Optically informed searches of high-energy neutrinos from interaction-powered supernovae

ABSTRACT

The interaction between the ejecta of supernovae (SNe) of Type IIn and a dense circumstellar medium can efficiently generate thermal ultraviolet/optical radiation and lead to the emission of neutrinos in the 1–10³ TeV range. We investigate the connection between the neutrino signal detectable at the IceCube Neutrino Observatory and the electromagnetic signal observable by optical wide-field, high-cadence surveys to outline the best strategy for upcoming follow-up searches. We outline a semi-analytical model that connects the optical light-curve properties to the SN parameters and find that a large peak luminosity (⁠|${L_{\rm {peak}}\gtrsim 10^{43}{-}10^{44}\, \mathrm{erg \, s^{-1}}}$|⁠) and an average rise time (t_rise ≳ 10−40 d) are necessary for copious neutrino emission. Nevertheless, the most promising L_peak and t_rise can be obtained for SN configurations that are not optimal for neutrino emission. Such ambiguous correspondence between the optical light-curve properties and the number of IceCube neutrino events implies that relying on optical observations only, a range of expected neutrino events should be considered (e.g. the expected number of neutrino events can vary up to two orders of magnitude for some among the brightest SNe IIn observed by the Zwicky Transient Facility up to now, SN 2020usa and SN 2020in). In addition, the peak in the high-energy neutrino curve should be expected a few t_rise after the peak in the optical light curve. Our findings highlight that it is crucial to infer the SN properties from multiwavelength observations rather than focusing on the optical band only to enhance upcoming neutrino searches.

acceleration of particles, neutrinos, radiation mechanisms: non-thermal, shock waves, circumstellar matter

1 INTRODUCTION

Astrophysical neutrinos with TeV–PeV energy are routinely observed by the IceCube Neutrino Observatory (Ahlers & Halzen 2018; Abbasi et al. 2021a; Halzen & Kheirandish 2022). While the sources of the observed neutrino flux are not yet known (Mészáros 2017; Vitagliano, Tamborra & Raffelt 2020), a number of follow-up programs aims to link the observed neutrinos to their electromagnetic counterparts. In this context, the All-Sky Automated Survey for SuperNovae (ASAS-SN, Shappee et al. 2014; Kochanek et al. 2017) the Zwicky Transient Facility (ZTF, Bellm et al. 2019; Dekany et al. 2020) and the Panoramic Survey Telescope and Rapid Response System 1 (Pan-STARRS1, Chambers et al. 2016) perform dedicated target-of-opportunity searches for optical counterparts of neutrino events (Kankare et al. 2019; Necker et al. 2022; Stein et al. 2023), and vice versa the IceCube Neutrino Observatory looks for neutrinos in the direction of the sources discovered by optical surveys (see e.g. Abbasi et al. 2021b; Abbasi et al. 2023). The importance of such multimessenger searches will be strengthened as large-scale transient facilities come online, such as the Rubin Observatory (Hambleton et al. 2022).

The putative coincidence of the high-energy neutrino event IC200530A with the candidate superluminous supernova (SLSN) AT2019fdr (Pitik et al. 2022).¹ makes searches of high-energy neutrinos from SNe timely. SLSNe are |$\mathcal {O}(10$|–100) times brighter than standard core-collapse SNe (Moriya, Sorokina & Chevalier 2018; Gal-Yam 2019), with kinetic energy sometimes larger than 10⁵¹ erg (Rest et al. 2011; Nicholl et al. 2020). SLSNe are broadly divided into two different spectral types: the ones with hydrogen emission lines (SLSNe II) and those without (SLSNe I, see e.g. Gal-Yam 2012). The majority of SLSNe II displays strong and narrow hydrogen emission lines similar to those of the less luminous SNe IIn (Ofek et al. 2007; Smith et al. 2008; Rest et al. 2011) and often dubbed SLSNe IIn. Type IIn SNe are a sub-class of core-collapse SNe (Gal-Yam et al. 2007; Smith et al. 2011) characterized by bright and narrow Balmer lines of hydrogen in their spectra which persist for weeks to years after the explosion (Schlegel 1990; Filippenko 1997; Gal-Yam 2017). Type IIn SNe are expected to have a dense circumstellar material (CSM) surrounding the exploding star. The large luminosity of SNe IIn and the evidence of slowly moving material ahead of the ejecta indicate an efficient interaction of the ejecta with the CSM, which has long been considered a major energy source of the observed optical radiation (Blinnikov 2017; Smith 2017). Given the similarities of the spectral characteristics, SLSNe IIn are deemed to be extreme cases of SNe IIn, albeit it is unclear whether SLSNe IIn are just the most luminous SNe IIn or they represent a separate population.

The collision between the expanding SN ejecta and the dense CSM gives rise to the forward shock, propagating in the dense SN environment, and the reverse shock moving backward in the SN ejecta. The plasma heated by the forward shock radiates its energy thermally in the ultraviolet (UV)/X-ray band. Depending on the column density of the CSM, energetic photons can be reprocessed through photoelectric absorption and/or Compton scattering downwards into the visible waveband, producing the observed optical light curve. Alongside the thermal population, a non-thermal distribution of protons and electrons can be created via diffusive shock acceleration.

Once accelerated, the relativistic protons undergo inelastic hadronic collisions with the non-relativistic protons of the shocked CSM, possibly leading to copious production of high-energy neutrinos and gamma rays (Murase et al. 2011; Zirakashvili & Ptuskin 2016; Petropoulou et al. 2017; Sarmah et al. 2022). While gamma rays are absorbed and reprocessed to a large extent in the dense medium (see e.g. Sarmah et al. 2022), neutrinos stream freely and reach Earth without absorption (Katz, Sapir & Waxman 2011; Murase et al. 2011; Cardillo, Amato & Blasi 2015; Zirakashvili & Ptuskin 2016; Brose, Sushch & Mackey 2022; Kheirandish & Murase 2022; Sarmah et al. 2022, 2023). If detected, neutrinos with energies ≳ 100 TeV from an interacting SN would represent a smoking gun of acceleration of cosmic rays up to PeV energies (Bell 2013; Blasi 2013; Cristofari, Blasi & Amato 2020; Cristofari 2021).

In this paper, we consider SNe IIn and SLSNe IIn as belonging to the same population, distinguished primarily by the ejecta energetics and CSM density. We investigate how neutrino production depends on the characteristic quantities describing interaction-powered SNe and connect the main features of the optical light curve to the observable neutrino signal in order to optimize joint multimessenger search strategies.

This work is organized as follows. Section 2 outlines the SN model. As for the CSM structure, we mostly focus on the scenario involving SN ejecta propagating in an extended envelope surrounding the progenitor with a wind-like density profile; we then extend our findings to the case involving SN ejecta propagating into a shell of CSM material with uniform density, which might result from a violent eruption shortly before the death of the star. In Section 3, we introduce the scaling relations for the SN light-curve properties. Section 4 focuses on investigating the dependence of the maximum proton energy on the SN model parameters. In Section 5, after introducing the method adopted to compute the neutrino spectral energy distribution, the dependence of the total energy emitted in neutrinos is investigated as a function of the SN model parameters. Section 6 outlines the detection prospects of neutrinos by relying on two benchmark SLSNe IIn observed by ZTF and discusses the most promising strategies to detect neutrinos by relying on optical observations as well multimessenger follow-up programs. Finally, our findings are summarized in Section 7. In addition, the dependence of the SN light-curve properties and maximum proton energy on the SN model parameters are discussed in Appendices  A and  B, respectively. Moreover, details on the constant density scenario are provided in Appendix  C.

2 MODEL FOR INTERACTION-POWERED SUPERNOVAE

In this section, we present the theoretical framework of our work. First, we describe the CSM configurations. Then, we focus on the modelling of the interaction between the SN ejecta and the CSM, leading to the observed electromagnetic radiation. We also outline the SN model parameters and the related uncertainty ranges adopted in this work.

2.1 Modelling of the circumstellar medium

Observational data and existing theoretical models indicate that the matter envelope surrounding massive stars could be spherical in shape or exhibit bipolar shells, disks or clumps, with non-trivial density profiles. This is the result of steady or eruptive mass-loss episodes, as well as binary interactions of the progenitor prior to the explosion (Smith 2017). To this purpose, we consider two CSM configurations: a uniform shell extended up to a radius R_{CSM, s} from the centre of the explosion and a spherically symmetric shell with a wind radial profile extending smoothly from the progenitor surface up to an external radius (R_{CSM, w}), as sketched in Fig. 1. Henceforth, we name the former ‘shell scenario’ (s) and the latter ‘wind scenario’ (w).

Figure 1.

Schematic representation of an interaction-powered SN, under the assumption of spherical symmetry. The central compact object (in black) is surrounded by the SN ejecta (brown) and a compact shell extended up to R_{CSM, s} (R_{CSM, w}) from the centre of explosion for the shell scenario on the left (and the wind scenario, on the right). For the wind density profile, the colour gradient tracks the density gradient (from darker to lighter hues as the density decreases). The region of interaction marked through the yellow-white circle represents the forward shock R_sh that propagates radially outwards. The solid olive line marks the position of the breakout radius (R_bo), where the first light leaks out, and the shock becomes collisionless. The dashed dark green line marks the location of the deceleration radius of the ejecta (R_dec). The latter is located at radii smaller than R_CSM (as in this sketch) for a relatively large CSM mass compared to the ejecta mass or radii larger than R_CSM for massive ejecta and rarefied CSM; note that we could have R_dec < R_bo for extremely large M_CSM/M_ej. The dashed bordeaux line represents the photospheric radius R_ph, where the radiation decouples from the CSM matter and stream in the outer space freely.

We assume that the CSM has a mass M_CSM, radial extent R_CSM, and it is spherically distributed around the SN with a density profile described by a power-law function of the radius:

$$\begin{eqnarray} n_{\rm {CSM}}(R)=\frac{\rho _{\rm {CSM}}(R)}{m}=\frac{(3-s) M_{\rm {CSM}}}{4\pi R^{3}_{\rm {CSM}}}\bigg (\frac{R}{R_{\rm {CSM}}}\bigg)^{-s}\equiv BR^{-s}\ , \end{eqnarray}$$

(1)

where m = μm_H, with μ = 1.3 (Lodders 2019) being the mean molecular weight for a neutral gas of solar abundance. We neglect the density dependence on the inner radius of the CSM and consider it to be the same as the progenitor radius |$R_{\star }=10^{13}\, \mathrm{cm} \ll R_{\rm {CSM}}$|⁠. The case s = 2 represents the stellar wind scenario, whereas s = 0 denotes a shell of uniform density. We assume that the density external to the CSM shell (R > R_CSM) is much smaller than the one at R < R_CSM.

2.2 Shock dynamics

After the SN explodes, and the shock wave passes through the stellar layers, the ejecta gas evolves to free homologous expansion. Relying on numerical simulations (e.g. Matzner & McKee 1999), we assume that during this phase the outer part of the SN ejecta has a power-law density profile (Chevalier & Fransson 1994; Moriya et al. 2013):

$$\begin{eqnarray} \rho _{\rm {ej}}(R,t) = g_{n}t^{n-3}R^{-n}\ , \end{eqnarray}$$

(2)

with

$$\begin{eqnarray} g_{n} = \frac{1}{4\pi (n-\delta)}\frac{[2(5-\delta)(n-5)E_{\rm {k}}]^{(n-3)/2}}{[(3-\delta)(n-3)M_{\rm {ej}}]^{(n-5)/2}}\ , \end{eqnarray}$$

(3)

where E_k is the total SN kinetic energy, M_ej is the total mass of the SN ejecta, n is the density slope of the outer part of the ejecta, and δ the slope of the inner one. The parameter n depends on the progenitor properties and the nature (convective or radiative) of the envelope; n ≃ 12 is typical of red supergiant stars (Matzner & McKee 1999), while lower values are expected for more compact progenitors. In this work, we adopt n = 10 and δ = 1, following Suzuki, Moriya & Takiwaki (2020).

The interaction between the SN ejecta and the CSM results in a forward shock moving in the CSM and a reverse shock propagating back in the stellar envelope. For our purposes, only the forward shock is relevant. It is indeed estimated that the contribution of the reverse shock to the electromagnetic emission, as well as its efficiency in accelerating particles during the time-scales of interest, is significantly lower than the one of the forward shock (Ellison et al. 2007; Patnaude & Fesen 2009; Schure et al. 2010; Slane et al. 2015; Zirakashvili & Ptuskin 2016; Sato et al. 2018; Suzuki et al. 2020).

