It has been recently proposed that the boosted dark matter BDM by supernova neutrinos inlineformulammlmath displayinlinemmlmrowmmlmiSNmmlmimmlmimmlmimmlmrowmmlmathinlineformula from SN1987a or from the next Galactic supernova SN can serve as a novel component to probe nonvanishing interaction between dark matter DM and the standard model leptons Y.H. Lin et al. Phys. Rev. Lett. 130 111002 2023PRLTAO0031900710.1103PhysRevLett.130.111002 and Y.H. Lin et al. Phys. Rev. D 108 083013 2023PRVDAQ2470001010.1103PhysRevD.108.083013. In this Letter we extend this concept and evaluate the presentday diffuse flux of inlineformulammlmath displayinlinemmlmiSNmmlmimmlmimmlmimmlmathinlineformula BDM originated from all galaxies at higher redshifts. We show that by considering this diffuse BDM DBDM component the best sensitivity on the product of the energyindependent DMinlineformulammlmath displayinlinemmlmimmlmimmlmathinlineformula and DMelectron cross sections inlineformulammlmath displayinlinemmlmrowmmlmsqrtmmlmrowmmlmsubmmlmrowmmlmimmlmimmlmrowmmlmrowmmlmimmlmimmlmimmlmimmlmrowmmlmsubmmlmsubmmlmrowmmlmimmlmimmlmrowmmlmrowmmlmimmlmimmlmiemmlmimmlmrowmmlmsubmmlmrowmmlmsqrtmmlmommlmommlmi mathvariantscriptOmmlmimmlmo stretchyfalsemmlmommlmsupmmlmrowmmlmn10mmlmnmmlmrowmmlmrowmmlmommlmommlmn37mmlmnmmlmrowmmlmsupmmlmo stretchyfalsemmlmommlmtext mmlmtextmmlmtext mmlmtextmmlmsupmmlmrowmmlmicmmmlmimmlmrowmmlmrowmmlmn2mmlmnmmlmrowmmlmsupmmlmrowmmlmathinlineformula for subMeV DM can be obtained with largesize neutrino experiments such as SuperKamiokande or HyperKamiokande surpassing the estimated inlineformulammlmath displayinlinemmlmiSNmmlmimmlmimmlmimmlmathinlineformula BDM bound from SN1987a. We also examine the impact due to the presence of DM spikes around the supermassive black holes in galaxies on inlineformulammlmath displayinlinemmlmiSNmmlmimmlmimmlmimmlmathinlineformula BDM and DBDM. Our results suggest that both the DBDM and the inlineformulammlmath displayinlinemmlmiSNmmlmimmlmimmlmimmlmathinlineformula BDM probes are robust 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.
['Cosmology', 'Astrophysics', 'and Gravitation', 'High Energy Physics - Phenomenology', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena']
SupernovaNeutrinoBoosted Dark Matter from All Galaxies
{'Supernova-Neutrino-Boosted Dark Matter from All Galaxies': "Yen-Hsun Lin 1, ∗ and Meng-Ru Wu 1, 2, 3, †  \n1 Institute of Physics, Academia Sinica, Taipei 115, Taiwan  \n2 Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 106, Taiwan  \n3 Physics Division, National Center for Theoretical Sciences, Taipei 106, Taiwan  \nIt has been recently proposed that the boosted dark matter (BDM) by supernova neutrinos (SN ν ) from SN1987a or from the next Galactic supernova (SN) can serve as a novel component to probe nonvanishing interaction between dark matter (DM) and the standard model leptons [Y.-H. Lin et al., Phys. Rev. Lett 130 , 111002 (2023) and Y.-H. Lin et al., Phys. Rev. D 108 , 083013 (2023)]. In this Letter, we extend this concept and evaluate the present-day diffuse flux of SN ν BDM originated from all galaxies at higher redshifts. We show that by considering this diffuse BDM (DBDM) component, the best sensitivity on the product of the energy-independent DMν and DMelectron cross sections, √ σ χν σ χe ≃ O (10 -37 ) cm 2 for sub-MeV DM, can be obtained with large-size neutrino experiments such as Super-Kamiokande or Hyper-Kamiokande, surpassing the estimated SN ν 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 ν BDM and DBDM. Our results suggest that both the DBDM and the SN ν BDM probes are robust 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.  \nIntroductionDespite 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 and indirect detections [1-11]. Therefore, light DM with mass m χ ≲ O (MeV) has gained much attention recently, with a plethora of theoretical models being proposed and examined [12], including those allowing for DM interacting with leptons [12-26].  \nBoosted DM (BDM), when upscattered by known high-energy cosmic particles, offers a viable method to probe light DM in large underground detectors [27-63], such as Super-Kamiokande [64], Hyper-Kamiokande [65], DUNE [66] and JUNO [67]. In [68], the authors proposed that BDM from supernova neutrinos (SN ν ) 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 χ . The follow-up study [69] further explored the dependency of this probe on the location of the SN in the Galactic disk, and applied it to gauged U (1) L µ -L τ model.  \nIn this Letter, we further extend this concept by considering the contribution of the SN ν 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 ν background (DSNB) [70, 71] 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 HyperKamiokande (HK), it can readily offer improved model-  \ndependent sensitivities on the product of the DMν and DM-electron cross sections, surpassing the existing constraint from the SN ν BDM associated with SN1987a.  \nIn addition to the evaluation of DBDM flux and sensitivity, we also consider the consequence on SN ν BDM and DBDM due to the potential presence of the DM spike [72, 73] 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 ν 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 sensitivity for the case without DM self-annihilation.  \nAveraged SN ν BDM spectrumConsidering the same approach used in Ref. [68] to describe the DMν interaction, the energy spectrum of the total amount of SN ν 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  \ndN χ ( R ) dT χ = (2 π ) 2 τ ∫ d cos θd cos θ c dℓℓ 2 j χ ( ℓ, θ, θ c , T χ ) , (1)  \nwhere T χ is the BDM kinetic energy, τ = 10 s is the characteristic SN ν emission time, and j χ is the local BDM emissivity, which encodes the information of the DM halo profile in that galaxy and the DMν cross section σ χν assumed to be energy-independent and isotropic in centerof-mass frame. See [74] for parameter definitions and details.  \nAssuming supernovae (SNe) occur near the disk midplane, we can compute the averaged SN ν BDM spectrum by averaging over its baryonic mass distribution Σ b pro-  \ncted onto the plane by  \nd ¯ N dT χ = ∫ dA dN χ ( R ) dT χ Σ b ( R ) /∫ dA Σ b ( R ) , (2)  \nwhere the integrations are performed over the galaxy's differential area dA in the midplane.  \nAs galaxies of different sizes have different DM halo profiles and baryonic distributions, which affect the exact amount of SN ν 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 DM ≃ ηM G with η = 50 [75, 76]. Second, for the halo mass distribution without DM spike, we use the Navarro-Frenk-White (NFW) profile [77, 78], ρ NFW χ , Eq. (S13) in [74], with ρ s = 184 MeV cm -3 and r s = 24 . 2 kpc for a Milky Way (MW) size galaxy and M G = M MW = 5 . 29 × 10 10 M ⊙ . For other galaxies, we take a simple scaling relation r 3 s ∝ M G to compute the corresponding ρ NFW χ . The DM profile with spike is calculated following [79] For the DM spike density ρ spike χ that depends on the central SMBH mass and age, we apply the relation M BH ≈ 7 × 10 7 M ⊙ × [ M DM / (10 12 M ⊙ )] 4 / 3 [7982] for the mass. We assume that t BH can be approximated by the age of the Universe at redshift z , computed with [83] for the DBDM calculation. For the SN ν BDM from MW and SN1987a, we take t BH ≃ 10 Gyr. Third, for the surface baryonic density Σ b , we consider the bulge and the disk components given in [84] for a MW-size galaxy to compute the projected Σ MW b . 