#  Effects of Neutrino-Ultralight Dark Matter Interaction on
the Cosmic Neutrino Background

Pablo Martínez-Miravé<EMAIL_ADDRESS>Niels Bohr International Academy
and DARK, Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, 2100, Copenhagen, Denmark  Yuber F. Perez-Gonzalez
<EMAIL_ADDRESS>Institute for Particle Physics
Phenomenology, Durham University, South Road DH13EL, Durham, United Kingdom
Manibrata Sen<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany

###### Abstract

Ultralight dark matter interacting with sterile neutrinos would modify the
evolution and properties of the cosmic neutrino background through active-
sterile neutrino mixing. We investigate how such an interaction would induce a
redshift dependence in neutrino masses. We highlight that cosmological
constraints on the sum of neutrino masses would require reinterpretation due
to the effective mass generated by neutrino-dark matter interactions.
Furthermore, we present an example where such interactions can alter the mass
ordering of neutrinos in the early Universe, compared to what we expect today.
We also address the expected changes in the event rates in a PTOLEMY-like
experiment, which aims to detect the cosmic neutrino background via neutrino
capture and discuss projected constraints.

††preprint: IPPP/24/28

##  I Introduction

The origin of neutrino masses and the nature of dark matter constitute, at
present, the two main open questions in our understanding of the Universe. On
the one hand, neutrino oscillation data is consistent with the existence of
three active neutrinos, being — at least two of them – massive particles at
present times [1, 2, 3]. Additional sterile neutrino species could exist and
explain several experimental anomalies [4, 5, 6, 7, 8, 9]. However, regions of
the parameter space determined by their mass and mixing with the three active
neutrinos are severely constrained from neutrino disappearance experiments as
well as cosmological observables [10, 11, 12, 13]. On the other end, a
plethora of dark matter candidates have been hypothesised and their signatures
are the object of many direct and indirect experimental searches. Nonetheless,
a positive signal remains elusive.

In this context, we explore the phenomenology of scalar dark matter candidates
with masses $\sim 10^{-22}-10^{-10}\,$eV, which are often referred to as
ultralight dark matter, fuzzy dark matter or wave-like dark matter [14, 15,
16]. Such candidates were initially proposed as an explanation to the small-
scale cosmological puzzles such as the too-big-to-fail problem, the missing-
satellites riddle or the cup-vs-core puzzle [17]. In the light of recent data,
the initial motivation has – at least partially – vanished [18]. Nevertheless,
ultralight dark matter exhibits a very rich phenomenology, testable in a wide
variety of experiments - ranging from precision atomic experiments to
gravitational wave observations. From a theoretical point of view, ultralight
scalars or pseudoscalars can arise in various minimal extensions of the
Standard Model (SM), for instance from spontaneous lepton number
violation[19], in the Nelson-Barr solution to the CP-problem [20] or in the
Peccei-Quinn mechanism [21].

From a phenomenological point of view, coupling the ultralight dark matter
(ULDM) to SM fermions can give rise to a variety of signatures, such as
enhanced stellar cooling, additional contributions to electric and magnetic
dipole moments, or lepton flavour violating decays, among others [22]. Recent
studies have focused on the signatures of time-variations of neutrino masses,
which result from neutrinos coupling to such ULDM scalar field [23, 24, 25,
26, 27, 28, 29, 30, 31, 32, 33, 28, 34, 35, 36, 37, 38, 39, 40, 41]. However,
such a scenario can also modify neutrino free-streaming or lead to redshift-
dependent neutrino masses. These, in turn, could be in conflict with
observations of the cosmic microwave background and large-scale structures
[42]. A simple way to evade this constraint is to couple the ULDM to sterile
neutrinos, which are SM singlets (see e.g. [31, 43]). Note that, due to the
mixing between active and sterile neutrinos, this interaction could manifest
as an effective coupling between ultralight dark matter and active neutrinos,
which could leave an imprint in oscillation experiments, beta-decay
measurements and neutrinoless double-beta decay searches. This portal would
simultaneously avoid the direct coupling between dark matter and charged
fermions, which has been tightly constrained from the non-observation of time
variations in the electron mass, see e.g. [44].

