# Probing sterile neutrino dark matter in the PTOLEMY-like experiment

Ki-Young Choi  Erdenebulgan Lkhagvadorj  and Seong Moon Yoo

###### Abstract

We study the prospect to detect the cosmic background of sterile neutrinos in
the Tritium $\beta$-decay, such as PTOLEMY-like experiments. The sterile
neutrino with mass between 1 eV - 10 keV may contribute to the local density
as warm or cold DM component. In this study, we investigate the possibility
for searching them in the models with different production in the early
Universe, without assuming sterile neutrino as full dark matter component. In
these models, especially with low-reheating temperature or phase transition,
the capture rate per year can be greatly enhanced to be $\mathcal{O}(10)$
without violating other astrophysical and cosmological observations.

##  1 Introduction

The neutrino oscillation data observed in the solar, atmospheric, reactor, and
accelerator experiments requires non-vanishing mass [1], which is absent in
the standard model (SM) of particle physics. Furthermore, it seems impossible
that any particle in SM can serve as a candidate for dark matter (DM) to
explain the missing mass in the Universe. One of the simplest ways to explain
both problems is to introduce right-handed neutrinos or sterile neutrinos [2,
3].

With the light sterile neutrino around eV, its mixing with the active
neutrinos may explain the anomalies in the neutrino oscillations [4, 5, 6, 7,
8, 9]. For the mass around keV, the sterile neutrinos may contribute
significantly to DM density [10, 11, 12]. The first analytical estimation of
the relic energy density of sterile neutrinos was made by Dodelson and Widrow
[10]. They assumed a negligible lepton number asymmetry, and sterile neutrinos
are produced by thermal scatterings induced via active-sterile neutrinos
oscillation in the early Universe. Since the mass is around keV, the sterile
neutrinos could be a warm dark matter (WDM) candidate. If there is a large
lepton asymmetry, sterile neutrinos can be produced resonantly, as has been
proposed by Shi and Fuller [13], with a non-thermal spectrum.

The light sterile neutrino exists as background in the whole Universe similar
to the background of active neutrinos or cosmic microwave. If its mass becomes
larger than around eV, they can cluster inside Galaxy due to the gravity and
the local density is enhanced compared to the average density in the whole
Universe [14, 15].

The direct detection of the cosmic neutrino background (C$\nu$B) can be done
in the capture of the electron neutrino on the radioactive $\beta$-decaying
nuclei with resultant peak in the electron spectrum. For this purpose, the
PTOLEMY experiment proposes to use 100g Tritium as a target coated on graphene
[16, 17]. Its energy resolution is the order of the neutrino mass scale
$\Delta\sim 0.15$ eV. In this experiment, it is expected that around 4 events
and 8 events per year could be detected on this target for Dirac neutrinos and
Majorana neutrinos, respectively. Although the direct detection of signal of
C$\nu$B seems challenging with small neutrino masses, several phenomenological
studies and sensitivity estimates have been done in the Literature [18, 19,
20, 21, 22, 23, 24].

Since the sterile neutrino can have mixing with electron neutrino, the cosmic
sterile neutrino background (CS$\nu$B) also can be measured on the radioactive
$\beta$-decaying nuclei. The resulting electron spectrum can show a peak which
is distinguishable from the ordinary $\beta$-decay spectrum or from C$\nu$B.
Probing sterile neutrino cosmic background was studied in [25, 26, 27, 28],
with assuming that the most of the DM was sterile neutrino with mass around
keV scale. If sterile neutrinos compose 100% of DM, the mass-mixing parameter
space is constrained by astrophysical observations such as phase space bounds
[29, 30], Lyman-$\alpha$ [31, 32], and X-ray emissions [33, 34], with a viable
mass between $1\ {\rm keV}<m_{s}<50\ {\rm keV}$ and mixing
$10^{-13}<|U_{e4}|^{2}<10^{-7}$. In that case, the number of event can be
$\mathcal{O}(1)$ per year with 10 kg Tritium or 10 ton Ru.

In this paper, we study the prospects for detecting CS$\nu$B using PTOLEMY-
like experiments without assuming 100% of DM with sterile neutrino. We
consider models which predict different abundance of the sterile neutrinos and
evaluate the capture rate in the beta-decaying experiment.

We find that in the models with low-reheating temperature [35, 36, 37, 38, 39]
or late phase transition [40] the production of the sterile neutrino is
suppressed compared to that in the standard Dodelson-Widrow mechansim, and
thus a large mixing angle is needed to compensate that. Therefore, the larger
mixing angle can enhance the capture rate up to $\mathcal{O}(10)$ per year
around the mass 10 - 100 eV of sterile neutrino without violating other
astrophysical and cosmological observations. .

The paper is organized as follows. In section 2, we focus on the PTOLEMY-like
experiment and review the detection of the standard C$\nu$B. In section 3, we
discuss capture rate of cosmic relic sterile neutrinos which could be sub-
dominant component of DM and clustered near the Earth. We introduce a new
fitting function of clustering effect, so we could obtain higher number
density of sterile neutrinos. In section 4, we review the present constraints
on sterile neutrinos including oscillation experiments, $\beta$-decay
experiments, X-ray telescopes, phase space bound, Lyman-$\alpha$, BBN and CMB
bounds in the early universe. In section 5, we calculate the number density of
clustered sterile neutrinos and capture rate of them with clustering effect in
the standard Dodelson-Widrow mechanism. In section 6, we discuss a model which
is based on low reheating temperature or late phase transition in the hidden
sector. In those models, the production of the sterile neutrino is suppressed
and large mixing is needed to generate the same amount of the sterile
neutrinos, which weakens the astrophysical and cosmological constraints. With
large mixing, we find that the capture rate of the CS$\nu$B is also enhanced
and it could be $\mathcal{O}(10)$ per year within the other constraints.
Finally, we present our conclusions and outlooks in section 7.

