# Interacting supernovae as high-energy multimessenger transients

Kohta Murase  Department of Physics; Department of Astronomy & Astrophysics;
Center for Multimessenger Astrophysics, Institute for Gravitation and the
Cosmos, The Pennsylvania State University, University Park, Pennsylvania
16802, USA  School of Natural Sciences, Institute for Advanced Study,
Princeton, New Jersey 08540, USA  Center for Gravitational Physics and Quantum
Information, Yukawa Institute for Theoretical Physics, Kyoto, Kyoto, 606-8502,
Japan

(submitted 24 December 2023)

###### Abstract

Multiwavelength observations have revealed that dense, confined circumstellar
material (CCSM) commonly exists in the vicinity of supernova (SN) progenitors,
suggesting enhanced mass losses years to centuries before their core collapse.
Interacting SNe, which are powered or aided by interaction with the CCSM, are
considered to be promising high-energy multimessenger transient sources. We
present detailed results of broadband electromagnetic emission, following the
time-dependent model proposed in the previous work on high-energy SN neutrinos
[Murase, Phys. Rev. D 97, 081301(R) (2018)]. We investigate electromagnetic
cascades in the presence of Coulomb losses, including inverse-Compton and
synchrotron components that significantly contribute to MeV and high-frequency
radio bands, respectively. We also discuss the application to SN 2023ixf.

##  I Introduction

Multiwavelength observations of supernovae (SNe) in optical, radio and x-ray
bands have provided cumulative evidences for the existence of dense
circumstellar material (CSM) surrounding their progenitors Smith (2017); Maeda
(2022), which has led to a paradigm shift in the stellar evolution study Smith
(2014). A good fraction of hydrogen-rich (Type II) SNe show signatures of
confined CSM (CCSM), which include narrow emission lines Khazov  _et al._
(2016); Yaron  _et al._ (2017); Boian and Groh (2020); Terreran  _et al._
(2022); Bruch  _et al._ (2021, 2023) and optical light curves at early times
Morozova  _et al._ (2017); Das and Ray (2017); Förster  _et al._ (2018). The
radiative output of interaction-powered SNe, which are often classified as
Type IIn SNe, is dominated by the shock interaction between the SN ejecta and
the CSM Smith and McCray (2007); Immler  _et al._ (2008); Miller  _et al._
(2009)). Variable activities of progenitors or even outbursts that occur
months to years before explosion are also observed Ofek  _et al._ (2013,
2014a); Margutti  _et al._ (2014); Jacobson-Galán  _et al._ (2022). The
interaction with such CCSM can be important even in Type Ibc SNe, as evidenced
from transrelativistic SNe Campana  _et al._ (2006); Nakar and Sari (2012),
some fast blue optical transients like AT 2018cow Margutti  _et al._ (2019);
Fox and Smith (2019), and interacting SNe like SN 2014C Margutti  _et al._
(2017). The diversity of such “interacting SNe, whether the optical emission
is mainly powered by the CSM interaction or not, suggests the importance of
understanding the final phase in the evolution of massive stars Woosley  _et
al._ (2007); Moriya and Tominaga (2012); Quataert and Shiode (2012); Chevalier
(2012); Fuller (2017), and the progenitors of core-collapse SNe may commonly
experience significant mass losses ranging from $\sim 10^{-3}$ to $\sim
1~{}M_{\odot}~{}{\rm yr}^{-1}$ that occur $\sim 0.1-100$ yr before the
explosion Smith (2014); Moriya  _et al._ (2014); Tsuna  _et al._ (2021a). This
picture has also been supported by late-time radio observations of core-
collapse SNe Stroh  _et al._ (2021). The recent discovery of SN 2023ixf at a
distance of $d\sim 7$ Mpc provided another golden example of CCSM-interacting
SNe II with various signatures at different wavelengths Yamanaka  _et al._
(2023); Jacobson-Galan  _et al._ (2023); Bostroem  _et al._ (2023); Teja  _et
al._ (2023); Hiramatsu  _et al._ (2023); Grefenstette  _et al._ (2023);
Chandra  _et al._ (2023); Zimmerman  _et al._ (2023); Li  _et al._ (2023).

The collision between SN ejecta and dense CSM creates a rich tapestry of
physical processes and will lead to various high-energy signatures. Large CSM
masses inevitably cause efficient dissipation of the kinetic energy of the SN
ejecta via shocks Chevalier and Fransson (2017). The formation of
collisionless shocks will be accompanied by the onset of diffusive shock
acceleration (DSA) of cosmic rays (CRs) and resulting nonthermal emission
Murase  _et al._ (2011); Katz  _et al._ (2012). Murase et al. Murase  _et al._
(2011) proposed that interaction-powered SNe like Type IIn SNe are promising
sources of high-energy neutrinos and gamma rays. Murase Murase (2018)
(hereafter M18) first investigated high-energy neutrino emission from various
types of SNe considering interaction with CCSM and showed that Galactic SNe
are promising multienergy neutrino transients for neutrino detectors such as
IceCube and Super-Kamiokande. Detection of high-energy SN neutrinos has been
of interest for studying particle physics Wen  _et al._ (2023). Hadronic
gamma-ray emission from interacting SNe has also been investigated both
theoretically Murase  _et al._ (2011); Kashiyama  _et al._ (2013); Murase  _et
al._ (2019); Cristofari  _et al._ (2022); Sarmah  _et al._ (2022); Brose  _et
al._ (2022); Guarini  _et al._ (2023); Sarmah (2023) and observationally
Ackermann  _et al._ (2015), and analyses with the Fermi data enable us to
obtain meaningful constraints on the CR acceleration from SN 2010jl Murase
_et al._ (2019) and find the possible signal from iPTF14hls Yuan  _et al._
(2018). This CCSM interaction scenario does not rely on an additional central
engine source, and it is different from the scenario assuming a pulsar
embedded in SN ejecta to be a CR accelerator Berezinskii and Ginzburg (1987);
Gaisser and Stanev (1987); Murase  _et al._ (2009); Kotera  _et al._ (2013);
Murase  _et al._ (2015). Interacting SNe can be super-PeVatrons Murase  _et
al._ (2014); Inoue  _et al._ (2021); Diesing (2023), which may also contribute
to Galactic CRs Sveshnikova (2003); Zirakashvili and Ptuskin (2016); Kawanaka
and Lee (2021). Revealing high-energy nonthermal emission from such
interacting supernovae will not only expand our understanding of the late
stages of stellar evolution but also shed light on the properties of particle
acceleration.

In this work, we investigate high-energy nonthermal emission from various
classes of SNe considering early CSM interactions lasting from days to months,
and provide details of numerical calculations with the Astrophysical
Multimessenger Emission Simulator (AMES). In particular, following M18 that
focuses on high-energy SN neutrino afterglows, we present the results on
broadband electromagnetic emission, taking into account synchrotron and
inverse-Compton (IC) processes and accompanied electromagnetic cascades. The
results will be useful for multimessenger modeling of nearby SNe and distant
SNe that could be found by neutrino-triggered followup observations.
Throughout this work, we use $Q\equiv 10^{x}Q_{x}$ in CGS units.

Table 1: CSM parameters for various types of SNe considered in this work and
M18.

Class | $D_{*}$ |  $\dot{M}_{w}$ [$M_{\odot}$ yr-1] |  $V_{w}$ [km s-1] |  $R_{*}$ [cm] |  $R_{\rm cs}$ [cm] | $\delta$ |  $t_{\nu\rm max}$ [s]
---|---|---|---|---|---|---|---
II (CCSM) | ${10}^{-2}-1$ | ${10}^{-3}-{10}^{-1}$ | $100$ | $6\times{10}^{13}$ | $(0.4-1.0)\times{10}^{15}$ | 12 | ${10}^{5.8}-{10}^{6.5}$
IIn | ${10}^{-2}-1$ | ${10}^{-3}-{10}^{-1}$ | $100$ | ${10}^{13}$ | ${10}^{16}$ | 10 | ${10}^{5.8}-{10}^{7.5}$
II-P | $1.34\times{10}^{-4}$ | $2\times{10}^{-6}$ | $15$ | $6\times{10}^{13}$ | - | 12 | ${10}^{5.4}$
II-L/IIb | ${10}^{-3}$ | $3\times{10}^{-5}$ | $30$ | $6\times{10}^{12}$ | - | 12 | ${10}^{4.6}$
Ibc | ${10}^{-5}$ | ${10}^{-5}$ | $1000$ | $3\times{10}^{11}$ | - | 10 | ${10}^{3.8}$

##  II Physical Model

###  II.1 Dynamics

Let us consider SN ejecta with an ejecta mass of $M_{\rm
ej}=10~{}M_{\odot}M_{\rm ej,1}$, which interact with a CCSM (that can also be
an extended stellar envelope) with a density profile of

$\varrho_{\rm cs}=DR_{\rm cs}^{-2}{\left(\frac{r}{R_{\rm cs}}\right)}^{-w},$
(1)

where $R_{\rm cs}$ is the outer radius of the CCSM and $w$ is a CSM density
slope. We use a wind profile (i.e., $w=2$), which is reasonable and sufficient
for the purpose of this work, although a shallower profile has been discussed
both theoretically Tsuna  _et al._ (2021b) and observationally Stroh  _et al._
(2021). In the wind case, the CSM parameter is written as

$D\equiv 5\times{10}^{16}~{}{\rm g~{}{\rm
cm}^{-3}}~{}D_{*}=\frac{\dot{M}_{w}}{4\pi f_{\Omega}V_{w}},$ (2)

where ${\dot{M}_{w}}$ is the wind mass-loss rate, $V_{w}$ is the wind
velocity, and $f_{\Omega}\equiv\Delta\Omega/(4\pi)$ is the covering factor of
the CSM, which can be lower than unity if the CSM is aspherical and/or clumpy.
For example, early flash spectroscopy for SN 2013fs indicate
$D_{*}\sim{10}^{-2}$ and $R_{\rm cs}\sim 4\times 10^{14}$ cm Yaron  _et al._
(2017). Modeling of SN 2020tlf (II-P) suggests $D_{*}\sim 0.6$ and $R_{\rm
cs}\sim 10^{15}$ cm Chugai and Utrobin (2022) (see also Ref. Jacobson-Galán
_et al._ (2022)). As an example of SNe IIn, observations of SN 2010jl infer
$D_{*}\sim 6$ and $R_{\rm cs}\sim 10^{16}$ cm Ofek  _et al._ (2014b); Fransson
_et al._ (2014). Ref. Boian and Groh (2020) shows that interacting SNe have a
range of parameters from $D_{*}\sim{10}^{-2}-1$. The number density of
nucleons is $n_{N}=\varrho_{\rm cs}/m_{H}$ and that in the shocked region
$n^{\prime}_{N}$ is larger by the compression ratio. Note that the total CSM
mass ($M_{\rm cs}$) is also obtained by performing the volume integral.

A faster component of the SN ejecta is decelerated earlier, and the forward
shock evolution is obtained by solving the equation for the conservation of
momentum Chevalier (1982); Nadezhin (1985). In the thin shell approximation,
the radius ($R_{s}$) and velocity ($V_{s}$) of the shell are determined by
Moriya  _et al._ (2013); Li  _et al._ (2023)

$\displaystyle M_{s}\frac{dV_{s}}{dt}=4\pi f_{\Omega}R_{s}^{2}[\varrho_{\rm
ej}{(V_{\rm ej}-V_{s})}^{2}-\varrho_{\rm cs}{(V_{s}-V_{w})}^{2}],$ (3)

where $\varrho_{\rm ej}\propto t^{-3}{(r/t)}^{-\delta}$ (where $\delta\geq
6.67$) is the the outer ejecta profile, $M_{s}$ is mass of the shell
consisting of the shocked SN and CSM, and $V_{\rm ej}$ is the ejecta velocity.
We adopt $\delta=12$ ($\delta=10$) for supergiant stars with a radiative
envelope (Wolf-Rayet-like compact stars with a convective envelope) Matzner
and McKee (1999), although our conclusions are largely insensitive to this
assumption. Typical progenitors of SNe II-P and II-L/IIb are thought to be red
supergiants (RSGs) and yellow supergiants, respectively Smith (2014).
Practically, a power-law solution of Eq. (3), which is obtained with $V_{\rm
ej}=R_{s}/t$ and $V_{s}\gg V_{w}$, is valid until the shock radius reaches
$R_{\rm s}=V_{t}t_{t}$, where $V_{t}={[10(\delta-5){\mathcal{E}}_{\rm
ej}/3/(\delta-3)/M_{\rm ej}]}^{1/2}$ Moriya  _et al._ (2013). The deceleration
of the whole ejecta would happen if $M_{\rm ej}\lesssim M_{\rm cs}$.

