#  High energy particles from young supernovae: gamma-ray and neutrino
connections

Prantik Sarmah  Sovan Chakraborty  Irene Tamborra  and Katie Auchettl

###### Abstract

Young core-collapse supernovae (YSNe) are factories of high-energy neutrinos
and gamma-rays as the shock accelerated protons efficiently interact with the
protons in the dense circumstellar medium. We explore the detection prospects
of secondary particles from YSNe of Type IIn, II-P, IIb/II-L, and Ib/c. Type
IIn YSNe are found to produce the largest flux of neutrinos and gamma-rays,
followed by II-P YSNe. Fermi-LAT and the Cherenkov Telescope Array (IceCube-
Gen2) have the potential to detect Type IIn YSNe up to $10$ Mpc ($4$ Mpc),
with the remaining YSNe Types being detectable closer to Earth. We also find
that YSNe may dominate the diffuse neutrino background, especially between
$10$ TeV and $10^{3}$ TeV, while they do not constitute a dominant component
to the isotropic gamma-ray background observed by Fermi-LAT. At the same time,
the IceCube high-energy starting events and Fermi-LAT data already allow us to
exclude a large fraction of the model parameter space of YSNe otherwise
inferred from multi-wavelength electromagnetic observations of these
transients.

##  1 Introduction

The IceCube Neutrino Observatory routinely detects high-energy neutrinos of
astrophysical origin, whose origin remains unknown [1, 2, 3, 4]. Various
source classes have been proposed to explain the IceCube neutrino flux [5, 6,
7, 8], such as galaxy clusters, (low-luminosity or chocked) gamma-ray bursts,
tidal distruption events, star-forming galaxies, and supernovae (SNe) [9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. However, none of these sources
can fully explain the IceCube data, hinting that more than one source class
contributes to the overall observed flux [22, 23]. In addition to the diffuse
background of high-energy neutrinos, neutrino events have been observed in
likely associations with blazars [24, 25, 26, 27, 28, 29], tidal distruption
events [30, 31], and a hydrogen-rich super-luminous SN [32]. Moreover, ongoing
electromagnetic follow-up searches, see e.g. [33, 27, 34], promise to provide
key insight to disentangle the origin of the ever growing number of IceCube
neutrino events of astrophysical origin.

The high-energy neutrino events observed by the IceCube Neutrino Observatory
are believed to have a correspondent counterpart in gamma-rays possibly
observable by Fermi-LAT, see e.g. [35, 13, 36, 37, 28, 38, 39, 40, 41, 42,
43]. This is because such high energy particles originate from the decay of
charged and neutral pions produced in hadronic interactions such as proton-
proton ($pp$) collisions or photo-hadronic ($p\gamma$) interactions [44, 45].
In this respect special attention has been devoted to the possible common
origin of the IceCube diffuse flux of high-energy neutrinos [46] and the
extragalactic diffuse gamma-ray background observed by Fermi-LAT [47].
Evidence suggests that the latter originates from the superposition of
unresolved extragalactic sources, such as blazars, star-forming galaxies,
gamma-ray bursts, etc. [48, 49, 50, 51, 52, 53, 54, 55, 47, 56, 57, 13, 58,
59].

Supernovae constitute a particularly interesting source class, possibly
emitting high-energy neutrinos and gamma-rays through inelastic $pp$
collisions between the relativistic protons accelerated at the SN shock and
the low energy protons of the circumstellar medium (CSM) [60, 61, 62, 63, 64,
65, 66]. In fact, the existence of dense CSM around massive stars has been
confirmed by multi-wavelength observations of a wide range of phenomena, e.g.,
[67, 68]. The CSM is made up of matter deposited through stellar winds further
enriched as the progenitor star loses mass via wind and/or violent outbursts
(e.g., see Ref. [67] and references therein). The CSM was previously thought
to originate from stars losing their mass through steady stellar winds.
However, recent SN observations have challenged this traditionally accepted
picture (see e.g. [67]). It is now clear that substantial and impulsive mass
losses occur in at least $10\%$ of massive stars within one year from the core
collapse.

The stellar progenitor of SN2009ip, whose repeated eruptions occurred in the
years before explosion, is the best studied example among Hydrogen (H)-rich
massive stars [69]. The progenitor of this SN was a Luminous Blue Variable
(LBV) star. An LBV is a massive star that can have sporadic, violent mass loss
events and exhibits mass loss rate as high as $10$ $\rm M_{\odot}yr^{-1}$ [70,
71, 72, 73]. LBVs were previously thought to be a transitional phase of
stellar evolution. However, several recent works have shown that they can be
progenitors of Type IIn SNe (see e.g. [74] and references therein). The
interaction between the CSM and the SN ejecta is evident from the observation
of narrow, symmetric emission lines in “flash” spectroscopy of SNe. This might
be due to the photo-ionization of the CSM occurring during the progenitor mass
loss phase before explosion, see e.g. [75, 76]. UV/optical excesses observed
in SNe at early times also hint towards such mass loss phenomena, see e.g.
[77].

In most cases, the interaction of the SN shock with the CSM is dominant during
the early stages of the SN evolution, for a few months to a few years [78].
After this preliminary phase, the interaction weakens due to the fall in the
CSM density [79]. We focus on the high-energy particle emission during this
initial SN phase and refer to it as “Young Supernova (YSN)”. Note that, a
clear distinction between YSNe and SN remnants (SNRs) is not straightforward
[80, 81, 64]. In a SNR, the SN ejecta interacts with and sweeps up the far
away CSM or the Interstellar Medium (ISM). For a sufficiently old SNR
($1~{}\rm kyr$), the medium is significantly less dense and has a density of
$1-10^{3}$ $\rm cm^{-3}$. While the dense CSM during the YSN phase can have
densities in the scale of $10^{9}-10^{12}$ $\rm cm^{-3}$ [59, 82, 83, 19].
Thus the largest contribution to the secondary particle emission is reasonably
considered to originate from the YSN phase.

The high-energy neutrinos emitted from SNe are possibly detectable by the
IceCube Neutrino Observatory [84, 19, 39, 63]. In addition, gamma-rays could
be observed from SNe by Fermi-LAT [85, 50, 47] and the Cerenkov Telescope
Array (CTA) [86]. Recently, Fermi-LAT detected gamma-rays from the direction
of a peculiar supernova iPTF14hls [87]. Such discovery is however uncertain
because of the presence of a blazar in the detection error circle.

Varying mass and metallicity of stars may give rise to different SN Types [88,
89]. By considering YSNe of Type IIn, II-P, II-L and Ib/c, we explore the
related gamma-ray and neutrino emission and investigate their detection
prospects with Fermi-LAT [85, 47] and CTA [86] as well as IceCube [90, 3],
IceCube-Gen2 [91], and Km3NeT [92].

While neutrinos produced in YSNe propagate undisturbed to Earth, gamma-rays
undergo energy losses as they interact with low energy photons
($\gamma\gamma\longrightarrow e^{-}e^{+}$) and ambient matter ($\gamma
N\longrightarrow Ne^{-}e^{+}$, where $N$ is nucleus) in the source [93]. In
addition, over cosmic scales ($\sim$Gpc), the Extra-galactic Background Light
(EBL) absorption further affects the gamma-ray flux expected at Earth [94,
52], becoming significant for the diffuse SN emission, but negligible for
point source detection in the local universe (i.e., below $\sim 10$ Mpc).

This paper is organized as follows. In Sec. 2, we introduce our YSN model with
interacting CSM and the related production of secondary particles: neutrinos
and gamma-rays. Different Types of YSNe and their properties are discussed in
Sec. 3. The dependence of the gamma-ray and neutrino emission on the YSN Types
is explored in Sec. 4. The diffuse gamma-ray and neutrino backgrounds from
YSNe are presented in Sec. 5 together with a discussion on the model parameter
uncertainties and the detection prospects. The detection prospects of YSNe in
the local universe with present and upcoming gamma-ray and neutrino telescopes
are analysed in Sec. 6. Finally, we summarize our findings in Sec. 7. An
overview on the characteristic time scales for particle acceleration and
energy losses of protons is provided in Appendix A.

##  2 Neutrino and gamma-ray production in young supernovae

In this Section, we model YSNe interacting with the CSM in the early phase
after the explosion. We focus on YSNe about a year old and develop a model for
the shock-CSM interaction, describing the creation of secondary particles:
neutrinos and anti-neutrinos 111Note that we do not distinguish between
particles and antiparticles in the following and use “neutrinos” to indicate
both species. and gamma-rays.

###  2.1 Model setup

The material expelled by the massive star at the end of its life forms the
CSM. The CSM consists of shells of mostly light elements like H or helium
(He). The massive star eventually explodes in the form of a core-collapse SN
222We focus on core-collapse SNe as the vast majority of Type Ia SNe do not
exhibit a dense CSM (with the notable exception of e.g., [95]).. The SN shock
and the expanding ejecta interact with the surrounding CSM. Charged particles
(electrons and protons) in the shocked CSM are accelerated to relativistic
energies via Fermi’s diffusive shock acceleration [96, 97, 98]. In particular,
we are interested in high energy processes with dominant neutrino production,
hence we focus on inelastic $pp$ collisions. Here, relativistic protons
collide with the non-relativistic protons (inelastic $pp$ collisions) in the
CSM and produce a large number of $\pi$ and $\eta$ mesons. Secondary
neutrinos, electrons and gamma-rays are produced from the decay of $\pi$ and
$\eta$ mesons [99, 100, 44, 83, 101, 19]. High energy protons can also undergo
photo-hadronic ($p\gamma$) interactions with low energy photons in the CSM.
However, the energy loss due to this process is negligble and hence neglected
(see Appendix Acknowledgments).

The CSM shell is assumed to be spherically symmetric with an effective inner
radius $r_{\rm in}$ and outer radius $r_{\rm out}$. The inner radius is
defined as $r_{\rm in}=\max[r_{\rm bo},r_{\rm e}]$ [84]. The shock breakout
radius333The shock breakout radius corresponds to the optical depth,
$\tau=c/v_{\rm sh}$, of the material through which the shock propagates, where
$v_{\rm sh}$ is the shock velocity introduced in the following [102]., $r_{\rm
bo}$, corresponds to the beginning of shock acceleration, and $r_{\rm e}$ is
the radius of the stellar envelope. Thus $r_{\rm in}$ may differ according to
the SN Type (see Sec. 3 for details). The outer radius or size of the CSM
shell also depends on the SN Type. Given the assumption of spherical symmetry,
the radial dependence of the CSM mass density can be modelled in the following
way [64]:

$\rho_{\rm CSM}(r)=\frac{\dot{M}_{\rm W}}{4\pi r_{\rm in}^{w}v_{\rm
w}}\left(\frac{r_{\rm in}}{r}\right)^{w}.$ (2.1)

Here $w$ is the power-law index of the CSM density profile (e.g., $w=2$ for a
wind-like CSM—see, e.g., Ref. [78]), $\dot{M}_{\rm W}$ is the mass loss rate
of the progenitor star and $v_{\rm w}$ is the wind velocity. The radial
dependence is provided by the propagation of the shock, $r$ being the shock
radius. The number density of the protons in the CSM ($n_{\rm
CSM}(r)=\rho_{\rm CSM}(r)/m_{\rm p}$) also varies accordingly. Thus, the CSM
at the inner radius is $n_{\rm in,CSM}={\rho_{\rm CSM}(r_{\rm
in})}/{m_{p}}={\dot{M}_{\rm W}}/{(4\pi m_{p}r_{\rm in}^{2}v_{\rm w})}$, where
$m_{p}$ is the mass of the proton.