Following Chevalier (1982); Moriya et al. (2013), we assume that the thickness of the shocked region is much smaller than its radius, R_sh. As long as the mass of the SN ejecta is larger than the swept-up CSM mass, which we define as the ejecta-dominated phase (or free expansion phase), the expansion of the forward shock radius is described by (Moriya et al. 2013):

$$\begin{eqnarray} R_{\rm {sh}}(t)=R_{\star }+\bigg [\frac{(3-s)(4-s)}{(n-4)(n-3)}\frac{g_{n}}{\rm B}\bigg ]^{\frac{1}{n-s}}t^{\frac{n-3}{n-s}}\ , \end{eqnarray}$$

(4)

with B defined as in equation (1), and hereafter we assume that the interaction starts at t = 0.

When the swept-up CSM mass becomes comparable to the SN ejecta mass, the ejecta start to slow down, entering the CSM-dominated phase. This happens at the deceleration radius, defined as the radius R_dec at which |$\int ^{R_{\rm {dec}}}_{R_{\star }}4\pi R^{2}\rho _{\rm {CSM}}{\rm d}R=M_{\rm {ej}}$|⁠, namely

$$\begin{eqnarray} R_{\rm {dec}} = \bigg [\frac{3-s}{4\pi B}M_{\rm {ej}} + R_{\star }^{3-s}\bigg ]^{\frac{1}{3-s}}\ . \end{eqnarray}$$

(5)

According to the relative ratio between M_ej and M_CSM, the deceleration can occur inside or outside the CSM shell (where a dilute stellar wind surrounds the collapsing star). After this transition, the forward shock evolves as (Suzuki et al. 2020):

$$\begin{eqnarray} R_{\rm {sh}}(t) = R_{\rm {dec}} \bigg (\frac{t}{t_{\rm {dec}}}\bigg)^{\frac{2}{5-s}}\ . \end{eqnarray}$$

(6)

Differentiating equations (4) and (6), we obtain the forward shock velocity as a function of time:

$$\begin{eqnarray} v_{\rm {sh}}(t)=\frac{{\rm d}R_{\rm {sh}}(t)}{{\rm d}t}=\left\lbrace \begin{array}{@{}l@{\quad }l@{}}\frac{n-3}{n-s}\bigg [\frac{(3-s)(4-s)}{(n-4)(n-3)}\frac{g_{n}}{\rm B}\bigg ]^{\frac{1}{n-s}}t^{\frac{s-3}{n-s}}\quad R\lt R_{\rm {dec}}\\ \frac{2 }{5-s}R_{\rm {dec}}\big (\frac{t}{t_{\rm {dec}}}\big)^{\frac{s-3}{5-s}}\,\, \quad \quad \quad \ \ R\ge R_{\rm {dec}}\ . \end{array}\right. \end{eqnarray}$$

(7)

We consider the dynamical evolution under the assumption that the shock is adiabatic for two reasons. First, we want to compare our results with the literature on the properties of the SN light curves extrapolated by relying on semi-analytic models for the adiabatic expansion – see e.g. Suzuki et al. (2020). Secondly, it has been shown that, in the radiative regime, R_sh has the same temporal dependence as the self-similar solution ∝ t^{(n − 3)/(n − s)} in the free expansion phase with radiative losses having a strong impact on the evolution of the shock (Moriya et al. 2013).

While the shock propagates in the CSM, the ejecta kinetic energy is dissipated in the interaction and converted into thermal energy. The shock-heated gas behind the forward shock front cools by emitting thermal energy in the form of free-free radiation (thermal bremsstrahlung). However, if the CSM ahead of the shock is optically thick, such radiation is trapped and remains confined until the shock breakout, which occurs at the breakout radius (R_bo). The latter is computed by solving the following equation for the Thomson optical depth (due to photon scattering on electrons)²:

$$\begin{eqnarray} {\tau _{T}=\int _{R_{\rm {bo}}}^{R_{\rm {CSM}}} \rho _{\rm {CSM}}(R) \kappa _{\rm {es}} {\rm d}R = \frac{c}{v_{\rm {sh}}}}\ , \end{eqnarray}$$

(8)

where κ_es is the electron-scattering opacity, c is the speed of light, and v_sh is defined in equation (7). If R_bo ≤ R_⋆, R_bo = R_⋆.

We make use of the assumption of constant opacity, valid for electron Compton scattering. The value of κ_es, which depends on the composition, typically ranges from |$\kappa _{\rm {es}}\sim 0.2\, \mathrm{cm^{2} g^{-1}}$| for hydrogen-free matter to |$\kappa _{\rm {es}} \sim 0.4\, \mathrm{cm^{2} g^{-1}}$| for pure hydrogen. We consider solar composition of the CSM, namely |$\kappa _{\rm {es}} = 0.2 (1 + X_{\rm {H}}) \simeq 0.34\, \rm {cm}^{2}~\rm {g}^{-1}$| (Rybicki & Lightman 1986), where X_H = 0.73 is the hydrogen mass fraction (Lodders 2019).

As long as τ_T ≫ c/v_sh, the shock is radiation-mediated (energy density of the radiation is larger than the energy density of the gas) and radiation pressure rather than plasma instabilities mediate the shock. In this regime, non-thermal particle acceleration is inefficient, since a shock width much larger than the particle gyro-radius hinders standard Fermi acceleration (Weaver 1976; Levinson & Bromberg 2008; Katz et al. 2011; Murase et al. 2011). Furthermore, diffusion can be neglected. When τ_T < c/v_sh, the shock becomes collisionless, and efficient particle acceleration begins.

2.3 Interaction-powered supernova emission

When the forward shock propagates in the region with τ_T < c/v_sh, the gas immediately behind the shock is heated to a temperature T_sh. Assuming electron–ion equilibrium, such a temperature can be obtained by the Rankine–Hugoniot conditions:

$$\begin{eqnarray} k_{\rm B}T_{\rm {sh}}=2\frac{(\gamma -1)}{(\gamma +1)^{2}}\tilde{\mu } m_{\rm {H}} v^{2}_{\rm {sh}}\approx 118\, {\rm {keV}}\, \bigg (\frac{v_{\rm {sh}}}{10^{9}~ {\rm {cm~s}^{-1}}}\bigg)^{2}\ , \end{eqnarray}$$

(9)

where γ = 5/3 is the adiabatic index of the gas. We adopt mean molecular weight |$\tilde{\mu }=0.6$|⁠; such a choice is appropriate for fully ionized CSM with solar composition as it is the case for the matter right behind the shock (this is different from equation (1) where the CSM is assumed to be neutral). The thermal emission properties of the shock-heated material can be fully characterized by the shock velocity v_sh and the other parameters characterizing the CSM (Margalit, Quataert & Ho 2022).

The observational signatures of the SN light curve and spectra depend on the radiative processes, which shape the thermal emission. The main photon production mechanism is free-free emission of the shocked electrons, whose typical time-scale is (Draine 2011):

$$\begin{eqnarray} t_{\rm {cool}}=\frac{3 k_{\rm B} T_{\rm {sh}}}{2n_{\rm {sh}} \Lambda (T)}\ , \end{eqnarray}$$

(10)

where k_B is the Boltzmann constant, n_sh = 4n_CSM is the density of the shocked region. The factor 4 comes from the Rankine–Hugoniot jump conditions across a strong non-relativistic shock. Λ(T) is the cooling function (in units of |$\rm {erg}\, \rm {cm}^{3}\, \rm {s}^{-1}$|⁠) that captures the physics of radiative cooling (Chevalier & Fransson 1994):

$$\begin{eqnarray} \Lambda (T)=\left\lbrace \begin{array}{@{}l@{\quad }l@{}}6.2\times 10^{-19} \, T^{-0.6}\quad 10^{5} \lt T \lesssim T_{\ast }\\ 2.5\times 10^{-27} \, T^{0.5}\, \, \, \, \,\quad \quad \quad T\gt T_{\ast }\ . \end{array}\right. \end{eqnarray}$$

(11)

The temperature T_* = 4.7 × 10⁷ K represents the transition from the regime where free-free emission is dominant (T ≳ T_*) to the one where line emission becomes relevant (T ≲ T_*). If the free-free cooling time-scale is shorter than the dynamical time, the shock becomes radiative. In this regime, particles behind the shock cool within a layer of width (t_cool/t_dyn)R_sh.

Although the radiation created during the interaction could diffuse from the CSM, the presence of dense pre-shock material causes the emitted photons to experience multiple scattering episodes before they reach the photosphere (defined as the surface where τ_T = 1):

$$\begin{eqnarray} R_{\rm {ph}}=\bigg [\frac{s-1}{\kappa _{\rm {es}}B}+R_{\rm {CSM}}^{1-s}\bigg ]^{\frac{1}{1-s}}\ . \end{eqnarray}$$

(12)

The dominant mechanisms responsible for the photon field degradation in the medium are photoelectric absorption and Compton scattering, that generate inelastic energy transfer from photons to electrons during propagation. The result of such energy losses is that the bulk of thermal X-ray photons (see equation 9) is absorbed and reprocessed via continuum and line emission in the optical. This phenomenon is strongly dependent on the CSM mass and extent, as well as on the stage of the shock evolution.

Alongside bremsstrahlung photons, a collisionless shock may produce non-thermal radiation from a relativistic population of electrons accelerated through diffusive shock acceleration. Synchrotron emission of these electrons is mainly expected in the radio band; it has been shown that the CSM mass and radius play an important role in defining the radio peak time and luminosity (see e.g. Petropoulou, Kamble & Sironi 2016).

2.4 Supernova model parameters

The parameters characterizing SNe/SLSNe of Type IIn carry large uncertainties. For our benchmark SN model, we take into account uncertainty ranges for the SN energetics, CSM and ejecta masses, as well as the CSM radial extent as summarized in Table 1. A number of other uncertainties can significantly impact the observational features, for example, the composition and geometry of the stellar environment or the stellar structure.

Table 1.

Supernova model parameters for the SN wind and shell scenarios. The reference values adopted for our benchmark SN model are provided together with the uncertainty range for the most uncertain parameters.

Parameter	Symbol	Benchmark value	Parameter range
Accelerated proton energy fraction	εp	0.1	−
Magnetic energy density fraction	εB	3 × 10−4	−
Proton spectral index	k	2	−
External ejecta density slope	n	10	−
Internal ejecta density slope	δ	1	−
Kinetic energy	Ek	1051 erg	1050–1053 erg
Ejecta mass	Mej	10 M⊙	1–70 M⊙
CSM mass	MCSM	10 M⊙	1–70 M⊙
CSM radius	RCSM	1016 cm	5 × 1015–1017 cm

The electromagnetic emission of SLSNe IIn can be explained invoking a massive CSM shell with enough inertia to decelerate and dissipate most of the kinetic energy of the ejecta: |$M_{\rm {CSM}}\gtrsim 40\, {\rm M}_{\odot }$|⁠, |$M_{\rm {ej}}\gtrsim 50\, {\rm M}_{\odot }$|⁠, and E_k ≳ 10⁵² erg have been invoked for SLSNe in the tail of the distribution (see e.g. Drake et al. 2011; Nicholl et al. 2020), consistent with pair-instability SN models. On the other hand, SNe IIn may result from the interaction with a less dense surrounding medium, or simply fall in the class of less powerful explosions, with |$M_{\rm {CSM}}\lesssim 5\, {\rm M}_{\odot }$|⁠, E_k ∼ a few 10⁵¹ erg, and |$M_{\rm {ej}}\lesssim 50\, {\rm M}_{\odot }$| (see e.g. Chatzopoulos et al. 2013).

To encompass the wide range of SN properties and the related uncertainties, we consider the space of parameters summarized in Table 1. In the following, we systematically investigate the dependence of the light-curve features, such as the rise time and the peak luminosity on the SN parameters. For the sake of completeness, we choose generous uncertainty ranges, albeit most of the observed SN events do not require kinetic energies larger than 10⁵² erg or CSM masses larger than 50 M_⊙, for example.

3 SCALING RELATIONS FOR THE PHOTOMETRIC SUPERNOVA PROPERTIES

In this section, we introduce the scaling relations for the peak luminosity and the rise time of an SN light cuve powered by shock interaction. Such relations connect these two observable quantities to the SN model parameters.

We are interested in the shock evolution after shock breakout, when τ_T < c/v_sh. During this regime, the light curve is powered by continuous conversion of the ejecta kinetic energy – see e.g. Chatzopoulos, Wheeler & Vinko (2012), Ginzburg & Balberg (2012), and Moriya et al. (2013). Such a phase, however, reproduces the decreasing-flat part of the SN light curve at later times (see Fig. 2), while the initial rising part of the optical signal can be explained considering photon diffusion in the optically thick region – see e.g. Chevalier & Irwin (2011) and Chatzopoulos et al. (2012).

Figure 2.

Sketch of the SN luminosity evolution (in arbitrary units) resulting from the interaction of the SN shock with the dense CSM. The origin (t = 0) coincides with the SN explosion time. Note that t_rise is defined from the time of the shock breakout.

