# Echoes of darkness: Supernova-neutrino-boosted dark matter from all galaxies

Yen-Hsun Lin<EMAIL_ADDRESS>Institute of Physics, Academia Sinica,
Taipei 115, Taiwan  Meng-Ru Wu<EMAIL_ADDRESS>Institute of Physics,
Academia Sinica, Taipei 115, Taiwan  Institute of Astronomy and Astrophysics,
Academia Sinica, Taipei 106, Taiwan  Physics Division, National Center for
Theoretical Sciences, Taipei 106, Taiwan

###### Abstract

It has been recently proposed that the boosted dark matter (BDM) by supernova
neutrinos (SN$\nu$) from SN1987a or from the next galactic SN can serve as a
novel component to probe nonvanishing interaction between DM and the Standard
Model leptons Lin  _et al._ (2023a, b). In this work, we extend this concept
and evaluate the present-day _diffuse_ flux of SN$\nu$ BDM originated from all
galaxies at higher redshifts. We show that by considering this diffuse BDM
(DBDM) component, the best model-independent sensitivity on the product of the
DM-$\nu$ and DM-electron cross sections, $\sqrt{\sigma_{\chi\nu}\sigma_{\chi
e}}\simeq\mathcal{O}(10^{-37})$ cm2 for sub-MeV DM, can be obtained with
large-size neutrino experiments such as Super-Kamiokande or Hyper-Kamiokande,
surpassing the estimated SN$\nu$ BDM bound from SN1987a. We also examine the
impact due to the presence of DM spikes around the supermassive black holes in
galaxies on SN$\nu$ BDM and DBDM. Our results suggest that both the DBDM and
the SN$\nu$ BDM probes are insensitive to the uncertain properties of DM
spikes, unless the next galactic SN happens to occur at a location extremely
close to or right behind the galactic center along the SN line of sight.

##  I Introduction

Despite the abundance of compelling evidence for dark matter (DM) in the
Universe, its nature remains a puzzling question in fundamental physics. The
Weakly Interacting Massive Particle was widely perceived as a possible
scenario for DM, but its parameter space has been tightly constrained by
modern DM direct (DD) and indirect detections Aguilar  _et al._ (2015); Akerib
_et al._ (2017a); Ackermann  _et al._ (2017); Akerib  _et al._ (2017b); Agnese
_et al._ (2018); Ambrosi  _et al._ (2017); Aprile  _et al._ (2018, 2019a,
2019b); Abramoff  _et al._ (2019); John  _et al._ (2023). Therefore, light DM
with mass $m_{\chi}\lesssim\mathcal{O}({\rm MeV})$ has gained much attention
recently, with a plethora of theoretical models proposed Holdom (1986); He
_et al._ (1991); Davoudiasl  _et al._ (2012); Battaglieri  _et al._ (2017);
Chang  _et al._ (2018); Foldenauer (2019); Escudero  _et al._ (2019); Croon
_et al._ (2021); Lin and Lin (2021); Workman  _et al._ (2022).

Boosted DM (BDM), when upscattered by known high-energy cosmic particles,
offers a viable method to probe light DM in large underground detectors
Bringmann and Pospelov (2019); Ema  _et al._ (2019); Cappiello and Beacom
(2019); Dent  _et al._ (2020); Wang  _et al._ (2020); Zhang (2022); Guo  _et
al._ (2020a); Ge  _et al._ (2021); Cao  _et al._ (2021); Jho  _et al._ (2020);
Cho  _et al._ (2020); Jaeckel and Yin (2021); Lei  _et al._ (2020); Guo  _et
al._ (2020b); Xia  _et al._ (2021); Dent  _et al._ (2021); Ema  _et al._
(2021); Flambaum  _et al._ (2020); Jho  _et al._ (2021); Das and Sen (2021);
Bell  _et al._ (2021); Chao  _et al._ (2021); Ghosh  _et al._ (2022); Feng
_et al._ (2022); Wang  _et al._ (2021); Xia  _et al._ (2022); Wang  _et al._
(2022); Abe  _et al._ (2023); Cui  _et al._ (2022); Xu  _et al._ (2022);
Granelli  _et al._ (2022); Cline  _et al._ (2023); Xia  _et al._ (2023);
Cappiello  _et al._ (2023); Carenza and De la Torre Luque (2023); Wang  _et
al._ (2023); Adhikari  _et al._ (2023); Xia  _et al._ (2024), such as Super-
Kamiokande Abe  _et al._ (2016), Hyper-Kamiokande Abe  _et al._ (2018), DUNE
Abi  _et al._ (2020) and JUNO Abusleme  _et al._ (2022). In Ref. Lin  _et al._
(2023a), the authors proposed that BDM from supernova neutrinos (SN$\nu$) can
provide competitive tests for light DM interacting with known leptons, and
carries time-of-flight information that potentially enables a direct
measurement of $m_{\chi}$. In a follow-up work Lin  _et al._ (2023b), the
authors explored the dependency of this probe on the location of the SN in the
galactic disk, as well as applied the method to a specific gauged
$U(1)_{L_{\mu}-L_{\tau}}$ model.

In this work, we further extend this concept by considering the contribution
of the SN$\nu$ BDM from galaxies that hosted all past SN explosions at
different redshifts to the present-day _diffuse_ flux. Such a diffuse BDM
(DBDM) component parallels the well known diffuse SN$\nu$ background (DSNB)
Horiuchi  _et al._ (2009); Beacom (2010) and thus represents a persistent and
isotropic BDM flux. We will demonstrate that by analyzing the DBDM signals at
large-size neutrino experiments such as Super-Kamiokande (SK) or Hyper-
Kamiokande (HK), it can readily offer improved model-independent sensitivities
on the product of the DM-$\nu$ and DM-electron cross sections, surpassing the
existing constraint from the SN$\nu$ BDM associated with SN1987a.

In addition to the evaluation of DBDM flux and sensitivity, we also consider
the consequence on SN$\nu$ BDM and DBDM due to the potential presence of the
DM spike Gondolo and Silk (1999); Ullio  _et al._ (2001) in the inner halo of
a galaxy, resulting from the accretion of DM by the central supermassive black
hole (SMBH). We will show that for both the SN$\nu$ BDM and DBDM, the
associated sensitivities or constraints are not sensitive to the uncertain
properties of the DM spike. Only when the next galactic SN happens to occur
extremely close to or right behind the galactic center along the SN line of
sight, the large DM spike density may substantially enhance the projected
SN$\nu$ BDM sensitivity for the case without DM self-annihilation.

