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tokens: signatures associated with relativistic protons. In this study, we have demonstrated how high-energy neutrino observations may be used to assess p at SN shocks propagating in dense CSM. In particular, the IceCubes measurement of the diffuse neutrino flux already constrains p 0.3(0.04/) or p 0.2(0.04/) when including the up-going muon neutrino sample from the Northern Sky (Section 3.1 ). Even more stringent constraints on the acceleration efficiency may be placed by direct observations of close-by SNe IIn with neutrino telescopes (Section 3.2 ). The neutrino flux from an SN IIn depends on its distance, shock velocity, mass-loading parameter and proton acceleration efficiency (Section 3.2 ). All parameters, except for the latter, can be, in principle, inferred from radio and optical observations of SNe IIn. For example, v sh is routinely inferred from the width of optical emission lines, while K w can be derived by the fitting of radio observations (Chevalier 1998 ; Chevalier, Fransson Nymark 2006 ; Chandra etal. 2015 ). Besides the intrinsic source parameters that affect the neutrino luminosity, as exemplified through scenarios S1 and S2, the actual distance of the source has the strongest impact on the IceCube expected neutrino rate (see equation 12 ). The IceCube event number expected within the first year in both scenarios discussed in Section 3.2 is plotted in Fig. 9 (bottom panel) as a function of the source distance. Different types of lines correspond to three values of the proton acceleration efficiency marked on the plot. The curves are obtained after scaling the total number of muon neutrinos obtained in S1 and S2 (see Table 2 ) with the source distance and p . The top panel in Fig. 9 shows the all-sky number of SNe IIn expected over 8years up to a distance D L (black thick line); the number is derived using the CC SN rate of HB06 and 0.04. For p values allowed by the diffuse flux measurements, i.e. p 0.2(0.04/), only S1-like SNe IIn at distances 18Mpc are strong enough neutrino emitters to constrain less efficient acceleration scenarios, in agreement with the predictions by Murase etal. ( 2011 ). If p 0.2, IceCube would be capable of claiming a neutrino detection from an S1-like SN IIn exploding at 10Mpc. For SNe IIn with lower mass-loading parameter K w , as discussed in S2, the neutrino rate decreases significantly, even for p 0.2. This restricts the accessible distance to no more than 4Mpc. Figure 9. Bottom panel: Expected IceCube event number as a function of luminosity distance D L for the two SN scenarios discussed in the text (S1: blue lines; S2: cyan lines). For the second scenario, the number of neutrinos expected only within the first year is shown. For both scenarios, the results for three different values of the proton acceleration efficiency are presented (see the legend). The horizontal black and red lines indicate the number of neutrinos required to reach IceCubes sensitivity and 5 discovery potential,