<EMAIL_ADDRESS>
# The Sun: light dark matter and sterile neutrinos

Ilídio Lopes${}^{1},^{2}$ ID [ 1Centro de Astrofísica e Gravitação - CENTRA,
Departamento de Física, Instituto Superior Técnico - IST, Universidade de
Lisboa - UL, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal  2 Institut
d’Astrophysique de Paris, UMR 7095 CNRS, Université Pierre et Marie Curie, 98
bis Boulevard Arago, Paris F-75014, France

###### Abstract

Next-generation experiments allow for the possibility to testing the neutrino
flavor oscillation model to very high levels of accuracy. Here, we explore the
possibility that the dark matter in the current universe is made of two
particles, a sterile neutrino and a very light dark matter particle. By using
a 3+1 neutrino flavor oscillation model, we study how such a type of dark
matter imprints the solar neutrino fluxes, spectra, and survival probabilities
of electron neutrinos. The current solar neutrino measurements allow us to
define an upper limit for the ratio of the mass of a light dark matter
particle $m_{\phi}$ and the Fermi constant $G_{\phi}$, such that
$G_{\phi}/m_{\phi}$ must be smaller than $10^{30}\;G_{\rm F}eV^{-1}$ to be in
agreement with current solar neutrino data from the Borexino, Sudbury Neutrino
Observatory, and Super-Kamiokande detectors. Moreover, for models with a very
small Fermi constant, the amplitude of the time variability must be lower than
$3\%$ to be consistent with current solar neutrino data. We also found that
solar neutrino detectors like Darwin, able to measure neutrino fluxes in the
low energy-range with high accuracy, will provide additional constraints to
this class of models that complement the ones obtained from the current solar
neutrino detectors.

###### Subject headings:

The Sun – Solar neutrino problem – Solar neutrinos – Neutrino oscillations –
Neutrino telescopes – Neutrino astronomy

##  1\. Introduction

The origin of dark matter has been a fundamental problem in physics for almost
six decades, during which most of the proposed solutions assumed that a single
massive particle that interacts weakly with baryons makes all the dark matter
observed in the universe (e.g., Wang et al., 2016). Recently, research has
emerged where more sophisticated solutions have been proposed to solve the
dark matter problem. One of these is the possibility of the dark matter being
a composite of two light particles: a light dark matter (LDM) particle $\phi$
and a sterile neutrino $\nu_{s}$.

The existence of such an LDM field can by identified with a dilation field of
an extradimensional extension of the Standard Model or/and a CP-violating
pseudo-Goldstone boson of a spontaneously broken global symmetry. For some of
these models, $\phi$ couples to the Standard Model fields, and as such it
induces periodic time variation in particle masses and couplings. In such
theories the gauge invariance suggests that the $\phi$ should possess an
identical coupling constant to charged leptons, in which case scalar
interactions with the electrons provide a good opportunity for detection
through atomic clocks (e.g., Arvanitaki et al., 2016), accelerometers (e.g.,
Arvanitaki et al., 2018), and gravitational wave detectors (e.g., Lopes and
Silk, 2014; Graham et al., 2016).

Similarly, this $\phi$ field can couple to neutrinos. Once again, these types
of interactions generically result in time-varying corrections to the neutrino
masses, neutrino mass differences, and mixing angles, which can be searched
for in the neutrino flux signals on present and future experimental neutrino
detectors (Aharmim et al., 2013; Abe et al., 2016; Borexino Collaboration et
al., 2018; Aalbers et al., 2020). If $\phi$ couples weakly to the neutrinos,
over a large range of masses, it can significantly modify the neutrino
oscillations probabilities leading to a distorted survival electron neutrino
probability function (Berlin, 2016; Krnjaic et al., 2018).

The motivation for such a model comes from the possibility of this composite
particle physics model resolving two observational problems:

  1. 1.

The classical cold dark matter model leads to several inconsistencies with the
cosmological observational data, such as the missing satellite problem and the
cusp problem (e.g., Primack, 2009). Light dark matter resolves such problems
if dark matter is totally or partially made of light scalar particles with a
mass of the order of $10^{-22}{\rm eV}$ (Hu et al., 2000; Peebles, 2000). In
hierarchical models of structure formation, such type of dark matter is able
to explain the flatness observed on the profiles of the distribution of gas
and stars in halos and filaments (Mocz et al., 2019).

  2. 2.

Although the standard three neutrino flavor model produces a reasonable good
global fit to all the neutrino data (Esteban et al., 2019), there are now many
hints that point out to the possibility of the existence of a fourth neutrino.
This one does not have any other interaction than gravity and for that reason
is known as a sterile neutrino (Diaz et al., 2019). It was found that a flavor
oscillation model made of the 3 active neutrinos plus a sterile neutrino could
explain some of the observed anomalies found on the Short baseline neutrino
oscillation experiments (Giunti et al., 2012, 2013), Liquid Scintillator
Neutrino Detector (Aguilar et al., 2001), and MiniBooNE Short-Baseline
Neutrino Experiment (MiniBooNE Collaboration et al., 2018), as well as the
anomalies related with GALLEX and SAGE solar-neutrino detectors – the so-
called Gallium anomalies (Kostensalo et al., 2019). For instance, Kostensalo
et al. (2019) found that the data favour a $3+1$ neutrino flavor model with
$m_{4}=\;1.1eV$ and mixing matrix element $U_{e4}=0.11$.

One possibility to resolve both problems (neutrino anomalies and structure
formation) is to consider that dark matter is made of a light scalar field
that couples to a sterile neutrino (e.g., Farzan, 2019). The interactions of
$\phi$ and $\nu_{s}$ could impede the oscillations in the universe and thereby
improve the agreement between the structure formation and cosmological
observations (e.g., Dasgupta and Kopp, 2014; Hannestad et al., 2014).

If such $\phi$ and $\nu_{s}$ particles exist today, they were produced
abundantly in the early universe. For instance, sterile neutrinos can be
produced via mixing with active neutrinos (Dodelson and Widrow, 1994), in some
scenarios such neutrino production is being enhanced by the oscillations
between active and sterile neutrinos (Bezrukov et al., 2019, 2020; de Gouvêa
et al., 2020) or by the lepton asymmetry (Shi and Fuller, 1999). The
production of light dark matter can take many forms, such as vector bosons by
parametric resonance production (Dror et al., 2019). For instance, some models
predict a sterile neutrino abundance of
$\Omega_{s}^{2}h^{2}=0.12\left({\sin^{2}{(2\theta_{s})}}/{3.5\times
10^{-9}}\right)\left({m_{\nu_{s}}}/{7keV}\right)$ where $m_{\nu_{s}}$ and
$\theta_{s}$ is the sterile neutrino mass and mixing angle (Kusenko, 2009).
For the light dark matter field some authors obtained
$\Omega_{\phi}^{2}=0.1\left({a_{o}}/{10^{17}\;{\rm
GeV}}\right)^{2}\left({m_{a}}/{10^{-22}\;{\rm eV}}\right)^{1/2}$, where
$a_{o}$ is a parameter that relates to the initial misalignment angle of the
axion, and $m_{a}$ is the axion mass (Hui et al., 2017; Niemeyer, 2019).
Conveniently, we will assume that in the present-day universe the total dark
matter abundance is given by

$\displaystyle\Omega_{\rm
DM}h^{2}=\Omega_{\phi}^{2}h^{2}+\Omega_{\nu_{s}}^{2}h^{2}$ (1)

where the $\Omega_{\phi}^{2}h^{2}$ and $\Omega_{s}^{2}h^{2}$ are the total
$\phi$ and $\nu_{s}$ densities in the present universe, respectively. For
future reference, we assume that the present-day total dark matter abundance
$\Omega_{\rm DM}h^{2}=0.12$ (Planck Collaboration et al., 2018), and the dark
matter density in the solar neighborhood is $\rho^{\odot}_{\rm
DM}=0.39\;GeV\;cm^{-3}$ (Catena and Ullio, 2010).

In this paper, we study the impact that this light dark matter field has in
the 3+1 neutrino flavor model. Specifically, we discuss how the light dark
matter field modifies the neutrino flavor oscillations, and by using the
current sets of solar neutrino data, we also put constraints in the parameters
of such models and make predictions for the future neutrino experiments.

The article is organized as follows. In section 2, we discuss how the light
dark matter drives the 3+1 neutrino flavor oscillations. In section 3, we
present the neutrino flavor oscillation model in the presence of a cosmic
light dark matter field. In section 4, we compute the survival electron
neutrino probabilities for the electron neutrinos produced in the proton-
proton (PP) chain and carbon-nitrogen-oxygen (CNO) cycle solar nuclear
reactions. In section 5, we discuss the results in relation to current
experiments and future ones. Finally, in section 6, we present the conclusion
and a summary of our results.

If not stated otherwise, we work in natural units in which $c=\hbar=1$. In
these units all quantities are measured in GeV, and we make use of the
conversion rules $1m=5.068\times 10^{15}GeV^{-1}$, $1kg=5.610\times
10^{26}GeV$ and $1sec=1.519\times 10^{24}GeV^{-1}$.

##  2\. Light Dark Matter and Sterile Neutrinos in the Universe

We assume that in the present universe, the dark matter is composed of two
fundamental particles: a light scalar boson $\phi$ and sterile neutrinos
$\nu_{s}$, where $m_{\phi}$ and $m_{\nu_{s}}$ are their respective masses
(Hannestad et al., 2014). The LDM field $\phi$ couples with the active
neutrinos and the sterile neutrino by a Yukawa interaction
$g_{\phi}\phi\nu_{s}\nu_{s}$ where $g_{\phi}$ is a dimensionless coupling
(Farzan, 2019). To illustrate this effect, consider an LDM scalar $\phi$ with
a Yukawa coupling to active neutrinos. Then the relevant part of the
Lagrangian reads

$\displaystyle{\cal L}\supset-(m_{\nu}+g\phi)\nu\nu+H.c.,$ (2)

where for convenience of representation the flavor indices have been
suppressed. We also assume that the dimensionless coupling is very small
($g\ll 1$). From the Euler-Lagrange equations of $\nu$ and $\phi$, it is
possible to show that the effect of $\phi$ on the propagation of the neutrino
is equivalent to changing the neutrino mass from $m_{\nu}$ to $m_{\nu}+\delta
m_{\nu}$. As we will see later, this $\delta m_{\nu}$ perturbation will induce
time variations in the mass-squared differences and mixing angles of all
neutrino flavors through $\phi$. In principle the Yukawa couplings can have
any structure in the neutrino flavor space. In this work, we will focus on two
convenient scenarios of great interest to neutrino detectors: mass-square
differences and mixing angles (e.g., Ding and Feruglio , 2020). Moreover, we
will also assume that $\delta m_{\nu}=g_{\phi}\phi$ (e.g., Smirnov and Xu,
2019).

###  2.1. Dark Matter Time-dependent Variation

The hypothesis that dark matter in the local universe is made of very light
particles leads to the following description: the LDM field $\phi$ in the dark
matter halo of the Milky Way is represented by a group of plane waves with
frequency $\omega_{\phi}$, such that
$\omega_{\phi}=m_{\phi}(1+v_{\phi}^{2}/2)$ where $v_{\phi}$ is the virial
velocity of the particles in the dark halo. A population of such light
particles will smooth inhomogeneities in the dark matter distribution on
scales smaller than the de Broglie wavelength $\lambda_{\rm dB}$ of these LDM
particles. For any particle, we compute $\lambda_{\rm dB}$ using the relation
$\lambda_{\rm dB}=1.24\times 10^{22}\left(10^{-23}{\rm
eV}/m_{\phi}\right)\left(10^{-3}/v_{\phi}\right)\;{\rm cm}$.

We notice that the kinetic term on $\omega_{\phi}$ is neglected once the
virial velocity $v_{\phi}\sim 10^{-3}$ is very small (e.g., Blas et al.,
2017). Therefore, we dropped the corrections related with $v_{\phi}$ for the
equation 2. Accordingly, the general form of this LDM field reads

$\displaystyle\phi(\vec{r},t)=\phi_{o}\cos{(m_{\phi}t+\epsilon_{o})}\approx\phi_{o}\cos{(m_{\phi}t)},$
(3)

where $\phi_{o}$ and $\epsilon_{o}=\vec{v}_{\phi}\cdot\vec{r}$ are the
amplitude and phase of the wave $\phi(t)$, respectively. In this work we
consider $\epsilon_{o}\approx 0$. Moreover, the energy momentum of a free
massive oscillating field has a density given by
$\rho_{\phi}=\phi_{o}m_{\phi}/2$ and a pressure given by
$p_{\phi}=-\rho_{\phi}\cos{(2m_{\phi}t)}$. Although formally $\rho_{\phi}$ has
an oscillating part proportional to $\phi(t)$, because this component is very
small we neglected its contribution in this analysis (Khmelnitsky and Rubakov,
2014).