##  II Diffuse SN$\nu$ BDM (DBDM)

###  II.1 Averaged SN$\nu$ BDM spectrum from a galaxy

Considering the same model independent approach used in Ref. Lin  _et al._
(2023a) to describe the DM–$\nu$ interaction, the energy spectrum of the
_total_ amount of SN$\nu$ BDM (number per unit energy) for a single SN that
explodes at a distance $R$ away from the center of a galaxy is given by

$\frac{dN_{\chi}(R)}{dT_{\chi}}=(2\pi)^{2}\tau\int d\cos\theta
d\cos\theta_{c}d\ell\ell^{2}j_{\chi}(\ell,\theta,\theta_{c},T_{\chi}),$ (1)

where $T_{\chi}$ is the BDM kinetic energy, $\tau=10$ s is the characteristic
SN$\nu$ emission time, $\theta$ and $\ell$ are the zenith angle and the
distance defined in Fig. 1, $\theta_{c}$ is the scattering angle of the BDM in
the center-of-mass (CM) frame, and

$j_{\chi}(\ell,\theta,\theta_{c},T_{\chi})=cn_{\chi}(r)\frac{d\sigma_{\chi\nu}}{d\Omega_{c}}\frac{dn_{\nu}}{dE_{\nu}}\frac{dE_{\nu}}{dT_{\chi}}\frac{v_{\chi}}{c}$
(2)

is the local BDM emissivity. In Eq. (2), $n_{\chi}(r)=\rho_{\chi}(r)/m_{\chi}$
is the DM number density at the location of $\nu$–DM scattering ($\mathsf{B}$
in Fig. 1) with $\rho_{\chi}(r)$ the DM mass density, and the differential
DM–$\nu$ interaction cross section in the CM frame
$d\sigma_{\chi\nu}/d\Omega_{c}=\sigma_{\chi\nu}/4\pi$ is assumed to be
isotropic in the CM frame where $d\Omega_{c}$ is the differential solid angle.
For the SN$\nu$ energy spectrum $dn_{\nu}/dE_{\nu}$, we use exactly the same
form as in Ref. Lin  _et al._ (2023a, b). The term $dE_{\nu}/dT_{\chi}$ in Eq.
(2) is the Jacobian factor that converts $dE_{\nu}$ to $dT_{\chi}$,

$\frac{dE_{\nu}}{dT_{\chi}}=\frac{1}{2c_{2}}\left(1+\frac{1+c_{2}m_{\chi}/T_{\chi}}{\sqrt{2c_{2}m_{\chi}/T_{\chi}+1}}\right),$
(3)

with $c_{2}=\cos^{2}(\theta_{c}/2)$ and
$v_{\chi}/c=\sqrt{T_{\chi}(2m_{\chi}+T_{\chi})}/(m_{\chi}+T_{\chi})$ is the
BDM velocity.

Figure 1:  Schematic plot of DM boosted by SN$\nu$ within an expanding
spherical shell. The SN occurs at $\mathsf{S}$, the galactic center is located
at $\mathsf{G}$ while DM is boosted at $\mathsf{B}$.

For the averaged SN$\nu$ BDM spectrum from the a galaxy, it can be computed
approximately by averaging over the galaxy’s baryonic mass distribution
$\Sigma_{b}$ projected onto the disk plane (assuming SNe occur near the disk
mid-plane),

$\frac{d\bar{N}}{dT_{\chi}}=\left.\int
dA\frac{dN_{\chi}(R)}{dT_{\chi}}\Sigma_{b}(R)\right/\int dA\,\Sigma_{b}(R),$
(4)

where the integrations are performed over the galaxy’s differential area $dA$
in the mid-plane.

###  II.2 DM and baryon density profiles in different galaxies

As galaxies of different sizes have different DM halo profiles and baryonic
distributions, which affect the exact amount of SN$\nu$ BDM, certain
assumptions are needed to model their contribution to DBDM. First, we assume
that for a galaxy with a known stellar mass $M_{G}$, its halo mass follows
$M_{\rm DM}\simeq\eta M_{G}$ with $\eta=50$ Moster  _et al._ (2010); Girelli
_et al._ (2020). Second, for the halo mass distribution without DM spike, we
use the Navarro-Frenk-White (NFW) profile Navarro  _et al._ (1996, 1997),
$\rho^{\rm NFW}_{\chi}$ in Eq. (16) of Appendix A, with $\rho_{s}=184$ MeV
cm-3 and $r_{s}=24.2$ kpc for a Milky Way (MW) size galaxy with $M_{G}=M_{\rm
MW}=5.29\times 10^{10}$ $M_{\odot}$. For other galaxies, we take a simple
scaling relation $r_{s}^{3}\propto M_{G}$ to compute the corresponding
$\rho^{\rm NFW}_{\chi}$. The DM profile with spike is calculated following
Ref. Cline and Puel (2023) (see Appendix A for details). For the DM spike
density $\rho^{\rm spike}_{\chi}$ that depends on the central SMBH mass and
age, we apply the relation $M_{\rm BH}\approx 7\times
10^{7}M_{\odot}\times[M_{\rm DM}/(10^{12}M_{\odot})]^{4/3}$ Ferrarese (2002);
Di Matteo  _et al._ (2003); Baes  _et al._ (2003); Cline and Puel (2023) for
the mass. We assume that $t_{\rm BH}$ can be approximated by the age of the
Universe at redshift $z$, computed with cos  for the DBDM calculation. For the
SN$\nu$ BDM from MW and SN1987a, we take $t_{\rm BH}\simeq 10$ Gyr. Third, for
the surface baryonic density $\Sigma_{b}$, we consider the bulge and the disk
components given in McMillan (2016) for a MW-size galaxy to compute the
projected $\Sigma_{b}^{\rm MW}$. For other galaxies, we assume that their
bulge-to-disk mass ratios are the same as the MW value, and apply the scaling
relations that the scale bulge radius and the scale disk length parameters,
$r_{\rm cut}$ and $R_{d}$ in Eqs. (1) and (3) of McMillan (2016), are
proportional to the cubic root of the bulge mass and the square root of the
disk mass, respectively, to estimate the corresponding $\Sigma_{b}$. Given
these assumptions, one can then use Eqs. (1)–(4) to compute
$d\bar{N}/dT_{\chi}$ for any given $M_{G}$ of a galaxy.

###  II.3 DBDM flux

Knowing the averaged SN$\nu$ BDM energy spectrum from a galaxy, the DBDM flux
at redshift $z=0$ consisting of the SN$\nu$ BDM contributions from all
galaxies at different redshifts can be written as

$\frac{d\Phi_{\chi}}{dT_{\chi}}=\frac{v_{\chi}}{H_{0}}\int_{0}^{z_{\rm
max}}\frac{dz}{\varepsilon(z)}\int dM_{G}\frac{d\Gamma_{{\rm
SN}}(z)}{dM_{G}}\frac{d\bar{N}_{\chi}(M_{G})}{dT_{\chi}^{\prime}},$ (5)

where $H_{0}=c/(4280\,{\rm Mpc})$ is the Hubble constant,
$\varepsilon(z)=[\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}]^{1/2}$ with
$(\Omega_{m},\Omega_{\Lambda})\approx(0.3,0.7)$,
$T_{\chi}^{\prime}=(1+z)T_{\chi}$ is the BDM kinetic energy at the source. In
Eq. (5), the term $d\Gamma_{\rm SN}/dM_{G}$ is the SN rate per comoving volume
per galaxy mass at $z$. We assume that this rate is approximately proportional
to $M_{G}$ as well as to the known star formation rate per volume
$\dot{\rho}_{*}(z)$, and can be formulated as

$\frac{\Gamma_{\rm
SN}(z)}{dM_{G}}=\frac{dn_{G}(z)}{dm}\frac{\dot{\rho}_{*}(z)}{\dot{\rho}_{*}(0)}\frac{R_{\rm
SN,0}}{M_{\rm MW}},$ (6)

where $m\equiv\log_{10}(M_{G}/M_{\odot})$ and $R_{{\rm SN},0}\approx
0.01~{}{\rm yr}^{-1}$ is the SN rate of the MW Rozwadowska  _et al._ (2021).
The term $dn_{G}(z)/dm$ represents the number density of galaxies per comoving
volume per galaxy mass at $z$ and can be parameterized as Conselice  _et al._
(2016)