A fraction of the protons injected in the shocked region will be accelerated
to high energies and in turn produce secondary particles while interacting
with the CSM protons. To model this, the SN shock is assumed to expand
spherically in the CSM and has a power law dependence on the radius. The
velocity of the shock ($v_{\rm sh}$) expanding in the CSM varies slowly with
radius [78]:

$v_{\rm sh}(r)=v_{\rm in}\left(\frac{r}{r_{\rm in}}\right)^{\alpha}\ ,$ (2.2)

where $v_{\rm in}$ is the shock velocity at $r_{\rm in}$ and $\alpha$
characterises the shock velocity profile. The index $\alpha$ can be expressed
in terms of the power law indices of the CSM and SN ejecta density profiles
and is usually found to be negligible, i.e. the shock velocity is nearly
constant over the length of the CSM shell during the early phase of the YSN.
For more details on the choice of $\alpha$, see Ref. [101]. In our YSN
computations, we assume $\alpha=0$ implying $v_{\rm sh}=v_{\rm in}$.

To estimate the secondary particle population, we need the injection spectra
of the accelerated protons in the shocked region. We assume a power law
distribution between the maximum ($E_{\rm p,max}$) and minimum ($E_{\rm
p,min}$) proton energy for the injection spectra [83]:

$N_{\rm p}^{\rm inj}(E_{\rm p},r)\propto E_{\rm p}^{-\alpha_{\rm
p}}\exp\left(-\frac{E_{\rm p}}{E_{\rm p,max}(r)}\right)\ ,$ (2.3)

where $E_{\rm p}$ is the proton energy, ${\alpha_{\rm p}}$ is the power law
index and it is assumed to be equal to 2 unless otherwise mentioned. The
choice ${\alpha_{\rm p}}=2$ is motivated by diffusive shock acceleration
theory [98]. The normalization of the injected spectra is obtained from the
energy budget of the shock [$E_{\rm K}=(9\pi/8)m_{\rm p}v_{\rm
sh}^{2}r^{2}n_{\rm CSM}(r)$, energy per unit radius]. A fraction of this
energy, $\epsilon_{\rm p}$, is transferred to the charged particles (protons)
confined in the shocked region due to the presence of a strong magnetic field
and thus accelerates protons to high energies. The fraction $\epsilon_{\rm p}$
is a free parameter of the problem and may depend on the SN Type. This is due
to the fact that CSM density, ejecta kinetic energy and progenitor mass vary
for different SN Types [103]. The minimum proton energy $E_{\rm p,min}$ is
taken as the proton rest mass $m_{\rm p}$. The maximum energy ($E_{\rm
p,max}$) of the accelerated protons depends on the different loss processes
competing with the acceleration time scale ($t_{\rm acc}=6E_{\rm p}c/eBv_{\rm
sh}^{2}$) [104, 83], $B$ being the magnetic field strength. The magnetic field
is inversely proportional to the shock radius $r$ and it is
$B=3/2(4\pi\epsilon_{\rm B}m_{\rm p}n_{\rm in,CSM}v_{\rm sh}^{2})^{1/2}(r_{\rm
in}/r)$, where $\epsilon_{\rm B}$ is the fraction of the post shock thermal
energy that goes into the magnetic field [83, 105, 106]. A detailed analysis
on the effects of magnetic field on $E_{\rm p,max}$ can be found in Ref.
[107].

The energy losses suffered by relativistic protons mainly come from two
competing processes, due to the adiabatic expansion of the shock shell and the
$pp$ interactions with the CSM protons. The adiabatic loss time scale or the
dynamical time scale is $t_{\rm ad}\sim t_{\rm dyn}=r/v_{\rm sh}$ and the $pp$
collision time scale is $t_{\rm pp}=[4\kappa_{\rm pp}\sigma_{\rm pp}n_{\rm
CSM}(r)c]^{-1}$, where $\kappa_{\rm pp}=0.5$ is the proton inelasticity and
$\sigma_{pp}$ is the $pp$ interaction cross-section. We assume $\sigma_{\rm
pp}\approx 3\times 10^{-26}$ $\rm cm^{2}$, neglecting the energy dependence of
$\sigma_{\rm pp}$ since it hardly affects $t_{\rm pp}$ [83, 19] (however, as
specified later, we include the energy dependence in $\sigma_{\rm pp}$ for the
calculation of the secondaries). In addition, protons undergo other energy
loss processes, as detailed in Appendix A. Thus, $E_{\rm p,max}$ can be
estimated from $t_{\rm acc}=6E_{\rm p,max}c/eBv_{\rm
sh}^{2}=\mathrm{min}[t_{\rm pp},t_{\rm ad}]$ and changes according to the
radial evolution of the background environment.

The radial evolution of the steady state proton number, $N_{\rm p}(E_{\rm
p},r)$ is described by the following equation [108, 96, 109, 83, 19]:

$\frac{\partial N_{\rm p}(E_{\rm p},r)}{\partial r}+\frac{N_{\rm p}(E_{\rm
p},r)}{v_{\rm sh}t_{\rm pp}(r)}-\frac{\partial}{\partial E_{\rm
p}}\left[\frac{E_{\rm p}N_{\rm p}(E_{\rm p},r)}{r}\right]=N_{\rm p}^{\rm
inj}(E_{\rm p},r)\ ,$ (2.4)

where the second term on the left hand side corresponds to energy losses via
$pp$ collisions and the third term stands for adiabatic losses. This primary
steady state proton number drives the production of the secondary neutrinos
and gamma-rays. The time evolution of the fluxes can be probed from the
relation betweeen the shock radius and the shock velocity: $r=v_{\rm sh}t$.

###  2.2 Gamma-ray production

Gamma-rays can be produced through $pp$ interactions via decay of neutral
$\pi$ and $\eta$ mesons, with the following decay channels:
$\pi^{0}\longrightarrow 2\gamma$, $\eta\longrightarrow 2\gamma$,
$\eta\longrightarrow 3\pi^{0}$, $\eta\longrightarrow\pi^{+}\pi^{-}\pi^{0}$,
and $\eta\longrightarrow\pi^{+}\pi^{-}\gamma$ [44]. The radial evolution of
gamma-rays with energy $E_{\gamma}$ produced in $pp$ collisions can be
computed as follows [93, 110]:

$\frac{\mathrm{d}N_{\gamma}(E_{\gamma},r)}{\mathrm{d}r}+\frac{N_{\gamma}(E_{\gamma},r)}{v_{\rm
sh}t_{\rm esc}}=N^{\rm inj}_{\gamma}(E_{\gamma},r),$ (2.5)

where $t_{\rm esc}=r/4c$ is the escape time of secondary particles. The factor
$4$ corresponds to the compression of the CSM due to shock pressure [96]. The
source term, $N^{\rm inj}_{\gamma}(E_{\gamma},r)$, below $E_{\rm p}=0.1$ TeV
is given by [44]

$N^{\rm
inj}_{\gamma}(E_{\gamma},r)=2\frac{\tilde{\eta}}{K_{\pi}}\frac{4cn_{\rm
CSM}(r)m_{\rm e}}{v_{\rm sh}m_{\rm p}}\int_{E_{\rm
min}}^{\infty}\frac{\mathrm{d}E_{\pi}}{\sqrt{E_{\pi}^{2}-m_{\pi}^{2}}}\sigma_{\rm
pp}\left(E_{\rm p}\right)N_{\rm p}\left(E_{\rm p}\right)\ ,$ (2.6)

where $E_{\rm p}=m_{\rm p}+{E_{\pi}}/{K_{\pi}}$ and $E_{\rm
min}=E_{\gamma}+{m_{\pi}^{2}}/{E_{\gamma}}$. Above $E_{\rm p}=0.1$ TeV,

$N^{\rm inj}_{\gamma}(E_{\gamma},r)=\frac{4cn_{\rm CSM}(r)m_{\rm e}}{v_{\rm
sh}m_{\rm p}}\int_{m_{\rm p}}^{\infty}\mathrm{d}E_{\rm p}\frac{\sigma_{\rm
pp}(E_{\rm p})}{E_{\rm p}}N_{\rm p}(E_{\rm
p},r)F_{\gamma}\left(\frac{E_{\gamma}}{E_{\rm p}}\right)\ ,$ (2.7)

with $F_{\gamma}\left({E_{\gamma}}/{E_{\rm p}}\right)$ being the gamma-ray
production rate, and $E_{\gamma}$ the gamma-ray energy. The parameters
$K_{\pi}$ and $\tilde{\eta}$ are free parameters, used for connecting the
injection rates above energy $E_{\rm p}=1$ TeV.

The total gamma-ray flux observed at Earth from a source at luminosity
distance $d_{\rm L}$ is given by:

$\displaystyle E_{\gamma,\rm obs}^{2}\Phi_{\gamma}(E_{\gamma,\rm obs})$
$\displaystyle=\frac{e^{-\tau_{\gamma,\rm EBL}(E_{\gamma},z)}}{4\pi d_{\rm
L}^{2}}E_{\gamma}^{2}\phi_{\gamma}^{\rm s}(E_{\gamma})$
$\displaystyle=\frac{e^{-\tau_{\gamma,\rm EBL}(E_{\gamma},z)}}{4\pi d_{\rm
L}^{2}}\int_{r_{\rm in}}^{r_{\rm
max}}\mathrm{d}r\frac{E_{\gamma}^{2}N_{\gamma}(E_{\gamma},r)e^{-\left(\tau_{\gamma\gamma}+\tau_{\rm
BH}\right)}}{m_{\rm e}c^{2}t_{\rm esc}v_{\rm sh}}\ ,$ (2.8)

where $E_{\gamma}=(1+z)E_{\gamma,\rm obs}$, $z$ is the cosmological redshift
and $\phi^{\rm s}_{\gamma}$ is the gamma-ray flux at source. The maximum shock
radius, $r_{\rm max}$ corresponds to the end of particle production (see Sec.
2.3 for details). Here, $\tau_{\gamma\gamma}$ and $\tau_{\rm BH}$ are the
optical depths for gamma-gamma and Bethe-Heitler interactions, respectively;
their modeling is introduced in the following. $\tau_{\gamma,\rm
EBL}(E_{\gamma},z)$ is the optical depth for the interaction of gamma-rays
with the EBL [111, 94].

The gamma-rays produced in the hadronic processes can interact with the low
energy thermal photons present in the CSM [93, 110, 83]. In these
interactions, electron-positron pairs can be produced. Electron-positron pairs
can also be produced in Bethe-Heitler processes, where high-energy photons
interact with nuclei (mostly protons) in the ejecta [110].

The attenuation of the gamma-ray spectra due to photon-photon pair production
is taken into account through the factor $e^{-\tau_{\gamma\gamma}}$ in Eq.
2.8; the optical depth $\tau_{\gamma\gamma}$ of the CSM thermal photons is
computed as follows

$\tau_{\gamma\gamma}=r_{\rm max}\int_{0}^{\infty}n_{\rm
ph}(\epsilon)\sigma_{\gamma\gamma}(E_{\gamma},\epsilon)\mathrm{d}\epsilon\ ,$
(2.9)

where $\epsilon$ is the energy of thermal photons and
$\sigma_{\gamma\gamma}(E_{\gamma},\epsilon)$ is the photon-photon annihilation
cross section [112, 110]. The number density of thermal photons follows a
black-body spectrum,

$n_{\rm ph}(\epsilon)\propto\frac{\epsilon^{2}}{\exp(\epsilon/T)+1}\ ,$ (2.10)

where $T$ is the temperature of the thermal photons and can be defined in
terms of average energy, $\epsilon_{\rm av}$ of photons as $T=\epsilon_{\rm
av}/3.15$. The constant of proportionality is such that

$\int_{0}^{\infty}\mathrm{d}\epsilon n_{\rm ph}(\epsilon)=U_{\rm ph,av}\ .$
(2.11)

The average energy density of thermal photons in the CSM is:

$U_{\rm ph,av}=\frac{1}{r_{\rm max}}\int_{r_{\rm in}}^{r_{\rm
max}}\mathrm{d}r\frac{L_{\rm SN,pk}}{4\pi cr^{2}}\ ,$ (2.12)

where $L_{\rm SN,pk}$ is the SN peak luminosity. In general, the energy
density of thermal photons scales as $1/r^{2}$ with the shock radius, $r$. We
take the average energy density of the thermal photons in the interaction
zone, i.e. between $r_{\rm in}\leq r<r_{\rm max}$, to make a reasonable
estimate of the absorption of gamma-rays caused by the thermal photons.