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 cut and R d in Eqs. (1) and (3) of [84], are proportional to the cubic root of the bulge mass and the square root of the disk mass, respectively, to estimate the corresponding Σ b . Given these assumptions, one can then use Eqs. (1)-(2) to compute d ¯ N/dT χ for any given M G of a galaxy. See [74] for discussions on DM halo and MW baryonic density profiles.  \nDBDM fluxKnowing the averaged SN ν BDM energy spectrum from a galaxy, the DBDM flux at redshift z = 0 consisting of the SN ν BDM contributions from all galaxies at different redshifts can be written as  \nd Φ χ dT χ = v χ H 0 ∫ z max 0 dz ε ( z ) ∫ dM G d Γ SN ( z ) dM G d ¯ N χ ( M G ) dT ' χ , (3)  \nwhere H 0 = c/ (4280 Mpc) is the Hubble constant, v χ = c √ T χ (2 m χ + T χ ) / ( m χ + T χ ) is the BDM velocity, ε ( z ) = [Ω m (1+ z ) 3 +Ω Λ ] 1 / 2 with (Ω m , Ω Λ ) ≈ (0 . 3 , 0 . 7) and T ' χ = (1 + z ) T χ is the BDM kinetic energy at the source. In Eq. (3), the term d Γ 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 ˙ ρ ∗ ( z ), and can be formulated as  \nd Γ SN ( z ) dM G = dn G ( z ) dM G ˙ ρ ∗ ( z ) ˙ ρ ∗ (0) M G M MW R SN , 0 , (4)  \nFIG. 1. The five-year DBDM fluence (red) and the total fluence of SN ν BDM from SN1987a (green) as well as from the next SN in the MW center (orange) for m χ = 1 keV and σ χν = 10 -35 cm 2 . The dotted, solid, and dashed lines denote different DM profile scenarios.  \n<!-- image -->  \nwhere R SN , 0 ≈ 0 . 01 yr -1 is the SN rate of the MW [85] and ˙ ρ ∗ ( z ) is the star formation rate per comoving volume [74]. The term dn G ( z ) /dM G represents the number density of galaxies per comoving volume per galaxy mass at z and can be parametrized as [86]  \ndn G ( z ) dm = ϕ 0 ln 10 × 10 ( m -M c )(1+ γ ) e -10 m -Mc , (5)  \nwhere M c is the characteristic mass in log-10 base, m = log 10 ( M G /M ⊙ ), ϕ 0 the normalization constant and γ the slope for fainter and lower mass galaxies. They are fitted to observational data at different z [74].  \nWe compute the DBDM flux given by Eq. (3) and show in Fig. 1 the corresponding fluence by multiplying the flux with an assumed exposure time t exp = 5 yr taking m χ = 1 keV and σ χν = 10 -35 cm 2 for cases considering the DM spikes with different thermally averaged DM self-annihilation cross section ⟨ σv ⟩ , 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 their spike radii.  \nFor comparison, we also compute the SN ν BDM flux using the same set of m χ and σ χν for a SN that explodes in the MW center following [68] 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, ∼ 1 . 75 kpc, of SN1987a from the LMC center [74].  \nThe resulting SN ν BDM fluences integrated over the entire duration before the BDM flux vanishes are also shown in Fig. 1 for SN1987a and for the MW center case. Without spikes, the 5-yr DBDM fluence can be comparable to the total SN ν BDMfluence from SN1987a, but several orders of magnitude smaller than that from the next galactic SN at the MW center. The energy spectrum of  \nFIG. 2. Reevaluated constraint from SN1987a SN ν 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.  \n<!-- image -->  \nDBDM is somewhat steeper than those from SN ν BDM, because the DBDM flux is dominated by the contribution at z ≃ 1 -2, which gets redshifted when arriving at the Earth. It is also evident that although the presence of the DM spike without self-annihilation can drastically enhance the SN ν BDMflux if the SN explodes at the center of a galaxy, a small displacement from the center like SN1987a will substantially reduce its effect. Moreover, a sizable DM annihilation rate ⟨ σv ⟩ = 3 × 10 -26 cm 3 s -1 can also largely undermine the impact of spike due to the suppressed spike densities.  \nSensitivityWithin an exposure time t exp , the DBDM events N DBDM in a neutrino or DM detector with N e electron targets for a nonvanishing interaction cross section σ χe between DM and the electrons can be calculated by  \nN DBDM = t exp × ∫ T max T min dT χ d Φ χ dT χ N e σ χe . (6)  \nTaking HK for instance, we set N e ≈ 7 . 3 × 10 34 , t exp = 5 yr and ( T min , T max ) = (5 , 100) MeV to estimate N DBDM . To achieve a 2 σ detection significance, it requires N DBDM > N s with N s satisfying 2 = N s / √ N s + N b and N b ≃ 5 . 8 × 10 5 , which is the estimated background events in HK in 5 yr also used in [69]. The background events are mainly composed of solar and atmospheric neutrinos, while other radioactive sources are subdominant [87-89]. For the reevaluated SN1987 bounds, we derive the corresponding 90% confidence level constraint by utilizing the data from Kamiokande and SK as in Ref. [68].  \nWe show in Fig. 2 the projected DBDM sensitivities and the reevaluated SN1987a limits on √ σ χν σ χe as functions of m χ for scenarios without DM spikes (red dotted),  \nwith DM spikes and ⟨ σv ⟩ = 0 (red solid), and with DM spikes and ⟨ σv ⟩ = 3 × 10 -26 cm 3 s -1 (red dashed), respectively. Similar to the fluence shown in Fig. 1 for m χ = 1 keV, the DBDM sensitivity and the SN1987a constraint here are nearly unaffected by the property of DM spikes for m χ ≳ 0 . 03 keV. For m χ ≲ 0 . 03 keV, selfannihilating DM slightly weakens both limits due to the reduced inner halo densities. For all scenarios, the projected 5-yr DBDM sensitivity from HK can clearly result in better constraint on √ σ χν σ χe than the SN1987a bound for most of the relevant m χ range. This is mainly because for SN1987a, nearly all or a significant amount of BDM arrives at 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 √ σ χν σ χe despite that the DM fluences are comparable (see Fig. 1). 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-yr HK curve, due to the much longer exposure time compensating for the smaller volume. At m χ ≃ 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 √ σ χν σ χe ∝ √ m χ for m χ ≲ 10 MeV due to N DBDM ∝ n χ ∝ 1 /m χ . For the SN1987a curve, it is steeper than √ m χ for lower m χ . The underlying reason is that for the SN ν BDM, the duration for nonvanishing flux is proportional to m χ , given that the SN distance is known [69]. For lighter m χ , it results in lower total background number within the considered exposure time. As a result, taking a smaller m χ gives rise to a better limit than the value inferred from a simple √ m χ scaling, and therefore leads to steeper dependence of √ σ χν σ χe on m χ .  \nFor 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 yr [90] of exposure time with HK. Consistent with what shown in Fig. 1, Fig. 2 shows that it can provide an improved sensitivity over the SN1987a and DBDM limits by ∼ O (10 -100) for the case without DM spike or the scenario of selfannihilating DM with spike. Without self-annihilation, the presence of DM spike can hugely boost the projected sensitivity by another factor of ∼ O (100).  \nHowever, 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 ν BDM events for SNe at different locations on the MW disk midplane for the scenario with DM spike and ⟨ σv ⟩ = 0 as well as that without DM spike. Fig. 3 shows the map of the SN ν BDM events taking m χ = 1 keV and √ σ χν σ χe = 10 -35 cm 2 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  \nFIG. 3. Map of the projected SN ν BDM events N χ at HK for SNe at different locations in the MW disk, taking m χ = 1 keV and √ σ χν σ χe = 10 -35 cm 2 for the scenarios considering DM spikes without self-annihilation. 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 · line outside the map. The cyan solid and dotted contours denote the event number ratio N spike χ /N w / o χ between scenarios with and without DM spikes for N spike χ /N w / o χ = 2 and 1.5, respectively.  \n<!-- image -->  \nor right behind the center along the line of sight.  \nGiven the expected event distribution, we can compute the expected average SN ν BDM events, by integrating over the occurrence probability distribution of Galactic SNe (proportional to MW's baryonic mass surface distribution Σ MW b ) for R < 8 kpc. For the background estimation, we conservatively take the largest exposure time among all locations for each m χ , 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. 2, which confirm that most likely the very uncertain properties of DM spikes will not affect the expected SN ν BDM sensitivity from the next Galactic SN.  \nFig. 4 compares the SN1987a limit, the projected sensitivity from 5-yr 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 ⟨ σv ⟩ = 0), to the existing bounds on √ σ χν σ χe from the DSNB BDM [91] and from the stellarν BDM [44], those on σ χe from the cosmic-ray BDM [29], solar reflection [93], and from direct searches [5, 8-10], as well as constraints on σ χν from blazar [79] and MW satellite and subhalo populations [92] (see also [94-99]). We note that the latter two categories do not directly constrain √ σ χν σ χe so the naive comparison with our re-  \nWith spike and  \nv  \n=0  \nFIG. 4. Comparison of the derived constraint and sensitivities on √ σ χν σ χe from this work for the scenario considering DM spike without self-annihilation to other existing bounds on √ σ χν σ χe [44, 91], σ χν [79, 92], and σ χe [5, 8-10, 29, 93].  \n<!-- image -->  \nsults only makes sense when assuming σ χν ∼ σ χe . 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.  \nWith the above cautions in mind, we see that both the SN ν BDM or the DBDM clearly probe √ σ χν σ χe better than the DSNB BDM and the stellarν BDM [100] by several orders of magnitudes. Assuming σ χν ∼ σ χe , DBDM and the SN ν BDM can also provide competitive or even dominating bounds when compared to those that directly probe σ χe for m χ ≲ O (0 . 1) MeV. As for the comparison with the existing σ χν bounds, taking the surface values, the SN1987a and the DBDM limit can readily give rise to complementary 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 ⟨ σv ⟩ = 0, then the SN ν BDM may independently provide similar constraints on σ χν as that from [79, 92, 94-99], given σ χν ∼ σ χe .  \nSummaryIn this Letter, we have further explored the framework of utilizing SN ν BDM to probe nonvanishing interaction between DM and leptons. For the first time, we evaluate the present-day flux of DBDM, which represents the diffuse component of SN ν 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 bound on √ σ χν σ χe . Specifically, we have estimated that with the consideration of 5-yr exposure time in a neutrino experiment similar to HK, the resulting sensitivity will be  \nbetter than the SN ν BDM constraint based on SN1987a from [68], probing √ σ χν σ χe to the level of O (10 -36 ) cm 2 for m χ ≲ O (1) MeV. This result implies that a reanalysis of SK data accumulated over the past three decades could allow to place a similarly leading limit on √ σ χν σ χe .  \nWe have also considered the impact due to the presence of DM spikes around the SMBHs in galaxies on DBDM and SN ν BDM from SN1987a and from the next Galactic SN. For the SN1987a SN ν BDM, we reevaluated the constraint by taking into account the potentially small displacement of SN1987a from the LMC center. 