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Blazar constraints on neutrino-dark matter scattering

Neutrino emission in coincidence with gamma rays has been observed from the blazar TXS 0506+056 by the IceCube telescope. Neutrinos from the blazar had to pass through a dense spike of dark matter (DM) surrounding the central black hole. The observation of such a neutrino implies new upper bounds on the neutrino-DM scattering cross section as a function of DM mass. The constraint is stronger than existing ones for a range of DM masses, if the cross section rises linearly with energy. For constant cross sections, competitive bounds are also possible, depending on details of the DM spike.


INTRODUCTION
The possible interactions of dark matter (DM) with ordinary matter have been constrained in many ways. The most challenging category is DMneutrino interactions, due to the difficulty of observing neutrinos. A promising strategy is to consider astrophysical sources of high-energy neutrinos, that could accelerate light DM particles to energies that would make them detectable in ground-based DM and neutrino search experiments [1][2][3]. This only works if, in addition to DM-ν interactions, there can also be scattering of DM from nuclei or electrons in the detector.
A more model independent strategy is to use the fact that a high energy neutrino has been observed by the IceCube experiment, which is identified as coming from the blazar TXS 0506+056 [4]. Ref. [5] set limits on the DM-ν scattering cross section using the fact that the neutrino had to pass through cosmological and galactic DM between the blazar and the Earth. In this work, we derive stronger limits, using the fact that the neutrino had to also traverse the dense DM spike surrounding the supermassive black hole powering TXS 0506+056.
IceCube has also observed a high-energy neutrino coming from the direction of another blazar, PKS 502+106. Ref. [6] finds that the observation could be consistent with theoretical expectations of neutrino flux from the blazar, but we do not include this data in our considerations. In addition, IceCube reported a statistical excess of lower energy neutrinos prior the 2017 flare of TXS 0506+056 [7], but the claimed excess is too large to be explained by theoretical blazar models [8][9][10][11], and we do not include it in the present analysis.

EXPECTED NEUTRINO EVENTS
We start by describing the theoretical models of neutrino emission from blazars and the expected flux from TXS 0506+056. The observed spectra of electromagnetic emission from blazars is well described by lepto-hadronic models [10][11][12][13][14], in which protons and electrons are shock-accelerated to create a relativistic jet, in a magnetized region that produces synchrotron radiation. The jet extends to distances ∼ 10 11 km [11,13], around 1000 times smaller than the extent of the DM spike to be described in Section 3. Proton-photon interactions in the jet produce pions, whose decays are the source of high-energy neutrinos.
Purely hadronic models are also able to fit the combined electromagnetic spectra at optical, X-ray and gamma-ray frequencies, but they lead to a neutrino flux which is too small to explain the observed IC-170922A event [13]; hence we focus on the leptohadronic models. Under the steady state approximation, Ref. [13] predicts a neutrino flux between E ν ∼ 100 TeV and 10 EeV, that peaks at a value E ν ∼ 10 PeV, which is orders of magnitude higher than IC-170922A. We find that the probability of observing a neutrino with energy 300 TeV is ∼ 3 % in this model. Hence we consider it to be disfavored for explaining IC-170922A.
On the other hand, the neutrino flux predicted by Ref. [11], based on a fully time-dependent approach, peaks near E ν = 100 TeV, and is compatible with the observation. It is well-fit by the formula with F 0 = 13.22, F 1 = 1.416, F 2 = −0.00262, F 3 = 3.826, and x = log 10 (E ν /TeV) ∈ [−2.2, 3.7]. The expected number of neutrino events observed at Ice-Cube is given by where t obs is the time interval of observation, Φ ν is the predicted neutrino flux from the blazar, and A eff is an effective area for detection, which depends on the geometry of the source direction and E ν , and encodes the probability for a neutrino to convert to a muon through weak interactions. Data for A eff from TXS 0506+056 is provided by Ice-Cube [15]. 1 For the campaign IC86c during JD (Julian day) 57161 − 58057 that observed the event IC-170922A, t obs = 898 d, and the reconstructed energy was E ν = 290 TeV for the event. We find that N pred = 2.6 from the flux (1), in rough agreement with the observed event. We adopt this as the input model for constraining the DM-ν cross section below.

