# Best-case scenarios for neutrino capture experiments

Kyrylo Bondarenko  Alexey Boyarsky  Josef Pradler  Anastasia Sokolenko

###### Abstract

A direct discovery of the cosmic neutrino background would bring to a closure
the searches for relic left-over radiation predicted by the Hot Big Bang
cosmology. Recently, the KATRIN experiment put a limit on the local relic
neutrino overdensity with respect to the cosmological predicted average value
at $\eta\lesssim 10^{11}$ [Phys. Rev. Lett. 129, 011806 (2022)]. In this work,
we first examine to what extent such values of $\eta$ are conceivable. We show
that even under cavalier assumptions, a cosmic origin of $\eta\gtrsim 10^{4}$
seems out of reach (with the caveat of forming bound objects under a new
force,) but find that a hypothetical local source of low-energy neutrinos
could achieve $\eta\sim 10^{11}$. Second, when such values are considered, we
point out that the experimental signature in KATRIN and other neutrino-capture
experiments changes, contrary to what has hitherto been assumed. Our results
are model-independent and maximally accommodating as they only assume the
Pauli exclusion principle. As intermittent physics target in the quest for
C$\nu$B detection, we identify an experimental sensitivity to $\eta\sim
10^{4}$ for which conceivable sources exist; to resolve the effect of a
degenerate Fermi gas for such overdensity an energy resolution of 10 meV is
required.

##  1 Introduction

The standard cosmological model predicts the existence of a primordially
generated relic neutrino background with the average number density today of
$n_{0}=56\text{ cm}^{-3}$ per spin and flavor degree of freedom and six times
that number in total [1, 2]. This number sets a lower bound on the abundance
of neutrinos in the Universe if the Standard Model of particle physics and
cosmology is valid. The primordial neutrino background plays an important role
in the formation of the first elements and affects the timing of the relic
photon decoupling, providing abundant indirect evidence of its existence, see
[3, 4, 5] and references therein. However, because of the extremely small
interaction cross section of low-energy neutrinos with ordinary matter, direct
detection of the relic neutrino background remains an extraordinarily
difficult experimental challenge.

The direct experimental searches for neutrino mass serve a double purpose in
that they are also sensitive to the relic neutrino background. The state-of-
the-art experimental facility KATRIN recently put a limit on the local relic
neutrino background overdensity $n<\eta n_{0}$ with $\eta\simeq 10^{11}$
[6],111Purely gravitational probes in our Solar system put bounds of the same
order, see [7]. i.e. many orders of magnitude away from the standard
value.222Gravitational clustering of non-relativistic SM neutrinos is not
expected to yield local numbers in excess of $\sim 20n_{0}$ [8]. A next-
generation neutrino mass experiment Project 8 is expected to have improved
sensitivity although numbers have not yet been given [9]. Finally, there is an
ambitious proposal PTOLEMY [10], that aims to detect the relic neutrino
background at $\eta=O(1)$.

The purpose of this paper is to explore to which extent current and future
experiments can put sensible limits on a relic neutrino overdensity. Clearly,
the neutrino number density _can_ be larger than in a standard cosmological
history. Dark Matter or a fraction thereof may decay into neutrinos [11, 12,
13, 14] and even Dark Energy has been considered as a source of additional
present-day radiation degrees of freedom [15]. Finally, an interesting
possibility to enhance the local neutrino asymmetry by matter-interference has
been recently noted in [16, 17]. However, even if such possibilities exist,
the fermionic nature of neutrinos calls into question if previously considered
overdensities of $\eta=O(10^{10})$ or even orders of magnitudes smaller are
tenable in what is assumed to be low-energy (sub-eV) neutrino background.

In order to provide a quantitative answer we shall take the most accommodating
assumptions that are aimed at saturating the permissible number of
cosmological, galactic, and local neutrino number densities.333The
experimental rate is determined by the _flux_ of particles times the neutrino
capture cross section $\Gamma\sim nv\times\sigma$. However, owing to the
independence of the product $\sigma v$ to the relative velocity $v$ (for the
range of energies considered), it suffices for us to simply discuss the
maximum permissible number densities of neutrinos, irrespective of the
neutrino velocity. For this purpose, the Pauli exclusion principle is our only
guiding input. The success of using such general considerations such as Pauli-
blocking and/or the Liouville theorem is best exemplified by the prominence of
the Gunn-Tremaine bound [18]; see also [19, 20] for a more recent application.
We find that under such agnostic and model-independent assumptions, $\eta\sim
10^{11}$ is attainable only locally in the solar neighborhood, through largely
hypothetical and presumably not very credible sources. Galactically and
extragalactically sourced number densities are limited to $\eta\lesssim
10^{4}$. Although much smaller, the latter value can nevertheless serve as a
clear target. In addition, we show that the present KATRIN analysis is in fact
sensitive to the local neutrino phase space distribution as the endpoint
region of the beta spectrum is subject to Pauli blocking effects.

The paper is organized as follows: in Sec. 2 we discuss the physics of the
neutrino capture process and our simplified treatment of electron neutrinos.
In Sec. 3 we discuss model-independent upper limits on the neutrino number
density, both cosmological and local. In Sec. 4 we discuss experimental
signatures for the KATRIN-like experiments. Finally, in Section 5 we summarize
our results and conclude.

