# The Late-Time Light Curves of Type Ia Supernovae: Confronting Models with
Observations

Vishal Tiwari,1,2 Or Graur,3,4 Robert Fisher,1 Ivo Seitenzahl,5 Shing-Chi
Leung,6 Ken’ichi Nomoto,7 Hagai Binyamin Perets8 and Ken Shen9
1Department of Physics, University of Massachusetts Dartmouth, 285 Old
Westport Road, North Dartmouth, MA 02740, USA
2Center for Relativistic Astrophysics, School of Physics, Georgia Institute of
Technology, Atlanta, GA 30332, USA
3Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth,
PO1 3FX, UK
4Department of Astrophysics, American Museum of Natural History, Central Park
West and 79th Street, New York, NY 10024, USA
5School of Science, University of New South Wales, Australian Defence Force
Academy, Canberra, ACT 2600, Australia
6TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA
91125, USA
7Kavli Institute for the Physics and Mathematics of the Universe (WPI), The
University of Tokyo Institutes for Advanced Study, The University of Tokyo,
Kashiwa, Chiba 277-8583, Japan
8Technion - Israel Institute of Technology, Physics Department, Haifa, Israel
32000
9Astronomy Department, University of California Berkeley, Berkeley, CA 94720,
USA
E-mail<EMAIL_ADDRESS>(VT)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Type Ia supernovae (SNe Ia) play a crucial role as standardizable candles in
measurements of the Hubble constant and dark energy. Increasing evidence
points towards multiple possible explosion channels as the origin of normal
SNe Ia, with possible systematic effects on the determination of cosmological
parameters. We present, for the first time, a comprehensive comparison of
publicly-available SN Ia model nucleosynthetic data with observations of late-
time light curve observations of SN Ia events. These models span a wide range
of white dwarf (WD) progenitor masses, metallicities, explosion channels, and
numerical methodologies. We focus on the influence of 57Ni and its isobaric
decay product 57Co in powering the late-time ($t>1000$ d) light curves of SNe
Ia. 57Ni and 57Co are neutron-rich relative to the more abundant radioisotope
56Ni, and are consequently a sensitive probe of neutronization at the higher
densities of near-Chandrashekhar (near-$M_{\rm Ch}$) progenitor WDs. We
demonstrate that observations of one SN Ia event, SN 2015F is only consistent
with a sub-$M_{\rm Ch}$ WD progenitor. Observations of four other events (SN
2011fe, SN 2012cg, SN 2014J, SN2013aa) are consistent with both near-$M_{\rm
Ch}$ and sub-$M_{\rm Ch}$ progenitors. Continued observations of late-time
light curves of nearby SNe Ia will provide crucial information on the nature
of the SN Ia progenitors.

###### keywords:

supernovae:general – white dwarfs – nuclear reactions, nucleosynthesis,
abundances – ISM: supernova remnants – hydrodynamics

††pubyear: 2021††pagerange: The Late-Time Light Curves of Type Ia Supernovae:
Confronting Models with Observations–The Late-Time Light Curves of Type Ia
Supernovae: Confronting Models with Observations

##  1 Introduction

Type Ia supernovae (SNe Ia) are thought to be white dwarf stars (WDs) composed
primarily of carbon and oxygen, which undergo explosive nuclear burning. SNe
Ia are important across many astrophysical domains, serving as standardizable
candles for cosmology (Rust, 1974; Pskovskii, 1977; Phillips, 1993), sources
of cosmic rays (Baade & Zwicky, 1934; Ginzburg & Syrovatskii, 1964; Drury,
2012), turbulence (Elmegreen & Scalo, 2004), enriched isotopes for the
interstellar medium (Nomoto et al., 1984; Thielemann et al., 1986), and
endpoints of binary evolution.

An isolated WD is inherently stable. Therefore, virtually all SNe Ia explosion
models call for a companion star.111One exception being the pycnonuclear-
driven model of Chiosi et al. (2015) However, the nature of the companion star
and the explosion mechanism of SNe Ia is still unclear. The most frequently-
discussed possibilities for the companion are a main-sequence or a red-giant
star in the single-degenerate (SD) channel (Whelan & Iben, 1973), or another
white dwarf in the double-degenerate (DD) channel (Webbink, 1984; Iben &
Tutukov, 1984; Nomoto, 1982b).

In this paper, we explore how late-time ($t>1000$ d) observations of the light
curves of SNe Ia constrain their stellar progenitors and explosion mechanisms.
An important distinction between explosion channels originates from the
importance of electron capture reactions in the dense cores of near-
Chandrashekhar (near-$M_{\rm Ch}$) WDs originating in the SD channel. The
process of neutronization leads to an overall greater prevalence of neutron-
rich isotopes in near-$M_{\rm Ch}$ SD events, including the radioisotopes
${}_{28}^{57}\textrm{Ni}$ and ${}_{27}^{55}\textrm{Co}$ (Nomoto et al., 1984;
Thielemann et al., 1986; Seitenzahl et al., 2013b). These neutron-rich
radioisotopes decay over much longer timescales than the well-known 56Ni chain
energizing early SNe Ia light curves (Pankey, 1962), and consequently power
the SN Ia light curves at late times. The $A=57$ isobar decay chain has a
half-life of just over nine months:
${}_{28}^{57}\textrm{Ni}\;\overset{\mathrm{1.5d}}{\longrightarrow}\;_{27}^{57}\textrm{Co}\;\overset{\mathrm{272d}}{\longrightarrow}\;_{26}^{57}\textrm{Fe}$.
In addition, the $A=55$ isobar radioisotopes decay to 55Mn over several years:
${}_{27}^{55}\textrm{Co}\;\overset{\mathrm{18h}}{\longrightarrow}\;_{26}^{55}\textrm{Fe}\;\overset{\mathrm{3y}}{\longrightarrow}\;_{25}^{55}\textrm{Mn}$.
In practice, the observational challenge of making accurate late-time light
curve measurements of SNe Ia beyond 1000 days limits the use of this method to
nearby well-studied SNe Ia followed up with Hubble Space Telescope (HST). To
date, the late-time light curve technique has been applied to six SN Ia
events: SN 2011fe (Shappee et al., 2017; Dimitriadis et al., 2017; Kerzendorf
et al., 2017), SN 2012cg (Childress et al., 2015; Graur et al., 2016), SN
2013aa (Jacobson-Galán et al., 2018), SN 2014J (Graur, 2018; Yang et al.,
2018b; Li et al., 2019), ASASN-14lp (Graur et al., 2018b) and SN 2015F (Graur
et al., 2018a). Abundance ratios of 57Ni/56Ni have been obtained for five of
these (SNe 2011fe, 2012cg, 2013aa, 2014J and 2015F), and ratios of 55Fe/57Co
for three (SNe 2011fe, 2013aa, and 2014J).

