# Neutrino propagation in the galactic dark matter halo

P. F. de Salas Instituto de Física Corpuscular – CSIC/U. Valencia,
Parc Científic, calle Catedrático José Beltrán 2, E-46980 Paterna, Spain
R. A. Lineros Instituto de Física Corpuscular – CSIC/U. Valencia,
Parc Científic, calle Catedrático José Beltrán 2, E-46980 Paterna, Spain
M. Tórtola Instituto de Física Corpuscular – CSIC/U. Valencia,
Parc Científic, calle Catedrático José Beltrán 2, E-46980 Paterna, Spain

(August 27, 2024)

###### Abstract

Neutrino oscillations are a widely observed and well established phenomenon.
It is also well known that deviations with respect to flavor conversion
probabilities in vacuum arise due to neutrino interactions with matter. In
this work, we analyze the impact of new interactions between neutrinos and the
dark matter present in the Milky Way on the neutrino oscillation pattern. The
dark matter-neutrino interaction is modeled by using an effective coupling
proportional to the Fermi constant $G_{F}$ with no further restrictions on its
flavor structure. For the galactic dark matter profile we consider an
homogeneous distribution as well as several density profiles, estimating in
all cases the size of the interaction required to get an observable effect at
different neutrino energies. Our discussion is mainly focused in the PeV
neutrino energy range, to be explored in observatories like IceCube and
KM3NeT. The obtained results may be interpreted in terms of a light
$\mathcal{O}$(sub-eV–keV) or WIMP-like dark matter particle or as a new
interaction with a mediator of $\mathcal{O}$(sub-eV–keV) mass.

neutrino oscillation, dark matter

###### pacs:

14.60.Pq, 95.35.+d

††preprint: IFIC/16-01

##  I Introduction

Neutrinos have been observed over a quite wide energy range, from eV to
hundreds of TeV, beyond the most energetic part of the atmospheric neutrino
flux. The detection of even higher energetic neutrinos is now possible thanks
to experiments like IceCube Aartsen  _et al._ (2013, 2014) and the future
KM3NeT Coniglione (2015), in the southern and northern hemisphere
respectively. It is also well known that neutrinos are massive and mixed, and
as a consequence they experience flavor oscillations Olive  _et al._ (2014).
Moreover, it was noticed that these oscillations get modified in presence of
matter due to the Mikheyev-Smirnov-Wolfenstein (MSW) effect Wolfenstein
(1978); Mikheev and Smirnov (1985). On the other hand, a great amount of
evidences indicate that most of the matter in our Universe is in the form of
Dark Matter (DM), being our galaxy embedded in a DM halo Peter (2012); Iocco
_et al._ (2015). From the point of view of low-energy neutrinos (below
$1\,\mathrm{TeV}$), the effect of ordinary matter in neutrino oscillations
through the MSW effect has been well studied. However, if neutrinos interact
with DM, their oscillations may also be affected, specially in the less
explored case of ultra-high-energy neutrinos. In this paper we are going to
explore this possibility, studying the possible effect of DM on neutrino
oscillations. We will focus on very-high-energy neutrinos, like the ones
detected by IceCube.

Similar hypothesis have been discussed in previous works. Most of them,
however, consider neutrinos as the main component of the galactic DM Horvat
(1999); Lunardini and Smirnov (2000) or the effect due to the interaction on
the Cosmic Neutrino Background Diaz and Klinkhamer (2016). The more recent
analysis in Ref. Miranda  _et al._ (2015) follows a closer approach, although
it does not account for the effect of the DM halo density profile of our
galaxy. In this work we show that the effect of the DM profile may be very
important, specially for neutrinos generated near the Galactic Center (GC).
This occurs because DM density variations can lead to a resonant behavior in
neutrino oscillations, as it happens in the Sun Wolfenstein (1978); Mikheev
and Smirnov (1985). A further motivation for our analysis are the recent
results from the IceCube Collaboration The IceCube Collaboration  _et al._
(2015) on the flavor composition of very high-energy neutrinos. IceCube data
indicate that the preferred flavor composition of the neutrino flux is far
from the expected region obtained assuming standard astrophysical neutrino
sources Bustamante  _et al._ (2015); Mena  _et al._ (2014); Palomares-Ruiz
_et al._ (2015), although the statistical significance of the discrepancy is
small. Indeed, if we just consider neutrinos coming from neutron decay, the
theoretical expectations and the observed flavor composition disagree at the
$3.7\sigma$ level. However, if the observed astrophysical neutrinos have been
produced from pion decay, the tension with the experimental data is reduced to
$1\sigma$. In any case, and provided that a better estimation of the flavor
composition will be done in the near future, it is important to understand the
origin of the discrepancy. A modification in the oscillatory neutrino pattern
induced by DM, for instance, can help explaining the disagreement between the
predicted and observed flavor composition of the neutrino flux. Furthermore,
this hypothesis may imply new consequences in future high-energy neutrino
observations. In particular, and given the anisotropic character of the
phenomenon due to the non-central position of the solar system inside the
Milky Way, the presence of galactic DM may predict different neutrino flavor
composition for KM3NeT and IceCube, sensitive to different regions of the sky.

The origin of very-high-energy neutrinos is not completely understood.
However, there are some estimations suggesting that, for energies larger than
60 TeV, 40% of the total neutrino flux has a galactic origin while the rest is
extragalactic Palladino and Vissani (2016). Even though galactic neutrinos are
less abundant at these energies, the present study is focused in such
component to analyze the effect of the DM halo of the Milky Way on the
neutrino propagation. The case of extragalactic neutrinos will be studied in a
future work F. de Salas  _et al._ (2016).

The paper is organized as follows. In Section II, we discuss the presence of
DM in our galaxy. In Section III, we review the main aspects of neutrino
oscillations, discussing how they are affected by the presence of dark matter.
This effect is analyzed in a model independent way, parameterizing the
interaction of neutrinos and dark matter in terms of an effective potential.
We show the correlations between the flavor composition of the neutrino flux
at Earth and the neutrino production point. The implications of our results
are discussed in detail in Section IV. Finally, our conclusions are summarized
in Section V.

##  II Dark matter distribution and candidates

Dark matter together with dark energy are the two largest components of the
energy budget of the Universe, controlling many aspects of its evolution. Dark
energy drives the accelerated expansion of the Universe while dark matter is
responsible for the structure formation at different scales including the
galaxy formation. The latest Cosmic Microwave Background (CMB) observations
done by the Planck Collaboration Planck Collaboration  _et al._ (2015) set the
present DM abundance in the $\Lambda$CDM model to

$\Omega_{\rm DM}h^{2}=0.1198\pm 0.0015\,,$ (1)

where $h=H_{0}/(100\,\rm{km}\,\rm{s}^{-1}\rm{Mpc}^{-1})=0.678\pm 0.009$ is the
scale factor for the Hubble expansion rate Planck Collaboration  _et al._
(2015).

From observations of galaxy rotation curves Rubin  _et al._ (1976a, b) and
N-body numerical simulations Moore  _et al._ (1998); Ghigna  _et al._ (1998),
one may conclude that the dark matter distribution in galaxies follows a
_universal_ profile. For the scope of this work, we describe the DM
distribution in the Milky Way by using two extreme choices: the isothermal
profile Bahcall and Soneira (1980) and Navarro-Frenk-White (NFW) profile
Navarro  _et al._ (1996, 1997). Both parameterizations are generically
described by the functional form

$\rho_{\rm
DM}(r,r_{s},\alpha,\beta,\gamma)=\rho_{\oplus}\left(\frac{r_{\oplus}}{r}\right)^{\gamma}\,\left(\frac{1+(r_{\oplus}/r_{s})^{\alpha}}{1+(r/r_{s})^{\alpha}}\right)^{(\beta-\gamma)/\alpha}\,,$
(2)

where spherical symmetry is assumed and the origin, $r=0$, corresponds to the
galactic center. We consider the solar system is located at
$r_{\oplus}=8.5\leavevmode\nobreak\ {\rm kpc}$ from the galactic center and
the local DM energy density is $\rho_{\oplus}=0.4\leavevmode\nobreak\ {\rm
GeV}/{\rm cm}^{3}$, as indicated by several studies Catena and Ullio (2010);
Pato  _et al._ (2015); Xia  _et al._ (2015); McKee  _et al._ (2015). The
isothermal DM profile is then described by the choice

$\rho_{\rm iso}(r)=\rho_{\rm DM}(r,5\leavevmode\nobreak\ {\rm kpc},2,2,0)\,,$
(3)

while the NFW profile is given by

$\rho_{\rm NFW}(r)=\rho_{\rm DM}(r,20\leavevmode\nobreak\ {\rm kpc},1,3,1)\,,$
(4)

where the main difference between both distributions is the presence of a cusp
at $r=0$ in the NFW profile.