The quantity $\phi_{o}$ is a slowly varying function of the position.
Conveniently, the amplitude of $\phi(t)$ can be written as
$\phi_{o}=\sqrt{2\rho_{\phi}(r)}/m_{\phi}$ where
$\rho_{\phi}(r)=\rho^{\odot}_{\rm DM}\left(\Omega_{\phi}/\Omega_{\rm
DM}\right)$ is the fraction of dark matter density in $\phi$ particles at the
space-time coordinate $\vec{r}$. Accordingly, the Sun immersed in this light
dark matter halo will experience a periodic perturbation due to the action of
the $\phi(\vec{r},t)$, which by the presence of a Yukawa coupling $g_{\phi}$
will exert a temporal variation on the propagation of all neutrinos. We
estimate the dark matter density number $n_{\phi}$ in the solar neighborhood
as follows: if we consider that the main contribution arises from a single
dark matter particle with mass $m_{\phi}$, then the relevant density in our
case will take the value $n_{\phi}=\rho_{\phi}/m_{\phi}$. If we assume that
all dark matter is made of $\phi$ bosons, we have
$\rho_{\phi}=\rho^{\odot}_{\rm DM}=0.39\;GeV\;cm^{-3}$ (Catena and Ullio,
2010) and $m_{\phi}=10^{-22}\;{\rm eV}$ then $n_{\phi}=3.9\times
10^{30}\;cm^{-3}$ (particles per centimeter cubed). This value is only 2
orders of magnitude smaller than the density of electrons in the Sun’s core,
$n_{e}\sim 6\;10^{31}\;cm^{-3}$ (Lopes and Turck-Chièze, 2013). Since these
particles are very light, we assume that there is no accretion of these
particles in the Sun’s core during its evolution in the main sequence until
the present age.

###  2.2. Neutrino Time-dependent Dark-matter-induced Oscillations

In the presence of the LDM field $\phi$, the neutrino mass $m_{\nu}$,
according to equation 2 (Ding and Feruglio , 2020), will receive a
contribution $\delta m_{\nu}=g\phi$, such that from equation 3, we obtain

$\displaystyle\frac{\delta
m_{\nu}}{m_{\nu}}=\epsilon_{\phi}\cos{(m_{\phi}t)},$ (4)

where $\epsilon_{\phi}$ is the amplitude

$\displaystyle\epsilon_{\phi}=\frac{g_{\phi}\sqrt{2\rho_{\phi}}}{m_{\phi}m_{\nu}}=\frac{g_{\phi}\sqrt{2\rho^{\odot}_{\rm
DM}}}{m_{\phi}m_{\nu}}\;\left(\frac{\Omega_{\phi}}{\Omega_{\rm
DM}}\right)^{1/2}.$ (5)

If not stated otherwise, we will assume that all dark matter in the present
universe is made of only LDM particles such that $\Omega_{\phi}=\Omega_{\rm
DM}$. We observe that $\epsilon_{\phi}$ is a relevant factor even if $\phi$ is
a small fraction of the dark matter halo. In particular, $\phi$ will affect
the oscillation parameters of all neutrino flavors, including the sterile
sterline neutrinos. If we only take into account the first order perturbation,
thus, the neutrino mass-squared difference can be written as

$\displaystyle\Delta m_{ij}^{2}(t)=m_{i}^{2}-m_{j}^{2}\approx\Delta
m^{2}_{ij,o}[1+2\epsilon_{\phi}\cos{(m_{\phi}t)}]$ (6)

where $\Delta m^{2}_{ij,o}$ is the standard (undistorted) value and $\Delta
m^{2}(t)$ evolves through $\phi(t)$ (see Equation 3), with an amplitude
$\epsilon_{\phi}$ (see Equation 5), and a frequency $m_{\phi}$. The mass-
squared difference $\Delta m^{2}_{ij}$ between neutrinos of different flavors
follows the usual convection (e.g., Lopes, 2017) such that $\Delta
m^{2}_{i1}=m^{2}_{i}-m^{2}_{1}$ ($i=$2, $3$, $4$). In particular for the
sterile neutrino, we have $\Delta m^{2}_{41}=m^{2}_{4}-m^{2}_{1}$ where
$m_{4}$ is the mass of the sterile neutrino. Similarly, the mixing angles
variation is written as

$\displaystyle\theta_{ij}(t)\approx\theta_{ij,o}+\epsilon_{\phi}\cos{(m_{\phi}t)},$
(7)

where $\theta_{ij,o}$ is the standard (undistorted) mixing angle. The indexes
$i$ and $j$ in $\theta_{ij}$ follow a convention identical but not equal for
the mass-squared differences (see Lopes , 2018a, and references therein).
Therefore, as first suggested by Krnjaic et al. (2018), the LDM $\phi(t)$
impacts the neutrino flavor oscillations through the modified expressions for
the mass-squared differences (Equation 6) and mixing angles (Equation 7).

##  3\. Light Dark Matter and the Sterile Neutrino Model

In the following section, we consider a 3+1 neutrino flavor oscillation model
to describe the propagation of active neutrinos ($\nu_{e}$, $\nu_{\tau}$,
$\nu_{\mu}$) plus a sterile neutrino $\nu_{s}$ through the solar plasma.
Following the usual notation $(\nu_{e},\nu_{\tau},\nu_{\mu},\nu_{s})$
corresponds to the neutrino flavors, ($\nu_{1}$, $\nu_{2}$, $\nu_{3}$,
$\nu_{4}$) are the mass neutrino eigenstates, and ($m_{1}$, $m_{2}$, $m_{3}$,
$m_{4}$) are the neutrino masses. The evolution of neutrinos propagating in
matter is described by the see

$\displaystyle i\frac{d\Psi}{dr}={\cal
H}\Psi=\frac{1}{2E}\left(\mathbf{U}M^{2}\mathbf{U}^{\dagger}+2E{\cal
V}\right)\Psi,$ (8)

where ${\cal H}$ is the Hamiltonian and
$\Psi=(\nu_{e},\nu_{\tau},\nu_{\mu},\nu_{s})^{T}$. $M^{2}$ is a neutrino mass
matrix, $\mathbf{U}$ is a ($4\times 4$) unitary matrix describing the mixing
of neutrinos and ${\cal V}$ is the diagonal matrix of Wolfenstein potentials
(Kuo and Pantaleone, 1989). $M^{2}$ is defined as
$M^{2}=\mathrm{{diag}}\\{0,\Delta m^{2}_{21},\Delta m^{2}_{31},\Delta
m^{2}_{41}\\}$. The first term of the Hamiltonian describes the neutrino
propagation through vacuum and the second term incorporates the matter effects
or Mikheyev-Smirnov-Wolfenstein (MSW) effects (Wolfenstein, 1978; Mikheyev and
Smirnov, 1985). In general, the Hamiltonian ${\cal H}$ that drives the
evolution of neutrino flavor must include the Wolfenstein potentials related
with $\phi(t)$ (Brdar et al., 2018).

In most studies of three-neutrino flavor models, the authors are solely
interested in the modulation coming from the square mass differences $\Delta
m^{2}_{ij}(t)$ (by Equation 6) and mixing angles $\theta_{ij}(t)$ (by Equation
7). For that reason, all neutrinos are assumed to couple $\phi(t)$. As a
consequence, their contribution to ${\cal V}$ cancels out. Hence, it is
correct to neglect the contribution of $\phi(t)$ to the Wolfenstein potential
(Dev et al., 2020). Nevertheless, here in this 3+1 neutrino flavor model, as
we will discuss later, we include the contribution of $\phi(t)$ in ${\cal V}$.

This 3+1 neutrino flavor model with dark matter is identical to the standard
(undistorted) three-neutrino flavor model (see Equation 8). However, in this
model we included a sterile neutrino, and the Wolfenstein potentials in ${\cal
V}$ are modified to take into account the new LDM field $\phi$ (Miranda et
al., 2015).

###  3.1. Neutrino Matter-induced Oscillations

Figure 1.— The survival probability electron neutrinos $P_{ee}(E,\phi)$ and
$\langle P_{ee}(E)\rangle$ in a 3+1 neutrino flavor oscillation model in which
neutrino couple to the light dark matter field $\phi$ with fixed value of
$m_{\phi}$ and $\epsilon_{\phi}$ (see main text). The figure shows $\langle
P_{ee}(E)\rangle$ (with $\epsilon_{\phi}\approx 0\%$) for the following LDM
models with $G_{\phi}/m_{\phi}$ ratios: $10^{29}\;{\rm G_{\rm F}\,eV^{-1}}$
(blue curve), $10^{30}\;{\rm G_{\rm F}\,eV^{-1}}$ (green curve),
$10^{31}\;{\rm G_{\rm F}\,eV^{-1}}$ (orange curve). The figure also shows a
model with $G_{\phi}\approx 0$ (red curve) and an ensemble of $P_{ee}(E,\phi)$
corresponding to different time-varying mass-square differences and angles for
$\epsilon_{\phi}=1.5\%$ (pink band). See main text for details.

In the standard three-neutrino flavor model111In this model, the intermediate
particle is an heavy boson, specifically the $Z$ or $W^{\pm}$ bosons., the
matter potential ${\cal V}$ takes into account the interaction of active
neutrinos ($\nu_{e},\nu_{\mu},\nu_{\tau}$) with the ordinary fermions of the
solar plasma, for which the ${\cal V}=\mathrm{{diag}}\\{V_{cc}+V_{\rm
nc},V_{\rm nc},V_{\rm nc}\\}$ where $V_{\rm cc}$ corresponds to the weak
charged current ($\rm cc$) that takes into account the forward scattering of
$\nu_{e}$ with electrons, and $V_{\rm nc}$ is the weak neutral current ($\rm
nc$) that corresponds to the scattering of the active neutrinos with the
ordinary fermions of the solar plasma (e.g., Xing, 2020). $V_{\rm nc}$ can be
expressed as $V_{\rm nc}=V^{e}_{\rm nc}+V^{p}_{\rm nc}+V^{n}_{\rm nc}$ where
$V^{j}_{\rm nc}$ with $j=$ $e$, $p$, $n$ are the contributions coming from
electrons, protons, and neutrons, respectively. However due to the electrical
neutrality of the solar plasma, the contribution of $V^{e}_{\rm nc}$ and
$V^{p}_{\rm nc}$ canceled out such that $V_{\rm nc}=V^{n}_{\rm nc}$.
Accordingly, $V_{\rm cc}=\sqrt{2}G_{\rm F}\;n_{e}(r)$ and $V_{\rm
nc}=V^{n}_{\rm nc}=G_{\rm F}/\sqrt{2}\;n_{n}(r)$. Here $G_{\rm F}$ is the
Fermi constant and $n_{e}(r)$ and $n_{n}(r)$ are the number density of
electrons and neutrons inside the Sun. Nevertheless, since $V_{\rm nc}$ is an
universal term for all active neutrino flavors, and as such does not change
the flavor oscillations pattern, conveniently we write ${\cal
V}=\mathrm{{diag}}\\{V_{\rm cc}+0,0,0\\}$. Now, the inclusion of sterile
neutrinos in the neutrino flavor model alters ${\cal V}$ (from Equation 8) by
incorporating a new degree of freedom, as a consequence ${\cal
V}=\mathrm{{diag}}\\{V_{\rm cc}+V_{\rm nc},V_{\rm nc},V_{\rm nc},0\\}$ (e.g.,
Giunti and Li, 2009; Maltoni and Smirnov, 2016; Xing, 2020).

Finally, in our 3+1 neutrino flavor model, we include the interaction of
active and sterile neutrinos with the dark matter field $\phi$ by means of an
intermediate heavy boson $I$222 We assume the boson $I$ has a mass $m_{I}$
identical to $Z$ and $W^{\pm}$ bosons.. These interactions result from the
forward scattering of these neutrinos through the LDM field $\phi$, thus
${\cal V}=\mathrm{{diag}}\\{V_{\rm cc}+V_{\rm nc}+V_{\nu_{e}\phi},V_{\rm
nc}+V_{\nu_{\mu}\phi},V_{\rm nc}+V_{\nu_{\tau}\phi},V_{\rm
nc}+V_{\nu_{s}\phi}\\}$, where $V_{\nu_{i}\phi}$ (with
$\nu_{i}=\nu_{e},\nu_{\mu},\nu_{\tau},\nu_{s}$) relates to the neutrino
$\nu_{i}$. This ${\cal V}$ corresponds to a generalization of the Wolfenstein
potentials found in the literature, for which most neutrino flavor models only
take into account the scattering of the sterile neutrinos on heavy dark matter
(Capozzi et al., 2017; Lopes , 2018a; Lopes and Silk, 2019).

In our model, we opt to assume that all active neutrinos experience the same
interaction with the LDM field $\phi$, such that their dark matter potentials
are the same, such that $V_{\nu_{j}\phi}=V_{\nu_{a}\phi}$ (with
$j=e,\mu,\tau$), it follows ${\cal V}=\mathrm{{diag}}\\{V_{\rm cc}+V_{\rm
nc}+V_{\nu_{a}\phi},V_{\rm nc}+V_{\nu_{a}\phi},V_{\rm
nc}+V_{\nu_{a}\phi},V_{\rm nc}+V_{\nu_{s}\phi}\\}$. Now, if we subtract the
common term $V_{\rm nc}+V_{\nu_{a}\phi}$ to the diagonal matrix ${\cal V}$,
the latter takes the simple form: ${\cal V}=\mathrm{{diag}}\\{V_{\rm
cc},0,0,V_{\nu_{s}\phi}-V_{\nu_{a}\phi}-V_{\rm nc}\\}$.