Comparing model predictions against the abundances inferred from late-time
light curves allows us to address key outstanding questions surrounding the
nature of SNe Ia. These questions include: Are the stellar progenitors of SNe
Ia sub-$M_{\rm Ch}$ or near-$M_{\rm Ch}$ WDs? What is the explosion mechanism
of SNe Ia? In the context of near-$M_{\rm Ch}$ WDs, how do the WDs ignite and
subsequently detonate? In the context of sub-$M_{Ch}$ WDs, do only relatively
massive $M\sim$ 1 $M_{\odot}$ CO WDs successfully detonate, or can lower mass
WDs also detonate, possibly through thin helium layers?

SNe Ia exhibit a diverse range of luminosities and nucleosynthetic yields,
both of the key radioactive isotope 56Ni which powers early light curves, as
well as of the trace radioisotopes 55Co and 57Ni which dominate late-time
light curves. In this paper, we will explore the full diversity within all
major channels of SNe Ia, ranging from sub-Chandrasekhar (sub-$M_{\rm Ch}$)
WDs detonating through double detonations and double-degenerate mergers,
through near-$M_{\rm Ch}$ WDs detonating within the SD channel, using a large
dataset compiled from the literature (see table 2 for a summary). In several
instances, we have included similar physical models computed by different
groups using different simulation codes, in order to address the robustness of
our conclusions to different numerical methodologies.

While previous work confronted a single supernova with a few models, this is
the first paper to confront the entire sample of late-time SN Ia observations
with a comprehensive range of models. With this work, we address the question
of how theoretical models of SN Ia cope with an ensemble that represents a
swath of the whole normal SN Ia population.222In this paper, we broadly
consider a wide range of explosion channels and computational methodologies
for which detailed nucleosynthetic yields are publicly available. It should be
noted that there are some additional explosion channels considered in the
literature, most notably the core degenerate channel (Kashi & Soker, 2011;
Soker, 2019), for which detailed nucleosynthetic yields have not yet been
obtained. Further, we also address the WD stellar progenitor parameters and
explosion channels which can be inferred from late-time observations.

The format of the paper is as follows. In section 2, we discuss how the
isobaric ratio of 57Ni/56Ni (57/56) nuclides can be used to distinguish
between near-$M_{\rm{Ch}}$ and sub-$M_{\rm{Ch}}$ progenitors. We also
investigate how key stellar parameters, including the WD stellar progenitor
metallicity, progenitor central density, and ignition can affect this 57/56
isobaric ratio. In section 3, we describe the details of the numerical
simulations of the openly available models used and observational events used
in this work. We present our results comparing the late-time data against
models in section 4. In section 5, we discuss the conclusions of our findings.

##  2 57Ni/56Ni vs 56Ni Theory

Constraining the parameters of individual SN Ia from the late-time light curve
is not trivial because of the very high-dimensional parameter space spanned by
explosion models. We reduce the dimensionality of the model output space to
only two parameters: 56Ni and 57Ni/56Ni, under the assumption that the late-
time light curve is mainly powered by the radioactive decay of 57Ni chain
nuclides.

In the following section, we discuss how the ratio 57Ni/56Ni varies within the
SN Ia parameter space, including the nature of the WD progenitor (sub-$M_{\rm
Ch}$ or near-$M_{\rm Ch}$), the stellar progenitor metallicity, central
density and ignition in case of a near-$M_{\rm{Ch}}$ progenitor. We illustrate
these effects on 57Ni/56Ni using the near-$M_{\rm Ch}$ models from Dave et al.
(2017) (Figure 2) and sub-$M_{\rm Ch}$ models from Shen et al. (2018a)(Figure
1).

###  2.1 WD Progenitors: sub-$M_{\rm Ch}$ and near-$M_{\rm Ch}$

The electron capture rate is higher and more sensitive in near-$M_{\rm{ch}}$
environments as compared to sub-$M_{\rm{Ch}}$ environments (Bravo, 2019).
Thus, we expect the ratio of 57/56 for near-$M_{\rm Ch}$ progenitors to be
much higher than for the sub-$M_{\rm{Ch}}$ progenitors. This distinction can
be qualitatively seen in Figure 2 and Figure 1, where the near-$M_{\rm{Ch}}$
models (shown in red in Fig 2) typically have on average a higher 57/56 ratio
as compared to the sub-$M_{\rm{Ch}}$ models (shown in blue in Fig 1). This
distinction in the 57/56 ratio shows how the the late-time light curves can be
used as a probe to constrain the parameters of the progenitors of SNe Ia.