##  2 Neutrino capture

Before we start the stock-taking of possible neutrino flux contributions, here
we recall the generalities of neutrino capture detection through the electron
flavor $\nu_{e}$. For the purpose of this paper, we shall consider only
incoherent ensembles of neutrino mass eigenstates and neglect oscillations
altogether. This is certainly a perfect assumption for cosmological fluxes but
may lead to $O(1)$ changes for local fluxes.

Neutrinos propagate as mass-eigenstates $\nu_{i}$ and enter as such the
detector in an incoherent mixture of flavor states as they have traveled
astronomical distances from the source. For tritium-based detectors, neutrinos
in mass-eigentstates $\nu_{i}$ will undergo the reaction $\nu_{e}+\rm
T\to\,^{3}{\rm He}+e^{-}$ according to their electron-flavor content
$|U_{ei}|^{2}$. The capture rate in a detector of total target mass $M_{\rm
T}$ is then given by

$\Gamma=\frac{M_{\rm T}}{m_{\text{T}}}\sum_{i}|U_{ei}|^{2}\int
dE_{\nu,i}\,(\sigma\,v_{\nu,i})\frac{dn_{\nu,i}}{dE_{\nu_{i}}}\approx\frac{M_{\rm
T}}{m_{\text{T}}}(\sigma v)_{0}\sum_{i=1}^{3}|U_{ei}|^{2}n_{\nu,i}.$ (2.1)

Here, ${dn_{\nu,i}}/{dE_{\nu_{i}}}$ is the (total) energy differential number
density of $\nu_{i}$, $m_{\text{T}}$ is the mass of one tritium atom, and
$v_{\nu,i}$ is the neutrino velocity in the detector frame. The last relation
follows on the account of velocity-independence of $(\sigma\,v_{\nu,i})$. For
a tritium target $(\sigma v)_{0}\approx 3.8\times 10^{-45}\ \text{ cm}^{2}$
[21],444More precisely, left-helical neutrino states interact with strength
$(1+v_{\nu,i})$ while right-helical states are weighted by a factor of
$(1-v_{\nu,i})$ in the cross section [21]. This leads to a factor of two
differences in the rate depending on the occupation number of these states and
their kinematic regime. We shall ignore this model-dependence in (2.1) because
it does not change the overall argument. independent of neutrino energy for
$E_{\nu,i}\lesssim 10\ {\rm keV}$. It will be the working hypothesis of this
paper that we saturate the neutrino densities at the lowest energies, either
relativistically or non-relativistically. Hence, the local number density
alone determines the overall rate.

For better orientation, we recall the electron flavor content of SM neutrinos:
$|U_{e1}|^{2}\simeq 0.68$, $|U_{e2}|^{2}\simeq 0.30$, $|U_{e3}|^{2}\simeq
0.02$ [22]; in the absence of further sterile states with $i>3$, $U$ becomes
the PMNS matrix. More generally, reserving $i=1,2,3$ for SM neutrinos and
$i>3$ for any other light sterile states we may discriminate two principal
scenarios,

$\displaystyle\text{active }\nu\text{ dominated rate:}$
$\displaystyle\quad\sum_{i=1}^{3}|U_{ei}|^{2}n_{i}\gg\sum_{i>3}|U_{ei}|^{2}n_{i},$
(2.2) $\displaystyle\text{sterile }\nu\text{ dominated rate:}$
$\displaystyle\quad\sum_{i=1}^{3}|U_{ei}|^{2}n_{i}\ll\sum_{i>3}|U_{ei}|^{2}n_{i}.$
(2.3)

Unitarity of $U$ imposes the constraint $\sum_{i}|U_{ei}|^{2}=1$. The
experimentally established non-violation of unitarity of the PMNS matrix puts
constraints on the entries $U_{ei}|_{i>3}$ and they are generally found to be
at the percent or permille level [23]. Therefore, a “sterile dominated rate”
requires a significant excess over SM states.

For the purpose of this paper, we shall assume that the capture rate is
dominated by active neutrinos. Since we deal with numbers that span many
orders of magnitude we make the identification

$\displaystyle n_{\nu_{e}}\equiv\sum_{i=1}^{3}|U_{ei}|^{2}n_{i},$ (2.5)

where $n_{\nu_{e}}$ stands as a proxy for the electron neutrino number density
that will be informative of the overall achievable rate.

##  3 Census of maximum neutrino occupation numbers

As discussed above, neutrino capture experiments are only sensitive to the
electron neutrino number density. Using the simplified “one-flavor model”
(2.5) we shall write $m_{\nu_{e}}$ for the mass of the neutrino-eigenstate
that dominates in electron flavor and assume it to constitute one degree of
freedom, $g_{\nu_{e}}=1$.

Neutrinos obey the Pauli exclusion principle. It means that their number
density is constrained by,

$n_{{\nu_{e}}}\leq\frac{V_{p}}{(2\pi)^{3}},$ (3.1)

where $V_{p}=\frac{4\pi}{3}p_{{\nu_{e}},\max}^{3}$ is a spherical volume in
momentum space with a maximal momentum equal to $p_{{\nu_{e}},\max}$. The
inequality is saturated for a fully degenerate neutrino gas for which
$p_{{\nu_{e}},\max}=k_{F}$ with $k_{F}$ being the Fermi momentum. In what
follows we shall consider the permissible values of $p_{{\nu_{e}},\max}$,
i.e., seek the maximal occupation numbers.