Since we are interested in exploring a broad space of SN model parameters, we rely on semi-analytical expressions for the characteristic quantities that describe the optical light curve, namely the bolometric luminosity peak L_peak and the rise time to the peak t_rise (see Fig. 2). By performing 1D radiation-hydrodynamic simulations for a large region of the space of parameters, Suzuki et al. (2020) fitted the output of their numerical simulations with semi-analytical scaling relations, investigating the relation between L_peak and t_rise. In this way, it is possible to analyze the dependence of the light curve properties on the parameters characterizing the SN interaction, i.e. the kinetic energy of the ejecta (E_k), the mass of the ejecta (M_ej), the mass of the CSM (M_CSM), and the extent of the CSM (R_CSM). Suzuki et al. (2020) found that the semi-analytical scaling relations describe relatively well the numerical results, once accounting for some calibration factors. In this section, we review the scaling relations we adopt throughout our work.

As the forward shock propagates in the CSM, the post-shock thermal energy per unit radius coming from the dissipation of the kinetic energy is given by

$$\begin{eqnarray} \mathcal {E}_{\rm {k}}(R)=\frac{{\rm d}E_{\rm {k}}}{{\rm d}R}=\frac{9}{8}\pi R^{2}v_{\rm {sh}}^{2}(R)\rho _{\rm {CSM}}(R)\ . \end{eqnarray}$$

(13)

where we have used the Rankine–Hugoniot jump conditions across a strong non-relativistic shock that provide a compression ratio ≃ 4.

We define the bolometric peak luminosity as the kinetic power of the shock at breakout:

$$\begin{eqnarray} L_{\rm {peak}}=\frac{{\rm d}E_{\rm {k}}}{{\rm d}t}\bigg |_{R_{\rm {bo}}}=\frac{9}{8}\pi R_{\rm {bo}}^{2}v_{\rm {sh}}^{3}(R_{\rm {bo}})\rho _{\rm {CSM}}(R_{\rm {bo}})\ . \end{eqnarray}$$

(14)

When the shock is still crossing the CSM envelope, the radiated photons undergo multiple scatterings before reaching the photosphere. The diffusion coefficient is D(R) ∼ cλ(R), with λ(R) = 1/κ_esρ_CSM(R) being the photon mean free path. The time required to diffuse from R_bo to the photosphere R_ph represents the rise time of the bolometric light curve (Ginzburg & Balberg 2012)³:

$$\begin{eqnarray} t_{\rm {rise}}\approx \int _{R_{\rm {\rm {bo}}}}^{R_{\rm {ph}}}\frac{{\rm d}(R-R_{\rm {bo}})^{2}}{D(R)}\sim \int _{R_{\rm {\rm {bo}}}}^{R_{\rm {ph}}} \frac{2(R-R_{\rm {bo}})\kappa _{\rm {es}}\rho _{\rm {CSM}}(R){\rm d}R}{c}\ . \end{eqnarray}$$

(15)

Furthermore, after the forward shock breaks out from the optically thick part of the CSM at R_ph, its luminosity is expected to be primarily emitted in the UV/X-ray region of the spectrum, and not in the optical (Ginzburg & Balberg 2012). Hence, we consider the photospheric radius as the radius beyond which the optical emission is negligible. Distinguishing the free-expansion regime (FE, M_ej ≫ M_CSM) and the blast-wave regime (BW, M_ej ≪ M_CSM) (Suzuki et al. 2020)⁴, the kinetic energy dissipated during the shock evolution in the optically thick region is:

$$\begin{eqnarray} E_{\rm {diss, thick}}=\left\lbrace \begin{array}{@{}l@{\quad }l@{}}\int _{R_{\star }}^{R_{\rm {ph}}} \mathcal {E}_{\rm {k}}(R) {\rm d}R \quad \quad \quad \mathrm{for\, \, FE}\\ \\ \frac{(3-s)(\gamma +1)}{3+9\gamma -2s-2\gamma s}E_{\rm {k}} \, \, \quad \quad \ \mathrm{for\, \, BW}\ . \end{array}\right. \end{eqnarray}$$

(16)

Part of this energy is converted into thermal energy and radiated. The fraction radiated in the band of interest depends on multiple factors, including the cooling regime of the shock during the evolution, as well as the ionization state and CSM properties. We parametrize these unknowns by introducing the fraction ε_rad of the total dissipated energy E_{diss, thick} that is emitted in the optical band. We note that we adopt a definition of the rise time which differs from the Arnett’s rule employed in Suzuki et al. (2020), leading to comparable results, except for extremely low values of R_CSM (∼10¹⁵ cm), which we do not consider in this work. In Appendix  A, we provide illustrative examples of the dependence of L_peak, t_rise, and E_{diss, thick} on the parameters characterizing the SN light curve for the wind CSM configuration (s = 2).

Fig. 3 shows L_peak as a function of t_rise, obtained by adopting the semi-analytic modelling in the FE and BW regimes. We note that the largest dispersion in the peak luminosity for long-lasting SNe/SLSNe IIn is obtained by varying the ejecta kinetic energy (left-hand panel). For fixed kinetic energy, we see that the SN models corresponding to different ejecta mass (middle panel) all converge to approximately similar peak luminosity for longer t_rise, which corresponds to the region where the shock evolution is in the BW regime. This means that there is an upper limit on L_peak for a certain t_rise, and the only way to overcome this limit is by increasing the ejecta energy. Changes in R_CSM (right-hand panel) lead to the smallest dispersion in L_peak among all the considered parameters. It is the variation of the kinetic energy that causes the largest spread in L_peak. Our findings are in agreement with the ones of Suzuki et al. (2020).

Figure 3.

Bolometric peak luminosity as a function of the rise time, for fixed M_ej and R_CSM (left-hand panel, and varying E_k), fixed E_k and R_CSM (middle panel, and varying M_ej), and fixed E_k and M_ej (right-hand panel, and varying R_CSM). In each panel, the arrow points in the direction of increasing values of the parameter indicated on top of the plot (e.g. in the left-hand panel, the highest curve is obtained with the the largest kinetic energy, 10⁵² erg). For each curve, the colour hues mark the variation of M_CSM. The longest rise times and brightest light curves are obtained for large kinetic energies (left-hand panel), the low ejecta mass (middle panel), large CSM mass and small CSM extension (right-hand panel). Models with intermediate t_rise can reach the largest peak luminosities. The largest dispersion of long-lasting interaction-powered SNe can be achieved by increasing the kinetic energy. By keeping E_k fixed, an upper limit on L_peak is expected for different M_ej and R_CSM.

4 MAXIMUM PROTON ENERGY

In order to estimate the number of neutrinos and their typical energy during the shock evolution in the CSM, we first need to examine the energy gain and loss mechanisms that determine the maximum energy up to which protons can be accelerated. We assume first-order Fermi acceleration, which takes place at the shock front with the accelerating particles gaining energy as they cross the shock front back and forth.

In the Bohm limit, where the proton mean free path is equal to its gyroradius r_g = γ_pm_pc²/eB, the proton acceleration time-scale is |${t_{\rm {acc}}\sim 6\gamma _{\rm {p}} m_{\rm {p}} c^{3}/eB v^{2}_{\rm {sh}}}$| (see e.g. Protheroe & Clay 2004; Tammi & Duffy 2009; Caprioli & Spitkovsky 2014), where |$B=\sqrt{9\pi \varepsilon _{\rm B} v_{\rm {sh}}^{2}\rho _{\rm {CSM}}}$| is the turbulent magnetic field in the post-shock region, whose energy density is assumed to be a fraction ε_B of the post-shock thermal energy |${U_{\rm {th}} = (9/8) v^{2}_{\rm {sh}} \rho _{\rm {CSM}} }$|⁠.

The maximum energy up to which protons can be accelerated is determined by the competition between particle acceleration and energy loss mechanisms, such that t_acc ≤ t_{p, cool}, with t_{p, cool} being the total proton cooling time. The relevant cooling times are the advection time (t_adv ∼ ΔR_acc/v_sh, with ΔR_acc being the width of the acceleration region) and the proton–proton interaction time (t_pp = (4k_ppσ_ppn_CSMc)⁻¹, where we assume constant inelasticity k_pp = 0.5 and energy-dependent cross-section σ_pp(E_p) (Zyla et al. 2020).

As pointed out in Fang et al. (2020), taking ΔR_acc ∼ R_sh may be appropriate for adiabatic shocks only. If the shock is radiative, particles in the post-shock region cool via free-free emission within a layer of width ∼(t_cool/t_dyn)R_sh (see Section 2.3), making the gas far from the shock quasi-neutral, and thus hindering the magnetic field amplification crucial in the acceleration mechanism (Bell 2004). Hence, we adopt ΔR_acc = (t_cool/t_dyn)R_sh for |$t_{\rm {\rm {cool}}}\lt t_{\rm {dyn}}$|⁠, and ΔR_acc = R_sh otherwise.

The total proton cooling time can thus be written as |${ t^{-1}_{\rm {p,cool}} = t^{-1}_{\rm {pp}} + \textrm {max}[t^{-1}_{\rm {dyn}},t^{-1}_{\rm {cool}}] }$|⁠. It is important to note that relativistic protons in the shocked region may also interact with the ambient photons via pγ interactions. However, we ignore such an energy loss channel, by relying on the findings of Murase et al. (2011) and Fang et al. (2020) that showed that pγ interactions can be neglected for a wide range of SN parameters.

Fig. 4 shows contours of E_{p, max} for the wind scenario. The black solid lines mark the edges of the interaction region, hence Fig. 4 also provides an idea of the the typical interaction duration. Fixing three of the SN model parameters to their benchmark values (see Table 1), the shortest period of interaction is obtained for small R_CSM and large M_CSM, or small M_ej and large E_k. In fact, in both cases, the shock breakout is delayed. The maximum proton energy increases with the radius, and the largest E_{p, max} can be obtained in the late stages of the shock evolution, hinting that high-energy neutrino production should be favoured at later times after the bolometric peak.

Figure 4.

Contour plots of the maximum proton energy for the wind scenario in the plane spanned by the distance from the central engine and M_CSM (top left-hand panel), M_ej (top right-hand panel), R_CSM (bottom left-hand panel), and E_k (bottom right-hand panel), while the remaining three SN model parameters are fixed to their benchmark values. In each panel, the dashed line marks the deceleration radius, after which E_{p, max} decreases. The maximum proton energy increases with the radius (and therefore with time). Indeed, the largest E_{p, max} is obtained in the late stages of the shock evolution. Large R_CSM and M_ej, and small M_CSM and E_k lead to the longest interaction times. This statement is not true when M_ej ≪ M_CSM (see left upper and bottom right-hand panels). The black solid lines define the edges of the interaction region, R_bo ≤ R ≤ R_CSM.

The breaks observed in the contour lines in the upper and lower right-hand panels of Fig. 4 represent the transition between the regimes where free-free and line-emission dominate. From the two upper panels, we see that E_{p, max} reaches its maximum value at R_dec, and declines later. But this is not always the case; as shown in Appendix  B, when the proton energy loss times are longer than the dynamical time, the maximum proton energy decreases throughout the evolution.

5 EXPECTED NEUTRINO EMISSION FROM INTERACTION-POWERED SUPERNOVAE

In this section, the spectral energy distribution of neutrinos is introduced. We then present our findings on the dependence of the expected number of neutrinos on the SN model parameters and link the neutrino signal to the properties of the SN light curves.

5.1 Spectral energy distribution of neutrinos

A fraction ε_p of the dissipated kinetic energy of the shock (equation 13) is used to accelerate protons swept-up from the CSM; we adopt ε_p = 0.1, assuming that the shocks accelerating protons are parallel or quasi-parallel and therefore efficient diffusive shock acceleration occurs (Caprioli & Spitkovsky 2014). However, lower values of ε_p would be possible for oblique shocks, with poorer particle acceleration efficiency. Given the linear dependence of proton and neutrino spectra on this parameter, it is straightforward to rescale our results.

Assuming a power-law energy distribution with spectral index k = 2, the number of protons injected per unit radius and unit Lorenz factor is

$$\begin{eqnarray} Q_{\rm {p}}(\gamma _{\rm {p}},R)=A(R)\gamma _{\rm {p}}^{-2}\log ^{-1}\bigg (\frac{\gamma _{\rm {p, max}}}{\gamma _{\rm {p, min}}}\bigg)\ , \end{eqnarray}$$

(17)

for γ_{p, min} < γ < γ_{p, max}, and zero otherwise. We set the minimum Lorentz factor of accelerated protons γ_{p, min} = 1, while γ_{p, max} is obtained by comparing the acceleration and the energy-loss time-scales at each radius during the shock evolution, as discussed in Section 4. The normalization factor A(R) is

$$\begin{eqnarray} A(R)&=&\frac{9}{8}\pi \varepsilon _{\rm {p}} R^{2}v_{\rm {sh}}^{2}(R)\rho _{\rm {CSM}}(R)\\&\propto& \left\lbrace \begin{array}{@{}l@{\quad }l@{}}R^{\frac{2n-sn+5s-12}{n-3}}\quad \textrm {for}\, R\le R_{\rm {dec}}\\ R^{-1}\quad \quad \quad \quad \, \, \textrm {for}\, R\gt R_{\rm {dec}}\ . \end{array}\right.\nonumber\\ \end{eqnarray}$$

(18)

The injection rate of protons in the deceleration phase does not depend on the SN density structure nor the CSM density profile. Since we aim to compute the neutrino emission, we track the temporal evolution of the proton distribution in the shocked region between the shock breakout radius R_bo and the outer radius R_CSM.