While the solution of Eq. (3) is used in our numerical calculations Murase
(2018), analytical estimates would be useful for understanding the physics.
The shock radius is approximately given by

$R_{s}\simeq\left\\{\begin{array}[]{lr}2.4\times{10}^{14}~{}{\rm
cm}~{}D_{*,-2}^{-1/10}{\mathcal{E}}_{\rm ej,51}^{9/20}M_{\rm
ej,1}^{-7/20}t_{5.5}^{9/10}&\mbox{($\delta=12$)}\\\ 1.7\times{10}^{14}~{}{\rm
cm}~{}D_{*}^{-1/8}{\mathcal{E}}_{\rm ej,51}^{7/16}M_{\rm
ej,1}^{-5/16}t_{5.5}^{7/8}&\mbox{($\delta=10$)}\end{array}\right.,$ (4)

and the corresponding velocity $V_{s}=dR_{s}/dt$ is

$V_{s}\simeq\left\\{\begin{array}[]{lr}6.9\times{10}^{8}~{}{\rm cm}~{}{\rm
s}^{-1}~{}D_{*,-2}^{-1/10}{\mathcal{E}}_{\rm ej,51}^{9/20}M_{\rm
ej,1}^{-7/20}t_{5.5}^{-1/10}\\\ \mbox{(for $\delta=12$)}\\\
4.6\times{10}^{8}~{}{\rm cm}~{}{\rm s}^{-1}~{}D_{*}^{-1/8}{\mathcal{E}}_{\rm
ej,51}^{7/16}M_{\rm ej,1}^{-5/16}t_{5.5}^{-1/8}\\\ \mbox{(for
$\delta=10$)}\end{array}\right..$ (5)

A fraction of the shell kinetic energy that is dissipated at a shock is
converted into internal energy, magnetic fields, and CRs. The kinetic
luminosity, $L_{s}=2\pi f_{\Omega}\varrho_{\rm cs}V_{s}^{3}R_{s}^{2}$, is
estimated to be

$L_{s}\simeq\left\\{\begin{array}[]{lr}1.0\times{10}^{42}~{}{\rm erg}~{}{\rm
s}^{-1}~{}f_{\Omega}D_{*,-2}^{7/10}{\mathcal{E}}_{\rm ej,51}^{27/20}M_{\rm
ej,1}^{-21/20}t_{5.5}^{-3/10}\\\ \mbox{(for $\delta=12$)}\\\
3.1\times{10}^{43}~{}{\rm erg}~{}{\rm
s}^{-1}~{}f_{\Omega}D_{*}^{5/8}{\mathcal{E}}_{\rm ej,51}^{21/16}M_{\rm
ej,1}^{-16/15}t_{5.5}^{-3/8}\\\ \mbox{(for $\delta=10$)}\end{array}\right..$
(6)

For the demonstrative purpose of this work, it is sufficient to assume
$f_{\Omega}=1$. In reality, the observed flux is related to $L_{s}$, and
uncertainty in $f_{\Omega}$ degenerates with uncertainties in the other
parameters.

###  II.2 CR acceleration and secondary production

It is believed that GeV–PeV CRs originate from SN remnants with an age of
$\sim 10^{3}-10^{4}$ yr (that is comparable to the deceleration time of the
whole SN ejecta). They are established as efficient particle accelerators Funk
(2015), where both ions and electrons are accelerated by the DSA mechanism
Drury (1983); Caprioli (2023) that is one of the Fermi acceleration processes
Fermi (1949). A shell caused by SN shocks had not been thought as promising
high-energy neutrino and gamma-ray emitters. First, most of the SN ejecta
freely expands just after the SN explosion, so that energy carried by CRs via
DSA is small especially if the shock propagates in the interstellar medium
Berezinskii and Ginzburg (1987). Second, DSA at a shock inside a star is
inefficient while the shock is radiation mediated Weaver (1976). This would
still be the case around shock breakout when the material has a steep density
profile and the shock is subject to radiative acceleration Katz  _et al._
(2012); Murase  _et al._ (2019); Tsuna  _et al._ (2023).

However, the existence of CCSM has changed prospects for high-energy neutrino
and gamma-ray emission, as studied in M18. Efficient DSA may start once the SN
shock leaves a star, for which the condition is given by $R_{*}=R_{s}(t_{*})$
for $V_{s}<V_{s,\rm max}$ (where $V_{s,\rm max}$ is the maximum velocity
Matzner and McKee (1999)), and we have

$\displaystyle t_{*}\simeq\left\\{\begin{array}[]{ll}6.8\times{10}^{4}~{}{\rm
s}~{}D_{*,-2}^{1/9}M_{\rm ej,1}^{7/18}{\mathcal{E}}_{\rm
ej,51}^{-1/2}R_{*,13.78}^{10/9}\\\ \mbox{(for $\delta=12$)}\\\
1.3\times{10}^{4}~{}{\rm s}~{}D_{*}^{1/7}M_{\rm ej,1}^{5/14}{\mathcal{E}}_{\rm
ej,51}^{-1/2}R_{*,13}^{8/7}\\\ \mbox{(for $\delta=10$)}\\\
\end{array}\right..$ (11)

If the CSM is too dense, the shock is initially radiation mediated, in which
the shock jump is smeared out by radiation from the downstream and low-energy
CRs gain little energy Murase  _et al._ (2014). However, the formation of
collisionless shocks (mediated by plasma instabilities) is unavoidable
especially for a wind or shallower density profile Katz  _et al._ (2012);
Kashiyama  _et al._ (2013); Murase  _et al._ (2014), and the condition is
given by $\tau_{T}\approx\sigma_{T}{\varrho}_{\rm
cs}R_{s}/(\mu_{e}m_{H})\lesssim c/V_{s}$, where $\tau_{T}$ is the optical
depth to the Thomson scattering with the cross section, $\sigma_{T}\approx
6.7\times{10}^{-25}~{}{\rm cm}^{2}$. This coincides with the photon breakout
time Chevalier and Irwin (2011), which is.

$\displaystyle t_{\rm bo}\simeq 5.8\times{10}^{3}~{}{\rm
s}~{}D_{*,-2}\mu_{e}^{-1}.$ (12)

Considering these two necessary criteria, the onset time of CR acceleration is
estimated by Murase (2018)

$t_{\rm onset}\simeq{\rm max}[t_{\rm bo},t_{*}].$ (13)

For $D_{*}\lesssim 0.1$, which is the case for most SNe II, we expect $t_{\rm
onset}\sim t_{*}$, although $t_{\rm onset}\sim t_{\rm bo}$ when the CSM is
denser. The CSM interaction ends when the shock reaches the edge of the CCSM,
and its time scale is given by

$\displaystyle t_{\rm
end}\approx\left\\{\begin{array}[]{ll}1.5\times{10}^{6}~{}{\rm
s}~{}D_{*,-2}^{1/9}M_{\rm ej,1}^{7/18}{\mathcal{E}}_{\rm ej,51}^{-1/2}R_{\rm
cs,15}^{10/9}.&\mbox{($\delta=12$)}\\\ 3.4\times{10}^{7}~{}{\rm
s}~{}D_{*}^{1/7}M_{\rm ej,1}^{5/14}{\mathcal{E}}_{\rm ej,51}^{-1/2}R_{\rm
cs,16}^{8/7}.&\mbox{($\delta=10$)}\end{array}\right..\,\,\,$ (16)

The CR acceleration time is given by $t_{\rm
acc}\approx\eta\varepsilon_{p}/(eBc)$, where $\eta=(20/3)(c^{2}/V_{s}^{2})$
for a nonrelativistic shock whose normal is parallel to the magnetic field in
the Bohm limit Drury (1983). The magnetic field is parametrized as

$U_{B}=\frac{B^{2}}{8\pi}=\varepsilon_{B}\frac{3L_{s}}{4\pi
f_{\Omega}R_{s}^{2}V_{s}}$ (17)

where $U_{B}$ is the magnetic energy density. Observations of SNe and
numerical simulations suggest $\varepsilon_{B}\sim{10}^{-3}-{10}^{-2}$ Maeda
(2012); Caprioli and Spitkovsky (2014), and we adopt
$\varepsilon_{B}={10}^{-2}$ as a fiducial value. The magnetic field strength
is estimated to be

$\displaystyle B\simeq\left\\{\begin{array}[]{ll}39~{}{\rm
G}~{}\varepsilon_{B,-2}^{1/2}D_{*,-2}^{1/2}t_{5.5}^{-1}.&\mbox{($\delta=12$)}\\\
380~{}{\rm
G}~{}\varepsilon_{B,-2}^{1/2}D_{*}^{1/2}t_{5.5}^{-1}.&\mbox{($\delta=10$)}\end{array}\right..$
(20)

CR acceleration is limited by the age or particle escape if energy losses are
irrelevant. In the escape-limited case Ohira  _et al._ (2010), for a CR proton
with energy $\varepsilon_{p}$, the maximum energy is

$\displaystyle\varepsilon_{p}^{\rm max-
esc}\simeq\left\\{\begin{array}[]{ll}9.7\times{10}^{6}~{}{\rm GeV}~{}{(l_{\rm
esc}/R_{s})}\varepsilon_{B,-2}^{1/2}D_{*,-2}^{3/10}\\\
\times{\mathcal{E}}_{\rm ej,51}^{9/10}M_{\rm
ej}^{-7/10}{t}_{5.5}^{-1/5}\,\,\,\,\,\,\,\,\,\mbox{(for $\delta=12$)}\\\
4.4\times{10}^{7}~{}{\rm GeV}~{}{(l_{\rm
esc}/R_{s})}\varepsilon_{B,-2}^{1/2}D_{*}^{1/4}\\\ \times{\mathcal{E}}_{\rm
ej,51}^{7/8}M_{\rm
ej}^{-5/8}{t}_{5.5}^{-1/4}\,\,\,\,\,\,\,\,\,\,\,\,\mbox{(for
$\delta=10$)}\end{array}\right.,\,\,\,$ (25)

where $l_{\rm esc}$ is the upstream escaping boundary that can be determined
by plasma processes including the magnetic field amplification by
(nonresonant) CR streaming instabilities Inoue  _et al._ (2021); Cristofari
_et al._ (2022); Diesing (2023) and neutral-ion damping Murase  _et al._
(2011). In this work, for simplicity, we assume that the escape boundary is
comparable to the system size, which is sufficient for the purpose of this
work because electromagnetic cascades make the results on photon spectra
insensitive to the CR maximum energy. In general, other energy losses such as
the photomeson production and the Bethe-Heitler pair production process
($p\gamma\rightarrow e^{-}e^{+}$) can play roles. In our setup, the inelastic
$pp$ interaction is the most relevant cooling process and we obtain

$\displaystyle\varepsilon_{p}^{{\rm
max}-{pp}}\simeq\left\\{\begin{array}[]{ll}1.2\times{10}^{7}~{}{\rm
GeV}~{}\varepsilon_{B,-2}^{1/2}D_{*,-2}^{-9/10}\\\ \times{\mathcal{E}}_{\rm
ej,51}^{9/5}M_{\rm ej}^{-7/5}{t}_{5.5}^{3/5}\,\,\,\mbox{(for $\delta=12$)}\\\
2.5\times{10}^{5}~{}{\rm GeV}~{}\varepsilon_{B,-2}^{1/2}D_{*}^{-1}\\\
\times{\mathcal{E}}_{\rm ej,51}^{7/4}M_{\rm
ej}^{-5/4}{t}_{5.5}^{1/2}\,\,\,\mbox{(for $\delta=10$)}\end{array}\right..$
(30)

For CR injections, we assume a CR spectrum to be a power-law, i.e.,

$\frac{dn_{\rm cr}}{dp_{\rm cr}}\propto p_{\rm cr}^{-s_{\rm cr}}e^{-p_{\rm
cr}/p_{\rm cr}^{\rm max}},$ (31)

where $s_{\rm cr}$ is the CR spectral index and $p_{\rm cr}$ is the CR
momentum. We assume proton CRs, and the spectrum is normalized via the CR
energy density $U_{\rm cr}$ as

$U_{\rm cr}=\int dp_{p}\varepsilon_{p}\frac{dn_{\rm
cr}}{dp_{p}}=\epsilon_{p}\frac{1}{2}\varrho_{\rm cs}V_{s}^{2},$ (32)

where $\epsilon_{\rm cr}$ is the energy fraction carried by CRs and
$\epsilon_{\rm cr}\sim 0.1$ is consistent with observations of Galactic CRs
Murase and Fukugita (2019) and fast nova shocks from RS Ophiuchi De Wolf  _et
al._ (2022). Although $\epsilon_{\rm cr}$ may be as low as $\epsilon_{\rm
cr}\sim 0.01$ for radiative shocks, as inferred from observations of classical
novae Li  _et al._ (2017) and constraints from SN 2010jl Murase  _et al._
(2011), the shock is adiabatic for modest values of $D_{*}$ including the
cases without CCSM and the CR acceleration could also be modified by the
preacceleration of CRs at the CSM eruption Murase  _et al._ (2014). We use
$\epsilon_{\rm cr}=0.1$, which is sufficient for the purpose of this work.
Although we assume $s_{\rm cr}=2.0-2.2$, we present the results for $s_{\rm
cr}=2.2$ throughout this work because Galactic CR data prefer $s_{\rm cr}\sim
2.2-2.4$ rather than $s_{\rm cr}=2.0$ Murase and Fukugita (2019). However, we
note that our results are largely insensitive to $s_{\rm cr}$ thanks to
electromagnetic cascades that make photon energy spectra “flat”. See also M18
for the impacts on the detectability of high-energy neutrinos.

CR ions interact with cold nucleons in the CSM and lead to the $pp$ production
of mesons (mostly pions), which generates a flux of high-energy neutrinos via
decay processes like
$\pi^{+}\to\mu^{+}\nu_{\mu}\to\nu_{\mu}\bar{\nu}_{\mu}\nu_{e}e^{+}$ and gamma
rays via $\pi^{0}\rightarrow 2\gamma$. The typical neutrino energy is
$\varepsilon_{\nu}\sim(0.03-0.05)\varepsilon_{p}$ Kelner  _et al._ (2006). The
approximate cross section and proton inelasticity of $pp$ collisions are
$\sigma_{pp}\approx 3\times{10}^{-26}~{}{\rm cm}^{2}$ and $\kappa_{pp}\approx
0.5$, respectively. Using Eqs. (4) and (5), the effective optical depth to
inelastic $pp$ interactions, $f_{pp}\approx\kappa_{pp}\sigma_{pp}(\varrho_{\rm
cs}/m_{H})R_{s}(c/V_{s})$, is estimated to be Murase (2018)

$\displaystyle
f_{pp}\simeq\left\\{\begin{array}[]{ll}0.82~{}D_{*,-2}^{6/5}M_{\rm
ej,1}^{7/10}{\mathcal{E}}_{\rm
ej,51}^{-9/10}t_{5.5}^{-4/5}.&\mbox{($\delta=12$)}\\\ 170~{}D_{*}^{5/4}M_{\rm
ej,1}^{5/8}{\mathcal{E}}_{\rm
ej,51}^{-7/8}t_{5.5}^{-3/4}.&\mbox{($\delta=10$)}\end{array}\right.$ (35)

This gives the transition time, at which the system becomes effectively
optically thin for confined CR ions,

$\displaystyle t_{f_{pp}=1}\simeq\left\\{\begin{array}[]{ll}2.4\times
10^{5}~{}{\rm s}~{}D_{*,-2}^{3/2}M_{\rm ej,1}^{7/8}{\mathcal{E}}_{\rm
ej,51}^{-9/8}&\mbox{($\delta=12$)}\\\ 3.1\times 10^{8}~{}{\rm
s}~{}D_{*}^{5/3}M_{\rm ej,1}^{5/6}{\mathcal{E}}_{\rm
ej,51}^{-7/6}&\mbox{($\delta=10$)}.\end{array}\right.$ (38)

One sees that the dense CSM allows us to naturally expect that early SNe have
the bright phase in high-energy neutrinos and gamma rays.