##  2 PTOLEMY-like experiment

In nature, the beta decay is a spontaneous process, which does not have energy
barrier. For example, the Tritium (${}^{3}\text{H}$) can decay with its half-
life 12.32 years, to Helium (${}^{3}\text{He}$), electron ($e^{-}$) and $i$-th
mass eigenstate of anti-neutrino ($\bar{\nu}_{i}$),
${}^{3}\text{H}\rightarrow{}^{3}\text{He}+e^{-}+\bar{\nu}_{i}$. In its inverse
beta decay, the neutrino can be captured by Tritium, and produce
${}^{3}\text{He}$ and $e^{-}$:

$\nu_{i}+{}^{3}\text{H}\rightarrow{}^{3}\text{He}+e^{-}.$ (2.1)

This process can happen even with non-relativistic neutrinos and their energy
is converted to the kinetic energy of the electron in the final stage
$K_{e}=E_{e}-m_{e}$. This energy is displaced from the Tritium beta decay
endpoint $K_{\rm end}$ [18]

$\begin{split}K_{\rm
end}=\frac{(m_{{}^{3}\mathrm{H}}-m_{e})^{2}-(m_{{}^{3}\mathrm{He}}+m_{\nu})^{2}}{2m_{{}^{3}\mathrm{H}}}.\end{split}$
(2.2)

For $m_{{}^{3}\mathrm{H}}\simeq m_{{}^{3}\mathrm{He}}\gg m_{e}\gg m_{\nu}$,
the electron energy fromm $C\nu B$ is

$\begin{split}K_{e}^{C\nu B}\simeq K_{\rm end}+2m_{\nu}\simeq K_{\rm
end,0}+m_{\nu},\end{split}$ (2.3)

where $K_{\rm end,0}\simeq 18.5988\,{\rm keV}$ is $K_{\rm end}$ for massless
neutrino.

PTOLEMY-like experiment [16, 17] has been proposed to probe the background
neutrinos using the inverse beta decay by measuring precisely the energy
spectrum of the final electrons. The energy resolution of $\Delta\sim 0.15$ eV
is expected to be obtained with a 100 g sample of Tritium, which is the order
of the neutrino mass.

Considering the energy resolution $\Delta$, the electron spectrum is
convoluted with a Gaussian envelope of FWHM $\Delta=2\sqrt{2\ln 2}\sigma\simeq
2.35\sigma$, with $\sigma$ the standard deviation of the Gaussian. The
observed spectrum after convolution [18],

$\frac{d\tilde{\Gamma}}{dE_{e}}=\frac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^{+\infty}dE_{e}^{\prime}\frac{d\Gamma}{dE_{e}^{\prime}}(E_{e}^{\prime})\exp\left[-\frac{(E_{e}^{\prime}-E_{e})^{2}}{2\sigma^{2}}\right],$
(2.4)

where $\frac{d\Gamma}{dE_{e}^{\prime}}(E_{e}^{\prime})$ is the true spectrum
of electron. This has two contributions from the beta decay and the cosmic
neutrino contributions. The sterile neutrino background can give additional
contribution to these, which will be the main subject in this paper.

The beta decay spectrum is given by

$\frac{d\Gamma_{\beta}}{dE_{e}}(E_{e})=\sum_{j=1}^{3}|U_{ej}|^{2}\frac{\bar{\sigma}}{\pi^{2}}H(E_{e},m_{\nu_{j}})N_{T},$
(2.5)

where $U_{ej}$ is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [41],
$\bar{\sigma}$ is the capture cross section of the electron neutrino defined
in [18] and for non-relativitic limit

$\bar{\sigma}\simeq\sigma_{e}v_{\nu}\simeq 3.834\times
10^{-45}\,{\mathrm{\,{\rm cm}}}^{2},$ (2.6)

and

$\left.H\left(E_{e},m_{\nu_{j}}\right)=\frac{1-m_{e}^{2}/\left(E_{e}m_{{}^{3}\mathrm{H}}\right)}{\left(1-2E_{e}/m_{{}^{\mathrm{H}}}+m_{e}^{2}/m_{3_{\mathrm{H}}}^{2}\right)^{2}}\sqrt{y\left(y+\frac{2m_{\nu_{j}}m_{{}^{3}\mathrm{He}}}{m_{{}^{\mathrm{H}}}}\right.}\right)\left[y+\frac{m_{\nu_{j}}}{m_{{}^{3}\mathrm{H}}}\left(m_{{}^{3}\mathrm{He}}+m_{\nu_{j}}\right)\right].$
(2.7)

Here $y=m_{e}+K_{\rm end}-E_{e}$ and $N_{T}=m_{T}/m_{{}^{3}\mathrm{H}}$ is the
approximate number of the nuclei in the sample.

The electron spectrum from the cosmic neutrino background is given by

$\frac{d\Gamma_{C\nu B}}{dE_{e}}=\Gamma_{C\nu B}\,\delta[E_{e}-(E_{\rm
end}+2m_{\nu})],$ (2.8)

where for the non-relativistic neutrinos, the rate $\Gamma_{C\nu B}$ is given
by [18] for Dirac neutrinos,

$\begin{split}\Gamma_{C\nu B}^{\rm D}=\bar{\sigma}n_{0}N_{T},\end{split}$
(2.9)

and for Majorana neutrinos,

$\begin{split}\Gamma_{C\nu B}^{\rm M}=2\bar{\sigma}n_{0}N_{T},\end{split}$
(2.10)

with the cosmological average of the neutrino number density $n_{0}=56\,{\rm
cm}^{-3}$. Here we assumed that clustering effects for the neutrino is
negligible and also used the unitarity of the PMNS matrix
$\sum_{i}|U_{ei}|^{2}=1$.

In Fig. 1, the expected event rate is shown with the dashed line for the beta
decay and the blue solid line for the inverse beta decay of background
neutrino. To distinguish the signal of the neutrino capture from the Tritium
beta decay, the energy resolution $\Delta$ should be smaller than half of the
neutrino mass, $\Delta\lesssim m_{\nu}/2$.