However, more than the issue of the DM distribution inside our galaxy, the key
point to deal with DM is its true nature, what constitutes one of the biggest
puzzles in physics nowadays. The most popular candidates to DM are generically
known as Weakly Interactive Massive Particles (WIMPs) Peter (2012); Steigman
and Turner (1985); Drees and Gerbier (2012) and they appear in many models
beyond the Standard Model (SM). The WIMP relic abundance arises from the epoch
when WIMPs and SM particles were in thermal equilibrium. Due to the expansion
of the Universe, WIMPs cannot remain in thermal equilibrium, _freezing out_
the WIMP abundance and forming the current DM abundance. In the context of the
thermal freeze-out mechanism, the observed relic abundance is then obtained
when the thermally averaged WIMP cross section is $\langle\sigma
v\rangle\simeq 3\times 10^{-26}\,{\rm cm}^{3}/{\rm s}$ at the freeze-out
temperature $T_{\rm f.o.}\simeq m_{\rm WIMP}/20$. This mechanism leads to a
WIMP DM mass in the range of GeV–TeV and sets the order of magnitude of the DM
interaction with the SM particles. This class of candidates opens the
possibility to look for DM annihilation at celestial objects (indirect
detection) as well as DM recoils on super sensitive detectors located at
underground laboratories (direct detection).

Asymmetric DM Petraki and Volkas (2013); Zurek (2014) is another popular
candidate. In this case, the DM abundance is generated via (model-dependent)
mechanisms similar to baryogenesis Petraki and Volkas (2013) leading to a DM
mass of the order of GeV. The main implication arising from this scenario is a
DM content of the Universe made of DM particles without any DM antiparticle.
In consequence, there are no DM annihilations to be detected through indirect
detection signals. Direct searches are still possible due to the interaction
of DM with normal matter.

Possible candidates to DM particles include also sterile neutrinos, majorons
and axions. Sterile neutrinos and majorons may be connected with the
generation of neutrino masses through different mechanisms Lattanzi  _et al._
(2014) while axions are related to the strong CP problem Duffy and van Bibber
(2009). In general, all these candidates have their mass in the sub-MeV range
although one can find candidates with even lighter mass Hu  _et al._ (2000).
Note that the zoology of DM models is quite broad Zhang and Zhao (2015);
Suárez  _et al._ (2014) and the ones discussed above provide a brief sample of
the most popular candidates.

As summarized above, the masses of most of the proposed DM candidates can
range from eV to TeV. For the sake of simplicity, along this work we will
assume a generic value for the mass of the DM particle as well as a generic
value for its coupling to neutrinos. In the discussion of our results in the
next section we will consider three different benchmark cases with particular
values for these parameters.

##  III Neutrino oscillations in dark matter

Neutrinos coming from the Milky Way, from sources outside the solar system,
may be affected by the DM halo of the galaxy. In particular, their flavor
oscillations might be modified due to the presence of DM in analogy with the
MSW effect in ordinary matter Wolfenstein (1978); Mikheev and Smirnov (1985).
Since the nature of DM is still unknown, one can assume the most general case
where its interaction with neutrinos can violate flavor universality. This can
give rise to a sizable effect when compared to neutrino oscillations in
vacuum.

It is well established that neutrino oscillations happen because mass and weak
eigenstates do not coincide:

$|\nu_{\alpha}\rangle=\sum_{k}U^{*}_{\alpha k}|\nu_{k}\rangle\,,$ (5)

where $\alpha$ is a flavor index (e, $\mu$, $\tau$), $k$ refers to neutrino
mass eigenstates with mass $m_{k}$, and $U$ is the unitary matrix which
diagonalizes the neutrino mass matrix.

When neutrinos travel across a medium like the Earth or the Sun, their
interaction with the medium modifies the vacuum oscillation pattern. This
effect introduces a new term in the total hamiltonian describing the evolution
of neutrinos in the medium

$\mathcal{H}_{\rm tot}=\mathcal{H}_{\rm vac}+\mathcal{V}\,,$ (6)

where $\mathcal{H}_{\rm vac}$ is the hamiltonian in vacuum in the flavor basis

$\mathcal{H}_{\rm vac}=\frac{1}{2E}\,U\left(\begin{array}[]{ccc}0&0&0\\\
0&\Delta m^{2}_{21}&0\\\ 0&0&\Delta
m^{2}_{31}\end{array}\right)U^{\dagger}\,.$ (7)

Here $\Delta m^{2}_{ij}=m^{2}_{i}-m^{2}_{j}$ are the mass squared differences
between the neutrino mass eigenstates $\nu_{i}$ and $\nu_{j}$ resulting after
the substraction of the global phase $m_{1}^{2}$. $U$ is the neutrino mixing
matrix Olive  _et al._ (2014) and $\mathcal{V}$ is the effective potential
that accounts for the neutrino interactions with matter. For the neutrino
mixing angles and mass squared differences we use the best fit values in Ref.
Forero  _et al._ (2014) for normal and inverse hierarchy. The CP violation
phase $\delta$ has been set to zero.

The effective potential describing the interaction between neutrinos and DM in
the flavor basis can be parametrized as

$\mathcal{V}_{\alpha\beta}=\lambda_{\alpha\beta}\,G_{F}\,N_{\chi}\,,$ (8)

where $\lambda_{\alpha\beta}$ is a hermitian matrix encoding the effective
couplings between neutrinos and DM, $G_{F}$ is the Fermi constant, and
$N_{\chi}$ is the DM number density. The number density is related to the
energy density by

$N_{\chi}=\frac{\rho_{\rm DM}}{m_{\rm DM}}\,.$ (9)

The parameterization used in Eq. (8) is well motivated in scenarios with
fermion asymmetric DM, where DM (and no anti-DM) is present in the Universe,
as well as in the case of scalar/vector DM candidates. Possible realizations
are presented in Fig. 1. In the simplest scenario, we require a mediator
particle that connects the DM and neutrino sectors without mixing them. This
is shown in diagram a) for a fermion asymmetric DM.

Figure 1:  Examples of processes leading to the parameterization in Eq. (8).
Diagram a) corresponds to the case of asymmetric DM while diagrams in b)
represent the case of scalar DM.

Since neutrino oscillations are blind to global phases, we can reduce the
number of free parameters in the effective potential by subtracting a term
proportional to the identity matrix. In our case we subtract
$\mathcal{V}_{\tau\tau}\,\mathbb{I}$. We also consider only real entries in
the $\lambda$-matrix. Then, the effective potential is reduced to

$V=\mathcal{V}-\mathcal{V}_{\tau\tau}\mathbb{I}=G_{F}N_{\chi}\left(\begin{array}[]{ccc}\lambda_{11}&\lambda_{12}&\lambda_{13}\\\
\lambda_{12}&\lambda_{22}&\lambda_{23}\\\
\lambda_{13}&\lambda_{23}&0\end{array}\right)\,,$ (10)

with five free parameters describing the effective interactions between DM and
neutrinos. Let us remark that a relative sign may appear in $\mathcal{V}$
modelling the effect for neutrinos and antineutrinos, although this sign would
depend on the nature of DM candidate. In our study, we consider all possible
signs in the entries of the potential to cover the DM effect including both
species without distinction. This is motivated by the fact that neutrino
telescopes are unable to distinguish between neutrinos and antineutrinos. Even
though there are small differences in the $\nu/\bar{\nu}$ cross sections with
the target material Formaggio and Zeller (2012); Adrian-Martinez  _et al._
(2016), the implications of such differences at the detection level are beyond
the scope of this work.

