results
are larger (up to about a factor of 3 at 1 TeV) or smaller than
ours. In any case, secondary electrons produced in the inter-
stellar medium are unimportant as a foreground. Secondary
electrons produced in dense sources could have a larger flux,
but likely still well below the primary CR electron spectrum.
Another foreground — one that we have not seen consid-
ered in this context — is astrophysical gamma rays, which
induce electromagnetic showers in Earth’s atmosphere that
are virtually indistinguishable from those created by CR elec-
trons. Any quasi-isotropic gamma-ray flux would define a
“floor” below which it would be hard to improve sensitivity
to the CR electron flux. At the highest energies shown in
Fig. 6, extragalactic gamma rays are heavily attenuated but
Milky Way gamma rays are only partially attenuated [153–
155]. We detail the possibilities in the next subsection.
If CR electrons are detected at very high energies, it will
be important to test for the presence of anisotropies, as men-
tioned in Sec. III. With many sources, the dipole anisotropies
from each may average out but, as the energy increases, it
is more likely that there is only a single source. As shown

9
in Fig. 5, LHAASO’s estimates [74] of their sensitivity to a
dipole anisotropy are quite encouraging.
B. Range of Specific Possibilities
At TeV–PeV energies, the Milky Way’s diffuse gamma-ray
emission should be morphologically similar to that observed
in the GeV range by Fermi, which found that while the bright-
est emission is from the disk, there is some emission from all
directions. The primary cause is pion-producing hadronic CR
collisions with gas, where the intensity in a given direction is
proportional to the product of the CR density, the gas density,
and the length of the line of sight. In making measurements
of the isotropic CR electron signal, LHAASO will avoid the
Milky Way plane and thus the brightest gamma-ray emission,
but a quasi-isotropic component will remain at high latitudes.
The spectrum of this emission follows that of the CR hadrons,
i.e., ∼E−2.7. It is possible that emission from sources could
be increasingly important at very high energies, due to sources
having harder spectra, e.g., ∼E−2.2; some sources could be at
moderately high latitudes.
Figure 6shows, via the yellow shaded region, a broad
range of possible intensity values for the gamma-ray floor.
As part of that, we show scales for guaranteed but uncertain
diffuse gamma-ray emission. The higher dashed band is set
by LHAASO observations of the Milky Way plane, and the
lower dashed band is set by a rough estimate of how much
dimmer the high-latitude emission could be. We estimate this
component based on the Fermi data, for which the averaged
high-latitude intensity (|b|>20◦) is smaller than the lower-
latitude intensity by approximately a factor of 10 [156–158].
As Fermi high-latitude emission has extragalactic component,
which is absent in the LHAASO band, this treatment might
slightly overestimate the floor.
In addition to these expected components of the gamma-
ray emission, there is some possibility of surprises. While
IceCube’s quasi-isotropic diffuse neutrino flux is likely extra-
galactic, some have argued that it could instead arise, at least
in part, from emission in the Milky Way halo [159–162] (See
also Ref. [163] for a relevant model). This could produce a
quasi-isotropic diffuse gamma-ray flux as well that can be es-
timated by assuming that both the neutrinos and the gamma
rays ultimately arise from pion decays. This flux level also
sets a scale, we show by a dot-dashed line in Fig. 6. Interest-
ingly, the direct upper limits on the isotropic gamma-ray flux
from HAWC and KASCADE are already close to this line,
and LHAASO should soon provide better sensitivity.
As a next step, more detailed theoretical efforts will be
needed to understand the Milky Way’s high-latitude gamma-
ray emission. This would require multi-messenger work, in-
cluding further understanding of diffuse neutrino emission.
V. AUXILIARY MULTI-MESSENGER STRATEGIES
In this section, we briefly discuss multi-messenger strate-
gies to complement direct observations of CR electrons. As
shown in Fig. 1, if we observe an apparent end to the CR elec-
tron spectrum, there are several possible causes. And even if
the CR electron spectrum is detected to very high energies, we
still will not know the details of the various factors that lead
to the observed flux. Therefore, other types of data are needed
to comprehensively disentangle the details of acceleration, es-
cape, propagation, and detection.
Figure 7outlines some likely fruitful approaches.
A. Sources
The maximum energies of electrons accelerated in indi-
vidual sources is most directly probed by observations of
the emission in X-rays (through synchrotron radiation) and
gamma rays (through inverse-Compton scattering). Existing
data indicate that pulsars are efficient accelerators of elec-
trons, reaching energies of at least 1 PeV for the Crab [164,
165]. LHAASO sources seem to contain many other lep-
tonic PeV accelerators [166–168]; understanding them is key
to probing the maximum electron energies accelerated by
sources. Future gamma-ray observations with LHAASO and
SWGO [169] will be especially important due to their sensi-
tivity at very high energies.
It is also important to test particle escape from sources.
Future gamma-ray observations with LHAASO will be espe-
cially important for probing TeV halos, which have low in-
tensities due to their large angular extent. As a complement,
future gamma-ray observations with CTA [170] will allow
studying the morphology of emitting regions, probing parti-
cle escape processes in detail.
B. Propagation
CR hadron data offer great insights about CR propagation.
We expect the propagation of CR hadrons and electrons in
the galactic magnetic field to be virtually identical, except for
the rate of cooling, which is much higher for CR electrons.
The CR hadron data have the advantage that the fluxes are
large, even for subdominant components such as nuclei (both
stable and unstable) and antiprotons. Secondary-to-primary
ratios are important for measuring the global diffusion coef-
ficient and the column densities for interaction and escape.
Future measurements with LHAASO, which is also a cosmic-
ray detector, will be important for their increased precision.
The diffuse emission of gamma rays and neutrinos due to CR
hadron interactions in the interstellar medium is also impor-
tant for studying propagation effects. It may also be possi-
ble to study CR electron propagation through the diffuse syn-
chrotron emission that they produce.
CR anisotropies are another rich source of information and
are important for testing isotropic versus anisotropic diffusion.
For hadrons, the amplitude, phase, and energy dependence of
the dipole anisotropy has been measured [171–175]. The re-
sults are difficult to reconcile with the standard isotropic dif-
fusion models; investigating this problem has provided sig-
nificant insights into the nearby sources and properties of lo-