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(1) at B is, following the definition in [68],  \nj χ ( ℓ, θ, θ c , T χ ) = cn χ ( r ( ℓ )) dσ χν d Ω c dn ν dE ν ( dE ν dT χ v χ c ) , (S1)  \nwhere r 2 ( ℓ ) = ℓ 2 + R 2 -2 ℓR cos θ is the distance from B to GC, θ c is the scattering angle of the BDM in the centerof-mass (CM) frame and v χ /c = √ T χ (2 m χ + T χ ) / ( m χ + T χ ) is the BDM velocity. In Eq. (S1), n χ ( r ) = ρ χ ( r ) /m χ is the DM number density at B with ρ χ ( r ) the DM mass density (discussed in the next section), and the differential DMν interaction cross section in the CM frame dσ χν /d Ω c = σ χν / 4 π is assumed to be isotropic in the CM frame where d Ω c is the differential solid angle. For the neutrino number density within the SN ν shell, dn ν /dE ν , we use the same form as in [68, 69],  \ndn ν dE ν = ∑ i L ν i 4 πℓ 2 ⟨ E ν i ⟩ E 2 ν f ν i ( E ν ) , (S2)  \nwhere L ν i = L ν, tot / 6 is the luminosity of each flavor ( ν e , ν µ , ν τ and their antineutrinos). The average energy ⟨ E ν e ⟩ , ⟨ E ¯ ν e ⟩ , and ⟨ E ν x ⟩ ( ν x ∈ { ν µ , ν τ , ¯ ν µ , ¯ ν τ } ) are taken to be 11, 16, 25 MeV, respectively [102]. We assume a Fermi-Dirac distribution f ν i with a pinch parameter η ν i ≡ µ ν i /T ν i = 3, such that T ν i ≈ ⟨ E ν i ⟩ / 3 . 99 [102]. The term dE ν /dT χ in Eq. (S1) is the Jacobian factor that converts dE ν to dT χ ,  \ndE ν dT χ = 1 2 c 2 ( 1 + 1 + c 2 m χ /T χ √ 2 c 2 m χ /T χ +1 ) , (S3)  \nwith c 2 = cos 2 ( θ c / 2).', 'B. Milky Way baryon density profile': "We follow [84] to model the Milky Way (MW) stellar density distribution by considering the bulge component ρ b ,  \nρ b ( R,z ) = ρ 0 ,b [1 + r ' ( R,z ) /r 0 ] α exp [ -r ' 2 ( R,z ) r 2 cut ] (S4)  \nand the stellar disc component ρ d ,  \nρ d ( R,z ) = Σ 0 2 z d exp ( -| z | z d -R R d ) , (S5)  \nwhere R and z are the radial distance and height to MW center in cylindrical coordinate. For the bulge, r ' = √ R 2 +( z/q ) 2 with α = 1 . 8, r 0 = 0 . 075 kpc, r cut = 2 . 1 kpc, q = 0 . 5 and ρ 0 ,b = 9 . 93 × 10 10 M ⊙ kpc -3 . For the stellar disc component, it can be further divided into thin and thick ones. The corresponding parameters are listed in Table S1. We ignore the contribution from gas as it is insignificant when estimating the average SN position. MW  \nTo obtain the MW area density Σ b ( R ), we integrate over the vertical height z by  \nΣ MW b ( R ) = 2 ∫ z max 0 dz [ ρ b ( R,z ) + ρ d ( R,z )] (S6)  \nwhere factor 2 accounts for symmetric ρ b,d ( R,z ) under z → -z and Σ MW b ( R ) is insensitive to the choice of z when z ≫ z max . In this work, we have z max = 10 kpc.", 'SN ν shell': 'FIG. S1. Schematic plot of DM boosted by SN ν within an expanding spherical shell. The SN occurs at S , the galactic center is located at G while DM is boosted at B .  \n<!-- image -->  \nTABLE S1. Values for parameters describing stellar dics.', 'II. DM DENSITY PROFILE WITH SPIKE': "The DM density in the presence of a spike in a galaxy has been extensively exploited in [72, 73, 79] and can be conveniently parametrized by [57, 79]  \nρ spike χ ( r ) =              0 , r < 4 R S , ρ α ( r ) ρ c ρ α ( r ) + ρ c , 4 R S ≤ r < R spike , ρ NFW χ ( r ) ρ c ρ NFW χ ( r ) + ρ c , r ≥ R spike , (S7)  \nwhere r is the radius from the galactic center, ρ c = m χ / ( ⟨ σv ⟩ t BH ) is the saturation density considering the effect of DM self-annihilation, ⟨ σv ⟩ denotes the thermally averaged DM annihilation cross section, R S = 2 GM BH /c 2 is the Schwarzschild radius of the central SMBH, M BH is the SMBH mass, and t BH is the SMBH age. The function ρ α denotes the spike profile inside the radius R spike without DM self-annihilation. For ρ α , the index parameter α determines the slope of the spike profile in the inner halo. In [57, 79], two values of α = 7 / 3 and 3 / 2 were assumed. α = 7 / 3 represents the original value proposed in [72, 73]. However, the lower dynamical relaxation resulting from gravitational scattering between DM and stars may reduce α to 3 / 2, leading to a less cuspy profile [103]. We conservatively take α = 3 / 2 in this work. The explicit expressions of ρ 3 / 2 and the corresponding R spike are given by [79]  \nρ 3 / 2 ( r ) ≃        ρ N ( 1 -r i r ) 3 ( r h r ) 3 / 2 , r i ≤ r < r h ρ ' N ( R spike r ) 7 / 3 , r > r h (S8)  \nwhere ρ N = N r -3 / 2 h , ρ ' N = ρ N ( r h /R spike ) 7 / 3 , r i = 4 R S and r h = GM BH /σ 2 ⋆ is the SMBH influence radius. The stellar velocity dispersion σ ⋆ in the galaxy bulge can be estimated by [104]  \nlog 10 ( M BH M ⊙ ) = 8 . 