###  3.1 Maximal cosmological density

The number of relativistic degrees of freedom in the early Universe prior to
CMB decoupling is well constrained, implying that any significant
overabundance of neutrinos $\eta\gg 1$ must have formed at a later epoch.
Candidate sources are processes that transfer mass and energy density from the
DM and Dark Energy (DE) reservoirs. Indeed, the late-time cosmic energy budget
remains in place for as long as the product $\langle E_{\nu}\rangle
n_{\nu}\ll\rho_{c}$ where $\langle E_{\nu}\rangle$ is the typical energy of
extra neutrinos and $\rho_{c}=3H_{0}^{2}M_{P}^{2}$ is the critical energy
density. This holds on the account of the similarity of $\Omega_{m}$ and
$\Omega_{\Lambda}$; $M_{P}$ is the reduced Planck mass.

Equating the neutrino energy density to the critical density saturates the
maximum permissible cosmological number density of neutrinos today and gives
us as figure of merit $\eta<10^{2}\ {\rm eV}/{\langle E_{\nu}\rangle}$. Of
course, $\eta$ cannot be amplified arbitrarily by considering non-relativistic
neutrinos of minuscule mass. Considering a Fermi gas at zero temperature and
taking the massless limit as the most accommodating case, the condition
$\rho_{\nu_{e}}<\rho_{c}$ yields the Fermi momentum

$k_{F}<\sqrt[4]{8\pi^{2}\rho_{c}}\approx 7.5\text{ meV}.$ (3.2)

From (3.1) with $p_{{\nu_{e}},\max}=k_{F}$ we may conclude $\eta\lesssim
2\times 10^{4}$ as the maximal permissible cosmological overdensity.

The most readily conceivable scenario is that light bosonic DM particles,
sufficiently populated in number, decay into neutrinos. Gravitational probes
of DM decay constrain the dominant DM lifetime $\tau_{\rm DM}\gtrsim 35t_{0}$
where $t_{0}$ is the age of the Universe [24]; see also [25, 26, 27]. In the
limit of $\tau_{\rm DM}\gg t_{0}$ the estimate on the number density is
$n_{\nu_{e}}\approx N_{\nu}\rho_{c}\Omega_{\rm DM}t_{0}/(m_{\rm DM}\tau_{\rm
DM})$ where $m_{DM}$ is the DM mass and $N_{\nu}$ is the effective
multiplicity of neutrinos in the decay. This translates into $\eta\approx
N_{\nu}(35t_{0}/\tau_{\rm DM})({\rm eV}/m_{\rm DM})$ which shows that the
maximum cosmological density is not readily achieved through DM decay (unless
one optimizes DM mass and neutrino multiplicity in the decay.)

A scenario that may actually achieve $\eta\gg 1$ has been considered in [15]:
a rolling scalar dark energy field generates different forms of dark radiation
through friction, which may ultimately result in a sizable population of
active neutrinos. Neutrino capture experiments hence offer an attractive
pathway for probing such dark energy dynamics.

###  3.2 Maximal galactic neutrino number density

In the previous section, we have estimated the maximal coarse-grained
cosmological electron neutrino number density. However, Earth is located in
the overdensity created by the gravitational field of our Galaxy and
surrounded by a dark matter halo. In this section, we will estimate the
maximal electron neutrino density in the Galaxy from general principles,
without assuming a specific dark matter density profile and neutrino
production mechanism.

####  3.2.1 Accumulation of neutrinos

Let us consider neutrinos that accumulate in our Galaxy because of some
process, e.g. dark matter decay into neutrinos, such that the decay products
remain gravitationally bounded. Such neutrinos necessarily need to be non-
relativistic and hence we may take
$p_{{\nu_{e}},\max}=m_{{\nu_{e}}}v_{\text{esc}}$, where $v_{\text{esc}}$ is
the escape velocity in our Galaxy. Using Eq. (3.1) this results in (see also
[28])

$\eta\leq
240\left(\frac{m_{{\nu_{e}}}}{1~{}\text{eV}}\right)^{3}\left(\frac{v_{\text{esc}}}{550\text{
km}/\text{s}}\right)^{3},$ (3.3)

where for the estimate we take the neutrino mass on the order of our current
laboratory constraints [29] and used a typical value for the escape velocity
of our galaxy [30, 31, 32, 33].

It should be noted that such overdensities are in principle easily achievable.
Comparing the implied neutrino energy density to the local DM density
$\rho_{0}\simeq 0.3\ {\rm GeV/cm^{3}}$ yields $\rho_{\nu}/\rho_{0}\sim
10^{-5}({m_{{\nu_{e}}}}/{1~{}\text{eV}})^{4}$. In other words, even if DM
decays with a lifetime much in excess of the age of the Galaxy or Universe,
(3.3) can be reached. The (severe) fine-tuning is in the kinematic decay
conditions as such neutrinos would need to be injected non-relativistically
below the escape speed.