Central densities also play a crucial role in the production of
$\rm{{}^{55}Co}$. For densities $\rho\gtrsim 2\times 10^{8}$ g cm-3, yields
reach normal NSE freeze-out because of lower entropy, which in turn leads to a
higher abundance of $\rm{{}^{55}Co}$ (Thielemann et al., 1986; Bravo &
Martínez-Pinedo, 2012). On the other hand, densities $\rho\lesssim 2\times
10^{8}$ g cm-3 lead to an alpha-rich freeze-out due to higher entropy and
$\rm{{}^{55}Co}$ doesn’t survive the freeze-out. Therefore we expect higher
fractions of $\rm{{}^{55}Co}$ (which is parent nuclei of stable monoisotope
$\rm{{}^{55}Mn}$) in the denser near-$M_{\rm{Ch}}$ progenitors as compared to
the lighter sub-$M_{\rm{Ch}}$ progenitors (Figure 5).

Figure 1: Plot showing the ratio
($\rm{{}^{57}Ni}$+$\rm{{}^{57}Co}$)/($\rm{{}^{56}Ni}$+$\rm{{}^{56}Co}$)
against $M(\rm{{}^{56}Ni})$ $M_{\odot}$ for the sub-$M_{\rm{Ch}}$ model of
Shen et al. (2018a) with varying metallicity and progenitor mass. The shade of
blue represents the metallicity, and we see that the ratio 57/56 increases
with an increasing metallicity. The horizontal blue line marks the upper bound
on 57/56 ratio among all the sub-$M_{\rm{Ch}}$ models, and the red horizontal
line shows the lower bound on all the near-$M_{\rm{Ch}}$ models considered in
this work. All the solar and sub-solar metallicity models of Shen et al.
(2018a) lie below the red-line, the "only sub-$M_{\rm{Ch}}$ region," while
some super solar metallicity models are in the "overlap region."  Figure 2:
Plot showing the ratio
($\rm{{}^{57}Ni}$+$\rm{{}^{57}Co}$)/($\rm{{}^{56}Ni}$+$\rm{{}^{56}Co}$)
against $M(\rm{{}^{56}Ni})$ for the near-$M_{\rm{Ch}}$ model of Dave et al.
(2017) with varying metallicity (Z), central density ($\rho_{c}$), C/O
fraction, bubble offset and bubble radius. The bubble offset and bubble radius
are the location of the ignition from the center of the WD and the radius of
the ignition bubble respectively. The models in green oval are high-density
DDT and GCD models with $\rho_{c}$ = 6 $\times$ 109 g cm-3, while the models
in red ovals are lower density models with $\rho_{c}$ = 2.2 $\times$ 109 g
cm-3. The DDT models in thick solid green ovals have unequal C/O fraction of
30/70 with centrally ignited detonation (bubble offsets = 0km) and an off-
center detonation (bubble offset of 200km). The GCD models in dotted green
oval have unequal C/O fraction of 30/70 and an off-center detonation with
bubble offset of 200km. The thin solid green ovals are GCD models with an
equal fraction of C/O (50/50) and a bubble offset of 100km. The dashed and
dotted red models show GCD models with a C/O fraction of 30/70 and 50/50,
respectively. As shown, red oval groups have models with 50km and 100km bubble
offsets. The horizontal blue line marks the upper bound on 57/56 ratio among
all the sub-$M_{\rm{Ch}}$ models, and the red horizontal line shows the lower
bound on all the near-$M_{\rm{Ch}}$ models considered in this work. Around
half of the models of Dave et al. (2017) lie in the "only near-$M_{\rm{Ch}}$
region," while the other half lies in the "overlap region."

Figure 3: Plots showing the ratio
($\rm{{}^{57}Ni}$+$\rm{{}^{57}Co}$)/($\rm{{}^{56}Ni}$+$\rm{{}^{56}Co}$)
against $M(\rm{{}^{56}Ni})$ for the near-$M_{\rm{Ch}}$ models of Leung &
Nomoto (2018). Plot (a) shows DDT models with central density
$\rho_{c}=3.0\times 10^{9}$ g cm-3 and metallicities ranging from Z = 0.0 Z⊙
to 3.0 Z⊙. Plot (b) shows DDT models with metallicity Z = 1.0 Z⊙ and central
density ranging from $\rho_{c}=1.0\times 10^{9}$ g cm-3 \- $5.0\times 10^{9}$
g cm-3. The horizontal blue line marks the upper bound on 57/56 ratio among
all the sub-$M_{\rm{Ch}}$ models, and the red horizontal line shows the lower
bound on all the near-$M_{\rm{Ch}}$ models considered in this work. All these
sets of models lie in the "overlap region".

###  2.2 Stellar Progenitor Metallcities

The radioactive decay of 56Ni is the major energy source of the early time SN
Ia light curve and the metallicity of the progenitor WD plays an important
role in its production. Timmes et al. (2003) analytically showed that the
yield of 56Ni drops with increased metallicity. Late-time light curves powered
by the decay of 57Ni are also affected by metallicity. The ratio 57/56
increases with an increasing metallicity, because a metal-rich WD stellar
progenitor has a lower electron fraction (i.e. a lower Ye) and thus extra
neutrons, as compared to a metal-poor WD stellar progenitor. This lower Ye
shifts the nuclear statistical equilibrium from the dominant 56Ni yield to
more neutron rich nuclides like 57Ni and 55Co.

The effect of progenitor metallicity can be seen in the sub-$M_{\rm{Ch}}$
models of Shen et al. (2018a) in Figure 1, where increasing the metallicity
for a fixed WD mass yields a higher value of the 57/56 ratio (darker shades
have a higher 57/56 ratio) (See also Figures 28 and 29 in Leung & Nomoto
(2020)). This effect can also be seen in the near-$M_{\rm{Ch}}$ models of Dave
et al. (2017) in Figure 2 and in the models of Leung & Nomoto (2018) in Figure
3a. (See also Figures 16 and 19 in Leung & Nomoto (2018)).

In near-$M_{\rm{ch}}$ progenitors, the convective simmering phase (Chamulak et
al., 2008; Piro & Bildsten, 2008) in the core of the WD sets a lower bound on
the neutron excess, before the onset of a detonation. Martínez-Rodríguez et
al. (2016) showed that the excess neutrons due to convective simmering is a
very small fraction compared to the stellar progenitor metallicity for Z
$\gtrsim$ 1/3 Z⊙.