The evolution of the proton distribution is given by (Sturner et al. 1997; Finke & Dermer 2012; Petropoulou et al. 2016)

$$\begin{eqnarray} \frac{\partial N_{\rm {p}}(\gamma _{\rm {p}},R)}{\partial R}-\dfrac{\partial }{\partial \gamma _{\rm {p}}}\bigg [\frac{\gamma _{\rm {p}}}{R} N_{\rm {p}}(\gamma _{\rm {p}}, R)\bigg ] + \frac{N_{\rm {p}}(\gamma _{\rm {p}}, R)}{v_{\rm {sh}}(R) t_{\rm {pp}}(R)} = Q_{\rm {p}}(\gamma _{\rm {p}},R)\ , \nonumber\\ \end{eqnarray}$$

(19)

where N_p(γ_p, R) represents the total number of protons in the shell at a given radius R with Lorentz factor between γ_p and |$\gamma _{\rm {p}} + \rm {d}\gamma _{\rm {p}}$|⁠. The second term on the left-hand side of equation (19) takes into account energy losses due to the adiabatic expansion of the SN shell, while pp collisions are treated as an escape term (Sturner et al. 1997). Other energy loss channels for protons are negligible (Murase et al. 2011). Furthermore, in equation (19), the diffusion term has been neglected since the shell is assumed to be homogeneous.

The neutrino production rates, |$Q_{\nu _{i}+\bar{\nu }_{i}} \, [\rm {GeV}^{-1}\rm {s}^{-1}]$|⁠, for muon and electron flavor (anti)neutrinos are given by (Kelner, Aharonian & Bugayov 2006)

$$\begin{eqnarray} Q_{\nu _{\mu } + \bar{\nu }_{\mu }}(E_{\nu }, R) &=& 4 n_{\rm {CSM}}(R) m_{\rm {p}} c^{3} \int _{0}^{1} dx \frac{\sigma _{\rm {pp}}(E_{\nu }/x)}{x} \nonumber\\ & \times & N_{\rm {p}} \bigg (\frac{E_{\nu }}{x m_{\rm {p}} c^{2}}, R\bigg)\bigg (F^{(1)}_{\nu _{\mu }}(E_{\nu }, x) + F^{(2)}_{\nu _{\mu }}(E_{\nu }, x)\bigg),\nonumber\\ \end{eqnarray}$$

(20)

$$\begin{eqnarray} Q_{\nu _{e} + \bar{\nu }_{e}}(E_{\nu }, R) &=& 4 n_{\rm {CSM}}(R) m_{\rm {p}} c^{3} \int _{0}^{1} dx \frac{\sigma _{\rm {pp}}(E_{\nu }/x)}{x}\nonumber\\ & \times & N_{\rm {p}} \bigg (\frac{E_{\nu }}{x m_{\rm {p}} c^{2}}, R\bigg) F_{\nu _{e}}(E_{\nu }, x) , \end{eqnarray}$$

(21)

where x = E_ν/E_p. The functions |$F^{(1)}_{\nu _{\mu }}$|⁠, |$F^{(2)}_{\nu _{\mu }}$|⁠, and |$F_{\nu _{e}}$| follow the definitions in Kelner et al. (2006). Equations (20) and (21) are valid for E_p > 0.1 TeV, corresponding to the energy range under investigation. Note that, for the parameters we use in this work, the synchrotron cooling of charged pions and muons produced via pp interactions is negligible. Therefore, the neutrino spectra are not affected by the cooling of mesons.

5.2 Energy emitted in neutrinos

The total energy that goes in neutrinos in the energy range [E_{ν, 1}, E_{ν, 2}] during the entire interaction period is given by

$$\begin{eqnarray} \mathcal {E}_{\nu +\bar{\nu }}= \int _{t_{\rm {BO}}}^{t_{\rm {CSM}}}{\rm d}t\int ^{E_{\nu ,2}}_{E_{\nu ,1}}{\rm d}E_{\nu } E_{\nu } [Q_{\nu _{\mu } + \bar{\nu }_{\mu }}(E_{\nu }, R)+Q_{\nu _{e} + \bar{\nu }_{e}}(E_{\nu }, R)], \nonumber\\ \end{eqnarray}$$

(22)

where t_BO and t_CSM are expressed in the progenitor reference frame.

In order to connect the observed properties of the SN light curve to the neutrino ones (e.g. the total energy that goes in neutrinos or their typical spectral energy), for each configuration of SN model parameters we integrate the neutrino production rate between t_BO and t_CSM, for |$E_{\nu }\ge 1\, \mathrm{TeV}$|⁠, as in equation (22). The results are shown in Fig. 5, where we fix two of the SN parameters at their benchmark values (see Table 1) and investigate |$\mathcal {E}_{\nu +\bar{\nu }}$| in the plane spanned by the remaining two. Note that we do not consider the regions of the SN parameter space with the maximum achievable proton energy (⁠|$E^\ast _{\rm {p,max}}$|⁠, see Appendix  B for more details) smaller than 10 TeV since they would lead to neutrinos in the energy range dominated by atmospheric events in IceCube (see Section 6). If we were to integrate the neutrino rate for |$E_{\nu ,1}\gt 1\, \mathrm{TeV}$| (equation 22), the contour lines for |$\mathcal {E}_{\nu +\bar{\nu }}$| would be shifted to the left. Isocontours of the maximum achievable proton energy |$E^{\ast }_{\rm {p,max}}$| (first row), the rise time t_rise (second row), and the bolometric peak L_peak (third row) are also displayed on top of the |$\mathcal {E}_{\nu +\bar{\nu }}$| colourmap in Fig. 5.

Figure 5.

Contour plots of the total neutrino energy |$\mathcal {E}_{\nu +\bar{\nu }}$| integrated for |$E_{\nu }\ge 1\, \mathrm{TeV}$| through the evolution of the shock in the CSM, as a function of M_CSM and E_k (left-hand panels), M_ej (middle panels), and R_CSM (right-hand panels) for the wind scenario. In order to highlight the dependence on the SN properties, isocontours of the maximum proton energy |$E^{\ast }_{\rm {p,max}}$| (double-dot dashed contours, top row), t_rise (dashed contours, middle row), and L_peak (dot dashed contours, bottom row) are displayed. All quantities are expressed in the SN reference frame. The white regions represent parts of the parameter space with |$E_{\rm {p,max}}^{\ast }\lesssim 10$| TeV excluded from our investigation. Our benchmark SN model is marked with an orange star. The SN configurations leading to the largest |$\mathcal {E}_{\nu +\bar{\nu }}$| are given by large SN kinetic energies (E_k ≳ 10⁵¹ erg), small ejecta masses (⁠|$M_{\rm {ej}}\lesssim 10 \, {\rm M}_{\odot }$|⁠), intermediate CSM masses with respect to M_ej (i.e. 1 ≲ M_CSM ≲ 30 M_⊙), and relatively large CSM extent (R_CSM ≳ 10¹⁶ cm).

In all panels of Fig. 5, |$\mathcal {E}_{\nu +\bar{\nu }}$| increases with M_CSM, due to the larger target proton number. Nevertheless, such a trend saturates once the critical n_CSM (corresponding to a critical M_CSM) is reached, where either pp interactions or the cooling of thermal plasma significantly limit the maximum proton energy, thus decreasing the number of neutrinos produced with high energy. For masses larger than the critical CSM mass, neutrinos could be abundantly produced either appreciably increasing the kinetic energy (left-hand panel), or decreasing the ejecta mass (middle panel), or increasing the CSM radius (right-hand panel). From the contour lines in each panel, we see that the optimal configuration for what concerns neutrino production results from large E_k and small M_ej, which lead to large shock velocities v_sh, large R_CSM, and not extremely large M_CSM, compared to a fixed M_ej. Nevertheless, the panels in the upper row of Fig. 5 indicate that the configurations with the largest proton energies (and thus spectral neutrino energies) always prefer a balance between E_k, M_ej, and R_CSM with M_CSM.

It is important to observe the peculiar behaviour resulting from the variation of R_CSM (right-hand panels of Fig. 5). For fixed M_CSM, |$\mathcal {E}_{\nu +\bar{\nu }}$| increases, then saturates at a certain R_CSM, and decreases thereafter. For very small R_CSM, the CSM density is relatively large, and the shock becomes collisionless close to R_CSM, probing a low fraction of the total CSM mass and thus producing a small number of neutrinos. This suppression is alleviated by increasing R_CSM. Nevertheless, a large R_CSM for fixed M_CSM leads to a low CSM density, and thus the total neutrino energy drops. For increasing M_CSM, such inversion in |$\mathcal {E}_{\nu +\bar{\nu }}$| happens at larger R_CSM. We also see that the largest |$\mathcal {E}_{\nu +\bar{\nu }}$| is obtained in the upper right corner of the right-hand panels. This is mainly related to the duration of the shock interaction. The longer the interaction time, the larger the CSM mass swept-up by the collisionless shock.

The panels in the middle row of Fig. 5 show how the neutrino energy varies as a function of t_rise. Large neutrino energy is obtained for slow rising light curves. In particular, given our choice of the parameters for these contour plots, the most optimistic scenarios for neutrinos lie in the region with |$10\,\lesssim t_{\rm {rise}}\lesssim 50\, \mathrm{d}$|⁠. Such findings hold for a wide range of parameters for interacting SNe. Extremely large t_rise, on the other hand, are expected to be determined by very large M_CSM, which can substantially limit the production of particles in the high-energy regime.

The bottom panels of Fig. 5 illustrate how |$\mathcal {E}_{\nu +\bar{\nu }}$| is linked to L_peak. In particular, L_peak closely tracks |$\mathcal {E}_{\nu +\bar{\nu }}$|⁠. However, L_peak can increase with M_CSM to larger values than what neutrinos do, given its linear dependence on the CSM density (see equation 14). Overall, the regions where the largest |$\mathcal {E}_{\nu +\bar{\nu }}$| (and hence number of neutrinos) is obtained are also the regions where L_peak is the largest. It is not always true the opposite. Hence, large L_peak is a necessary, but not sufficient condition to have large |$\mathcal {E}_{\nu +\bar{\nu }}$|⁠.

To summarize, a large |$\mathcal {E}_{\nu +\bar{\nu }}$| is expected for large SN kinetic energy (E_k ≳ 10⁵¹ erg), small ejecta mass (⁠|$M_{\rm {ej}}\lesssim 10 \, {\rm M}_{\odot }$|⁠), intermediate CSM mass with respect to M_ej (⁠|$1\, \lesssim M_{\rm {CSM}}\lesssim 30\, {\rm M}_{\odot }$|⁠), and relatively extended CSM (R_CSM ≳ 10¹⁶ cm). These features imply large bolometric luminosity peak (L_peak ≳ 10⁴³–10⁴⁴ erg) and average rise time (t_rise ≳ 10–20 d). On the other hand, it is important to note that degeneracies are present in the SN parameter space (see also Pitik et al. 2022) and comparable L_peak and t_rise can be obtained for SN model parameters (E_k, M_ej, R_CSM, and M_ej) that are not optimal for neutrino emission.

It is important to stress that in this section we have considered |$\mathcal {E}_{\nu +\bar{\nu }}$| as a proxy of the expected number of neutrino events that is investigated in Section 6. Moreover, we have compared |$\mathcal {E}_{\nu +\bar{\nu }}$| to the bolometric luminosity expected at the peak and not to the luminosity effectively radiated, L_{peak, obs}.

6 NEUTRINO DETECTION PROSPECTS

In this section, we investigate the neutrino detection prospects. In order to do so, we select two especially bright SNe observed by ZTF, SN 2020usa, and SN 2020in. On the basis of our findings, we also discuss the most promising strategies for neutrino searches and multimessenger follow-up programs.