The evolution equation of neutrinos passing through a medium is given by

$i\frac{\partial\Psi}{\partial t}=\mathcal{H}_{\rm tot}\Psi,$ (11)

where $\mathcal{H}_{\rm tot}$ is the hamiltonian in the flavor basis as given
in Eq. (6) and $\Psi$ the neutrino field. The hamiltonian in the mass basis
will be obtained after rotating $\mathcal{H}_{\rm tot}$ with the neutrino
mixing matrix $U$,

$\mathcal{H}_{\mathrm{tot}}^{m}=U^{\dagger}\mathcal{H}_{\mathrm{tot}}U=\frac{1}{2E}\left(\begin{array}[]{ccc}0&0&0\\\
0&\Delta m_{21,\mathrm{eff}}^{2}&0\\\ 0&0&\Delta
m_{31,\mathrm{eff}}^{2}\end{array}\right)\,,$ (12)

where $\Delta m_{21,\mathrm{eff}}^{2}$, and $\Delta m_{31,\mathrm{eff}}^{2}$
are the effective mass squared splittings in the presence of an effective
potential.

In the case of a constant effective potential and starting with an initial
flavor content $f^{0}=\left(f^{0}_{e},f^{0}_{\mu},f^{0}_{\tau}\right)$, the
final averaged flavor state will be given by

$f_{\beta}=\sum_{\alpha=e,\mu,\tau}\left(\sum_{i=1}^{3}\left|U_{\beta
i}U^{*}_{\alpha i}\right|^{2}f^{0}_{\alpha}\right)\,,$ (13)

which is valid for distances much larger than the characteristic oscillation
wavelength.

Figure 2:  $R_{e}$ map in the plane $V_{11}$ – $V_{22}$ starting with a flavor
content (1:0:0) for $E_{\nu}=1\,{\rm MeV}$ (top left panel), $E_{\nu}=1\,{\rm
TeV}$ (top right panel) and $E_{\nu}=1\,{\rm PeV}$ (bottom panel).

In a more realistic scenario, the effective potential will depend on the
neutrino position. In our case, the spatial dependence arises from the DM
distribution that modifies $N_{\chi}$. The solution to the flavor evolution
equation requires the diagonalization of the hamiltonian in matter at every
instant, depending on the value of $N_{\chi}$ at the neutrino position. We
describe that position in terms of the line of sight distance, $l$, with
respect to the solar system and the angle, $\phi$, with respect to the
galactic center. The galactocentric radius $r$ is then simply described by

$r^{2}=r_{\oplus}^{2}+l^{2}-2\,l\,r_{\oplus}\,\cos{\phi}\,.$ (14)

In this scheme, the evolution of the flavor states is given by

$\displaystyle f_{\beta}$ $\displaystyle(l_{n+1},\phi)=$
$\displaystyle\sum_{\alpha=e,\mu,\tau}\left(\sum_{i=1}^{3}\left|U_{\beta
i}(l_{n},\phi)U^{*}_{\alpha
i}(l_{n},\phi)\right|^{2}f_{\alpha}(l_{n},\phi)\right),$ (15)

where the initial state corresponds to $f_{\alpha}(l_{0},\phi)=f^{0}_{\alpha}$
with $l_{0}$ being the distance to the source. $U(l_{n},\phi)$ is the matrix
that diagonalizes the hamiltonian $\mathcal{H}_{\rm tot}$ evaluated at the
neutrino position $(l_{n},\phi)$, and $f_{\beta}(l=0,\phi)=f^{\oplus}_{\beta}$
is the final state at Earth. The distances involved in these scenarios are of
the order of 1 kpc, while the largest oscillation wavelength is much smaller
for neutrino energies of TeV–PeV. Therefore, we can safely assume that the
neutrino flavor evolution may be described by an averaged flavor oscillation.

On the other hand, if neutrino propagation is adiabatic, the numerical
integration of the evolution described above can be further simplified. In
that case, the neutrino flavor evolution would only depend on the initial and
final DM densities. As we have checked in Appendix A, for the DM densities
analyzed the adiabaticity is satisfied in the neutrino propagation so we can
safely calculate the final neutrino flavor composition under this assumption.
We have numerically checked that this is the case.

In what follows, we will consider two different approaches to evaluate the
effect of neutrino interactions with DM on the observed signal at experiments.
As a first approximation, we will consider the case of a homogeneous DM
distribution. Next, we will proceed including the more realistic case of a
varying DM density profile assuming either a NFW or an isothermal DM profile.

###  III.1 Case I: homogeneous DM halo

Figure 3:  $R_{\beta}$ ($\beta$ = e, $\mu$, $\tau$) maps in the plane $V_{11}$
– $V_{22}$ for $E_{\nu}=1\,{\rm PeV}$ and an initial flavor content of
(1:2:0).

In this section we will search for values of the effective potential $V$
leading to measurable effects on the neutrino oscillation pattern. As a first
approximation to the problem, we will focus on the effects of a homogeneous DM
halo. This is equivalent to use a constant value of the effective potential
and therefore the final flavor state is obtained from Eq. (13). The deviations
between the DM-modified final state, $f_{\beta}^{\rm DM}$, and the expected
final state in vacuum, $f_{\beta}^{\rm vac}$, at Earth are then described by

$R_{\beta}(V,E)=\frac{f_{\beta}^{\rm DM}-f_{\beta}^{\rm vac}}{f_{\beta}^{\rm
vac}}\,,$ (16)

where the effective potential $V$ is given at Eq. (10), $E$ is the neutrino
energy and $\beta=(e,\mu,\tau)$ is the neutrino flavor.

We calculate the deviations in the neutrino flavor content for different
configurations of the effective potential (i.e. varying different entries of
the potential $V_{ij}$) and we find that their effect qualitatively shows
similar results. Therefore, for simplicity, we consider that only the entries
$V_{11}$ and $V_{22}$ are different from zero. We then found regions where
$V_{ij}$ gives rise to strong modifications in the behavior of neutrino
oscillations, with abrupt changes in the sign of $\partial R/\partial V_{ij}$.
These regions are a clear hint for a resonant behavior at the neutrino
oscillations. The transition can be observed in Fig. 2, where the sudden
changes in color are related to the presence of a resonance.

Due to the nature of neutrino oscillations in a medium, an increase in
$E_{\nu}$ would be equivalent to an increase in $V_{ij}$. Therefore, the
effects of DM on neutrino oscillations will become more relevant at higher
neutrino energies. In addition, the ratio $R_{\beta}$ in Eq. (16) reaches a
saturation value when $E_{\nu}$ grows and $V_{ij}$ is fixed. In Fig. 2, we
present the values of $R_{e}$ in the plane $V_{11}$ – $V_{22}$ starting from
the flavor configuration $f^{0}$=(1:0:0) at the source. Comparing the three
panels in the figure, we observe a linear change in the size of the regions
due to the linear dependence on $E_{\nu}$. The resonant behavior of $R_{e}$
around $V_{11}=10^{-10}\,{\rm eV}$ for $E=1\,{\rm MeV}$ is shifted to
$V_{11}=10^{-19}\,{\rm eV}$ for $E=1\,{\rm PeV}$. This resonant pattern and
its shift in energies also manifest in $R_{\mu}$ and $R_{\tau}$. In
consequence, the DM-neutrino interaction could explain a non-standard flavor
composition in the high-energy neutrino flux observed at Earth, as the case of
recent IceCube data The IceCube Collaboration  _et al._ (2015) without
compromising lower energy observations.

We also study the case of an astrophysical source of neutrinos with initial
flavor content equal to (1:2:0). In Fig. 3, we present the $R_{e,\mu,\tau}$
maps in the plane $V_{11}-V_{22}$ for $E_{\nu}=1\,\mathrm{PeV}$, an energy
value relevant for IceCube. We have explored the effective potential $V_{ii}$
in the range from $10^{-23}$ to $10^{-16}\,{\rm eV}$. In our scan, the lower
limit of $V_{ii}$ does not produce appreciable deviations from the vacuum
solution (i.e $R_{\beta}\simeq 0$), while the upper limit saturates the flavor
oscillations beyond the resonance and then $R_{\beta}$ does not change for
larger values of $V_{ii}$.

At this level, we can compare the patterns shown in Figs. 2 and 3. To
understand the difference we need to consider the definition of $R_{\beta}$ in
Eq. (16). The averaged oscillations in vacuum for an initial flavor content of
(1:2:0) lead to a final flavor composition

$f^{\rm vac}=(0.331:0.347:0.322)\,.$ (17)

In the region where both $\left|V_{11}\right|$ and $\left|V_{22}\right|$ are
larger than $10^{-18}\,{\rm eV}$, the initial flavor content remains
unchanged. This is because the effective potential dominates the neutrino
hamiltonian and then flavor oscillations are suppressed. For an initial flavor
composition of (1:2:0), we obtain

$f^{\rm DM}=(0.333:0.667:0)\,$ (18)

in the presence of DM, which is equivalent to

$(R_{e}:R_{\mu}:R_{\tau})\approx(6\times 10^{-3}:0.92:-1)\,.$ (19)

This explains the values of the deviations $R_{\beta}$ at the corners of the
three panels in Fig. 3.