The potential $V_{\nu_{i}\phi}$ (with $i=a,s$) is given by
$V_{\nu_{i}\phi}=G_{\nu_{i}\phi}n_{\phi}$ where $G_{\nu_{i}\phi}$ is the
equivalent of the Fermi constant and $n_{\phi}$ is the distribution of dark
matter inside the Sun (Smirnov and Xu, 2019). Equally, $V_{\nu_{i}\phi}$
relates directly with the local density of dark matter $\rho^{\odot}_{\rm DM}$
by the expression:
$V_{\nu_{s}\phi}=\left(G_{\nu_{s}\phi}/m_{\phi}\right)\left(\rho^{\odot}_{\rm
DM}\Omega_{\phi}/\Omega_{\rm DM}\right)$, where we assume the ratio
$G_{\nu_{s}\phi}/m_{\phi}$ is a free parameter of the LDM model. The
generalized Fermi constant is defined as
$G_{\nu_{i}\phi}=g_{\nu_{i}}g_{\phi}/m_{I}^{2}$ where $g_{\nu_{i}}$ represents
the coupling constant of the corresponding neutrino $\nu_{i}$, and $m_{I}$ is
the mass of the intermediate boson $I$ (Miranda et al., 2015). This expression
for the potential $V_{\nu_{i}\phi}$ is valid since we assume that
$m_{I}^{-1}\ll R_{\odot}$ where $R_{\odot}$ is the solar radius333As an
example, if we consider $m_{I}\approx m_{Z}\approx 90GeV$ where $m_{Z}$ is the
mass of the $Z$ boson, then the propagation of neutrinos (like of the neutral
current) verifies the condition $m_{I}^{-1}\ll R_{\odot}$. (Smirnov and Xu,
2019). In general, we could expect that the contribution of $\phi(t)$ to
$V_{\nu_{i}\phi}$ could lead to a time-dependent relation, however, as
discussed previously (in section 2.1) and mentioned for the first time by
Khmelnitsky and Rubakov (2014), this is because the oscillatory component on
the local density relates with $v_{\phi}^{2}$. This term is minimal, and
therefore we neglected it.

In this preliminary study, without loss of generality, we choose to simplify
${\cal V}$ further: since the term $V_{\nu_{s}\phi}-V_{\nu_{a}\phi}-V_{\rm
nc}$ has two Wolfenstein potentials ( $V_{\nu_{a}\phi}$ and $V_{\nu_{s}\phi}$)
that effectively correspond to two new degrees of freedom, both of these have
an identical impact on the neutrino flavor oscillation model. We choose to
simplify the model by assuming that $V_{\nu_{a}\phi}$ is much smaller than
$V_{\rm nc}$. Consequently, ${\cal V}$ takes the simplified form: ${\cal
V}\approx\mathrm{{diag}}\\{V_{\rm cc},0,0,V_{\nu_{s}\phi}-V_{\rm nc}\\}$. For
reference, we note that in the Sun’s core $V_{\rm nc}$ is always smaller than
$V_{\rm cc}$, once $n_{e}$ is more than twice as lager as $n_{n}$ (e.g. Lopes,
2018b). This potential is identical to others found in the literature, for
instance in Capozzi et al. (2017) and Lopes  (2018a). Therefore, the matter
potential $V_{\nu_{s}\phi}$ reads $V_{\nu_{s}\phi}=G_{\nu_{s}\phi}n_{\phi}$
where for convenience of analysis, we choose to define the generalized Fermi
constant as $G_{\nu_{s}\phi}=4\sqrt{2}G_{\phi}G_{\rm F}$ where $G_{\phi}$ is
our free parameter. Since these dark matter particles have a mass much smaller
than $4\;{\rm GeV}$, the solar plasma conditions do not allow the accretion of
dark matter by the Sun (e.g., Lopes and Lopes, 2019), therefore we will assume
that the distribution of dark matter inside the star is equal to the value
measured for the solar neighborhood $n_{\phi}$ (see section 2.1).

Figure 2.— The survival probability electron neutrinos $P_{ee}(E,\phi)$ and
the corresponding $\langle P_{ee}(E)\rangle$ for some of the models shown in
figure 1 for which $\epsilon_{\phi}=1.5\%$ : $G_{\phi}=10^{20}\;G_{\rm F}$
(blue band and dark blue curve), $G_{\phi}=10^{21}\;G_{\rm F}$ (green band and
dark green curve) and $G_{\phi}=10^{22}\;G_{\rm F}$ (orange band and dark red
curve). The figure also shows $\langle P_{ee}(E)\rangle$ for the 3+1 model
with no time dependence ($G_{\chi}\approx 0$) and $\epsilon_{\phi}\approx 0$)
as a thin dashed black curve. The latter curve corresponds to the red curve in
figure 1.

###  3.2. Neutrino Flavor Oscillation Model and the Survival Probability of
Electron Neutrinos

If we adopt as reference the current experimental set of parameters for the
active neutrinos (e.g., Esteban et al., 2019), the propagation neutrinos in
the solar interior are completely adiabatic. The same is valid for the 3+1
neutrino flavor oscillation model coupled to an LDM field $\phi$ considered in
this study. Conveniently, the propagation of neutrinos away from resonances is
well represented by a two neutrino flavor oscillation model. The motivation
for such approximation can be found in Lopes (2018b) and references therein.
In such a case, the electron neutrino flavor oscillation is dominated by the
$(\nu_{1},\nu_{2})$ mass eigenstates and is only slightly affected by the
decoupled $(\nu_{3},\nu_{4})$ eigenstates, since the associated mixing angles
for the latter pair are very small (Kuo and Pantaleone, 1986). Moreover,
$\nu_{3}$ and $\nu_{4}$ evolve independent of each other and are completely
independent of the doublet $(\nu_{1},\nu_{2})$. In this limit, as proposed by
several authors (e.g, Palazzo, 2011; Blennow and Smirnov, 2013), the split of
the 3+1 neutrino flavor model into a dominant two neutrino flavor model
($\nu_{e},\nu_{\mu}$) with additional corrections for $\nu_{\tau}$ and
$\nu_{s}$ significantly simplified the calculation and allowed us to obtain an
analytical solution (e.g., Kuo and Pantaleone, 1989).

Among the many expressions available in the literature to compute the survival
probability of electron neutrinos $P_{e}$ (e.g., Lunardini and Smirnov, 2000;
Miranda et al., 2015) in a 3+1 neutrino flavor model developed in the
approximate scenario of a two-flavor neutrino model (e.g., Kuo and Pantaleone,
1989), we opted to choose the expression obtained by Capozzi et al. (2017) for
the case in which $V_{\rm cc}E/\Delta m_{31}^{2}\ll 1$ (and $s_{34}=0$) which
has a better numerical accuracy than others. In that case the survival
probability of electron neutrinos, i.e., $P_{e}$ $[\equiv
P(\nu_{e}\rightarrow\nu_{e}]]$, reads

$\displaystyle
P_{ee}(E,\phi)=s_{13}^{4}+c_{13}^{4}c_{24}^{4}c_{14}^{4}+a_{m}+b_{m},$ (9)

where $c_{ij}=\cos{\theta_{ij}}$ and $s_{ij}=\sin{\theta_{ij}}$. The functions
$a_{m}$ and $b_{m}$ are dependent on the internal structure of the Sun and are
given by the expressions:

$\displaystyle a_{m}=C_{1}(s_{m}c_{14}-c_{m}s_{14}s_{24})^{2}$ (10)

and

$\displaystyle b_{m}=C_{2}(c_{m}c_{14}+s_{m}s_{14}s_{24})^{2}$ (11)

where $c_{m}=\cos{\theta_{m}}$, $s_{m}=\sin{\theta_{m}}$,
$C_{1}=c_{13}^{4}(c_{14}s_{12}-c_{12}s_{14}s_{24})^{2}$ and
$C_{2}=c_{13}^{4}(c_{12}c_{14}+s_{12}s_{14}s_{24})^{2}$. The angle
$\theta_{m}$ is obtained for the present-day Sun (i.e., the standard solar
model, see details of this model in Lopes and Silk, 2013) using the expression
(Capozzi et al., 2017): $\cos{(2\theta_{m})}={\cal M}_{x}({\cal
M}_{y}^{2}+{\cal M}_{x}^{2})^{-1/2},$ where ${\cal
M}_{x}\equiv\cos{(2\theta_{12})}-\eta_{\nu}V_{x}$ and ${\cal
M}_{y}\equiv|\sin{(2\theta_{12})}+\eta_{\nu}V_{y}|$. $\eta_{\nu}$ is the ratio
of the energy of the neutrino $E$ in relation to $\Delta m^{2}_{21}$ given by
$\eta_{\nu}(E)=4E/\Delta m_{21}^{2}$. The functions $V_{x}$ and $V_{y}$ are
given by

$\displaystyle V_{x}=\frac{1}{2}\left[V_{\rm
cc}c^{2}_{13}(c^{2}_{14}-s_{14}^{2}s_{24}^{2})+V_{s}(s_{14}^{2}-c_{14}^{2}s_{24}^{2})\right],$
(12) $\displaystyle V_{y}=(V_{s}-V_{\rm cc}c_{13}^{2})c_{14}s_{14}s_{24}$ (13)

and

$\displaystyle V_{s}=V_{\nu_{s}\phi}-V_{\rm nc}.$ (14)

Figure 3.— The survival probability of the electron neutrinos for several
solar nuclear reactions: $pp$ (yellow diamond), ${}^{7}Be$ (green square), 7Be
(red upward triangle), $pep$ (blue downward triangle) and 8B HER (salmon
circle), 8B HER-I (orange circle), ${}^{8}B$ HER-II (magenta circle) and
${}^{8}B$ (cyan square). The diamond, triangle, and circles points use data
from the Borexino (Borexino Collaboration et al. , 2020; Agostini et al.,
2019; Borexino Collaboration et al., 2018; Bellini et al., 2010) and the two
square data points correspond to data obtained from SNO (Aharmim et al., 2013)
and Super-Kamiokande (Abe et al., 2016; Cravens et al., 2008) detectors. Each
color band corresponds to an ensemble of $P_{ee}(E,\phi)$ (Equation 9) in a
3+1 neutrino model with $G_{\phi}\approx 0$ and $\epsilon_{\phi}\approx 0$
(dashed curve), 1.5% (green), 3.0% (orange) and 6.0% (light blue). These
survival probabilities are computed using an high-Z SSM (see the main text)
and the experimental values from the different neutrino experiments.

###  3.3. Light Dark Matter Impact on Solar Neutrinos

The survival probability of electron neutrinos (Equation 9) is a time-
dependent function through equations (6), (7) and (3). Conveniently we define
an effective oscillation probability $\langle P_{ee}(E)\rangle$ that
corresponds to an ensemble average of all the $P_{ee,}(E,\phi)$ (Equation 9),
as such

$\displaystyle\langle
P_{ee}(E)\rangle=\int_{0}^{\tau_{\phi}}P_{ee}(E,\phi)\frac{dt}{\tau_{\phi}},$
(15)

where $\tau_{\phi}=2\pi/m_{\phi}$ is the period of the LDM field $\phi(t)$.

The ability of a solar neutrino detector to measure the impact of the time-
dependent LDM field $\phi(t)$ on the survival probability $P_{ee}(E,\phi)$
(Equation 9) depends on three characteristic time scales: the neutrino flight
time $\tau_{\nu}$, the time between two consecutive neutrino detections
$\tau_{ev}$, and the total run time of the experiment $\tau_{ex}$. The
neutrino flight time is proportional to the Earth-Sun distance $d_{\earth}$
such that $\tau_{\nu}=d_{\earth}/c\approx 8.2\;{\rm min}$ where $c$ is the
speed of light. The number of events measured by a detector varies strongly
from one to another.

The next generation of experiments will have $\tau_{ev}$ much larger than the
pioneer Homestake experiment that only detects a few events per year (Bahcall
and Davis, 1976). The forthcoming Jiangmen Underground Neutrino Observatory
(JUNO; Adam et al., 2015) experiment expects to measure a few tens of
neutrinos per day (for instance 200 events per day or $\tau_{ev}\approx{\rm
7\;min}$). The total experimental run time for most solar neutrino detectors
is of the order of a few decades (for instance $\tau_{ex}\approx{\rm
10\;yr}$), and future experiments will also have significant running times.
Hence for all models considered in this study, we assume that solar neutrino
detectors will run for long periods and will collect a large number of events,
therefore we assume that $\tau_{ev}$ and $\tau_{ex}$ have sufficient small and
large values, respectively.

In such conditions, the solar neutrino spectra time modulation by $\phi(t)$
depends on the period $\tau_{\phi}$ of the LDM field in comparison to the
flight time of solar neutrinos $\tau_{\nu}$. Since these neutrinos have a
$\tau_{\nu}\approx 8.2\;{\rm min}$, it is possible to find the value of
$m_{\phi}$ for which $\tau_{\nu}=\tau_{\phi}$ which occurs for
$m_{\phi,c}=8.3\;10^{-18}\;{\rm eV}$. Accordingly, we can define two regimes
for the time modulation of survival probability of electron neutrinos :

  1. 1.

For $\tau_{\phi}\geq\tau_{\nu}$ (low-frequency regime or low LDM mass), the
time modulation of $P_{ee}(E,\phi)$ occurs when the period of $\phi(t)$ is
larger than $\tau_{\nu}$. In this case a temporal variation of the neutrino
signal may be observed. This corresponds to a LDM field with a mass such that
$m_{\phi}\leq m_{\phi,c}$. Therefore, the LDM field can induce an observable
time variation in neutrino oscillation measurements as periodicity in the
solar neutrino fluxes (Berlin, 2016). Obviously, if $\tau_{\phi}$ becomes very
large, the modulation of $P_{ee}(E,\phi)$ becomes indistinguishable from the
standard scenario (undistorted case), since the running time of the experiment
is not sufficient to observe this phenomena. Nevertheless, in our study, the
LDM field has always an $m_{\phi}\geq 10^{-23}\;{\rm eV}$ or a period
$\tau_{\phi}\geq 13\;{\rm yr}$. Therefore, it is always possible to probe such
a model with current experimental running times.