Figure 1: The expected event rate in terms of observed electron energy for
sterile neutrinos using benchmark values of $m_{s}=10$ eV and
$|U_{e4}|^{2}=10^{-3}$ assuming $n_{s,\mathrm{loc}}\simeq 6.9\times 10^{4}\
\rm{cm}^{-3}$ in PTOLEMY with 100 g of Tritium and energy resolution
$\Delta=150$ meV. For comparison, we plot $\beta$-decay background and
Majorana C$\nu$B signal with $m_{\nu}=0.2$ eV.

##  3 Capture rate of Sterile neutrino Dark Matter

The sterile neutrinos are produced in the early Universe and comprise a
component of hot, warm or cold dark matter. Once the sterile neutrino exists
as a background in our Milky Way, it can be captured by the Tritium through
mixing with the electron neutrino $U_{e4}$. This small mixing suppresses the
capture rate of the sterile neutrino compared to the active ones and makes it
more difficult to probe. However massive sterile neutrinos can cluster and
enhance the local density around the Earth. The relic density of the sterile
neutrino also can be modified depending on the production models in the early
Universe and the cosmological constraints can be relaxed. In this section, we
summarise the capture rate of the sterile neutrino and the relevant
constraints on them in the next section.

The capture rate of the sterile neutrino can be obtained using the equations
for the active neutrinos, except the mixing angle and the relic number
density. The capture rate $\Gamma_{C\nu B}$ in Eq. (2.8) should be modified to
the rate for Majorana sterile neutrino $\Gamma_{C\nu_{s}B}$ with

$\begin{split}\Gamma_{C\nu_{s}B}=N_{T}|U_{e4}|^{2}\int
dE_{\nu_{4}}\sigma_{e}v_{\nu_{4}}\frac{dn_{\nu_{4}}}{dE_{\nu_{4}}}\simeq
N_{T}|U_{e4}|^{2}\bar{\sigma}n_{s,\text{loc}},\end{split}$ (3.1)

where we assumed that $\sigma_{e}v_{\nu_{4}}$ is energy-independent for low
velocity and approximates to be $\bar{\sigma}$ and $n_{s,\text{loc}}=\int
dE_{\nu_{4}}\frac{dn_{\nu_{4}}}{dE_{\nu_{4}}}$ is the local number density of
the sterile neutrino near the Earth.

For massive sterile neutrinos, their number density near the Sun in the Milky
is larger than that of the cosmological relic density due to the gravitational
clustering. The clustering effect near the Earth is often parameterized with a
parameter $f_{c}$ given by

$\begin{split}n_{s,\text{loc}}=(1+f_{c})n_{s}=n_{s}+n_{s,\text{cls}},\end{split}$
(3.2)

where $n_{s}$ is the global number density before clustering. The number
density due to the clustering should be smaller than the bound from the local
phase space constraint, which will be discussed later in Sec. 4.4.

The clustering $f_{c}$ may depend on the mass and momentum distribution of the
sterile neutrino. In the case that they are produced thermally in the early
Universe, the distribution function can be the same form as the thermally
produced particles such as active neutrinos [14, 42, 15, 23, 43]. In this
case, we set a new fitting function of clustering factor given by

$\begin{split}f_{c}(m_{s})=f_{c,\text{DM}}\left[1+{\left(a\>\frac{\,{\rm
keV}}{m_{s}}\right)^{b}}\right]^{-2.21/b},\end{split}$ (3.3)

where $f_{c,\text{DM}}=2.4\times 10^{5}$ is clustering factor of cold dark
matter, and $a=0.038$, and $b=2.45$ obtained by the optimization method.

We use a parameter $\omega$ to denote the fraction of the energy density of
the sterile neutrino in the total dark matter. For the fraction in the global
cosmological energy density

$\begin{split}\omega_{s}=\frac{\rho_{s}}{\rho_{\rm
DM}}=\frac{m_{s}n_{s}}{m_{\rm DM}n_{\rm DM}}.\end{split}$ (3.4)

For the fraction in the local DM density near the Earth, we use

$\begin{split}\omega_{s,\rm{loc}}=\frac{\rho_{s,\mathrm{loc}}}{\rho_{\rm
DM,\mathrm{loc}}}=\omega_{s}\frac{f_{c}}{f_{c,\text{DM}}}.\end{split}$ (3.5)

In Fig. 1, the electron peak spectrum is also shown in red line corresponding
to the sterile neutrino capture rate which is around 5 events per year with
$m_{s}=10\ \rm{eV}$ and $|U_{e4}|^{2}=10^{-3}$ assuming
$n_{s,\mathrm{loc}}\simeq 6.9\times 10^{4}\ \rm{cm}^{-3}$. From the endpoint
of the Tritium beta decay spectrum, there is a larger separation to give a
significant signal for the future PTOLEMY-like detector.

##  4 Constraints on Sterile neutrino

In this paper, we are mainly focused on the mixing of the sterile neutrino
with the electron neutrinos to see the possibility for the sterile neutrino
capture. This mixing can also affect the neutrino oscillation, $\beta$-decay
experiments, X-ray constraints, and cosmological observations. In this
section, we summarize the possible constraints on the mixing of the sterile
neutrinos depending on its mass.

###  4.1 Oscillation experiments

The neutrino oscillation experiments to measure the appearance and
disappearance of the neutrinos can constrain the mixings of the sterile
neutrino to the active neutrinos. The Daya Bay and Bugey-3 reactor experiments
provide an upper limit of $\sin^{2}2\theta_{14}\lesssim 0.06$ at 90% C.L.
around $\Delta m^{2}_{41}\approx 1.75\ \text{eV}^{2}$ [5, 4]. In the short-
baseline approximation [7], this bound corresponds to $|U_{e4}|^{2}\lesssim
0.015$.

To explain the $\nu_{e}$ disappearance anomaly and the NEOS/Daya Bay and DANSS
overlapped analysis, non-zero mixing is preferred at about $\sim$ 3 $\sigma$
with best-fit point of $m_{41}^{2}\approx 1.3\ \text{eV}^{2}\ \text{and}\
|U_{e4}|^{2}\approx 0.012$ [9, 6, 8].