$\frac{dn_{G}(z)}{dm}=\phi_{0}\ln 10\times
10^{(m-M_{c})(1+\gamma)}e^{-10^{m-M_{c}}},$ (7)

where $M_{c}$ is the characteristic mass in $\log$-10 base, $\phi_{0}$ the
normalization constant and $\gamma$ the slope for fainter and lower mass
galaxies. They are fitted to observational data at different $z$ and are
documented in Ref. Conselice  _et al._ (2016) and references therein.111We
took the parametrizations from Refs. Fontana  _et al._ (2004, 2006) for $z<4$
and Ref. Song  _et al._ (2016) for $4\leq z<8$. They are labeled as
Fontana+04, Fontana+06 and Song+16, respectively, in the Table I of Ref.
Conselice  _et al._ (2016). For $\dot{\rho}_{*}(z)$, we take

$\dot{\rho}_{*}(z)=\dot{\rho}_{0}\left[(1+z)^{a\eta}+\left(\frac{1+z}{B}\right)^{b\eta}+\left(\frac{1+z}{C}\right)^{c\eta}\right]^{1/\eta}$
(8)

from Yuksel  _et al._ (2008), with $\dot{\rho}_{0}=0.0178M_{\odot}~{}{\rm
yr}^{-1}~{}{\rm Mpc}^{-3}$, $(a,b,c,\eta)=(3.4,-0.3,-3.5,-10)$, and the
redshift break constants $B=(1+z_{1})^{1-a/b}$ and
$C=(1+z_{1})^{(b-a)/c}(1+z_{2})^{1-b/c}$ at $z_{1}=1$ and $z_{2}=4$,
respectively.

Figure 2:  The five-year DBDM fluence (green) and the total fluence of SN$\nu$
BDM from SN1987a (orange) as well as from the next SN in the MW center (blue)
for $m_{\chi}=1$ keV and $\sigma_{\chi\nu}=10^{-35}~{}{\rm cm^{2}}$. The
dotted, solid, and dashed lines denote different DM profile scenarios.

We compute the DBDM flux given by Eq. (5) and show in Fig. 2 the corresponding
fluence by multiplying the flux with an assumed exposure time $t_{\rm exp}=5$
years taking $m_{\chi}=1$ keV and $\sigma_{\chi\nu}=10^{-35}$ cm2 for cases
considering the DM spikes with different thermally averaged DM self-
annihilation cross section $\langle\sigma v\rangle$, as well as the case
without DM spikes. It shows that the DBDM flux is nearly unaffected by the
presence of DM spikes. This is mainly because the presence of DM spikes only
affects a small fraction of SNe that occur very close to the center of
galaxies within $\sim R_{\rm spike}$.

For comparison, we also compute the SN$\nu$ BDM flux using the same set of
$m_{\chi}$ and $\sigma_{\chi\nu}$ for a SN that explodes in the MW center
following Ref. Lin  _et al._ (2023a) and for SN1987a in the Large Magellanic
Cloud (LMC), considering cases with and without spikes. Note that here we have
taken into account the potentially slight displacement of SN1987a from the LMC
center.222Note that although the adopted distances and coordinates of LMC
center and SN1987a result in a displacement of SN1987a from the LMC center by
1.75 kpc, the associated uncertainties are relatively large so that the
SN1987a can be much closer to the LMC center. For SN1987a, it occurs at 51.4
kpc Panagia (2005) away with right ascension (RA) of
$05^{h}\,35^{m}\,27.8733^{s}$ and declination (DEC) of
$-69^{\circ}\,16^{\prime}\,10.478^{\prime\prime}$ nas . The center of LMC was
estimated to be around 49.97 kpc away Pietrzyński  _et al._ (2013), with RA of
$05^{h}\,23^{m}\,34.5264^{s},$ and DEC of
$-69^{\circ}\,45^{\prime}\,22.053^{\prime\prime}$ nas . The LMC’s spike
profile is calculated using Eq. (10) by taking $M_{\rm BH}=10^{7}M_{\odot}$
Boyce  _et al._ and replacing $\rho_{\chi}^{\rm NFW}$ with the Hernquist halo
profile used in Lin  _et al._ (2023a).

The resulting SN$\nu$ BDM fluences integrated over the entire duration before
the BDM flux vanishes are also shown in Fig. 2 for SN1987a and for the MW
center case. Without spikes, the 5-year DBDM fluence can be comparable to the
total SN$\nu$ BDM fluence from SN1987a, but several orders of magnitude
smaller than that from the next galactic SN at the MW center. The energy
spectrum of DBDM is somewhat steeper than those from SN$\nu$ BDM, because the
DBDM flux is dominated by the contribution at $z\simeq 1-2$, which gets red-
shifted when arriving at the Earth. It is also evident that although the
presence of the DM spike without self-annihilation can dramatically enhance
the SN$\nu$ BDM flux if the SN explodes at the center of a galaxy, a small
displacement from the center like SN1987a will substantially reduce the effect
of spike. Moreover, a sizable DM annihilation rate $\langle\sigma
v\rangle=3\times 10^{-26}$ cm3 s-1 can also largely undermine the impact of
spike due to the suppressed spike densities (see Appendix A).

##  III Sensitivity and constraint

Figure 3:  Reevaluated constraint from SN1987a SN$\nu$ BDM (green), the
projected sensitivities from DBDM (red), the next galactic SN at MW center
(orange), and the averaged projected sensitivity from the next galactic SN
(violet) for difference scenarios.

###  III.1 DBDM and SN$\nu$ BDM Sensitivity

Within an exposure time $t_{\rm exp}$, the DBDM events $N_{\rm DBDM}$ in a
neutrino or DM detector with $N_{e}$ electron targets for a non-vanishing
interaction cross section $\sigma_{\chi e}$ between DM and the electrons can
be calculated by

$N_{\rm DBDM}=t_{\rm exp}\times\int_{T_{\rm min}}^{T_{\rm
max}}dT_{\chi}\frac{d\Phi_{\chi}}{dT_{\chi}}N_{e}\sigma_{\chi e}.$ (9)

Taking HK as an example, we set $N_{e}\approx 7.3\times 10^{34}$, $t_{\rm
exp}=5$ years and $(T_{\rm min},T_{\rm max})=(5,100)$ MeV to estimate $N_{\rm
DBDM}$. To achieve a $2\sigma$ detection significance, it requires $N_{\rm
DBDM}>N_{s}$ with $N_{s}$ satisfying $2=N_{s}/\sqrt{N_{s}+N_{b}}$ and
$N_{b}\simeq 5.8\times 10^{5}$, which is the estimated background events in HK
in 5 years also used in Lin  _et al._ (2023b). The background events are
mainly composed of solar and atmospheric neutrinos, while other radioactive
sources are subdominant Battistoni  _et al._ (2005); Nakano (2017); Dutta and
Strigari (2019).

For the reevaluated SN1987 bounds, we take into account the potential
displacement of SN1987a from LMC center and the effect of spikes as in Sec.
II.3 to compute the SN$\nu$ BDM flux. We then follow the same procedure of Lin
_et al._ (2023a) by utilizing the data from Kamiokande and SK experiments to
derive the constraint corresponding to the 90% confidence level.