  2. 2.

For $\tau_{\phi}\leq\tau_{\nu}$ (high-frequency regime or high LDM mass), the
change of $P_{ee}(E,\phi)$ due to $\phi(t)$ is too fast to be observed as a
modulating signal like in the previous case. This regime occurs for LDM fields
with a mass such that $m_{\phi}\geq m_{\phi,c}$. Nevertheless, the time
average of the ensemble of oscillation probability $P_{ee}(E,\phi)$ can be
distorted in such a regime, hence the effect can be detected as $\langle
P_{ee}(E)\rangle$ which will deviate from the standard scenario (Krnjaic et
al., 2018). The net effect of averaging over time induces a shift in the
observed values of $P_{ee}(E,\phi)$ relative to its undistorted value.

Therefore, we can expect to study both regimes in a quite reasonable range of
LDM masses using data from the present and future solar neutrino experiments.
In fact, some of the current solar neutrino detectors have already large
statistics and high event rates that we can use to look for time modulations
in solar neutrinos. Some of these neutrino collaborations have already
searched for regular phenomena with periods varying from 10 minutes to 10 yr
(e.g., Yoo et al., 2003; Aharmim et al., 2010).

In this work, we will study models that will fall in these two regimes of time
modulation. Therefore to satisfy the conditions mentioned above, we decided to
analyze the impact of the LDM field in solar neutrino fluxes for $\phi(t)$
with a period $\tau_{\phi}$ varying from $4\;\mu{\rm s}$ to $13\;{\rm yr}$ or
equivalently with a $m_{\phi}$ varying from $10^{-9}\;{\rm eV}$ to
$10^{-23}\;{\rm eV}$, which is a range possible to be scanned by future
detectors like Deep Underground Neutrino Experiment (DUNE Collaboration et
al., 2015) and JUNO (An et al., 2016).

Figure 4.— The survival probability of the electron neutrinos for several LDM
models (see the main text and caption of figure 3 and 2 for details). The LDM
models are identical to the ones presented in Figure 2, with the following
changes: (i) $G_{\phi}=0$ and $\epsilon_{\phi}=1.5\%$ (red band); (ii)
$G_{\phi}=10^{20}G_{\rm F}$ and $\epsilon_{\phi}=3.0\%$ (light-blue band);
(iii) $G_{\phi}=10^{21}G_{\rm F}$ and $\epsilon_{\phi}=1.5\%$ (green band);
(iv) $G_{\phi}=10^{22}G_{\rm F}$ and $\epsilon_{\phi}=3.0\%$ (orange band).
The data points correspond to the same ones displayed in figure 3. However,
the data with the lowest energy corresponds to the expected precision to be
attained by the Darwin experiment in measuring $P_{ee}\pm\Delta P_{ee}$, for
which $\Delta P_{ee}$ could be as low as $\Delta P_{ee}=0.017$ (Aalbers et
al., 2020). The continuous dashed curve corresponds to a 3+1 neutrino model
with $G_{\phi}\approx 0$ and $\epsilon=0\%$.

##  4\. Light Dark Matter Impact on Electron Neutrino Spectra

Inside the Sun, the flux variation of neutrinos with different flavors due to
matter (including LDM) is strongly dependent of the local distributions of
electrons and neutrons, but also on the population of dark matter particles in
the solar neighbourhood. This new flavor mechanism (sterile neutrinos and LDM
field $\phi$) affects all electron neutrinos produced in the Sun’s core. A
detailed discussion about the neutrino sources inside the Sun, and their
specific solar properties, can be found in Lopes (2013, 2017). The average
survival probability of electron neutrinos for each nuclear reaction in the
solar interior, i.e., $P_{e,i}(E,\phi)$ is computed by

$\displaystyle
P_{ee,i}(E,\phi)=C_{i}\int_{0}^{R_{\odot}}P_{e}(E,\phi,r)S_{i}(r)4\pi\rho(r)r^{2}dr,$
(16)

where $C_{i}$
$\left(=\left[\int_{0}^{R_{\odot}}S_{i}(r)4\pi\rho(r)r^{2}\;dr\right]^{-1}\right)$
is a normalization constant and $S_{i}(r)$ is the electron neutrino emission
function for the $i$ solar nuclear reaction. $i$ corresponds to the following
solar neutrino sources (from the PP chain and CNO cycle nuclear reactions):
$pp$, $pep$, 8B, 7Be, 13N, 15O and 17F.

Moreover, since the survival probabilities $P_{ee,i}(E,\phi)$ (Equation 16)
are time dependent through $\phi$, these quantities also vary with time.
Therefore, the oscillation probability $\langle P_{ee}(E)\rangle$ (Equation
15) is generalized for each specific nuclear reaction $i$:

$\displaystyle\langle
P_{ee,i}(E)\rangle=\int_{0}^{\tau_{\phi}}P_{e,i}(E,\phi)\frac{dt}{\tau_{\phi}}.$
(17)

The LDM field $\phi$ can lead to different temporal imprints on the neutrino
oscillation measurements. The specific impact depends on the mass of the LDM
particle. In the following, we compute the spectra of neutrinos from any
specific nuclear reaction that we know to be essentially independent of the
properties surrounding solar plasma. Since in the 3+1 neutrino flavor model
new processes exist to change the survival probability of electron neutrinos,
this will modify the solar neutrino spectra measured on Earth. These new
processes will alter the conversion rates of $\nu_{e}$ to other flavors
($\nu_{\mu}$, $\nu_{\tau}$ and $\nu_{s}$) and vice versa. Accordingly, the
electron neutrino spectrum of the nuclear reaction $i$ inside the core is
defined as $\Phi_{i}$, and $\Phi_{i\odot}$ is the electron neutrino spectrum
arriving on Earth (Lopes, 2018b) such that:

$\displaystyle\Phi_{i,\odot}(E)=P_{ee,i}(E,\phi)\Phi_{i}(E)$ (18)

where $P_{ee,i}(E)$ is the average survival probability of electron neutrinos
for reactions in the solar interior as given by equation 16. Equally if we
take the time average of equation 18, we obtain the following averaged
spectrum for each nuclear reaction $i$:

$\displaystyle\Phi_{i,\odot}(E)=\langle P_{ee,i}(E)\rangle\Phi_{i}(E),$ (19)

where $\langle P_{ee,i}(E)\rangle$ is the average survival probability of
electron neutrinos as given by equation 17.

Figure 5.— The ${}^{8}B$ solar electron neutrino spectra measured on Earth
detectors. The blue curve corresponds to the standard (undistorted) neutrino
spectra, the cyan curve to the averaged (distorted) electron spectra, and the
red band corresponds to the ensemble of spectra related with the time-
dependent flavor parameters. This LDM model has a $G_{\phi}\approx 0$ and an
$\epsilon_{\phi}=3.0\%$. In the calculation of the 8B spectrum we use a high-Z
SSM (see the main text). Figure 6.— The $pp$ solar electron neutrino spectra
measured on Earth detectors. The LDM model and color scheme is the same one as
in Figure 5. The error bar curves (yellow curves) on the averaged (distorted)
electron spectra (cyan curve) corresponds to the expected precision to be
attained by the Darwin experiment (Aalbers et al., 2020). See Figure 4 and
main the text.

##  5\. The Sun: Light Dark Matter and Sterile Neutrinos

Here, we will study the impact of the theoretical model presented in the
previous sections, specifically we compute the survival probability of
electron neutrinos (as given by equations (9), (15), (16) and (17)) in the
case of a standard solar model with low-Z (e.g., Lopes and Silk, 2013; Capelo
and Lopes, 2020).

In the parameterization for the 3+1 neutrino flavor oscillation model, we opt
to adopt the recent values obtained in the data analysis of the standard
three-neutrino flavor oscillation model obtained by de Salas et al. (2020),
and for the sterile neutrino additional fiducial parameters we used the values
obtained by Gariazzo et al. (2015). Accordingly, for a parameterisation with a
normal ordering of neutrino masses, the mass-square difference and the mixing
angles have the following values: $\Delta
m^{2}_{21}=7.50^{+0.22}_{-0.20}\times 10^{-5}{\rm eV^{2}}$,
$\sin^{2}{\theta_{12}}=0.318\pm 0.016$, and
$\sin^{2}{\theta_{13}}=0.02250^{+0.00055}_{-0.00078}$. Although, $\Delta
m^{2}_{31}=2.56^{+0.0003}_{-0.0004}\times 10^{-3}{\rm eV^{2}}$ and
$\sin^{2}{\theta_{23}}=0.56^{+0.016}_{-0.022}$, we mention them here for
reference (de Salas et al., 2020). These new parameters are consistent with
previous estimations (Esteban et al., 2019; Gonzalez-Garcia et al., 2016). For
the sterile neutrino, we choose the following fiducial values for the mass-
square difference and mixing angles (Gariazzo et al., 2015, 2016; Capozzi et
al., 2017): $\Delta m^{2}_{41}=1.6\ {\rm eV^{2}}$,
$\sin^{2}{\theta_{14}}=0.027$, $\sin^{2}{\theta_{42}}=0.014$ and the other
mixing angle for the sterile neutrinos are fixed to zero. Moreover, we assume
that all phases ($\delta_{13,14,34}$) and other angles related to the sterile
neutrino are equal to zero.

The present-day internal structure of the Sun corresponds to an up-to-date
standard solar model (SSM) that has a better agreement with neutrino fluxes
and helioseismic data sets. This solar model was obtained from a one-
dimensional stellar evolution code allowed to evolve in time until the
present-day solar age, $4.57$ Gyr, having been calibrated to the values of
luminosity and effective temperature of the present Sun, of $3.8418\times
10^{33}$ erg s-1 and $5777$ K, respectively, as well as the observed abundance
ratio at the Sun’s surface: ($Z_{\text{s}}/X_{\text{s}}$)${}_{\odot}=0.0181$,
where $Z_{s}$ and $X_{s}$ are the metal and hydrogen abundances at the surface
of the star (Turck-Chieze and Lopes, 1993; Bahcall et al., 1995, 2006). This
stellar model was computed with the release version 12115 of the stellar
evolution code MESA (Paxton et al., 2011, 2019). The details about the physics
of this standard solar model in which we use the AGSS09 (low-Z) solar
abundances (Asplund et al., 2009) are described in Lopes and Silk (2013) and
Capelo and Lopes (2020).

Figures 1 and 2 show the impact of the time-dependent mass-square difference
(Equation 6) and mixing angles (Equation 7) on the averaged electron survival
probability $\langle P_{ee}(E)\rangle$ (Equation 15) for which the LDM field
$\phi$ has a fixed amplitude (Equation 5): $\epsilon_{\phi}=0$ or
$\epsilon_{\phi}=1.5\%$. We also show LDM models for which the sterile
neutrino couples to $\phi$ with strength $G_{\phi}$.

The overall shape of the curve $\langle P_{ee}(E)\rangle$ depends on
$G_{\phi}$ times $n_{\phi}$ in the potential $V_{\nu_{s}\phi}$ or the ratio
$G_{\phi}/m_{\phi}$ as previously mentioned. For instance, in a LDM model in
which we fix $m_{\phi}=10^{-9}\;{\rm eV}$ (or $n_{\phi}$), an increase of
$G_{\phi}$ from $10^{20}$ to $10^{22}\;G_{\rm F}$ leads $\langle
P_{ee}(E)\rangle$ to vary significantly, as shown in Figure 1. As expected
this change in $\langle P_{ee}(E)\rangle$ is more pronounced for high energy
neutrinos where the MSW effect is more significant. If we choose higher values
of $m_{\phi}$ the results will somehow be similar (see Figure 1).

Evidently, for an LDM model in which $m_{\phi}$ decreases by a certain amount
($n_{\phi}=\rho_{\phi}/m_{\phi}$), the constancy of $G_{\phi}/m_{\phi}$ in
$V_{\nu_{s}\phi}$ implies that $G_{\phi}$ can increase by the same order of
magnitude to obtain the same MSW effect on the $\langle P_{ee}(E)\rangle$
curve (see Figure 1). For instance, an LDM model with $m_{\phi}=10^{-23}\;{\rm
eV}$ and $G_{\phi}=10^{7}G_{\rm F}$ will have $\langle P_{ee}(E)\rangle$
identical to an LDM model with $m_{\phi}=10^{-9}\;{\rm eV}$ and
$G_{\phi}=10^{21}G_{\rm F}$ , since in both LDM models we have the same ratio:
$G_{\phi}/m_{\phi}\approx 10^{30}$. The same argument explains the reason why
the coupling constant between sterile neutrinos and more massive dark matter
particles is much smaller in those models than in the present study. For
instance, this is the case for particles captured from the dark matter halo by
the Sun. Since over time, the star accreted a significant amount of dark
matter (e.g., Lopes , 2018a), for these models $G_{\phi}$ is significantly
smaller than the value found for the present study.