###  2.3 Near-$M_{\rm{Ch}}$ WD Progenitors: Central Densities

The central densities of WDs play an important role in the production of iron
group elements in near-$M_{\rm{Ch}}$ progenitors. Progenitor WDs with higher
central densities facilitate higher electron capture rates and thus higher
neutronization. But higher central densities can indirectly reduce the level
of neutronization - during the deflagration phase, higher densities allow for
higher flame speeds. Due to high flame speed more fuel is burned and thus
releases more energy, lead to a greater pre-expansion of the WD. This pre-
expansion suppresses electron captures and degree of neutronization.

The effect of the central density of a near-$M_{\rm{Ch}}$ WD on the
neutronization of SN Ia yields are illustrated the gravitationally confined
detonation (GCD) models presented by Dave et al. (2017) in Figure 2. When the
authors only varied the central density of the WDs with all other parameters
fixed, the models with lower central density $\rho_{c}=2.2\times 10^{9}$ g
cm-3 produced a higher ratio of 57/56 as compared to a denser model with
$\rho_{c}=6\times 10^{9}$ g cm-3. This relation of central density and 57/56
ratio suggests a lower neutronization in a more centrally dense WD because of
a higher degree of pre-expansion. Contrary to this trend, when Dave et al.
(2017) reduced the C/O fraction to less than one, they found that the denser
WD progenitors yielded more neutron rich isotopes because a lower carbon
fraction reduces the flame speed leading to a lower energy release during
deflagration. A lower energy release leads to a lower Atwood number and
therefore less buoyancy and a lower growth exponential of the Rayleigh-Taylor
instability. This causes the flame to spend more time at high density regions
that facilitates more electron captures, and in turn produces a higher ratio
of 57/56.

Figure 3b shows DDT models with the same metallicity of Z=1.0Z⊙ and equal
carbon/oxygen ratio (C/O=1.0) from Leung & Nomoto (2018) with central
densities ranging from $1.0\times 10^{9}$ g cm-3 to $5.0\times 10^{9}$ g cm-3.
In contrast to Dave et al. (2017) GCD models in Figure 2, higher central
density produces a higher 57/56 ratio, even with C/O=1.0.

The central densities at the carbon ignition of the accreting WDs depend on
the accretion rate, the initial mass of the WD, and the angular momentum of
the WD (Nomoto, 1982a; Benvenuto et al., 2015; Nomoto & Leung, 2018). Such
high central densities as $\rho_{c}=5-6\times 10^{9}$ g cm-3 can be realized
in the following cases of accreting WDs.

  1. 1.

The accreting WD becomes cold if the accretion rate is so low that the
radiative cooling exceeds the compressional heating.

  2. 2.

If the accretion rate is high, and if the WD is initially already cold and
relatively massive, the central region remains cold until carbon ignition
because there is not enough time for energy inflow from outer layers to heat
up the central region of the WD.

  3. 3.

If the accreting WD gets enough angular momentum from the accretion disk
materials to rotate fast and uniformly, the WD mass reaches $\sim
1.43~{}M_{\odot}$ (Benvenuto et al., 2015), which is very close to
$M_{\rm{Ch}}$, without igniting carbon due to the centrifugal force. If such a
rotating WD loses angular momentum, e.g., via magnetic winds, the WD contracts
and ignites carbon at rather high central densities, Then carbon burning grows
into thermonuclear runaway at $\rho_{c}=5-6\times$ 109 g cm-3 as calculated by
Benvenuto et al. (2015).

###  2.4 Near-$M_{\rm{Ch}}$ WD Progenitors: Ignition

The ignition offset inside the near-$M_{\rm{Ch}}$ WD progenitor plays an
important role in the level of neutronization in the SN Ia yields. The
initialization of the ignition flame in the convective core of a
near-$M_{\rm{Ch}}$ WD is stochastic in nature, with ignition offset going out
to 100 km with an expectation value of 50 km, as shown with 3D simulations by
Zingale et al. (2011).

With a single ignition bubble, Fisher & Jumper (2015) analytically showed that
an ignition originating beyond 20 km from the center is buoyancy-driven and
burns only a small fraction of the fuel as compared to a central ignition,
which originates within 20 km of the center of the WD. This smaller energy
release in the buoyancy-driven ignition leads to a smaller degree of pre-
expansion, which results in the production of more neutronized isotopes during
the detonation phase as compared to a centrally ignited deflagration flame,
and thus leading to a higher ratio of 57/56.

The effect of ignition offset can be seen in the DDT models presented by Dave
et al. (2017) in Figure 2, where the models with central density
$\rho_{c}=6\times 10^{9}$ g cm-3 and C/O = 30/70 are ignited centrally, i.e.,
with offset 0 km and others with a large offset of 200 km. The models with 0
km offset produce a lesser ratio of 57/56 as compared to models with offset
200km, which suggests a high degree of energy release during the deflagration
phase in the centrally ignited models. This higher energy release leads to a
higher degree of pre-expansion and less neutronization during the detonation
phase as compared to the high offset models.

##  3 Simulation Models and Observations

The following section describes the various near-$M_{\rm{Ch}}$ and
sub-$M_{\rm{Ch}}$ simulation models. The models presented vary greatly – both
in terms of the explosion channels simulated as well as in the numerical
methods employed. The WDs also vary in the stellar progenitor metallicity and
their masses, or equivalently their central densities. The detonation
scenarios, primary WD central densities, and stellar progenitor metallicities
are presented in Table 3.

We also present the estimates of 56Ni and 57Ni/56Ni for the five SNe Ia,
listed in the Table 2 and a brief summary of how they were estimated from the
late-time light curves.