DARK MATTER SPIKE
Since the overdensity of DM surrounding the central black hole plays a crucial role in our result, we review in this section what is known about it. The possibility of adiabatic accretion of DM around the black hole (BH) was first considered by Gondolo and Silk in Ref. [16]. They derived an inner radius for the spike of r i = 4R S , where R S = 2GM BH is the BH Schwarzschild radius, and an outer profile ρ (r) = N r −α with α = (9−2γ)/(4−γ) ∈ [2.25, 2.5], depending on the inner cusp of the initial DM halo density, ρ ∼ r −γ , with 0 ≤ γ ≤ 2. The normalization N of ρ can be determined using the finding that the mass of the spike is of the same order as M BH [17], 4πN ro ri dr r 2 ρ ∼ = M BH , within a radius of typical size r o ∼ = 10 5 R S [18]. The BH mass of the blazar TXS 0506+056 is estimated to be 3.09×10 8 M [19]. In Ref. [20], it was argued that gravitational scattering of DM with stars in the central region would lead to dynamical relaxation to a less cuspy profile with α = 3/2; hence we also consider this possibility below.
The spike density is reduced relative to these initial profiles if there is subsequent DM annihilation, leading to a maximum density of ρ c = m χ /( σ a v t BH ), where m χ is the DM mass, σ a v is an effective annihilation cross section, and t BH is the age of the BH. The spike density then becomes ρ χ = ρ c ρ /(ρ c + ρ ). The quantity σ a v is "effective" in the sense that it could be negligible even if the actual annihilation cross section is large. This would be the case for asymmetric dark matter, in which the symmetric component has completely annihilated away in the early universe. Then annihilations would have no effect at later times, when the DM spike is formed. To illustrate the range of possible outcomes from varying σ a v , we follow Ref. [3] by considering three benchmark models BM1-BM3, in which σ a v = (0, 0.01, 3) × 10 −26 cm 3 /s, respectively, and t BH = 10 9 yrs. These models assumed α = 7/3 in ρ ∼ r −α . We also consider models BM1 -BM3 using the less cuspy value α = 3/2.
The important quantity for neutrinos to scatter from DM in the spike is the DM column density, ( We have recomputed it for the benchmark values, using the above formulas. One finds that Σ/m χ can be accurately fit by a power law, Σ/m χ = A Σ (MeV/m χ ) BΣ , with B Σ = 1 for the case of σ a v = 0, and a fractional power when annihilation occurs. We list the parameters A Σ , B Σ for the   benchmark models in table I. 2 Although the DM spike does not extend to arbitrary distances, the integral in (3) converges around 10 R S in the case of no DM annihilation, and at larger radii ∼ (10 5 −10 7 ) R S for the cases with annihilation.