The most important feature of such a class of LDM models is the time
dependence of the dark matter field $\phi(t)$ and its imprint in the flavor
oscillation parameters’ mass-square differences (Equation 6) and mixing angles
(Equation 7). As predicted by equation 9, there are many $P_{ee}(E,\phi)$ with
near similar behavior. Figure 1 shows an ensemble of time-dependent
$P_{ee}(E,\phi)$ as a pink band. The difference between curves relates to the
dependence of the oscillation parameters on time. In this LDM model it is
assumed there is a very negligible interaction between sterile neutrinos and
$\phi$ (for which $G_{\phi}\approx 0$). The figure also shows $\langle
P_{ee}(E)\rangle$ (red curve) the time-averaged of the ensemble of
$P_{ee}(E,\phi)$ curves that we compute using equation 15.

Although there are several parameters that contribute for the time variability
of $P_{ee}(E,\phi)$ (Equation 9), the main contributions come from
$\theta_{21}$ and $\theta_{13}$. The variability related $\theta_{24}$ is
relevant for the high energies. We notice that the contributions coming from
$\Delta m^{2}_{21}$ and $\theta_{32}$ are much smaller than all the parameters
mentioned above. The amplitude of the $P_{ee}(E,\phi)$ pink band is defined by
the value of $\epsilon_{\phi}$ for which we adopt the fiducial value of
$\epsilon_{\phi}=1.5\%$. It is worth pointing out that the $P_{ee}(E,\phi)$
band is much larger for low-energy than for higher-energy values. Moreover,
the averaged value of this ensemble given by $\langle P_{ee}(E)\rangle$ is
identical to $\langle P_{ee}(E)\rangle$ with $\epsilon_{\phi}\approx 0$.
Figure 2 shows the variability of $P_{ee}(E,\phi)$ for a LDM with different
$G_{\phi}$ values. These results are identical to the model in figure 1.
Nevertheless, the LDM model with the largest $G_{\phi}$ has (an orange) band
with a smaller amplitude around $\langle P_{ee}(E)\rangle$. Once again, the
band thickness decreases for neutrinos with higher energy for all these
models.

Figures 3 and 4 compare our predictions with current solar neutrino data
(e.g., Aalbers et al., 2020). These figures show that LDM models with
relatively low values of $\epsilon_{\phi}$ and $G_{\phi}$ are compatible
overall with current solar neutrino data coming from Borexino, Super-
Kamiokande, and SNO. Clearly, this analysis has also shown that the precision
of our current solar neutrino experiments is not able to distinguish between
some of these LDM models. Nevertheless, it is already possible to put some
constraints on these LDM models. For instance, we found that LDM models with
$G_{\phi}\approx 0$ must have a $\epsilon_{\phi}$ smaller than $3\%$ to be
consistent with all data, including $pp$ measurements of the Borexino detector
(Borexino Collaboration et al. , 2020) (see Figure 3); and any LDM models must
have a ratio $G_{\phi}/m_{\phi}$ smaller than $10^{30}$, otherwise they become
inconsistent with $pp$ and 7Be measurements for several solar detectors
(Borexino Collaboration et al. , 2020; Agostini et al., 2019; Borexino
Collaboration et al., 2018; Bellini et al., 2010) (see figure 4). Figure 4
shows a LDM model with a $m_{\phi}=10^{-9}\;{\rm eV}$ and
$G_{\phi}=10^{22}\;G_{\rm F}$ with a ratio $G_{\phi}/m_{\phi}$ of the order of
$10^{31}\;G_{\rm F}$ (see Figure 4); This ratio is one order of magnitude
larger than the critical $G_{\phi}/m_{\phi}$ value of $10^{30}$ discussed in
the previous section. Figure 4 also shows that the variability related with
time dependence on $P_{ee}(E,\phi)$ decreases for large values of $G_{\phi}$.

There is another important effect that also contributes to the time
variability of $P_{ee}(E,\phi)$. The PP chain and CNO cycle, nuclear reactions
occur at different distances from the center of the Sun and each nuclear
reaction emits neutrinos in a well-defined energy range. As a consequence, the
electron neutrinos produced in each specific nuclear reaction will be affected
differently by the MSW effect. As such, this effect will also contribute to
the overall variability of electron neutrinos $P_{ee,i}(E,\phi)$ (see Equation
16) and their time-averaged $\langle P_{ee,i}(E,\phi)\rangle$ (see Equation
17).

The time-dependent electron neutrino survival probability will have a
significant impact on the neutrino spectra of the different nuclear reactions.
Accordingly, figures 5 and 6 show the spectra correspond to two neutrino
types: $pp$ and ${}^{8}B$ neutrinos. An essential difference between these two
spectra relates to the thickness of the $\epsilon_{\phi}$ band for a fixed
value since thickness decreases with neutrino energy. Therefore the
$\epsilon_{\phi}$ band is more significant for a $pp$ spectrum than for a
${}^{8}B$ neutrino spectrum. This is an effect identical to the one discussed
previously for the $P_{ee}(E,\phi)$ functions. Therefore, the measurement of
solar neutrino fluxes and solar neutrino spectrum in the energy range below
$0.2\;MeV$ will provide the strongest constraint for such a class of dark
matter models. Figure 5 and 6 show the spectra of 8B and $pp$, if we assume
the precision expected to be attained by the Darwin experiment (Aalbers et
al., 2020). Figure 6 also shows the precision expected for the Darwin
experiment.

##  6\. Conclusion

This article focuses on the impact of LDM on solar neutrino fluxes, spectra,
and survival probabilities of electron neutrinos, specifically a dark matter
model made of two particles: a sterile neutrino and an LDM particle. In
particular, we describe how the 3+1 neutrino flavor model is affected by this
type of LDM particles, with an emphasis on how the LDM affects the Wolfenstein
potentials. We also study how the dark matter models affect the survival
probability functions of electron neutrinos related to the different nuclear
reactions occurring in the solar interior, and we compute the spectra of two
relevant solar neutrino sources: $pp$ and ${}^{8}B$ neutrino nuclear
reactions.

By studying a large range of dark matter particle masses (from $10^{-9}$ to
$10^{-23}\;{\rm eV}$) we found that depending on the mass of these LDM
particles and the value of the generalized Fermi constant, the shape of
electron neutrino survival probability and their spectra can vary with time.
We establish that for LDM particles with low masses (low-frequency regime),
the solar neutrino detectors can observe the electron neutrino survival
probability changing with time. Conversely, for dark matter particles with
higher masses (high-frequency regime), this impact can be determined by
measuring the time-averaged electron neutrino survival probability.

It was possible to establish, using data of current solar neutrino
measurements, that those models with a $G_{\phi}/m_{\phi}$ ratio smaller than
$10^{30}\;G_{\rm F}eV^{-1}$ agree with current solar neutrino data from the
Borexino, SNO and Super-Kamiokande detectors. We also found that for models
with a near-zero constant, the time-variability amplitude must be smaller than
3%. Such a constraint is equivalent to the condition
${g_{\phi}\sqrt{2\rho^{\odot}_{\rm
DM}}}/{(m_{\phi}m_{\nu})}\;\left({\Omega_{\phi}}/{\Omega_{\rm
DM}}\right)^{1/2}\leq 0.03$.

Finally, we also found that the precision expected in the measurements to be
made by the Darwin detector will allow us to put powerful constraints to this
class of models.

The author is grateful to the MESA team for having made their code publicly
available. The author also thanks the anonymous referee for the revision of
the manuscript. I.L. thanks the Fundação para a Ciência e Tecnologia (FCT),
Portugal, for the financial support to the Center for Astrophysics and
Gravitation (CENTRA/IST/ULisboa) through the Grant Project No. UIDB/00099/2020
and grant No. PTDC/FIS-AST/28920/2017.