In addition to $\gamma\gamma$ pair production losses, Bethe-Heitler pair
production losses can also affect the gamma-ray spectra. The loss due to this
process can be estimated by the factor $e^{-\tau_{BH}}$ in Eq. 2.8, where
$\tau_{BH}$ is the optical depth. The latter depends on the mass and
composition of the ejecta and we model it as in Ref. [110].

The electrons (positrons) produced in these processes may lose energy via
synchrotron radiation due to the presence of magnetic fields in the shocked
CSM. They also lose energy due to inverse Compton scattering with low-energy
photons in the CSM. Electrons and positrons can then annihilate and produce
low-energy gamma-rays, which in turn modify the observed gamma-ray spectra.
This cascading of electromagnetic (EM) interactions modifying the gamma-ray
spectrum can be estimated with a broken power law and is given by [113],

$\phi_{\gamma,\rm
cascade}(E_{\gamma})\propto\begin{cases}E_{\gamma}^{-1.5}&(E_{\gamma}<E_{\gamma,\rm
b})\\\ E_{\gamma}^{-\alpha_{\gamma}}&(E_{\gamma,\rm b}<E_{\gamma}<E_{\rm
cut})\end{cases}$ (2.13)

where $E_{\rm cut}$ is the cut-off energy defined as
$\tau_{\gamma\gamma}(E_{\gamma}=E_{\rm cut}=E_{\rm max})=1$, $E_{\gamma,\rm
b}$ is the break energy given by $E_{\gamma,\rm b}={4E_{\rm
cut}^{2}}/{3(2m_{\rm e}c^{2})^{2}}\epsilon_{\rm av}$ and the power law index
$\alpha_{\gamma}=2$. This cascaded flux is normalized to the total energy lost
above $E_{\rm cut}$ due to pair production. This suggests that the larger the
absorption, the larger the cascaded flux. Therefore, harder spectra would
produce large cascaded gamma-ray flux.

The primary electrons accelerated in the shock could also produce gamma-rays
via inverse Compton. The gamma-ray contribution of $pp$ collisions (i.e.,
$\pi^{0}\longrightarrow 2\gamma$) generally dominates in the high-energy range
(GeV), hence the gamma ray contribution from the primary electrons can be
ignored, see e.g. [93].

###  2.3 Neutrino production

Neutrinos are produced through the decay of charged pions created in $pp$
interactions, $\pi\longrightarrow\mu\nu_{\mu}$ and $\mu\longrightarrow
e\nu_{e}\nu_{\mu}$. The decay of $\eta$ mesons also produces neutrinos, but
their contribution is smaller than the one of pions [44]. The evolution of
neutrinos at different radii is governed by the following equation [19]:

$\frac{\mathrm{d}N_{\nu_{\rm f}}(E_{\nu},r)}{\mathrm{d}r}+\frac{N_{\nu_{\rm
f}}(E_{\nu},r)}{v_{\rm sh}t_{\rm esc}(r)}=N^{\rm inj}_{\nu_{\rm
f}}(E_{\nu},r)\ ,$ (2.14)

where $t_{\rm esc}(r)=r/4c$ is the time required for neutrinos to escape the
CSM (see comments below Eq. 2.5). of flavour $f$ (${f}=e,\mu$ at the source)
with neutrino energy $E_{\nu}$. The neutrino injection term $N^{\rm
inj}_{\nu_{\rm f}}(E_{\nu},r)$ is obtained by following Ref. [44].

For $E_{\rm p}<0.1$ TeV:

$\displaystyle N^{\rm inj}_{\nu_{e}}(E_{\nu},r)$
$\displaystyle=2\frac{\tilde{\eta}}{K_{\pi}}\frac{4cn_{\rm CSM}(r)m_{\rm
e}}{v_{\rm sh}m_{\rm p}}\int_{E_{\rm
min}}^{\infty}\frac{\mathrm{d}E_{\pi}}{\sqrt{E_{\pi}^{2}-m_{\pi}^{2}}}\sigma_{\rm
pp}\left(E_{\rm p}\right)N_{\rm p}\left(E_{\rm
p},r\right)f_{\nu_{e}}\left(\frac{E_{\nu}}{E_{\pi}}\right)$ $\displaystyle
N^{\rm inj}_{\nu_{\mu}}(E_{\nu},r)$
$\displaystyle=2\frac{\tilde{\eta}}{K_{\pi}}\frac{4cn_{\rm CSM}(r)m_{\rm
e}}{v_{\rm sh}m_{\rm p}}\int_{E_{\rm
min}}^{\infty}\frac{\mathrm{d}E_{\pi}}{\sqrt{E_{\pi}^{2}-m_{\pi}^{2}}}\sigma_{\rm
pp}\left(E_{\rm p}\right)N_{\rm p}\left(E_{\rm p},r\right)$
$\displaystyle\hskip
170.71652pt\times\left[f_{\nu_{\mu}}^{(1)}\left(\frac{E_{\nu}}{E_{\pi}}\right)+f_{\nu_{\mu}}^{(2)}\left(\frac{E_{\nu}}{E_{\pi}}\right)\right]$
(2.15)

where $E_{\rm p}=m_{\rm p}+{E_{\pi}}/{K_{\pi}}$, and $E_{\pi}$ is the energy
of pion and $m_{\rm e}$ is mass of electron. The minimum energy of pion is
given by $E_{\rm min}\simeq E_{\nu}+{m_{\pi}^{2}}/{4E_{\nu}}$. The parameters
$K_{\pi}$ and $\tilde{\eta}$ are free parameters and used for connecting the
injection rates above energy $E_{\rm p}=1$ TeV.

For $E_{\rm p}\geq 0.1$ TeV:

$\displaystyle N^{\rm inj}_{\nu_{e}}(E_{\nu},r)$ $\displaystyle=\frac{4cn_{\rm
CSM}(r)m_{\rm e}}{v_{\rm sh}m_{\rm p}}\int_{m_{\rm
p}}^{\infty}\mathrm{d}E_{\rm p}\frac{\sigma_{\rm pp}(E_{\rm p})}{E_{\rm
p}}N_{\rm p}(E_{\rm p},r)F_{\nu_{e}}\left(\frac{E_{\nu}}{E_{\rm p}}\right)$
$\displaystyle N^{\rm inj}_{\nu_{\mu}}(E_{\nu},r)$
$\displaystyle=\frac{4cn_{\rm CSM}(r)m_{\rm e}}{v_{\rm sh}m_{\rm
p}}\int_{m_{\rm p}}^{\infty}\mathrm{d}E_{\rm p}\frac{\sigma_{\rm pp}(E_{\rm
p})}{E_{\rm p}}N_{\rm p}(E_{\rm p},r)$ $\displaystyle\hskip
170.71652pt\times\left[F_{\nu_{\mu}}^{(1)}\left(\frac{E_{\nu}}{E_{\rm
p}}\right)+F_{\nu_{\mu}}^{(2)}\left(\frac{E_{\nu}}{E_{\rm p}}\right)\right]\
.$ (2.16)

Here, $F_{\nu_{e}}$, $F_{\nu_{\mu}}^{(1)}$, $F_{\nu_{\mu}}^{(2)}$ and
$f_{\nu_{e}}$, $f_{\nu_{\mu}}^{(1)}$, $f_{\nu_{\mu}}^{(2)}$ are the neutrino
distributions from pion decay valid for $E_{p}\geq 0.1$ TeV and $E_{p}<0.1$
TeV, respectively [44]. The energy dependent cross-section $\sigma_{\rm
pp}(E_{\rm p})$ is taken from Ref. [44]. The neutrino flux of flavour $f$ at
source is given by,

$\displaystyle E_{\nu}^{2}\phi^{\rm s}_{\nu_{\rm f}}(E_{\nu})=\int_{r_{\rm
in}}^{r_{\rm max}}\frac{\mathrm{d}r}{v_{\rm sh}}\frac{E_{\nu}^{2}N_{\nu_{\rm
f}}(E_{\nu},r)}{m_{\rm e}c^{2}t_{\rm esc}}.$ (2.17)

The all flavour flux at source is $E_{\nu}^{2}\phi_{\nu}^{\rm s}=\sum_{\rm
f}E_{\nu}^{2}\phi^{\rm s}_{\nu_{\rm f}}(E_{\nu})$. During propagation to
Earth, the flavour combination may change due to flavour mixing [114], leading
to the following flavour composition at Earth: $\nu_{\rm
e}:\nu_{\mu}:\nu_{\tau}=1:1:1$.

$E_{\nu,\rm obs}^{2}\Phi_{\nu}(E_{\nu,\rm obs})=\frac{\sum_{\rm
f}E_{\nu}^{2}\phi^{\rm s}_{\nu_{\rm f}}(E_{\nu})}{4\pi d_{\rm L}^{2}},$ (2.18)

where $E_{\nu}=(1+z)E_{\nu,\rm{obs}}$ and $z$ is the cosmological redshift.

Note that the neutrino (gamma-ray) production from $pp$ interactions ceases to
be significant as the shock radius reaches the maximum radius, $r_{\rm max}$.
This is determined by $\min[r_{\rm dec},r_{\rm out}]$. The deceleration
radius, $r_{\rm dec}$, of the SN shock corresponds to the radius when the
ejecta mass, $M_{\rm ej}$, sweeps up the same amount of CSM mass and is
defined as $r_{\rm dec}=M_{\rm ej}v_{\rm w}/\dot{M}_{\rm W}=r_{\rm in}+{M_{\rm
ej}}/{(4\pi mn_{\rm in,CSM}r_{\rm in}^{2})}$. The region $r<r_{\rm dec}$ is
known as the free expansion phase of the CSM. Particle acceleration and
production are dominant in this phase. Once the shock reaches the maximum
radius i.e., $r_{\rm max}$, the interaction of the shock with the ISM begins,
which is beyond the scope of this work444In the case of old SNR ($~{}1$ kyr),
the flux due to the Sedov-Taylor phase ($r>r_{\rm dec}$) decreases by several
(more than $3$–$4$) orders of magnitude due to the very low target (CSM or
ISM) density. Thus, the total flux from the Sedov-Taylor phase will be
negligible compared to the early YSN phase. .

Most SNe show signs of interaction within a few years of their evolution with
rarer cases showing interaction on timescales of $\mathcal{O}(10)$ years, see
e.g. Refs. [115, 116, 117, 118, 119, 120]. Since the maximum time, $t_{\rm
max}$ (corresponding to $r_{\rm max}$) depends on the extent of the CSM, YSNe
with extended CSM could have $t_{\rm max}\sim\mathcal{O}(10)$ years. However,
this extended CSM is less dense than the CSM of $t_{\rm max}\simeq 1$ year.
Therefore, the neutrino and gamma-ray fluxes for this extended CSM are similar
to that of YSNe with CSM of 1 year (results not shown here). Therefore, we
take $r_{\rm max}=r_{\rm out}$ (corresponds to $t_{\rm max}=1$ year)
throughout this work.

Different classes of YSNe produce different fluxes of secondaries (gamma-rays
and neutrinos) due to wide differences in these input parameters. In the
following, we discuss the different Types of YSNe and their impact on the
secondary production.

##  3 Young supernova types

Supernovae are classified as Type-I and Type-II based on the presence of H
lines in the observed spectra. Type-I SNe do not show H lines, whereas Type-II
do. Supernovae are further classified into different sub-Types of Type-I (Ia,
Ib and Ic) and Type-II (IIn, II-P, IIb, II-L) SNe depending on different
observed characteristics. Type Ia SNe show Silicon II (Si II) absorption lines
in their spectra. Presence of He lines in Type Ib SNe differentiates them from
Type Ic. Type IIn and IIb exhibit narrow and broad H lines respectively. The
spectra of Type II-P SNe show a plateau whereas Type II-L SNe show linearly
declining spectra (for details on SN classification, see e.g. Refs. [88, 89]).
Except for SNe Ia, all other SN classes are powered by the collapse of the
stellar core.