In the region where $\left|V_{11}\right|\geq 10^{-18}\,{\rm eV}$ and
$\left|V_{22}\right|\leq 10^{-18}\,{\rm eV}$, only electron neutrino
oscillations are suppressed, leaving $f^{\rm DM}_{e}\simeq f^{0}_{e}$ (i.e.
$R_{e}\simeq 0$). However, oscillations for muon and tau neutrinos are still
active. Since the atmospheric mixing angle $\theta_{23}$ in vacuum is almost
maximal, the initial content of muon neutrinos, $f_{\mu}^{0}=0.667$, is
equally distributed between muon and tau neutrinos. This leads to $f^{\rm
DM}\approx(0.33:0.33:0.33)$, what explains the slightly negative value of
$R_{\mu}$ and the slightly positive one of $R_{\tau}$ and $R_{e}$ obtained
after the comparison with the vacuum expectations in Eq. (17).

The final saturation region, where $V_{11}$ is small and $V_{22}$ is large,
prevents muon neutrinos to oscillate. The vacuum oscillations between electron
and tau neutrinos are mainly controlled by the reactor mixing angle
$\theta_{13}\simeq 9^{0}$ . This implies that just a small part of the initial
electron neutrino content, $f_{e}^{0}=0.333$, will be transferred to tau
neutrinos. Consequently $R_{e}$ remains small, but negative, while $R_{\tau}$
will have a negative value close to $-1$.

Figure 4: Flavor triangle for $E_{\nu}=1$ PeV for the initial neutrino
compositions (1:0:0), (0:1:0), (0:0:1) and (1:2:0). Color regions include the
possible flavor neutrino compositions at Earth predicted in the presence of
DM, whereas the points correspond to the expected flavor composition in
vacuum. Left panel results from the scan using only pure initial neutrino
states and diagonal terms in the potential while the right panel has been
obtained scanning over all entries, including the non-diagonal ones.

A more general analysis, where all the entries of the effective potential
$V_{ij}$ are free to vary, is summarized in Fig. 4. The regions in color cover
all the possible flavor neutrino compositions at Earth predicted in the
presence of DM. In the same figure, the three points indicate the predictions
assuming neutrino oscillations in vacuum. We present our results for four
different initial states: (1:0:0), (0:1:0), (1:2:0) and (0:0:1). The first
three cases are motivated by astrophysical processes as neutron decay, damped
muon source and pion decay, respectively. The last case, (0:0:1), is shown for
comparison. A compendium of possible astrophysical neutrino sources are
described in Ref. Bustamante  _et al._ (2015). In the left panel, we only vary
the diagonal terms of the effective potential keeping the non-diagonal equal
to zero, while in the right panel, all terms are allowed to vary, showing the
maximum deviation area. In both cases, $\left|V_{ij}\right|$ take values in
the range $10^{-23}$ – $10^{-13}\,{\rm eV}$. From the figure we see that there
is no difference between the regions obtained for (0:1:0) and (0:0:1) in the
two panels. On the contrary, for the initial flavor content (1:0:0), the
diagonal-only scan can not cover all the region obtained in the right panel.
Our results agree with the analysis presented in Ref. Arguelles  _et al._
(2015) which considers possible New Physics scenarios.

In this part, we have analyzed the order of magnitude of the effective
potential which may give a sizable deviation from the neutrino oscillation
pattern in vacuum for an homogeneous DM profile. However, one may expect that
the effective potential will take different values depending on the neutrino
position with respect to the DM halo. This more realistic scenario will be
analyzed in the next subsection.

###  III.2 Case II: DM halo profile

xxBenchmarkxx |  xxCase Axx |  xxCase Bxx |  xxCase Cxx
---|---|---|---
$V_{11}^{\oplus}$ [$10^{-21}$ eV] | 4 | 20 | 40
Table 1:  Benchmark cases A, B and C, defined by the effective potential value
at Earth.

In this subsection we will consider realistic profiles for the galactic DM
density distribution. This will imply a spacial dependence in the effective
matter potential along the neutrino path that may produce similar effects to
the ones explaining the solar neutrino problem. In section III.1, we estimated
the values of $V_{ij}$ for which the neutrino flavor composition at Earth is
different from the case of oscillations in vacuum. Here we combine that
information with the spatial distribution of DM in the Milky Way assuming
homogeneously produced neutrinos up to $r_{\rm max}=20$ kpc on the galactic
plane. We consider three benchmark cases: A, B and C (see Table 1 for
details). The values of each benchmark are chosen to give a very small value
of $R_{\beta}$ in the homogeneous DM case. However, thanks to the DM
distribution profile, the effective potential for neutrinos is larger in
regions closer to the galactic center compared to the outskirts of the galaxy.
In Fig. 5 we show how the effective potential changes with the DM
distribution. We consider the isothermal and NFW profiles described in Eqs.
(3) and (4) to analyze the impact of the DM distribution on neutrino
oscillations. In addition, the spatial variation of the potential would
produce resonances in the neutrino flavor conversion enhancing the deviations
with respect to the oscillation pattern in vacuum.

Figure 5:  Effective potential $V_{11}$ as a function of the distance from the
Earth, $l$, for a path crossing the galactic center. Different curves
correspond to the benchmark cases A, B and C in Tab. 1 for a NFW DM profile.
The benchmark case C with an isothermal profile is also shown. Figure 6:
$R_{\beta}$ maps for different neutrino production position. $l=0$ corresponds
to the Earth’s position. In all plots, the initial flavor state is (1:0:0),
$E_{\nu}=1\,{\rm PeV}$ and a NFW profile. Columns from left to right
correspond to $R_{e}$, $R_{\mu}$ and $R_{\tau}$ respectively, and rows from
top to bottom are benchmark cases A, B and C.

In Fig. 6 we show the color map of the deviation parameter $R_{\beta}$ for an
initial state (1:0:0) in terms of the neutrino production point in the
galactic plane for the NFW profile. We consider neutrinos with energies of 1
PeV and the benchmark cases in Table 1. For simplicity, only
$V_{11}^{\oplus}$, the 11-entry of the effective potential resulting from the
dark matter-neutrino interaction at Earth $V_{ij}(r_{\oplus})$, has been
chosen different from zero. Each column of plots, from left to right,
represents $R_{e}$, $R_{\mu}$ and $R_{\tau}$, respectively, while each row
corresponds to a different benchmark point. We observe that the impact of the
spatial dependence is very important and the effect is different for each
$R_{\beta}$. The use of different benchmarks has also an impact on the
$R_{\beta}$ maps. As expected, one can notice that the benchmark C presents
the largest deviations over every $R_{\beta}$ if compared with benchmarks A
and B.

In Fig. 7 we present the $R_{\beta}$ map using the benchmark C and the
isothermal profile. The direct comparison of Figs. 6 and 7 shows that the
flavor conversion is enhanced towards the galactic center. One can notice
that, thanks to the inner cusp, the NFW case is rather different from the
isothermal one only in the $\sim 1$ kpc region surrounding the GC. This
promising feature might indicate that a DM–MSW effect can reveal relevant
information about the innermost GC. We also observe that the obtained values
of $R_{\beta}$ are symmetrical with respect to the GC.

We show in Fig. 8 the predicted flavor compositions at Earth for neutrinos
originally produced as (1:2:0) and (1:0:0). As initial neutrino position we
have considered all points inside a 20 kpc radius from the GC. In the left
panel, we compare the overall region for a constant potential (reddish area)
with the region resulting from the NFW profile (blueish region). Also, we
include the uncertainty bands as well as the best fit point obtained from the
IceCube results on neutrino flavor composition The IceCube Collaboration  _et
al._ (2015). The values of all entries of the potential $|V_{ij}^{\oplus}|$
range from $10^{-24}$ eV to $10^{-16}$ eV. The main difference between the
blueish and reddish areas arises because the DM distributions are different
near the GC and in the outskirts of the galaxy. The strongest impact of the DM
interaction on neutrino oscillations on these production zones broadens the
blueish area beyond the expected region for a constant potential.