###  3.1 Simulations Models

Historically, near-$M_{\rm Ch}$ progenitors have been the favored model to
explain SNe Ia. But recent observational evidence - absence of a main-sequence
or red giant star in pre-explosion (Li et al., 2011; Kelly et al., 2014; Graur
et al., 2014; Graur & Woods, 2019; Do et al., 2021) and post-explosion data
(Maoz & Mannucci, 2008; Edwards et al., 2012; Hernández et al., 2012;
Kerzendorf et al., 2012; Schaefer & Pagnotta, 2012; Kerzendorf et al., 2014),
among others have led to a fall in favour of this model in the astrophysics
community. In recent years, the sub-$M_{Ch}$ progenitor model has been gaining
in popularity because the explosion of sub-$M_{Ch}$ WD can produce the light
curves and spectra of normal SNe Ia (Sim et al., 2010; Maoz et al., 2014).
Moreover, the recent discovery of three hypervelocity WDs in the second
release of the Gaia dataset (Shen et al., 2018b) supports the detonation of a
sub-$M_{\rm Ch}$ WD. Moreover, the lack of hydrogen in the nebular phases
pointed against the near-$M_{\rm Ch}$ progenitor Leonard (2007), but recent
observations have found evidence for H-alpha lines in the late nebular phases
for sub-luminous fast-declining SN Ia (Kollmeier et al., 2019; Vallely et al.,
2019; Prieto et al., 2020; Elias-Rosa et al., 2021). Despite the fall in
favour of the near-$M_{\rm Ch}$ channel, a few systems like SNR 3C 397
(Yamaguchi et al., 2015; Ohshiro et al., 2021; Zhou et al., 2021) suggest
strong observational evidence of a near-$M_{\rm Ch}$ progenitor.

We consider two SD explosion models:

  * •

The deflagration to detonation transition (DDT) presented by Dave et al.
(2017), Leung & Nomoto (2018), Seitenzahl et al. (2013a) and Ohlmann et al.
(2014). In the DDT model, the WD accretes from a main-sequence or a red giant
star and increases its mass near the Chandrasekhar limit (Leung & Nomoto,
2018). The growth in mass raises the temperature and pressure near the core,
which ignites a deflagration flame near the center of the WD, which in the
process of rising to the surface transitions to a supersonic detonation and
leads to a SN Ia.

  * •

The gravitational confined detonation (GCD) presented by Dave et al. (2017)
and Seitenzahl et al. (2016). In the GCD model, the deflagration doesn’t
transition to detonation, but rather rises to the surface and wraps around the
WD, leading to the detonation of carbon after the merger of the deflagration
fronts coming from the opposite sides.

We also consider:

  * •

Pure detonations presented in Sim et al. (2010) and Shen et al. (2018a) \- in
which a detonation is manually ignited in a sub-$M_{\rm Ch}$ WD.

  * •

Violent mergers presented by Pakmor et al. (2010), Pakmor et al. (2012),
Kromer et al. (2013) and Kromer et al. (2016) \- where two sub-$M_{\rm Ch}$
WDs lose energy by gravitational radiation and would gradually come close
enough for the secondary to be tidally disrupted and merge with the primary.
This merger increases the density and temperature of the primary, leading to a
SN Ia.

  * •

The head-on collision of two sub-$M_{\rm Ch}$ WDs presented by Papish & Perets
(2016) \- where the eccentricity of a binary WD in a hierarchical triple
system can be driven close to unity via Kozai-Lidov mechanism(Lidov, 1962;
Kozai, 1962), leading to a head-on collision. This collision will trigger
shocks causing a thermonuclear explosion resulting in SN Ia (Katz & Dong,
2012; Kushnir et al., 2013).

  * •

The double detonation model presented by Sim et al. (2012), Zenati et al.
(2018), Perets et al. (2019) and Leung & Nomoto (2020) \- where the WDs have
helium shells on their surfaces, and the detonation of the helium layer wraps
around the primary and sends shock waves inwards, which leads to the
detonation of carbon. We have omitted the models from Leung & Nomoto (2020)
with metallicities greater than 3 Z⊙.

  * •

We have omitted the double detonation models from Sim et al. (2012) whose WD
CO progenitors with masses less than 0.58$M_{\odot}$ which yielded very low
$\rm{}^{56}Ni$ ($\sim 0.1M_{\odot}$), and therefore such models would produce
very dim transients. We note that these double detonation models had very high
mass ratios for 57/56 because of silicon burning (Woosley et al., 1973). As an
example, the model with the maximum 57/56 = 0.17, yielded M($\rm{}^{56}Ni$)=
0.00282 M⊙.

To model these different channels, the simulations usually follow a general
pipeline - creating the initial conditions of a single WD using the WD
density, temperature, composition profile either by evolving a main-sequence
star to a WD in MESA (Paxton et al., 2010; Paxton et al., 2013, 2015) or by
using an appropriate equation of state of a hydrostatic WD. These profiles are
then mapped to an Eulerian grid and evolved for that particular detonation
model with tracer particles. These tracer particles are then post-processed
through a large nuclear reaction network to get the final nucleosynthetic
yields. Channels with binary WDs start by initializing and evolving a smooth
particle hydrodynamics simulation (SPH) of the system. The first snapshot that
meets a certain condition (for example a threshold temperature is reached) is
mapped onto the grid to follow the detonation fronts and tracer particles.

The Eulerian grid codes used in the simulation models are FLASH (Fryxell et
al., 2000; Dubey et al., 2009; Dubey et al., 2014), LEAFS (Reinecke et al.,
1998; Reinecke et al., 2002), PROMETHEUS (Fryxell et al., 1989), and the code
presented by Leung et al. (2015). The smooth particle hydrodynamics code in
the merger models is GADGET-2 (Springel, 2005). The nuclear reaction network
codes used for post-processing are TORCH (Timmes, 1999), MESA’s nuclear
reaction module (Paxton et al., 2010; Paxton et al., 2013, 2015) and the code
presented in Travaglio et al. (2004).