The diverse classes of YSNe differ in parameters like mass loss rate, wind
velocity, size of the CSM, and efficiency of shock energy transfer to protons.
Below we report about these characteristic parameters as from electromagnetic
observations for different Types of YSNe.

  * •

Type IIn SNe are found to have very dense CSM as a result of heavy mass loss
($10^{-4}$–$10^{1}$ $\rm\dot{M}_{\odot}yr^{-1}$) before explosion [67, 121].
The wind velocity, $v_{\rm w}$, for such SNe can typically vary between
$30$–$600~{}\rm kms^{-1}$ [67]. Their progenitors are not well known and have
been found to be connected to LBV, Red Supergiant (RGB) and Yellow Hypergiant
(YHG) stars (see e.g., [122, 123, 124, 125] and references therein). These
stars have large mass loss rates ($\dot{M}_{\rm W}\sim 10^{-3}$-$10^{1}~{}\rm
M_{\odot}yr^{-1}$) and moderate wind velocity ($v_{\rm w}\sim 100~{}\rm
kms^{-1}$) [67, 121]. Therefore, Type IIn SNe show a dense CSM ($n_{\rm
in,CSM}\sim 10^{11}$ $\rm cm^{-3}$ [67]). Their shock velocity, $v_{\rm sh}$,
ranges between $10^{3}$ and $10^{4}$ $\rm km\ s^{-1}$ [19, 78, 101] (see Eq.
2.1). The fraction of kinetic energy, $\epsilon_{\rm p}$, that goes into high
energy protons is estimated to be $10^{-2}$–$1.5\times 10^{-1}$ [103].
Therefore, we choose the range of $\epsilon_{\rm p}$ as $10^{-2}$–$10^{-1}$.
The fraction of post-shock magnetic energy, $\epsilon_{\rm B}$, can be
constrained from radio observations of SNe and is in the range $\epsilon_{\rm
B}\sim 10^{-4}$–$10^{-2}$ [78, 126, 81, 127].

  * •

Type II-P SNe show a plateau in their light curves up to a few months ($\sim
20$–$100$ days) after the explosion [128, 129]. This is due to the abundance
of hydrogen in the SN ejecta. This SN class usually arises from red supergiant
(RSG) stars [67, 130, 131, 132, 133] and they are found to have large CSM
density. Eruptive mass losses of such stars just before [$\mathcal{O}(1)$
year] the SN explosion are believed to be the reason of their large CSM
densities [128, 134]. The typical mass loss rate during this phase has been
reported to be $\mathcal{O}(10^{-3})~{}\rm\dot{M}_{\odot}$yr-1 [134, 135, 136]
with wind velocity of $\mathcal{O}(10)~{}\rm km\ s^{-1}$ [137]. Larger mass
loss rates [$\mathcal{O}(1)\rm\dot{M}_{\odot}yr^{-1}$] with wind velocity
$100~{}{\rm km\ s^{-1}}$ have also been predicted, see [129, 128]; however,
this might be an exception (see e.g., [138, 75, 139]) 555Ref. [129]
investigated early light curves of some Type II SNe. Their results suggest the
need for local CSM ($r<2\times 10^{14}$ cm) in order to reproduce the rapid
rise time and brighter emission at peak observed in some SNe. This is an
exception and we therefore focus on the average mass loss over a longer period
of time. . Interestingly, our calculation only depends on the ratio
$\dot{M}_{\rm W}/{v_{\rm W}}$ (see Eq. 2.1); hence, given the evidence of
large mass loss rates, we adopt $10^{-2}$ $\rm\dot{M}_{\odot}yr^{-1}$ as our
typical mass loss rate with wind velocity $\mathcal{O}(100)$ $\rm km\ s^{-1}$
[134, 129]. This choice is equivalent to assume a mass loss rate
$10^{-3}~{}\rm\dot{M}_{\odot}yr^{-1}$ with wind velocity $10~{}\rm km\
s^{-1}$.

The eruptive mass loss is responsible for a dense CSM which is found to exist
up to $\sim 10^{15}~{}\rm cm$ [135]. The CSM density beyond this radius is
smaller and is due to the stellar wind of RSG stars [67]. A typical RSG wind
has a mass loss rate in the range $10^{-6}$–$10^{-5}$ $\rm\dot{M}_{\odot}$
yr-1 with wind velocities of about $10$–$20$ $\rm km/s$ [67]. We distinguish
these two different phases of the CSM by naming them “eruptive” (for the large
mass loss rates) and “normal” (the smaller RSG winds). The shock velocity for
this YSN class is $\mathcal{O}(10^{4})$ $\rm km/s$ [136].

  * •

Type II-L SNe have linear light curves, while Type II-b show broad He lines in
their spectra. Their progenitors have similar mass loss rates, which lie in
the range $10^{-5}$–$10^{-4}$ $\rm\dot{M}_{\odot}{\rm yr}^{-1}$ and have wind
velocity between $2\times 10^{1}$–$10^{2}$ $\rm km\ s^{-1}$ [140, 67, 141,
142, 143, 144, 75]. The shock velocity for these Types of SNe is found to be
of the order of $10^{4}$ $\rm km/s$ [145].

  * •

Type Ib/c SNe are rich in helium (He) and usually have lower mass loss rates
($10^{-7}$–$10^{-4}$ $\rm\dot{M}_{\odot}~{}yr^{-1}$) than IIn and II-P SNe
[67, 146, 147, 77]. The wind velocity for Ib/c SNe is very large ($v_{\rm
w}\sim 10^{3}$ $\rm km~{}s^{-1}$). Moreover, the inner radius of the CSM is
small ($r_{\rm in}\simeq 10^{12}$ cm, corresponds to the size of the
progenitor). This implies that the volume of the CSM shell in the vicinity of
$r_{\rm in}$ is also small and therefore the total number of proton targets is
quite low. Thus, the neutrino and gamma-ray emission from such YSNe is
negligible (as most of the secondaries are produced near $r_{\rm in}$).
However, late time observations of SN 2014C (which was initially classified as
Ib/c SN) highlight the presence of interaction of the SN ejecta with the
hydrogen rich CSM at $6\times 10^{16}$ cm [137]. The CSM is found to extend up
to $2.5\times 10^{17}$ cm [148]; hence, we assume $r_{\rm out}=10^{17}$ cm.
This corresponds to the beginning of shock interaction at about $1.5$ years
after the explosion and substantial reduction of the dense CSM interaction at
about $3$ years. Thus the duration of emission is considered to be about $1.5$
years. This dense CSM is possibly created by large mass loss rate of
$\mathcal{O}(1)~{}\rm M_{\odot}yr^{-1}$ with wind velocity in the range
($10^{1}-10^{3}~{}\rm km\ s^{-1}$) [137]. However, smaller mass loss rate of
$\mathcal{O}(10^{-3})~{}{\rm M_{\odot}yr^{-1}}$ with velocity $100~{}{\rm km\
s^{-1}}$ has also been proposed [149]. Hence, we assume a mass loss rate of
$\mathcal{O}(10^{-2})~{}{\rm M_{\odot}yr^{-1}}$ and wind velocity $100~{}{\rm
km\ s^{-1}}$ to be conservative. The exact mechanism of this large mass loss
is not well understood. Therefore, we assume a wind profile in order to obtain
a conservative estimate of the CSM density. The deceleration radius, $r_{\rm
dec}~{}(9\times 10^{16}~{}{\rm cm})$ of the shock moving forward in the CSM is
comparable to $r_{out}$. SN 2004dk and SN2019yvr are similar examples of Ib/c
SN with late time interaction after explosion [150, 151]. Reference [137]
investigated a sample of $183$ Ib/c SNe and found that $10\%$ of the sample
has late time interaction like SN2014C, which is about $2.6\%$ of the total
core-collapse SNe. Hence, we include late time interaction of Ib/c SNe in our
analysis. We choose the SN 2014C as a representative of the class Type Ib/c
late time (LT) SN [149, 152, 137, 93].

In Table 1, we list the typical values adopted for each of our modelling
parameters for all SN Types introduced above. While we chose to be
conservative in our choice of the characteristic parameters, we also consider
a range of variability for the most uncertain model parameters and the latter
defines a band in the final results (see Sec. 5 and 2).

Table 1: Characteristic parameters for different YSN Types. We consider two different phases of the CSM of II-P SNe namely, eruptive (created by the large mass loss occurring a few years prior to explosion) and normal (due to the usual RSG wind during stellar evolution).  Parameters | IIn | II-P | IIb/II-L | Ib/c | Ib/c (LT)
---|---|---|---|---|---
| ($1$ yr) | Eruptive | Normal | ($1$ yr) | ($1$ yr) | (1.5 yrs)
|  | ($\leq 12$) | ($>12$ days) |  |  |
$n_{\rm in,CSM}$ ($\rm cm^{-3}$) | $10^{11}$ | $10^{12}$ | $10^{9}$ | $6\times 10^{11}$ | $2.4\times 10^{12}$ | $2\times 10^{6}$
$\dot{\rm M}_{W}(\rm M_{\odot}~{}yr^{-1}$ ) | $10^{-1}$ | $10^{-2}$ | $2\times 10^{-6}$ | $3\times 10^{-5}$ | $10^{-5}$ | $2\times 10^{-2}$
$v_{\rm w}$ ($\rm kms^{-1}$) | $10^{2}$ | $10^{2}$ | $1.5\times 10^{1}$ | $3\times 10^{1}$ | $10^{3}$ | $10^{2}$
$r_{\rm out}$ ($\rm cm$) | $2\times 10^{16}$ | $10^{15}$ | $3\times 10^{16}$ | $3\times 10^{16}$ | $6\times 10^{16}$ | $10^{17}$
$v_{\rm sh}$ ($\rm kms^{-1}$) | $7\times 10^{3}$ | $10^{4}$ | $10^{4}$ | $10^{4}$ | $2\times 10^{4}$ | $10^{4}$
$r_{\rm in}$ ($\rm cm$) | $4\times 10^{14}$ | $6\times 10^{13}$ | $6\times 10^{13}$ | $6\times 10^{12}$ | $3\times 10^{11}$ | $5\times 10^{16}$
$\epsilon_{\rm p}$ | $10^{-2}$ | $10^{-1}$ | $10^{-1}$ | $10^{-1}$ | $10^{-1}$ | $5\times 10^{-2}$
$\epsilon_{\rm B}$ | $10^{-2}$ | $10^{-2}$ | $10^{-2}$ | $10^{-2}$ | $10^{-2}$ | $10^{-2}$

##  4 Dependence of the gamma-ray and neutrino emission on the young supernova
type

The YSN emission properties largely vary across different SN Types. In this
Section, we first characterize the gamma-ray and neutrino emission for our
benchmark YSN IIn and then compare the particle production across different
YSN classes. Throughout this Section, the emission of secondaries is computed
for $1$ year for all YSN Types, except for Ib/c (LT) YSNe for which we
consider an emission period of $1.5$ years.

###  4.1 Characteristic gamma-ray and neutrino emission

Figure 1: Top panels: Luminosity of gamma-rays (left) and all-flavour
neutrinos (right) of our benchmark YSN IIn (see Table 1) as a function of
time. The temporal evolution at the source are shown for three representative
energies ($E_{\gamma}=10^{0},10^{2}$, and $10^{4}$ GeV for gamma-rays and
$E_{\nu}=10^{3},10^{4}$, and $10^{5}$ GeV for neutrinos). Bottom panels:
Corresponding gamma-ray (left) and all-flavour neutrino (right) fluences at
Earth as functions of the observed particle energy. The distance of our YSN
IIn from Earth is chosen to be $10$ Mpc. Both fluences for neutrinos and
gamma-rays are time integrated over the first year of the SN evolution. Gamma-
rays are shown without any attenuation due to energy losses (see discussion of
Fig. 2 for details). Therefore, the spectra of both species from $pp$
collision follow a similar behaviour of growing with particle’s energy and
falling rapidly above $10^{5}$ GeV.