We show in Fig. 3 the derived DBDM sensitivities and the reevaluated SN1987a
limits on $\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}$ as functions of $m_{\chi}$
for scenarios without DM spikes (red-dotted), with DM spikes and
$\langle\sigma v\rangle=0$ (red-solid), and with DM spikes and $\langle\sigma
v\rangle=3\times 10^{-26}$ cm3 s-1 (red-dashed), respectively. Similar to the
fluence shown in Fig. 2 for $m_{\chi}=1$ keV, the DBDM sensitivity and the
SN1987a constraint here are nearly unaffected by the property of DM spikes for
$m_{\chi}\gtrsim 0.03$ keV. For $m_{\chi}\lesssim 0.03$ keV, self-annihilating
DM slightly weakens both limits, due to the reduced inner halo densities (see
Appendix A). For all scenarios, the projected 5-year DBDM sensitivity from HK
can clearly result in better constraint on $\sqrt{\sigma_{\chi\nu}\sigma_{\chi
e}}$ than the SN1987a BDM bound for most of the relevant $m_{\chi}$ range.
This is mainly because for SN1987a, nearly all or a significant amount of BDM
arrives the Earth within a few years after the explosion when only Kamiokande
was operating. As a result, the much larger volume of HK can easily provide
improved limits on $\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}$ despite that the
DM fluences are comparable (see Fig. 2). We also note here that although we do
not perform the DBDM analysis using the SK data, we fully expect that such an
analysis can yield similar DBDM constraint as our projected 5-year HK curve,
due to the much longer exposure time compensating for the smaller volume. At
$m_{\chi}\simeq 1$ eV, the SN1987a bound is slightly better than the projected
DBDM sensitivity. This is because for DBDM, the sensitivity curve exhibits a
perfect scaling of $\sqrt{\sigma_{\chi\nu}\sigma_{\chi
e}}\propto\sqrt{m_{\chi}}$ for $m_{\chi}\lesssim 10$ MeV due to $N_{\rm
DBDM}\propto n_{\chi}\propto 1/m_{\chi}$. For the SN1987a curve, it is steeper
than $\sqrt{m_{\chi}}$ for lower $m_{\chi}$. The underlying reason is that for
the SN$\nu$ BDM, the duration for nonvanishing flux is proportional to
$m_{\chi}$, given that the SN distance is known Lin  _et al._ (2023b). For
lighter $m_{\chi}$, it results in lower total background number within the
considered exposure time. As a result, taking a smaller $m_{\chi}$ gives rise
to a better limit than the value inferred from a simple $\sqrt{m_{\chi}}$
scaling, and therefore leads to steeper dependence of
$\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}$ on $m_{\chi}$.

Figure 4: Map of the projected SN$\nu$ BDM events $N_{\chi}$ at HK for SNe at
different locations in the MW disk, taking $m_{\chi}=1$ keV and
$\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}=10^{-35}~{}{\rm cm^{2}}$ for the
scenarios considering DM spikes without self-annihilation (see text for
details). MW center is at the origin of the coordinates. The concentric
circles indicate the distances to the MW center in the unit of kpc. Earth is
located along the $0^{\circ}$ line outside the map. The cyan solid and dotted
contours denote the event number ratio $N_{\chi}^{\rm spike}/N_{\chi}^{\rm
w/o}$ between scenarios with and without DM spikes for $N_{\chi}^{\rm
spike}/N_{\chi}^{\rm w/o}=2$ and 1.5, respectively.

For the projected sensitivities from the next galactic supernova in MW, we
perform two sets of calculations as follows. This first set is to assume that
the SN is right at the MW center and consider up to 35 years333Note again that
the duration of having non-vanishing SN$\nu$ BDM flux is typically much less
than $\mathcal{O}(1)$ year for $m_{\chi}\lesssim\mathcal{O}(1)$ keV. of
exposure time with HK. Consistent with what shown in Fig. 2, Fig. 3 shows that
it can provide an improved sensitivity over the SN1987a and DBDM limits by
$\sim\mathcal{O}(10-100)$ for the case without DM spike or the scenario of
self-annihilating DM with spike. Without self-annihilation, the presence of DM
spike can hugely boost the projected sensitivity by another factor of
$\sim\mathcal{O}(100)$.

However, as inferred from the reevaluated SN1987a cases, the large enhancement
from DM spike is not expected for a SN occurring away from the MW center. To
quantify this, we compute the expected total numbers of SN$\nu$ BDM events for
SNe at different locations on the MW disk mid-plane for the scenario with DM
spike and $\langle\sigma v\rangle=0$ as well as that without DM spike. Fig. 4
shows the map of the SN$\nu$ BDM events taking $m_{\chi}=1$ keV and
$\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}=10^{-35}$ cm2 for the case with DM
spike. Also shown are the contours delineating constant event ratio of $1.5$
and $2$ between the two scenarios. This plot shows clearly that the DM spike
only plays a significant role if the next galactic SN occurs very close to the
MW center or right behind the center along the line of sight.

Given the expected event distribution, we can compute the expected average
SN$\nu$ BDM events, by integrating over the occurrence probability
distribution of galactic SNe (proportional to MW’s baryonic mass surface
distribution $\Sigma_{b}^{\rm MW}$ as used in Sec. II.2) for $R<8$ kpc. For
the background estimation, we conservatively take the largest exposure time
among all locations for each $m_{\chi}$, which is defined by the SN location
with $R=8$ kpc behind the MW center. These allow us to compute the “averaged”
projected sensitivities from the next galactic SN. The resulting limits are
also shown in Fig. 3, which confirm that most likely the very uncertain
properties of DM spikes will not affect the expected SN$\nu$ BDM sensitivity
from the next galactic SN.

Figure 5:  Comparison of the derived constraint and sensitivities on
$\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}$ from this work (thick solid lines)
for the scenario considering DM spike without self-annihilation to other
existing bounds on $\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}$ Jho  _et al._
(2021); De Romeri  _et al._ (2024), $\sigma_{\chi\nu}$ Akita and Ando (2023);
Cline and Puel (2023), and $\sigma_{\chi e}$ Cappiello and Beacom (2019); An
_et al._ (2018); Aprile  _et al._ (2019b, a); Abramoff  _et al._ (2019);
Agnese  _et al._ (2018).

###  III.2 Comparison with other constraints

Fig. 5 compares the SN1987a limit, the projected sensitivity from five-year of
DBDM search in HK, and the sensitivity from the next galactic SN at the MW at
center as well as the averaged one derived in earlier sections (with DM spikes
and $\langle\sigma v\rangle=0$), to the existing model-independent bounds on
$\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}$ from the DSNB BDM De Romeri  _et al._
(2024) and from the stellar-$\nu$ BDM Jho  _et al._ (2021), those on
$\sigma_{\chi e}$ from the cosmic-ray BDM Cappiello and Beacom (2019), solar
reflection An  _et al._ (2018), and from direct searches Aprile  _et al._
(2019b, a); Abramoff  _et al._ (2019); Agnese  _et al._ (2018), as well as
constraints on $\sigma_{\chi\nu}$ from the Lyman-$\alpha$ Akita and Ando
(2023) and blazar Cline and Puel (2023). We note that the latter two
categories do not directly constrain $\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}$
so the naive comparison with our results only makes sense when assuming
$\sigma_{\chi\nu}\sim\sigma_{\chi e}$. We would also like to caution that all
these constraints involve different energy scales in interactions, so the
conclusion shown in this plot cannot be directly carried over to scenarios
where the cross sections are energy dependent.