6.1 Expected number of neutrino events at Earth

The neutrino and antineutrino flux (⁠|$F_{\nu _{\alpha } +\bar{\nu }_{\alpha }}$| with α = e, μ, τ) at Earth from a SN at redshift z and as a function of time in the observer frame is [GeV⁻¹s⁻¹cm⁻²]

$$\begin{eqnarray} F_{\nu _{\alpha }+\bar{\nu }_{\alpha }}(E_{\nu }, t) \!=\! \frac{(1 \!+\! z)^2}{4 \pi d^{2}_{\rm L}(z)} v_{\rm {sh}}\underset{\beta }{\sum }\! P_{\nu _{\beta } \rightarrow \nu _{\alpha }}Q_{\nu _{\beta } +\bar{\nu }_{\beta }}\bigg (\!E_{\nu _{\alpha }}(1\! + \!z),\frac{v_{\rm {sh}} t}{1\! +\! z}\!\bigg)\!,\nonumber\\ \end{eqnarray}$$

(23)

where |$Q_{\nu _{\beta } +\bar{\nu }_{\beta }}$| is defined as in equations (20) and (21). Neutrinos change their flavor while propagating, hence the flavor transition probabilities are given by (Anchordoqui et al. 2014)

$$\begin{eqnarray} P_{\nu _{e}\rightarrow \nu _{\mu }} = P_{\nu _{\mu }\rightarrow \nu _{e}} = P_{\nu _{e}\rightarrow \nu _{\tau }} = \frac{1}{4}\sin ^{2}2\theta _{12}\ , \end{eqnarray}$$

(24)

$$\begin{eqnarray} P_{\nu _{\mu }\rightarrow \nu _{\mu }} = P_{\nu _{\mu }\rightarrow \nu _{\tau }} = \frac{1}{8}(4-\sin ^{2}2\theta _{12})\ , \end{eqnarray}$$

(25)

$$\begin{eqnarray} P_{\nu _{e}\rightarrow \nu _{e}}= 1-\frac{1}{2}\sin ^{2}2\theta _{12}\ , \end{eqnarray}$$

(26)

with θ₁₂ ≃ |$33{_{.}^{\circ}}5$| (Esteban et al. 2020), and |$P_{\nu _{\beta }\rightarrow \nu _{\alpha }} = P_{\bar{\nu }_{\beta }\rightarrow \bar{\nu }_{\alpha }}$|⁠. The luminosity distance d_L(z) is defined in a flat lambda cold dark matter cosmology:

$$\begin{eqnarray} d_{\rm L}(z) = (1+z) \frac{c}{H_{0}} \int _{0}^{z} \frac{{\rm d}z^\prime }{\sqrt{\Omega _{\Lambda }+\Omega _{\rm M}(1+z^\prime)^3}}\ , \end{eqnarray}$$

(27)

where Ω_M = 0.315, Ω_Λ = 0.685, and the Hubble constant is H₀ = 67.4 km s⁻¹ Mpc⁻¹ (Aghanim et al. 2020).

Due to the better angular resolution of muon-induced track events compared to cascades, we focus on muon neutrinos and antineutrinos. Therefore, the event rate expected at the IceCube Neutrino Observatory, after taking into account neutrino flavor conversion, is

$$\begin{eqnarray} \dot{N}_{\nu _{\mu }+\bar{\nu }_{\mu }}= \int _{E_{\nu ,1}}^{E_{\nu ,2}} {\rm d}E_{\nu } A_{\rm {eff}}(E_{\nu }, \alpha) F_{\nu _{\mu }+\bar{\nu }_{\mu }}(E_{\nu }, t)\ , \end{eqnarray}$$

(28)

where A_eff(E_ν, α) is the detector effective area (IceCube Collaboration et al. 2021) for a SN at declination α.

6.2 Expected number of neutrino events for SN 2020usa and SN 2020in

To investigate the expected number of neutrino events, we select two among the brightest sources observed by ZTF whose observable properties are summarized in Table 2: SN2020usa⁵ and SN2020in.⁶ We retrieve the photometry data of the sources in the ZTF-g and ZTF-r bands, and correct the measured fluxes for Galactic extinction (Schlafly & Finkbeiner 2011). Using linear interpolation of the individual ZTF-r and ZTF-g light curves, we perform a trapezoid integration between the respective centre wavelengths to estimate the radiated energy at each time of measurement. The resulting light curve is interpolated with Gaussian process regression (Ambikasaran et al. 2015) and taken as a lower limit to the bolometric SN emission. From such pseudo-bolometric light curve, the rise time and peak luminosity are determined. The rise time is defined as the difference between the peak time and the estimated SN breakout time. The latter is determined by taking the average between the time of the first detection in ZTF-g or ZTF-r bands and the last non-detection in either band. In what follows, we consider the radiative efficiency ε_rad = L_{peak, obs}/L_peak as a free parameter in the range ε_rad ∈ [0.2, 0.7] (see e.g. Villar et al. 2017). Furthermore, we assume that ε_rad = E_{rad, obs}/E_{diss, thick} also holds.

Table 2.

Characteristic properties of our representative SLSNe, SN 2020usa, and SN 2020in.

	Redshift	trise, obs (d)	Lpeak, obs (⁠|$\mathrm{erg\, s^{-1}}$|⁠)	Erad, obs (erg)	tdur, obs (d)	Declination (°)
SN 2020usa	0.26	65	8 × 1043	1.3 × 1051	350	−2.3
SN 2020in	0.11	42	3 × 1043	3.3 × 1050	413	20.2

For both SNe, we perform a scan over the SN model parameters (E_k, M_ej, M_CSM, and R_CSM) which fulfill the following conditions:

• t_rise ∈ [1, 1.5] × t_{rise, obs}, namely we allow an error up to 50 per cent on the estimation of t_rise.

• L_peak ≥ L_{peak, obs}.

• E_k > E_{rad, obs}. We narrow the investigation range to E_k ∈ [10⁵¹, 2 × 10⁵²] erg for SN2020usa and E_k ∈ [4 × 10⁵⁰, 5 × 10⁵¹] erg for SN2020in, assuming that at least |$\sim 10~{{\ \rm per\ cent}}$| and at most 80 per cent of the total energy E_k is radiated.

• t_{dur, th} ≥ t_{dur, obs}. Here, t_{dur, obs} is the observational temporal window available for each SN event, while t_{dur, th} = t(R_ph) − t(R_bo) is the time that the shock takes to cross the optically thick part of the CSM envelope after breakout (as mentioned in Section 2.3, the shock is expected to peak in the X-ray band once out of the optically thick part).

Fig. 6 shows the total number of muon neutrino and antineutrino events, integrated over the duration of the interaction in the CSM for E_ν ≥ 1 TeV, expected at IceCube in the the wind scenario, for E_k selected to maximize the space of parameters compatible with the conditions mentioned above. Similarly to Fig. 5, the regions with the largest number of neutrino events are those with lower M_ej and larger R_CSM, for fixed M_CSM. It is important to note that, for given observed SN properties (L_{peak, obs} and t_{rise, obs}), the expected number of neutrino events is not unique; in fact, as shown in Section 5, there is degeneracy in the SN model parameters that leads to the same L_{peak, obs} and t_{rise, obs}.

Figure 6.

Contour plot of the number of muon neutrino and antineutrino events expected at the IceCube Neutrino Observatory (for the wind scenario and integrated over the duration of CSM interaction) in the SN model parameter space compatible with the observation of SN2020usa (top panels) and SN2020in (bottom panels). Only a fraction of the SN parameter space is compatible with the optical data. Importantly, for fixed L_{peak, obs} and t_{rise, obs}, a different number of neutrino events could be obtained according to the specific combination of M_ej, M_CSM, R_CSM, E_k compatible with the observed optical properties.

Fig. 7 represents the muon neutrino and antineutrino event rate expected at IceCube for SN2020usa and SN2020in, each as a function of time for two energy ranges, and for the most optimistic scenario. Fig. 8 displays the corresponding cumulative neutrino number of events for both most optimistic and pessimistic scenarios. The two cases are selected after scanning over ε_rad. The smaller is ε_rad, the higher E_k is needed to explain the observations, and since we adopt a fixed fraction of the shock energy ε_p that goes into acceleration of relativistic protons, the best case for neutrino production is the one with the lowest ε_rad. Choosing ε_{rad, min} = 0.2, we only select the SN parameters that satisfy the following conditions: t_rise ∈ [1, 1.5] × t_{rise, obs}, L_peak ∈ [1, 1.5] × L_{peak, obs}/ε_{rad, min}, and E_rad ∈ [1, 1.5] × E_{rad, obs}/ε_{rad, min}, hence considering an error on L_{peak, obs} and E_{rad, obs} of at most 50 per cent. After an initial rise, the neutrino event rate for both considered energy ranges (100 GeV–100 TeV and 100 TeV–1 PeV) decreases with time, with a steeper rate for the high-energy range, where the slow increase of E_{p, max} does not compensate the drop in the CSM density. Note that the cumulative number of neutrino events is relatively small because, although the SN 2020usa and SN 2020 in have large L_{peak, obs}, they occurred at relatively large distance from Earth (∼ Gpc), as evident from Table 2. If other SNe exhibiting similar photometric properties should be observed at smaller z, then the expected neutrino flux should be rescaled with respect to the results shown here by the SN distance squared (see Section 6.4 and Fig. 10).

Figure 7.

Muon neutrino and antineutrino event rates predicted for SN 2020usa and SN 2020in at the IceCube Neutrino Observatory as functions of time in the observer frame, after the shock breakout, assuming ε_rad = 0.2. The SN model parameters have been chosen to optimize neutrino production [|$M_{\rm {ej}}=5.5\, {\rm M}_{\odot }$|⁠, |$M_{\rm {CSM}}=48\, {\rm M}_{\odot }$|⁠, |$R_{\rm {CSM}}=5.5\times 10^{16}\, \mathrm{cm}$|⁠, |$E_{\rm {k}}=10^{52}\, \mathrm{erg}$| for SN2020usa; |$M_{\rm {ej}}=5\, {\rm M}_{\odot }$|⁠, |$M_{\rm {CSM}}=46 \, {\rm M}_{\odot }$|⁠, |$R_{\rm {CSM}}= 10^{17}\, \mathrm{cm}$|⁠, |$E_{\rm {k}}=5\times 10^{51}\, \mathrm{erg}$| for SN2020in]. The event rate increases slightly more slowly in the high-energy band (100 TeV–1 PeV) with respect to the low energy one at early times, and it declines after peak because of the decreasing trend of v_sh as a function of time. The grey vertical lines indicate the time at which the shock reaches the photospheric radius R_ph (solid and dashed for SN 2020 usa and SN 2020in, respectively).

Figure 8.

Cumulative number of muon neutrino and antineutrino events for SN 2020usa and SN2020in, as functions of time in the observer frame. The solid and dashed lines correspond to the the most optimistic and pessimistic cumulative number of events in the indicated energy range, respectively. The SN model parameters for the most optimistic scenario are the same as the ones in Fig. 7, while the parameters leading to the most pessimistic conditions for neutrino production are |$M_{\rm {ej}}=1\, {\rm M}_{\odot }$|⁠, |$M_{\rm {CSM}}=25\, {\rm M}_{\odot }$|⁠, |$R_{\rm {CSM}}=9\times 10^{15}\, \mathrm{cm}$|⁠, |$E_{\rm {k}}=2\times 10^{51}\, \mathrm{erg}$| for SN 2020usa, and |$M_{\rm {ej}}=1.6\, {\rm M}_{\odot }$|⁠, |$M_{\rm {CSM}}=10\, {\rm M}_{\odot }$|⁠, |$R_{\rm {CSM}}=9\times 10^{15}\, \mathrm{cm}$|⁠, |$E_{\rm {k}}=7\times 10^{50}\, \mathrm{erg}$|⁠, for SN2020in. In both cases, ε_rad = 0.7. Neutrinos in the the energy range [100 TeV, 1 PeV] are not produced in the pessimistic scenarios. The grey vertical lines indicate the time at which the shock reaches the photospheric radius R_ph.

Fig. 9 shows, for the most optimistic SN model parameter configuration, a comparison between |$L_{\nu _{\mu }+\bar{\nu }_{\mu }}$| (obtained taking into account flavor oscillation) and the optical luminosity for SN 2020usa and SN 2020in. Besides the difference in the intrinsic optical brightness, the two SNe display comparable evolution in the neutrino luminosity, with the neutrino luminosity peak being ∼3 times brighter for SN 2020usa than SN 2020in. This is due to the fact that t_rise and L_peak for both SNe are such to lead to similar SN model parameters for what concerns the most optimistic prospects for neutrino emission. Note that an investigation that also takes into account the late evolution of the optical light curve might have an impact on this result, but it out of the scope of this work.

Figure 9.