As a primary example to illustrate the YSN neutrino and gamma-ray emission we
consider Type IIn SNe (see Sec. 2 and Table 1) as they have the highest mass
loss rate in the SN sample presented in Table 1. Unless otherwise specified,
we assume that our benchmark YSN is located at $10$ Mpc from Earth. Most SNe
are detected between $1$ and $100$ Mpc, with the majority being discovered at
$\mathcal{O}(90)$ Mpc, see e.g. Fig. 2 of Ref. [153]. Hence, our choice of the
typical SN distance is somewhat optimistic, yet compatible with observations.

The top and bottom left (right) panels of Fig. 1 show the gamma-ray (neutrino)
luminosity and fluence. The top left panel shows the gamma-ray light curve at
$E_{\gamma}=$1, $10^{2}$, $10^{4}$ GeV, while the gamma-ray fluence produced
by $pp$ collisions in IIn YSNe during its first year is displayed in the
bottom left panel. The fluence in the bottom left panel of Fig. 1 does not
include any attenuation at the source. Gamma-rays above $100$ GeV can interact
with low energy photons and ambient matter in the source and may get lost.
This corresponds to $\tau_{\gamma\gamma}$ and $\tau_{\rm BH}$, respectively,
in Eq. 2.8 and are discussed below (see discussion of Fig. 2).

The top panel on the right shows the neutrino light curve (blue) for three
different energies ($E_{\nu}=10^{3}$, $10^{4}$, $10^{5}$ GeV) for the first
year of the YSN. The light curves start around $7$ days after the explosion.
This delay corresponds to the inner radius, $r_{\rm in}$ (i.e., when the
interaction between the shock wave accelerated protons and the CSM protons has
begun). The inner radius, $r_{\rm in}$, also depends on the class properties
of each SN Type. The plot in the bottom right panel shows the energy spectra
of the YSN emitted neutrinos for the first year. The neutrino flux (top right
plot) falls rapidly above energy $E_{\nu}\sim 10^{5}$ GeV. This is due to the
maximum energy of protons $E_{\rm p,max}$, which determines the exponential
cut-off of the proton spectra. This clearly suggests that the SN properties
driving $E_{\rm p,max}$ will influence the cut-off, and the latter would be
different for different YSN Types. Note that these interpretations also hold
for the gamma-rays in the left panel.

Figure 2: Gamma-ray fluence at Earth after including the relevant absorption
processes and EM cascades at the source as a function of the gamma-ray energy
for our benchmark YSN IIn (see Table 1) at a distance of $10$ Mpc. The red-
dashed curve represents the fluence obtained via $pp$ collisions. The purple-
dotted curve is the fluence after absorption due to $\gamma\gamma$ pair
production loss and Bethe-Heitler loss in the CSM. The gamma-ray fluence due
to EM cascades is shown by the dot-dashed magenta curve. The thick red solid
curve represents total flux at the source after absorption and EM cascades.
The gamma-ray fluence between $10^{2}$ to $10^{4}$ GeV is heavily attenuated
due to the absorption processes occurring in the source. The final gamma-ray
flux (thick red curve) is a combination of the flux surviving to absorption
and the EM cascaded flux.

The fluence in the bottom left panel of Fig. 1 will be affected by absorption
processes. Fig. 2 shows the effect of absorption and EM cascades on the gamma-
rays produced via $pp$ interactions for Type IIn YSNe for a thermal energy
distribution of photons with average energy $\epsilon_{\rm av}=1$ eV and SN
peak luminosity $L_{\rm SN,pk}=10^{41}$ $\rm erg/s$ respectively [154, 83]. We
show gamma-rays without any absorption (red-dashed curve), gamma-rays with
absorption due to photon-photon pair production and BH pair production
(purple-dotted curve), and the EM cascade (magenta dot-dashed curve). It is
interesting to note that the initial flux above $200$ GeV mostly disappears
except a little tail at higher energies due to photon-photon pair production
losses. This is due to the fact that the photon-photon cross section
$\sigma_{\gamma\gamma}$ falls rapidly at higher energies and the thermal
photons, $n_{\rm ph}(\epsilon)$, have a black-body spectrum; the product of
both gives rise to a optical depth, $\tau_{\gamma\gamma}$ creating a well-
shaped attenuation factor $e^{-\tau_{\gamma\gamma}}$. The cascaded gamma-ray
flux also falls sharply at $200$ GeV because the photon-photon pair loss
factor $e^{-\tau_{\gamma\gamma}}$ hardly allows for anything to survive above
$200$ GeV.

Bethe-Heitler losses generally have a tiny effect on the gamma-ray energy
distribution. For Type IIn SNe, the amount of attenuation is the small gap
between the red-dashed and the purple dotted curve below $200$ GeV. Thus the
final gamma-ray flux (thick red curve) is a combination of the flux surviving
to absorption (purple dotted) and EM cascaded flux (magenta dot-dashed). The
EM cascades are responsible for a boost in the gamma-ray spectra below $200$
GeV. Propagation to Earth is not important at $10$ Mpc but important for large
distances ($\sim 1~{}\rm Gpc$) as losses due to EBL will take place, for
example for the diffuse flux (see Sec. 5).

###  4.2 Dependence on the young supernova type

Because of the intrinsic differences among the properties of the YSN Types
introduced in Sec. 3, we should expect a wide variation in the emission of
secondary particles. Below we discuss a few examples of these variations for
the secondary gamma-rays and neutrinos for a source at $10$ Mpc and
$\alpha_{\rm p}=2.0$.

Figure 3: Gamma-ray (left) and all-flavour neutrino (right) fluences
integrated for a year at Earth from different classes of YSN, as listed in
Table 1, for a source located at $10$ Mpc. For both gamma-rays and neutrinos,
Type IIn YSNe have the largest fluence, followed by II-P and Ib/c (LT) YSNe.

We show the gamma-ray fluences produced by YSNe in the top left panel of Fig.
3 for the benchmark parameters shown in Table 1. The fluxes are integrated
over a year. Type IIn YSNe produce the largest gamma-ray fluence. The other
Types of YSNe create smaller fluences and the fluence of Type Ib/c YSNe is
extremely small relative to the one seen from the Type IIn YSNe. The sharp
fall and rise of these fluences at higher energies (above $10$ GeV) is due to
the fact that gamma-rays suffer losses due to $\gamma\gamma$ pair production,
as discussed in Sec. 2.2. The fall and rise of the fluences occur at different
energies according to the YSN class. The amount of losses are also different
for different YSNe. This is due to the variation of the parameters, i.e.
average energy ($\epsilon_{\rm av}$) and peak luminosity ($L_{\rm SN,pk}$) of
thermal photons expected from different Types of YSNe. The parameter
$\epsilon_{\rm av}$ is assumed to be: $1$ eV for IIn YSNe, $4.5$ eV for II-P
YSNe, $1$ eV for IIb/II-L and Ib/c YSNe, and $0.05$ eV for Ib/c (LT) YSNe. The
peak luminosities are assumed to be $10^{41}$ erg/s for IIn SNe, $\sim 5\times
10^{40}$ erg/s for II-P and Ib/c (LT) SNe, and $10^{42}$ erg/s for IIb/II-L
and Ib/c SNe [83, 135, 155, 145, 93, 156, 157]. The fluence of Type Ib/c (LT)
YSN falls at higher energy than that of other SN Types due to its small
average energy of thermal photons. The average energy is small because of the
lower temperature of thermal photons as the CSM is located very far away (see
Table 1) from the stellar envelope. The large distance to the CSM also affects
the density of the thermal photons that results in less absorption of gamma-
rays for Type Ib/c (LT) YSNe. Type IIn and II-P YSNe have similar losses as
the peak luminosities of thermal photons are quite similar. Heavy losses of
gamma-rays are seen for Type IIb/II-L and Ib/c YSNe because their thermal
photon luminosity is quite large which implies large thermal photon density.

The right panel of Fig. 3 shows the corresponding neutrino fluence integrated
over a year for the different YSN Types. Due to the dense CSM, Type IIn YSNe
show the largest fluence falling rapidly around $E_{\nu}\sim 10^{6}$ GeV. The
fluence of Type Ib/c (LT) YSNe starts to dominate above $E_{\nu}\sim 3\times
10^{6}$ GeV. The reason is that the maximum proton energy ($E_{\rm p,max}$)
strongly depends on the shock velocity and shock radius. The shock velocity
and shock radius for Ib/c (LT) YSNe are larger than those of Type IIn YSNe,
resulting in greater $E_{\rm p,max}$. The fluence of Type II-P666 The fluences
of Type II-P (gamma-rays and neutrinos) are dominated by the emission from the
eruptive phase (see Table 1), the contribution from the normal phase is
negligible. and Ib/c (LT) YSNe are comparable at lower energies due to their
dense CSM. The fluences of Type IIb/II-L and Type Ib/c YSNe are quite small in
spite of the large initial CSM density, $n_{\rm in,CSM}$. This is due to the
small size of their inner radius ($r_{\rm in}$) of the CSM; moreover, most of
the secondaries are only produced in the vicinity of $r_{\rm in}$.

The estimated neutrino fluences for the various YSNe classes are comparable to
the ones reported in Ref. [84] except for the ones of Type II-P YSNe. In this
case, our YSNe-IIP fluence is larger 777Note that the large mass loss rates of
II-P SNe discussed in Ref. [129] may be responsible for a fluence comparable
to the one of IIn YSNe [158, 159]. However, such mass loss rates might be an
exception, as we discussed in Sec. 3 and therefore we do not consider them.
since it relies on model parameters extrapolated from more recent observations
of II-P SNe, as discussed in Sec. 3; specifically, the mass loss rate that we
adopt is one order of magnitude larger than the one assumed in Ref. [84]. On
the other hand, a CSM density smaller than our benchmark case in Table 1 would
result in a smaller fluence [84]. In addition, the neutrino emission from Ib/c
(LT) SNe is explored for the first time in this paper.

Figure 4:  Total energy emitted by each source in gamma-rays (left) and
neutrinos (right). The duration of emission of secondary particles is taken to
be a year and integrated in the range $10^{-1}$–$10^{8}$ GeV. The total energy
emitted in gamma-rays is generally smaller than that of neutrinos; this is due
to the attenuation of gamma-rays in the source. The largest energy is emitted
by Type IIn YSNe, followed by Type Ib/c (LT) and II-P YSNe.

Fig. 4 shows the total energy emitted in gamma-rays (left) and neutrinos
(right) for each YSN Type, in order to favour a comparison among their overall
energetics. We have integrated the gamma-ray and neutrino fluxes in the energy
range $10^{-1}$–$10^{8}$ GeV for a year. Both for gamma-rays and neutrinos,
the largest energy is emitted by Type IIn YSNe, followed by Type Ib/c (LT) and
II-P YSNe. By looking at the fluxes of neutrinos and gamma-rays in Fig. 3, one
might have the impression that the total energy emitted in gamma-rays is
smaller than that of neutrinos due to the large absorption of gamma-rays (the
dips). However this is not the case, most of the attenuated high energy gamma-
ray flux has reappeared at lower energies due to the EM cascades (see Fig. 2
and discussion) determining an overall small amount of loss. If there is no
absorption of gamma-rays, the total energy of neutrinos is larger by a factor
of about $3/2$ than that of gamma-rays. This corresponds to the product of
charged to neutral pion production ratio ($2$) and the fraction of energy
carried by the neutrinos in a charged pion decay ($3/4$).

##  5 Diffuse gamma-ray and neutrino backgrounds from young supernovae

In this Section, we first introduce the procedure adopted to compute the
diffuse backgrounds of high-energy particles. Then, we present our results on
the gamma-ray and neutrino diffuse backgrounds from YSNe. A discussion on the
model uncertainties and comparison with existing data follows.