###  4.2 $\beta$-decay experiments

The $\beta$-decay of Tritium can produce the sterile neutrino through the
mixing if its mass is small, and the electron spectrum is distorted. The
present bound from non-observation of its distortion is given on the mixing
parameter as |$\mathrm{U}_{\mathrm{e}4}|^{2}\lesssim 10^{-2}-10^{-3}$ for the
sterile neutrino mass around 10 eV - 1 keV [44, 45, 46, 47, 48].

In the future, the distortion from the PTOLEMY experiment with 100 gram of
Tritium would probe the mixing $|\mathrm{U}_{\mathrm{e}4}|^{2}\sim
10^{-4}-10^{-6}$ for the sterile neutrino mass 0.1 - a few eV [17]

###  4.3 X-ray telescope

The dominant decay mode of the sterile neutrino is the decay into 3 active
neutrinos mediated by $Z^{0}$-boson and the decay rate is given by [3]

$\begin{split}\Gamma(\nu_{s}\rightarrow\nu_{\alpha}+\nu_{\beta}+\bar{\nu}_{\beta})\simeq\frac{2G_{F}^{2}m_{s}^{5}}{192\pi^{3}}\sum_{\alpha=e}^{\mu}|U_{\alpha
s}|^{2},\end{split}$ (4.1)

for $m_{s}\gg m_{i}$. The corresponding lifetime is

$\begin{split}\tau\simeq 1.44\times 10^{27}\,\textrm{s}{\left(\frac{1\,{\rm
keV}}{m_{s}}\right)}^{5}\frac{10^{-8}}{\sum|U_{\alpha s}|^{2}}.\end{split}$
(4.2)

This should be larger than the age of the Universe to be probed in the
PTOLEMY-like experiments.

The sterile neutrino can decay also into the photon and neutrino through one-
loop, and its decay rate [3] is

$\begin{split}\Gamma(\nu_{s}\rightarrow\nu_{a}+\gamma)=4.4\times
10^{-29}\,{\rm
sec}^{-1}{\left(\frac{\sin^{2}2\theta_{s}}{10^{-8}}\right)}{\left(\frac{m_{s}}{1\,{\rm
keV}}\right)}^{5}.\end{split}$ (4.3)

The resulting photon could be observed in the X-ray telescope. Several
observations have been made to detect the X-ray light from sterile neutrino
DM. However, only the upper bound on the decay rate and the abundance of
sterile neutirno DM is given. In our study, we used the results from M31 by
Chandra X-ray observatory[49] and the galactic bulge observation of NuSTAR[33,
34].

The conventional bound on the sterile neutrino mixing to the active neutrino
assumes that the sterile neutrino explains whole DM. In our study, since we
don’t assume that, we rescale with the corresponding relic density of the
sterile neutrino. Therefore, the new bound can be written as

$\begin{split}|U_{e4}|^{2}_{\omega_{s}<1}=\left(\frac{\Omega_{\rm
DM,local}}{\Omega_{\text{s,local}}}\right)|U_{e4}|^{2}_{\omega_{s}=1}.\end{split}$
(4.4)

###  4.4 Phase space bound

Identical fermionic particles cannot occupy the same state. Therefore for a
given escape velocity (or a momentum), there exists a maximum number density
when the particles occupy the energy states form the lowest level [29, 24].
This gives the bound on the relic density of fermionic DM.

For degenerate fermion up to the maximum momentum $p_{\rm max}$, the number
density is bounded from the value with the distribution function $f_{s}=1$

$\begin{split}n_{s}=g_{s}\int_{0}^{p_{\rm
max}}f_{s}\frac{d^{3}p}{(2\pi)^{3}}\lesssim g_{s}\frac{p_{\rm
max}^{3}}{6\pi^{2}},\end{split}$ (4.5)

with the degrees of freedom of $g_{s}$.

Near the Earth of the Milky, $p_{\rm max}=m_{s}v_{esc}$ with the escape
velocity $v_{esc}\simeq 550$km/s for non-relativistic DM, the maximum number
density of clustered sterile neutrino is bounded

$n_{s,\text{cls}}\leq g_{s}\frac{(m_{s}v_{esc})^{3}}{6\pi^{2}}.$ (4.6)

###  4.5 Constraints from early Universe

Even though the sterile neutrinos are non-relativistic in the present
Universe, they can be relativistic in the early Universe, since their momentum
redshifts with the expansion of the Universe. The relativistic component may
affect the the big bang nucleosynthesis (BBN), cosmic microwave background
(CMB), and the small scale of the structure formation in the early Universe.

The extra relativistic component is usually parameterized by $\Delta
N_{\text{eff}}$ from the relation

$\begin{split}\rho_{s}=\Delta
N_{\text{eff}}\,\frac{7}{8}{\left(\frac{4}{11}\right)}^{4/3}\rho_{\gamma},\end{split}$
(4.7)

where $\rho_{\gamma}=\frac{\pi^{2}}{15}T_{\gamma}^{4}$. The upper limit from
BBN and CMB [50] gives $\Delta N_{\text{eff}}<0.180$ (2$\sigma$). However the
extension of the $\Lambda$CDM parameters may broaden this constraint to
$\Delta N_{\text{eff}}<0.5$ [51]. Below we use the maximum value as $\Delta
N_{\text{eff}}^{\text{max}}=0.5$.