[t]  Model-Dataset Explosion Mechanism $\rho_{c}$ (g cm-3) $Z/Z_{\odot}$ Dave
et al, 2017 near-$M_{\rm Ch}$-GCD $2.2\times 10^{9}-6\times 10^{9}$ 0-3 Dave
et al, 2017 near-$M_{\rm Ch}$-DDT $6\times 10^{9}$ 0-3 Leung+Nomoto, 2018
near-$M_{\rm Ch}$ DDT $1\times 10^{9}-5\times 10^{9}$ 0-5 HESMA1 near-$M_{\rm
Ch}$ GCD $2.9\times 10^{9}$ 1 HESMA1 near-$M_{\rm Ch}$ DDT $1.0\times
10^{9}-5.5\times 10^{9}$ 0.01-1 HESMA1 sub-$M_{\rm Ch}$ Pure DET $1.0\times
10^{7}-2.0\times 10^{8}$ 0-3 HESMA1 sub-$M_{\rm Ch}$ Merger $2\times
10^{6}-1.4\times 10^{7}$ 0.01 - 1 HESMA1 sub-$M_{\rm Ch}$ Double Det $2\times
10^{6}-1.4\times 10^{7}$ 0.01 - 1 Shen et al, 2018 sub-$M_{\rm Ch}$ Pure DET
N/A 0-2 Papish+Perets, 2015 sub-$M_{\rm Ch}$ Collision $3.4\times
10^{6}-1.0\times 10^{7}$ 0 Zenati+Perets, 2019 sub-$M_{\rm Ch}$ Hybrid Double
DET $3.5\times 10^{6}-1.2\times 10^{7}$ 1 Leung+Nomoto, 2020 sub-$M_{\rm Ch}$
Double DET $1.7\times 10^{7}-1.5\times 10^{8}$ 0-5

Table 1: Table of Simulation Runs.

  * 1

HESMA - Heidelberg Supernova Model Archive - Seitenzahl et al. (2013a)
,Ohlmann et al. (2014) , Seitenzahl et al. (2016) , Sim et al. (2010) ,
Marquardt et al. (2015) , Pakmor et al. (2010) , Pakmor et al. (2012) , Kromer
et al. (2013) , Kromer et al. (2016) Sim et al. (2012)

###  3.2 Observations

Determining the 57Ni/56Ni ratios from the late-time light curves is quite
challenging because multiple processes can produce late-time light curves
tails. Processes like freeze-out(Fransson & Jerkstrand, 2015; Kerzendorf et
al., 2017), partial positron trapping(Dimitriadis et al., 2017; Kushnir &
Waxman, 2020), delayed decay of 56Ni or luminosity from a surviving
companion(Shen & Schwab, 2017) can mimic 57Co.

Light echoes have been observed to slow down the light curves of several SNe
Ia starting at 500 days (e.g., Schmidt et al. (1994); Sparks et al. (1999);
Cappellaro et al. (2001); Quinn et al. (2006); Wang et al. (2008); Drozdov et
al. (2015)), and so could mimic the late-time light curves of the SNe
discussed here. This source of systematic uncertainty can be removed by
studying the color evolution of the SNe. Because light echoes are created by
the dispersion of the original SN light by dust sheets in space, the light
that reaches our apertures will be bluer than the original SN light (e.g.,
Patat (2005); Rest et al. (2012)). Furthermore, the light echo will be
dominated by the light of the SN at peak, when it was most luminous. Hence,
the colors of a SN contaminated by a light echo will be bluer than the colors
of the SN at maximum light. The colors of SNe 2012cg, ASASSN-14lp, 2014J, and
2015F did not obey this rule; they were found to be bluer than peak in B-V but
redder than peak in V-R (Graur et al., 2018a; Graur et al., 2018b; Graur,
2018; Yang et al., 2018a). The color evolution of these SNe, as well as SN
2013aa (Jacobson-Galán et al., 2018) was also shown to be very similar to that
of the well-studied SN 2011fe, which exploded in a very clean environment and
has exhibited no light echoes (Graham et al., 2015; Shappee et al., 2017). We
note that SN 2014J has known, resolved light echoes (Crotts, 2015; Yang et
al., 2017), but Yang et al. (2018a) and Graur (2018) both note that the PSF of
the SN within the aperture used for photometry is not contaminated by an
unresolved echo.

Values of 56Ni and 57Ni/56Ni for the five SNe Ia, listed in Table 2, are
presented in the following subsection. Childress et al. (2015) estimated the
mass of 56Ni for thirty one SNe Ia, including the five used in this work by
looking at the [Co III] nebular spectral line of multiple SNe Ia, and showing
that the line flux is proportional to the mass of 56Co at that time. Childress
et al. (2015) note that [Co III] line flux measurement could be affected by
contamination from neighboring nebular emission, undetected light echos, or
residual host galaxy light. The decay of 56Co emits positrons and gamma rays.
Under the assumption that positrons deposit their energy equally to all
species in the ejecta and the ejecta being optically thin to gamma rays,
Childress et al. (2015) show that the [Co III] line flux is proportional to
the square of the mass of cobalt at that time. The estimated M(56Ni) for SN
2011fe, SN 2012cg, SN 2014J, SN 2015F and SN 2013aa are
0.500${}^{+0.026}_{-0.026}$, 0.477${}^{+0.048}_{-0.048}$,
0.838${}^{+0.176}_{-0.176}$, 0.4${}^{+0.05}_{-0.05}$ and
0.732${}^{+0.151}_{-0.151}$ M⊙ respectively.

The determination of 57Ni/56Ni were made as follows: firstly the pseudo-
bolometric light curves of the events in the optical range were calculated,
using the observed fluxes from the HST. These pseudo-bolometric fluxes were
then converted to luminosities by taking the distances to the SNe Ia or the
host galaxy into account. Finally, the Bateman equation (Seitenzahl et al.,
2014) is fit to the observed luminosities, to get the estimates for the yields
and ratios of the nuclide. We would like to make a note that these
observations are prone to systematic uncertainties, like missing multi-band
observations for some SNe events, the uncertainties of the distance to the SNe
would introduce errors to the calculated luminosities.