In this article, we test interactions between ULDM and active neutrinos using
the sea of relic neutrinos from the Big Bang that permeates the Universe. This
sea of relic neutrinos – aptly known as the cosmic neutrino background (${\rm
C}\nu{\rm B}$) – is a crucial prediction of the $\Lambda$CDM model of
cosmology, and positive detection of this background can be used to test
fundamental properties of neutrinos [45, 46, 47, 48, 49, 50, 51, 52, 53, 29,
54, 55, 56, 57]. A number of ideas has been proposed for the detection of this
neutrino background [45, 58, 59, 60, 61, 62, 63, 64], however, the most
feasible one till date is that of neutrino capture on a beta-decaying nuclei
like tritium, as put forward by Weinberg [45] and currently the major focus of
the PTOLEMY collaboration [65]. From neutrino oscillation experiments, it is
expected that at least two generations of neutrinos composing this ${\rm
C}\nu{\rm B}$ are non-relativistic at present times. However, this picture
might change if the neutrino masses acquired an additional contribution from
their interaction with the ULDM. Such an effective neutrino mass, sourced from
ULDM, alters the evolution of the ${\rm C}\nu{\rm B}$ in the late Universe.
Not only that, the effects of these interactions can show up in a PTOLEMY-like
experiment, aimed at detecting the ${\rm C}\nu{\rm B}$. Here, we propose to
use the ${\rm C}\nu{\rm B}$ as a laboratory to test ULDM interactions with
neutrinos and set novel constraints on the parameter space defined by ULDM
mass and coupling.

The paper is organised as follows. In Section II, we present the framework in
which we discuss dark matter-neutrino coupling and the general
phenomenological implications. In Section III, we describe how this coupling
would induce neutrino masses that grow with redshift. Indirect probes of
neutrino masses can constrain their redshift evolution. We also comment on the
possibility of altering the hierarchy of neutrino masses at early times. In
Section IV, we show the observable imprint of this scenario in a PTOLEMY-like
experiment, based on the process of neutrino capture. Finally, in Section V,
we outline the main conclusions and discuss future work along these lines. We
use the natural unit system where $\hbar=c=k_{\rm B}=1$, and define the Planck
mass to be $m_{\rm PL}=1/\sqrt{8\pi G}$, with $G$ being the gravitational
constant, throughout this manuscript.

##  II THEORETICAL FRAMEWORK

In the early Universe, a scalar field $\Phi$ with mass $m_{\Phi}$ evolves
according to the following equation of motion,

$\displaystyle\ddot{\Phi}+3H(t)\dot{\Phi}+m^{2}_{\Phi}\Phi=0\,,$ (1)

where the dot represents the derivative with respect to proper time and $H$ is
the Hubble rate. Then, the relic density of an ultralight dark matter $\Phi$
can be determined through the misalignment mechanism [66, 67, 68]. This
mechanism involves the initial displacement of $\Phi$ from the potential’s
minimum, where the field remains static due to Hubble friction – i.e. the
second term in Equation 1. At this point, the DM density is frozen to its
initial value. At later times, when the temperature drops below a critical
value $T_{H}$ – defined by the equation $m_{\Phi}=3H(T_{H})$ – $\Phi$ begins
oscillating. As the temperature decreases further ($T<T_{H}$), the DM
transitions to a non-relativistic state, causing its energy density to vary as
$T^{3}$ [28, 37]. This behavior is roughly described by

$\rho_{\Phi}(T)=\rho_{\Phi}(T_{0})\left(\frac{g(T_{0})}{g(T)}\right)\left(\frac{{\rm
min}(T,T_{H})}{T_{0}}\right)^{3}\,,$ (2)

with $\rho_{\Phi}(T_{0})=10^{-5}\rho_{\Phi,\textrm{local}}$ the current DM
density, scaled by the local DM overdensity factor, $g(T)$ denotes the entropy
effective relativistic degrees of freedom, and $T_{0}=2.72\,$K represents the
current temperature of the cosmic microwave background (CMB). Consequently,
the energy density of DM is constant at early times until $m_{\phi}=3H$, after
which it starts decreasing following the temperature evolution. Fig. 1
displays the evolution of the ultralight dark matter field and its energy
density in a flat radiation-dominated Friedmann-Lemaître-Robertson-Walker
background (see e.g. [69] for more details).