NEUTRINO ATTENUATION BY DM
One can make an initial estimate for the maximum DM-ν scattering cross section σ νχ as being inverse to the column density Σ/m χ of the DM spike surrounding the central BH of TXS 0506+056. To be more quantitative, we recompute the expected number of IceCube events from the 2017 flare that led to the observed event, taking into account the attenuation from scattering on DM. The analogous computation for scattering of neutrinos by galactic DM has been considered in Ref. [31]. The evolution of the flux due to scattering is described by a form of  Fig. 1. The displayed bounds are: (cyan) CMB and baryon acoustic oscillations [21], (pink) Lyman-α preferred model [22], (dark violet, blue) diffuse supernova neutrinos [2], (orange) stellar neutrinos [23], (yellow) supernova SN1987A [24], (green) previous bound on TXS 0506+056 by IceCube [5], (slate blue) solar reflection [25], (brown) Super-K for DM boosted by cosmicray electrons, (turquoise) blazar BL Lacertae for BM3 model [3], (gold) muon-DM scattering in neutron stars which is stronger than e-DM contribution in the region of interest and less model-dependent [26], (gray) direct detection for light DM interacting with electrons [27][28][29][30]. See text for description of previous limits. The limits labelled with involve only DM-electron scattering.
the Boltzmann equation known as cascade equation, where τ = Σ(r)/m χ = r dr ρ χ /m χ is the accumulated column density. The second term represents the effect of neutrino energies being redistributed, rather than simply being lost from the beam. To proceed, we must make an assumption about the energy dependence of the cross section. In section 6 we will discuss particle physics models that predict σ νχ (E ν ). A particularly simple and wellmotivated choice is linear energy-dependence, taking the reference energy E 0 = 290 TeV to be that of the observed event. Approximating the scattering as being isotropic in the center of mass frame, one can show that dσ νχ /dE ν = σ νχ /E ν = σ 0 /E 0 . The cascade equation can be discretized, choosing equal logarithmic intervals ∆x in x = log 10 (E ν /TeV). Defining a dimensionless column density y = (m χ /Σ χ ) τ , it takes the form where A = (Σ χ /m χ )(σ 0 /Ê 0 ),Ê i = 10 xi is the energy in TeV units,Ê 0 = 290 and y ∈ [0, 1]. To solve Eq. (6), one can either evolve the initial condition from y = 0 to y = 1 by incrementing in y, or use the algorithm presented in Ref. [31]. We have checked that both methods give the same results, obtaining the conservative limit by demanding the number of events giving a neutrino of energy E ν ≥ 290 TeV be greater than 0.1. The corresponding constraints in the plane of σ 0 versus m χ are plotted in Fig. 1 for the six DM spike benchmark models. The constraint (7) can be expressed as σ 0 < 1.5 m χ /Σ χ , in agreement with the initial estimate. We reiterate that σ 0 is defined as the cross section at the reference energy E 0 , assuming that σ νχ rises linearly with energy. The effects of relaxing this assumption are considered in Section 6.

COMPARISON TO PREVIOUS LIMITS
Limits on the elastic scattering cross section for DM and neutrinos are harder to derive than for other DM scatterings with visible matter because of the weakly interacting nature of neutrinos. A modelindependent way to probe such interactions is to look for suppression in the primordial density fluctuations at temperatures ∼ 1 eV, which would produce detectable effects in the cosmic microwave background (CMB) and matter power spectrum [21,22,[32][33][34][35]. For a constant scattering cross section, Ref. [35] derived a limit of σ νχ 10 −36 (m χ /MeV) cm 2 for massless neutrinos, which becomes weaker by about five orders of magnitude if a neutrino mass of ∼ 0.06 eV is properly included [21]. A more recent analysis using Lyman-α forest data found a mild preference for DM interacting with massive neutrinos, which requires confirmation [22].
Besides its effect on cosmology, DM-ν scattering can also be probed in direct detection experiments and neutrino observatories, if further assumptions about the DM interaction with either leptons or nucleons are made. A prominent example involves boosting DM within our galaxy by astrophysical neutrinos such as those coming from stars [23,36], diffuse supernovae [2,[37][38][39] or from supernova SN1987A [24], leading to larger energy deposition than could occur for light DM particles. Alternative ways to probe DM scattering with neutrinos is via attenuation of neutrino fluxes from supernovae [32,40] and the galactic centre [41], delayed neutrino propagation [42], and through effects in the extragalactic distribution and spectra of PeV neutrinos [43,44]. Fig. 2 shows a compilation of the most stringent bounds on σ νχ after rescaling them to the common energy scale E 0 = 290 TeV, assuming eq. (5). Here we include also constraints on DM-electron scattering, since it is natural for neutrinos and electrons to interact with DM with the same strength, as discussed in the next section. DM-e scattering can be probed in a variety of ways. It would alter the CMB anisotropies, the shape of the matter power spectrum and the abundance of Milky-Way satellites [45][46][47], cause CMB spectral distortions [48,49], and heat or cool the gas in dwarf galaxies [50]. Similarly to the neutrino case, DM particles can be boosted by cosmic rays , particles in the solar interior [25] or in the relativistic jets of blazars [1,3] and be directly detected. Standard direct detection constraints on light DM particles can apply [27][28][29][30][72][73][74][75][76][77][78][79][80][81]. DM-electron scattering can alter the cosmic ray spectrum [82], and heat neutron stars [26,83,84] or white dwarfs [85].
The new blazar limits on σ νχ shown in Fig. 2, assuming σ νχ ∝ E ν , are several orders of magnitude stronger than existing ones for sub-GeV DM, when the latter are rescaled to the blazar neutrino energy. In the case of light mediators that could lead to a constant-in-energy cross section, we lose this advantage, as shown in Fig. 1, except in the most optimistic case of the spike model BM1.