###  II.3 Thermal emission

Figure 1: Energy fluxes of thermal optical (thick) and x-ray (thin) emission
used in this work. For thermal bremsstrahlung emission, spectra without/with
external matter attenuation are shown with upper/lower curves. A Galactic SN
at $d=10$ kpc is considered an $D_{*}=0.01$ is used for a SN II (CCSM).

AMES focuses on the calculations of nonthermal emission but thermal emission
has to be modeled consistently especially for the studies of SNe. This is
because thermal photons typically overwhelm nonthermal photons in the optical
and x-ray bands, and they provide target photons for high-energy gamma rays
through $\gamma\gamma\rightarrow e^{+}e^{-}$. We approximately implement time-
dependent thermal spectra in the following physical manner.

The kinetic energy of SN ejecta is dissipated through shocks. The forward
shock is more important for our ejecta profiles Chevalier and Fransson (2017)
and CSM parameters ($M_{\rm cs}<M_{\rm ej}$) considered in this work (but see,
e.g., Ref. Murase  _et al._ (2011) for reverse shock contributions). Then, the
post-shock temperature ${\mathcal{T}}^{\prime}_{\rm cs}$ is

$\displaystyle k{\mathcal{T}}^{\prime}_{\rm cs}$ $\displaystyle=$
$\displaystyle\frac{3}{16}m_{H}\mu V_{s}^{2}$ (39) $\displaystyle\simeq$
$\displaystyle\left\\{\begin{array}[]{ll}57~{}{\rm
keV}~{}D_{*,-2}^{-1/5}M_{\rm ej,1}^{-7/10}{\mathcal{E}}_{\rm
ej,51}^{9/10}t_{5.5}^{-1/5}\\\ \mbox{(for $\delta=12$)}\\\ 26~{}{\rm
keV}~{}D_{*}^{-1/4}M_{\rm ej,1}^{-5/8}{\mathcal{E}}_{\rm
ej,51}^{7/8}t_{5.5}^{-1/4}\\\ \mbox{(for $\delta=10$)}\end{array}\right.,$
(44)

which is typically in the hard x-ray range, and $\mu$ is the mean molecular
weight and $m_{H}$ is the hydrogen mass. Note that the immediate downstream
temperature can be reduced if the Compton cooling is faster than plasma
heating processes such as the Coulomb heating from ions to electrons Murase
_et al._ (2011); Katz  _et al._ (2012); Chevalier and Irwin (2012). The
thermal bremsstrahlung luminosity is

$\displaystyle L_{\rm bre}$ $\displaystyle=$ $\displaystyle 4\pi R^{2}\Delta
R^{\prime}(\Lambda_{\rm ff}n^{\prime}_{e}n^{\prime}_{H})$ (45)
$\displaystyle\simeq$ $\displaystyle 2.9\times{10}^{41}~{}{\rm erg}~{}{\rm
s}^{-1}~{}\mu_{H}^{-1}\mu_{e}^{-1}D_{*,-2}^{2}t_{5.5}^{-1},$

where $\Lambda_{\rm ff}$ is the cooling function. For moderate values of
$D_{*}\lesssim 0.03$, including SN 2013fs-like cases, the shock is expected to
be adiabatic. If $L_{\rm bre}>L_{s}$, where the shock would be radiative, we
limit the bremsstrahlung luminosity by $L_{s}$.

In dense environments, photons are reprocessed both in the downstream and
upstream, and a significant energy fraction of the radiation should be
released as SN emission in the optical band. In this work, following M18, we
implement a gray body spectrum with thermal luminosity $L_{\rm
sn}=\epsilon_{\rm rad}L_{s}$ for a component originating from the interaction
with CSM. The photon temperature is set by ${\mathcal{T}}_{\rm sn}={\rm
max}[{\mathcal{T}}_{\rm bb},{\mathcal{T}}_{\rm rec}]$, where
${\mathcal{T}}_{\rm bb}$ is the black body temperature and ${\mathcal{T}}_{\rm
rec}={10}^{4}$ K, and $\epsilon_{\rm rad}=1/4$ is used. In addition, there can
be an ordinary SN component from the photosphere, which can be energized by
radioactive nuclei, shocks propagating in the progenitor and resulting cooling
envelope emission. We also add this external component with luminosity $L_{\rm
ph}$ to discuss the impacts. As light-curve templates, optical luminosities of
SN 1999bm for Type II-P Bersten and Hamuy (2009) and SN 2004aw for Type Ibc
Mazzali  _et al._ (2017) are considered in our baseline calculations.

###  II.4 Numerical Method

For given dynamics and CR acceleration, AMES allows us to numerically
calculate neutrino and gamma-ray spectra through solving kinetic equations in
a time-dependent manner as in M18. For SN dynamics, we employ the equation of
motion of the shocked shell for parameters listed in Table 1, and we evaluate
$\varepsilon_{p}^{\rm max}$. The SN module of AMES can be used for arbitrary
$R_{s}$, $V_{s}$, $\varrho_{\rm cs}$, $L_{\rm sn}$, CR distributions, and the
magnetic field as a function of time. Although AMES includes various processes
for particle production, $pp$ interactions, $p\gamma$ interactions, nuclear
photodisintegration, and Bethe-Heitler pair production, in this work it is
sufficient to consider $pp$ interactions. See Ref. Zhang and Murase (2023) for
details on the other processes. Compared to previous works, in AMES we have
updated treatments on $pp$ interactions, Coulomb losses, bremsstrahlung, and
free-free absorption.

Distributions of gamma rays and electrons/positrons are obtained by solving
the following partial differential equations (see Supplementary Material of
M18 for details) via the implicit method,

$\displaystyle\dot{n}_{\varepsilon_{e}}^{e}$ $\displaystyle=$
$\displaystyle\dot{n}_{\varepsilon_{e}}^{(\gamma\gamma)}-\frac{\partial}{\partial\varepsilon_{e}}[(P_{\rm
IC}+P_{\rm syn}+P_{\rm ad}+P_{\rm bre}+P_{\rm Cou})n_{\varepsilon_{e}}^{e}]$
$\displaystyle+\dot{n}_{\varepsilon_{e}}^{\rm inj}$
$\displaystyle\dot{n}_{\varepsilon_{\gamma}}^{\gamma}$ $\displaystyle=$
$\displaystyle-\frac{n_{\varepsilon_{\gamma}}^{\gamma}}{t_{\gamma\gamma}}-\frac{n_{\varepsilon_{\gamma}}^{\gamma}}{t_{\rm
esc}}-\frac{n_{\varepsilon_{\gamma}}^{\gamma}}{t_{\rm
matter}}+\dot{n}_{\varepsilon_{\gamma}}^{(\rm
IC)}+\dot{n}_{\varepsilon_{\gamma}}^{(\rm
syn)}+\dot{n}_{\varepsilon_{\gamma}}^{(\rm bre)}$
$\displaystyle+\dot{n}_{\varepsilon_{\gamma}}^{\rm inj}$

where $n_{\varepsilon_{i}}^{i}\equiv dn^{i}/d\varepsilon_{i}$ and
$\varepsilon_{i}$ is particle energy for a particle species with $i$. Energy
loss rates of electrons/positrons for the IC radiation ($P_{\rm IC}$),
synchrotron radiation ($P_{\rm syn}$), adiabatic cooling ($P_{\rm ad}$),
relativistic bremsstrahlung ($P_{\rm bre}$), and Coulomb collisions ($P_{\rm
Cou}$), respectively, are Blumenthal and Gould (1970); Schlickeiser (Springer,
Berlin, Germany 2002)

$\displaystyle P_{\rm IC}$ $\displaystyle=$ $\displaystyle\int
d\varepsilon_{\gamma}\,\,\varepsilon_{\gamma}\int
d\varepsilon_{t}\,\,n_{\varepsilon_{t}}^{\gamma}\int\frac{d\cos\theta}{2}\,\,\tilde{c}\frac{d\sigma_{\rm
IC}}{d\varepsilon_{\gamma}},$ $\displaystyle P_{\rm syn}$ $\displaystyle=$
$\displaystyle\frac{1}{6\pi}\sigma_{T}c\frac{p_{e}^{2}}{m_{e}^{2}c^{2}}B^{2},$
$\displaystyle P_{\rm bre}$ $\displaystyle=$
$\displaystyle\left(\sum_{i}\frac{Z_{i}^{2}}{A_{i}}x_{i}+\frac{1}{\mu_{e}}\right)\frac{3\alpha_{\rm
em}}{2\pi}\sigma_{T}cn^{\prime}_{N}\left(\ln[2\gamma_{e}]-\frac{1}{3}\right)\varepsilon_{e},$
$\displaystyle P_{\rm Cou}$ $\displaystyle=$
$\displaystyle\frac{3}{4}\sigma_{T}m_{e}c^{3}n^{\prime}_{N}\mu_{e}^{-1}\left(74.3+\ln[\gamma_{e}\mu_{e}/n^{\prime}_{N}]\right),$
(47)

where $\tilde{c}=(1-\cos\theta)c$ (where $\theta$ is the angle between two
particles), $d\sigma_{\rm IC}/d\varepsilon_{\gamma}$ is the differential IC
cross section for target photon energy $\varepsilon_{t}$ Blumenthal and Gould
(1970), $Z_{i}$ is the nuclear charge, $A_{i}$ is the nuclear mass number,
$x_{i}$ is the nuclear mass fraction, and $\mu_{e}^{-1}$ is the number of
electrons per baryon. The weak shielding limit is assumed for bremsstrahlung
and collisions with electrons in the shell are considered, and contributions
from both nuclei and electrons are included Schlickeiser (Springer, Berlin,
Germany 2002). High-energy gamma rays interact with other photons via
$\gamma\gamma\rightarrow e^{+}e^{-}$, and we take into account the gamma-ray
attenuation and subsequent regeneration. The two-photon annihilation rate and
cross section are

$\displaystyle t_{\gamma\gamma}^{-1}$ $\displaystyle=$ $\displaystyle\int
d\varepsilon_{t}\,\,n_{\varepsilon_{t}}^{\gamma}\int\frac{d\cos\theta}{2}\,\,\tilde{c}\sigma_{\gamma\gamma},$
$\displaystyle\sigma_{\gamma\gamma}$ $\displaystyle=$
$\displaystyle\frac{3}{16}\sigma_{T}(1-\beta_{\rm cm}^{2})\left[2\beta_{\rm
cm}(\beta_{\rm cm}^{2}-2)\right.$ (48) $\displaystyle+$
$\displaystyle\left.(3-\beta_{\rm cm}^{4})\ln[(1+\beta_{\rm cm})/(1-\beta_{\rm
cm})]\right],$

respectively, where $\beta_{\rm cm}=\sqrt{(1-4m_{e}^{2}c^{4}/S)}$, $S$ is the
Mandelstam variable, and $t_{\rm esc}=R_{s}/c$ is the photon escape time.
Differential particle generation rate densities are

$\displaystyle\dot{n}_{\varepsilon_{e}}^{(\gamma\gamma)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int
d\varepsilon_{\gamma}\,\,n_{\varepsilon_{\gamma}}^{\gamma}\,\int
d\varepsilon_{t}\,\,n_{\varepsilon_{t}}^{\gamma}\,\int\frac{d\cos\theta}{2}\,\,\tilde{c}\frac{d\sigma_{\gamma\gamma}}{d\varepsilon_{e}},$
$\displaystyle\dot{n}_{\varepsilon_{\gamma}}^{(\rm IC)}$ $\displaystyle=$
$\displaystyle\int d\varepsilon_{e}\,\,n_{\varepsilon_{e}}^{e}\,\int
d\varepsilon_{t}\,\,n_{\varepsilon_{t}}^{\gamma}\,\int\frac{d\cos\theta}{2}\,\,\tilde{c}\frac{d\sigma_{\rm
IC}}{d\varepsilon_{\gamma}},$
$\displaystyle\dot{n}_{\varepsilon_{\gamma}}^{(\rm syn)}$ $\displaystyle=$
$\displaystyle\int
d\varepsilon_{e}\,\,n_{\varepsilon_{e}}^{e}\,\frac{1}{\varepsilon_{\gamma}}\frac{dP_{\rm
syn}}{d\varepsilon_{\gamma}},$
$\displaystyle\dot{n}_{\varepsilon_{\gamma}}^{(\rm bre)}$ $\displaystyle=$
$\displaystyle\int
d\varepsilon_{e}\,\,n_{\varepsilon_{e}}^{e}\,\frac{1}{\varepsilon_{\gamma}}\frac{dP_{\rm
bre}}{d\varepsilon_{\gamma}},$ (49)

where
$d\sigma_{\gamma\gamma}/d\varepsilon_{e}=2\sigma_{\gamma\gamma}\delta(\varepsilon_{e}-\varepsilon_{\gamma}/2)$
is the simplified differential two-photon annihilation cross section with
$\varepsilon_{e}=\varepsilon_{\gamma}/2$ is used as in Ref. Murase  _et al._
(2015), and $dP_{\rm syn}/d\varepsilon_{\gamma}$ and $dP_{\rm
bre}/d\varepsilon_{\gamma}$ are synchrotron and bremsstrahlung (Blumenthal and
Gould, 1970; Zhang  _et al._ , 2021) powers per photon energy, respectively.