Figure 7:  $R_{\beta}$ maps for different neutrino production position. Same
initial state as in Fig. 6 but DM distribution corresponds to an isothermal
profile. Panels from left to right correspond to $R_{e}$, $R_{\mu}$,
$R_{\tau}$ respectively and all plots are for the benchmark case C.

The right panel of Fig. 8 shows the possible flavor compositions produced from
an initial flavor (1:0:0) and a NFW DM profile with different values of
$V_{ij}^{\oplus}$. Each colored area arises as the result of a scan with a
maximum value of $|V_{ij}^{\oplus}|$. We observe that values of
$V_{ij}^{\oplus}$ as larger as $10^{-17}$ eV can almost cover the full flavor
triangle. However, as soon as the maximum value of $V_{ij}^{\oplus}$ is
reduced, the obtained areas tend to converge to the vacuum solution. Maximum
values of $V_{ij}^{\oplus}$ smaller than $10^{-21}$ eV might not be enough to
explain the 3$\sigma$ region of IceCube. We observe that, for the same range
of $V_{ij}^{\oplus}$, the area produced by a homogeneous dark matter profile
(Fig. 4) is much more restricted than the one obtained from a NFW profile.

The best fit point and allowed bands obtained from the latest IceCube data are
qualitatively compatible with the constant potential and the DM profile
explanations111A more detailed analysis including full energy range and cuts
will be required in order to check the agreement in a quantitative way..
Nevertheless, we note that for the constant DM case the values of the
potential $V_{ij}^{\oplus}$ required to explain the observed flavor
composition in IceCube are larger than the ones in the DM profile case. This
is due to the effect of the variable DM profile density over neutrino
oscillations. The change in the effective potential felt by neutrinos will
depend on their arrival direction, e.g. neutrinos coming from regions near the
GC would feel larger modifications in their effective potential. This
indicates that flavor neutrino composition might be angularly correlated and
then, it may depend on the part of the sky accessible to IceCube.

##  IV Discussion

In the previous sections we have analyzed the effect of DM on neutrino
oscillations using an effective potential parametrized in Eq.(10). In a more
generic framework, considering New Physics beyond the Standard Model, the
interaction between both particles can also be expressed as

$V_{ij}=\lambda^{\prime}_{ij}\,G^{\prime}_{F}\,\frac{\rho_{\rm DM}}{m_{\rm
DM}}\,.$ (20)

This equation allows the reinterpretation of the effective potential in terms
of a new interaction strength $G^{\prime}_{F}$ as well as on the DM mass,
$m_{\rm DM}$. We highlight that Eqs. (10) and (20) are parameterizations of
the potential where the structure of the $\lambda_{ij}(\lambda_{ij}^{\prime})$
parameters will depend on the choice of a given particle physics model. For
simplicity, we will assume that the interaction between neutrinos and DM
particles happens via the interchange of a $Z^{\prime}$-like boson. In this
case, the _primed_ Fermi constant is related to the standard $G_{F}$ by

$G^{\prime}_{F}=\frac{m^{2}_{Z}}{m^{2}_{Z^{\prime}}}\,G_{F}\,,$ (21)

with $m_{Z}\simeq 91$ GeV. Here, the mass of the mediator and the interaction
strength are tightly related. This scaling is valid only for the coherent
scattering regime if there is no momentum transferred by the mediator, or if
the mediator is so heavy that it can be integrated out.

It is important to mention that, besides the effect of forward coherent
scattering encoded in the DM potential $V$, neutrinos might actually scatter
on the DM halo. This could disrupt the effect of neutrino oscillations,
modifying also their arrival directions and energy spectrum Davis and Silk
(2015). To this end, we have to ensure that the mean free path due to the
neutrino-DM scattering cross section at Earth,

$l_{\nu}=\left(\sigma_{\nu\chi}\frac{\rho_{\rm DM}}{m_{\rm
DM}}\right)^{-1}=\left(\frac{\sigma_{\nu\chi}}{8.1\times 10^{-22}{\rm
cm}^{2}}\right)^{-1}\left(\frac{m_{\rm DM}}{\rm GeV}\right){\rm kpc},$ (22)

is large enough, allowing neutrinos to cross the galaxy and being affected
only at the level of oscillations.

Figure 8:  Flavor triangles for the initial states (1:2:0) (left) and (1:0:0)
(right). Left panel shows the flavor area covered by a homogeneous DM profile
(reddish area) and the one covered by the effect of a NFW profile (blueish
area). Right panel shows the areas covered by imposing a maximum value for
$V_{ij}^{\oplus}$ and a NFW profile. We observe that for smaller maximum
values the area is closer to the solution in vacuum. The best fit point and
68% and 95% C.L. allowed regions from IceCube data are also shown.

For the case of galactic neutrinos, we set $l_{\nu}$ at Earth to be 50 kpc
which corresponds to a cross section $\sigma_{\nu\chi}=1.62\times
10^{-23}\,\left(m_{\rm DM}/{\rm GeV}\right){\rm cm}^{2}$. This value
guarantees that, apart from the effect of coherent forward scattering,
neutrinos would rarely scatter along any trajectory, including the ones
passing through the GC for a NFW profile. A complementary study for ultra-
light DM, in the regime where neutrino-DM scattering cross section has a
relevant role for the neutrino propagation is analyzed in Ref. Reynoso and
Sampayo (2016).

In a more extreme case, we also consider the bound
$\sigma_{\nu\chi}<10^{-33}\,\left(m_{\rm DM}/{\rm GeV}\right){\rm cm}^{2}$,
which comes from the CMB analysis when DM-neutrino interactions are allowed
Wilkinson  _et al._ (2014). This bound corresponds to $l_{\nu}>10^{6}$ Gpc at
Earth. Let us remark that this bound applies for DM-neutrino cross sections at
the MeV scale and therefore its value might be different for the neutrino
energies considered in this work, depending on the particle physics model
considered. In what follows, we will use these two values of the neutrino mean
free path as benchmarks to discuss the dependence of the effective potential
on the remaining parameters, $\lambda^{\prime}$, $G_{F}^{\prime}$ and
$m_{Z^{\prime}}$. We will also consider a particle physics scenario with
unconstrained mean free path in order to better understand the role of the
parameters.

In Table 2, we present six different choices for the involved parameters that
can reproduce the three selected values for the effective potential at Earth.
These values have been chosen in the range from $10^{-21}$ to $10^{-17}$ eV, a
representative range of the potential that might produce sizable effects on
the neutrino oscillation flavor content as it is shown in Fig. 8. All over the
table we have considered that the neutrino-DM cross section takes the form

$\sigma\propto{\lambda^{\prime}}^{2}G_{F}^{\prime}\simeq{\lambda^{\prime}}^{2}\,\left(\frac{m_{Z^{\prime}}}{\rm
GeV}\right)^{-2}\,3.75\times 10^{-29}\,{\rm cm}^{2}\,,$ (23)

well motivated for the case $\sqrt{s}\gg m_{Z^{\prime}},m_{\rm DM}$. At this
point, the functional relation between $V^{\oplus}$, $l_{\nu}$ and the rest of
the parameters ($m_{\rm DM}$, $\lambda^{\prime}$ and $G_{F}^{\prime}$) allows
us to describe the effect on neutrino oscillations by fixing only three of
them.

The scenarios described in the table are the following:

  * •

Weak scale.- We assume for all cases that $G_{F}^{\prime}=G_{F}$. If we
further impose $\lambda^{\prime}=1$ (case a), the DM mass has to lie in the
mass range from $10^{-12}$ to $10^{-8}$ eV. This corresponds to an extremely
light DM particle which is in the spirit of Axion Bose-Einstein Condensates
(BEC) Sikivie and Yang (2009), although the mass we obtain is (at best) two
orders of magnitude smaller than the BEC case. One can find models with such
extremely light candidates like in scalar (wave) dark matter models Matos and
Urena-Lopez (2001). As the result of the chosen $\lambda^{\prime}$ and
$G_{F}^{\prime}$, the neutrino mean free path is in the sub–pc range. This
small value of $l_{\nu}$ might imply that high energy neutrinos are screened
due to their interaction with the DM halo Davis and Silk (2015). If we now
impose $l_{\nu}=50$ kpc (case b), as discussed above, the values of
$\lambda^{\prime}$ are reduced to the range between $10^{-7}$ and $10^{-11}$,
and the DM mass becomes even lighter, laying on the range from $10^{-15}$ to
$10^{-23}$ eV.