Muon neutrino and antineutrino (taking into account flavor oscillation, in blue and orange) and optical luminosities (after interpolation, in green) for SN 2020usa (solid lines) and SN 2020in (dashed lines) as functions of time in the source frame. The two selected SNe exhibit a comparable evolution of the total neutrino luminosity (blue lines) because t_rise and L_peak for both SNe are such to lead to similar parameters for what concerns the most optimistic prospects for neutrino emission. The blue curves have been obtained by considering the 100 GeV–1 PeV energy range. The orange lines represent the neutrino luminosity in the high-energy range 100 TeV–1 PeV and show how the peak of the high-energy neutrinos is shifted [up to |$\mathcal {O}(\rm {100\, days})$|] with respect to the optical peak.

6.3 Characteristics of the detectable neutrino signal

The neutrino luminosity curve does not peak at the same time as the optical light curve, as visible from Fig. 9. In fact, the position of the optical peak is intrinsically related to propagation effects of photons in the CSM, and thus to the CSM properties, as discussed in Section 3 and Appendix  A. The peak in the neutrino curve, instead, solely depends on the CSM radial density distribution and the evolution of the maximum spectral energy. Because of this, the neutrino event rate as well as the neutrino luminosity in the high-energy range (100 TeV–1 PeV) peak at |$t|_{E^{\ast }_{\rm {p,max}}}$|⁠, namely the time at which the maximum proton energy is reached (see Appendix  B for E_{p, max} and Fig. 11 for the trend of the neutrino flux at Earth).

Figure 10.

Number of muon neutrino and antineutrino events expected at the IceCube Neutrino Observatory (solid lines) and IceCube Gen2 (dashed lines) as functions of the redshift for a benchmark SN with the same properties of SN 2020usa but located at declination α = 0 and variable z. The number of neutrino events is obtained integrating up to 200 d to optimize the signal discrimination with respect to the background. The redshift of SN 2020usa is marked with a dashed orange line to guide the eye. The core-collapse SN rate is plotted as a dot–dashed line (see y-axis scale on the right), in order to compare the expected number of neutrino events with the probability of finding SNe at a given z; the core-collapse SNe rate should be considered as an upper limit of the rate of interaction-powered SNe and SLSNe (see the main text for details). We expect |$N_{\nu _\mu +\bar{\nu }_\mu } =10$| at z ≃ 0.002 (d_L ≥ 9 Mpc) for IceCube and z ≃ 0.003 (d_L ≥ 13 Mpc) for IceCube-Gen2.

Figure 11.

Contour plot of the muon neutrino and antineutrino energy flux expected at Earth for SN2020usa in the best-case scenario and in the plane spanned by the arrival time of neutrinos and the neutrino energy. At low energies, the neutrino flux decreases with time after the breakout. At high energies (E_ν ≳ 100 TeV), instead, it increases with time, peaks at around 100 d, and then decreases. This is related to the time of maximum E_{p, max} (see also Fig. 9). The white colour marks the regions where the flux is zero.

The most favorable time window for detecting energetic neutrinos (≳ 100 TeV) would be a few times t_rise around the electromagnetic bolometric peak, which corresponds to |$\mathcal {O}(100\, \rm {days})$| days for E_k ≲ 10⁵² erg, M_ej ≲ 10 M_⊙, M_CSM ≲ 20 M_⊙, and R_CSM ≲ few × 10¹⁶ cm (see Fig. B1). Interestingly, the IceCube neutrino event IC200530A associated with the candidate SLSN event AT2019fdr was detected about 300 d after the optical peak (Pitik et al. 2022), in agreement with our findings.⁷

Fig. 10 shows the dependence of the number of neutrino events expected in IceCube and IceCube-Gen2 in a temporal window of 200 d and as functions of the redshift for a benchmark SN with the same properties of SN 2020usa, but placed at declination α = 0° and redshift z. We consider the number of neutrino events expected in a time window of 200 d in order to optimize the signal over background classification (see Section 6.5). One can see that IceCube expects to detect |$N_{\nu _\mu +\bar{\nu }_\mu } \gtrsim 10$| for SNe at distance ≲ 9 Mpc (z ≲ 0.002); while |$N_{\nu _\mu +\bar{\nu }_\mu } \gtrsim 10$| should be detected for SNe at a distance ≲ 13 Mpc (z ≲ 0.003) for IceCube-Gen2 (Aartsen et al. 2021).

In order to compare the expected number of neutrino events with the likelihood of finding SNe at redshift z, Fig. 10 also shows the core-collapse SN rate (Yuksel et al. 2008; Vitagliano et al. 2020) for reference. Note that the rate of interaction-powered SNe is very uncertain and it is not clear whether their redshift evolution follows the star formation rate (Smith et al. 2011); hence the core-collapse SN rate should be considered as an upper limit of the rate of interaction-powered SNe and SLSNe, under the assumption that the latter follow the same redshift evolution.

The evolution of the energy flux of neutrinos is displayed in Fig. 11. One can see that for E_ν ≳ 100 TeV, the energy flux increases up to around 100 d, and then decreases. This trend can be explained considering the evolution of E_{p, max} (see also Fig. 9).

6.4 Follow-up strategy for neutrino searches

Our findings in Section 5.1 suggest that a large L_peak and average t_rise are necessary, but not sufficient, to guarantee large neutrino emission. This is due to the large degeneracy existing in the SN model parameter space that could lead to SN light curves with comparable properties in the optical, but largely different neutrino emission.

Despite the degeneracy in the SN properties leading to comparable optical signals, the semi-analytical procedure outlined in this work allows to restrict the range of E_k, M_ej, M_CSM, and R_CSM that matches the measured t_rise and L_peak. This procedure then forecasts an expectation range for the number of neutrino events detectable by IceCube to guide upcoming follow-up searches (see Section 6.2 for an application to two SNe detected by ZTF), also taking into account the unknown radiative efficiency ε_rad.

For measured t_rise and L_peak, through the method outlined in this paper, it is possible to predict the largest expected number of neutrino events. On the other hand, if an interaction-powered SN should be detected in the optical, and no neutrino should be observed, this would imply that the SN model parameters compatible with the measured t_rise and L_peak are not optimal for neutrino production.

Our findings highlight the need to carry out multiwavelength SN observations to better infer the SN properties and then optimize neutrino searches through the procedure presented in this work. In fact, relying on radio and X-ray all-sky surveys, one could narrow down the values of M_ej, M_CSM, and R_CSM compatible with the data (Chevalier & Fransson 2017; Margalit et al. 2022). Because CSM interaction signatures appear clearly in the UV, early SN observations by future UV satellites, such as ULTRASAT (Shvartzvald et al. 2023), will be critical to provide insights into the CSM properties. Further information on the CSM can be obtained in the X-ray regime (Margalit et al. 2022), for example, through surveys such as the extended ROentgen Survey with an Imaging Telescope Array (eROSITA; Predehl et al. 2010). In addition, the Vera Rubin Observatory (LSST Science Collaboration et al. 2009) will detect numerous SNe providing a large sample for a neutrino stacking analysis.

6.5 Multimessenger follow-up programs

There are two ways to search for neutrinos from SNe.

• One can compile a catalogue of SNe detected by electromagnetic surveys and use archival all-sky neutrino data to search for an excess of neutrinos from the catalogued sources. Such a search is most sensitive when a stacking of all sources is applied (see e.g. Abbasi et al. 2023). The stacking requires a weighting of the sources relative to each other. Previous searches assumed that all sources are neutrino standard candles, i.e. the neutrino flux at Earth would scale with the inverse of the square of the luminosity distance, or used the optical peak flux as a weight. This work indicates that neither of those assumptions is justified. Modelling of the multiwavelength emission can yield a source-by-source prediction of the neutrino emission, which can be used as a weight.

  Another important analysis choice is the time window to consider for the neutrino search. A too-long time window increases the background of atmospheric neutrinos, while a too-short time window cuts parts of the signal. The prediction of the temporal evolution of the neutrino signal by our modelling allows to optimize the neutrino search time window. Finally, also the spectral energy distribution of neutrinos from SNe can be used to optimize the analysis in terms of background rejection.

• One can utilize electromagnetic follow-up observations of neutrino alerts released by neutrino telescopes (see e.g. Aartsen et al. 2017). Also here, defining a time window in order to assess the coincidence between an electromagnetic counterpart and the neutrino alert will be crucial. Once a potential counterpart is identified further follow-up observations (e.g. spectroscopy and multiple wavelength) can be scheduled to ensure classification of the source as SN and allow for a characterization of the CSM properties.

  In order to forecast the expected neutrino signal reliably and better guide neutrino searches, in addition to optical data, input from X-ray and radio surveys would allow to characterize the CSM properties (see Section 6.4). In addition, it would be helpful to guide neutrino searches relying on the optical spectra at different times to characterize the duration of the interaction.

In summary, the modelling of particle emission from SNe presented in this paper will be crucial to guide targeted multimessenger searches.

7 CONCLUSIONS

Supernovae and SLSNe of Type IIn show in their spectra strong signs of circumstellar interaction with a hydrogen-rich medium. The interaction between the SN ejecta and the CSM powers thermal UV/optical emission as well as high-energy neutrino emission. This work aims to explore the connection between the energy emitted in neutrinos detectable at the IceCube Neutrino Observatory (and its successor IceCube-Gen2) and the photometric properties of the optical signals observable by wide-field, high-cadence surveys. Our main goal is to outline the best follow-up strategy for upcoming multimessenger searches.

We rely on a semi-analytical model that connects the optical light curve observables to the SN properties and the correspondent neutrino emission, we find that the largest energy emitted in neutrinos and antineutrino is expected for large SN kinetic energy (E_k ≳ 10⁵¹ erg), small ejecta mass (⁠|$M_{\rm {ej}}\lesssim 10 \, {\rm M}_{\odot }$|⁠), intermediate CSM mass (⁠|$1\, {\rm M}_{\odot }\lesssim M_{\rm {CSM}}\lesssim 30\, {\rm M}_{\odot }$|⁠), and extended CSM (R_CSM ≳ 10¹⁶ cm). Such parameters lead to large bolometric peak luminosity (L_peak ≳ 10⁴³–10⁴⁴ erg) and average rise time (t_rise ≳ 10–40 d). However, these light curve features are necessary but not sufficient to guarantee ideal conditions for neutrino detection. In fact, different configurations of the SN model parameters could lead to comparable optical light curves, but vastly different neutrino emission.

The degeneracy between the optical light-curve properties and the SN model parameters challenges the possibility of outlining a simple procedure to determine the expected number of IceCube neutrino events by solely relying on SN observations in the optical. While our method allows to foresee the largest possible number of neutrino events for given L_peak and t_rise, the eventual lack of neutrino detection for upcoming nearby SNe could hint towards SN properties that are different with respect to the ones maximizing the neutrino signal, therefore constraining the SN model parameter space compatible with neutrino and optical observations.

We also find that the peak of the neutrino curve does not coincide with the one of the optical light curve. Hence, one should consider a time window of a few t_rise around L_peak when looking for neutrinos. The time window should indeed be optimized to guarantee a fair signal discrimination with respect to the background.

Our findings suggest that previous neutrino stacking searches that assumed all SNe as neutrino standard candles, or used weights based on optical peak flux, might have not been optimal, as they do not take into account the diversity in the SN properties leading to a large variation in the number of neutrinos expected at Earth. Importantly, multiwavelength observations are necessary to break the degeneracy between the optical light curve and the SN properties and will be essential to forecast the expected neutrino signal and optimize multimessenger searches.

ACKNOWLEDGEMENTS

We are grateful to Takashi Moriya for insightful discussions, Steve Schulze for discussions on the ZTF light-curve data, and Jakob van Santen for exchanges concerning the IceCube-Gen2 effective areas. We acknowledge support from the Villum Foundation (Project No. 13164), the Carlsberg Foundation (CF18-0183), as well as the Deutsche Forschungsgemeinschaft through Sonderforschungbereich SFB 1258 ‘Neutrinos and Dark Matter in Astro- and Particle Physics’ (NDM) and the Collaborative Research Center SFB 1491 ‘Cosmic Interacting Matters – from Source to Signal’. TP also acknowledges support from Fondo Ricerca di Base 2020 (MOSAICO) of the University of Perugia.

DATA AVAILABILITY

Data can be shared upon reasonable request to the authors.

Footnotes

1

Note that the identification of the nature AT2019fdr is still under debate; it has been suggested that its properties might be compatible with the ones of a tidal disruption event (Reusch et al. 2022).

2

Note that we do not adopt the common approximation R_bo ≡ (κ_esKv_sh)/c, valid only when R_bo ≪ R_CSM and v_sh independent of R (Chevalier & Irwin 2011).

3

This definition of the rise time is valid as long as the CSM is dense enough to cause shock breakout in the CSM wind. If this is not the case, the breakout occurs at the surface of the collapsing star; the CSM masses responsible for this scenario are not considered in our investigation.

4

Note that this distinction should not be confused with the ejecta/CSM-dominated phases introduced in Section 2.2.