Photons lose their energies due to interactions with matter during their
escape from the shocked shell, and $t_{\rm matter}^{-1}=(\kappa_{\rm
BH}\sigma_{\rm BH}+\kappa_{\rm Comp}\sigma_{\rm Comp})n_{N}c$ is the energy
loss time and $n_{N}$ is the average nucleon density. The attenuation cross
section of the Bethe-Heitler pair production process can be approximated by

$\kappa_{\rm BH}\sigma_{\rm BH}=\frac{x-2}{x}\sigma_{\rm BH},$ (50)

where $x\equiv\varepsilon_{\gamma}/(m_{e}c^{2})$. Ignoring the contribution
from electron-positron annihilation, we use inelasiticity $\kappa_{\rm BH}$
and cross section $\sigma_{\rm BH}$ obtained with the Born approximation
Chodorowski  _et al._ (1992). The attenuation cross section of the Compton
scattering process is Murase  _et al._ (2015)

$\displaystyle\kappa_{\rm Comp}\sigma_{\rm Comp}$ $\displaystyle=$
$\displaystyle\frac{3}{4}\sigma_{T}\left[\frac{2{(1+x)}^{2}}{x^{2}(1+2x)}-\frac{1+3x}{{(1+2x)}^{2}}\right.$
(51) $\displaystyle+$
$\displaystyle\frac{(1+x)(2x^{2}-2x-1)}{x^{2}{(1+2x)}^{2}}+\frac{4x^{2}}{3{(1+2x)}^{3}}$
$\displaystyle+$
$\displaystyle\left.\left(\frac{1+x}{x^{3}}-\frac{1}{2x}+\frac{1}{2x^{3}}\right)\ln(1+2x)\right],$

where the Klein-Nishina effect is fully taken into account.

The differential injection rate densities via $pp$ interactions are calculated
by

$\displaystyle\dot{n}_{\varepsilon_{\nu}}^{\rm
inj}=\frac{d\sigma_{pp}\xi_{\nu}}{d\varepsilon_{\nu}}\frac{cM_{\rm
cs}(<R_{s})}{m_{H}\mathcal{V}}\int dp_{\rm cr}\,\,\,\frac{dn_{{\rm
cr}}}{dp_{\rm cr}}$ $\displaystyle\dot{n}_{\varepsilon_{\gamma}}^{\rm
inj}=\frac{d\sigma_{pp}\xi_{\gamma}}{d\varepsilon_{\gamma}}\frac{cM_{\rm
cs}(<R_{s})}{m_{H}\mathcal{V}}\int dp_{\rm cr}\,\,\,\frac{dn_{{\rm
cr}}}{dp_{\rm cr}}$ $\displaystyle\dot{n}_{\varepsilon_{e}}^{\rm
inj}=\frac{d\sigma_{pp}\xi_{e}}{d\varepsilon_{e}}\frac{cM_{\rm
cs}(<R_{s})}{m_{H}\mathcal{V}}\int dp_{\rm cr}\,\,\,\frac{dn_{{\rm
cr}}}{dp_{\rm cr}},$ (52)

where $\xi_{\nu}$, $\xi_{\gamma}$ and $\xi_{e}$ are multiplicities of
neutrinos, gamma rays and electrons/positrons, respectively, $\mathcal{V}$ is
the volume, and $M_{\rm cs}(<R_{s})$ is the CSM mass swept up by the forward
shock. In AMES, treatments on low-energy neutrinos and gamma rays are improved
by interpolating simulation results of Geant4 Agostinelli  _et al._ (2003)
with those from differential spectra of secondary particles following the
parametrization by Ref. Kelner  _et al._ (2006) and the $pp$ cross section by
Ref. Kafexhiu  _et al._ (2014). The resulting secondary spectra are consistent
with the previous works Kamae  _et al._ (2006); Kafexhiu  _et al._ (2014).

Intrinsic energy fluxes of neutrino and gamma rays reaching Earth are given by

$\displaystyle EF_{E}=\frac{\varepsilon L_{\varepsilon}}{4\pi
d_{L}^{2}}=\frac{(\varepsilon^{2}n_{\varepsilon})\mathcal{V}}{t_{\rm esc}},$
(53)

where $E=\varepsilon/(1+z)$ is particle energy on Earth, $d_{L}$ is the
luminosity distance. For neutrinos, the flavor mixing is additionally taken
into account. For photons, we implement the synchrotron self-absorption (SSA)
process by multiplying $\tau_{\rm sa}^{-1}(1-e^{-\tau_{\rm sa}})$, where

$\tau_{\rm sa}=R_{s}\int d\gamma_{e}\,n_{\gamma_{e}}^{e}\sigma_{\rm sa},$ (54)

is the SSA optical depth in the emission region, and the SSA cross section is
(e.g., Chiaberge and Ghisellini, 1999; Ghisellini and Svensson, 1991).

$\sigma_{\rm sa}=\frac{1}{8\pi
m_{e}\nu^{2}\gamma_{e}p_{e}}\frac{\partial}{\partial\gamma_{e}}\left[\gamma_{e}p_{e}\frac{dP_{\rm
syn}}{d\nu}\right],$ (55)

where $\nu=\varepsilon_{\gamma}/h$ is frequency.

Photons leaving the source are further reprocessed in the external screen
region, which corresponds to unshocked CSM in our setup. We treat such
external attenuation by the suppression factor, $f_{\rm sup}^{\gamma\gamma-\rm
ex}=e^{-\tau_{\gamma\gamma}}$. This is reasonable when photons are
significantly reprocessed to optical emission in the CSM. For the photons
coming from the photosphere this treatment is more approximate. For matter
attenuation of photons, as in Ref. Murase  _et al._ (2019), we implement

$f_{\rm sup}^{\rm mat-ex}={\rm max}[e^{-\tau_{\rm mat}},f_{\rm esc}].$ (56)

where $\tau_{\rm mat}$ consists of all processes, Bethe-Heitler pair
production, Compton, photoelectric absorption and free-free absorption. We
also introduce $f_{\rm esc}$ as the effective escape fraction, which can be
significant when the CSM is clumpy or aspherical.

At sufficiently high energies above the electron-positron pair production
threshold, the Bethe-Heitler pair production process is dominant. The Bethe-
Heitler cross section on a nucleus scales as $\sigma_{\rm BH}=Z^{2}\sigma_{\rm
BH}^{(p)}$, where $\sigma_{\rm BH}^{(p)}$ is the cross section on a proton.
Taking into account contributions from both nuclei and electrons, we have

$\displaystyle\kappa_{\rm BH}\tau_{\rm BH}$ $\displaystyle=$
$\displaystyle\left(\sum_{i}\frac{Z_{i}^{2}}{A_{i}}x_{i}+\frac{1}{\mu_{e}}\right)\frac{D\kappa_{\rm
BH}\sigma_{\rm BH}^{(p)}}{m_{H}R_{s}}$ (57) $\displaystyle=$
$\displaystyle(\tilde{Z}+1)\kappa_{\rm BH}\sigma_{\rm BH}^{(p)}n_{e}R_{s},$

where $\tilde{Z}\equiv\Sigma_{i}(Z_{i}^{2}/A_{i})x_{i}\mu_{e}$, which depends
on chemical composition of the ejecta. We consider $x_{\rm H}=0.7$ and $x_{\rm
He}=0.25$ and $x_{\rm CNO}=0.05$, leading to $\tilde{Z}\simeq 0.97$ and
$\mu_{e}\simeq 0.85$.

Energy losses of x-ray and gamma-ray photons via Compton scattering is
included as effective attenuation considering the inelasticity. The effective
optical depth is given by

$\kappa_{\rm Comp}\tau_{\rm Comp}=\frac{D\kappa_{\rm Comp}\sigma_{\rm
Comp}}{\mu_{e}m_{H}R_{s}},$ (58)

where $\kappa_{\rm Comp}\sigma_{\rm Comp}$. In addition, in the x-ray band, we
implement photoelectric absorption with its optical depth, $\tau_{\rm
pe}=K_{\rm pe}\varrho_{\rm cs}R_{s}$, where we use $K_{\rm pe}=0.024~{}{\rm
g}^{-1}~{}{\rm cm}^{2}~{}{(\varepsilon/1~{}\rm keV)}^{-3}$ above 10.2 keV for
simplicity. Nonthermal emission can overwhelm thermal emission at sufficient
high energies, where the photoelectric absorption is not much important.

Free-free absorption can be crucial at the radio band. The free-free optical
depth is (Mezger and Henderson, 1967)

$\displaystyle\tau_{\rm ff}=\int_{R_{s}}dR\,\alpha_{\rm ff}$
$\displaystyle\approx$
$\displaystyle\frac{1}{(2w-1)}\sum_{i}8.4\times{10}^{-28}~{}\frac{\hat{Z}_{i}^{2}}{A_{i}}x_{i}$
(59) $\displaystyle\times$ $\displaystyle{\mathcal{T}}_{\rm
cs,4}^{-1.35}\nu_{10}^{-2.1}n_{e}n_{N}R_{s},\,\,\,\,\,\,\,\,\,\,\,\,$

where ${\mathcal{T}}_{\rm cs}$ is the upstream CSM temperature and $\hat{Z}$
is the effective charge considering the ionization. Radio waves may be
absorbed by cold plasma in the upstream and we also include the Razin effect
in an approximately manner with an exponential cutoff, where the Razin-
Tsytovich frequency is given by $\nu_{\rm RT}\equiv 2ecn_{e}/B$.

Figure 2: Left panel: Light curves of high-energy neutrinos (at
$\varepsilon_{\nu}=1$ TeV) for various types of SNe. Right panel: Energy
fluences of
$\nu_{e}+\bar{\nu}_{e}+\nu_{\mu}+\bar{\nu}_{\mu}+\nu_{\tau}+\bar{\nu}_{\tau}$
(thick) and generated gamma rays (thin) integrated over $t_{\nu\rm max}$. A
Galactic SN at $d=10$ kpc is considered and $D_{*}=0.01$ is used for a SN II
(CCSM). In both panels, $s_{\rm cr}=2.2$ is assumed.

##  III Light curves and spectra

We calculate multimessenger spectra and light curves that are obtained by
numerical calculations with AMES. We also present analytical expressions of
multimessenger spectra to understand the physical situation, assuming
$\delta=12$. Throughout this work, the total SN kinetic energy (i.e.,
integrated over velocities) and SN ejecta mass are set to ${\mathcal{E}}_{\rm
ej}=10^{51}~{}{\rm erg}~{}{\mathcal{E}}_{\rm ej,51}$ and $M_{\rm
ej}=10~{}M_{\odot}~{}{M}_{1}$, respectively.

###  III.1 Neutrinos

The differential neutrino luminosity (for the sum of all flavors) above $\sim
1$ GeV is approximated to be

$\displaystyle\varepsilon_{\nu}L_{\varepsilon_{\nu}}$ $\displaystyle\approx$
$\displaystyle\frac{1}{2}{\rm min}[f_{pp},1]\frac{\epsilon_{\rm
cr}L_{s}}{{\mathcal{R}}_{\rm cr10}}{\left(\frac{\varepsilon_{\nu}}{0.4~{}{\rm
GeV}}\right)}^{2-s_{\nu}}$ (60) $\displaystyle\simeq$ $\displaystyle
5.0\times{10}^{39}~{}{\rm erg}~{}{\rm s}^{-1}~{}{\rm min}[f_{pp},1]f_{\Omega}$
$\displaystyle\times$ $\displaystyle\epsilon_{\rm cr,-1}{\mathcal{R}}_{\rm
cr10,1}^{-1}{\left(\frac{\varepsilon_{\nu}}{0.4~{}{\rm
GeV}}\right)}^{2-s_{\nu}}$ $\displaystyle\times$ $\displaystyle
D_{*,-2}^{7/10}{\mathcal{E}}_{\rm ej,51}^{27/20}M_{\rm
ej,1}^{-21/20}t_{5.5}^{-3/10},$

where the factor $1/2$ comes from the facts that the $\pi^{\pm}/\pi^{0}$ ratio
is $\approx 2$ in $pp$ interactions and neutrinos carry $3/4$ of the pion
energy in the decay chain. We also introduce ${\mathcal{R}}_{\rm
cr10}\equiv\epsilon_{\rm cr}L_{s}/(\varepsilon_{p}^{2}d\dot{N}_{\rm
cr}/d\varepsilon_{p})|_{10~{}{\rm GeV}}$ that is a spectrum dependent factor
that converts the bolometric luminosity to the differential luminosity, and
$s\approx s_{\rm cr}$ for $f_{pp}\gtrsim 1$ and $s\approx s_{\rm cr}-0.1$ for
$f_{pp}\lesssim 1$. In the latter case, the secondary spectra are somewhat
harder than the CR spectrum because of the weak energy dependence of $pp$
interactions.

Given that $pp$ interactions are dominant, the neutrino flux would obey the
following scaling,

$E_{\nu}F_{E_{\nu}}\propto\left\\{\begin{array}[]{lr}\varepsilon_{\nu}^{2-s_{\rm
cr}}t^{-0.3}&\mbox{($t\leq t_{f_{pp}=1}$)}\\\ \varepsilon_{\nu}^{1.9-s_{\rm
cr}}t^{-1.1}&\mbox{($t>t_{f_{pp}=1}$)}\end{array}\right.,$ (61)

where $f_{pp}$ is given by Eq. (35). For neutrino detection, the time
evolution of the energy fluence is important as long as the atmospheric
background is negligible Murase (2018). Eq. (61) suggests that the neutrino
fluence has a peak around $t_{f_{pp}=1}$, which gives a characteristic time
scale of high-energy neutrino emission.

In Fig. 2 left, we show neutrino light curves at $\varepsilon_{\nu}=1$ TeV.
Thick curves represent CCSN-interacting SNe, while thin curves do SNe II-P
without CCSM. We use $R_{\rm cs}=4\times{10}^{14}$ cm for $D_{*}=0.01$ and
$D_{*}=0.1$ as in M18, while we consider $R_{\rm cs}={10}^{15}$ cm and
$D_{*}=1$ to cover the most optimistic cases (see also Ref. Kheirandish and
Murase (2023)). Our model predicts that the neutrino luminosity typically
ranges from $L_{\nu}\sim 10^{39}-{10}^{40}~{}{\rm erg}~{}{\rm s}^{-1}$ with
durations of $\sim 1-30$ d. With CCSM, the system is nearly calorimetric so
that the neutrino luminosity scales as $D^{7/10}$. The temporal slope changes
at the time when $f_{pp}$ becomes less than unity, which is consistent with
Eq. (61). The onset time is significantly longer for $D_{*}=1$ because the
shock propagating in the CSM is radiation mediated at early times. For Type
IIn SNe, characteristic time windows are $\sim 0.1-1$ yr after the explosion
Murase  _et al._ (2011). Without CCSM, $t_{\rm onset}$ almost coincides with
shock breakout from the progenitor. We note that as shown in M18 high-energy
neutrinos from Galactic SNe are detectable for IceCube, KM3Net Adrian-Martinez
_et al._ (2016), Baikal-GVD Avrorin  _et al._ (2018), P-ONE Agostini  _et al._
(2020), TRIDENT Ye  _et al._ (2022) and especially IceCube-Gen2 Aartsen  _et
al._ (2021) even without CCSM.