  * •

100 GeV DM.- Here we fix $l_{\nu}$ at Earth to the two values previously
discussed. For $l_{\nu}=50$ kpc (case a), we obtain the same values of
$\lambda^{\prime}$ as before. In this case, $m_{Z^{\prime}}$ takes sub-eV
values, $10^{-2}$ to $10^{-6}$ eV, which may be in the line of models with
light mediators in the neutrino sector Farzan and Hannestad (2015). On the
other hand, for $l_{\nu}=10^{6}$ Gpc (case b), the values of
$\lambda^{\prime}$ are in the range from $10^{-17}$ to $10^{-21}$,
significantly smaller than for larger values of $l_{\nu}$. The same happens
for $m_{Z^{\prime}}$ that now takes values within the range from $10^{-7}$ to
$10^{-11}$ eV, well below the predictions of case (a), In general terms,
larger values of the mean free path imply lower values of $\lambda^{\prime}$
and $m_{Z^{\prime}}$.

  * •

1 keV DM.- Here, the lower value of the DM mass results in an increase value
of the number density of DM particles, $N_{\chi}$. Depending on the value of
$V^{\oplus}$, this could lead to a mediator mass of the order of few eV, in
agreement with models including light mediators. The values of
$\lambda^{\prime}$ remain unchanged with respect to the 100 GeV DM case.

MM$V_{11}^{\oplus}$ [eV] MM |  M $10^{-21}$ M |  M $10^{-19}$ M |  M $10^{-17}$ M
---|---|---|---
Weak scale (a) assumptions: |  $G_{F}^{\prime}=G_{F}$, $\lambda_{11}=1$
$m_{\mathrm{DM}}$ [eV] | $10^{-8}$ | $10^{-10}$ | $10^{-12}$
$l_{\nu}$ [pc] | $10^{-2}$ | $10^{-4}$ | $10^{-6}$
Weak scale (b) assumptions: |  $G_{F}^{\prime}=G_{F}$, $l_{\nu}=50$ kpc
$\lambda_{11}$ | $10^{-7}$ | $10^{-9}$ | $10^{-11}$
$m_{\mathrm{DM}}$ [eV] | $10^{-15}$ | $10^{-19}$ | $10^{-23}$
100 GeV DM (a) assumptions: |  $m_{\mathrm{DM}}=100$ GeV, $l_{\nu}=50$ kpc
$\lambda_{11}$ | $10^{-7}$ | $10^{-9}$ | $10^{-11}$
$m_{Z^{\prime}}$ [eV] | $10^{-2}$ | $10^{-4}$ | $10^{-6}$
100 GeV DM (b) assumptions: |  $m_{\mathrm{DM}}=100$ GeV, $l_{\nu}=10^{6}$ Gpc
$\lambda_{11}$ | $10^{-17}$ | $10^{-19}$ | $10^{-21}$
$m_{Z^{\prime}}$ [eV] | $10^{-7}$ | $10^{-9}$ | $10^{-11}$
1 keV DM (a) assumptions: |  $m_{\mathrm{DM}}=1$ keV, $l_{\nu}=50$ kpc
$\lambda_{11}$ | $10^{-7}$ | $10^{-9}$ | $10^{-11}$
$m_{Z^{\prime}}$ [eV] | $10^{2}$ | $1$ | $10^{-2}$
1 keV DM (b) assumptions: |  $m_{\mathrm{DM}}=1$ keV, $l_{\nu}=10^{6}$ Gpc
$\lambda_{11}$ | $10^{-17}$ | $10^{-19}$ | $10^{-21}$
$m_{Z^{\prime}}$ [eV] | $10^{-3}$ | $10^{-5}$ | $10^{-7}$
Table 2:  Particle physics interpretations for three given values of the
effective potential at Earth, $V_{11}^{\oplus}$. Six different scenarios are
displayed, with the corresponding assumptions shown in the upper row of each
block. For each case, second, third and fourth columns show the values of the
remaining parameters for the corresponding values of $V_{11}^{\oplus}$.
Implications of each case are described in the text.

From the model-building point of view, this type of interaction presents
tantalizing insights between DM and neutrinos. For instance, models connecting
neutrino masses with dark matter candidates might produce flavor ratios at
Earth outside the expectations in vacuum. These particle model scenarios might
imply interesting consequences and motivate further analyses.

All over our calculations, we have parameterized the DM-neutrino interaction
in such a way that DM interacts with both neutrinos and antineutrinos. Let us
remark that the interaction with neutrinos and antineutrinos is intrinsically
related to the nature of DM and the particle mediator. For instance, in models
of asymmetric DM, we might expect that DM affects only neutrinos or
antineutrinos. On the other hand, this could also happen in symmetric DM
scenarios (WIMP-like), where the nature of the mediator might be responsible
for the non-symmetric effect, e.g. coupling DM to antineutrinos and anti-DM to
neutrinos. Let us highlight that this asymmetry is already present in the
standard MSW effect, since most of the media are mainly composed by matter and
no antimatter. In a more general scenario, the predicted neutrino flavor
composition at Earth would be different for neutrinos and antineutrinos.

Another aspect to be considered is the relation between the neutrino arrival
direction and the observed flavor composition. The DM distribution molds the
observed neutrino flavor composition depending on its production point within
the Milky Way, as seen in Subsection III.2. This implies that the location of
the neutrino observatory plays an important role on the observation of the
neutrino flavor content. For instance, KM3NeT and IceCube will have access to
different parts of the sky. One will be able to observe neutrinos coming from
directions towards the galactic center while the other one may observe
neutrinos coming from the Milky Way outskirts. In consequence, the averaged
flavor composition observed at each experiment may be different. Therefore, it
will be important to include the arrival direction information in future
analyses in order to disentangle a possible correlation between neutrino
flavor and DM distribution.

Although it has not been discussed in this work, the effect of DM on neutrino
oscillations might also be relevant for extragalactic neutrinos. In
consequence, PeV neutrinos coming from regions above or below the galactic
plane can also present different flavor compositions with respect to the
expectations in vacuum. In this case, the effect would depend on the
properties of the DM halo where the neutrino source is located and on how it
compares with the one of the Milky Way. Besides that, a larger $l_{\nu}$ might
be required, depending on the size and density of the departing DM halo. The
distance to the source would also be relevant in this case, due to the loss of
coherence in the neutrino propagation.

##  V Conclusions

The observation of PeV neutrinos opens a new window to explore astrophysical
processes and neutrino sources. Independently of the source location,
neutrinos must travel across the galactic DM halo. Depending on the distance
to the source, we can observe deviations with respect to neutrino oscillations
in vacuum due to the interaction between DM and neutrinos. In this work, we
have assumed the most generic interaction between dark matter and neutrinos,
as described in Eq. (10). We have studied the effects of a uniform effective
potential by assuming a homogeneous DM halo. We have obtained that the flavor
composition at Earth is rather different with respect to the expected
composition in vacuum for values of the effective potential larger than
$10^{-20}$ eV. In the analysis, we have used different initial flavor
compositions inspired by astrophysical processes. We have scanned on a range
of effective potential values $V_{ij}$ between $10^{-23}$ and $10^{-16}\,{\rm
eV}$, finding different covered areas in the flavor triangle. We have checked
that these areas can not be enlarged by considering larger values of the
interaction potential. In particular, we have found that the full triangle is
not accessible for the case of homogeneous DM and a fixed type of neutrino
source. Along this work, we have also checked that our results hold for both,
normal and inverted hierarchy.

In a more realistic scenario, we have included a radial density distribution
for the DM halo of the Milky Way by using NFW and isothermal DM profiles. As a
consequence, the effective potential acquires a spatial dependence and
therefore the neutrino propagation equation has to be solved numerically taken
into account the DM distribution profile222In the case of adiabatic neutrino
propagation, however, this calculation gets simplified and only the DM density
at the initial and final points of the neutrino path are relevant.. In
addition, we have considered that neutrinos are produced homogeneously on the
whole galactic plane up to 20 kpc from the GC. The covered flavor composition
areas depend on the considered DM distribution and we found that the more
realistic case with a DM profile produces a larger area in the flavor triangle
than the homogeneous case. This implies a richer scenario than other New
Physics cases (see Ref. Bustamante  _et al._ (2015) and references within),
e.g. violation of Lorentz symmetry Arguelles  _et al._ (2015).