With the above cautions in mind, we see that both the SN$\nu$ BDM or the DBDM
clearly probe $\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}$ better than the DSNB
BDM and the stellar-$\nu$ BDM444The authors of Jho  _et al._ (2021) have shown
preliminary results in several workshops on improved constraints by
considering the stellar-$\nu$ BDM from earlier galaxies; see e.g., PPC . by
several orders of magnitudes. Assuming $\sigma_{\chi\nu}\sim\sigma_{\chi e}$,
DBDM and the SN$\nu$ BDM can also provide competitive or even dominating
bounds when compared to those directly probe $\sigma_{\chi e}$ for
$m_{\chi}\lesssim\mathcal{O}(0.1)$ MeV. As for the comparison with the
existing $\sigma_{\chi\nu}$ bounds, taking the surface values, the SN1987a and
the DBDM limit can readily give rise to complimentary constraints to the
blazar one, and further improvement are likely to be obtained from the next
galactic SN indicated by the MW averaged curve. In the most extreme case, if
the next galactic SN occurs very close to the MW center and if the DM spike
density can be largely enhanced in a way similar to the profile from taking
$\langle\sigma v\rangle=0$, then the SN$\nu$ BDM may independently provide
similar constraints on $\sigma_{\chi\nu}$ as that from the cosmological
consideration (Lyman $\alpha$), given $\sigma_{\chi\nu}\sim\sigma_{\chi e}$.

##  IV Summary and prospect

In this work, we have further explored the framework of utilizing SN$\nu$ BDM
to probe non-vanishing interaction between DM and SM leptons. For the first
time, we evaluate the present-day flux of DBDM, which represents the diffuse
component of SN$\nu$ BDM from galaxies at all redshifts. We have shown that
the presence of this intriguing component, conceptually similar to the DSNB,
can readily be used to derive the strongest model-independent bound on
$\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}$. Specifically, we have estimated that
with the consideration of five-year exposure time in a neutrino experiment
similar to HK, the resulting sensitivity will be better than the SN$\nu$ BDM
constraint based on SN1987a from Lin  _et al._ (2023a), probing
$\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}$ to the level of
$\mathcal{O}(10^{-36})$ cm2 for $m_{\chi}\lesssim\mathcal{O}(1)$ MeV. This
result implies that a reanalysis of Super-Kamiokande data accumulated over the
past three decades could allow to place a similarly leading limit on
$\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}$.

Figure 6:  MW DM profile for $m_{\chi}=1$ eV (left) and $100$ keV (right). The
solid lines indicate DM profiles containing spikes for scenario without
annihilation, the dashed lines for DM profiles including spikes but with
$\langle\sigma v\rangle=3\times 10^{-26}$ cm3 s-1, and the dotted lines the
original NFW profiles.

We have also considered the impact due to the presence of DM spikes around the
SMBHs in galaxies on DBDM and SN$\nu$ BDM from SN1987a and from the next
galactic SN. For the SN1987a SN$\nu$ BDM, we reevaluated the constraint by
taking into account the potentially small displacement of SN1987a from the LMC
center. Taking three specific scenarios of the spike profiles corresponding to
cases with and without DM self-annihilation, we have shown that both the DBDM
sensitivity and the reevaluated SN1987a bound are not sensitive to the
uncertain profile of DM spikes. For the next galactic SN, we have observed
that the presence of spike can significantly enhance the SN$\nu$ BDM flux and
the associated sensitivity only if the SN occurs very close to the MW center
or right behind the center along the line of sight. For most region on the MW
disk where the SN can occur, the resulting SN$\nu$ BDM sensitivity is also
insensitive to the presence of spike. Given these, the next galactic SN will
likely offer further improved sensitivity on
$\sqrt{\sigma_{\chi\nu}\sigma_{\chi e}}$ by another factor of 10–100 over the
SN1987a or the DBDM limits.

Our results highlight the rich phenomena and the significant discovery
potential associated with SN$\nu$ BDM. We expect that further improved limits
and bounds on specific particle physics models can be deduced, similarly to
what demonstrated in Ref. Lin  _et al._ (2023b). We leave such explorations to
future work.

##  Appendix A DM density profile with spike

The DM density in the presence of a spike in a galaxy has been extensively
exploited in Refs. Gondolo and Silk (1999); Ullio  _et al._ (2001); Cline and
Puel (2023) and can be conveniently parametrized by Cline  _et al._ (2023);
Cline and Puel (2023)

$\rho_{\chi}^{\rm spike}(r)=\begin{dcases}0,&r<4R_{S},\\\
\frac{\rho_{\alpha}(r)\rho_{c}}{\rho_{\alpha}(r)+\rho_{c}},&4R_{S}\leq
r<R_{\rm spike},\\\ \frac{\rho_{\chi}^{\rm NFW}(r)\rho_{c}}{\rho_{\chi}^{\rm
NFW}(r)+\rho_{c}},&r\geq R_{\rm spike},\end{dcases}$ (10)

where $r$ is the radius from the galactic center,
$\rho_{c}=m_{\chi}/(\langle\sigma v\rangle t_{\rm BH})$ is the saturation
density considering the effect of DM self-annihilation, $\langle\sigma
v\rangle$ denotes the thermally averaged DM annihilation cross section,
$R_{S}=2GM_{\rm BH}/c^{2}$ is the Schwarzschild radius of the central SMBH,
$M_{\rm BH}$ is the SMBH mass, and $t_{\rm BH}$ is the SMBH age. The function
$\rho_{\alpha}$ denotes the spike profile inside the radius $R_{\rm spike}$
without DM self-annihilation. For $\rho_{\alpha}$, the index parameter
$\alpha$ determines the slope of the spike profile in the inner halo. In Refs.
Cline  _et al._ (2023); Cline and Puel (2023), two values of $\alpha=7/3$ and
$3/2$ were assumed. $\alpha=7/3$ represents the original value proposed in
Refs. Gondolo and Silk (1999); Ullio  _et al._ (2001). However, the lower
dynamical relaxation resulting from gravitational scattering between DM and
stars may reduce $\alpha$ to $3/2$, leading to a less cuspy profile Gnedin and
Primack (2004). We conservatively take $\alpha=3/2$ in this work. The explicit
expressions of $\rho_{3/2}$ and the corresponding $R_{\rm spike}$ are given by
Cline and Puel (2023)

$\rho_{3/2}(r)\simeq\begin{cases}\rho_{N}\left(1-\frac{r_{i}}{r}\right)^{3}\left(\frac{r_{h}}{r}\right)^{3/2},&r_{i}\leq
r<r_{h}\\\ \rho_{N}^{\prime}\left(\frac{R_{\rm
spike}}{r}\right)^{7/3},&r>r_{h}\end{cases}$ (11)

where $\rho_{N}=\mathcal{N}r_{h}^{-3/2}$,
$\rho_{N}^{\prime}=\rho_{N}(r_{h}/R_{\rm spike})^{7/3}$, $r_{i}=4R_{S}$ and
$r_{h}=GM_{\rm BH}/\sigma_{\star}^{2}$ is the SMBH influence radius. The
stellar velocity dispersion $\sigma_{\star}$ in the galaxy bulge can be
estimated by McConnell  _et al._ (2011)