For the constraints from the Lyman-$\alpha$ observations, we use the
constraints on the thermal warm dark matter (WDM) [32], where the constraints
are given on the parameter space of the WDM mass $m_{w}$ and the relic density
$\Omega_{w}$,

$\begin{split}m_{w}\gtrsim m_{w}^{L-\alpha}\equiv 7.2\,{\rm
keV}\left(\frac{\Omega_{w}}{\Omega_{\rm DM}}-0.1\right).\end{split}$ (4.8)

For the same relic density of the sterile neutrino as WDM, we can find the
corresponding mass of sterile neutrino which gives the same free-streaming
scale, and then that is the lower bound for the given relic density from the
Lyman-$\alpha$ observation [31]. Note that the free-streaming scale is
determined by the background temperature $T_{NR}$ when the sterile neutrino
becomes non-relativistic and $T_{NR}$ can be written in terms of its mass and
the temperature of the sterile neutrino as

$\begin{split}T_{NR}=T_{s,NR}{\left(\frac{T_{NR}}{T_{s,NR}}\right)}\simeq\frac{m_{s}}{3}{\left(\frac{T}{T_{s}}\right)},\end{split}$
(4.9)

where $T_{s,NR}$ is the temperature of the sterile neutrino when it becomes
non-relativistic, which is $T_{s,NR}=m_{s}/3$ for thermally produced case, and
$T_{NR}$ is the temperature of the background plasma at the same time. In the
second equality we used that the temperature ratio of the sterile neutrino and
the background does not change after it becomes non-relativistic. Finally, by
equating $T_{s,NR}=T_{w,NR}$, we can find the relation

$\begin{split}\frac{m_{s}}{T_{s}}=\frac{m_{w}}{T_{w}}.\end{split}$ (4.10)

Using the known relations

$\begin{split}\Omega_{w}h^{2}={\left(\frac{T_{w}}{T_{\nu}}\right)}^{3}\frac{m_{w}}{94\,{\rm
eV}},\qquad\frac{T_{s}}{T_{\nu}}={\left(\frac{10.75}{g_{*}(T_{NR})}\right)}^{1/3},\end{split}$
(4.11)

we obtain the one-to-one correspondence between the lower bound on $m_{s}$ and
the lower bound on $m_{w}$ as

$m_{s}^{L-\alpha}=4.46\
\text{keV}\left(\frac{m_{w}^{L-\alpha}}{\text{keV}}\right)^{4/3}\left(\frac{10.75}{g_{*}}\right)^{1/3}\left(\frac{0.12}{\Omega_{s}h^{2}}\right)^{1/3},$
(4.12)

where $m_{w}^{L-\alpha}$ is given in Eq. (4.8).

##  5 Production with Dodelson-Widrow mechanism

In this section, we consider a popular model for the production of sterile
neutrino called Dodelson-Widrow mechanism [10], which demonstrates that
oscillations between active and sterile neutrinos can yield a sterile neutrino
population abundant enough to comprise all or part of the dark matter. In this
mechanism, the sterile neutrinos are produced when the active neutrinos are in
thermal equilibrium ($T\gg$ MeV). The Boltzmann equation for the evolution of
the distribution function of the sterile neutrino $f_{s}(E,t)$ is given by
[10, 52]

$\frac{\partial f_{s}(E,t)}{\partial t}-HE\frac{\partial f_{s}(E,t)}{\partial
E}=\frac{1}{4}\sin^{2}(2\theta_{M})\Gamma_{\alpha}(f_{\alpha}-f_{s})$ (5.1)

where $H=\sqrt{\pi^{2}g_{*}/30}T^{2}/M_{\text{pl}}$ is Hubble parameter and in
the radiation-dominated Universe, with reduced Planck mass
$M_{\text{pl}}=2.4\times 10^{19}\text{GeV}$ and $g_{*}$ the effective degrees
of freedom of the relativistic particles in the thermal equilibrium. Here, the
effective mixing angle in the matter between the sterile and electron neutrino
is given by [53, 54, 55]

$\sin^{2}(2\theta_{M})=\frac{\sin^{2}(2\theta)}{\sin^{2}(2\theta)+[\cos(2\theta)-2E\>V_{T}(T)/m_{s}^{2}]^{2}},\qquad\Gamma_{\alpha}\approx
1.27\times G_{F}^{2}T^{4}E,$ (5.2)

with mixing angle $\theta$ in the vacuum and

$V_{T}=-BT^{4}E,\quad\textrm{and}\quad B\sim\begin{cases}10.88\times
10^{-9}\>\text{GeV}^{-4}&T>2m_{e}\\\ 3.04\times
10^{-9}\>\text{GeV}^{-4}&T<2m_{e}\end{cases}.$ (5.3)

By using relations $y\equiv E/T$ and $t=1/(2H)$ for radiation dominated era,
the equation is simplified as

$HT\left(\frac{\partial f_{s}(y,T)}{\partial T}\right)_{y\equiv
E/T}\simeq-\frac{1}{4}\sin^{2}(2\theta_{M})\Gamma_{\alpha}(f_{\alpha}-f_{s}),$
(5.4)

where the partial derivative about $T$ in the left-hand side is evaluated
assuming constant $g_{*}$. The distribution function of the active neutrino
$f_{\alpha}$ is assumed to be in the thermal equilibrium and constant with
time for fixed $E/T$ as $f_{\alpha}=(\exp(y)+1)^{-1}$.

We can find the general solution for $f_{s}$ by redefining
$f_{s}=f_{\alpha}(1-e^{-f_{s,0}/f_{\alpha}})$ in Eq. (5.4) and solving
differential equation for $f_{s,0}$ which is given by

$HT\left(\frac{\partial f_{s,0}(y,T)}{\partial T}\right)_{y\equiv
E/T}\simeq-\frac{1}{4}\sin^{2}(2\theta_{M})\Gamma_{\alpha}f_{\alpha}.$ (5.5)

The integral solution for $f_{s,0}$ is

$f_{s,0}(y,T)\simeq-
f_{\alpha}\int^{T}_{\infty}\frac{1}{4HT}\sin^{2}(2\theta_{M})\Gamma_{\alpha}dT.$
(5.6)

In fact, $f_{s}$ approaches to $f_{s,0}$ for $f_{s}\ll f_{\alpha}$, which
corresponds to the case when the sterile neutrinos mixing is small enough.
Here, we count $g_{*}$ of the thermal particles in the standard model and
neglect the contribution from the sterile neutrino, which is subdominant. We
checked that our result is consistent with that using the program LASAGNA
[56].