PARTICLE PHYSICS MODELS
The simplest models for DM-ν scattering involve the exchange of a vector boson Z between DM and the standard model lepton doublets L i , where i is the flavor index. 3 We assume flavor universality, and a universal coupling g between the mediator and χ or L i . As an example, we consider the DM to be a complex scalar, although by dimensional analysis, the results are expected to be insensitive to the spin of the DM. (Exact expressions for σ(E) in various models can be found in the appendix of Ref. [31].) At energies E ν m χ , the cross section goes as where s ∼ = 2m χ E ν , equally for scattering of neutrinos or electrons on DM. For m 2 Z > m χ E ν 1 GeV 2 (considering m χ as low as 1 keV), σ νχ rises linearly with E ν by expanding the logarithm to second order in s/m 2 Z , while for E ν m 2 Z /m χ , σ νχ saturates to a constant value. The corresponding differential cross section that appears in the second term of the cascade equation (4) is 3 To cancel gauge anomalies, the Z should also couple to right-handed leptons and neutrinos. A scalar-mediated interaction would require insertions of the Higgs field to satisfy SU(2) L gauge invariance, which would lead to inelastic scattering involving Higgs bosons at high energy.  This model is similar to that in Eq. (5) in having σ νχ ∝ E ν at low energy, but it is physically distinct because the differential scattering implied by (9) is not isotropic. One can show that its behavior in the cascade equation is determined by just two (dimensionless) parameters, that we take to be With this choice, A plays the same role of A in Eq. (6) in the low-energy regime where σ νχ ∼ g 4 m χ E ν /(4πm 4 Z ). By solving the cascade equation on a grid of values in the A -B plane, again demanding at least 0.1 predicted IceCube events above 290 TeV, we obtain the constraint shown in Fig. 3. For small B 0.1, the model is in the regime of σ νχ ∼ E ν , and the constraint is given by A < 0.0072. As B increases, the cross section transitions to its large-E ν constant behavior, and the limit on A relaxes. We translate the A versus B bound into the microscopic model parameters, g 2 versus m Z in Fig. 4, for some choices of the DM spike models and DM masses.

SUMMARY AND CONCLUSIONS
It is not disputed that dark matter accumulates in the vicinity of supermassive black holes that power active galactic nuclei, but there are significant uncertainties from astrophysics and from the unknown annihilation properties of the dark matter, in the density of the ensuing DM spike. Despite the orders of magnitude uncertainty, we find strong constraints on the elastic scattering cross section σ νχ for DM-neutrino scattering, so long as the IceCube event IC-170922A indeed came from the blazar TXS 0506+056 during its 2017 flare, as is widely believed.
Since the single event has a unique neutrino energy E 0 , our constraint applies to σ νχ at that energy. A natural hypothesis is that such interactions arise from exchange of a massive mediator, which leads to the prediction of linear energy dependence, σ νχ = σ 0 E ν /E 0 at sufficiently low energies. Under that assumption, we compared our limit to previous ones in the literature, which are set at much lower energies.
Even in the least optimistic case (models BM3-BM3 ), our limits improve on the existing ones by several orders of magnitude, for sub-GeV DM masses. The stronger of our constraints (BM1-BM1 ) are likely to be applicable in the case of asymmetric DM, where the effective annihilation cross section is essentially zero, due to the negligible proportion of a symmetric component that is necessary to have annihilation. Our constraints are weakened if the mediator mass is sufficiently small, which causes the cross section to stop rising with energy at a scale of order m 2 Z /m χ , becoming constant at higher energies, and thereby reducing the leverage of our bound coming from the 290 TeV scale.
A further natural assumption, motivated by SU(2) L gauge symmetry in the standard model, is that charged leptons should have an equal cross section with DM relative to neutrinos, allowing us to compare to existing electron-DM scattering constraints. Here too our constraints improve on previous limits, for linearly rising cross sections.
We look forward to future observations by neutrino telescopes that may confirm the multimessenger signals from blazars, and perhaps lead to refined constraints on lepton-DM scattering.