5

https://lasair-ztf.lsst.ac.uk/objects/ZTF20acbcfaa/

6

https://lasair-ztf.lsst.ac.uk/objects/ZTF20aaaweke/

7

AT2019fdr occurred at z ≃ 0.27, and the optical light curve displayed t_rise = 98 d and L_peak = 2.1 × 10⁴⁴ erg s^–1, considering a radiative efficiency of 18–35 per cent, Pitik et al. (2022) estimated that about 4.6 × 10⁻² muon neutrino and antineutrino events were expected assuming excellent discrimination of the atmospheric background.

APPENDIX A: DEPENDENCE OF THE SUPERNOVA LIGHT-CURVE PROPERTIES ON THE MODEL PARAMETERS

In this appendix, we investigate the dependence of the parameters characteristic of the light curve on the SN model properties. Fig. A1 displays how the rise time t_rise (defined in Section 3) of the bolometric luminosity depends on the SN parameters of interest. For any fixed combination of E_k, M_ej, and R_CSM, the rise time increases with M_CSM, since a denser CSM extends the photon diffusion time. In the left-hand panel, we see that the larger the kinetic energy, the shorter t_rise. This is explained by the fact that large shock velocities cause the breakout to happen later and shorten the time that photons take to reach the photosphere. The same trend is expected for decreasing M_ej, as shown in the middle panel, where a mild trend in this direction is noticeable. Furthermore in the BW regime (which corresponds to the breaks in the curves) we see that t_rise is independent of M_ej (the curves for low M_ej saturate at the same value), a trend confirmed by the numerical simulations presented in (Suzuki et al. 2020). In the right-hand panel of Fig. A1, one can observe that for large CSM masses, there is a transition region shifting towards larger R_CSM where the trend of t_rise is reversed. The reason of this inversion is to be found in the dependence of the photospheric radius on R_CSM (see equation 12), which for fixed M_CSM increases and saturates at a certain R_CSM, to turn and decrease for larger CSM radii.

Figure A1.

Rise time of the bolometric light curve (top panels), bolometric peak luminosity (middle panels), dissipated energy in the optically thick part of the CSM envelope (bottom panels) as functions of the CSM mass and E_k (left-hand panels, for fixed M_ej and R_CSM), the ejecta mass (middle panels, for fixed E_k and R_CSM), and R_CSM (right-hand panels, for fixed E_k and M_ej), respectively. In each panel, the arrow indicates the direction of increase of the parameter under investigation, for example, in the left-hand panel of the first row, t_rise decreases for increasing E_k, for fixed M_CSM; t_rise increases with M_CSM since a denser CSM envelope increases the optical depth and delays the photon escape. From the top left-hand and middle panels, we see that for increasing E_k or decreasing M_ej, the diffusion time becomes shorter. Indeed both the increase of E_k and the decrease of M_ej are responsible for an increase of the shock velocity, which in turn causes the radius where the photon diffusion velocity exceeds the shock velocity to shift outwards. In the top right-hand panel, we observe that for small R_CSM an initial increase of t_rise, which declines again for larger CSM radii. This transition region is related to the photosphere dependence on M_CSM and R_CSM. For what concerns L_peak, in all three middle panels, we see that L_peak initially increases with M_CSM in the FE regime. When transitioning to the BW regime (indicated by the breaks in the curves), a saturation of the radiated energy occurs and this, together with the increase of t_rise, causes L_peak to drop as M_CSM increases. Larger E_k and smallerM_ej are responsible for a larger shock velocity, and thus an increase of L_peak, as it can be seen in the left-hand and middle panels. In the right middle panel, we observe that small R_CSM, for fixed M_CSM, make the medium denser and therefore it is easier to dissipate the ejecta energy, leading to an increase of L_peak. Similarly, E_{diss, thick} increases with M_CSM, and saturates to a constant fraction of E_k in the BW regime. The dots are coloured according to the M_CSM value, as shown in the colour bar.

The middle panels of Fig. A1 show that an increase in M_CSM makes L_peak larger in all cases, since a larger M_CSM causes more kinetic energy to be dissipated and radiated. This is true as long as the shock is in the FE regime. In the BW regime, L_peak declines with M_CSM. The left-hand and middle panels show that the peak luminosity increases with larger ejecta energy and smaller ejecta masses, since both make the shock velocity larger and thus more energetic. In the BW regime, the peak luminosity becomes independent of the ejecta mass, as confirmed by the saturation to the same branch for low M_ej. The right-hand panel shows that the brightest light curves are obtained when the CSM is more compact, i.e. for the smallest R_CSM (apart from the transition region visible for large M_CSM, due to the transition into the BW regime).

The bottom panels show the trend of E_{diss, thick}. The dissipated energy in the optically thick part of the CSM increases with M_CSM, is very large for small M_ej and R_CSM, since the first allows for high shock velocity, and the second for very compact region, and thus high densities.

APPENDIX B: DEPENDENCE OF THE MAXIMUM PROTON ENERGY ON THE SUPERNOVA MODEL PARAMETERS

In this appendix, we analyze the dependence of the maximum E_{p, max} on the SN model parameters. To do so, we first highlight the dependence on the SN parameters of the main time-scales entering the problem. From equations (10) and (11), we see that the plasma cooling time-scale scales as

$$\begin{eqnarray} t_{\rm {cool}}\propto \frac{1}{n_{\rm {sh}}}\times \left\lbrace \begin{array}{@{}l@{\quad }l@{}}v_{\rm {sh}}^{16/5}\quad \, \textrm {if}\quad 10^{5} \lt T \lesssim 4.7\times 10^{7}~\rm {K}\\ v_{\rm {sh}}\quad \, \, \textrm {if} \quad T\gt 4.7\times 10^{7}~\rm {K}\ . \end{array}\right. \end{eqnarray}$$

(B1)

For the wind scenario, it becomes

• for R < R_dec:

  $$\begin{eqnarray} t_{\rm {cool}}\propto \bigg (\frac{R_{\rm {CSM,w}}}{M_{\rm {CSM,w}}}\bigg)\times \left\lbrace \begin{array}{@{}l@{\quad }l@{}}R^{54/35}\quad \, \textrm {if}\quad 10^{5} \lt T \lesssim 4.7\times 10^{7}~\rm {K}\\ R^{13/7}\quad \quad \quad \textrm {if} \quad T\gt 4.7\times 10^{7}~\rm {K}\ . \end{array}\right. \end{eqnarray}$$

  (B2)
• for R > R_dec:

  $$\begin{eqnarray} t_{\rm {cool}}\propto \bigg (\frac{R_{\rm {CSM,w}}}{M_{\rm {CSM,w}}}\bigg)\times \left\lbrace \begin{array}{@{}l@{\quad }l@{}}R^{2/5}\quad \, \textrm {if}\quad 10^{5} \lt T \lesssim 4.7\times 10^{7}~\rm {K}\\ R^{3/2}\quad \quad \quad \textrm {if} \quad T\gt 4.7\times 10^{7}~\rm {K}\ . \end{array}\right. \end{eqnarray}$$

  (B3)

For the shell scenario, it is

• for R < R_dec:

  $$\begin{eqnarray} t_{\rm {cool}}\propto \bigg (\frac{R_{\rm {CSM,s}}^{3}}{M_{\rm {CSM,s}}}\bigg)\times \left\lbrace \begin{array}{@{}l@{\quad }l@{}}R^{-48/35}\quad \, \textrm {if}\quad 10^{5} \lt T \lesssim 4.7\times 10^{7}~\rm {K}\\ R^{-3/7}\quad \quad \quad \textrm {if} \quad T\gt 4.7\times 10^{7}~\rm {K}\ . \end{array}\right. \end{eqnarray}$$

  (B4)
• for R > R_dec:

  $$\begin{eqnarray} t_{\rm {cool}}\propto \bigg (\frac{R_{\rm {CSM,s}}^{3}}{M_{\rm {CSM,s}}}\bigg)\times \left\lbrace \begin{array}{@{}l@{\quad }l@{}}R^{-24/5}\quad \, \textrm {if}\quad 10^{5} \lt T \lesssim 4.7\times 10^{7}~\rm {K}\\ R^{-3/2}\quad \quad \quad \textrm {if} \quad T\gt 4.7\times 10^{7}~\rm {K}\ . \end{array}\right. \end{eqnarray}$$

  (B5)

The acceleration time-scales as |$t_{\rm {acc}}\propto E_{\rm {p}} v_{\rm {sh}}^{-3}n_{\rm {sh}}^{-1/2}$|⁠, given |${B\propto v_{\rm {sh}}n_{\rm {sh}}^{1/2}}$|⁠. For the wind scenario, it is

$$\begin{eqnarray} t_{\rm {acc}}\propto \bigg (\frac{R_{\rm {CSM,w}}}{M_{\rm {CSM,w}}}\bigg)^{1/2} E_{\rm {p}}\times \left\lbrace \begin{array}{@{}l@{\quad }l@{}}R^{10/7}\quad \, \textrm {if}\quad R\lt R_{\rm {dec}}\\ R^{5/2}\quad \textrm {if} \quad R\gt R_{\rm {dec}}\ ; \end{array}\right. \end{eqnarray}$$

(B6)

while for the shell scenario, it is

$$\begin{eqnarray} t_{\rm {acc}}\propto \bigg (\frac{R_{\rm {CSM,s}}^{3}}{M_{\rm {CSM,s}}}\bigg)^{1/2} E_{\rm {p}}\times \left\lbrace \begin{array}{@{}l@{\quad }l@{}}R^{9/7}\quad \, \textrm {if}\quad R\lt R_{\rm {dec}}\\ R^{9/2}\quad \textrm {if} \quad R\gt R_{\rm {dec}}\ . \end{array}\right. \end{eqnarray}$$

(B7)

The proton–proton interaction time t_pp = (cn_shσ_pp)⁻¹ is

$$\begin{eqnarray} t_{\rm {pp}}\propto \left\lbrace \begin{array}{@{}l@{\quad }l@{}}\frac{R_{\rm {CSM,w}}}{M_{\rm {CSM,w}}}\times R^{2}\quad \textrm {for~the~wind}\\ \frac{R_{\rm {CSM,s}}^{3}}{M_{\rm {CSM,s}}}\quad \quad \quad \ \ \textrm {for~the~shell}\ . \end{array}\right. \end{eqnarray}$$

(B8)

Using the relations above, we can investigate how E_{p, max} depends on the SN model parameters and how it evolves with the shock radius. If t_cool is the min[t_cool, t_dyn, t_pp], the maximum proton energy is determined by t_acc = t_cool. For the wind scenario,

• for R < R_dec:

  $$\begin{eqnarray} E_{\rm {p,max}}\!\propto\! \bigg (\!\frac{R_{\rm {CSM,w}}}{M_{\rm {CSM,w}}}\!\bigg)^{1/2}\!\times \! \left\lbrace\! \begin{array}{@{}l@{\quad }l@{}}R^{4/35}\quad \, \textrm {if}\quad 10^{5} \lt T \lesssim 4.7\!\times \! 10^{7}~\rm {K}\\ R^{3/7}\quad \, \, \textrm {if} \quad T\gt 4.7\!\times \! 10^{7}~\rm {K}\ ; \end{array}\right.\nonumber\\ \end{eqnarray}$$

  (B9)
• for R > R_dec:

  $$\begin{eqnarray} E_{\rm {p,max}}\!\propto\! \bigg (\!\frac{R_{\rm {CSM,w}}}{M_{\rm {CSM,w}}}\!\bigg)^{1/2}\!\times \! \left\lbrace\! \begin{array}{@{}l@{\quad }l@{}}R^{-21/10}\quad \, \textrm {if}\quad 10^{5} \lt T \lesssim 4.7\!\times \! 10^{7}~\rm {K}\\ R^{-1}\quad \quad \quad \textrm {if} \quad T\gt 4.7\!\times \! 10^{7}~\rm {K}\ . \end{array}\right.\!\!\!\!\!\nonumber\\ \end{eqnarray}$$

  (B10)

For the shell scenario, instead, it is

• for R < R_dec:

  $$\begin{eqnarray} E_{\rm {p,max}}\!\propto\! \bigg (\!\frac{R_{\rm {CSM,s}}^{3}}{M_{\rm {CSM}}}\!\bigg)^{1/2}\!\times \! \left\lbrace\! \begin{array}{@{}l@{\quad }l@{}}R^{-93/35}\, \, \textrm {if}\quad 10^{5} \lt T \lesssim 4.7\!\times \! 10^{7}~\rm {K}\\ R^{-12/7}\quad \, \, \textrm {if} \quad T\gt 4.7\!\times \! 10^{7}~\rm {K}\ . \end{array}\right.\nonumber\\ \end{eqnarray}$$