In Fig. 2 right, we show energy fluences of neutrinos and generated gamma rays
at $t_{\nu{\rm max}}$, which is the time when the significance of neutrino
emission becomes the maximum. One of the ultimate goals is to reveal CR ion
acceleration through multimessenger observations, for which simultaneous
observations between neutrinos and photons are critical. We show energy fluxes
at $t_{\nu{\rm max}}$ as default, and the values are shown in Table 1. High-
energy SN neutrino emission is long lasting with optimized time windows from
days to months for SNe II, which can be regarded as high-energy neutrino
afterglows in contrast to SN neutrino bursts in the MeV range.

###  III.2 Pionic Gamma Rays

Figure 3: Light curves of high-energy gamma rays at $\varepsilon_{\gamma}=1$
GeV (left panel), MeV gamma rays at $\varepsilon_{\gamma}=1$ MeV (middle
panel), and radio waves at $\nu=100$ GHz (right panel) for various types of
SNe. Note that radio emission without external matter attenuation is shown. In
all three panels, $s_{\rm cr}=2.2$ is assumed.  Figure 4: Energy fluxes of
nonthermal electromagnetic emission from a SN at $d=10$ kpc, corresponding to
Fig. 2. Spectra with/without external matter attenuation (thick/thin) are
shown. For a SN II (CCSM), $D_{*}=0.01$ is used.

Figure 5: Time scales relevant for electromagnetic cascades for CCSM-
interacting SNe II without $L_{\rm ph}$ (left panel), CCSM-interacting SNe II
with $L_{\rm ph}$ (middle panel) and SNe IIn (right panel). Considered energy
loss processes for electrons and positrons are synchrotron radiation (syn),
inverse-Compton scattering (IC), bremmstrahlung emission (brems), Coulomb
scattering (Cou) and adiabatic losses whose time scale is comparable to the
dynamical time (dyn). For photons, the interaction time of two-photon
annihilation ($\gamma\gamma$) and the light crossing time (lc) are shown.

Energy fluxes of generated gamma rays and neutrinos are related by the
multimessenger connection. The differential luminosities of neutrinos and
pionic gamma rays (from $\pi^{0}\rightarrow 2\gamma$) are approximately
related as Murase  _et al._ (2013)

$\varepsilon_{\gamma}L_{\varepsilon_{\gamma}}^{\pi^{0}}\approx\frac{2}{3}{[\varepsilon_{\nu}L_{\varepsilon_{\nu}}]}_{\varepsilon_{\nu}=\varepsilon_{\gamma}/2}f_{\rm
sup},$ (62)

where $f_{\rm sup}=f_{\rm sup}(\varepsilon_{\gamma})$ is the energy-dependent
suppression fraction, which considers both emission and screen regions. Eq.
(62) suggests that the energy fluxes of neutrinos and generated gamma rays are
comparable, which is an unavoidable consequence of inelastic $pp$
interactions. The differential gamma-ray luminosity is

$\displaystyle\varepsilon_{\gamma}L_{\varepsilon_{\gamma}}^{\pi^{0}}$
$\displaystyle\approx$ $\displaystyle\frac{f_{\rm sup}}{3}{\rm
min}[f_{pp},1]\frac{\epsilon_{\rm cr}L_{s}}{{\mathcal{R}}_{\rm
cr10}}{\left(\frac{\varepsilon_{\gamma}}{0.8~{}{\rm GeV}}\right)}^{2-s}$ (63)
$\displaystyle\simeq$ $\displaystyle 3.3\times{10}^{39}~{}{\rm erg}~{}{\rm
s}^{-1}~{}{\rm min}[f_{pp},1]f_{\rm sup}f_{\Omega}$ $\displaystyle\times$
$\displaystyle\epsilon_{\rm cr,-1}{\mathcal{R}}_{\rm
cr10,1}^{-1}{\left(\frac{\varepsilon_{\gamma}}{0.8~{}{\rm
GeV}}\right)}^{2-s_{\gamma}}$ $\displaystyle\times$ $\displaystyle
D_{*,-2}^{7/10}{\mathcal{E}}_{\rm ej,51}^{27/20}M_{\rm
ej,1}^{-21/20}t_{5.5}^{-3/10}.$

which agrees with the numerical curves shown in Fig. 3 left. Photon spectra
resulting from spectra of generated gamma rays (see Fig. 2 right) are shown at
$t_{\nu{\rm max}}$ in Fig. 4, where electromagnetic cascades are included.

The suppression factor for gamma rays mainly consists of two parts, $f_{\rm
sup}^{\rm BH-ex}$ and $f_{\rm sup}^{\gamma\gamma-\rm ex}$. For analytical
estimates on the Bethe-Heitler suppression, one may use a simpler formula

$\sigma_{\rm BH}\approx Z^{2}\frac{3\alpha_{\rm
em}}{8\pi}\sigma_{T}\left[\frac{28}{9}\ln(2x)-\frac{218}{27}\right],$ (64)

where $\alpha_{\rm em}$ is the fine structure constant and we see $\sigma_{\rm
BH}\sim Z^{2}{10}^{-26}~{}{\rm cm}^{2}$ at GeV energies. Then, the Bethe-
Heitler optical depth for gamma rays with $\varepsilon_{\gamma}=1$ GeV is
estimated to be

$\displaystyle\tau_{\rm BH}$ $\displaystyle\approx$ $\displaystyle
0.031~{}[(\tilde{Z}+1)/2]\tau_{T}$ (65) $\displaystyle\simeq$ $\displaystyle
0.025~{}(\tilde{Z}+1)\mu_{e}^{-1}D_{*,-2}^{11/10}{\mathcal{E}}_{\rm
ej,51}^{-9/20}M_{\rm ej,1}^{7/20}t_{5.5}^{-9/10},\,\,\,\,\,\,\,\,\,$

and the transition time for the system to be optically thin to GeV gamma rays
is

$t_{\tau_{\rm BH}=1}\simeq 5.3\times{10}^{3}~{}{\rm
s}~{}{(\tilde{Z}+1)}^{10/9}\mu_{e}^{-10/9}D_{*,-2}^{11/9}{\mathcal{E}}_{\rm
ej,51}^{-1/2}M_{\rm ej,1}^{7/18}.$ (66)

As inferred in Fig. 3 left, we see the rising of GeV light curves until $\sim
t_{\tau_{\rm BH}=1}$ for $D_{*}\gtrsim 0.1$. Without CCSM, as indicated by the
thin curves, the system is optically thin to the Bethe-Heitler pair production
process just after the breakout from a progenitor. For gamma-ray spectra shown
in Fig. 4, the Bethe-Heitler attenuation is no longer important. Even with
CCSM, given that CR acceleration at $\tau_{T}\lesssim c/V_{s}$, the Bethe-
Heitler pair production can be important only at earlier times, and it is
negligible for high-velocity shocks with $V_{s}\gtrsim 10^{4}~{}{\rm
km}~{}{\rm s}^{-1}$.

Note that only sufficiently high-energy gamma rays interact with SN photons,
and the two-photon annihilation becomes important at

$\displaystyle\tilde{\varepsilon}_{\gamma\gamma-{\rm
sn}}\approx\frac{m_{e}^{2}c^{4}}{\varepsilon_{\rm sn}}\simeq 260~{}{\rm
GeV}~{}{(\varepsilon_{\rm sn}/1~{}{\rm eV})}^{-1},$ (67)

where $\varepsilon_{\rm sn}\sim 3k{\mathcal{T}}_{\rm sn}$. The optical depth
to $\gamma\gamma\rightarrow e^{+}e^{-}$ is estimated to be

$\displaystyle\tau_{\gamma\gamma}\approx\frac{3}{16}\sigma_{\gamma\gamma}n_{\gamma}R_{s}$
$\displaystyle\simeq$ $\displaystyle 8200~{}{\rm max}[1,\tau_{T}]L_{\rm
sn,42.5}{(\varepsilon_{\rm sn}/1~{}{\rm eV})}^{-1}$ (68) $\displaystyle\times$
$\displaystyle D_{*,-2}^{1/10}{\mathcal{E}}_{\rm ej,51}^{-9/10}M_{\rm
ej,1}^{7/20}t_{5.5}^{-9/10},\,\,\,\,\,\,\,\,\,\,$

where $n_{\gamma}$ is the photon number density. The interaction time,
$t_{\gamma\gamma}$, is shown in Fig. 5. The ratio of the light crossing time
to the interaction time has a local maximum around
$\tilde{\varepsilon}_{\gamma\gamma-{\rm sn}}$, and the corresponding
break/cutoff energy is expected to be $\sim 10-100$ GeV below
$\tilde{\varepsilon}_{\gamma\gamma-{\rm sn}}$. The transition time for the
system to be optically thin for TeV gamma rays would occur much later around

$\displaystyle t_{\tau_{\gamma\gamma}=1}$ $\displaystyle\simeq$ $\displaystyle
7.0\times{10}^{9}~{}{\rm s}~{}L_{\rm sn,42.5}^{9/10}{(\varepsilon_{\rm
sn}/1~{}{\rm eV})}^{-9/10}$ (69) $\displaystyle\times$ $\displaystyle
D_{*,-2}^{1/9}{\mathcal{E}}_{\rm ej,51}^{-1/2}M_{\rm ej,1}^{7/18},$

which suggests that gamma rays at TeV and higher energies are effectively
absorbed inside and outside the system. This can be clearly seen in gamma-ray
spectra shown in Fig. 4. In Fig. 6 left, we show light curves of 100 GeV gamma
rays (thin curves). For SNe II with CCSM, such very high-energy gamma-ray
emission is suppressed during the main interaction phase and escape after the
shock enters the ordinary RSG wind medium. Light curves for SNe IIn with
different values of $D_{*}$ are also shown in Fig. 6. One sees that the 100
GeV gamma-ray fluxes at early times anticorrelate with $D$, implying that
observations within appropriate time windows are critical to utilize very
high-energy gamma rays as a promising probe of SNe IIn. Note that the two-
photon annihilation with x-ray photons from thermal bremsstrahlung is not
significant in our cases. However, for SNe Ibc, x-ray photons from shock
breakout can provide important target photons for $\gamma\gamma\rightarrow
e^{+}e^{-}$ as well as photomeson production, as shown in Ref. Kashiyama  _et
al._ (2013).

Analytically, the pionic gamma-ray flux at GeV energies obeys the following
scaling,

$E_{\gamma}F_{E_{\gamma}}^{\pi^{0}}\propto e^{-\tau_{\rm
BH}(t)}\left\\{\begin{array}[]{lr}\varepsilon_{\gamma}^{2-s_{\rm
cr}}t^{-0.3}&\mbox{($t<t_{f_{pp}=1}$)}\\\ \varepsilon_{\gamma}^{1.9-s_{\rm
cr}}t^{-1.1}&\mbox{($t>t_{f_{pp}=1}$)}\end{array}\right.,$ (70)

which is similar to Eq. (61) but with matter attenuation at early times (see
Fig. 3 left).

Figure 6: Left panel: Light curves of high-energy gamma rays at
$\varepsilon_{\gamma}=1$ GeV (thick) and $\varepsilon_{\gamma}=100$ GeV
(thin). Middle panel: Light curves of soft gamma rays at
$\varepsilon_{\gamma}=1$ MeV (thick) and hard x rays at
$\varepsilon_{\gamma}=10$ keV (thin). Right panel: Light curves of radio waves
at $\nu=100$ GHz with (thick) and without (thin) external matter attenuation.
In all three panels, we include $L_{\rm ph}$ for CCSM-interacting SNe II, in
which $D_{*}=0.01$ is used.

###  III.3 Inverse-Compton (cascade) emission

Figure 7: Energy fluxes of nonthermal electromagnetic emission from a Galactic
SN II (CCSM) for $D_{*}=0.01$ at $d=10$ kpc. Thick/thin curves represent
spectra with/without external matter attenuation, and $s_{\rm cr}=2.2$ is
used. One sees that synchrotron emission is suppressed in the presence of
ordinary SN emission with $L_{\rm ph}$ from the photosphere.

Secondary electrons and positrons as well as primary electrons lose their
energies via various cooling processes. In our setup, there are three
characteristic injection energies. The hadronic injection occurs through the
decay of charged pions and the two-photon annihilation process. In the former
case, the injection Lorentz factor of electrons and positrons produced via
$pp$ interactions is

$\gamma_{e,h}\approx\frac{m_{\pi}}{4m_{e}}\simeq 68.$ (71)

In the latter case, the injection Lorentz factor of electrons and positrons is

$\gamma_{e,\gamma\gamma}\approx\frac{\tilde{\varepsilon}_{\gamma\gamma-{\rm
sn}}}{2m_{e}c^{2}}\simeq 2.5\times{10}^{5}~{}{(\varepsilon_{\rm sn}/1~{}{\rm
eV})}^{-1}.$ (72)

In addition, leptonic injection occurs through primary electron acceleration.
Observations of SN remnants and CR spectra on Earth suggest
$\epsilon_{e}\sim{10}^{-4}-10^{-3}$, and it has been shown that the hadronic
component is dominant when $\epsilon_{e}\lesssim 0.2\epsilon_{\rm cr,-1}$ in
the calorimetric limit Murase  _et al._ (2014). Although the leptonic
injection is available in AMES, it is negligible during the main interaction
phase for CCSM parameters considered in this work.