## References

  * Aalbers et al. (2020) Aalbers, J., and 166 colleagues 2020, ”Solar Neutrino Detection Sensitivity in DARWIN via Electron Scattering”,  arXiv e-prints, arXiv:2006.03114.
  * Abe et al. (2016) Abe, K., and 169 colleagues 2016, ”Solar neutrino measurements in Super-Kamiokande-IV”,  Phys. Rev. D, 94, 052010.
  * Adam et al. (2015) Adam, T., and 396 colleagues 2015, ”JUNO Conceptual Design Report”,  arXiv e-prints, arXiv:1508.07166.
  * Agostini et al. (2019) Agostini, M., and 107 colleagues 2019, ”Simultaneous precision spectroscopy of p p , 7Be, and p e p solar neutrinos with Borexino Phase-II”,  Phys. Rev. D, 100, 082004.
  * Aharmim et al. (2010) Aharmim, B., and 124 colleagues 2010, ”Searches for High-frequency Variations in the 8B Solar Neutrino Flux at the Sudbury Neutrino Observatory”,  ApJ, 710, 540-548.
  * Aharmim et al. (2013) Aharmim, B., and 122 colleagues 2013, ”Combined analysis of all three phases of solar neutrino data from the Sudbury Neutrino Observatory”,  Phys. Rev. C, 88, 025501.
  * Aguilar et al. (2001) Aguilar, A., and 25 colleagues 2001, ”Evidence for neutrino oscillations from the observation of $\bar{\nu}_{e}$ appearance in a $\bar{\nu}_{\mu}$ beam”, Phys. Rev. D, 64, 112007.
  * Arvanitaki et al. (2016) Arvanitaki, A., S. Dimopoulos, and K. Van Tilburg 2016, ”Sound of Dark Matter: Searching for Light Scalars with Resonant-Mass Detectors”,  Phys. Rev. Lett., 116, 031102.
  * Arvanitaki et al. (2018) Arvanitaki, A., P. W. Graham, J. M. Hogan, S. Rajendran, and K. Van Tilburg 2018, ”Search for light scalar dark matter with atomic gravitational wave detectors”,  Phys. Rev. D, 97, 075020.
  * Asplund et al. (2009) Asplund, M., N. Grevesse, A. J. Sauval, and P. Scott 2009, ”The Chemical Composition of the Sun”,  ARA&A, 47, 481-522.
  * An et al. (2016) An, F., and 227 colleagues 2016, ”Neutrino physics with JUNO”,  Journal of Physics G Nuclear Physics, 43, 030401.
  * Bahcall and Davis (1976) Bahcall, J. N. and R. Davis 1976, ”Solar Neutrinos: A Scientific Puzzle”,  Science, 191, 264-267.
  * Bahcall et al. (1995) Bahcall, J. N., M. H. Pinsonneault, and G. J. Wasserburg 1995, ”Solar models with helium and heavy-element diffusion”,  Reviews of Modern Physics, 67, 781-808.
  * Bahcall et al. (2006) Bahcall, J. N., A. M. Serenelli, and S. Basu 2006, ”10,000 Standard Solar Models: A Monte Carlo Simulation”,  ApJS, 165, 400-431.
  * Bellini et al. (2010) Bellini, G., and 86 colleagues 2010, ”Measurement of the solar B8 neutrino rate with a liquid scintillator target and 3 MeV energy threshold in the Borexino detector”,  Phys. Rev. D, 82, 033006.
  * Berlin (2016) Berlin, A. 2016, ”Neutrino Oscillations as a Probe of Light Scalar Dark Matter”,  Phys. Rev. Lett., 117, 231801.
  * Bezrukov et al. (2019) Bezrukov, F., A. Chudaykin, and D. Gorbunov 2019, ”Induced resonance makes light sterile neutrino dark matter cool”, Phys. Rev. D, 99, 083507.
  * Bezrukov et al. (2020) Bezrukov, F., A. Chudaykin, and D. Gorbunov 2020, ”Scalar induced resonant sterile neutrino production in the early Universe”, Phys. Rev. D, 101, 103516.
  * Blas et al. (2017) Blas, D., D. L. Nacir, and S. Sibiryakov 2017, ”Ultralight Dark Matter Resonates with Binary Pulsars”,  Phys. Rev. Lett., 118, 261102.
  * Blennow and Smirnov (2013) Blennow, M. and A. Y. Smirnov 2013, ”Neutrino Propagation in Matter”,  arXiv e-prints, arXiv:1306.2903.
  * Borexino Collaboration et al. (2020) Borexino Collaboration et al., Agostini, M., and 98 colleagues 2020, ”Improved measurement of 8B solar neutrinos with 1.5 kt .y of Borexino exposure”,  Phys. Rev. D, 101, article id.062001.
  * Borexino Collaboration et al. (2018) Borexino Collaboration, M. Agostini, and 108 colleagues 2018, ”Comprehensive measurement of pp-chain solar neutrinos”,  Nature, 562, 505-510.
  * Brdar et al. (2018) Brdar, V., J. Kopp, J. Liu, P. Prass, and X.-P. Wang 2018, ”Fuzzy dark matter and nonstandard neutrino interactions”,  Phys. Rev. D, 97, 043001.
  * Catena and Ullio (2010) Catena, R. and P. Ullio 2010, ”A novel determination of the local dark matter density”,  Journal of Cosmology and Astroparticle Physics, 2010, 004.
  * Capelo and Lopes (2020) Capelo, D. and I. Lopes 2020, ”The impact of composition choices on solar evolution: age, helio- and asteroseismology, and neutrinos”,  MNRAS, 498, 1992-2000.
  * Capozzi et al. (2017) Capozzi, F., I. M. Shoemaker, and L. Vecchi 2017, ”Solar neutrinos as a probe of dark matter-neutrino interactions”,  Journal of Cosmology and Astroparticle Physics, 2017, 021.
  * Cravens et al. (2008) Cravens, J. P., and 144 colleagues 2008, ”Solar neutrino measurements in Super-Kamiokande-II”,  Phys. Rev. D, 78, 032002.
  * Dasgupta and Kopp (2014) Dasgupta, B. and J. Kopp 2014, ”Cosmologically Safe eV-Scale Sterile Neutrinos and Improved Dark Matter Structure”, Phys. Rev. Lett., 112, 031803.
  * de Gouvêa et al. (2020) de Gouvêa, A., M. Sen, W. Tangarife, and Y. Zhang 2020, ”Dodelson-Widrow Mechanism in the Presence of Self-Interacting Neutrinos”, Phys. Rev. Lett., 124, 081802.
  * Dev et al. (2020) Dev, A., P. A. N. Machado, and P. Martínez-Miravé 2020, ”Signatures of Ultralight Dark Matter in Neutrino Oscillation Experiments”,  arXiv e-prints, arXiv:2007.03590.
  * de Salas et al. (2020) de Salas, P. F., D. V. Forero, S. Gariazzo, P. Martínez-Miravé, O. Mena, C. A. Ternes, M. Tórtola, and J. W. F. Valle 2020, ”2020 Global reassessment of the neutrino oscillation picture”,  arXiv e-prints, arXiv:2006.11237.
  * Diaz et al. (2019) Diaz, A., C. A. Argüelles, G. H. Collin, J. M. Conrad, and M. H. Shaevitz 2019, Where Are We With Light Sterile Neutrinos?, arXiv e-prints, arXiv:1906.00045.
  * Ding and Feruglio (2020) Ding , G-J., F. Feruglio Testing Moduli and Flavon Dynamics with Neutrino Oscillations, arXiv e-prints, arXiv:2003.13448.
  * Dodelson and Widrow (1994) Dodelson, S. and L. M. Widrow 1994, ”Sterile neutrinos as dark matter”, Phys. Rev. Lett., 72, 17-20.
  * Dror et al. (2019) Dror, J. A., K. Harigaya, and V. Narayan 2019, ”Parametric resonance production of ultralight vector dark matter”, Phys. Rev. D, 99, 035036.
  * DUNE Collaboration et al. (2015) DUNE Collaboration, R. Acciarri, and 804 colleagues 2015, ”Long-Baseline Neutrino Facility (LBNF) and Deep Underground Neutrino Experiment (DUNE) Conceptual Design Report Volume 2: The Physics Program for DUNE at LBNF”,  arXiv e-prints, arXiv:1512.06148.
  * Esteban et al. (2019) Esteban, I., M. C. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni, and T. Schwetz 2019, ”Global analysis of three-flavour neutrino oscillations: synergies and tensions in the determination of $\theta$23, $\delta$CP, and the mass ordering”,  Journal of High Energy Physics, 2019, 106.
  * Farzan (2019) Farzan, Y. 2019, ”Ultra-light scalar saving the 3 + 1 neutrino scheme from the cosmological bounds”, Physics Letters B, 797, 134911.
  * Graham et al. (2016) Graham, P. W., D. E. Kaplan, J. Mardon, S. Rajendran, and W. A. Terrano 2016, ”Dark matter direct detection with accelerometers”,  Phys. Rev. D, 93, 075029.
  * Gariazzo et al. (2016) Gariazzo, S., C. Giunti, M. Laveder, Y. F. Li, and E. M. Zavanin 2016, ”Light sterile neutrinos”,  Journal of Physics G Nuclear Physics, 43, 033001.
  * Gariazzo et al. (2015) Gariazzo, S., C. Giunti, M. Laveder, Y. F. Li, and E. M. Zavanin 2015, ”Light sterile neutrinos”,  Journal of Physics G Nuclear Physics, 43, 033001.
  * Gonzalez-Garcia et al. (2016) Gonzalez-Garcia, M. C., M. Maltoni, and T. Schwetz 2016, ”Global analyses of neutrino oscillation experiments”,  Nuclear Physics B, 908, 199-217.
  * Giunti and Li (2009) Giunti, C. and Y. F. Li 2009, ”Matter effects in active-sterile solar neutrino oscillations”,  Phys. Rev. D, 80, 113007.
  * Giunti et al. (2012) Giunti, C., M. Laveder, Y. F. Li, Q. Y. Liu, and H. W. Long 2012, ”Update of short-baseline electron neutrino and antineutrino disappearance”, Phys. Rev. D, 86, 113014.
  * Giunti et al. (2013) Giunti, C., M. Laveder, Y. F. Li, and H. W. Long 2013, ”Short-baseline electron neutrino oscillation length after the Troitsk experiment”, Phys. Rev. D, 87, 013004.
  * Hannestad et al. (2014) Hannestad, S., R. S. Hansen, and T. Tram 2014, ”How Self-Interactions can Reconcile Sterile Neutrinos with Cosmology”, Phys. Rev. Lett., 112, 031802.
  * Hu et al. (2000) Hu, W., R. Barkana, and A. Gruzinov 2000, ”Fuzzy Cold Dark Matter: The Wave Properties of Ultralight Particles”,  Phys. Rev. Lett., 85, 1158-1161.
  * Hui et al. (2017) Hui, L., J. P. Ostriker, S. Tremaine, and E. Witten 2017, ”Ultralight scalars as cosmological dark matter”,  Phys. Rev. D, 95, 043541.
  * Khmelnitsky and Rubakov (2014) Khmelnitsky, A. and V. Rubakov 2014, ”Pulsar timing signal from ultralight scalar dark matter”,  Journal of Cosmology and Astroparticle Physics, 2014, 019.
  * Krnjaic et al. (2018) Krnjaic, G., P. A. N. Machado, and L. Necib 2018, ”Distorted neutrino oscillations from time varying cosmic fields”,  Phys. Rev. D, 97, 075017.
  * Kuo and Pantaleone (1986) Kuo, T. K. and J. Pantaleone 1986, ”Solar-neutrino problem and three-neutrino oscillations.”,  Phys. Rev. Lett., 57, 1805-1808.
  * Lopes (2013) Lopes, I. 2013, ”Probing the Sun’s inner core using solar neutrinos: A new diagnostic method”,  Phys. Rev. D, 88, 045006.
  * Lopes (2017) Lopes, I. 2017, ”New neutrino physics and the altered shapes of solar neutrino spectra”,  Phys. Rev. D, 95, 015023.
  * Lopes (2018a) Lopes, I. 2018, ”The Sterile-Active Neutrino Flavor Model: The Imprint of Dark Matter on the Electron Neutrino Spectra”,  ApJ, 869, 112.
  * Lopes (2018b) Lopes, I. 2018, ”The spectroscopy of solar sterile neutrinos”,  European Physical Journal C, 78, 327.
  * Lopes and Lopes (2019) Lopes, J. and I. Lopes 2019, ”Asymmetric Dark Matter Imprint on Low-mass Main-sequence Stars in the Milky Way Nuclear Star Cluster”,  ApJ, 879, 50.
  * Lopes and Silk (2019) Lopes, I. and J. Silk 2019, ”Dark matter imprint on 8B neutrino spectrum”,  Phys. Rev. D, 99, 023008.
  * Lopes and Silk (2014) Lopes, I. and J. Silk 2014, ”Helioseismology and Asteroseismology: Looking for Gravitational Waves in Acoustic Oscillations”,  ApJ, 794, 32.
  * Lopes and Turck-Chièze (2013) Lopes, I. and S. Turck-Chièze 2013, ”Solar Neutrino Physics Oscillations: Sensitivity to the Electronic Density in the Sun’s Core”,  ApJ, 765, 14.
  * Lopes and Silk (2013) Lopes, I. and J. Silk 2013, ”Planetary influence on the young Sun’s evolution: the solar neutrino probe”,  MNRAS, 435, 2109-2115.
  * Lunardini and Smirnov (2000) Lunardini, C. and A. Y. Smirnov 2000, ”The minimum width condition for neutrino conversion in matter”,  Nuclear Physics B, 583, 260-290.
  * Kuo and Pantaleone (1989) Kuo, T. K. and J. Pantaleone 1989, ”Neutrino oscillations in matter”,  Reviews of Modern Physics, 61, 937-980.
  * Kostensalo et al. (2019) Kostensalo, J., J. Suhonen, C. Giunti, and P. C. Srivastava 2019, ”The gallium anomaly revisited”, Physics Letters B, 795, 542-547.
  * Kusenko (2009) Kusenko, A. 2009, ”Sterile neutrinos: The dark side of the light fermions”, Phys. Rep., 481, 1-28.
  * Maltoni and Smirnov (2016) Maltoni, M. and A. Y. Smirnov 2016, ”Solar neutrinos and neutrino physics”,  European Physical Journal A, 52, 87.
  * Mikheyev and Smirnov (1985) Mikheyev, S. P. and A. Y. Smirnov 1985, ”Resonance enhancement of oscillations in matter and solar neutrino spectroscopy”,  Yadernaya Fizika, 42, 1441-1448.
  * MiniBooNE Collaboration et al. (2018) MiniBooNE Collaboration, A. A. Aguilar-Arevalo, and 47 colleagues 2018, arXiv e-prints, arXiv:1805.12028.
  * Miranda et al. (2015) Miranda, O. G., C. A. Moura, and A. Parada 2015, ”Sterile neutrinos, dark matter, and resonant effects in ultra high energy regimes”,  Physics Letters B, 744, 55-58.
  * Mocz et al. (2019) Mocz, P., and 12 colleagues 2019, ”First Star-Forming Structures in Fuzzy Cosmic Filaments”, Phys. Rev. Lett., 123, 141301.
  * Niemeyer (2019) Niemeyer, J. C. 2019, ”Small-scale structure of fuzzy and axion-like dark matter”, arXiv e-prints, arXiv:1912.07064.
  * Palazzo (2011) Palazzo, A. 2011, ”Testing the very-short-baseline neutrino anomalies at the solar sector”,  Phys. Rev. D, 83, 113013.
  * Paxton et al. (2011) Paxton, B., L. Bildsten, A. Dotter, F. Herwig, P. Lesaffre, and F. Timmes 2011, ”Modules for Experiments in Stellar Astrophysics (MESA)”,  ApJS, 192, 3.
  * Paxton et al. (2019) Paxton, B., and 16 colleagues 2019, ”Modules for Experiments in Stellar Astrophysics (MESA): Pulsating Variable Stars, Rotation, Convective Boundaries, and Energy Conservation”,  ApJS, 243, 10.
  * Planck Collaboration et al. (2018) Planck Collaboration, N. Aghanim, and 181 colleagues 2018, ”Planck 2018 results. VI. Cosmological parameters”, arXiv e-prints, arXiv:1807.06209.
  * Peebles (2000) Peebles, P. J. E. 2000, ”Fluid Dark Matter”, ApJ, 534, L127-L129.
  * Primack (2009) Primack, J. R. 2009, ”Cosmology: small-scale issues”, New Journal of Physics, 11, 105029.
  * Smirnov and Xu (2019) Smirnov, A. Y. and X.-J. Xu 2019, ”Wolfenstein potentials for neutrinos induced by ultra-light mediators”,  Journal of High Energy Physics, 2019, 46.
  * Shi and Fuller (1999) Shi, X. and G. M. Fuller 1999, ”New Dark Matter Candidate: Nonthermal Sterile Neutrinos”, Phys. Rev. Lett., 82, 2832-2835.
  * Turck-Chieze and Lopes (1993) Turck-Chieze, S. and I. Lopes 1993, ”Toward a Unified Classical Model of the Sun: On the Sensitivity of Neutrinos and Helioseismology to the Microscopic Physics”,  ApJ, 408, 347.
  * Wang et al. (2016) Wang, B., E. Abdalla, F. Atrio-Barandela, and D. Pavón 2016, ”Dark matter and dark energy interactions: theoretical challenges, cosmological implications and observational signatures”,  Reports on Progress in Physics, 79, 096901.
  * Wolfenstein (1978) Wolfenstein, L. 1978, ”Neutrino oscillations in matter”,  Phys. Rev. D, 17, 2369-2374.
  * Yoo et al. (2003) Yoo, J., and 130 colleagues 2003, ”Search for periodic modulations of the solar neutrino flux in Super-Kamiokande-I”,  Phys. Rev. D, 68, 092002.
  * Xing (2020) Xing, Z.-. zhong . 2020, ”Flavor structures of charged fermions and massive neutrinos”,  Phys. Rep., 854, 1-147.


# A study on the effect of input data length on deep learning based magnitude
classifier

Megha Chakraborty  Frankfurt Institute for Advanced Studies, 60438 Frankfurt
am Main, Germany  Institute of Geosciences, Goethe-University Frankfurt, 60438
Frankfurt am Main, Germany  Wei Li  Frankfurt Institute for Advanced Studies,
60438 Frankfurt am Main, Germany  Johannes Faber  Frankfurt Institute for
Advanced Studies, 60438 Frankfurt am Main, Germany  Institute for Theoretical
Physics, Goethe Universität, 60438 Frankfurt am Main, Germany  Georg Rümpker
Frankfurt Institute for Advanced Studies, 60438 Frankfurt am Main, Germany
Institute of Geosciences, Goethe-University Frankfurt, 60438 Frankfurt am
Main, Germany  Horst Stoecker  Frankfurt Institute for Advanced Studies, 60438
Frankfurt am Main, Germany  Institute for Theoretical Physics, Goethe
Universität, 60438 Frankfurt am Main, Germany  Xidian-FIAS international Joint
Research Center, Giersch Science Center, D-60438 Frankfurt am Main, Germany
GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany
Nishtha Srivastava

###### Abstract

The rapid characterisation of earthquake parameters such as its magnitude is
at the heart of Earthquake Early Warning (EEW). In traditional EEW methods the
robustness in the estimation of earthquake parameters have been observed to
increase with the length of input data. Since time is a crucial factor in EEW
applications, in this paper we propose a deep learning based magnitude
classifier and, further we investigate the effect of using five different
durations of seismic waveform data after first P-wave arrival– 1s, 3s, 10s,
20s and 30s. This is accomplished by testing the performance of the proposed
model that combines Convolution and Bidirectional Long-Short Term Memory units
to classify waveforms based on their magnitude into three classes– "noise",
"low-magnitude events" and "high-magnitude events". Herein, any earthquake
signal with magnitude equal to or above 5.0 is labelled as high-magnitude. We
show that the variation in the results produced by changing the length of the
data, is no more than the inherent randomness in the trained models, due to
their initialisation.