####  3.2.2 Escaping neutrinos

Let us now consider the case when neutrinos are created in our Galaxy but
carry enough kinetic energy to leave the galactic gravitational potential.
Taking again the most accommodating position to arrive at a figure of merit,
one may saturate the number by assuming that escaping neutrinos at the Sun’s
orbital radius in the Galaxy should not carry away energy larger than the
total enclosed mass within this radius,

$\langle
E_{{\nu_{e}},\text{esc}}\rangle\frac{dN_{{\nu_{e}},\text{prod}}}{dt}t_{\text{U}}<M_{\text{encl}},$
(3.4)

where $\langle E_{{\nu_{e}},\text{esc}}\rangle$ is the average escaping
neutrino energy, $dN_{{\nu_{e}},\text{prod}}/dt$ is neutrino production rate,
that we assume constant, $t_{\text{U}}$ is the age of the Universe, and
$M_{\text{encl}}$ is an enclosed mass within Sun’s radius.

The neutrino production rate can be related to the escaping neutrino number
density $n_{{\nu_{e}},\text{esc}}$ at the Sun’s orbit radius from the
condition that the total production rate of neutrino should be equal to the
total escape rate,

$\frac{dN_{{\nu_{e}},\text{prod}}}{dt}=\frac{dN_{{\nu_{e}},\text{esc}}}{dt}\sim
4\pi R_{\text{Sun}}^{2}n_{{\nu_{e}},\text{esc}}v_{{\nu_{e}},\text{esc}},$
(3.5)

where $R_{\text{Sun}}=8.3$ kpc is the distance from Sun to the galactic
center. This gives a constraint on the escaping neutrino number density,

$n_{{\nu_{e}},\text{esc}}<\frac{M_{\text{encl}}}{4\pi
R_{\text{Sun}}^{2}t_{\text{U}}\langle E_{{\nu_{e}},\text{esc}}\rangle
v_{{\nu_{e}},\text{esc}}}=\frac{V_{\text{Sun}}^{2}}{4\pi
GR_{\text{Sun}}t_{\text{U}}\langle p_{{\nu_{e}},\text{esc}}\rangle},$ (3.6)

where in the last equality we used relation
$p_{{\nu_{e}}}=v_{{\nu_{e}}}E_{{\nu_{e}}}$, that works for both relativistic
and non-relativistic particles, and the enclosed mass is related with the
Sun’s circular velocity as $V_{\text{Sun}}\simeq 240$ km/s [34] as
$M_{\text{encl}}=V_{\text{Sun}}^{2}R_{\text{Sun}}/G$. This constraint can be
released for the low average neutrino momentum $\langle
p_{{\nu_{e}},\text{esc}}\rangle$.

For low neutrino momentum, we must account for Pauli blocking. For the sake of
the argument, we may take that half of the escaping neutrinos have momenta
smaller than the average one. Hence, even if all the quantum states below the
average momentum are occupied, we may adopt as an accommodating constraint on
the escaping neutrino number density [see Eq. (3.1)],

$\frac{n_{{\nu_{e}},\text{esc}}}{2}<\frac{\langle
p_{{\nu_{e}},\text{esc}}\rangle^{3}}{6\pi^{2}}.$ (3.7)

Comparing (3.6) and (3.7) we see that there is an optimal value of the average
neutrino momentum $\langle p_{{\nu_{e}},\text{esc}}^{\text{opt}}\rangle$, for
which the electron neutrino number density is maximized,

$\langle p_{{\nu_{e}},\text{esc}}^{\text{opt}}\rangle=\left(\frac{3\pi
V_{\text{Sun}}^{2}}{4GR_{\text{Sun}}t_{\text{U}}}\right)^{1/4}\approx 4\text{
meV},$ (3.8)

which gives the final constraint on the number density of escaping neutrino

$\eta<\frac{1}{3\pi^{2}n_{0}}\left(\frac{3\pi
V_{\text{Sun}}^{2}}{4GR_{\text{Sun}}t_{\text{U}}}\right)^{3/4}\approx 5\times
10^{3}.$ (3.9)

We see that the maximal conceivable neutrino overdensity of galactic origin is
in the same ballpark as the cosmological one obtained in Sec. 3.1. A more
detailed discussion on accumulating and escaping neutrino fluxes that
originate from DM decay in the galaxy can be found in [13].

Figure 1: The maximal conceivable overdensities from different sources
relative to the relic neutrino background number density. The blue line shows
the maximal cosmological average value (see section 3.1), green dashed line
shows the maximal neutrino number density that are gravitationally bounded in
our Galaxy for $m_{\nu_{e}}=1$ eV (lower mass corresponds to even lower
maximal number density, see Eq. (3.3)). Red long-dashed, orange dotted, and
cyan dashed-dotted lines show the maximal neutrino density that can be reached
by a constant neutrino flux generated by our Galaxy (Eq. (3.9)), Sun (Eq.
(3.10)) and Earth (Eq. (3.11)), respectively. The black dots are current
constraints from Los Alamos [35], Troitsk [36], and KATRIN [6] and the gray
dot is the 3 year projection from KATRIN [6].