Figure 1: Left panel: Evolution of the ultralight dark matter field in the
misalignment mechanism, according to its initial value $\Phi_{i}$. Radiation
is assumed to dominate the background evolution. The vertical dashed line
indicates when the mass and Hubble-friction terms in Equation 1 become
comparable. Right panel: Evolution of the energy density of the ULDM field in
a radiation-dominated background, given an initial value
$\rho_{i}=\frac{1}{2}m_{\Phi}^{2}\Phi^{2}$. The dotted lines indicate the
regimes for which the energy density is constant and when it scales as a
function of the temperature, namely $T^{3}$.

At present times, and due to the large occupation number, the ultralight
scalar field is well-described as a classical field,

$\displaystyle\Phi(t,z=0)\simeq\frac{\sqrt{2\rho_{\Phi,\textrm{local}}}}{m_{\Phi}}\cos\left(m_{\Phi}t\right)\,,$
(3)

being $\rho_{\Phi,\textrm{local}}$ the local dark matter density, and we have
neglected a phase in the time modulation. An elaborate discussion on why this
phase can be neglected can be found in [37].

Motivated by the feebly interacting nature of sterile neutrinos and dark
matter, we assume an interaction between both species that acts as a portal
between dark matter and the Standard Model due to active-sterile neutrino
mixing. In particular, let us consider a simplified case where a sterile
neutrino, $N$, couples to an ultralight dark matter scalar field, $\Phi$, and
mixes with only one family of leptons. Such a scenario is described by the
following Lagrangian,

$\displaystyle-\mathcal{L}\supset\,y_{D}\overline{l_{L}}\widetilde{h}N+\frac{1}{2}\left(m_{N}+y_{\Phi}\Phi\right)\overline{N^{c}}N+\frac{1}{2}m_{\Phi}\Phi^{2}+\frac{\kappa}{2}\left(\overline{l^{c}_{L}}\widetilde{H}^{*}\right)\left(\widetilde{H}^{\dagger}l_{L}\right)+\textrm{h.c.}\,.$
(4)

Here, $m_{N}$ is the mass of sterile neutrino, and $y_{\Phi}$ is the coupling
between both species. In addition, $h$ denotes the Higgs field and $l_{L}$
denotes the Standard Model lepton doublet $l_{L}\equiv(\nu,e)_{L}$. The Dirac
mass, $m_{D}=y_{D}v/\sqrt{2}$, and the active-neutrino Majorana mass,
$m_{\nu}=\kappa v^{2}/2$, results from the Higgs field getting a vacuum
expectation value, $v$. Note that the dimension-5 operator ensures that the
sterile neutrino mass and mixing angle can be chosen independently in our 1+1
scenario. Extensions to more generations will be considered in future work.

The neutrino mass matrix in this simplified framework with one active and one
sterile neutrino (1+1) is diagonalised by the mixing matrix, $U$,
parameterised by the angle $\theta_{14}$. We denote the corresponding mass
eigenstates as $m_{1}$ and $m_{4}$. Once we include a dark-matter–sterile
neutrino coupling, the mass matrix is written in the new basis as