The radiative cooling Lorentz factor of electrons and positrons is given by

$\displaystyle\gamma_{e,\rm rc}$ $\displaystyle\approx$
$\displaystyle\frac{6\pi m_{e}c}{\sigma_{T}B^{2}(1+Y)t}$ (73)
$\displaystyle\simeq$ $\displaystyle
1.6~{}\varepsilon_{B,-2}^{-1}D_{*,-2}^{-1}t_{5.5}{(1+Y)}^{-1}$

where $Y$ is the Compton Y parameter,

$\displaystyle Y$ $\displaystyle\approx$ $\displaystyle\frac{-1+\frac{U_{\rm
sn}}{U_{B}}+\sqrt{{\left(1+\frac{U_{\rm
sn}}{U_{B}}\right)}^{2}+\frac{2}{3}{\rm min}[f_{pp},1]\frac{\epsilon_{\rm
cr}V_{s}}{\varepsilon_{B}c}}}{2}$ (74) $\displaystyle\rightarrow$
$\displaystyle\frac{U_{\rm sn}}{U_{B}}\,\,\,\,\,\mbox{(for $\epsilon_{\rm
rad}\gg\epsilon_{\rm cr}$)}$

in the Thomson limit, where $U_{\rm sn}$ is the energy density of SN thermal
photons in the emission region. Different from the pulsar-driven SN scenario,
where the SN emission itself is attributed to the regeneration of nonthermal
synchrotron radiation Murase  _et al._ (2015), we expect that the Compton Y
parameter is governed by seed photons originating from the thermal radiation.
If the SN emission is powered by the CSM interaction, as in SNe IIn, we obtain

$\displaystyle Y$ $\displaystyle\simeq$ $\displaystyle 0.57~{}{\rm
max}[1,\tau_{T}]\epsilon_{\rm th,-0.6}\varepsilon_{B,-2}^{-1}D_{*,-2}^{-1/10}$
(75) $\displaystyle\times$ $\displaystyle{\mathcal{E}}_{\rm
ej,51}^{9/20}M_{\rm ej,1}^{-7/20}t_{5.5}^{-1/10},$

while for $L_{\rm sn}\approx L_{\rm ph}$ we have

$\displaystyle Y\approx\frac{{\rm max}[1,\tau_{T}]L_{\rm
ph}V_{s}}{\varepsilon_{B}L_{s}c}$ $\displaystyle\simeq$ $\displaystyle
7.2~{}{\rm max}[1,\tau_{T}]L_{\rm ph,42.5}\varepsilon_{B,-2}^{-1}$ (76)
$\displaystyle\times$ $\displaystyle D_{*,-2}^{-4/5}{\mathcal{E}}_{\rm
ej,51}^{-9/10}M_{\rm ej,1}^{7/10}t_{5.5}^{1/5}.\,\,\,\,\,$

Cooling time scales of electrons and positrons at $t_{\nu{\rm max}}$ are shown
in Fig. 5. One finds $\gamma_{e,\rm rc}<\gamma_{e,h},\gamma_{e,\gamma\gamma}$,
implying that the system is in the fast cooling regime for leptons. Ignoring
Coulomb losses, the transition time from the fast to slow cooling regimes is

$t_{f\rightarrow s}\simeq 1.3\times{10}^{7}~{}{\rm
s}~{}\varepsilon_{B,-2}D_{*,-2}(1+Y).$ (77)

Thus, we may expect the fast cooling in most of the interaction duration
$t_{\rm end}$ especially if $Y\gg 1$. In the Thomson limit, the ratio of
synchrotron to IC cooling time scales is $t_{\rm syn}/t_{\rm IC}\approx U_{\rm
sn}/U_{B}$, and the analytical estimates agree with the numerical results
well. In the limit that SN thermal emission in the optical band is powered by
CSM interaction, the synchrotron cooling is comparable to the IC cooling (see
Fig. 5 left and right). On the other hand, as expected in ordinary SNe II-P,
the IC cooling is dominant in the presence of SN optical photons from the
photosphere (see Fig. 5 middle). We also note that the Klein-Nishina effect is
important for leptons with $\gtrsim 10-100$ GeV energies.

However, there is one complication from the standard fast-cooling spectrum.
Due to strong Coulomb losses in dense CSM, the lepton distribution can be
modified and the cooling Lorentz factor does not have to be $\gamma_{e,\rm
rc}$. By comparing Coulomb losses with radiative losses, the “Coulomb break”
Lorentz factor in the fast cooling regime can be introduced as,

$\displaystyle\gamma_{e,\rm Cou}$ $\displaystyle\sim$
$\displaystyle{\left(\frac{270\pi m_{e}c^{2}n_{e}}{B^{2}(1+Y)}\right)}^{1/2}$
(78) $\displaystyle\simeq$ $\displaystyle
84~{}\varepsilon_{B,-2}^{-1/2}\mu_{e}^{-1/2}D_{*,-2}^{1/10}{\mathcal{E}}_{\rm
ej,51}^{-9/20}$ $\displaystyle\times$ $\displaystyle M_{\rm
ej,1}^{7/20}t_{5.5}^{1/10}{(1+Y)}^{-1/2}.\,\,\,\,\,\,\,\,\,\,$

This analytical estimate agrees with numerical results shown in Fig. 5, and
Coulomb losses are important in SNe II with CCSM. Note that this break Lorentz
factor is different from that in the slow cooling regime Uchiyama  _et al._
(2002), where it is defined by comparing the Coulomb cooling time with the
dynamical time. The transition to the standard fast cooling regime occurs at

$\displaystyle t_{\gamma_{e,\rm rc}=\gamma_{e,\rm Cou}}$ $\displaystyle\simeq$
$\displaystyle 2.5\times{10}^{7}~{}{\rm
s}~{}\varepsilon_{B,-2}^{5/9}\mu_{e}^{-5/9}D_{*,-2}^{11/9}$ (79)
$\displaystyle\times$ $\displaystyle{\mathcal{E}}_{\rm ej,51}^{-9/18}M_{\rm
ej,1}^{7/18}{(1+Y)}^{5/9}.$

In the fast cooling limit, the spectrum of electrons and positrons in the
quasisteady state is roughly expressed as

$\displaystyle
n_{\gamma_{e}}^{e}\propto\gamma_{e}^{-q}\propto\left\\{\begin{array}[]{lr}\gamma_{e}^{-2+\Delta
s}&\mbox{($\gamma_{e}<\gamma_{e,h}$)}\\\ \gamma_{e}^{-\tilde{s}-1+\Delta
s}&\mbox{($\gamma_{e,h}\leq\gamma_{e}$)}\end{array}\right.,$ (82)

where $\tilde{s}$ is the effective injection index, which is $\tilde{s}\approx
s$ for secondary injection from $pp$ interactions. If there is an additional
contribution from electromagnetic cascades through $\gamma\gamma\rightarrow
e^{+}e^{-}$, the energy spectrum is flatter, which may lead to $\tilde{s}\sim
2$. Here $\Delta s$ represents the possible spectral hardening effect due to
Coulomb losses. If the Coulomb cooling is more important than the radiative
cooling, i.e., $\gamma_{e}<\gamma_{e,\rm Cou}$, we have $\Delta s\approx 2$.
As a result, most of the energy comes from leptons with $\gamma_{e,b}={\rm
max}[\gamma_{e,h},\gamma_{e,\rm Cou}]$, and their characteristic energy of IC
emission from secondaries injected through $pp$ interactions is

$\displaystyle\varepsilon_{\rm IC}^{b}\approx
2\gamma_{e,b}^{2}\varepsilon_{\rm sn}$ $\displaystyle\simeq$ $\displaystyle
9.2~{}{\rm keV}~{}(\varepsilon_{\rm sn}/1~{}{\rm
eV}){(\gamma_{e,b}/\gamma_{e,h})}^{2},\,\,\,$ (83)

which is expected in the hard x-ray range. For the $\gamma\gamma\rightarrow
e^{+}e^{-}$ injection component, we obtain

$\displaystyle\varepsilon_{\rm IC}^{\gamma\gamma}\approx
2\gamma_{e,\gamma\gamma}^{2}\varepsilon_{\rm sn}$ $\displaystyle\simeq$
$\displaystyle 130~{}{\rm GeV}~{}{(\varepsilon_{\rm sn}/1~{}{\rm eV})}^{-1},$
(84)

which is expected in the gamma-ray range.

Assuming the fast cooling regime, the resulting IC luminosity from CR-induced
electrons and positrons is

$\displaystyle\varepsilon_{\gamma}L_{\varepsilon_{\gamma}}$
$\displaystyle\approx$ $\displaystyle\frac{Y}{2(1+Y)}\frac{f_{\rm sup}^{\rm
mat}}{6}{\rm min}[f_{pp},1]\frac{\epsilon_{\rm cr}L_{s}}{{\mathcal{R}}_{\rm
cr10}}g_{\rm Cou}{\left(\frac{\varepsilon_{\gamma}}{\varepsilon_{\rm
IC}^{b}}\right)}^{2-\beta_{\rm IC}}$ (85) $\displaystyle\simeq$ $\displaystyle
8.4\times{10}^{38}~{}{\rm erg}~{}{\rm s}^{-1}~{}{(Y/[1+Y])}g_{\rm Cou}$
$\displaystyle\times$ $\displaystyle{\rm min}[f_{pp},1]f_{\Omega}\epsilon_{\rm
cr,-1}{\mathcal{R}}_{\rm
cr10,1}^{-1}{\left(\frac{\varepsilon_{\gamma}}{\varepsilon_{\rm
IC}^{b}}\right)}^{2-\beta_{\rm IC}}$ $\displaystyle\times$ $\displaystyle
D_{*,-2}^{7/10}{\mathcal{E}}_{\rm ej,51}^{27/20}M_{\rm
ej,1}^{-21/20}t_{5.5}^{-3/10}e^{-\tau_{\rm Comp}-\tau_{\rm pe}},$

where $\beta_{\rm IC}=3/2$ or $\tilde{s}/2$ or $2/3$ for $\varepsilon_{\rm
IC}<\varepsilon_{\rm IC}^{b}$ and $\beta_{\rm IC}=(\tilde{s}+2)/2$ for
$\varepsilon_{\rm IC}^{b}\leq\varepsilon_{\rm IC}$, respectively, and $g_{\rm
Cou}={(\varepsilon_{\gamma}^{b}/\varepsilon_{\gamma}^{h})}^{(2-\tilde{s})/2}$.
Here $f_{\rm sup}^{\rm mat}$ is the suppression fraction of x rays, which is
dominated by Compton scattering and photoelectric absorption in the unshocked
CSM. As shown in Fig. 4, the energy spectrum of the CR-induced IC emission is
flat in $E_{\gamma}F_{E_{\gamma}}$ especially above $E_{\rm IC}^{b}$, which is
consistent with an analytical expectation by Eq. (85).

In order to demonstrate the impacts of SN emission from the photosphere, Fig.
7 shows the results with $L_{\rm ph}$. As analytically expected by considering
$Y\gg 1$, additional external photons enhance the IC cascade emission
especially at late times, whereas the energy flux below keV energies is
smaller because the synchrotron flux significantly contributes at these
energies (see also next subsection). The electromagnetic cascade makes overall
energy spectra very flat, and because of $\gamma_{e,\rm Cou}\sim\gamma_{e,h}$
in our cases the effects of Coulomb losses are hardly seen in the resulting
electromagnetic spectra. The IC cascade spectrum is extended from the x-ray
band to the gamma-ray energy range, being overwhelmed by the pionic gamma-ray
component above 0.1 GeV, as seen in Figs. 4 and 7. Although the break from
$\pi^{0}$ decay should exist, the pionic gamma-ray component may stand out by
only a factor of $\sim 3-5$. We also note that thermal bremsstrahlung emission
is dominant in the $1-100$ keV band (cf. Fig. 1), so higher energies
especially in the MeV–GeV bands are better to detect the IC cascade component.

At MeV energies, where the nonthermal component is likely to be dominant,
photons lose their energies mainly via Compton downscatterings, and the
Compton optical depth is estimated to be

$\tau_{\rm Comp}\simeq
0.025~{}(\tilde{Z}+1)\mu_{e}^{-1}D_{*,-2}^{11/10}{\mathcal{E}}_{\rm
ej,51}^{-9/20}M_{\rm ej,1}^{7/20}t_{5.5}^{-9/10},$ (86)

and the transition time for the system to be optically thin is

$t_{\tau_{\rm Comp}=1}\simeq 5.3\times{10}^{3}~{}{\rm
s}~{}{(\tilde{Z}+1)}^{10/9}\mu_{e}^{-10/9}D_{*,-2}^{11/9}{\mathcal{E}}_{\rm
ej,51}^{-1/2}M_{\rm ej,1}^{7/18}.$ (87)

Given that hadronic components from $pp$ interactions are dominant, the CR-
induced IC energy flux follows

$\displaystyle E_{\gamma}F_{E_{\gamma}}^{\rm IC}$ $\displaystyle\propto$
$\displaystyle e^{-\tau_{\rm Comp}(t)}Y(t){(1+Y(t))}^{-1}$ (88)
$\displaystyle\times$
$\displaystyle\left\\{\begin{array}[]{lr}\varepsilon_{\gamma}^{2-\beta_{\rm
IC}}\varepsilon_{\rm sn}^{\beta_{\rm
IC}-2}t^{-0.3}&\mbox{($t<t_{f_{pp}=1}$)}\\\ \varepsilon_{\gamma}^{2-\beta_{\rm
IC}}\varepsilon_{\rm sn}^{\beta_{\rm
IC}-2}t^{-1.1}&\mbox{($t>t_{f_{pp}=1}$)}\end{array}\right.,$ (91)

which agrees with the numerical curves shown in the middle panels of Figs. 3
and 6.