###  5.1 Diffuse flux and its ingredients

The differences among the different YSNe classes for what concerns the
emission of secondaries give rise to interesting questions regarding the
contribution of each YSN class to the diffuse emission of gamma-rays and
neutrinos. In addition to the individual YSN fluxes, the diffuse flux of
secondary particles embeds the contribution from the redshift dependence of
the various YSNe Types 888We assume a constant luminosity function based on
the benchmark parameters introduced in Table 1, in light of the existing
observational uncertainties [160, 161].:

$E_{\rm j,\rm obs}^{2}\phi_{\rm j,\rm diff}(E_{\rm j,\rm
obs})=\zeta\frac{c}{H_{0}}\int_{0}^{z_{\rm max}}\mathrm{d}z\frac{R_{\rm
CCSN}(z)E_{\rm j}^{2}\phi_{\rm j}^{\rm s}(E_{\rm
j})}{\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}}e^{-\tau_{\rm j,EBL}(E_{\rm
j},z)}\ ,$ (5.1)

where, ${\rm j}=\nu$ or $\gamma$, $E_{\rm j}=(1+z)E_{\rm j,obs}$ and
$\tau_{\rm j,EBL}(E_{\rm j},z)$ is optical depth of EBL. For neutrinos,
$\tau_{\nu,\rm EBL}=0$. For gamma-rays, the optical depth $\tau_{\gamma,\rm
EBL}$ is taken from Ref. [94]. We adopt the $\Lambda\mathrm{CDM}$ cosmological
model, with $\Omega_{m}=0.31$, $\Omega_{\Lambda}=0.69$, and $H_{0}=68\hskip
2.84544pt\rm{km}\hskip 2.84544pt\rm{s}^{-1}\rm{Mpc}^{-1}$ [162].

The fraction of different SN Types may vary with redshift due to the change in
density of stars and metallicity of the host galaxies seen at higher
redshifts. However, unfortunately, the redshift distribution of SNe of
different Types is quite uncertain, and limited information is available up to
$z=1$, for some SN Types, which is not sufficient for our purposes [161].
Hence, we assume that all SN Types follow the core-collapse SN rate as a
function of the redshift. In addition, in order to take into account that some
SN Types are more common than others, we follow Ref. [160] and assume that the
fraction of different core-collapse SN Types at $z=0$ ($\zeta$) holds at
higher $z$ as well. The fraction of different SN Types at $z=0$ is shown in
Fig. 5.

The rate of core-collapse SNe is given by [163, 164, 165]:

$R_{\rm CCSN}(z)=\int_{8M_{\odot}}^{125M_{\odot}}\mathrm{d}MR_{\rm SN}(z,M),$
(5.2)

where

$R_{\rm
SN}(z,M)=\frac{\eta(M)}{\int_{0.5M_{\odot}}^{125M_{\odot}}\mathrm{d}MM\eta(M)}R_{\rm
SFR}(z),$ (5.3)

with $\eta(M)\propto M^{-2.35}$ being the initial mass function (following the
Salpeter law) [166]. The star formation rate $R_{\rm SFR}$ is [167],

$R_{\rm
SFR}(z)=C_{0}\left[(1+z)^{p_{1}k}+\left(\frac{1+z}{5000}\right)^{p_{2}k}+\left(\frac{1+z}{9}\right)^{p_{3}k}\right]^{1/k},$
(5.4)

where $k=-10$, $p_{1}=3.4$, $p_{2}=-0.3$ and $p_{3}=-3.5$. The constant of
proportionality $C_{0}$ is determined by normalizing the SN rate to the local
SN rate as $\int_{8M_{\odot}}^{125M_{\odot}}\mathrm{d}MR_{\rm SN}(0,M)=1.25\pm
0.5\times 10^{-4}\hskip 5.69046pt\mathrm{Mpc}^{-3}\mathrm{yr}^{-1}$ [168].

Figure 5: Local rate of core-collapse SNe [160]. Type II-P SNe are the most
abundant ones at $z=0$. Type Ib/c and IIb/II-L SNe are also more frequent than
Type IIn SNe. We assume Type Ib/c (LT) SNe to be $10\%$ of SNe Ib/c [137]; the
total rate of SNe Ib/c (i.e., $26\%$) includes the one of Ib/c (LT) SNe.

###  5.2 Diffuse background of gamma-rays

The diffuse background of gamma-rays from YSNe can be computed by relying on
Eq. 5.1. Because of $\gamma\gamma$ interactions occurring between YSN gamma-
rays and the EBL, losses similar to the ones occurring in the source, and
discussed in Sec. 2.2, can take place. Substantial EBL losses can affect the
high-energy tail of the gamma-ray spectral distribution, while gamma-rays
travel over large distances. For example, a $300$ GeV gamma-ray photon would
need to travel about $1$ Gpc to be attenuated by an amount of $1/e$ [94].

In order to compute the amount of EBL absorption, one needs to know the
redshift dependence of the EBL for different energies [111, 169, 94, 170]. In
this paper, we model $\tau_{\gamma,\rm EBL}(E_{\gamma})$ (see Eq. 5.1)
following Ref. [94].

Fig. 6 shows the diffuse gamma-ray background for Type IIn YSNe ($\zeta\approx
0.9$, see Fig. 5) as a function of the observed photon energy. The red-dashed
curve shows the gamma-rays produced through $pp$ interactions (without any
energy loss). The purple dot-dashed curve shows the diffuse flux with source
absorption and EM cascade, where the small part of the flux at higher energies
(above $10^{3}$ GeV) for the purple dot-dashed curve is due to the
$\gamma\gamma$ cross-section as discussed in Sec. 2.2. The red solid curve
represents the gamma-ray flux after taking into account all the absorption
(source+EBL) processes and EM cascades. The purple dot-dashed and red solid
curves are larger than the dashed red curve (no absorption) below $100$ GeV
because of the additional cascaded flux.

Figure 6: Diffuse gamma-ray flux at Earth for our benchmark Type IIn YSN (see
Table 1). The red dashed curve corresponds to the diffuse gamma-ray flux
without any absorption. The purple dot-dashed curve shows the diffuse flux
after losses ($\gamma\gamma$ and Bethe-Heitler) and EM cascade at source. The
final gamma-ray flux at Earth after EBL absorption is represented by the thick
red solid curve. EBL absorption is more pronounced above $100$ GeV. Moreover,
the propagation loss in the diffuse flux has attenuated the higher energy tail
above $10^{4}$ GeV. Thus, the final diffuse flux peaks at around $10$ GeV and
ends abruptly around $100$ GeV.

The left panel of Fig. 7 displays the diffuse gamma-ray background for the
different YSN Types by considering the benchmark values introduced in Table 1
and for $\alpha_{\rm p}=2.0$. The largest contribution to the diffuse emission
comes from Type IIn YSNe. The diffuse flux of Type II-P YSNe falls at around
$20$ GeV, i.e. at lower energies than Type IIn YSNe because the thermal
photons of Type II-P YSNe have larger average energies than the ones of Type
IIn YSNe ($\epsilon_{av}=1$ eV for IIn YSNe vs. $\epsilon_{av}=4.6$ eV for
II-P YSNe). The diffuse flux of Type Ib/c (LT) YSNe falls at larger observed
gamma-ray energies than the one of Type IIn YSNe because of the average energy
of thermal photons. It has been found that Type II-P contributes small amount
to the diffuse background at lower energies (below $10$ GeV). Overall, the
contribution of Type IIb/II-L, and Ib/c (LT) YSNe to the total diffuse gamma-
ray background is negligible. However, a small diffuse flux of gamma-rays from
Ib/c (LT) YSNe may show up above $10^{2}$ GeV. Therefore, the total gamma-ray
background (thick red curve) is mostly dominated by the contribution from Type
IIn SNe.

Figure 7: Diffuse gamma-rays (left) and all-flavour neutrinos (right) fluxes
as functions of the observed particle energy for the different YSNe classes
(see Table 1 for our benchmark model parameters). For both gamma-rays and
neutrinos, Type IIn (black solid curves) YSNe contribute the most to the total
diffuse emission, followed by Type II-P (magenta dashed curves) YSNe at lower
energies. The contribution (despite being small) from Type Ib/c (LT) (blue
dot-dashed curves) YSNe may show up above $10^{2}$ GeV in gamma-rays, whereas
in neutrinos above $10^{7}$ GeV. Type IIb/II-L (orange dotted curves) YSNe
contribute negligibly. The flux of Type Ib/c YSNe is not shown since it lies
outside the plot range. The total diffuse gamma-ray and neutrino fluxes are
plotted as thick red and thick green curves. It can be seen that IIn, II-P and
Ib/c (LT) YSNe are the main contributors to the total diffuse spectra of both
gamma-ray and neutrinos.

###  5.3 Diffuse background of high-energy neutrinos

The diffuse background of high-energy neutrinos for the different YSN Types
can be computed by relying on Eq. 5.1. The crucial difference with respect to
the diffuse gamma-ray flux is the loss of the gamma-ray flux due to the
propagation.

The diffuse background of high-energy neutrinos is shown in the right panel of
Fig. 7 for our benchmark model parameters introduced in Table 1 and for
$\alpha_{\rm p}=2.0$. Similar to what observed for gamma-rays (see left panel
of Fig. 7), the contribution from Type IIn YSNe is larger than the one of Type
II-P and Ib/c (LT) YSNe below $E_{\nu}=10^{7}$ GeV. However, Type II-P YSNe
also have significant contribution and their flux is smaller than IIn YSNe by
about a factor of $2$. The diffuse flux of Type Ib/c (LT) is small at lower
energies but might show up above $E_{\nu}=10^{7}$ GeV, this is due to the fact
that the maximum proton energy of this YSN Type is large because of the larger
shock velocity and shock radius (see also Fig. 3). Also note that the
contribution of Type IIb/II-L YSNe to the total diffuse emission of high-
energy neutrinos is negligible as the source flux of Type IIb/II-L YSNe is
already quite small in comparison to the one of other Types of YSNe.

###  5.4 Model parameter uncertainties

Figure 8: Diffuse backgrounds of gamma-rays (in red) and all-flavour neutrinos
(dark green) for all YSNe Types as a function of the observed particle energy
for $\alpha_{\rm p}=2.0$ (solid) and $\alpha_{\rm p}=2.2$ (dashed). The
subscript $j$ represents $\nu$ and $\gamma$. Larger values of $\alpha_{\rm p}$
are responsible for softer energy spectra for both gamma-rays and neutrinos;
the dependence of the diffuse spectrum on $\alpha_{p}$ is more pronounced for
neutrinos than for gamma-rays.

The injection spectral index of protons crucially affects the high-energy
diffuse backgrounds. Fig. 8 shows the total diffuse gamma-ray and neutrino
backgrounds for $\alpha_{\rm p}=2.0$ (solid curves) and $\alpha_{\rm p}=2.2$
(dashed curves). Both diffuse backgrounds are larger for $\alpha_{\rm p}=2.0$
than that for $\alpha_{\rm p}=2.2$. This is due to the hardness of the energy
distribution for $\alpha_{\rm p}=2.0$ at higher energies. Moreover, for gamma-
rays, the EM cascaded flux of $\alpha_{\rm p}=2.0$ is larger than that
obtained by using $\alpha_{\rm p}=2.2$. The dependence on $\alpha_{\rm p}$ is
more pronounced for neutrinos than for gamma-rays. This is because the higher
energy part (above $100$ GeV) of gamma-rays is heavily attenuated.

Figure 9:  Left panel: Diffuse gamma-ray background from all Types of YSNe as
a function of the observed gamma-ray energy. The solid red line indicated our
benchmark diffuse emission for $\alpha_{\rm p}=2.0$ and the other model
parameters fixed as in Table 1. The band correspond to the uncertainties in
shock velocity $v_{\rm sh}$, power law $\alpha_{\rm p}$, fraction of kinetic
energy $\epsilon_{\rm p}$ and fraction of magnetic energy $\epsilon_{\rm B}$
(see Table 2). The uncertainty associated to the local SN rate is represented
by the thin band. Right panel: Correspondent diffuse all-flavour neutrino
background. The uncertainty from model parameters is larger than the one from
the SN rate.