This effect can also be used to probe particle physics models beyond the SM
for DM-neutrino interactions. We provide different interpretations in terms of
three simple scenarios: weak scale, implying an extremely light DM with mass
$\mathcal{O}(10^{-23}-10^{-15})\,{\rm eV}$; 100 GeV and 1 keV DM cases. The
latter two cases can be explained in terms of new very light mediators with
masses $\mathcal{O}($sub-eV $-$ keV$)$. Each of these scenarios might provide
very interesting insights in relation to models of neutrino masses Boehm  _et
al._ (2008).

A more detailed analysis considering the neutrino arrival direction might
further reveal the presence of the effect considered in this work. For
instance, depending on the available sky for observation, the averaged
neutrino flavor content might be different. This could be the case of IceCube
and KM3NeT. For the timescale of the IceCube observatory, the most optimistic
expectations indicate that 10 years of data taking would be enough to improve
the estimation of the neutrino flavor composition at Earth Bustamante  _et
al._ (2015); Shoemaker and Murase (2016). This would happen within the
timescale of future extensions of IceCube, as the proposed IceCube-Gen2
detector The IceCube-Gen2 Collaboration  _et al._ (2015). In this scenario,
our hypothesis could be tested by imposing constraints on the maximum value of
the effective potential due to the DM-neutrino interaction. During this
period, KM3NeT will be in operation as well Adrian-Martinez  _et al._ (2016)
and it hopefully will present its own results for the neutrino flavor
composition at Earth. The comparison of the results of both observatories will
allow us to test the hypothesis presented in this work in further detail.

###### Acknowledgements.

We acknowledge O. G. Miranda, S. Pastor, B. Panes, and N. Rojas for useful
discussions and comments. This work was supported by the Spanish MINECO under
grants FPA2014-58183-P, and MULTIDARK CSD2009-00064 (Consolider-Ingenio 2010
Programme); by Generalitat Valenciana grant PROMETEOII/2014/084, and Centro de
Excelencia Severo Ochoa SEV-2014-0398. P. FdS. is supported by the FPU
Scholarship FPU13/03729 (MINECO). R. L. is supported by a Juan de la Cierva
contract JCI-2012-12901 (MINECO). M. T. is supported by a Ramón y Cajal
contract (MINECO).

##  Appendix A Adiabaticity condition for neutrino propagation

The equation of motion for the neutrino weak eigenstates in matter is given by
Wolfenstein (1978)

Figure 9:  Adiabaticity parameter $\eta$ for a 1 PeV neutrino crossing the GC
as a function of the distance from the GC, $r$, given in units of the distance
to the Earth, $r_{\oplus}$. A NFW DM profile is assumed. The variation of
$\eta$ is shown for three different effective potentials at Earth,
$V_{11}^{\oplus}$, assuming normal (solid line) and inverted (dashed line)
neutrino mass hierarchy.

$i\,\dot{\nu}_{W}=\mathcal{H}_{\rm tot}\nu_{W},$ (24)

where $\nu_{W}$ is a vector containing the different neutrino flavour states,
and $\mathcal{H}_{\rm tot}$ is the total hamiltonian. The relation between
weak and mass eigenstates is given by

$\nu_{W}=U\nu_{m}\,,$ (25)

where $U$ is a unitary matrix that diagonalizes the total hamiltonian,

$\mathcal{D}=U^{\dagger}\mathcal{H}_{\mathrm{tot}}U\,,$ (26)

and $\nu_{m}$ is the vector containing the effective neutrino mass
eigenstates. If neutrinos travel through a medium with varying effective
potential, the mixing matrix $U$ will vary along the neutrino path and,
therefore, the equation of motion for the neutrino mass eigenstates will be

$i\,\dot{\nu}_{m}=\left(\mathcal{D}-iU^{\dagger}\dot{U}\right)\nu_{m}\,,$ (27)

where the term $U^{\dagger}\dot{U}$ vanishes for a constant effective
potential. However, whenever neutrinos travel through a varying density
medium, this term is different from zero and $\nu_{m}$ no longer describes the
mass eigenstates. Nevertheless, for slow changing medium density (with respect
to oscillation wavelength), $\nu_{m}$ can be approximated by true mass
eigenstates. In this case the process is said to be _adiabatic_. On the
contrary, in the non-adiabatic regime, where the medium density changes fast
enough, there is a probability of level crossing between neutrino mass states
known as Landau-Zener probability (See Refs. Landau (1932); Zener (1932)).

To check that adiabaticity is fulfilled along the neutrino path, we define the
adiabaticity parameter as

$\eta=\frac{\langle\left|U^{\dagger}\dot{U}\right|\rangle}{\langle\left|\mathcal{D}-\mathrm{Tr}\left(\mathcal{D}\right)/3\,\mathbb{I}\right|\rangle}\,,$
(28)

where $\langle\cdots\rangle$ stands for the average over the matrix indices.
Notice that the diagonal elements of $U^{\dagger}\dot{U}$ are zero and the
denominator is traceless. Since $U^{\dagger}\dot{U}$ has only non-diagonal
elements different from zero and $\mathcal{D}$ is a diagonal matrix, the non-
diagonal elements of ($\mathcal{D}-iU^{\dagger}\dot{U}$) in Eq. (27) come only
from the second term. Hence, a process is adiabatic if $\eta\ll 1$. This
condition ensures no mixing between the effective mass eigenstates induced by
the variation of the medium and, therefore, $\nu_{m}$ stay as the true energy
eigenstates of the hamiltonian along the neutrino path. In consequence, one
can evaluate the final neutrino flavor composition by considering only the
effective DM potential at the beginning and end of the neutrino trajectory.

In Fig. 9, we show the adiabaticity parameter for a 1 PeV neutrino crossing
the GC for the DM-neutrino interaction cases presented in Tab. 2 and a NFW
profile. The case with $V_{11}^{\oplus}=10^{-17}\;{\rm eV}$ corresponds to an
effective potential that saturates neutrino oscillations. Therefore, the value
of $\eta$ is very small along the neutrino path. The other values of the
potential shown in the figure, $V_{11}^{\oplus}=10^{-19}\;{\rm eV}$ and
$V_{11}^{\oplus}=10^{-21}\;{\rm eV}$, do not saturate oscillations at Earth
but they do it instead at the very center of the galaxy, due to the larger DM
density. This explains the peaks in $\eta$ followed by a decrease in the
function when $r$ approximates to zero. The peaks correspond to regions with
maximum change in the potential previous to reach the saturation values, i.e.
the regions where the effective oscillation parameters are more sensitive to
variations in the DM potential. For a further discussion about adiabaticity
see e.g. Section 8.3 in Kim and Pevsner (1993), where the adiabaticity
condition is studied under the two flavor neutrino oscillation approximation.