The pseudo-bolometric light curves of SNe 2012cg, 2013aa, ASASSN-14lp, 2014J,
and 2015F cover the optical wavelength range of 3500 to 10000 A. Shappee et
al. (2017) had near-infrared (NIR) as well as optical observations of SN
2011fe, which they used to construct a pseudo-bolometric light curve spanning
4000 to 17000 A. Graur et al. (2020) showed that even though normal SNe Ia
have varying light curve widths in the H band, their light curves converge
after 500 days and follow the same decline rate as in the optical. The SNe in
our sample cover a similar range of peak light curve widths (compare Graur et
al. (2018b) and Graur et al. (2020)). Hence, we assume that the SNe analysed
here have the same NIR light curve shape at 500-1000 days as in the optical.
After 1000 days, only SN 2011fe has NIR observations (Shappee et al., 2017).
These mirror the slow-down of the optical light curves at the same phases.
Since SN 2011fe is an archetypical normal SNe Ia and lies at the center of our
range of peak light curve widths, we assume that the rest of the SNe in our
sample had similar NIR light curves at >1000 days. This last point should be
tested by NIR observations of future, nearby SNe Ia. Thus, while some care
needs to be taken when producing pseudo-bolometric light curves in the phase
range 150-500 days, it is safe to assume that during the late times studied
here, $>$ 500 days past maximum light, the NIR and optical light curves behave
in a similar manner.

Shappee et al. (2017) used the light curve of SN 2011fe for 1840 days past
maximum light and estimated log(57Ni/56Ni) = -1.59${}^{+0.06}_{-0.07}$ and
also put an upper bound on the 55Fe/57Co $<0.22$, with 99% confidence. Using
these ratios, we place an upper bound of 55Fe/56Co $<0.006$ which we designate
as 2011fe†.

Tucker et al. (2021) made the observations for SN 2011fe for $\approx$ 2400
days past maximum light and estimated log(57Ni/56Ni) =
-1.71${}^{+0.12}_{-0.11}$ and made the first detection of 55Fe with
log(55Fe/57Ni)=-0.61${}^{+0.13}_{-0.15}$, from which we estimated 55Fe/56Ni =
0.00479${}^{+0.00195}_{-0.00208}$, noted as 2011feX.

Graur et al. (2016) used the light curve of SN 2012cg for 1055 days past
maximum light and found a mass ratio of M(57Co)/M(56Co) =
0.043${}^{+0.012}_{-0.011}$, noted as 2012cg†.

Li et al. (2019) followed the normal SN Ia 2014J to $\sim 900$ days after
maximum light, and from its late-time light curve fit for 56Ni, 57Ni, and 55Fe
to get the 57Ni/56Ni mass ratio, which they estimated to be
0.035${}^{+0.011}_{-0.011}$. They took the yields of the three isotopes as
free parameters and accounted for lepton escape to estimate M(56Ni) =
0.68${}^{+0.12}_{-0.12}$ $M_{\odot}$. Their 56Ni yields are consistent with
the estimates by Srivastav et al. (2016), Telesco et al. (2014), Churazov et
al. (2014), but are in contrast with the yield calculated by Childress et al.
(2015), which is M(56Ni) = 0.838${}^{+0.176}_{-0.176}$ M⊙. They note that
their 57Ni/56Ni estimates are half that of Yang et al. (2018b), as they did
not take lepton escape into account.

Graur et al. (2018a) fit the pseudo-bolometric light curves of SN 2011fe, SN
2012cg, SN 2014J and SN 2015F but only considered the contributions from 56Ni
and 57Ni to estimate the 57Ni/56Ni ratios as 0.043${}^{+0.004}_{-0.004}$
(2011fe∗), 0.072${}^{+0.002}_{-0.002}$ (2012cg∗), 0.129${}^{+0.021}_{-0.016}$
(2014J∗) and 0.004${{}^{+0.003}_{-0.002}}$ (2015F), respectively.

Jacobson-Galán et al. (2018) combined archival data at $<400$ days with one
epoch of HST observation at 1500 days after explosion of the over-luminous SN
2013aa. They estimated the mass of 56Ni = 0.732${}^{+0.151}_{-0.151}$ M⊙ by
fitting a straight line to the M(56Ni) values over SiFTO stretch values by
Childress et al. (2015). To estimate the mass ratio M(57Co)/M(56Co), they
fixed the estimated M(56Ni) and found a mass ratio M(57Co)/M(56Co) =
0.02${}^{+0.01}_{-0.02}$ (2013aa∗). The authors also allowed for variation in
56Ni for which they estimated the M(56Ni) = 0.631${}^{+0.015}_{-0.015}$ M⊙,
and from their estimates of M(57Co) and M(56Co) we compute the M(57Co)/M(56Co)
= 0.0095${}^{+0.00016}_{-0.00095}$ (2013aa†).