###  3.3 Maximal Solar System density

The analysis of the previous section for the Galaxy can be applied to any
gravitationally bound system, including the Solar System and Earth. Let us
consider the (hypothetical) case when neutrinos are massively produced in the
Sun or inside Earth. Since the escape velocity from Earth or from the Solar
System at Earth’s position is an order of magnitude smaller than the escape
velocity from our Galaxy, we need not consider the accumulation of neutrino
within the Solar System; see (3.3) above.555See [37] for a treatment of
accumulating particle populations in the solar neighborhood; in [13] those
ideas have been applied to the solar neutrino flux with negligible resulting
overdensities.

For escaping neutrinos the situation is markedly different. From Eq. (3.9) we
see that the bound scales as $V_{c}^{2}/R$, where $V_{c}$ is a circular
velocity of the orbit with radius $R$. This is just the gravitational
acceleration at the point where the experiment is located. Of course, the
gravitational acceleration from Sun or from Earth is orders of magnitude
stronger than from the Galaxy, hence in principle allowing for significantly
larger values of $\eta$.

Using well-known parameters for Sun and Earth we obtain the following limits
on the electron neutrino number density in the laboratory as

$\langle p_{{\nu_{e}},\text{esc}}^{\text{opt,Sun}}\rangle=\left(\frac{3\pi
g_{\text{Sun}}}{4Gt_{\text{Sun}}}\right)^{1/4}\approx 380\text{
meV}\quad\Rightarrow\quad\eta\lesssim 4\times 10^{9}.$ (3.10)

for neutrinos produced in the Sun, and

$\langle p_{{\nu_{e}},\text{esc}}^{\text{opt,Earth}}\rangle=\left(\frac{3\pi
g_{\text{Earth}}}{4Gt_{\text{Earth}}}\right)^{1/4}\approx 2.4\text{
eV}\quad\Rightarrow\quad\eta\lesssim 10^{12}.$ (3.11)

for neutrinos produced in the Earth. Here $t_{\text{Sun}}\approx
t_{\text{Earth}}=4.6\times 10^{9}$ yr are the ages of Earth and Sun, while
$g_{\text{Sun}}$ and $g_{\text{Earth}}$ are gravitational laboratory
acceleration due to the Sun and Earth’s masses, respectively.

Let us pause and recall again that the derivation of those extremely large
values of $\eta$ assumed an injection rate that implies that a fair fraction
of Earth and/or Sun mass has already been converted into neutrinos (under the
assumption of a constant rate) and hence relies on truly cataclysmic
assumptions that are, mildly put, observationally challenged. This puts into
perspective the constraint $\eta\lesssim 10^{11}$ derived by KATRIN [6] and
other preceding works [35, 36].

###  3.4 Neutrino clusters

We close our census of neutrino occupation number by mentioning the
possibility of bound objects that we may term neutrino clusters. A neutrino
gas in equilibrium, under the influence of gravity and supported by degeneracy
pressure will have a size of $R\sim R_{\rm
NS}(m_{n}/m_{\nu_{e}})^{2}\approx(0.5\ {\rm eV}/m_{\nu_{e}})^{2}{\rm Mpc}$ and
total mass $M\sim M_{\rm NS}(m_{n}/m_{\nu_{e}})^{2}$, where $R_{\rm NS}$
($M_{\rm NS}$) is the radius (mass) of a neutron star and $m_{n}$ is the
neutron mass [38]. Such an object is of cosmic span, outweighs the most
massive galaxy clusters, and has a central mass density that exceeds
$\rho_{c}$ by a factor $10^{4}(m_{\nu_{e}}/0.1\ {\rm eV})$. At face value,
this translates into a central overdensity of

$\displaystyle\eta\approx\frac{\rho_{\rm
NS}(0)}{n_{0}m_{n}}\left(\frac{m_{\nu_{e}}}{m_{n}}\right)^{3}\approx 6\times
10^{6}\left(\frac{m_{\nu_{e}}}{0.1\ {\rm eV}}\right)^{3},$ (3.12)

but it is hard to see how such an object could have come into existence in a
$\Lambda$CDM Universe and not be ruled out observationally through its
gravitational effect.

An alternative avenue to reduce the size and mass of a neutrino cluster is to
consider a new Yukawa force that binds neutrinos stronger than gravity. This
possibility has been studied in detail in [39] for both, non-relativistic and
relativistic bound objects. A maximal overdensity
$\eta<10^{7}(m_{\nu_{e}}/0.1\ {\rm eV})^{3}$ and minimal radius of $R\gtrsim
0.6\ {\rm km}(10^{-7}/y)(0.1\ {\rm eV}/m_{\nu_{e}})$ are found; $y$ is the
Yukawa coupling. Such clusters could form by fragmentation of the C$\nu$B at
some redshift $z_{\rm f}\lesssim 200$ yielding an overdensity today of
$\eta\sim z_{\rm f}^{3}$. The detection prospects will depend on the size of
the clusters: their effect may average out or, if situated in a large enough
(solar-system sized) structure, an enhancement of the signal rate, in the best
of cases by $\eta\sim 10^{7}$ is possible. For a detailed account, we refer
the reader to [39]; see also [40].