29 + 5 . 12 log 10 ( σ ⋆ 200 kms -1 ) . (S9)  \nThe spike radius R spike is given by  \nR spike = ( N ρ s r s ) 3 / 4 r 5 / 8 h , (S10)  \nwhere  \nf α ( r ) ≡ r -α ( r 3 3 -α + 3 r i r 2 α -2 -3 r 2 i r α -1 + r 3 i α ) . (S12)  \nWithout DM self-annihilation, ρ spike χ ( r ) relaxes to the conventional Navarro-Frenk-White profile [77, 78] at r > R spike ,  \nρ NFW χ ( r ) = ρ s r r s (1 + r r s ) 2 , (S13)  \nwhere r s and ρ s characterize the scale radius and the density of the halo. For self-annihilating DM, a nonzero ⟨ σv ⟩ reduces the spike density and produces a relatively cored profile that saturates at ρ c .  \nNumerical results of MW DM number density n χ ( r ) = ρ spike χ ( r ) /m χ for m χ = 1eV (left) and 100 keV (right) are shown in Fig. S2. Without annihilation, the spike profile simply takes the form given by Eq. (S8), which is independent of m χ in r < R spike . When taking non-zero ⟨ σv ⟩ , the DM self-annihilation can significantly reduce the central DM density, giving rise to cored profiles saturate at ρ c . For smaller m χ , the annihilation effect can also affect the radial range outside R spike , illustrated by the left panel of the figure.  \nWe note that even without a spike, the DM density in the inner halo region could be shallower than the NFW profile, as discussed in [105]. However, we expect that when taking into account the average over area density for the case of DBDM or the averaged MW SN ν sensitivity, the exact shape of the inner halo has minimal impact on our result.", 'III. ˙ ρ ∗ ( z ) AND FITTED VALUES FOR dn G /dm': 'We take the star formation rate per comoving ˙ ρ ∗ ( z ) in Eq. (6) by  \n˙ ρ ∗ ( z ) = ˙ ρ 0 [ (1 + z ) aη + ( 1 + z B ) bη + ( 1 + z C ) cη ] 1 /η (S14)  \nfrom [106], with ˙ ρ 0 = 0 . 0178 M ⊙ yr -1 Mpc -3 , ( a, b, c, η ) = (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. For the values of parameters used in dn G /dm , Eq. (7) of the main text, they are documented in the Table I of [86] and listed explicitly in Table S2.  \n<!-- image -->  \nFIG. S2. MWDMprofile for m χ = 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 ⟨ σv ⟩ = 3 × 10 -26 cm 3 s -1 , and the dotted lines the original NFW profiles.  \n<!-- image -->  \nwith the normalization constant  \nN = M BH 4 π [ f 3 / 2 ( r h ) -f 3 / 2 ( r i )] , (S11)  \nTABLE S2. Fitted values for parameters in Eq. (7) of the main text.', 'IV. SN1987A AND LMC': "To estimate the SN ν BDM coming from SN1987a, we acquire more accurate positioning data from [110]. For SN1987a, it occurs at 51.4 kpc [111] away with right ascension (RA) of 05 h 35 m 27 . 8733 s and declination (DEC) of -69 · 16 ' 10 . 478 '' [110]. The center of LMC was estimated to be around 49.97 kpc away [112], with RA of 05 h 23 m 34 . 5264 s , and DEC of -69 · 45 ' 22 . 053 '' [110]. The displacement between the two objects is approximate 1.75 kpc, however, the associated uncertainties are relatively large so that the SN1988a can be much closer to the LMC center. Furthermore, the LMC's spike profile is calculated using Eq. (S7) by taking M BH = 10 7 M ⊙ [113] and replacing ρ NFW χ with the Hernquist halo profile [68, 114]  \nρ Hernquist χ ( r ) = ρ s r r s (1 + r r s ) 3 , (S15)  \nwhere r s = 31 . 9 kpc and ρ s = 68 MeVcm -3 .", 'V. CODE AVAILABILITY': 'We provide a python package dukes on PyPI that can fully reproduce the DBDM results in this work. See its project page [115] for installation and usage.'}