###  III.4 Synchrotron (cascade) emission

Primary electrons and secondary electron-positron pairs emit synchrotron
emission while they are advected in the shock downstream. Sufficiently high-
energy pionic gamma rays create electron-positron pairs, which also contribute
to additional synchrotron emission though cascades. When the hadronuclear
component is dominant, their characteristic frequencies are given by

$\displaystyle\nu_{\rm syn}^{b}$ $\displaystyle\approx$
$\displaystyle\frac{3}{4\pi}\gamma_{e,b}^{2}\frac{eB}{m_{e}c}$ (92)
$\displaystyle\simeq$ $\displaystyle 760~{}{\rm
GHz}~{}\varepsilon_{B,-2}^{1/2}D_{*,-2}^{1/2}t_{5.5}^{-1}{(\gamma_{e,b}/\gamma_{e,h})}^{2},$

and

$\displaystyle\varepsilon_{\rm syn}^{\gamma\gamma}$ $\displaystyle\approx$
$\displaystyle\frac{3}{4\pi}\gamma_{e,{\gamma\gamma}}^{2}\frac{heB}{m_{e}c}$
(93) $\displaystyle\simeq$ $\displaystyle 44~{}{\rm keV}~{}{(\varepsilon_{\rm
sn}/1~{}{\rm eV})}^{-2}\varepsilon_{B,-2}^{1/2}D_{*,-2}^{1/2}t_{5.5}^{-1},$

respectively. Ref. Murase  _et al._ (2014) pointed out that secondary
synchrotron emission peaks in the high-frequency radio (submillimter or
millimeter) band, which provides a smoking gun of ion acceleration in
interaction-powered SNe or Type IIn SNe (see also Refs. Petropoulou  _et al._
(2016); Matsuoka  _et al._ (2019)). However, for more ordinary SNe, we find
that the CR-induced synchrotron emission can be suppressed by two effects.
First, as discused in the previous subsection, IC cooling can be more
efficient due to external SN photons, which is especially the case for Type
II-P SNe. The second is Coulomb scattering with electrons, through which
energy of nonthermal leptons can rather be used for plasma heating.

Assuming the fast cooling regime, the differential synchrotron luminosity is
analytically expressed as

$\displaystyle\nu L_{\nu}^{\rm syn}$ $\displaystyle\approx$
$\displaystyle\frac{1}{2(1+Y)}\frac{f_{\rm sup}}{6}{\rm
min}[f_{pp},1]\frac{\epsilon_{\rm cr}L_{s}}{{\mathcal{R}}_{\rm cr10}}g_{\rm
Cou}{\left(\frac{\nu}{\nu_{b}}\right)}^{2-\beta_{\rm syn}}$ (94)
$\displaystyle\simeq$ $\displaystyle 8.4\times{10}^{38}~{}{\rm erg}~{}{\rm
s}^{-1}~{}{(1+Y)}^{-1}g_{\rm Cou}$ $\displaystyle\times$ $\displaystyle{\rm
min}[f_{pp},1]f_{\Omega}\epsilon_{\rm cr,-1}{\mathcal{R}}_{\rm
cr10,1}^{-1}{\left(\frac{\nu}{\nu_{\rm syn}^{b}}\right)}^{2-\beta_{\rm syn}}$
$\displaystyle\times$ $\displaystyle D_{*,-2}^{7/10}{\mathcal{E}}_{\rm
ej,51}^{27/20}M_{\rm ej,1}^{-21/20}t_{5.5}^{-3/10}\frac{e^{-\tau_{\rm
ff}}}{1+\tau_{\rm sa}},$

where $\beta_{\rm syn}=3/2$ or $\tilde{s}/2$ or or $2/3$ for $\nu_{\rm
syn}<\nu_{\rm syn}^{b}$ and $\beta_{\rm syn}=(\tilde{s}+2)/2$ for $\nu_{\rm
syn}\geq\nu_{\rm syn}^{b}$, respectively, and $f_{\rm sup}$ is the suppression
factor that mainly consists of SSA and free-free absorption. In Figs. 4 and 7,
low-energy spectra originate from CR-induced synchrotron emission. Energy
fluxes of radio emission from secondary pairs are comparable to those of CR-
induced IC emission for interaction-powered SNe with $L_{\rm sn}\sim L_{s}$
because of $Y\lesssim 1$. However, for SNe with $L_{\rm sn}\sim L_{\rm ph}\gg
L_{s}$, we find that radio emission can be strongly suppressed by IC cooling,
as clearly indicated in Fig. 7. Note that in the last expression of Eq. (94)
we have assumed that the emission region is completely screened or embedded by
the CSM. In reality, the CCSM is clumpy or aspherical, in which a significant
fraction of radio emission may escape. The radio light curves without $f_{\rm
sup}^{\rm mat}$ are shown in the right panels of Figs. 3 and 6. The spectra
without $f_{\rm sup}^{\rm mat}$ are also shown in Figs. 4 and 7 with the thin
curves. The hadronic scenario predicts that the synchrotron emission below
$\nu_{\rm syb}^{b}$ has a hard spectrum with $\beta_{\rm syn}\lesssim 3/2$,
which is rather robust thanks to Coulomb losses that make the spectra harder.
If a softer spectrum is observed in the radio band, the existence of primary
electrons or additional processes such as reacceleration via turbulence may be
inferred.

The SSA optical depth is analytically estimated by (Murase  _et al._ , 2014)

$\tau_{\rm sa}(\nu)={\eta_{\rm sa}}\frac{3\tilde{f}_{e}L_{s}e}{4\pi
R_{s}V_{s}m_{p}c^{2}B\gamma_{n}^{5}}{\left(\frac{\nu}{\nu_{n}}\right)}^{-(q+4)/2}$
(95)

where $\tilde{f}_{e}\approx(1/6){\rm min}[1,f_{pp}]\epsilon_{\rm
cr}/{\mathcal{R}}_{\rm cr10}/(\tilde{s}-1)\equiv 0.045\tilde{f}_{e,*}$ for the
hadronic scenario, $\gamma_{e,n}\approx{\rm min}[\gamma_{e,h},\gamma_{e,c}]$,
$\nu_{n}=3\gamma_{e,n}^{2}eB/(4\pi m_{e}c)$ and $\eta_{\rm sa}$ is

$\eta_{\rm
sa}=\tilde{\xi}_{q}\frac{\pi^{\frac{3}{2}}2^{\frac{q}{2}}}{3^{\frac{3}{2}}}\frac{\Gamma(\frac{q}{4}+\frac{11}{6})\Gamma(\frac{q}{4}+\frac{1}{6})\Gamma(\frac{q}{4}+\frac{3}{2})}{\Gamma(\frac{q}{4}+2)},$
(96)

where $\tilde{\xi}_{q}=(q-1)$. Although the SSA optical depth can be affected
by Coulomb losses, as noted above, Eq. (78) suggests that $\gamma_{e,\rm Cou}$
is not too far from $\gamma_{e,h}$ and considering $\gamma_{e,\rm
rc}<\gamma_{e}<\gamma_{e,h}$ would give a conservative result. With $q=2$, the
SSA optical depth at $\nu_{\rm syn}^{h}$ is approximated to be

$\tau_{\rm sa}(\nu_{\rm syn}^{h})\simeq
0.020~{}\tilde{f}_{e,*}\varepsilon_{B,-2}^{-3/2}D_{*,-2}^{-3/5}{\mathcal{E}}_{\rm
ej,51}^{9/20}M_{\rm ej,1}^{-7/20}t_{5.5}^{9/10}{(1+Y)}^{-1},$ (97)

where we note that $\tilde{f}_{e,*}$ depends on $t$. Then, for $\nu_{\rm
sa}<\nu_{\rm syn}^{h}$, we obtain

$\nu_{\rm sa}\simeq 210~{}{\rm
GHz}~{}\tilde{f}_{e,*}^{1/3}D_{*,-2}^{3/10}{\mathcal{E}}_{\rm
ej,51}^{3/20}M_{\rm ej,1}^{-7/60}t_{5.5}^{-21/30}{(1+Y)}^{-1/3},$ (98)

and the transition time when the emission region is optically thin to SSA is

$\displaystyle t_{\tau_{\rm sa}=1}$ $\displaystyle\simeq$ $\displaystyle
1.7\times{10}^{5}~{}{\rm
s}~{}\tilde{f}_{e,*}^{10/21}D_{*,-2}^{3/7}{\mathcal{E}}_{\rm
ej,51}^{3/14}M_{\rm ej,1}^{-1/6}$ (99) $\displaystyle\times$
$\displaystyle{(1+Y)}^{-10/21}\nu_{11.5}^{-10/7}.$

On the other hand, the free-free optical depth in the unshocked CSM is
approximated to be

$\nu_{\rm ff}\simeq 2.2~{}{\rm
THz}~{}\tilde{Z}^{10/21}\mu_{e}^{-20/21}{\mathcal{T}}_{\rm
ext,5}^{-9/14}D_{*,-2}^{23/21}{\mathcal{E}}_{\rm ej,51}^{-9/14}M_{\rm
ej,1}^{1/2}t_{5.5}^{-9/7},$ (100)

in the fully ionized limit. This gives the radio breakout time against the
free-free absorption process,

$\displaystyle t_{\tau_{\rm ff}=1}$ $\displaystyle\simeq$ $\displaystyle
1.4\times{10}^{6}~{}{\rm
s}~{}\tilde{Z}^{10/27}\mu_{e}^{-20/27}{\mathcal{T}}_{\rm cs,5}^{-1/2}$ (101)
$\displaystyle\times$ $\displaystyle D_{*,-2}^{23/27}{\mathcal{E}}_{\rm
ej,51}^{-1/2}M_{\rm ej,1}^{7/18}\nu_{11.5}^{-7/9}.$

These results imply that the free-free absorption is more severe than the SSA
absorption in cases of the simplest spherical CSM. However, we note that the
radio escape fraction can be readily larger if the CSM is aspherical or
clumpy, and/or if the ionization fraction is lower in the unshocked CSM.

Finally, the CR-induced synchrotron flux from secondaries produced via
inelastic $pp$ interactions follows

$\displaystyle E_{\gamma}F_{E_{\gamma}}^{\rm syn}$ $\displaystyle\propto$
$\displaystyle e^{-\tau_{\rm ff}(t)}{[(1+Y(t))(1+\tau_{\rm sa}(t))]}^{-1}$
(102) $\displaystyle\times$
$\displaystyle\left\\{\begin{array}[]{lr}\nu^{2-\beta_{\rm
syn}}t^{1.7-\beta_{\rm syn}}&\mbox{($t<t_{f_{pp}=1}$)}\\\ \nu^{2-\beta_{\rm
syn}}t^{0.9-\beta_{\rm syn}}&\mbox{($t>t_{f_{pp}=1}$)}\end{array}\right.,$
(105)

which agrees with the numerical curves shown in Fig. 6 right. The cases
without $\tau_{\rm ff}$ are also shown in the right panels of Figs. 3 and 6.
The results imply that submillimeter or millimeter emission from CCSM-
interacting SNe II is promising only around $t_{\rm end}$ unless the escape
fraction is larger than $f_{\rm sup}^{\rm mat}$.

##  IV Implications

###  IV.1 Multiwavelength (gamma–x–radio) relation and testability of the
hadronic scenario

Figure 8: Characteristic time windows for observations of neutrinos, gamma
rays, x rays and radio waves for Type II SNe interacting with CCSM. Strong CSM
interaction accompanied by CR acceleration occurs from $t_{\rm onset}$ to
$t_{\rm end}$. See text for details.

Figure 9: Detection horizons of hadronic gamma rays at $E_{\gamma}=1$ GeV with
Fermi-LAT sensitivities (left panel) and $E_{\gamma}=50$ GeV with the CTA
sensitivity (middle panel). The right panel shows detection horizons of
hadronic radio emission at 100 GHz with ALMA and ngVLA sensitivities, where
upper limits are shown for SNe II (CCSM) and SNe IIn by ignoring external
matter attenuation. Note that for CCSM SNe II $D_{*}=0.01$ is used without
$L_{\rm ph}$.

AMES allows us to consistently calculate gamma-ray, x-ray and radio emissions,
considering electromagnetic cascades. In addition to the “multimessenger”
connection shown in Eq. (63), one may expect the following “multiwavelength”
relation among nonthermal luminosities,

$\frac{\varepsilon_{\gamma}L_{\varepsilon_{\gamma}}^{\pi^{0}}}{{\left(\frac{\varepsilon_{\gamma}}{0.8~{}{\rm
GeV}}\right)}^{2-s}}\sim\frac{4(1+Y)\nu L_{\nu}^{\rm
IC}}{Y{\left(\frac{\varepsilon_{\gamma}}{\varepsilon_{\rm
IC}^{b}}\right)}^{2-\beta_{\rm IC}}}\sim\frac{4(1+Y)\nu L_{\nu}^{\rm
syn}}{{\left(\frac{\nu}{\nu_{\rm syn}^{b}}\right)}^{2-\beta_{\rm syn}}},$
(106)

which essentially reflects the ratio between $\pi^{\pm}$ and $\pi^{0}$
produced via inelastic $pp$ interactions. Observationally, deviations from Eq.
(106) can be caused by contributions from other components by e.g., thermal
bremsstrahlung and nonthermal emission from primary electrons.

Simultaneous observations are important for testing the above relation. The
next Galactic SN would be ideal for detailed studies, and in the previous
sections the results on $EF_{E}$ were presented for a SNe at $d=10$ kpc. On
the other hand, Galactic SNe are rare and the core-collapse SN rate in the
Milky Way is estimated to be $\sim 3~{}{\rm century}^{-1}$ Adams  _et al._
(2013). The rates of Type II-P, II-L/IIb and IIn SNe are $\sim 50$% and $\sim
15-20$%, and $\sim 10$%, respectively Smith  _et al._ (2011); Li  _et al._
(2011). Thus, searching for extragalactic SNe is relevant especially for rarer
types of SNe including Type IIn SNe. Then, it would be useful to clarify
characteristic time windows, e.g., for neutrino-triggered optical followup
observations that have been suggested for extragalactic SNe Murase  _et al._
(2006); Yoshida  _et al._ (2022); Pitik  _et al._ (2023).