$\displaystyle\mathcal{M}_{\nu+\textrm{DM}}=U^{\dagger}\begin{pmatrix}m_{1}&0\\\
0&m_{4}\end{pmatrix}U+\begin{pmatrix}0&0\\\
0&y_{\Phi}\Phi\end{pmatrix}\,=\widetilde{U}^{\dagger}\begin{pmatrix}\widetilde{m}_{1}&0\\\
0&\widetilde{m}_{4}\end{pmatrix}\widetilde{U}\,,$ (5)

from which stems the definition of the effective masses, $\widetilde{m}_{1}$
and $\widetilde{m}_{4}$, and the effective mixing matrix, $\widetilde{U}$,
parameterised in terms of the mixing angle $\widetilde{\theta}_{14}$. As a
consequence of the mixing, the effective mass eigenstates acquire a time
dependence, namely

$\displaystyle\widetilde{m}_{1,4}=\frac{m_{1}+m_{4}+y_{\Phi}\Phi}{2}\pm\frac{1}{2}\sqrt{(m_{4}-m_{1})^{2}+(y_{\Phi}\Phi)^{2}+2(m_{4}-m_{1})y_{\Phi}\Phi\cos
2\theta_{14}}\,.$ (6)

Additionally, the effective mixing also becomes time-dependent,

$\displaystyle\tan 2\widetilde{\theta}_{14}=\frac{(m_{4}-m_{1})\sin
2\theta_{14}}{(m_{4}-m_{1})\cos 2\theta_{14}+y_{\Phi}\Phi}\,.$ (7)

Note that due to the self-interactions, the effective sterile neutrino mass
and mixing can be suppressed even though their “vacuum” values are large. This
can have important consequences on the sterile neutrino interpretation of the
short baseline anomalies, as explored in [31]. In the next section, we discuss
some of the implications of such an effective neutrino mass and mixing.

##  III Evolution of the effective neutrino mass and relevance to the ${\rm
C}\nu{\rm B}$

Due to the redshift evolution of the dark matter field, the effective masses
$\widetilde{m}_{1}$ and $\widetilde{m}_{4}$, and the effective mixing,
$\widetilde{\theta}_{14}$, would also change along with the evolution of the
Universe. This means that active neutrinos would get an effective mass, as a
consequence of mixing with sterile neutrinos. On the one hand, for a suitable
choice of parameters, in the present-day scenario, $y_{\Phi}\Phi\ll m_{4}$ can
be reached. It can be shown that this case would manifest as a time variation
in neutrino observables at terrestrial experiments. In the early Universe, due
to redshift variation, the condition could evolve to be $y_{\Phi}\Phi\gg
m_{4}$. In this limit, it has been shown that the induced neutrino mass in the
early Universe is small as long as $m_{4}\,\sin^{2}\theta_{14}$ is small [31].
This allows us to bypass the limits arising on the sum of neutrino masses from
cosmology [31]. This particular limit can not only lead to large neutrino mass
cosmology [51], but also give rise to scenarios where the neutrino mass
ordering in the early Universe can be different from what it is today and
modify the interpretation of cosmological probes of neutrino masses.

As a demonstration, let us consider a simple case in which the mixing and the
structure of the coupling are such that only $\nu_{2}$ gets an effective mass
due to the mixing with $N$. Let us also assume that at present times the mass-
ordering is the so-called normal and the lightest neutrino, $\nu_{1}$, is
massless, $m_{1}=0$. Then, according to oscillation data [1], one can
constrain neutrino masses to be $m_{2}(z=0)=\sqrt{\Delta m^{2}_{21}}=8.7$ meV
and $m_{3}(z=0)=\sqrt{\Delta m^{2}_{31}}=50$ meV. However, due to the
evolution of the dark-matter energy density, $\tilde{m}_{2}$ could have been
larger than $\tilde{m}_{3}$ in the early Universe, as shown in Fig. 2 for a
set of illustrative values. As long as the limit on the sum of neutrino masses
is not violated by such a transition, this kind of scenario is extremely
difficult to constrain cosmologically. For the example shown in Fig. 2, we see
that $\sum\tilde{m}_{i}$ remains virtually unchanged for redshifts up to
$z\sim\mathcal{O}(10^{3})$, and hence this is consistent with cosmological
limits. This can possibly be tested by using neutrino free-streaming arguments
[70], but a dedicated study is beyond of the scope of this work. One should
also notice that, in this particular case, the ordering of neutrino masses
changes but it does not correspond to the so-called inverted ordering, for
which $\nu_{3}$ is the lightest mass eigenstate.