The left panel of Fig. 9 shows the total diffuse background of gamma-rays,
including the contribution from all YSN Types. For reference, the diffuse
background estimated for benchmark model parameters as in Table 1 and
$\alpha_{\rm p}=2.0$ is represented by the solid red curve. The uncertainty
band results from the convolution of the uncertainties on the parameters
listed in Table 1. In particular, the model parameters mostly affecting the
spectral distribution are $v_{\rm sh}$, $\epsilon_{\rm p}$ and $\epsilon_{\rm
B}$; the upper and lower limits of the parameters are reported in Table 2. Our
choices on the upper limits for these model parameters are conservative
compared to observations [67, 135, 136]. The remaining model parameters (see
Table 1) are instead kept fixed. Note that we have not considered
uncertainties on the benchmark parameters of Type IIb/II-L YSNe because their
contribution to diffuse backgrounds is negligible. The lower limit of the
gamma-ray diffuse emission is instead obtained for $\alpha_{\rm p}=2.2$ and
shows a softer energy distribution. In addition, the uncertainty associated to
the local SN rate (see Sec. 5) might also have important consequences on the
diffuse background spectra of gamma-rays and neutrinos. This uncertainty is
included in our benchmark diffuse spectra and shown by the thin band.
Interestingly, the uncertainty in the diffuse background from the SN rate is
smaller than the one from the model parameters. Analogously, the right panel
of Fig. 9 shows the total diffuse background of high-energy neutrinos.

Table 2: Uncertainties on a selection of model parameters of different YSN Types (see Table 1 for the benchmark parameters). Uncertainties on the model parameters of Type IIb/II-L and Ib/c YSNe are not included because their contributions to diffuse backgrounds of high-energy particles is negligible.  Parameters | Type IIn | Type II-P | Type Ib/c (LT)
---|---|---|---
| Upper | Lower | Upper | Lower | Upper | Lower
$v_{\rm sh}$ ($\rm kms^{-1}$) | $9.5\times 10^{3}$ | $5\times 10^{3}$ | $2\times 10^{4}$ | $8\times 10^{3}$ | $2\times 10^{4}$ | $5\times 10^{3}$
$\epsilon_{\rm p}$ | 0.1 | 0.01 | 0.1 | 0.01 | 0.1 | 0.01
$\epsilon_{\rm B}$ | $10^{-2}$ | $3\times 10^{-4}$ | $3\times 10^{-2}$ | $10^{-3}$ | $3\times 10^{-2}$ | $10^{-2}$

###  5.5 Discussion

Fig. 10 summarizes our findings on the gamma-ray and neutrino diffuse emission
from YSNe. For reference, the data points of the Fermi-LAT Isotropic Gamma-ray
Background (IGRB) between $100$ MeV to $820$ GeV are shown [171]. Moreover,
the unexplained IGRB component in the range $50$–$1000$ GeV is plotted with
the purple dashed line [172, 39]. We also show the data points with error bars
corresponding to $7.5$ years by IceCube High-Energy Starting Event (HESE). The
best fit to the IceCube data is represented by the black-dashed line; the
related $68\%$ confidence level uncertainty is plotted in cyan [3]. The
diffuse neutrino flux sensitivity at $68\%$ confidence level for KM3NeT is
shown in light blue [173].

Figure 10: Total diffuse gamma-ray (in red) and all-flavour neutrino (in
green) backgrounds from YSNe as functions of the observed particle energy,
analogous to Fig. 9. The subscript $j$ stands for $\nu$ or $\gamma$. For
gamma-rays, the orange data points with error bars illustrate the diffuse
gamma-ray background measured by Fermi-LAT (IGRB) [171]. The purple dashed
curve shows the unexplained portion of the IGRB [47, 172, 82]. For neutrinos,
the black-dashed line shows the IceCube (HESE) diffuse flux best fit for $7.5$
years of data (black data points with error bars); the cyan band depicts the
uncertainty on the IceCube diffuse flux at $68\%$ confidence level [3]. The
diffuse flux sensitivity of the future neutrino experiment KM3NeT is also
shown by the light blue band [173]. It is evident that part of the parameter
space considered for YSNe is ruled out from multi-messenger constraints from
Fermi-LAT and IceCube. Nevertheless, our benchmark YSN parameters (Table 1)
can very well explain part of the IceCube diffuse flux without the
correspondent gamma-ray emission being in tension with the Fermi-LAT gamma-ray
data. KM3NeT will further probe the diffuse neutrino flux from YSNe in the
energy range $10^{4}$-$10^{6}$ GeV.

As for gamma-rays, a large fraction of the IGRB observed by Fermi-LAT may be
coming from blazars, although recent work shows evidence for star-forming
galaxies as the dominant contributors to the IGRB [52, 174, 55, 175, 59]. Our
findings are in agreement with this picture on the IGRB composition. In fact,
our benchmark YSN gamma-ray background (red solid line) is severely attenuated
above $100$ GeV and not in tension with blazar unexplained flux (purple dashed
line). Moreover, the gamma-ray diffuse emission from star-forming galaxies
should originate from the collisions of the SN accelerated protons with
molecular clouds (ISM) in these active galaxies [176, 177] and therefore
include the contribution of YSNe as well. However, the gamma-rays created in
YSNe undergo larger attenuation (due to the dense CSM environment) than gamma-
rays created in a thin ISM [59]. By comparing the diffuse gamma-ray emission
predicted in this work with the Fermi-LAT data in Fig. 9, it is evident that
our benchmark diffuse gamma flux is smaller than the Fermi-LAT IGRB and thus
might negligibly contribute to the total SBG flux.

As for high-energy neutrinos, our benchmark YSN neutrino background is in good
agreement with the IceCube HESE data below $10^{6}$ GeV. Intriguingly, the YSN
neutrino background at low energies ($10^{4}$–$10^{5}$ GeV) is significantly
large and can explain the HESE concentration at these energies, in alternative
to dark sources or hypernovae [178, 179, 180, 181, 176]. Within the YSN
interpration, the low-energy component of the neutrino diffuse background may
originate from Type IIn SNe, whereas the neutrino diffuse background above
$10^{5}$ GeV may also have contributions from II-P and Ib/c (LT) SNe. This
multi component interpretation, in addition to likely contributions to the
neutrino diffuse emission from other sources, can accommodate possible
different power laws in different energy ranges in future data fits of IceCube
data [3]. The KM3NeT sensitivity shows that it will be able to probe the
diffuse flux of neutrinos from YSNe in the energy range $10^{4}$–$10^{6}$ GeV.

The contribution of YSNe may be subleading (e.g. for our benchmark model
parameters) to the IGRB, however it could explain very well the observed
IceCube diffuse emission below $10^{6}$ GeV, relaxing the tension between
gamma-ray and neutrino data for hadronic sources invoked as motivation for
“hidden” sources [178]. Interestingly, part of the YSN parameter space allowed
by multi-wavelength electromagnetic observations (see Table 2) may overshoot
the Fermi-LAT and IceCube data, as shown by the bands in Fig. 10. This
suggests that YSNe with such extreme model parameters are not representative
of the YSN population.

The green and red solid curves in Fig. 10 have been obtained for
$\epsilon_{\rm p}=0.01$ for Type IIn SNe. The maximum value allowed by the
IGRB data is $\epsilon_{\rm p}\simeq 0.03$, for which the corresponding
neutrino background is enhanced by a factor of $3$. Kinetic simulations
predict $\epsilon_{\rm p}$ as large as $0.2$ [103], which may be in conflict
with the high-energy diffuse backgrounds, if representative of the whole Type
IIn YSN population.

The diffuse neutrino flux from Type IIn SNe reported in Ref. [19] relies on
$\epsilon_{\rm p}\sim 0.2$, which may be in tension with the Fermi-LAT IGRB.
On the other hand, the Monte-Carlo simulations of Ref. [19] for randomly
distributed $\epsilon_{\rm p}$ in the range $[0.01,0.1]$ produced a smaller
diffuse flux, contributing up to $10\%$ to the IceCube HESE flux. We find that
the IceCube HESE data can be accommodated by smaller values of $\epsilon_{\rm
p}$, thus remaining consistent with the IGRB data. In our case, the smaller
$\epsilon_{\rm p}$ is allowed as the total diffuse background also includes
non-negligible contributions from Type IIP and Ib/c (LT) YSNe. In particular,
Ib/c (LT) YSNe having the harder spectra dominate the higher energy tail above
$1$ PeV.

Our findings are also in agreement with the ones of Ref. [182], which
estimated the neutrino and gamma-ray emission from a range of non-relativistic
shock powered transients, concluding that the observed neutrino emission may
come from gamma-ray dim sources. However, their model is based on optical
observations of these transients, while we relied on YSN model parameters
coming from a wide range of multi-wavelength surveys.

Upper limits on the high-energy diffuse neutrino background from SNe have been
provided in Refs. [183, 184], which found that the diffuse neutrino background
is dominated by SNe II-P, followed by SNe IIn and Ib/c. These findings are in
contrast with ours because of the different choices of the local SN rates
(Ref. [183] assumes $\xi=52.4\%$ for SNe II-P, $6.4\%$ for SNe IIn, and $25\%$
for SNe Ib/c, while we considered the YSN fractions summarized in Fig. 5). In
addition, Refs. [183, 184] do not take into account gamma-ray constraints.

We note that the larger mass loss rates of SNe II-P reported in Refs. [129,
128] (see also Sec. 3) can produce a diffuse emission from YSN II-P comparable
to that of Type IIn SNe. This, in turn, may further constrain the parameters
characteristics of the SN IIn population.

##  6 Detection prospects of nearby young supernovae in gamma-rays and
neutrinos

Figure 11: Detection prospects of nearby supernovae using gamma-ray (left) and
neutrino telescopes (right). Top left panel: Gamma-ray energy fluxes from the
different YSN Types at $10$ Mpc as functions of the observed particle energy.
The one year Fermi-LAT sensitivity is shown by the thick light brown dashed
curve [85] and the thick green dotted curve represents the $100$ hour CTA
sensitivity [86]. Type IIn YSNe may be detected by both Fermi-LAT and CTA,
while all other sources will be too dim at 10 Mpc. Top right panel:
Corresponding muon neutrino energy fluxes. The sensitivities of IceCube,
IceCube-Gen2 and KM3NeT for point source detection are plotted (thick dashed
lines) in dark cyan (IceCube) [90], red (IceCube-Gen2) [91] and dark brown
(KM3NeT) [92]. The sensitivities of IceCube, IceCube-Gen2 and KM3NeT are
plotted for the declination angle $\delta=0^{o}$. All these neutrino
observatories will be able to detect YSNe at distances smaller than $10$ Mpc.
Bottom left panel: Gamma-ray YSN detection horizon for Fermi-LAT (light brown)
and CTA (green) as functions of the YSN Type. For each YSN Type, the error
band takes into account the model uncertainties (see Sec. 5.5). Fermi-LAT and
CTA could detect YSNe up to $10$ Mpc (see YSNe IIn); CTA could have better
sensitivity than Fermi-LAT and reach up to $2$ Mpc for YSNe Ib/c (LT). Bottom
right panel: Corresponding neutrino YSN detection horizon for IceCube (dark
cyan), IceCube-Gen2 (red) and KM3NeT (dark brown). IceCube-Gen2 will be able
to detect YSNe up to $\sim 4$ Mpc (see YSNe Type IIn).

As shown in the previous Section, the diffuse backgrounds of neutrinos and
gamma-rays from YSNe have large uncertainties due to the widely varying model
parameters. The detection of neutrinos and gamma-rays from nearby YSNe will
help to further constrain these model parameters and can potentially provide
complementary understanding of shock-CSM interactions. In the following, we
investigate the detection prospects of YSNe with present and future gamma-ray
telescopes and neutrino detectors.

Figure 11 illustrates the detection prospects for our benchmark YSNe in gamma-
rays and neutrinos. The top panels display the energy flux expected at Earth
for a YSN at $10$ Mpc. The top left panel shows that the gamma-ray energy flux
would be detectable for Type IIn YSN by Fermi-LAT [85] and CTA [86], while
other SN Types can be probed at distances smaller than $10$ Mpc.
Interestingly, due to gamma-ray attenuation, dips could appear in the gamma-
ray spectra between $100$–$1000$ GeV (see related discussion for Fig. 2), CTA
may be able to probe this feature for local YSNe. However, the detection of
such dips requires more detailed analysis [185, 186]. The top right panel
shows that corresponding neutrino flux from all YSNe are detectable in IceCube
[90] below $10$ Mpc, in agreement with Ref. [84]. Km3NeT [92] and IceCube-Gen2
[91] will also be able to detect neutrinos from these sources below $10$ Mpc.