Since the mixing in the matter is suppressed at high temperature, the
production rate of the sterile neutrino in the ratio
$d(n_{s}/n_{\alpha})/d\log T$ is maximum at a temperature $T_{\text{max}}$
[10]

$T_{\text{max}}\simeq 108\
\text{MeV}\left(\frac{m_{s}}{\text{keV}}\right)^{1/3},$ (5.7)

under an assumption of constant $g_{*}$, where $m_{s}$ is the mass of the
sterile neutrino. The numerical coefficient is slightly different from that in
[10] due to the change of the number in Eq. (5.3). Therefore, for $T\gg
T_{\text{max}}$, the abundance becomes independent of the temperature and the
relic density of the sterile neutrino at present is given by [11, 3]

$\Omega_{s}h^{2}\equiv\frac{n_{s}m_{s}}{\rho_{c}/h^{2}}\approx
0.1\left(\frac{\sin^{2}\theta}{3\times 10^{-9}}\right)\left(\frac{m_{s}}{3\
\text{keV}}\right)^{1.8},$ (5.8)

where $n_{s}=\frac{g_{s}T_{0}^{3}}{(2\pi)^{3}}\int f_{s}(y,T_{0})d^{3}y$ at
present temperature $T_{0}$ and $\rho_{c}\equiv 3M_{P}^{2}H_{0}^{2}\simeq
10^{-5}\,{\rm GeV}\,{\rm cm}^{-3}$ and the present Hubble parameter
$H_{0}=100\,h\,{\rm km}/(\sec\,{\rm Mpc})$.

Once the present relic density of the non-relativistic sterile neutrino is
given, we can estimate the number density in the early Universe when the
sterile neutrinos are relativistic,

$\begin{split}n_{s}(a)=\frac{\Omega_{s}\rho_{c}}{m_{s}}{\left(\frac{a_{0}}{a}\right)}^{3},\end{split}$
(5.9)

where $a$ is the scale factor and $a_{0}$ is its value at present. Since the
number density of the active neutrino can be obtained in the same way, we can
write $\Delta N_{\text{eff}}$ as

$\begin{split}\Delta
N_{\text{eff}}=\frac{n_{s}(a)}{n_{\nu}(a)}=\frac{\Omega_{s}h^{2}/m_{s}}{\Omega_{\nu}h^{2}/m_{\nu}}<\Delta
N_{\text{eff}}^{\rm max},\end{split}$ (5.10)

where $\Omega_{\nu}h^{2}=\frac{m_{\nu}}{94{\rm eV}}$ is the energy density of
the non-relativistic sigle flavor neutrino at present.

The local density of the sterile neutrinos near the Earth is enhanced due to
the clustering by the gravitational interaction. Using the clustering effect
in Eq. (3.3), the local number density of the sterile neutrino in the DW
mechanism is given by

$\begin{split}n_{s,{\text{loc}}}=&\frac{\Omega_{s}}{\Omega_{\text{DM}}}\frac{1+f_{c}(m_{s})}{f_{c,\text{DM}}}\frac{\rho_{\text{DM},\rm
local}}{m_{s}},\\\ =&\Delta N_{\text{eff}}(1+f_{c}(m_{s}))n_{\nu},\end{split}$
(5.11)

where $n_{\nu}=112/{\text{cm}}^{3}$ is the global number density of the active
neutrino in the present Universe. Therefore, the total event rate for the
sterile neutrino from Eq. (2.10) becomes

$\Gamma_{s}\simeq N_{T}\bar{\sigma}\Delta
N_{\text{eff}}|U_{e4}|^{2}(1+f_{c}(m_{s}))n_{\nu}.$ (5.12)

In Fig. 2, we show (Left) the fraction of local energy density $\omega_{s,{\rm
loc}}$ of sterile neutrino DM in the Dodelson-Widrow model near the Earth with
blue dashed lines for corresponding mixing
$\left|U_{e4}\right|^{2}=10^{-8},10^{-6},10^{-4}$, and (Right) contour of the
capture rate of the sterile neutrino DM in the PTOLEMY-like experiments for
the Dodelson-Widrow model on the plane of $m_{s}$ and
$\left|U_{e4}\right|^{2}$. The contour represents the number of captured event
per year with $100$ g Tritium. The Constraints described in Sec. 4 are shown:
the phase space bound (yellow), the Lyman-$\alpha$ bound (red) [32], the CMB
(dark cyan) [51], the X-ray (green) [32], the Tritium beta decay (grey) [44,
45, 57, 46, 47, 48], and Short Base Line and DayaBay+Bugey3 experiments
(olive). Blue star represents the best-fit point from neutrino oscillation
experiments [9, 6, 8]. We cut the large mixing $\left|U_{e4}\right|^{2}>0.1$
with black dashed region in the left figure due to current constraints. The
orange dashed line shows the expected sensitivity of PTOLEMY in the future by
detecting kink and distortion of the $\beta$ decay spectrum [17].

For the sterile neutrino produced in the standard Dodelson-Widrow mechanism,
the mixing is constrained mostly by the CMB and Lyman-$\alpha$. The most
probable capture rate is 0.01-0.1 per year for the mass of the sterile
neutrino around 1 eV - 100 eV, which is quite difficult to see in the real
experiments. However, this result may change in different production
mechanisms of the sterile neutrino, that we will be shown in the next Section.