Figure 2: Evolution of the effective mass as a function of redshift in the
scenario where the ULDM couples only with $\nu_{2}$. In this case,
$\tilde{m}_{2}$ redshifts and can become larger than $m_{3}$ at large
redshifts. The plot assumes normal mass ordering and the lightest neutrino
mass $m_{1}=0$.

Time and redshift variation in neutrino mass can also show up in the ${\rm
C}\nu{\rm B}$, which consists of neutrinos free-streaming since the time of
neutrino decoupling from the thermal plasma. Decoupling occurred when the weak
interactions of the SM neutrinos became less frequent compared to the
expansion rate of the Universe at a photon temperature of approximately
$T_{\gamma}\simeq 1~{}{\rm MeV}$. Since decoupling, these neutrinos have
gradually cooled down due to the expansion of the Universe. At present, they
are expected to exhibit a Fermi-Dirac distribution with a temperature of about
$T_{\nu}\simeq 1.95\,{\rm K}$. This temperature is slightly lower than the
photon temperature due to electron-positron annihilation at around $0.5\,{\rm
MeV}$ that heated up the photon bath approximately by a factor of
$(11/4)^{1/3}$. In this standard scenario, the number density of these
neutrinos can be estimated to be

$n_{\nu}=\frac{3}{4}\frac{\zeta(3)}{\pi^{2}}gT_{\nu}^{3}\,,$ (8)

where $g$ is the number of degrees of freedom. Consequently, the expected
number density of the ${\rm C}\nu{\rm B}$ is $112\,{\rm cm}^{-3}$ per flavor.

Due to mixing with sterile neutrinos, neutrinos from the ${\rm C}\nu{\rm B}$
will pick up a time-redshift-dependent contribution to the effective mass. As
a result, any experiment trying to detect the ${\rm C}\nu{\rm B}$ through the
effect of a non-zero neutrino mass would be sensitive to this signal.
Currently, one of the most promising avenues in the detection of the ${\rm
C}\nu{\rm B}$ is provided by the PTOLEMY collaboration. In the next section,
we will study how a PTOLEMY-like experiment can set very competitive
constraints on the coupling and masses of ULDM.

##  IV Capture rate in a PTOLEMY-like experiment

PTOLEMY is one of the most realistic proposals for the detection of the ${\rm
C}\nu{\rm B}$. It considers the production of an observable electron after the
neutrino capture by a neutron, $\nu_{e}+n\to p^{+}+e^{-}$. To make the capture
experimentally viable, PTOLEMY intends to use tritium as a target due to its
large lifetime and its $Q_{\beta}$ value, of about 12.3 years and $18.591$
keV, respectively [71]. The capture rate of the ${\rm C}\nu{\rm B}$ on a
target nuclei is generally given by [29]

$\displaystyle\Gamma_{{\rm C}\nu{\rm
B}}=N_{T}\,\overline{\sigma}\,\sum_{i=1}^{3}\left[n(\nu_{i,+1})\,{\cal
A}_{i}(+1)+n(\nu_{i,-1})\,{\cal A}_{i}(-1)\right]\,,$ (9)

where $N_{T}$ are the number of targets, $n(\nu_{i,\pm 1})$ represent the
${\rm C}\nu{\rm B}$ number densities for each helical state, and ${\cal A}(h)$
are spin-dependent factors that arise due to the mismatch between helicity and
chirality,

$\displaystyle{\cal A}_{i}(h)\equiv 1-hv_{i},$ (10)

being $v_{i}=|\vec{p}|/\sqrt{|\vec{p}|^{2}+m_{i}^{2}}$ the average neutrino
velocity, $h$ the helicity. The nucleus-dependent factor $\overline{\sigma}$
in the capture rate is the spin-averaged cross-section. Assuming tritium as
the target, we have that