Figure 2: Differential electron emission rate for a tritium mass of $100\mu$g
as a function of the electron kinetic energy $E_{e}$ measured relative to the
beta-decay endpoint energy $Q$; an illustration of the signatures of the large
neutrino and antineutrino overdensities near Earth for massless neutrino (left
panel) and neutrino mass $m_{\nu_{e}}=0.5$ eV (right panel). The green dotted
line shows the prediction for the standard tail of the beta-decay, the blue
line is its modification due to the presence of antineutrinos with an
overdensity $\bar{\eta}=10^{11}$, and the red line is a signal for the
neutrino capture rate for a neutrino overdensity $\eta=10^{11}$. We assume
that both neutrinos and antineutrinos are distributed as a Fermi gas at zero
temperature.

##  4 Detection of a local low-energy neutrino background

###  4.1 Different signatures of neutrino background

Figure 3: Differential electron rate with applied energy resolution for a
tritium mass $100\mu$g versus electron kinetic energy $E_{e}$ for massless
neutrinos (left panels) and massive neutrino of mass $m_{\nu_{e}}=0.5$ eV
(right panels). In the upper panels we adopt an energy uncertainty $\Delta
E=1$ eV, while for lower panels we chose $\Delta E=2.8$ eV. The green dotted
line shows the standard tail of the beta-decay, the blue line is its
modification due to the presence of antineutrinos with the overdensity
$\bar{\eta}=10^{11}$ for the Fermi sphere distribution. The red line is a
signal from the neutrino capture rate for overdensity $\eta=10^{11}$ for a
Fermi sphere, while the purple dashed line shows neutrino capture rate in the
assumption that all neutrino momenta are negligibly small (Dirac delta
function in momentum space, as it was assumed in [6]).

In previous sections, we discussed situations $\eta\gg 1$ in which the local
neutrino number density can be much larger than the average one. In this
section we discuss the observational signatures for a neutrino-capture capable
experiment such as KATRIN or PTOLEMY, ignoring backgrounds and/or potential
systematics. We focus on tritium as a target; results are easily generalized
to other candidate targets.

Figure 4: The number of expected signal events divided by square root of
background events (standard tail of beta-decay) for different energy bins for
massless neutrinos (left panel) and for massive neutrinos with
$m_{\nu_{e}}=0.5$ eV (right panel). The figure is shown for a tritium mass of
$13\mu$g, an experimental observation time of $36$ hours per bin, and an
energy resolution $\Delta E=2.8$ eV (parameters of the KNM2 data set of KATRIN
[6]). We consider that the neutrino background forms a Fermi sphere in
momentum space and show the case of antineutrino overdensity
$\bar{\eta}=10^{11}$ (blue line), neutrino overdensity $\eta=10^{11}$ (red
line), and the case of equal neutrino and antineutrino overdensity
$\eta=\bar{\eta}=10^{11}$ (magenta dotted line). For comparison, we also show
the assumption adopted in [6] that all neutrino momenta are negligibly small
(purple dashed line) for $\eta=10^{11}$.

The experimentally important process is neutrino capture (signal),

$\text{T}+\nu_{e}\to\,^{3}{\rm He}+e^{-},$ (4.1)

for which the rate depends on the local neutrino density $n_{\nu}=\eta n_{0}$,
see Section 2. It is enabled beta-decay instability of the target
(background),

$\text{T}\to\,^{3}{\rm He}+e^{-}+\bar{\nu}_{e}.$ (4.2)

Importantly, when considering large local overdensities, the endpoint region
of the beta-spectrum becomes affected by the presence of an antineutrino
background with number density $n_{\bar{\nu}}=\bar{\eta}n_{0}$ [41].

In the following we demonstrate the modifications by adopting a value of
$\eta=10^{11}$ (and $\bar{\eta}=10^{11}$) in line with the published work by
the KATRIN collaboration [6]. We assume a Fermi gas at zero temperature, i.e.,
all levels up to $E_{F}$ are filled.666We call Fermi energy
$E_{F}=\sqrt{k_{F}^{2}+m_{\nu_{e}}^{2}}-m_{\nu_{e}}$, where $k_{F}$ is the
Fermi momentum. For the capture reaction, such phase space distribution leads
to a monotonically rising trend of the electron kinetic energy spectrum from
$Q+m_{\nu_{e}}$ until $Q+m_{\nu_{e}}+E_{F}$ where $Q=18.56$ keV is the beta
endpoint energy. In turn, Pauli blocking prohibits the release of an anti-
electron neutrino with energy smaller than $E_{F}$.777We note in passing that
although we consider large neutrino asymmetries below, it does not modify the
dispersion relation of free neutrinos. Parametrically, the energy shift is of
order $\Delta E\sim G_{F}n_{0}(\eta-\bar{\eta})\sim 10^{-25}\ {\rm eV}$ for
$\eta=10^{11}$ and hence minute. This cuts off the beta spectrum for electron
kinetic energies in the interval $[Q-m_{\nu_{e}}-E_{F},Q-m_{\nu_{e}}]$. In
Fig. 2 we show examples of a signal from neutrino capture (red line) and
modification of a beta-spectrum because of an antineutrino background (blue
line) for two different neutrino masses. Details on the beta-spectrum and
neutrino-capture rates are given in Appendix A.

One should keep in mind that, in general, $\eta$ and $\bar{\eta}$ are
independent parameters. We consider three different scenarios below:

  1. 1.

Presence of large neutrino background with overdensity $\eta$, but no
significant antineutrino background, $\bar{\eta}\ll\eta$

  2. 2.