The bottom panels of Fig. 11 show the detection horizon of existing and
upcoming gamma-ray and neutrino telescopes for the different YSN Types, by
considering the model parameter uncertainties, see Table 2. The reach of
different telescopes is calculated by taking into account the integrated flux
of each YSN in the specific energy range of detectors’ best integral
sensitivity. For gamma-rays, the energy range [$10^{-1},10^{1}$] GeV is chosen
for Fermi-LAT as it has the best integral sensitivity in this range. However,
the whole energy range of CTA, i.e., [$2\times 10^{1},7\times 10^{4}$] GeV is
taken into account as the fluxes of most of YSNe are attenuated where CTA is
most sensitive. The maximum reach of Fermi-LAT and CTA for different YSNe is
shown in the bottom left panel of Fig. 11. Fermi-LAT and CTA could detect YSNe
up to $10$ Mpc (see YSNe IIn, for the benchmark parameters). However, Fermi-
LAT has comparatively better sensitivity to YSNe than CTA except for Ib/c (LT)
YSNe. This is due to the fact that CTA is more sensitive at higher energies
and gamma-rays at these energies are attenuated by the thermal photons in the
source. The exception of Ib/c (LT) YSNe is accounted for the small gamma-ray
attenuation in the source (see top left plot and the discussion of Fig. 3).
The detection horizon of CTA is affected by the amount of gamma-ray
attenuation by the source thermal photons as discussed above. Thus, the
detection of these sources in gamma-rays has the potential to unleash
information about particle production, emission and propagation in YSNe.

Similarly, for the neutrino telescopes, we consider the YSNe fluxes in the
energy range [$10^{5},10^{6}$] GeV as these detectors have best sensitivity to
the astrophysical neutrino flux in these energies (see the top right panel).
The detection horizons marked in the plot refer to the ones for which the
integrated flux is equal to the detectors’ integral sensitivity (see the top
panels). For example, for YSNe Type IIn at distances above $2$ Mpc, the
integrated neutrino flux in the energy range $10^{5}$–$10^{6}$ GeV would fall
below the IceCube sensitivity at $\delta=0^{o}$ declination. The error bars
represent the uncertainties in the model parameters. Our findings concerning
the detectability of neutrinos from YSNe IIn are in agreement with the ones of
Ref. [19, 187]. Considering the uncertainties in the model parameters,
IceCube-Gen2 will instead be able to detect Type IIn YSNe up to $\sim 20$ Mpc
($4$ Mpc for the benchmark parameters).

##  7 Conclusions

In this paper, we investigate the high energy neutrino and gamma ray signals
from different classes of young SNe (Type IIn, II-P, IIb/II-L and Ib/c YSNe),
up to one year after their explosion. During this time, shock accelerated
protons interact with the dense CSM around these objects, leading to the
production of secondary high-energy neutrinos and gamma-rays. In particular,
for the first time, we also investigate the late time emission of YSNe Ib/c,
coming from the interaction of the SN shock with a dense hydrogen rich CSM far
away from the stellar envelope.

Despite intense research activity, the origin of the bulk of the diffuse high-
energy neutrino background observed by the IceCube Neutrino Observatory as
well as the the Isotropic Gamma-Ray Background detected by Fermi-LAT is yet
unknown. Our benchmark (intermediate) diffuse neutrino emission is in
excellent agreement with the IceCube HESE data and suggests that YSNe could
constitute the bulk of the high-energy diffuse neutrino emission observed by
IceCube below $10^{6}$ GeV, while being dim enough in gamma-rays to avoid any
conflict with Fermi-LAT data.

The largest contribution to the diffuse neutrino emission mainly comes from
Type IIn YSNe, followed by II-P and Ib/c (LT) YSNe, with Type IIn YSNe
dominating the overall neutrino diffuse emission up to $10^{7}$ GeV. Type Ib/c
(LT) YSNe populate the diffuse neutrino emission above $10^{7}$ GeV. The Type
II-P YSN contribution is also significant, but smaller than the Type IIn SN
one. Similar findings also hold for the diffuse gamma-ray background, after
including all attenuation effects taking place both in the source and en route
to Earth.

Intriguingly, the uncertainty bands obtained for the diffuse high-energy
backgrounds by taking into account the uncertainties on the shock velocity,
injection spectral index of protons, the kinetic and magnetic energy
fractions, and the YSN rate suggest that a large fraction of the YSN parameter
space inferred from multi-wavelength electromagnetic observations of YSNe is
excluded by Fermi-LAT and IceCube HESE data.

The detection prospects of nearby YSNe with existing and upcoming gamma-ray
and neutrino telescopes have also been explored. Among all YSNe, Type IIn SNe
have the best discovery potential up to $\sim 10$ Mpc ($\sim 4$ Mpc) in gamma-
rays with Fermi-LAT and CTA (in neutrinos with IceCube, KM3NeT and IceCube-
Gen2). Interestingly, CTA may be able to distinguish gamma-ray attenuation
features in the spectral energy distributions for nearby transients. However,
a possible amplification of magnetic field, e.g. due to Bell non-resonant
streaming of cosmic rays, might influence proton acceleration and the gamma-
ray spectra [see e.g., 188, 189, 190]. Multimessenger observations of such
point sources will be able to probe these mechanisms and requires future
dedicated work.

To conclude, the high-energy neutrino emission (especially coming from Type
IIn, II-P and Ib/c (LT) SNe) can be a strong contender to explain the low-
energy IceCube HESE dataset. The corresponding diffuse gamma ray emission is
not in tension with the IGRB data from Fermi-LAT as YSNe gamma-rays are
heavily attenuated and effectively “hidden.” The detection of high-energy
particles from young supernovae will provide new insights on the processes
linked to particle acceleration in young SNe.

## Acknowledgments

We are grateful for helpful discussions with Madhurima Chakraborty, Jens
Hjorth, Raffaella Margutti, and especially Tetyana Pitik. S.C acknowledges the
support of the Max Planck India Mobility Grant from the Max Planck Society,
supporting the visit and stay at MPP during the project. S.C has also received
funding from DST/SERB projects CRG/2021/002961 and MTR/2021/000540. This
project has received funding from the Villum Foundation (Project No. 37358),
the Carlsberg Foundation (CF18-0183), the Deutsche Forschungsgemeinschaft
through Sonderforschungbereich SFB 1258 “Neutrinos and Dark Matter in Astro-
and Particle Physics” (NDM). Parts of this research were supported by the
Australian Research Council Centre of Excellence for All Sky Astrophysics in 3
Dimensions (ASTRO 3D), through project number CE170100013.

##  Appendix A Characteristic timescales for proton acceleration and radiative
processes

Shock accelerated protons lose energy through different processes such as $pp$
interactions, dynamic or adiabatic losses, proton synchrotron, inverse
Compton, Bethe-Heitler, etc. In this Appendix, we provide an overview of the
different energy loss time scales for high energy protons and compare them
with the acceleration time scales.

Figure 12: Acceleration time scale of protons and various proton energy loss
processes as functions of the proton energy for Type IIn YSNe (top panels) and
Type II-P SNe (bottom panels) using the parameters in 1 and Sec. 4.2. The left
panels show the interaction time for $r=r_{\rm in}$, while right panels
consider $r=10\ r_{\rm in}$. Proton cooling is dominated by $pp$ losses and
therefore the total cooling timescale is the same as the $pp$ collision
timescale.

### Acceleration time scale

The proton acceleration time scale is given by [104]:

$t_{\rm acc}=\frac{6E_{\rm p}c}{eBv_{\rm sh}^{2}}\ ,$ (A.1)

where the magnetic field $B$ is given in Sec. 2.1.

### Adiabatic loss time scales

The adiabatic loss time scale for protons is same as the dynamical time scale
and is given by:

$t_{\rm ad}\sim t_{\rm dyn}=\frac{r}{v_{\rm sh}}.$ (A.2)

### Proton-proton collision time scale

The $pp$ collision time scale is given by

$t_{\rm pp}=(\kappa_{\rm pp}\sigma_{\rm pp}(E_{\rm p})n_{\rm CSM}(r)c)^{-1},$
(A.3)

where $\kappa_{\rm pp}=0.5$ is the $pp$ collision inelasticity.

### Proton synchrotron time scale

High energy protons may lose energy due to proton synchrotron radiation. The
corresponding synchrotron time scale is given by [191, 192]:

$t_{\rm syn}=\frac{\gamma m_{\rm p}c^{2}}{P_{\rm syn}},$ (A.4)

where $\gamma=E_{\rm p}/m_{\rm p}c^{2}$ and $P_{\rm{syn}}$ is the synchrotron
power loss and is given by

$P_{\rm syn}=\frac{4}{3}\sigma_{T}c\left(\frac{m_{\rm e}}{m_{\rm
p}}\right)^{2}\frac{B^{2}(r)}{8\pi}\left(\frac{v_{\rm
sh}}{c}\right)^{2}\gamma^{2}\ ,$ (A.5)

where $\sigma_{\rm T}$ is the Thomson scattering cross-section.

### Inverse Compton loss time scales

Protons may lose energy to low energy SN photons via inverse Compton
scattering and the corresponding loss time scale is given by [191]

$t_{\rm IC}=\frac{\gamma m_{\rm p}c^{2}}{P_{\rm IC}},$ (A.6)

where the inverse Compton power $P_{\rm IC}$ is given by

$P_{\rm IC}=\frac{4}{3}\sigma_{\rm T}cU_{\rm
ph}(r)\gamma^{2}\left(\frac{v_{\rm sh}}{c}\right)^{2}\left(\frac{m_{\rm
e}}{m_{\rm p}}\right)^{2}\ ,$ (A.7)

where, $U_{\rm ph}(r)$ is the energy density of photons in CSM [83]:

$U_{\rm ph}(r)=\frac{{L_{\rm SN,pk}}}{{(4\pi cr_{\rm
in}^{2})}}\left(\frac{r_{\rm in}}{r}\right)^{2}.$ (A.8)

The SN peak luminosity $L_{\rm SN,pk}$ for different YSNe can be found in Sec.
4.2.

### Bethe-Heitler energy loss time scale

The Bethe-Heitler energy loss time scale is

$t_{\rm BH}=\tau_{\rm BH}\left(\frac{r}{r_{\rm in}}\right)^{2}$ (A.9)

where $\tau_{\rm BH}^{-1}={(\kappa_{\rm BH}\sigma_{\rm BH}cU_{\rm ph}(r_{\rm
in}))}/{\epsilon_{\rm av}}$ and $\kappa_{\rm BH}\sigma_{\rm BH}\approx 6\times
10^{-31}$ $\rm cm^{2}$ [83].

### Dominant energy loss processes

Fig. 12 shows the time scales introduced above as functions of the proton
energy for the dominant YSN classes: SNe IIn (top panel) and II-P (bottom
panel), see Table 1. Since the CSM density and SN thermal photon density
change with the shock radius, the acceleration timescale and different cooling
timescales also change. Therefore, we show the evolution of these timescales
for two different values of shock radius, i.e., $r=r_{\rm in}$ (left panel)
and $r=10r_{\rm in}$ (right panel). For both classes of SNe, the dominant
energy loss processes are $pp$ collisions and adiabatic losses for both shock
radii. Note that photo-hadronic ($p\gamma$) interactions are not important
here as the average energy of SN thermal photons is about $1$ eV; the
threshold energy of protons for this process is $\sim 10$ PeV and the SN flux
of protons above this energy is very small. Moreover, the cross-section of
$p\gamma$ interactions ($\sim 10^{-28}$ $\rm cm^{2}$ [45]) is much smaller
than that of $pp$ interaction ($\sim 10^{-26}$ $\rm cm^{2}$ [44]).

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