$\displaystyle\overline{\sigma}\approx 3.8\times 10^{-45}{\rm\ cm^{2}}.$ (11)

Although the number of events is expected to be large, $\sim 8~{}{\rm
yr^{-1}}$ for 100 g of tritium [29], the main challenge for the experimental
confirmation of the ${\rm C}\nu{\rm B}$ is the energy resolution [72, 73].
Since tritium is a $\beta$ emitter, it is necessary to have an excellent
energy resolution to distinguish the ${\rm C}\nu{\rm B}$-produced electrons
from those from $\beta$ decay. PTOLEMY is expected to have a resolution of
$\Delta=50-150$ meV, with $\Delta=\sqrt{8\ln 2}\sigma$ the full width at half
maximum (FWHM) of the Gaussian resolution [74, 73]. The spectra of ${\rm
C}\nu{\rm B}$ and $\beta$ decay events, our signal and background
respectively, are computed by including a Gaussian resolution function [29]

$\displaystyle\frac{d\Gamma_{\rm C\nu B}}{dE_{e}}$
$\displaystyle=\sum_{i}\frac{1}{\sqrt{2\pi}\sigma}\int
dE_{e}^{\prime}\,\Gamma_{{\rm C}\nu{\rm B}}^{i}\,\delta(E_{e}^{\prime}-E_{\rm
end}-2m_{i})\exp\left[-\frac{(E_{e}^{\prime}-E_{e})^{2}}{2\sigma}\right]$
(12a) $\displaystyle\frac{d\Gamma_{\beta}}{dE_{e}}$
$\displaystyle=\sum_{i}\frac{1}{\sqrt{2\pi}\sigma}\int
dE_{e}^{\prime}\,\frac{d\Gamma_{\beta}}{dE_{e}}(E_{e}^{\prime})\exp\left[-\frac{(E_{e}^{\prime}-E_{e})^{2}}{2\sigma}\right]$
(12b)

where $d\Gamma_{\beta}/dE_{e}$ is the $\beta$ spectrum from tritium decay, and
$E_{\rm end}$ is the beta decay endpoint energy,

$\displaystyle E_{\rm end}=K_{\rm end}+m_{e},$ (13)

with $m_{e}$ the electron’s mass and

$\displaystyle K_{\rm
end}=\frac{(m_{\rm{}^{3}H}-m_{e})-(m_{\rm{}^{3}H_{e}}+m_{i})^{2}}{2m_{\rm{}^{3}H}},$
(14)

the electron endpoint’s kinetic energy. In what follows, we use the $\beta$
decay spectrum given in Ref. [75].

Figure 3: Differential electron spectrum from beta decay of tritium and from
neutrino capture on tritium. The left and right panels correspond to energy
resolutions of $\Delta=10$ meV and $\Delta=$ 50 meV respectively. Dashed lines
represent the standard predictions and the solid lines are the predictions for
a coupling $y_{\Phi}=5\times 10^{-16}$ and an ultralight scalar mass
$m_{\Phi}=10^{-18}$ eV.

In our scenario of interest, where the neutrino masses vary over time due to
the interaction between the ${\rm C}\nu{\rm B}$ and the ULDM, we anticipate
modifications in the spectra measured by a PTOLEMY-like experiment due to
these novel interactions. The primary changes induced by neutrino-ULDM
interactions involve alterations in the ${\rm C}\nu{\rm B}$ capture rate,
which is directly influenced by neutrino mass values, and shifts in the
spectrum’s peak position. However, it is worth noting that the $\beta$
background is also influenced by mass variation, as the endpoint energies
directly correlate with neutrino masses. In Fig. 3, we display spectra for
both ${\rm C}\nu{\rm B}$ and $\beta$ decay electrons under two energy
resolution assumptions: $\Delta=10$ meV (left) and $\Delta=50$ meV (right).
Dashed lines represent the standard scenario, while full lines indicate the
inclusion of neutrino-dark matter (DM) interactions with $y_{\Phi}=5\times
10^{-16}$ for ultralight DM with $m_{\Phi}=10^{-18}$ eV. The spectra are
plotted against electron kinetic energy minus the value of the endpoint
kinetic energy assuming massless neutrinos $K_{\rm end}^{0}$. Both signal and
background show significant modifications due to novel interactions in both
panels. Remarkably, under the optimistic energy resolution of $\Delta=10$ meV,
the ${\rm C}\nu{\rm B}$ spectra widen, with its peak shifting away from the
neutrino mass value. Additionally, the $\beta$ decay spectrum shifts away from
the ${\rm C}\nu{\rm B}$ events, leading to a clearer distinction between
signal and background. For the more realistic energy resolution of $\Delta=50$
meV, observed shifts remain, albeit with smoother spectra. Consequently, these
spectra hint at PTOLEMY’s potential to test the mass-varying scenario.