## References

  * Aartsen  _et al._ (2013) M. G. Aartsen _et al._ (IceCube), Science 342, 1242856 (2013), arXiv:1311.5238 [astro-ph.HE] .
  * Aartsen  _et al._ (2014) M. G. Aartsen _et al._ (IceCube), Phys. Rev. Lett. 113, 101101 (2014), arXiv:1405.5303 [astro-ph.HE] .
  * Coniglione (2015) R. Coniglione (KM3NeT),  _Proceedings, 24th European Cosmic Ray Symposium (ECRS 2014)_ , J. Phys. Conf. Ser. 632, 012002 (2015).
  * Olive  _et al._ (2014) K. A. Olive _et al._ (Particle Data Group), Chin. Phys. C38, 090001 (2014).
  * Wolfenstein (1978) L. Wolfenstein, Phys. Rev. D17, 2369 (1978).
  * Mikheev and Smirnov (1985) S. P. Mikheev and A. Yu. Smirnov, Sov. J. Nucl. Phys. 42, 913 (1985), [Yad. Fiz.42,1441(1985)].
  * Peter (2012) A. H. G. Peter, ArXiv e-prints (2012), arXiv:1201.3942 [astro-ph.CO] .
  * Iocco  _et al._ (2015) F. Iocco, M. Pato, and G. Bertone, Nature Physics 11, 245 (2015), arXiv:1502.03821 .
  * Horvat (1999) R. Horvat, Phys. Rev. D59, 123003 (1999), arXiv:hep-ph/9812228 [hep-ph] .
  * Lunardini and Smirnov (2000) C. Lunardini and A. Yu. Smirnov, Nucl. Phys. B583, 260 (2000), arXiv:hep-ph/0002152 [hep-ph] .
  * Diaz and Klinkhamer (2016) J. S. Diaz and F. R. Klinkhamer, Phys. Rev. D93, 053004 (2016), arXiv:1512.00817 [hep-ph] .
  * Miranda  _et al._ (2015) O. Miranda, C. Moura, and A. Parada, Phys.Lett. B744, 55 (2015), arXiv:1308.1408 [hep-ph] .
  * The IceCube Collaboration  _et al._ (2015) The IceCube Collaboration, M. G. Aartsen, K. Abraham, M. Ackermann, J. Adams, J. A. Aguilar, M. Ahlers, M. Ahrens, D. Altmann, T. Anderson, and et al., ArXiv e-prints (2015), arXiv:1510.05223 [astro-ph.HE] .
  * Bustamante  _et al._ (2015) M. Bustamante, J. F. Beacom, and W. Winter, Physical Review Letters 115, 161302 (2015), arXiv:1506.02645 [astro-ph.HE] .
  * Mena  _et al._ (2014) O. Mena, S. Palomares-Ruiz, and A. C. Vincent, Phys. Rev. Lett. 113, 091103 (2014), arXiv:1404.0017 [astro-ph.HE] .
  * Palomares-Ruiz  _et al._ (2015) S. Palomares-Ruiz, A. C. Vincent, and O. Mena, Phys. Rev. D91, 103008 (2015), arXiv:1502.02649 [astro-ph.HE] .
  * Palladino and Vissani (2016) A. Palladino and F. Vissani, (2016), arXiv:1601.06678 [astro-ph.HE] .
  * F. de Salas  _et al._ (2016) P. F. de Salas _et al._ , (2016).
  * Planck Collaboration  _et al._ (2015) Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett, and et al., ArXiv e-prints (2015), arXiv:1502.01589 .
  * Rubin  _et al._ (1976a) V. C. Rubin, M. S. Roberts, J. A. Graham, W. K. Ford, Jr., and N. Thonnard, Astron. J. 81, 687 (1976a).
  * Rubin  _et al._ (1976b) V. C. Rubin, N. Thonnard, W. K. Ford, Jr., and M. S. Roberts, Astron. J. 81, 719 (1976b).
  * Moore  _et al._ (1998) B. Moore, F. Governato, T. Quinn, J. Stadel, and G. Lake, Astrophys. J. Lett. 499, L5 (1998), astro-ph/9709051 .
  * Ghigna  _et al._ (1998) S. Ghigna, B. Moore, F. Governato, G. Lake, T. Quinn, and J. Stadel, Mon. Not. R. Astron. Soc. 300, 146 (1998), astro-ph/9801192 .
  * Bahcall and Soneira (1980) J. N. Bahcall and R. M. Soneira, Astrophys. J. Suppl. Ser. 44, 73 (1980).
  * Navarro  _et al._ (1996) J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J. 462, 563 (1996), astro-ph/9508025 .
  * Navarro  _et al._ (1997) J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J. 490, 493 (1997), astro-ph/9611107 .
  * Catena and Ullio (2010) R. Catena and P. Ullio, J. Cosmology Astropart. Phys. 8, 004 (2010), arXiv:0907.0018 .
  * Pato  _et al._ (2015) M. Pato, F. Iocco, and G. Bertone, ArXiv e-prints (2015), arXiv:1504.06324 .
  * Xia  _et al._ (2015) Q. Xia, C. Liu, S. Mao, Y. Song, L. Zhang, R. J. Long, Y. Zhang, Y. Hou, Y. Wang, and Y. Wu, ArXiv e-prints (2015), arXiv:1510.06810 .
  * McKee  _et al._ (2015) C. F. McKee, A. Parravano, and D. J. Hollenbach, Astrophys. J. 814, 13 (2015), arXiv:1509.05334 .
  * Steigman and Turner (1985) G. Steigman and M. S. Turner, Nucl. Phys. B253, 375 (1985).
  * Drees and Gerbier (2012) M. Drees and G. Gerbier, ArXiv e-prints (2012), arXiv:1204.2373 [hep-ph] .
  * Petraki and Volkas (2013) K. Petraki and R. R. Volkas, International Journal of Modern Physics A 28, 1330028 (2013), arXiv:1305.4939 [hep-ph] .
  * Zurek (2014) K. M. Zurek, Phys. Rept. 537, 91 (2014), arXiv:1308.0338 [hep-ph] .
  * Lattanzi  _et al._ (2014) M. Lattanzi, R. A. Lineros, and M. Taoso, New Journal of Physics 16, 125012 (2014), arXiv:1406.0004 [hep-ph] .
  * Duffy and van Bibber (2009) L. D. Duffy and K. van Bibber, New Journal of Physics 11, 105008 (2009), arXiv:0904.3346 [hep-ph] .
  * Hu  _et al._ (2000) W. Hu, R. Barkana, and A. Gruzinov, Phys. Rev. Lett. 85, 1158 (2000), arXiv:astro-ph/0003365 [astro-ph] .
  * Zhang and Zhao (2015) Y. Zhang and Y. Zhao, Sci. Bull. 60, 986 (2015).
  * Suárez  _et al._ (2014) A. Suárez, V. H. Robles, and T. Matos,  _Proceedings of 4th International Meeting on Gravitation and Cosmology (MGC 4)_ , Astrophys. Space Sci. Proc. 38, 107 (2014), arXiv:1302.0903 [astro-ph.CO] .
  * Forero  _et al._ (2014) D. V. Forero, M. Tórtola, and J. W. F. Valle, Phys. Rev. D 90, 093006 (2014), arXiv:1405.7540 [hep-ph] .
  * Formaggio and Zeller (2012) J. A. Formaggio and G. P. Zeller, Rev. Mod. Phys. 84, 1307 (2012), arXiv:1305.7513 [hep-ex] .
  * Adrian-Martinez  _et al._ (2016) S. Adrian-Martinez _et al._ (KM3Net), (2016), arXiv:1601.07459 [astro-ph.IM] .
  * Arguelles  _et al._ (2015) C. A. Arguelles, T. Katori, and J. Salvado, Phys. Rev. Lett. 115, 161303 (2015), arXiv:1506.02043 [hep-ph] .
  * Davis and Silk (2015) J. H. Davis and J. Silk, (2015), arXiv:1505.01843 [hep-ph] .
  * Reynoso and Sampayo (2016) M. M. Reynoso and O. A. Sampayo, Astropart. Phys. 82, 10 (2016), arXiv:1605.09671 [hep-ph] .
  * Wilkinson  _et al._ (2014) R. J. Wilkinson, C. Boehm, and J. Lesgourgues, JCAP 1405, 011 (2014), arXiv:1401.7597 [astro-ph.CO] .
  * Sikivie and Yang (2009) P. Sikivie and Q. Yang, Physical Review Letters 103, 111301 (2009), arXiv:0901.1106 [hep-ph] .
  * Matos and Urena-Lopez (2001) T. Matos and L. A. Urena-Lopez, Phys. Rev. D63, 063506 (2001), arXiv:astro-ph/0006024 [astro-ph] .
  * Farzan and Hannestad (2015) Y. Farzan and S. Hannestad, ArXiv e-prints (2015), arXiv:1510.02201 [hep-ph] .
  * Boehm  _et al._ (2008) C. Boehm, Y. Farzan, T. Hambye, S. Palomares-Ruiz, and S. Pascoli, Phys. Rev. D 77, 043516 (2008), hep-ph/0612228 .
  * Shoemaker and Murase (2016) I. M. Shoemaker and K. Murase, Phys. Rev. D 93, 085004 (2016), arXiv:1512.07228 [astro-ph.HE] .
  * The IceCube-Gen2 Collaboration  _et al._ (2015) The IceCube-Gen2 Collaboration, :, M. G. Aartsen, K. Abraham, M. Ackermann, J. Adams, J. A. Aguilar, M. Ahlers, M. Ahrens, D. Altmann, and et al., ArXiv e-prints (2015), arXiv:1510.05228 [astro-ph.IM] .
  * Landau (1932) L. Landau, Phys. Z. Sowjetunion (1932).
  * Zener (1932) C. Zener, Proc. R. Soc. London (1932), 10.1098/rspa.1932.0165.
  * Kim and Pevsner (1993) C. Kim and A. Pevsner,  _Neutrinos in physics and astrophysics_, Contemporary concepts in physics (Harwood Academic, 1993).