$\log_{10}\left(\frac{M_{\rm
BH}}{M_{\odot}}\right)=8.29+5.12\log_{10}\left(\frac{\sigma_{\star}}{200\,{\rm
km\,s^{-1}}}\right).$ (12)

The spike radius $R_{\rm spike}$ is given by

$R_{\rm
spike}=\left(\frac{\mathcal{N}}{\rho_{s}r_{s}}\right)^{3/4}r_{h}^{5/8},$ (13)

with the normalization constant

$\mathcal{N}=\frac{M_{\rm BH}}{4\pi[f_{3/2}(r_{h})-f_{3/2}(r_{i})]},$ (14)

where

$f_{\alpha}(r)\equiv
r^{-\alpha}\left(\frac{r^{3}}{3-\alpha}+\frac{3r_{i}r^{2}}{\alpha-2}-\frac{3r_{i}^{2}r}{\alpha-1}+\frac{r_{i}^{3}}{\alpha}\right).$
(15)

Without DM self-annihilation, $\rho_{\chi}^{\rm spike}(r)$ relaxes to the
conventional Navarro-Frenk-White profile Navarro  _et al._ (1996, 1997) at
$r>R_{\rm spike}$,

$\rho_{\chi}^{\rm NFW}(r)=\frac{\rho_{s}}{(r/r_{s})(1+r/r_{s})^{2}},$ (16)

where $r_{s}$ and $\rho_{s}$ characterize the scale radius and the density of
the halo. For self-annihilating DM, a nonzero $\langle\sigma v\rangle$ reduces
the spike density and produces a relatively cored profile that saturates at
$\rho_{c}$.

Numerical results of MW DM number density $n_{\chi}(r)=\rho^{\rm
spike}_{\chi}(r)/m_{\chi}$ for $m_{\chi}=1\,$eV (left) and $100\,$keV (right)
are shown in Fig. 6. Without annihilation, the spike profile simply takes the
form given by Eq. (11), which is independent of $m_{\chi}$ in $r<R_{\rm
spike}$. When taking non-zero $\langle\sigma v\rangle$, the DM self-
annihilation can significantly reduce the central DM density due to
annihilation, giving rise to cored profiles saturate at $\rho_{c}$. For
smaller $m_{\chi}$, the annihilation effect can also affect the radial range
outside $R_{\rm spike}$, illustrated by the left panel of the figure.

##  Appendix B Code availability

We provide a python package dukes on PyPI that can fully reproduce the DBDM
results in this work. See its project page duk  for installation and usage.

###### Acknowledgements.

We are grateful to useful discussions with Yongsoo Jho and Jong-Chul Park.
Y.-H. L. and M.-R. W. acknowledge supports from the National Science and
Technology Council, Taiwan under Grant No. 111-2628-M-001-003-MY4, the
Academia Sinica under Project No. AS-CDA-109-M11, and Physics Division,
National Center for Theoretical Sciences of Taiwan. We would like to also
acknowledge the use of computational resources provided by the Academia Sinica
Grid-computing Center (ASGC).