To assess PTOLEMY’s capability in detecting neutrino-ULDM interactions, we
define the region of interest (ROI) as electron energies within the range of
$K_{e}-K_{\rm end}^{0}=[0,0.2]$ eV. This range encompasses the broadening of
the ${\rm C}\nu{\rm B}$ spectra and potential alterations to the $\beta$ decay
background. Events are computed by integrating over the electron’s energy in
both ${\rm C}\nu{\rm B}$ and $\beta$ spectra using a bin width of $\Delta$,

$\displaystyle
N_{i}=\int_{E_{e}^{i}}^{E_{e}^{i}+\Delta}dE_{e}\left\\{\frac{d\Gamma_{\rm C\nu
B}}{dE_{e}}+\frac{d\Gamma_{\beta}}{dE_{e}}\right\\}.$ (15)

Thus, the number of bins is determined by the energy resolution. We consider
the following Poissonian test statistics for our analysis,

$\displaystyle\chi^{2}=2\sum_{i={\rm bins}}N_{i}(y_{\Phi},m_{\Phi})-N_{i}^{\rm
SM}+N_{i}^{\rm SM}\ln\left(\frac{N_{i}(y_{\Phi},m_{\Phi})}{N_{i}^{\rm
SM}}\right),$ (16)

being $N_{i}^{\rm SM}$ and $N_{i}(y_{\Phi},m_{\Phi})$ the number of events for
the standard case and including neutrino-DM interactions, respectively. As for
the experimental exposure, we assume a setup with 100 g of tritium with an
observation time of 10 years. Note that we have not accounted for a local
overdensity of neutrinos due to clustering [76, 46, 77]. Such an effect would
further enhance the significance of the signal over the background, boosting
the sensitivity projections here discussed. However, a dedicated computation
of such overdensity for ultralight dark matter would be needed. For
simplicity, we do not consider this effect.

Figure 4: Sensitivity of a PTOLEMY-like experiment to neutrino-ULDM
interactions that produce a time-dependent neutrino mass on the coupling
$y_{\Phi}$ vs $m_{\Phi}$ plane. We present the capabilities to discriminate
the additional interaction for two different energy resolutions of $\Delta=50$
meV (full contour) and $\Delta=10$ meV (dashed contour) at the $95\%$ CL. We
also show the region excluded from ultra-faint white dwarfs heating in the
blue regions [78].

In Fig. 4, we illustrate PTOLEMY’s sensitivity to the parameter space spanned
by $y_{\Phi}$ and $m_{\Phi}$, assuming the same energy resolutions as
previously specified, namely $\Delta=10$ meV (dashed line) and $\Delta=50$ meV
(full line), at the $95\%$ CL. As expected, the optimistic value of
$\Delta=10$ meV offers better sensitivity for our scenario compared to the
more realistic energy resolution of $\Delta=50$ meV. Although the $\beta$
background significantly contributes to the event count in our ROI at higher
energy resolutions, sensitivity to the mass-varying effect improves with
better discrimination of the ${\rm C}\nu{\rm B}$ signal from the $\beta$
background. This is due to the significant dependence of the recoil electron
spectra on neutrino masses as previously observed in Fig. 3, should neutrino-
ULDM interactions exist. However, we observe that PTOLEMY’s capability for
setting constraints is only mildly dependent on the energy resolution. As a
comparison, we also show the constraints obtained from heating ultrafaint
dwarf galaxies [78]. Note that these are stronger than the ones from dwarf
galaxies, $m_{\Phi}>2.2\times 10^{-22}$ eV at 95% C.L. [79]. We find that
could set more stringent constraints on neutrino-ULDM coupling in part of the
parameter space. Model-dependent constraints can also arise from the condition
that the ULDM does not redshift like radiation and wash away satellite
galaxies of the Milky Way [28]. These constraints are competitive with the
ones we obtain from PTOLEMY.