##  1 Introduction

The earthquake magnitude, defined as a logarithmic measure of the relative
strength of an earthquake, is one of the most fundamental parameters in its
characterisation[1]. The complex nature of the geophysical processes affecting
earthquakes makes it very difficult to have a single reliable measure for its
size [2] and hence, magnitude values measured in different scales often differ
by more than 1 unit. This is especially true for larger events due to
saturation effects [3, 4]. Owing to above-mentioned reasons and the empirical
nature of majority of the magnitude scales, it is one of the most difficult
parameters to estimate [5, 6].

Some of the classical approaches to obtain first estimates of earthquake
magnitude have used empirical relations for parameters such as predominant
period $\tau_{p}^{max}$ [7, 8], effective average period $\tau_{c}$ [9, 10] in
the frequency domain and parameters such as peak displacement ($P_{d}$) [11,
10] in the amplitude domain calculated from the initial 1-3 seconds of
P-waves. These relations form the basis of existing Earthquake Early Warning
(EEW) systems in Japan, California, Taiwan etc. ([12] and the references
therein). The accuracy of such estimates have been shown to increase with the
duration of data used to calculate them [13].

The recent developments in the area of deep learning [14], combined with the
availability of affordable high-end computational power through GPUs, have led
to state-of-the-art results in image recognition [15, 16], speech recognition
[17, 18] and natural language processing [19, 20]. In fields such as
seismology, where the volume of available data has increased exponentially
over the last decades [21], deep learning has achieved great success in tasks
such as seismic phase picking [22, 23, 24], event detection [25, 26, 27],
magnitude estimation [1], event location characterisation [28, 29, 30], and
first motion polarity detection [31].

Considering that timeliness is of the essence in rapid earthquake
characterisation, it becomes important to find an optimum duration for the
input data, that can provide a reliable and statistically significant estimate
for various earthquake parameters while using minimum amount of P-wave data.
In this study, we present a deep learning model to perform time-series
multiclass classification [32, 33] that classifies seismic waveforms as –
"noise, "low-magnitude" or "high-magnitude". Here a local magnitude of 5.0 is
taken to be the boundary between the low-magnitude and high-magnitude classes.
We further investigate the effect of using different lengths of data on the
model performance. Please note, that the boundary of 5.0 is arbitrarily
chosen, and can be modified depending on the purpose of the model and the
local geology (which influences the correlation between earthquake magnitude
and intensity). The boundary in itself does not influence the model
performance. Unlike [34], which uses data from three seismic station to
characterise different earthquake parameters, the model discussed in this
paper only uses three-component data from a single station.

##  2 Methodology

###  2.1 Data Used

We use data from the STanford EArthquake Dataset (STEAD) [35] to train and
test our model. STEAD is a high-quality bench-marked dataset created for
machine learning and deep learning applications and contains seismic event and
noise waveforms of duration 1 minute recorded by over 2,500 seismic stations
across the globe. The waveforms have been detrended and filtered with a
bandpass filter between 1.0 to 40.0 Hz, followed by a resampling at 100Hz. A
metadata consisting of 35 attributes for earthquake traces and 8 attributes
for noise traces is provided by the authors.

To ensure consistency in magnitude we only use traces for which the magnitude
is provided in ‘ml’ scale (as this is the case for most of the traces in the
dataset). We also discard traces with signal-to-noise ratio less than 10dB for
quality control. We divide the noise and earthquake traces into training,
validation and test sets in the ratio 60:10:30. Care is taken to make sure
that the three aforementioned datasets are non-overlapping. This means, that
traces corresponding to a particular earthquake (represented by the
‘source_id’ attribute) but recorded at different stations are included in only
one of the three sets. For noise traces, recordings from a particular seismic
station are included in only one of the three sets.

In this paper, we propose a classifier model for rapid earthquake
characterisation. Furthermore, we investigate the effect of using different
lengths of data after the first P-arrival (1s, 3s, 10s, 20s and 30s) on the
performance of this classifier model. In each case the P-wave data is preceded
by 2.8-3.0 seconds of pre-signal noise, so the model can learn the noise
characteristics of the station [36]. The data labels 0, 1, and 2 are used to
denote the classes noise, low-magnitude and high-magnitude, respectively.

Figure 1: Original distribution of local magnitudes in the chunk of STEAD [35]
data used for training.

As mentioned earlier, we take a local magnitude 5.0 to be the decision
boundary between high-magnitude and low-magnitude events. However, the
training dataset originally has a magnitude distribution as shown in Figure 1;
this would lead to a high imbalance between the low-magnitude and high-
magnitude classes (a ratio of nearly 3300:1). It is widely agreed by the
Machine Learning community that most classifiers assume an equal distribution
between the different classes.[37] Although examples from some domains where
models perform reasonably well even in highly imbalanced datasets, show that
there are other factors at play, imbalanced datasets not only are a major
hindrance in the development of good classifiers but can also lead to
misleading evaluations of the accuracy of the model [37]. To tackle this
imbalance problem we apply resampling of the data [38] as follows:

  * •

Events with magnitude equal to or above 5.0 are represented 20 times in the
dataset, by using a shifting window starting from 300 samples to 280 samples
before the first P-arrival sample, the window being shifted by 2 samples for
each representation. Each of these traces are also flipped, i.e. their
polarity is reversed, since it does not affect the magnitude information of
the data. Such data augmentation techniques used for images have also been
found to be useful for time series data [37, 39].

  * •

For low-magnitude events the following strategy of random-undersampling is
adopted:

    1. 1.

All events with magnitude between 4.5 and 5.0 are used.

    2. 2.

1/3${}^{\text{rd}}$ of events with magnitude between 4.0 and 4.5 are used.

    3. 3.

1/50${}^{\text{th}}$ of events with magnitude between 2.0 and 4.5 are used.

    4. 4.

1/100${}^{\text{th}}$ of events with magnitude less than 2.0 are used.

  * •

1/25${}^{\text{th}}$ of the available noise traces are used.

Figure 2: The distribution of classes in the training dataset obtained by
under-sampling noise and low-magnitude data and applying data-augmentation to
high-magnitude events. Classes 0,1 and 2 represent ‘noise’, ‘low-magnitude’
and ‘high-magnitude’ data, respectively. A similar distribution of classes is
seen in the validation and test datasets as well. Figure 3: The architecture
of the model used to perform the 3-class classification. The input to the
model is 3-component seismic waveform data from a single station. The example
shown here corresponds to the case where 3 seconds of P-wave data is used (the
total length of data is, thus, 6 seconds). The 1D Convolution layers have a
kernel size of 4 and 8 filters each; the drop rate for each Dropout layer is
0.2, and each MaxPooling layer reduces the size of the data by a factor of 4;
the Bi-LSTM layers have dimensions of 256, 256 and 128, respectively. The
final layer is a Softmax layer, that outputs the probability of the trace
belonging to classes 0 (noise), 1 (low-magnitude) and 2 (high-magnitude),
represented here as $P_{0}$, $P_{1}$ and $P_{2}$ respectively. In this case a
probability of 0.9933 is assigned to class 2, for an event with magnitude 5.3;
thus, this is a case of correct classification.

Note that special care is taken to include more events close to the decision
boundary, so that the model can learn to differentiate between events of
magnitude say, 4.0 to 5.0 which is more difficult compared to differentiating
between events of magnitude say, 2.0 and 5.0. The corresponding distribution
of the different classes is shown in Figure 2. The validation and test
datasets follow a similar distribution. As one can see, in spite of the
resampling techniques employed, the high-magnitude class is still under-
respresented in the dataset, as compared to the other two classes. So we apply
a class-weight [38] of 1:1:10 (chosen, experimentally) for classes 0,1 and 2
while training the model. The data is used without instrument response
removal. Unlike [40] we do not normalise the data. Only the waveform
information is provided to the model.

###  2.2 Model Architecture and Model Training

The model architecture [41] consists of two sets of 1D Convolution [42],
Dropout [43] and MaxPooling[44], followed by three bidirectional Long-Short
Term Memory (LSTM) [45] layers; the final layer is a Softmax layer [46] which
gives a three-element array of the form $[P_{0},P_{1},P_{2}]$, where $P_{i}$
is the probability of the waveform belonging to the class $i$ (Figure 3).

The model is trained using Adam optimiser [47], Categorical Crossentropy [48]
loss and a batch size of 256. Early stopping [49] is used to prevent
overfitting, whereby the validation loss is monitored and the training stops
when there is no reduction in it for 20 consecutive epochs. We start with a
learning rate of $10^{-3}$ and reduce it by a factor of 10 if the validation
loss does not reduce for 15 consecutive epochs until it reaches $10^{-6}$. The
model for the epoch corresponding to the lowest validation loss is retained.

Figure 4: Examples of waveforms that have been correctly classified. In each
case the highest probability corresponds to the respective class.

##  3 Results

To analyse the effect of different lengths of data on the performance of the
classifier model, we use the metrics listed below to evaluate the model
performance. The metrics are calculated in terms of true positives (TP), true
negatives (TN), false positives (FP) and false negatives (FN).

  * •

Accuracy: The accuracy of a classifier is the proportion of testing samples
that are correctly classified. Mathematically, it can be defined as follows:

$\text{Accuracy}=\frac{TP+TN}{TP+FP+TN+FN}$ (1)

  * •

Precision: This is the ratio of the number of times the model correctly
predicts a class to the total number of times it predicts that class.
Mathematically it is defined as:

$\text{Precision}=\frac{TP}{TP+FP}$ (2)

  * •

Recall: This is the ratio of the number of times the model correctly predicts
a class to the total number occurences of that class in the dataset.
Mathematically it is defined as:

$\text{Recall}=\frac{TP}{TP+FN}$ (3)

Figure 5: Softmax probabilities for different input lengths of the same ,
predicted by the models trained on the corresponding lengths of data. The
waveform used here corresponds to an event of magnitude 2.8, although the
maximum probability corresponds to class 1, the values of these probabilities
are different for different data lengths, and there is no clear dependence
between the length of the data and this probability.

(a)

(b)

Figure 6: (a) Variation in classifier model performance when different
duration of P-wave data are used; (b) Variation in the classifier model
performance when the same model is re-trained on the same data (in this case 3
seconds of P-wave data used) five times. This shows that the variation in the
two cases are comparable.

Figure 4 shows three waveforms, (one from each class) that has been correctly
classified. The softmax probalities, as described in section 2.2, are also
shown. In each case the highest probability is predicted for the corresponding
class.

Figure 5 shows the softmax probalities, predicted by the model for different
lengths of the same waveform. Although the waveform is correctly classified in
each case, the predicted probabilities are different and show no dependence on
the length of input data. Figure 6(a) shows the variation in the model
performance with the duration of P-wave data used. We also look at the
randomness in the performance when the model is trained on the same data five
times (Figure 6(b)), as we do not tune a random seed during model training
[50, 51]. Thus, we can see that the variation in the results caused by
changing the length of data is comparable to the randomness in the results due
to random-initialisation upon re-training the model on the same data.

Figure 7 shows the classification statistics for one of the iterations of the
model trained on the 3 second data. One can see that the events classified as
noise tend to be of low magnitude, while the mis-classification of low-
magnitude events as high-magnitude and vice-versa, is most pronounced at the
decision boundary of 5.0. Another important observation is that the degree of
misclassification of low-magnitude events is much higher than the reverse
case; approximately 65% of events with magnitude between 4.5 and 5.0 and 35%
of events with magnitude between 4.0 and 4.5 get classified as high magnitude,
while less than 10% of events with magnitude between 5.0 and 5.5 are
classified as low-magnitude; this is intentional as a missed alarm is
considered more dangerous than a false alarm in this context [52] and is
achieved by giving the high-magnitude class more weight during model training.

(a)

(b)

Figure 7: The classification results for a model trained on the 3 second data.
(a) The confusion matrix [53] for a model trained and tested on the 3 second
data. (b) The mis-classification statistics for the same model, for different
magnitude values. Note how the highest degree of mis-classification happens
close to the decision boundary; the percentage of low-magnitude events
classified as high-magnitude is much higher than the percentage of high-
magnitude events classified as low-magnitude; this is a result of the class-
weights we used while training the model.

##  4 Conclusion

In this study, we present a deep learning model that classifies seismic
waveform into three-classes: noise, low-magnitude events and high-magnitude
events, with events having local magnitude equal to or above 5.0 categorised
as ‘high-magnitude’. We investigate the effect of using different duration of
P-wave data to perform the said task and demonstrate that changing the length
of the waveform has no significant effect on the model performance. We also
find that the model classifies most the data above a magnitude of 4.5 as high-
magnitude, even though the decision boundary is chosen at 5.0, due to the
higher class weight assigned to high-magnitude events. We obtain an overall
accuracy ranging between 90.04% and 93.86% (which is comparable to the
magnitude classification accuracy of 93.67% achieved by [34] using data from
three seismic stations).