## References

  * Lin  _et al._ (2023a) Y.-H. Lin, W.-H. Wu, M.-R. Wu, and H. T.-K. Wong, Phys. Rev. Lett. 130, 111002 (2023a), arXiv:2206.06864 [hep-ph] .
  * Lin  _et al._ (2023b) Y.-H. Lin, T.-H. Tsai, G.-L. Lin, H. T.-K. Wong, and M.-R. Wu, Phys. Rev. D 108, 083013 (2023b), arXiv:2307.03522 [hep-ph] .
  * Aguilar  _et al._ (2015) M. Aguilar _et al._ (AMS), Phys. Rev. Lett. 115, 211101 (2015).
  * Akerib  _et al._ (2017a) D. S. Akerib _et al._ (LUX), Phys. Rev. Lett. 118, 021303 (2017a), arXiv:1608.07648 [astro-ph.CO] .
  * Ackermann  _et al._ (2017) M. Ackermann _et al._ (Fermi-LAT), Astrophys. J. 840, 43 (2017), arXiv:1704.03910 [astro-ph.HE] .
  * Akerib  _et al._ (2017b) D. S. Akerib _et al._ (LUX), Phys. Rev. Lett. 118, 251302 (2017b), arXiv:1705.03380 [astro-ph.CO] .
  * Agnese  _et al._ (2018) R. Agnese _et al._ (SuperCDMS), Phys. Rev. Lett. 121, 051301 (2018), [Erratum: Phys.Rev.Lett. 122, 069901 (2019)], arXiv:1804.10697 [hep-ex] .
  * Ambrosi  _et al._ (2017) G. Ambrosi _et al._ (DAMPE), Nature 552, 63 (2017), arXiv:1711.10981 [astro-ph.HE] .
  * Aprile  _et al._ (2018) E. Aprile _et al._ (XENON), Phys. Rev. Lett. 121, 111302 (2018), arXiv:1805.12562 [astro-ph.CO] .
  * Aprile  _et al._ (2019a) E. Aprile _et al._ (XENON), Phys. Rev. Lett. 123, 251801 (2019a), arXiv:1907.11485 [hep-ex] .
  * Aprile  _et al._ (2019b) E. Aprile _et al._ (XENON), Phys. Rev. Lett. 123, 241803 (2019b), arXiv:1907.12771 [hep-ex] .
  * Abramoff  _et al._ (2019) O. Abramoff _et al._ (SENSEI), Phys. Rev. Lett. 122, 161801 (2019), arXiv:1901.10478 [hep-ex] .
  * John  _et al._ (2023) I. John, R. K. Leane, and T. Linden, (2023), arXiv:2311.16228 [astro-ph.HE] .
  * Holdom (1986) B. Holdom, Phys. Lett. B 166, 196 (1986).
  * He  _et al._ (1991) X.-G. He, G. C. Joshi, H. Lew, and R. R. Volkas, Phys. Rev. D 44, 2118 (1991).
  * Davoudiasl  _et al._ (2012) H. Davoudiasl, H.-S. Lee, and W. J. Marciano, Phys. Rev. D 85, 115019 (2012), arXiv:1203.2947 [hep-ph] .
  * Battaglieri  _et al._ (2017) M. Battaglieri _et al._ , in  _U.S. Cosmic Visions: New Ideas in Dark Matter_ (2017) arXiv:1707.04591 [hep-ph] .
  * Chang  _et al._ (2018) J. H. Chang, R. Essig, and S. D. McDermott, JHEP 09, 051 (2018), arXiv:1803.00993 [hep-ph] .
  * Foldenauer (2019) P. Foldenauer, Phys. Rev. D 99, 035007 (2019), arXiv:1808.03647 [hep-ph] .
  * Escudero  _et al._ (2019) M. Escudero, D. Hooper, G. Krnjaic, and M. Pierre, JHEP 03, 071 (2019), arXiv:1901.02010 [hep-ph] .
  * Croon  _et al._ (2021) D. Croon, G. Elor, R. K. Leane, and S. D. McDermott, JHEP 01, 107 (2021), arXiv:2006.13942 [hep-ph] .
  * Lin and Lin (2021) G.-L. Lin and Y.-H. Lin, Phys. Rev. D 104, 063021 (2021), arXiv:2102.11151 [hep-ph] .
  * Workman  _et al._ (2022) R. L. Workman _et al._ (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Bringmann and Pospelov (2019) T. Bringmann and M. Pospelov, Phys. Rev. Lett. 122, 171801 (2019), arXiv:1810.10543 [hep-ph] .
  * Ema  _et al._ (2019) Y. Ema, F. Sala, and R. Sato, Phys. Rev. Lett. 122, 181802 (2019), arXiv:1811.00520 [hep-ph] .
  * Cappiello and Beacom (2019) C. V. Cappiello and J. F. Beacom, Phys. Rev. D 100, 103011 (2019), [Erratum: Phys.Rev.D 104, 069901 (2021)], arXiv:1906.11283 [hep-ph] .
  * Dent  _et al._ (2020) J. B. Dent, B. Dutta, J. L. Newstead, and I. M. Shoemaker, Phys. Rev. D 101, 116007 (2020), arXiv:1907.03782 [hep-ph] .
  * Wang  _et al._ (2020) W. Wang, L. Wu, J. M. Yang, H. Zhou, and B. Zhu, JHEP 12, 072 (2020), [Erratum: JHEP 02, 052 (2021)], arXiv:1912.09904 [hep-ph] .
  * Zhang (2022) Y. Zhang, PTEP 2022, 013B05 (2022), arXiv:2001.00948 [hep-ph] .
  * Guo  _et al._ (2020a) G. Guo, Y.-L. S. Tsai, and M.-R. Wu, JCAP 10, 049 (2020a), arXiv:2004.03161 [astro-ph.HE] .
  * Ge  _et al._ (2021) S.-F. Ge, J. Liu, Q. Yuan, and N. Zhou, Phys. Rev. Lett. 126, 091804 (2021), arXiv:2005.09480 [hep-ph] .
  * Cao  _et al._ (2021) Q.-H. Cao, R. Ding, and Q.-F. Xiang, Chin. Phys. C 45, 045002 (2021), arXiv:2006.12767 [hep-ph] .
  * Jho  _et al._ (2020) Y. Jho, J.-C. Park, S. C. Park, and P.-Y. Tseng, Phys. Lett. B 811, 135863 (2020), arXiv:2006.13910 [hep-ph] .
  * Cho  _et al._ (2020) W. Cho, K.-Y. Choi, and S. M. Yoo, Phys. Rev. D 102, 095010 (2020), arXiv:2007.04555 [hep-ph] .
  * Jaeckel and Yin (2021) J. Jaeckel and W. Yin, JCAP 02, 044 (2021), arXiv:2007.15006 [hep-ph] .
  * Lei  _et al._ (2020) Z.-H. Lei, J. Tang, and B.-L. Zhang, (2020), arXiv:2008.07116 [hep-ph] .
  * Guo  _et al._ (2020b) G. Guo, Y.-L. S. Tsai, M.-R. Wu, and Q. Yuan, Phys. Rev. D 102, 103004 (2020b), arXiv:2008.12137 [astro-ph.HE] .
  * Xia  _et al._ (2021) C. Xia, Y.-H. Xu, and Y.-F. Zhou, Nucl. Phys. B 969, 115470 (2021), arXiv:2009.00353 [hep-ph] .
  * Dent  _et al._ (2021) J. B. Dent, B. Dutta, J. L. Newstead, I. M. Shoemaker, and N. T. Arellano, Phys. Rev. D 103, 095015 (2021), arXiv:2010.09749 [hep-ph] .
  * Ema  _et al._ (2021) Y. Ema, F. Sala, and R. Sato, SciPost Phys. 10, 072 (2021), arXiv:2011.01939 [hep-ph] .
  * Flambaum  _et al._ (2020) V. V. Flambaum, L. Su, L. Wu, and B. Zhu, (2020), arXiv:2012.09751 [hep-ph] .
  * Jho  _et al._ (2021) Y. Jho, J.-C. Park, S. C. Park, and P.-Y. Tseng, (2021), arXiv:2101.11262 [hep-ph] .
  * Das and Sen (2021) A. Das and M. Sen, Phys. Rev. D 104, 075029 (2021), arXiv:2104.00027 [hep-ph] .
  * Bell  _et al._ (2021) N. F. Bell, J. B. Dent, B. Dutta, S. Ghosh, J. Kumar, J. L. Newstead, and I. M. Shoemaker, Phys. Rev. D 104, 076020 (2021), arXiv:2108.00583 [hep-ph] .
  * Chao  _et al._ (2021) W. Chao, T. Li, and J. Liao, (2021), arXiv:2108.05608 [hep-ph] .
  * Ghosh  _et al._ (2022) D. Ghosh, A. Guha, and D. Sachdeva, Phys. Rev. D 105, 103029 (2022), arXiv:2110.00025 [hep-ph] .
  * Feng  _et al._ (2022) J.-C. Feng, X.-W. Kang, C.-T. Lu, Y.-L. S. Tsai, and F.-S. Zhang, JHEP 04, 080 (2022), arXiv:2110.08863 [hep-ph] .
  * Wang  _et al._ (2021) W. Wang, L. Wu, W.-N. Yang, and B. Zhu, (2021), arXiv:2111.04000 [hep-ph] .
  * Xia  _et al._ (2022) C. Xia, Y.-H. Xu, and Y.-F. Zhou, JCAP 02, 028 (2022), arXiv:2111.05559 [hep-ph] .