In Fig. 8, we show several time scales that may represent characteristic
observational time windows. For neutrinos which can leave the system without
attenuation, $t_{f_{pp}=1}$, which corresponds to the peak time of $tL_{\nu}$
or neutrino fluences, is shown (see Eq. 38). For electromagnetic emission, we
also consider various breakout times when the system is optically thin to
relevant attenuation or scattering processes. For pionic gamma-ray and IC
cascade components, the Bethe-Heitler pair production (Eq. 38) and Compton
(Eq. 38) processes are relevant as an effective attenuation process,
respectively. For radio waves, the SSA (Eq. 99) and free-free (Eq. 101)
absorption are considered at $\nu=10^{2.5}$ GHz. In addition, the range from
$t_{\rm onset}$ to $t_{\rm end}$ for SNe II with CCSM is depicted. For x rays
and gamma rays, we find that $t_{f_{pp}=1}$ is longer than the breakout time
scales in the range of $D_{*}\gtrsim 0.01$. This implies that the
characteristic time windows are essentially governed by $t_{f_{pp}=1}$. For
radio waves, if the free-free absorption process is critical,
$t_{\tau_{ff}=1}$ is the most relevant time scale for radio followup
observations. However, if the free-free attenuation is reduced by e.g.,
aspherical or clumpy CSM, the characteristic time for hadronic radio emission
may be either $t_{f_{pp}=1}$ or $t_{\tau_{\rm sa}=1}$.

Figure 10: Left panel: Light curves of high-energy neutrinos at
$\varepsilon_{\nu}=1$ TeV, high-energy gamma rays at $\varepsilon_{\gamma}=1$
GeV, soft gamma rays at $\varepsilon_{\gamma}=1$ MeV, and radio waves at
$\nu=100$ GHz, for SN 2023ixf with $D_{*}=0.003$ and $R_{\rm cs}={10}^{15}$ cm
(thick) and $D_{*}=0.1$ and $R_{\rm cs}=5\times{10}^{14}$ cm (thin). Middle
panel: Energy fluences of
$\nu_{e}+\bar{\nu}_{e}+\nu_{\mu}+\bar{\nu}_{\mu}+\nu_{\tau}+\bar{\nu}_{\tau}$
and cascaded gamma rays, integrated over $\approx 30$ d, from SN 2023ixf,
where $D_{*}=0.003$ and $R_{\rm cs}={10}^{15}$ cm (thick) and $D_{*}=0.1$ and
$R_{\rm cs}=5\times{10}^{14}$ cm (thin) are considered. Right panel: Energy
fluxes of nonthermal electromagnetic emission (at $t=10^{6}$ s) with (thick)
and without (thin) external matter attenuation. The preliminary gamma-ray
upper limit by Fermi-LAT is overlaid. In all three panels, $d=6.9$ Mpc and
$s_{\rm cr}=2.2$ are used, and $L_{\rm ph}$ is included based on Ref.
Hiramatsu  _et al._ (2023).

The detectability of high-energy neutrinos and gamma rays from extragalactic
interacting SNe have been studied (see Refs. Petropoulou  _et al._ (2016);
Murase  _et al._ (2019) for the earliest works). In Fig. 9 we presented
detection horizons for GeV gamma rays (left), sub-TeV gamma rays (middle), and
radio emission (right). For the detectability of GeV gamma rays by Fermi, both
the energy flux and Fermi sensitivity for each SN type are shown for the
optimized observation time, which is found by maximizing the ratio of the
signal flux to the sensitivity. One sees that detection of pionic gamma rays
is promising for not only Galactic SNe but also SNe at the Small Megellanic
Cloud at 60 kpc and the Large Magellanic Cloud at 50 kpc. In particular, SNe
II with CCSM are detectable up to $d\sim 5$ Mpc for $D_{*}\sim 0.01$ (SN
2013fs-like). See also Fig. 1 of Ref. Kheirandish and Murase (2023) for the
list of nearby galaxies (see also Ref. Valtonen-Mattila and O’Sullivan
(2023)). For the detectability of 50 GeV gamma rays by imaging atmospheric
Cherenkov telescopes (IACTs), we assume the Cherenkov Telescope Array (CTA)
north array with an integration time of 50 hr Hassan  _et al._ (2017). The
energy flux for each SN type is shown for the optimized observational time
that is often the peak time caused by gamma-ray suppression via the two-photon
annihilation process. Even for SNe II with CCSM and IIn, the detection is
possible up to $d\sim 3-10$ Mpc because the suppression due to
$\gamma\gamma\rightarrow e^{+}e^{-}$ is more important for brighter SNe
including SNe with stronger interaction with CCSM. We also note that future
MeV gamma-ray satellites such as AMEGO-X Caputo  _et al._ (2022) and
e-ASTROGAM De Angelis  _et al._ (2017) can play important roles in detecting
CR-induced IC cascade emission. For example, the e-ASTROGAM sensitivity for
$10^{6}$ s is $\sim{10}^{-11}~{}{\rm erg}~{}{\rm cm}^{-2}~{}{\rm s}^{-1}$ at 1
MeV, which allows us to hadronic MeV gamma-ray emission from SNe II with CCSM
at $d\lesssim 2-3$ Mpc. The detectability of radio emission is strongly
affected by free-free absorption even in the high-frequency band at $\sim 100$
GHz. For CCSM-interacting SNe II and IIn, we indicate the cases without the
free-free absorption as upper limits, and the integration time is assumed to
be 300 s for ALMA (Atacama Large Millimeter/submillimeter Array) ALM  and 1 hr
for ngVLA (next-generation Very Large Array) ngV , respectively. One sees that
CR-induced synchrotron emission can be detected more easily than pionic gamma
rays, although nondetection of radio signals is not surprising for SNe with
dense CSM. Note that for SNe with ordinary CSM (i.e., SNe II-P, II-L/IIb, and
Ibc without CCSM) the radio detectability is even conservative because it is
enhanced by synchrotron emission from primary electrons.

###  IV.2 Application to SN 2023ixf

A nearby Type II SN observed in the galaxy M101 at $d=6.9$ Mpc Jacobson-Galan
_et al._ (2023) provided unique opportunities to investigate the interaction
with CCSM. Early spectroscopic observations suggest $\dot{M}_{w}\sim
10^{-3}-10^{-2}~{}M_{\odot}~{}{\rm yr}^{-1}$ and $R_{\rm
cs}\sim(0.5-1)\times{10}^{15}$ cm Jacobson-Galan  _et al._ (2023); Bostroem
_et al._ (2023); Teja  _et al._ (2023). Although modeling of light curves may
favor large values of $\dot{M}_{w}$ Hiramatsu  _et al._ (2023), we adopt
$D_{*}=0.1$ and $R_{\rm cs}=5\times{10}^{14}$ cm Jacobson-Galan  _et al._
(2023) as the case motivated by optical observations. Detailed observations
indicate that the CSM structure is complex, and the CSM profile may be steeper
around $R_{\rm cs}\sim{\rm a~{}few}\times{10}^{14}$ cm Zimmerman  _et al._
(2023); Li  _et al._ (2023). This is supported by x-ray observations, whose
spectrum is consistent with thermal bremsstrahlung emission with a
photoelectric absorption feature Grefenstette  _et al._ (2023); Chandra  _et
al._ (2023), and we also adopt $D_{*}=0.003$ and $R_{\rm cs}=10^{15}$ cm. The
apparent discrepancies among different observations may suggest that the CCSM
is not spherical, which has been inferred by high-resolution spectroscopy and
polarization measurements Smith  _et al._ (2023); Vasylyev  _et al._ (2023).

In Fig. 10 left, we show neutrino and photon light curves for these two
parameter sets of $D_{*}$ and $R_{\rm cs}$. For $D_{*}=0.1$, the system is
mostly calorimetric during the interaction with CCSM, and the light curves of
neutrinos and gamma rays are nearly flat although gamma rays at early times
are subject to Compton or Bethe-Heitler attenuation. For $D_{*}=0.003$, the
light curves decline more rapidly because of $f_{pp}<1$ and the gamma-ray
attenuation is negligible from MeV to GeV energies. Radio emission is
suppressed mainly due to the free-free absorption process especially for
$D_{*}=0.1$, but it will break out at $t\sim 10-20$ d when the shock reaches
the CSM edge. The resulting synchrotron emission may explain the VLA detection
of radio waves around that time Matthews  _et al._ (2023).

Neutrino and gamma-ray fluences are presented in Fig. 10 middle. As shown in
Ref. Kheirandish and Murase (2023), the predicted numbers of throughgoing muon
events are approximately 0.0007 and 0.02 events for $D_{*}=0.003$ and
$D_{*}=0.1$, respectively. Thus, the CCSM-interaction scenario is consistent
with the null results reported by the IceCube fast response analysis (Thwaites
_et al._ , 2023) (see also Ref. Guetta  _et al._ (2023)). Neutrino and photon
fluxes at 30 d after the explosion are also shown in Fig. 10 right, where one
sees that CR-induced synchrotron emission is strongly suppressed due to
optical SN emission from the photosphere. Instead, IC (cascade) emission is
enhanced especially in the MeV range although in the x-ray range it is still
below the thermal bremsstrahlung flux. Furthermore, one sees that the
predicted gamma-ray fluxes for $\epsilon_{\rm cr}\sim 0.01-0.1$ are consistent
with the preliminary upper limit reported by Fermi Marti-Devesa (2023), which
leads to the constraint on the CR energy fraction, $\epsilon_{\rm cr}\lesssim
0.2-200$, reflecting the range of $D_{*}$. This limit is complementary to that
of SN 2010jl, $\epsilon_{\rm cr}\lesssim 0.05-0.1$ Murase  _et al._ (2019), in
that the results are obtained for Type II SNe that are more ordinary than SNe
IIn. Because ordinary SNe II (not SNe IIn) at $d\sim 7$ Mpc would still be too
far to detect neutrino and gamma-ray signals, these results strengthen the
importance of not missing nearby SNe at $d\lesssim 1-5$ Mpc. The core-collapse
SN rate in local galaxies is higher than in the rate estimated from the star-
formation rate in the continuum limit Ando  _et al._ (2005); Nakamura  _et
al._ (2016), and it can be as high as $\sim 0.8~{}{\rm yr}^{-1}$ for SNe
within $\sim 4-5$ Mpc. Such nearby SNe are promising targets for not only
gamma-ray detectors such as Fermi and IACTs but also the high-energy neutrino
detector network Kheirandish and Murase (2023) and Hyper-Kamiokande Abe  _et
al._ (2018).

##  V Summary

We investigated high-energy multimessenger emission from interacting SNe for
different classes of SNe. Considering the velocity distribution of SNe ejecta,
we presented detailed results on broadband multimessenger emission, including
light curves and spectra. The system can be nearly calorimetric for inelastic
$pp$ collisions, leading to promising GeV-TeV signals of high-energy neutrinos
and pionic gamma rays although TeV or higher-energy gamma rays may be
significantly affected by the two-photon annihilation process. Electromagnetic
cascades are developed in the presence of Coulomb losses, in which both IC and
synchrotron components are relevant in the x-ray/soft gamma-ray and radio
range, respectively. We also derived the analytical expressions that describe
electromagnetic fluxes from the interacting SNe, which are consistent with the
results of numerical calculations. The overall energy spectrum is expected to
be flat over wide energy ranges, especially from MeV to $\sim 10$ GeV energies
(although the low-energy break from neutral pion decay would still be
visible). Hadronic synchrotron emission predicts hard spectra with $\beta_{\rm
syn}\lesssim 3/2$ in the radio band, and Coulomb losses can make the spectra
harder. The synchrotron fluxes may also be significantly suppressed by
efficient IC cooling due to SN photons, and various absorption processes
affect the spectra in the radio band.

The hadronic scenario predicts the unique multimessenger relationship, which
can be tested with simultaneous observations of nearby SNe. The next Galactic
SN would be the most promising event that enable us to confront the theory
with observational data. However, detection of nonthermal signals from
extragalactic SNe may be challenging, and we discussed characteristic time
windows. For high-energy neutrinos and gamma rays, the time at which the
system is effectively transparent to inelastic $pp$ interactions is typically
the most important. The gamma-ray signal is detectable up to a few Mpc for
ordinary SNe II, although Type IIn SNe may be observed up to dozens of Mpc.
TeV or higher-energy gamma rays would significantly absorbed by SN photons,
and detection with $\sim 10-50$ GeV energies is relevant for IACTs such as
HESS, MAGIC, VERITAS and CTA. For radio waves and soft x rays, attenuation or
scattering processes affect the detectability of interacting signatures.
Hadronic radio emission is predicted to be dominate over leptonic radio
emission during the interaction with dense CCSM, but the radio signal at early
times is suppressed by SSA and free-free absorption so that the detectability
depends on details of the CSM structure. If the escape fraction is $\sim 1$%
due to clumpy or aspherical geometry, high-frequency radio emission from SNe
with CCSM may be detectable up to dozens of Mpc. We also discussed SN 2023ixf
as an example of interest. Nondetection of neutrinos and gamma rays is
consistent with our theoretical predictions. While CR-induced IC emission in
the x-ray range is overwhelmed by thermal bremsstrahlung emission, CR-induced
synchrotron emission may significantly contribute to the observed radio
emission. It is very important not to miss future nearby SNe within $\sim 3-5$
Mpc, and all-sky monitoring of the brightest SNe with followup observations
(by e.g., ASAS-SN, Tomo-e Gozen and ZTF) are necessary.

The method used in AMES for calculating multimessenger spectra of interacting
SNe can be applied to arbitrary evolution of $R_{s}$ and $V_{s}$ as well as
energy distributions of CRs and external SN radiation fields. Given that AMES
includes the photomeson production and photodisintegration processes Zhang and
Murase (2023), the SN module can be used for not only the SN reverse shock but
also other transients powered by shock interactions such as winds from compact
binary mergers Murase  _et al._ (2018) and tidal disruption events Murase  _et
al._ (2020). The results with a combination of CHIPS (Complete History of
Interaction-powered Supernovae) Takei  _et al._ (2022) will be presented in
the forthcoming paper.

###### Acknowledgements.

K.M. thanks Kunihito Ioka, Daichi Tsuna and B. Theodore Zhang for useful
discussions. The work of K.M. is supported by the NSF grants No. AST-1908689,
No. AST-2108466 and No. AST-2108467, and KAKENHI No. 20H01901 and No.
20H05852. The multimessenger spectral templates presented in this work and M18
are available on Github Git . The public version of the SN module in AMES will
be available upon request.

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