Figure 2: Left: The fraction of local energy density $\omega_{s,{\rm loc}}$ of
sterile neutrino DM in the Dodelson-Widrow model near the Earth with blue
dashed lines for corresponding mixing
$\left|U_{e4}\right|^{2}=10^{-8},10^{-6},10^{-4}$. Right: Contour of the
capture rate of the sterile neutrino DM in the PTOLEMY-like experiments for
the Dodelson-Widrow model on the plane of $m_{s}$ and
$\left|U_{e4}\right|^{2}$. The contour represents the number of captured event
per year. Constraints: The constraints described in Sec. 4 are shown: the
phase space bound (yellow), the Lyman-$\alpha$ bound (red) [32], the CMB (dark
cyan) [51], and the X-ray (green) [32], the Tritium beta decay (grey) [44, 45,
57, 46, 47, 48], and Short Base Line and DayaBay+Bugey3 experiments (olive).
Blue star represents the best-fit point from neutrino oscillation experiments
[9, 6, 8]. The orange line shows the expected sensitivity of PTOLEMY by
detecting kink and distortion of the $\beta$ decay spectrum [17].

##  6 Sterile neutrino DM in the model of low reheating temperature

When the temperature of the early Universe is lower than $T_{\text{max}}$, Eq.
(5.8) cannot be applied any more. In this case, the production of the sterile
neutrino is suppressed and the cosmological and astrophysical constraints can
be relaxed [35, 40, 36]. This can happen when the reheating temperature after
inflation is very low or the phase transition for generating the Majorana mass
of the sterile neutrino occurs very late.

###  6.1 Low reheating temperature

When the reheating temperature $T_{R}$ is smaller than $T_{\text{max}}$,
$T_{R}\ll T_{\text{max}}$, the abundance of the sterile neutrino cannot reach
the value in Eq. (5.8). By solving Eq. (5.4) up to the temperature $T_{R}\ll
T_{\rm max}$, the distribution function of sterile neutrinos can be obtained
as

$f_{s}(E,T_{0})=\int_{T_{R}}^{T_{0}}\frac{\partial f_{s}}{\partial T}dT\simeq
0.13\
|U_{e4}|^{2}\left(\frac{10.75}{g_{*}}\right)^{1/2}\left(\frac{T_{R}}{\text{MeV}}\right)^{3}\left(\frac{E}{T_{0}}\right)f_{\alpha}(E,T_{0}),$
(6.1)

wehre $|U_{e4}|^{2}\simeq\sin^{2}\theta$. The number density of sterile
neutrinos becomes

$n_{s}\simeq 51.2\
|U_{e4}|^{2}\left(\frac{10.75}{g_{*}}\right)^{1/2}\left(\frac{T_{R}}{5\
\text{MeV}}\right)^{3}n_{\nu},$ (6.2)

and the relic density is [35, 40]

$\Omega_{s}h^{2}\simeq
0.5\left(\frac{|U_{e4}|^{2}}{10^{-3}}\right)\left(\frac{10.75}{g_{*}}\right)^{1/2}\left(\frac{m_{s}}{\text{keV}}\right)\left(\frac{T_{R}}{5\
\text{MeV}}\right)^{3}.$ (6.3)

Compared to the standard Dodelson-Widrow relic density, a large mixing is
needed for small $T_{R}$ to obtain the given relic density of the sterile
neutrino. Accordingly, the cosmological and astrophysical constraints are also
relaxed in the $(m_{s},|U_{e4}|^{2})$ plane to the large mixing. Therefore,
the large mixing angle $|U_{e4}|^{2}\lesssim 10^{-3}$ now survives from the
constraints, and a large capture rate can be viable in the future PTOLEMY-like
experiment.

In Fig. 3, we show the local DM fraction of the sterile neutrino near the
Earth and the capture rate in the scenario of the low reheating temperature
with $T_{R}=5\,{\rm MeV}{}$ (upper window) and $T_{R}=10\,{\rm MeV}{}$ (lower
window), respectively. Due to the suppression of the production in the early
Universe, the fraction of DM and cosmological constraints appear at large
mixing angles, where the capture rate in the beta-decay experiment increases
for the same amount of DM fraction in the standard DW mechanism. We find that,
for each case, the capture rate increases up to 10 events, or a few events per
year, respectively, at the mass around 10 eV. The future experiments may probe
these models.

Figure 3: Same as Fig. 2 but for the low reheating temperature model with
$T_{R}=5\,{\rm MeV}{}$ (upper window) and $T_{R}=10\,{\rm MeV}{}$ (lower
window).

###  6.2 Late time phase transition in the hidden sector

In this section, we consider a hidden sector where the phase transition for
generating Majorana mass occurs very late after reheating. Before the phase
transition, the Majorana mass is vanishing and neutrinos comprise Dirac
fermion.

We consider a Lagrangian in addition to the standard model [40]

$\mathcal{L}=i\bar{N}\not{\partial}N+Y_{\nu}H\bar{\nu}_{\alpha}N+\frac{\lambda}{2}\phi\bar{N^{c}}N+h.c.,$
(6.4)

where $N$ is the right-handed (RH) neutrino with Yukawa interaction with Higgs
$H$ and the left-handed (LH) neutrino $\nu$, and also couples to the hidden
sector scalar $\phi$ which give Majorana mass to the RH neutrino after the
phase transition with $\langle\phi\rangle$. Before the phase transition and
after electro-weak symmetry breaking, the RH neutrino becomes Dirac fermion
with LH neutrino with Dirac mass $M_{D}=Y_{\alpha}\langle H\rangle$. After the
phase transition at temperature $T_{c}$, the hidden sector scalar $\phi$
develops VEV and gives a Majorana mass to the RH neutrino
$M=\lambda\langle\phi\rangle$.