Presence of large antineutrino background with overdensity $\bar{\eta}$, but
no significant neutrino background, $\bar{\eta}\gg\eta$;

  3. 3.

Presence of both neutrino and antineutrino background. For simplicity we
assume the same overdensity $\eta=\bar{\eta}$ in this case.888Matter
interference effects induce an asymmetry close to the Earth’s surface even
when starting from a symmetric situation [16, 17].

###  4.2 Toy-model estimate

Let us consider an experimental setup with the energy resolution $\Delta E$.
To model the finite experimental resolution we follow [10] and use Gaussian
smearing with full-width-at-half-maximum (FWHM) $\Delta E$ to obtain the
observed rate,

$\frac{d\widetilde{\Gamma}}{dE_{e}}(E_{e})=\frac{1}{\sqrt{2\pi}(\Delta
E/\sqrt{8\ln
2})}\int\frac{d\Gamma}{dE_{e}}(E^{\prime})\exp\left(-\frac{(E^{\prime}-E_{e})^{2}}{2(\Delta
E/\sqrt{8\ln 2})^{2}}\right)dE^{\prime}.$ (4.3)

The demand on the required energy resolution to resolve the capture peak(s)
from the beta-decay endpoint spectrum depends critically on the value of the
neutrino mass(es). The current limit on the sum of the neutrino masses from
cosmological structure formation, $\sum m_{\nu_{i}}<0.12\ {\rm eV}\ (95\
\%~{}{\rm C.L.})$ [5], implies a maximal value of $30(16)$ meV for the
lightest neutrino mass for normal (inverted) mass ordering when taking into
account the solar and atmospheric mass splittings [22]. This constitutes a
challenging [42] requirement of $\Delta E=O(10)$ meV for C$\nu$B
detection.999For a scenario where the cosmological neutrino mass bound is
relaxed to eV-scale, see [43]. The current KATRIN limit on the effective
neutrino mass defined through $m_{\beta}^{2}\equiv\sum
m_{\nu_{i}}^{2}|U_{ei}|^{2}$ stands at $m_{\beta}\leq 0.8\ {\rm eV}\ (90\%\
{\rm C.L.})$ [44] with eV-scale energy resolution.

In Fig. 3 we present the expected differential rates in electron kinetic
energy for neutrino capture (red line) and beta-decay (blue line). For
comparison, we also show the signal for neutrino capture if the momentum
distribution of neutrino were given by the Dirac delta function (purple dashed
line), as it was assumed by KATRIN in [6]. We see that one should not neglect
the effects of Pauli blocking, as it produces a sizable difference in
differential rates for $\Delta E\sim E_{F}$ or better.

Adopting the energy resolution $\Delta E=2.8$ eV we may directly compare the
differences between the KATRIN result which neglects the neutrino phase space
distribution and our results for various cases. Following [6], we assume that
the experiment counts the number of electrons with energy larger than some
minimal value $E_{e,\min}$ and adopt the same energy bins (values of the
minimal energies) as in the KNM2 run [6]. With a total mass of the tritium
target of $m_{\text{T}}=13~{}\mu$g we calculate the number of events in each
energy bin within a window of $36$ h of exposure.

The resulting signal-to-noise ratios are shown in Fig. 4.101010We call “noise”
events from $\beta$-electrons and “signal” the difference between the number
of events in the presence of a neutrino background and the case of its
absence. From this we see that the KATRIN experiment is sensitive to all three
scenarios. The hardest scenario for exclusion is when both neutrino and
antineutrino backgrounds are present, but even in this case the sensitivity of
the KATRIN experiment drops only by a factor of few. Also, it is worth
mentioning that the physically motivated Fermi-sphere distribution of neutrino
(red line) is easier to detect than the unphysical Dirac delta-function
distribution (purple dashed line). The difference between the two momentum
distributions is stronger for low neutrino mass.

##  5 Summary and conclusions

Motivated by recent experimental results of the KATRIN experiment [6] which
placed the limit $\eta\lesssim 10^{11}$, in this work we explore the viability
of such large overdensities of the low-energy neutrino background with respect
to the standard cosmological C$\nu$B prediction. We show that the Pauli
exclusion principle alone puts severe restrictions on the possible values of
$\eta$, even without detailed information about its formation mechanism or the
present neutrino momentum distribution.

We find that both, the cosmological average value and any generated density
(globally) within the Milky Way is restricted to $\eta\lesssim 10^{4}$. This
bound is derived under the most accommodating assumptions but proposals exist
that may saturate that number. An alternative avenue exists in the
consideration of bound neutrino clusters, where, under the influence of a new
force, $\eta\lesssim 10^{7}$ appears attainable. When one is willing to
entertain the hypothesis of Solar System local sources, a local neutrino
overdensity of $\eta\sim 10^{11}$ remains a theoretical possibility. A summary
of the various studied origins is given in Fig. 1.