  (B11)
• for R > R_dec:

  $$\begin{eqnarray} E_{\rm {p,max}}\!\propto\! \bigg (\!\frac{R_{\rm {CSM,s}}^{3}}{M_{\rm {CSM}}}\!\bigg)^{1/2}\!\times \! \left\lbrace \!\begin{array}{@{}l@{\quad }l@{}}R^{-93/10}\, \, \textrm {if}\quad 10^{5} \lt T \lesssim 4.7\!\times \! 10^{7}~\rm {K}\\ R^{-6}\quad \quad \textrm {if} \quad T\gt 4.7\!\times \! 10^{7}~\rm {K}\ . \end{array}\right.\nonumber\\ \end{eqnarray}$$

  (B12)

If t_pp corresponds to the min[t_cool, t_dyn, t_pp], then the maximum proton energy is determined by t_acc = t_pp and can be written for the wind scenario as

$$\begin{eqnarray} E_{\rm {p,max}}\propto \bigg (\frac{R_{\rm {CSM,w}}}{M_{\rm {CSM,w}}}\bigg)^{1/2}\times \left\lbrace \begin{array}{@{}l@{\quad }l@{}}R^{4/7} \quad \textrm {for}\quad R\lt R_{\rm {dec}}\\ R^{-1/2}\quad \textrm {for}\quad R\gt R_{\rm {dec}}\ , \end{array}\right. \end{eqnarray}$$

(B13)

and for the shell scenario as

$$\begin{eqnarray} E_{\rm {p,max}}\propto \bigg (\frac{R_{\rm {CSM,s}}^{3}}{M_{\rm {CSM,s}}}\bigg)^{1/2}\times \left\lbrace \begin{array}{@{}l@{\quad }l@{}}R^{-9/7} \quad \textrm {for}\quad R\lt R_{\rm {dec}}\\ R^{-9/2}\quad \textrm {for}\quad R\gt R_{\rm {dec}}\ . \end{array}\right. \end{eqnarray}$$

(B14)

Finally, if t_dyn corresponds to min[t_cool, t_dyn, t_pp], the maximum proton energy is determined by t_acc = t_dyn and for the wind scenario it is

$$\begin{eqnarray} E_{\rm {p,max}}\propto \bigg (\frac{M_{\rm {CSM,w}}}{R_{\rm {CSM,w}}}\bigg)^{1/2}\times \left\lbrace \begin{array}{@{}l@{\quad }l@{}}R^{-2/7} \quad \textrm {for}\quad R\lt R_{\rm {dec}}\\ R^{-1}\quad \textrm {for}\quad R\gt R_{\rm {dec}}\ , \end{array}\right. \end{eqnarray}$$

(B15)

while, for the shell scenario, it is

$$\begin{eqnarray} E_{\rm {p,max}}\propto \bigg (\frac{M_{\rm {CSM,s}}}{ R_{\rm {CSM,s}}^{3}}\bigg)^{1/2}\times \left\lbrace \begin{array}{@{}l@{\quad }l@{}}R^{1/7} \quad \textrm {for}\quad R\lt R_{\rm {dec}}\\ R^{-2}\quad \textrm {for}\quad R\gt R_{\rm {dec}}\ . \end{array}\right. \end{eqnarray}$$

(B16)

Note that we assume constant σ_pp ∼ 3 × 10⁻²⁶ cm² for the sake of simplicity in this appendix in order to obtain the above analytical relations.

We immediately see from the relations above that for the wind scenario, independently on the cooling mechanism, the maximum proton energy has a decreasing trend with R in the deceleration phase (R > R_dec). However, in the ejecta-dominated phase (R < R_dec), the maximum proton energy always increases, except for the case in which the adiabatic cooling is dominant (equation B15). Finally, in the shell scenario, E_{p, max} always decreases, apart from the case where t_cool and t_pp are too long compared to the dynamical time, and it slowly increases in the free-expansion phase.

We define R_cool as the radius where t_dyn = t_cool, and R_pp the radius where t_dyn = t_pp. The maximum value of E_{p, max}, denoted as |$E^{\rm {\ast }}_{\rm {p,max}}$|⁠, can be achieved at any of the following radii: R_bo, R_cool, R_pp, R_dec, or R_CSM. There are various configurations of such radii. If for example R_bo < R_cool < R_pp < R_CSM < R_dec, and both t_dyn < t_cool for R > R_cool and t_dyn < t_pp for R > R_pp, then the maximum E_{p, max} is obtained at R_pp.

Note that this procedure serves to inspect the dependence of the maximum proton energy analytically. However, the total cooling time is the sum of t_dyn and t_pp or t_cool and t_pp; since the energy dependence of t_pp increases slightly at higher energies, the value of |$E^{\rm {\ast }}_{\rm {p,max}}$| that we find is underestimated by a few per cent in the transition region t_dyn ∼ t_cool and at very large energies. Fig. B1 displays how |$E^{\ast }_{\rm {p,max}}$| depends on the SN parameters. The most promising configurations that allow to reach large |$E^{\ast }_{\rm {p,max}}$| are the ones with large E_k and low M_CSM (left-hand panel), or low M_ej and low M_CSM (middle panel), or high R_CSM and low M_CSM (right-hand panel), which maximize the acceleration rate and minimize the energy loss rate.

Figure B1.

Contour plots of the maximum proton energy |$E^{\ast }_{\rm {p,max}}$| reached throughout the evolution of the shock in the wind scenario, in the plane spanned by M_CSM and E_k (left-hand panel), M_ej (middle panel), and R_CSM (right-hand panel). The dotted contours mark isocontours of |$E^{\ast }_{\rm {p,max}}$| to guide the eye. The largest proton energies can be achieved with large E_k and small M_ej, both maximizing v_sh, and thus the acceleration rate; low M_CSM and/or large R_CSM, both making the CSM less dense, and thus the proton energy losses less severe. For each panel, the grey line represents |$\Delta t_{\rm {pk}}=t|_{E^{\ast }_{\rm {p,max}}}-t_{\rm {peak}}$|⁠, i.e. the time at which the maximum proton energy is reached with respect to the bolometric peak of the light curve. The solid grey lines correspond to Δt_pk = 0. From the dashed grey line, we can see that the largest time interval is expected for low E_k, and large M_ej and R_CSM. The parameter space between the solid and the dashed grey lines leads to 0 < Δt_pk < 400 d, which is the follow-up time window adopted for SNe. The orange star marks our benchmark scenario (see Table 1).

For the fiducial parameters adopted in each panel of Fig. B1, total energies ≳ 10⁵¹ erg, relatively low ejecta (≲ 20 M_⊙), CSM masses (≲ 10 M_⊙), and extended CSM envelopes (≳ 10¹⁶ cm) are required to obtain protons with ∼ PeV energy. Furthermore, as shown through the grey contour lines, which display |$t|_{E^{\ast }_{\rm {p,max}}}-t_{\rm {peak}}$| (where |$t|_{E^{\ast }_{\rm {p,max}}}$| is the time at which the maximum proton energy is reached), the maximum |$E^{\ast }_{\rm {p,max}}$| is achieved at relatively late times [|$\mathcal {O}(100\, \rm {days})$|] with respect to the peak time t_peak = t_bo + t_rise. Such longer time-scales are expected for low kinetic energies of the ejecta, and large M_ej and R_CSM. Only the configurations with large CSM mass, due to the onset of the decelerating phase, are expected to invert the increasing trend of |$E^{\ast }_{\rm {p,max}}$| before the light curve reaches its peak.

APPENDIX C: CONSTANT DENSITY SCENARIO

In this appendix, we explore the dependence of neutrino production in the scenario of a radially independent CSM mass distribution. We follow a similar approach to the wind-profile case discussed in Section 5.2. Specifically, we investigate the connection between the total energy in neutrinos (⁠|$\mathcal {E}_{\nu +\bar{\nu }}$|⁠, see equation 22) with |$E_{\nu }\ge 1\, \mathrm{TeV}$|⁠. The results are shown in Fig. C1.

Figure C1.

The same as in Fig. 5, but for the constant density shell scenario. The beige region has been excluded from our investigation since here the breakout of the shock does not occur in the CSM shell, but at the radius of the progenitor star. The white region, visible only in the lower right-hand corner of the third column, has instead been excluded because leading to |$E^{\ast }_{\rm {p,max}}\lt \rm {10 TeV}$|⁠. The SN configurations leading to the largest outcomes in neutrinos are similar to the ones in the wind case, and are given by large SN kinetic energies (E_k ≳ 10⁵¹ erg), small ejecta masses (⁠|$M_{\rm {ej}}\lesssim 10 \, {\rm M}_{\odot }$|⁠), intermediate CSM masses with respect to M_ej (1 ≲ M_CSM ≲ 30 M_⊙), and relatively large CSM extent (R_CSM ≳ 10¹⁶ cm).

We exclude from our investigation the region of the SN parameter space where the maximum achievable proton energy is |$E^\ast _{\rm {p,max}}\le 10$| TeV. Additionally, we disregard parameters that lead to a shock breakout at the surface of the progenitor star (R_bo ≡ R_⋆), as indicated by the beige region in the contour plots. In this work, our focus is on the parameter space that results in the shock breakout occurring inside the CSM envelope. This is the first difference with the wind case, where the much higher density at smaller radii cause the shock to occur inside the wind for all the considered parameters. Isocontours of |$E^{\ast }_{\rm {p,max}}$| (first row), the rise time t_rise (second row), and the bolometric peak L_peak (third row) are also displayed on top of the |$\mathcal {E}_{\nu +\bar{\nu }}$| colourmap in Fig. C1.

The dependence of |$\mathcal {E}_{\nu +\bar{\nu }}$| on the SN model parameters is analogous to the wind scenario. Indeed we see that in all panels of Fig. C1, |$\mathcal {E}_{\nu +\bar{\nu }}$| increases with M_CSM, namely with larger target proton numbers, and then saturates once the critical ρ_CSM is reached. Beyond such critical density, pp interactions or the cooling of thermal plasma becomes too strong, limiting the maximum achievable proton energy, and thus the neutrino outcome. From the contour lines in each panel, analogously to the wind case, we see that the optimal configuration for what concerns neutrino production, results from large E_k, M_CSM ≳ M_ej, and R_CSM larger as M_CSM increases.

We see from Fig. C1 that we do not have the same regions of the parameter space excluded as in the wind case (see Fig. 5) that lead to |$E^\ast _{\rm {p,max}}\le 10$| TeV. Indeed in a constant density shell the proton maximum energy has a rather different dependence especially on the radius as discussed in Appendix  B. This leads to overall higher values of |$E^\ast _{\rm {p,max}}$| in the parameter space, as well as the times at which they are achieved during the shock evolution. Most of the parameter space in all panels leads to |$\Delta t_{\rm {pk}}=t|_{E^{\ast }_{\rm {p,max}}}-t_{\rm {peak}} \lt 0$| (see Fig. B1 for the wind case). This means that in the constant density scenario most of the energetic neutrinos are produced earlier than the bolometric peak.

With respect to the wind scenario, another difference lies in the relation between t_rise and L_peak, as can be seen from the second and third row of Fig. C1. In the case of a constant density shell, the CSM density is considerably lower. Consequently, the shock breakout tends to occur earlier than in the wind scenario, resulting in significantly smaller peak luminosities across a significant portion of the parameter space. None the less, the lower CSM density leads to larger deceleration radii compared to the wind case. As a result, a larger M_CSM is required to enter the decelerating regime, delaying the transition to the decreasing trend of L_peak with M_CSM in the blast-wave regime (as observed in the wind case in Fig. A1). As for t_rise, lower CSM densities result in longer photon mean free paths, enabling faster diffusion through the CSM envelope. Furthermore, as shown in the second row of Fig. C1, t_rise increases with M_CSM, but remains independent on M_ej and E_k for most of the parameter space. This is explained because R_bo is significantly smaller than R_ph, making the diffusion time unaffected by the shock velocity.

In summary, similar to the wind scenario, large |$\mathcal {E}_{\nu +\bar{\nu }}$| is expected for large SN kinetic energy (E_k ≳ 10⁵¹ erg), small ejecta mass (⁠|$M_{\rm {ej}}\lesssim 10 \, {\rm M}_{\odot }$|⁠), and large CSM radii, R_CSM ≳ 10¹⁶ cm. Unlike in the wind case, a larger range of M_CSM leads to comparable predictions, even if scenarios with M_CSM ≫ M_ej would limit neutrino production. Such parameters imply large bolometric luminosity peak (L_peak ≳ 10⁴³–10⁴⁴ erg) and relatively long rise times (t_rise ≳ 10–90 d). In the shell case, large t_rise do not necessarily correspond to low |$\mathcal {E}_{\nu +\bar{\nu }}$|⁠, as it is the case for the wind scenario. Furthermore, energetic neutrinos are produced at early times. Hence, if neutrinos should be observed from long-rising optical light curves relatively soon with respect to the optical peak, this might hint towards a constant density of the CSM envelope.