10
Do sources !
accelerate electrons!
to TeV ?
x
Do -TeV electrons!
escape the source?
x
Do -TeV electrons!
propagate to the!
Earth?
x
Is the -TeV flux !
detectable?
x
Sources Propagation Detection
Acceleration Escape
Gamma-ray and X-ray Sources CR Hadrons,
Diffuse Emission
High-Latitude Emission
FIG. 7. Multi-messenger approach towards understanding the end of the CR electron spectrum. While direct detection of CR electrons is
critical, it will not be enough to separate the physics of these four steps.
cal diffusion tensor [133–138,176–179], which is a key to
understanding the CR electrons. Medium- and small-scale
anisotropies have also been detected in hadronic CR data.
These are useful for studying the properties of turbulent mag-
netic fields in the local interstellar medium [180,181].
C. Detection
The greatest unknown affecting detection is the level of the
astrophysical foregrounds, especially Milky Way gamma rays
at high latitude. To reduce uncertainties, a key step is to first
better understand the emission from the galactic plane at low
latitudes. We need a better separation between source and
diffuse fluxes, plus a consistent picture that relates the diffuse
fluxes of CR, gamma rays, and neutrinos over a wide range of
energies. With this in hand, similar studies are needed for the
high-latitude emission.
The possibility of high gamma-ray floor highlights the im-
portance of space-based, direct CR electron detection. While
existing facilities only can observe up to several TeV, the
proposed mission High Energy cosmic-Radiation Detection
(HERD) could extend the direct measurements beyond tens of
TeV [182], making the detection prospects quite encouraging.
VI. CONCLUSIONS
Questions about the origins of CRs are manifold and long-
standing, primarily because of magnetic deflections during
propagation obscure the directions of sources. For Milky Way
CRs, the questions associated with CR electrons are partic-
ularly challenging, in part because their fluxes are low com-
pared to those for CR hadrons. CR electrons have higher en-
ergy loss rates, which has the benefit that only nearby sources
contribute to the observed flux. In addition, recent rapid ad-
vances in detector sensitivity mean that CR electrons can be
studied with new reach and precision, both directly and with
auxiliary multi-messenger data.
Anticipating significant progress within the next several
years, here we focus on the high-energy end of the CR elec-
tron spectrum, a question that has received less attention than
it should. While CR electrons have only been observed up
to 5 TeV (possibly 20 TeV), it has recently been shown that
multiple Milky Way sources must accelerate electrons up to
at least 1 PeV. The fact that PeV CR electrons have not been
observed could be due to one of four steps that we explore: ac-
celeration, escape, propagation, and detection. If one or more
of these steps “fails,” then the CR electron spectrum can end
at energies well below 1 PeV. We show that for each of these
steps, there are significant but not unbounded uncertainties.
While we must make approximations to calculate the range
of possibilities for each step, an overall clear vista emerges
where decisive progress is within reach.
Determining the energy corresponding to the end of the CR
electron spectrum will provide new insights about many as-
pects of CRs. This may allow astronomy without directional-
ity if it can be shown that only one nearby object could reach
high enough energies. While this may be less decisive than we
might like, there are presently no clearly identified sources to
the CR flux at Earth.
At still higher energies, there are exciting opportunities
for exotic-physics searches, as no conventional astrophysical
source should be able to contribute due to the small CR elec-
tron horizon distance at very high energies. This should allow
excellent sensitivity for searches for new physics such as dark
matter annihilation or decay, which would be present at all dis-
tances from Earth. Such signals may be enhanced by nearby
dark-matter clumps. We leave detailed investigations of these
points to future work.
ACKNOWLEDGEMENTS
We are grateful for helpful discussions with Isabelle John,
Matt Kistler, Tim Linden, and Hasan Y¨
uksel. We thank Chris
Hirata for discussions on the superluminal diffusion problem.
T.S. was primarily supported by an Overseas Research Fel-
lowship from the Japan Society for the Promotion of Sci-
ence (JSPS), a JSPS PD Research Fellowship, and KAKENHI
Grant No. 23KJ2005. T.S. was partially supported by and
J.F.B. was supported by National Science Foundation Grant
No. PHY-2012955.

11
∗takahiro sudoh@icloud.com
†beacom.7@osu.edu
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