To get the sterile neutrino abundance in this model, we can integrate the
Boltzmann equation Eq. (5.4) about background temperature $T$, from $T_{R}$ to
the present temperature $T_{0}$. For the temperature $T<T_{c}<T_{R}$, the
sterile neutrino can be produced from the oscillation using the mixing angle
in Eq. (5.2). However, for $T_{c}<T<T_{R}$, there is no mixing angle since
$M=0$, and the sterile neutrino can be generated through the Dirac mass term.
Therefore the abundance of the sterile neutrino has two contributions

$\begin{split}\Omega_{s}=\Omega_{s,c}+\Omega_{s,R},\end{split}$ (6.5)

with $\Omega_{s,c}$ from the generation at the temperature between
$(T_{0},T_{c})$ and $\Omega_{s,R}$ between $(T_{c},T_{R})$. When $T_{c}\ll
T_{\rm max}$, we can approximate $\Omega_{s,c}h^{2}$ as in Eq. (6.3), with
replacing $T_{R}$ by $T_{c}$

$\Omega_{s,c}h^{2}\simeq
0.5\left(\frac{|U_{e4}|^{2}}{10^{-3}}\right)\left(\frac{10.75}{g_{*}(T_{c})}\right)^{1/2}\left(\frac{m_{s}}{\text{keV}}\right)\left(\frac{T_{c}}{5\
\text{MeV}}\right)^{3}.$ (6.6)

Figure 4: Same as Fig. 2 but for the late phase transition model with
$T_{c}=1\,{\rm MeV}$ and $T_{R}=10\,{\rm MeV}$.

For temperature $T_{c}<T<T_{R}$, there is no mixing term, however sterile
neutrino can be produced from the chirality flip in the Dirac mass term which
is quite suppressed since it is proportional to $M_{D}^{2}/p^{2}$. Therefore
the Boltzmann equation, Eq. (5.4) is now modified to

$\left(\frac{\partial f_{s}(E,T)}{\partial T}\right)_{y\equiv
E/T}\simeq-\frac{1}{2}\frac{M_{D}^{2}}{E^{2}}\frac{\Gamma_{\alpha}}{HT}f_{\alpha},\qquad\Gamma_{\alpha}\approx
1.27\times G_{F}^{2}T^{4}E,$ (6.7)

where we used $p\simeq E$ for the sterile neutrino since we focus $T_{c}\gg
M$. By integrating this equation, we obtain

$\begin{split}\frac{f_{s}}{f_{\alpha}}=&-\int_{T_{R}}^{T_{c}}\frac{1}{2}\frac{M_{D}^{2}}{E^{2}}\frac{\Gamma_{\alpha}}{HT}dT\approx
0.4\frac{G_{F}^{2}M_{\text{pl}}M_{D}^{2}}{\sqrt{g_{*}(T_{R})}}\frac{T_{R}}{y}\\\
\approx&\frac{8\times
10^{-7}|U_{e4}|^{2}}{y}\left(\frac{10.75}{g_{*}(T_{R})}\right)^{1/2}\left(\frac{m_{s}}{\text{keV}}\right)^{2}\left(\frac{T_{R}}{\text{5
MeV}}\right),\end{split}$ (6.8)

where we used $|U_{e4}|^{2}=M_{D}^{2}/m_{s}^{2}$. The number density of
sterile neutrino is obtained as

$\begin{split}n_{s,R}=&\frac{g}{2\pi^{2}}\int_{0}^{\infty}dyT^{3}y^{2}f_{s}\\\
\simeq&8\times
10^{-7}|U_{e4}|^{2}\left(\frac{10.75}{g_{*}}\right)^{1/2}\left(\frac{M_{s}}{\text{keV}}\right)^{2}\left(\frac{T_{R}}{\text{5
MeV}}\right){\left(\frac{g}{2\pi^{2}}\right)}T^{3}\int_{0}^{\infty}dyyf_{\alpha},\end{split}$
(6.9)

and thus the relic density of sterile neutrino is 111We find different result
from that in [40]. In our case, in the range $T_{c}<T<T_{R}$ the amount
$f_{s}/f_{\alpha}$ is still proportional to $T_{R}$, however Ref. [40] finds
that it is proportional to $T_{c}$.

$\Omega_{s,R}h^{2}\approx 4\times 10^{-9}\
{\left(\frac{|U_{e4}|^{2}}{10^{-3}}\right)}\left(\frac{10.75}{g_{*}(T_{R})}\right)^{1/2}\left(\frac{m_{s}}{\text{keV}}\right)^{3}\left(\frac{T_{R}}{\text{5
MeV}}\right).$ (6.10)

The abundance $\Omega_{s,R}$ generated between $T_{c}<T<T_{R}$ is quite
suppressed and subdominant to $\Omega_{s,c}$ for $T_{R}\lesssim 10^{3}\,{\rm
GeV}$ if $T_{c}=1\,{\rm MeV}$.

In Fig. 4, we show the numerical result for the fraction of the sterile
neutrino and the capture rate in this model with $T_{c}=1\,{\rm MeV}$, and
$T_{R}=10\,{\rm MeV}$. Now due to the suppression of the production, larger
mixing is needed and the cosmological constraints are hidden behind the
constraints from the terrestrial experiments. The maximum capture rate around
10 can be possible for the sterile neutrino mass $100$ eV and mixing
$|U_{e4}|^{2}\sim 10^{-2}$.

##  7 Conclusion

One of the natural way to explain the neutrino oscillation and the component
of dark matter is to introduce right-handed neutrinos. The light sterile
neutrinos are produced in the early Universe and can be stable enough to
survive up to the present Universe to make a cosmic background as hot or warm
dark matter.

In the PTOLEMY-like experiments, the radioactive $\beta$-decaying nuclei can
capture the background neutrino and emit electrons which make a peak in the
energy spectrum corresponding to the mass of the sterile neutrino.

In this paper, we studied the possibility to detect the cosmic sterile
neutrino background in the Tritium decay experiments for different models of
the sterile neutrino production in the early Universe, such as the low-
reheating temperature and the late phase transition. In both models, the
production is suppressed and large mixing between active and sterile neutrinos
is needed, where the capture rate enhanced. Furthermore, the smaller abundance
of the sterile neutrino than the active ones are compensated by the
gravitational clustering for the massive sterile neutrinos.

We find that in both models the capture rate can be $\mathcal{O}(10)$ per year
with 100 gram Tritium for the sterile neutrino mass around 100 eV and a mixing
around $|U_{e4}|^{2}\sim 10^{-2}$, which is probably accessible in the near
future.

## Acknowledgments

The authors were supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MEST) (NRF-2022R1A2C1005050).

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