We furthermore discuss experimental signatures in the presence of a large
neutrino/antineutrino background. We show that Pauli blocking effects cannot
be ignored, invalidating the assumptions on the electron neutrino distribution
function made in [6]. Although the overall sensitivity to $\eta$ remains
comparable, the expected event shape is altered. If an overdensity in anti-
electron neutrinos is present, $\bar{\eta}\gg 1$, the endpoint of the beta-
spectrum is additionally modified. Those signatures are observable if the
energy resolution of the experiment $\Delta E$ is in the ballpark of the Fermi
energy $E_{F}$. For the case of KATRIN, this condition is indeed fulfilled,
$\Delta E\sim E_{F}\sim 1~{}{\rm eV}$, with their present sensitivity on
$\eta$.

An improved sensitivity to more palatable values of $\eta,\bar{\eta}\lesssim
10^{4}$ of course requires significant experimental advances; we recall that a
detection of the standard ${\mathrm{C}\nu\mathrm{B}}$ demands sensitivity to
$\eta\lesssim 10$ when taking into account local clustering factors. Since
$k_{F}\sim\eta^{1/3}$ we find that for $\eta,\bar{\eta}\sim 10^{4}$,
observation of the imprints of Pauli blocking requires an energy resolution of
$\Delta E\sim 10\ {\rm meV}$.111111A detailed analysis of this case requires
going beyond the effective “one-neutrino” picture used in this work since
$\Delta E$ is at the level of the atmospheric neutrino mass splitting
$\sqrt{|\Delta m_{\rm atm}^{2}|}\simeq 50$ meV. It is clear though that such
an advance in neutrino capture experiments offers the prospect of observing a
spectacular signature of an otherwise most elusive dark radiation component in
our Universe and provides an important intermittent physics target in the
ultimate quest for a direct detection of the C$\nu$B.

##### Acknowledgements

This work was supported by the Research Network Quantum Aspects of Spacetime
(TURIS). KB is partly funded by the INFN PD51 INDARK grant. AB is supported by
the European Research Council (ERC) Advanced Grant “NuBSM” (694896). AS is
supported by the Kavli Institute for Cosmological Physics at the University of
Chicago through an endowment from the Kavli Foundation and its founder Fred
Kavli. This work has been supported by the Fermi Research Alliance, LLC under
Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of
High Energy Physics. Funded/Co-funded by the European Union (ERC, NLO-DM,
101044443). Views and opinions expressed are however those of the author(s)
only and do not necessarily reflect those of the European Union or the
European Research Council. Neither the European Union nor the granting
authority can be held responsible for them.

##  Appendix A Beta spectrum and neutrino capture for tritium

For the differential beta-spectrum we modified the formula from [45, 10]
taking into account that in the presence of antineutrinos that form a Fermi
sphere with Fermi momentum $k_{F}$, some decay channels are forbidden by Pauli
blocking principle; we simplified the formula to the case of only one neutrino
flavor. The resulting differential beta-decay rate is given by,

$\frac{d\Gamma_{\beta}}{dE_{e}}=\frac{(\sigma
v)_{0}}{\pi^{2}}N_{T}H(E_{e},m_{\nu_{e}})\Theta(Q-m_{\nu_{e}}-E_{F}-E_{e}),$
(A.1)

with

$\displaystyle H(E_{e},m_{\nu_{e}})$
$\displaystyle=\frac{1-m_{e}^{2}/(E_{e}m_{\text{T}})}{\left(1-2E_{e}/m_{\text{T}}+m_{e}^{2}/m_{\text{T}}^{2}\right)^{2}}\sqrt{y\left(y+2m_{\nu_{e}}\frac{m_{{}^{3}\text{He}}}{m_{\text{T}}}\right)}\times$
$\displaystyle\times\left[y+m_{\nu_{e}}\frac{m_{{}^{3}\text{He}}+m_{\nu_{e}}}{m_{\text{T}}}\right],$
(A.2)

and where $y=Q-E_{e}-m_{\nu_{e}}$. The capture cross section $(\sigma v)_{0}$
is given in Sec. 2 and $E_{F}=\sqrt{k_{F}^{2}+m_{\nu_{e}}^{2}}-m_{\nu_{e}}$,
where $k_{F}$ depend on local neutrino density by Eq. (3.1).

The differential neutrino capture can be obtained from (2.1) together with
energy conservation ($E_{e}=Q+E_{\nu_{e}}$),

$\frac{d\Gamma_{\text{capt}}}{dE_{e}}(E_{e})=\frac{M_{\text{T}}}{m_{\text{T}}}(\sigma
v)_{0}\frac{dn_{\nu_{e}}}{dE_{\nu_{e}}}(E_{e}-Q),$ (A.3)

where the neutrino energy distribution is

$\frac{dn_{{\nu_{e}}}}{dE_{\nu_{e}}}(E_{\nu_{e}})\simeq\frac{g_{\nu_{e}}}{2\pi^{2}}E_{\nu_{e}}\sqrt{E_{\nu_{e}}^{2}-m_{\nu_{e}}^{2}}\Theta(m_{\nu_{e}}+E_{F}-E_{\nu_{e}})$
(A.4)

with $E_{F}=\sqrt{k_{F}^{2}+m_{\nu_{e}}^{2}}-m_{\nu_{e}}$ and $g_{\nu_{e}}=1$
for SM neutrinos; $k_{F}$ depends on the local neutrino density by Eq. (3.1).
Integrating Eq. (A.4) over neutrino energy yields $n_{{\nu_{e}}}=\eta n_{0}$.

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