##  V Conclusion

A hypothetical interaction between neutrinos and an ultralight dark matter
scalar field would result in a plethora of experimental signatures. It is well
known that a direct coupling of neutrinos with ultralight dark matter can
result in the neutrino mass redshifting in past, which can be in tension with
the limits on the sum of the neutrino mass derived from the cosmic microwave
background. This can be evaded through a sterile neutrino portal, where the
dark matter couples directly with sterile neutrinos, and the active neutrinos
feel this coupling only through mixing. Such an interplay between ultralight
dark matter and active neutrinos can affect the evolution and detection of the
cosmic neutrino background. In this work, we have focused on direct and
indirect probes of such interacting dark matter with the cosmic neutrino
background.

On the one hand, active neutrinos would get a redshift-dependent effective
mass due to their interaction with the ultralight scalar field through mixing
with sterile neutrinos. The redshift evolution of the energy density of dark
matter would get imprinted in observables related to neutrino masses. In
particular, specific coupling structures and values of the active-sterile
mixing could lead to a change in the mass ordering of neutrinos between the
present time and the early Universe. The phenomenology related to this family
of models could also address a discrepancy between indirect – cosmological –
determinations of the absolute neutrino-mass scale and direct ones – such as
beta decay.

On the other hand, the interaction between neutrinos and dark matter could
also manifest in experiments aiming to directly measure the cosmic neutrino
background. Specifically, a PTOLEMY-like experiment would observe a distortion
in the spectrum from neutrino capture, which could be used to constrain the
mass of the dark matter candidate and its coupling to neutrinos. We have
presented the projected limits of such an experiment and highlighted the
importance of the energy resolution to separate the signal from neutrino
capture and beta decay of the target isotope.

The phenomenology and experimental probes here proposed are not unique to the
scenario of ultralight scalar dark matter coupling to active neutrinos via
mixing with a sterile neutrino. For instance, other scenarios in which
neutrinos acquire a redshift-dependent neutrino mass include couplings to dark
energy [80, 81], phase transitions [82] and topological defects [83]. In this
case, one has to ensure that the specific scenario is consistent with all
cosmological probes, i.e. cosmic microwave background, Big Bang
nucleosynthesis, and the observed large- and small-scale structures.
Similarly, one can consider scenarios in which neutrino mass shows a similar
time-modulation at the present time, or other time-dependent phenomena, for
instance, in the case of Lorentz Invariance Violation [84, 85, 86]. Finally,
other ultralight dark matter candidates include vector or tensor fields (e.g.
[87, 88]). In each of those cases, a coupling to neutrinos would also modify
the evolution of the cosmic microwave background and its detection prospects.
In each of the cases, the signatures differ from those of coupling to scalars
and hence are not discussed in this article. Nonetheless, the methodology and
part of the discussions here presented would also apply to them.

## Acknowledgements

We would like to thank Tim Herbermann, Joachim Kopp, and Alexei Smirnov for
useful discussions. PMM acknowledges support from the Carlsberg Foundation
(CF18-0183). The work of YFPG has been funded by the UK Science and Technology
Facilities Council (STFC) under grant ST/T001011/1. This project has received
funding/support from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No 860881-HIDDeN.
This work has made use of the Hamilton HPC Service of Durham University.

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