##  5 Acknowledgement

This research is supported by the “KINachwuchswissenschaftlerinnen" - grant
SAI 01IS20059 by the Bundesministerium für Bildung und Forschung - BMBF.
Calculations were performed at the Frankfurt Institute for Advanced Studies’
new GPU cluster, funded by BMBF for the project Seismologie und Artifizielle
Intelligenz (SAI).

## References

  * [1] S.. Mousavi and G.. Beroza  “A machine-learning approach for earthquake magnitude estimation.”  In _Geophysical Research Letters_ 47, e2019GL085976., 2020  URL: https://doi.org/10.1029/2019GL085976
  * [2] H. Kanamori and G.. Stewart  “Seismological aspects of the Guatemala Earthquake of February 4, 1976”  In _Journal of Geophysical Research: Solid Earth_ 83.B7, 1978, pp. 3427–3434  DOI: https://doi.org/10.1029/JB083iB07p03427
  * [3] B.F. Howell Jr  “On the saturation of earthquake magnitudes.”  In _Bulletin of the Seismological Society of America_ 71(5), 1981, pp. 1401–1422  URL: https://doi.org/10.1785/BSSA0710051401
  * [4] H. Kanamori  “Magnitude scale and quantification of earthquakes.”  In _Tectonophysics_ 93(3-4), 1983, pp. 185–199  URL: https://doi.org/10.1016/0040-1951(83)90273-1
  * [5] D.. Chung and D.. Bernreuter  “Regional relationships among earthquake magnitude scales”  In _Reviews of Geophysics_ 19.4, 1981, pp. 649–663  DOI: https://doi.org/10.1029/RG019i004p00649
  * [6] G. Ekström and A. Dziewonski  “Evidence of bias in estimations of earthquake size.”  In _Nature_ 332, 1988, pp. 319–323  DOI: https://doi.org/10.1038/332319a0
  * [7] Y. Nakamura  “On the Urgent Earthquake Detection and Alarm System (UrEDAS)”  In _9th world conference on earthquake engineering_ VII.B7, 1988, pp. 673–678
  * [8] R. Allen and H. Kanamori  “The Potential for Earthquake Early Warning in Southern California”  In _Science (New York, N.Y.)_ 300, 2003, pp. 786–789  DOI: 10.1126/science.1080912
  * [9] H. Kanamori  “REAL-TIME SEISMOLOGY AND EARTHQUAKE DAMAGE MITIGATION”  In _Annual Review of Earth and Planetary Sciences_ 33.1, 2005, pp. 195–214  DOI: 10.1146/annurev.earth.33.092203.122626
  * [10] X. Jin et al.  “Earthquake magnitude estimation using the $\tau_{c}$ and $P_{d}$ method for earthquake early warning systems”  In _Earthquake Science_ 26.es-26-1-23, 2013, pp. 23–31  DOI: 10.1007/s11589-013-0005-4
  * [11] Y.M. Wu and L. Zhao  “Magnitude estimation using the first three seconds P-wave amplitude in earthquake early warning”  In _Geophysical Research Letters_ 33.16, 2006  URL: https://doi.org/10.1029/2006GL026871
  * [12] R. Allen, P. Gasparini, O. Kamigaichi and M. Böse  “The Status of Earthquake Early Warning around the World: An Introductory Overview”  In _Seismological Research Letters_ 80, 2009, pp. 682–693  DOI: 10.1785/gssrl.80.5.682
  * [13] A. Ziv  “New frequency-based real-time magnitude proxy for earthquake early warning”  In _Geophysical Research Letters_ 41.16, 2014, pp. 7035–7040  URL: https://doi.org/10.1002/2014GL061564
  * [14] Y. LeCun, Y. Bengio and G. Hinton  “Deep learning.”  In _Nature_ 521, 2015, pp. 436–444  URL: https://doi.org/10.1038/nature14539
  * [15] A. Krizhevsky, I. Sutskever and G.. Hinton  “ImageNet Classification with Deep Convolutional Neural Networks”  In _Communications of the ACM_ 60.6, 2017, pp. 84–90  URL: https://doi.org/10.1145/3065386
  * [16] K. He, S. Ren, J. Sun and X. Zhang  “Deep Residual Learning for Image Recognition”  In _IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2016, pp. 770–778  DOI: 10.1109/CVPR.2016.90
  * [17] T. Mikolov et al.  “Strategies for training large scale neural network language models”  In _2011 IEEE Workshop on Automatic Speech Recognition Understanding_ , 2011, pp. 196–201  DOI: 10.1109/ASRU.2011.6163930
  * [18] G. Hinton et al.  “Deep Neural Networks for Acoustic Modeling in Speech Recognition: The Shared Views of Four Research Groups”  In _IEEE Signal Processing Magazine_ 29.6, 2012, pp. 82–97  DOI: 10.1109/MSP.2012.2205597
  * [19] M.. Peters et al.  “Deep contextualized word representations”  In _Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers)_ Association for Computational Linguistics, 2018, pp. 2227–2237  DOI: 10.18653/v1/N18-1202
  * [20] R. Collobert et al.  “Natural language processing (almost) from scratch”  In _Journal of machine learning research_ 12.ARTICLE, 2011, pp. 2493–2537  DOI: 10.5555/1953048.2078186
  * [21] Q. Kong et al.  “Machine Learning in Seismology: Turning Data into Insights”  In _Seismological Research Letters_ 90.1, 2018, pp. 3–14  URL: https://doi.org/10.1785/0220180259
  * [22] W. Zhu and G.. Beroza  “PhaseNet: a deep-neural-network-based seismic arrival-time picking method”  In _Geophysical Journal International_ 216.1, 2019, pp. 261–273  URL: https://doi.org/10.1093/gji/ggy423
  * [23] W.Y. Liao et al.  “ARRU Phase Picker: Attention Recurrent-Residual U-Net for Picking Seismic P- and S-Phase Arrivals”  In _Seismological Research Letters_ 92.4, 2021, pp. 2410–2428  DOI: 10.1785/0220200382
  * [24] W. Li et al.  “EPick: Multi-Class Attention-based U-shaped Neural Network for Earthquake Detection and Seismic Phase Picking”, 2021  URL: https://arxiv.org/abs/2109.02567
  * [25] J. Wang and T.L. Teng  “Artificial neural network-based seismic detector”  In _Bulletin of the Seismological Society of America_ 85.1, 1995, pp. 308–319  URL: https://doi.org/10.1785/BSSA0850010308
  * [26] S.M. Mousavi et al.  “Earthquake transformer—an attentive deep-learning model for simultaneous earthquake detection and phase picking”  In _Nature Communications_ 11.3952, 2020  URL: https://doi.org/10.1038/s41467-020-17591-w
  * [27] M.A. Meier et al.  “Reliable Real-Time Seismic Signal/Noise Discrimination With Machine Learning”  In _Journal of Geophysical Research: Solid Earth_ 124.1, 2019, pp. 788–800  DOI: https://doi.org/10.1029/2018JB016661
  * [28] T. Perol, M. Gharbi and M. Denolle  “Convolutional neural network for earthquake detection and location”  In _Science Advances_ 4.2, 2018, pp. e1700578  DOI: 10.1126/sciadv.1700578
  * [29] A. Panakkat and H. Adeli  “Recurrent neural network for approximate earthquake time and location prediction using multiple seismicity indicators”  In _Computer-Aided Civil and Infrastructure Engineering_ 24.4 Wiley Online Library, 2009, pp. 280–292  URL: https://doi.org/10.1111/j.1467-8667.2009.00595.x
  * [30] H.. Kuyuk and O. Susumu  “Real-time classification of earthquake using deep learning”  In _Proc. Comput. Sci_ 140, 2018, pp. 298–305  URL: https://doi.org/10.1016/j.procs.2018.10.316
  * [31] Z.. Ross, M.A. Meier and E. Hauksson  “P wave arrival picking and first-motion polarity determination with deep learning”  In _Journal of Geophysical Research: Solid Earth_ 123.6 Wiley Online Library, 2018, pp. 5120–5129  URL: https://doi.org/10.1029/2017JB015251
  * [32] H. Ismail Fawaz et al.  “Deep learning for time series classification: a review.”  In _Data Mining and Knowledge Discovery_ 33, 2019, pp. 917–963  DOI: https://doi.org/10.1007/s10618-019-00619-1
  * [33] M. Aly  “Survey on multiclass classification methods”  In _Neural Netw_ 19 Citeseer, 2005, pp. 1–9
  * [34] O.. Saad, A.. Hafez and M.. Soliman  “Deep Learning Approach for Earthquake Parameters Classification in Earthquake Early Warning System.”  In _IEEE Geoscience and Remote Sensing Letters_ , 2020, pp. 1–5  DOI: 10.1109/LGRS.2020.2998580
  * [35] S.. Mousavi, Y. Sheng, W. Zhu and G.. Beroza  “STanford EArthquake Dataset (STEAD): A Global Data Set of Seismic Signals for AI.”  In _IEEE Access_ 7, 2019, pp. 179464–179476  URL: https://doi.org/10.1109/ACCESS.2019.2947848
  * [36] J. Münchmeyer, D. Bindi, U. Leser and F. Tilmann  “The transformer earthquake alerting model: a new versatile approach to earthquake early warning”  In _Geophysical Journal International_ 225.1, 2020, pp. 646–656  DOI: 10.1093/gji/ggaa609
  * [37] G….. Batista, R.. Prati and M.. Monard  “A Study of the Behavior of Several Methods for Balancing Machine Learning Training Data”  In _SIGKDD Explorations Newsletter_ 6.1, 2004, pp. 20–29  DOI: 10.1145/1007730.1007735
  * [38] B. Krawczyk  “Learning from imbalanced data: open challenges and future directions”  In _Progress in Artificial Intelligence_ 5.4 Springer, 2016, pp. 221–232  URL: https://doi.org/10.1007/s13748-016-0094-0
  * [39] Q. Wen et al.  “Time Series Data Augmentation for Deep Learning: A Survey”  In _Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence_ International Joint Conferences on Artificial Intelligence Organization, 2021  DOI: 10.24963/ijcai.2021/631
  * [40] A. Lomax, A. Michelini and D. Jozinović  “An Investigation of Rapid Earthquake Characterization Using Single-Station Waveforms and a Convolutional Neural Network”  In _Seismological Research Letters_ 90, 2019, pp. 517–529  DOI: 10.1785/0220180311
  * [41] M. Chakraborty et al.  “Real Time Magnitude Classification of Earthquake Waveforms using Deep Learning”  In _EGU General Assembly 2021, online_ EGU21-15941, 2021  URL: https://doi.org/10.5194/egusphere-egu21-15941
  * [42] S. Kiranyaz et al.  “1D convolutional neural networks and applications: A survey”  In _Mechanical Systems and Signal Processing_ 151, 2021, pp. 107398  URL: https://doi.org/10.1016/j.ymssp.2020.107398
  * [43] N. Srivastava et al.  “Dropout: A Simple Way to Prevent Neural Networks from Overfitting”  In _Journal of Machine Learning Research_ 15.56, 2014, pp. 1929–1958  URL: http://jmlr.org/papers/v15/srivastava14a.html
  * [44] J. Nagi et al.  “Max-pooling convolutional neural networks for vision-based hand gesture recognition”  In _2011 IEEE International Conference on Signal and Image Processing Applications (ICSIPA)_ , 2011, pp. 342–347  DOI: 10.1109/ICSIPA.2011.6144164
  * [45] S. Hochreiter and J. Schmidhuber  “Long Short-Term Memory”  In _Neural Computation_ 9.8 Cambridge, MA, USA: MIT Press, 1997, pp. 1735–1780  DOI: 10.1162/neco.1997.9.8.1735
  * [46] I. Goodfellow, Y. Bengio and A. Courville  “Deep learning”  MIT press, 2016  URL: http://www.deeplearningbook.org
  * [47] D.. Kingma and J. Ba  “Adam: A Method for Stochastic Optimization”  In _International Conference on Learning Representations_ , 2014
  * [48] K.P. Murphy  “Machine learning: a probabilistic perspective”  MIT press, 2012
  * [49] L Prechelt  “Early Stopping — But When?”  In _Neural Networks: Tricks of the Trade: Second Edition_ Springer Berlin Heidelberg, 2012, pp. 53–67  DOI: 10.1007/978-3-642-35289-8_5
  * [50] Y. Bengio  “Practical Recommendations for Gradient-Based Training of Deep Architectures”  In _Neural Networks: Tricks of the Trade: Second Edition_ Berlin, Heidelberg: Springer Berlin Heidelberg, 2012, pp. 437–478  DOI: 10.1007/978-3-642-35289-8_26
  * [51] P. Madhyastha and R. Jain  “On Model Stability as a Function of Random Seed”, 2019  arXiv:1909.10447 [cs.LG]
  * [52] R.. Allen and D. Melgar  “Earthquake Early Warning: Advances, Scientific Challenges, and Societal Needs”  In _Annual Review of Earth and Planetary Sciences_ 47.1, 2019, pp. 361–388  DOI: 10.1146/annurev-earth-053018-060457
  * [53] K.. Ting  “Confusion Matrix”  In _Encyclopedia of Machine Learning and Data Mining_ Boston, MA: Springer US, 2017, pp. 260–260  DOI: 10.1007/978-1-4899-7687-1_50