  * Wang  _et al._ (2022) J.-W. Wang, A. Granelli, and P. Ullio, Phys. Rev. Lett. 128, 221104 (2022), arXiv:2111.13644 [astro-ph.HE] .
  * Abe  _et al._ (2023) K. Abe _et al._ (Super-Kamiokande), Phys. Rev. Lett. 130, 031802 (2023), arXiv:2209.14968 [hep-ex] .
  * Cui  _et al._ (2022) X. Cui _et al._ (PandaX-II), Phys. Rev. Lett. 128, 171801 (2022), arXiv:2112.08957 [hep-ex] .
  * Xu  _et al._ (2022) R. Xu _et al._ (CDEX), (2022), arXiv:2201.01704 [hep-ex] .
  * Granelli  _et al._ (2022) A. Granelli, P. Ullio, and J.-W. Wang, (2022), arXiv:2202.07598 [astro-ph.HE] .
  * Cline  _et al._ (2023) J. M. Cline, S. Gao, F. Guo, Z. Lin, S. Liu, M. Puel, P. Todd, and T. Xiao, Phys. Rev. Lett. 130, 091402 (2023), arXiv:2209.02713 [hep-ph] .
  * Xia  _et al._ (2023) C. Xia, Y.-H. Xu, and Y.-F. Zhou, Phys. Rev. D 107, 055012 (2023), arXiv:2206.11454 [hep-ph] .
  * Cappiello  _et al._ (2023) C. V. Cappiello, N. P. A. Kozar, and A. C. Vincent, Phys. Rev. D 107, 035003 (2023), arXiv:2210.09448 [hep-ph] .
  * Carenza and De la Torre Luque (2023) P. Carenza and P. De la Torre Luque, Eur. Phys. J. C 83, 110 (2023), arXiv:2210.17206 [astro-ph.HE] .
  * Wang  _et al._ (2023) W. Wang, W.-L. Xu, J. M. Yang, and R. Zhu, (2023), arXiv:2305.12668 [hep-ph] .
  * Adhikari  _et al._ (2023) G. Adhikari _et al._ (COSINE-100), (2023), arXiv:2306.00322 [hep-ex] .
  * Xia  _et al._ (2024) C. Xia, C.-Y. Xing, and Y.-H. Xu, (2024), arXiv:2401.03772 [hep-ph] .
  * Abe  _et al._ (2016) K. Abe _et al._ (Super-Kamiokande), Phys. Rev. D 94, 052010 (2016), arXiv:1606.07538 [hep-ex] .
  * Abe  _et al._ (2018) K. Abe _et al._ (Hyper-Kamiokande), (2018), arXiv:1805.04163 [physics.ins-det] .
  * Abi  _et al._ (2020) B. Abi _et al._ (DUNE), (2020), arXiv:2002.03005 [hep-ex] .
  * Abusleme  _et al._ (2022) A. Abusleme _et al._ (JUNO), Prog. Part. Nucl. Phys. 123, 103927 (2022), arXiv:2104.02565 [hep-ex] .
  * Horiuchi  _et al._ (2009) S. Horiuchi, J. F. Beacom, and E. Dwek, Phys. Rev. D 79, 083013 (2009), arXiv:0812.3157 [astro-ph] .
  * Beacom (2010) J. F. Beacom, Ann. Rev. Nucl. Part. Sci. 60, 439 (2010), arXiv:1004.3311 [astro-ph.HE] .
  * Gondolo and Silk (1999) P. Gondolo and J. Silk, Phys. Rev. Lett. 83, 1719 (1999), arXiv:astro-ph/9906391 .
  * Ullio  _et al._ (2001) P. Ullio, H. Zhao, and M. Kamionkowski, Phys. Rev. D 64, 043504 (2001), arXiv:astro-ph/0101481 .
  * Cline and Puel (2023) J. M. Cline and M. Puel, JCAP 06, 004 (2023), arXiv:2301.08756 [hep-ph] .
  * Gnedin and Primack (2004) O. Y. Gnedin and J. R. Primack, Phys. Rev. Lett. 93, 061302 (2004), arXiv:astro-ph/0308385 .
  * Navarro  _et al._ (1996) J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J. 462, 563 (1996), arXiv:astro-ph/9508025 .
  * Navarro  _et al._ (1997) J. F. Navarro, C. S. Frenk, and S. D. M. White, Astrophys. J. 490, 493 (1997), arXiv:astro-ph/9611107 .
  * Moster  _et al._ (2010) B. P. Moster, R. S. Somerville, C. Maulbetsch, F. C. v. d. Bosch, A. V. Maccio’, T. Naab, and L. Oser, Astrophys. J. 710, 903 (2010), arXiv:0903.4682 [astro-ph.CO] .
  * Girelli  _et al._ (2020) G. Girelli, L. Pozzetti, M. Bolzonella, C. Giocoli, F. Marulli, and M. Baldi, Astron. Astrophys. 634, A135 (2020), arXiv:2001.02230 [astro-ph.CO] .
  * Ferrarese (2002) L. Ferrarese, Astrophys. J. 578, 90 (2002), arXiv:astro-ph/0203469 .
  * Di Matteo  _et al._ (2003) T. Di Matteo, R. A. C. Croft, V. Springel, and L. Hernquist, Astrophys. J. 593, 56 (2003), arXiv:astro-ph/0301586 .
  * Baes  _et al._ (2003) M. Baes, P. Buyle, G. K. T. Hau, and H. Dejonghe, Mon. Not. Roy. Astron. Soc. 341, L44 (2003), arXiv:astro-ph/0303628 .
  * (79) https://www.astro.ucla.edu/~wright/CosmoCalc.html.
  * McMillan (2016) P. J. McMillan, Mon. Not. Roy. Astron. Soc. 465, 76 (2016), arXiv:1608.00971 [astro-ph.GA] .
  * Rozwadowska  _et al._ (2021) K. Rozwadowska, F. Vissani, and E. Cappellaro, New Astron. 83, 101498 (2021), arXiv:2009.03438 [astro-ph.HE] .
  * Conselice  _et al._ (2016) C. J. Conselice, A. Wilkinson, K. Duncan, and A. Mortlock, Astrophys. J. 830, 83 (2016).
  * Fontana  _et al._ (2004) A. Fontana _et al._ , Astron. Astrophys. 424, 23 (2004), arXiv:astro-ph/0405055 .
  * Fontana  _et al._ (2006) A. Fontana _et al._ , Astron. Astrophys. 459, 745 (2006), arXiv:astro-ph/0609068 .
  * Song  _et al._ (2016) M. Song _et al._ , The Astrophysical Journal 825, 5 (2016).
  * Yuksel  _et al._ (2008) H. Yuksel, M. D. Kistler, J. F. Beacom, and A. M. Hopkins, Astrophys. J. Lett. 683, L5 (2008), arXiv:0804.4008 [astro-ph] .
  * Panagia (2005) N. Panagia, Springer Proc. Phys. 99, 585 (2005), arXiv:astro-ph/0309416 .
  * (88) NASA/IPAC Extragalactic Database, https://ned.ipac.caltech.edu/.
  * Pietrzyński  _et al._ (2013) G. Pietrzyński _et al._ , Nature 495, 76 (2013), arXiv:1303.2063 [astro-ph.GA] .
  * (90) H. Boyce, N. Lützgendorf, R. P. van der Marel, H. Baumgardt, M. Kissler-Patig, N. Neumayer, and P. T. de Zeeuw, The Astrophysical Journal 846, 10.3847/1538-4357/aa830c.
  * Battistoni  _et al._ (2005) G. Battistoni, A. Ferrari, T. Montaruli, and P. R. Sala, Astropart. Phys. 23, 526 (2005).
  * Nakano (2017) Y. Nakano (Super-Kamiokande), J. Phys. Conf. Ser. 888, 012191 (2017).
  * Dutta and Strigari (2019) B. Dutta and L. E. Strigari, Ann. Rev. Nucl. Part. Sci. 69, 137 (2019), arXiv:1901.08876 [hep-ph] .
  * De Romeri  _et al._ (2024) V. De Romeri, A. Majumdar, D. K. Papoulias, and R. Srivastava, JCAP 03, 028 (2024), arXiv:2309.04117 [hep-ph] .
  * An  _et al._ (2018) H. An, M. Pospelov, J. Pradler, and A. Ritz, Phys. Rev. Lett. 120, 141801 (2018), [Erratum: Phys.Rev.Lett. 121, 259903 (2018)], arXiv:1708.03642 [hep-ph] .
  * Akita and Ando (2023) K. Akita and S. Ando, (2023), arXiv:2305.01913 [astro-ph.CO] .
  * (97) PPC 2023: XVI International Conference on Interactions between Particle Physics and Cosmology, https://indico.ibs.re.kr/event/540/.
  * McConnell  _et al._ (2011) N. J. McConnell, C.-P. Ma, K. Gebhardt, S. A. Wright, J. D. Murphy, T. R. Lauer, J. R. Graham, and D. O. Richstone, Nature 480, 215 (2011), arXiv:1112.1078 [astro-ph.CO] .
  * (99) GitHub: https://github.com/yenhsunlin/dukes and PyPI: https://pypi.org/project/dukes/.