[ht]

Table 2: Event Types

Event |  |  | M(56Ni) |  |  57Ni/56Ni |  | Wavelength Range (Å)
---|---|---|---|---|---|---|---
2011fe |  |  | 0.500${}^{+0.026}_{-0.026}$ | 0.043${}^{+0.004*}_{-0.004}$ | 0.026${}^{+0.0037\dagger}_{-0.0044}$ | 0.0195${}^{+0.0054X}_{-0.0049}$ | 4000-17000
2012cg |  |  | 0.477${}^{+0.048}_{-0.048}$ | 0.072${}^{+0.002*}_{-0.002}$ | 0.043${}^{+0.011\dagger}_{-0.012}$ |  | 3500-10000
2014J |  |  | 0.838${}^{+0.176}_{-0.176}$ | 0.129${}^{+0.021*}_{-0.016}$ | 0.035${}^{+0.011\dagger}_{-0.011}$ |  | 3500-10000
2015F |  |  | 0.400${}^{+0.05}_{-0.05}$ | 0.004${}^{+0.003}_{-0.002}$ |  |  | 3500-10000
2013aa |  |  | 0.732${}^{+0.151}_{-0.151}$ | 0.02${}^{+0.01*}_{-0.02}$ | 0.0095${}^{+0.0016\dagger}_{-0.0095}$ |  | 3500-10000

  * •

All of the SNe are normal, but that 2015F is subluminous and 2013aa is
overluminous. Events with multiple estimates of 57Ni/56Ni are marked with
superscript dagger($\dagger$) or asterisk($*$) or cross($X$).
2014J† \- Li et al. (2019) 2011fe† \- Shappee et al. (2017), 2012cg† \- Graur
et al. (2016),
2014J∗, 2011fe∗, 2012cg∗, 2015F - Graur et al. (2018a)
2013aa∗-Fit 3 , 2013aa†-Fit 2 Jacobson-Galán et al. (2018)
2011feX \- Tucker et al. (2021)

##  4 Results

###  4.1 WD Progenitors: sub-$M_{\rm Ch}$ and near-$M_{\rm Ch}$

To place constraints on the simulation models, we plot the abundance ratio of
the 57 isobar chain to the 56 isobar chain
($\rm{{}^{57}Ni}$+$\rm{{}^{57}Co}$)/($\rm{{}^{56}Ni}$+$\rm{{}^{56}Co}$)
against $M(\rm{{}^{56}Ni})$ in Figure 4. The plot shows the near-$M_{\rm Ch}$
models in red and the sub-$M_{\rm Ch}$ models in blue, with different shades
representing their metallicities. As we go up the
($\rm{{}^{57}Ni}$+$\rm{{}^{57}Co}$)/($\rm{{}^{56}Ni}$+$\rm{{}^{56}Co}$) axis
the degree of neutronized nuclides increases while going towards the
increasing $\rm{}^{56}Ni$ yields on the x-axis we would expect brighter
events. Further, we expect to see more neutronized isotopes in a much denser
environment because of higher electron captures, which are the signature of
dense near-$M_{\rm Ch}$ progenitors. From Figure 4, we see that the
near-$M_{\rm Ch}$ models have on average much higher 57/56 ratios due to their
higher central densities as compared to the sub-$M_{\rm Ch}$ progenitors. An
overlap region exists between the two channels, where sub-$M_{\rm Ch}$ WD
models with supersolar progenitor metalicities can have neutronizations
comparable to those of near-$M_{\rm Ch}$ models. This shows how the late-time
light curves can be used as a new probe to constrain the parameters of the
progenitors of SNe Ia.

Figure 4: Plot showing the ratio
($\rm{{}^{57}Ni}$+$\rm{{}^{57}Co}$)/($\rm{{}^{56}Ni}$+$\rm{{}^{56}Co}$)
against $M(\rm{{}^{56}Ni})$. The horizontal blue line marks the upper bound on
57/56 ratio among all the sub-$M_{\rm{Ch}}$ models, and the red horizontal
line shows the lower bound on all the near-$M_{\rm{Ch}}$ models considered in
this work. The near-$M_{\rm Ch}$ models(red) have on average much higher 57/56
ratios due to their higher central densities as compared to the sub-$M_{\rm
Ch}$(blue) progenitors. An overlap region exists between the two channels,
where sub-$M_{\rm Ch}$ WD models with supersolar progenitor metalicities can
have neutronizations comparable to those of near-$M_{\rm Ch}$ models. Only
sub-$M_{\rm{Ch}}$ models are consistent with event SN 2015F and SN 2011feX.
Only near-$M_{\rm{Ch}}$ models are consistent with SN 2012cg∗. The events SN
2012cg†, 2012cg∗, 2011fe†, 2014J†, 2013aa∗, 2013aa† match with both the
near-$M_{\rm{Ch}}$ and sub-$M_{\rm{Ch}}$ models.  Figure 5:
($\rm{{}^{55}Ni}$+$\rm{{}^{55}Co}$+$\rm{{}^{55}Fe}$)/($\rm{{}^{56}Ni}$+$\rm{{}^{56}Co}$)
versus $M(\rm{{}^{56}Ni}$) for near-$M_{\rm{Ch}}$(red) and
sub-$M_{\rm{Ch}}$(blue) models, with the event 2011feX, 2014J† and upper
bounds on the events 2011fe†, 2013aa∗ and 2013aa†. The black and yellow dashed
lines show the region where the yields of $\rm{}^{55}Mn$ are in the range of
0.002 M⊙ to 0.005 M⊙; most of the near-$M_{\rm{Ch}}$ models fall in this
range. The mass ratio of Mn/Fe of the SN remnant SNR 3C 397 is shown as the
orange bar, which supports a near-$\rm M_{Ch}$ progenitor for this SNR. To
validate the potential of using the late-time light curve channel, we also
plot the values for Mn/Fe from De los Reyes et al. (2020) for the dSph
Sculptor galaxy, shown as the green bar.

Further, one can also note the relation of metallicities in the Dave et al.
(2017) DDTs and GCDs, in which more metal-rich WDs have a higher 57/56 ratio
and lower yields of 56Ni. Similar trends can be seen in Leung & Nomoto (2018)
DDTs and Shen et al. (2018a) pure detonations of sub-$M_{\rm Ch}$ WDs.

We qualitatively compare the five near by observations with the models by
plotting them in Figure 4. We note that the events 2011fe, 2012cg and 2014J
each have multiple estimates, the first one calculated by fitting the Bateman
equation using the nuclides $\rm{}^{56}Ni$, $\rm{}^{57}Ni$ and $\rm{}^{55}Co$,
which are all marked with daggers, while the other estimates were computed
using only the nuclides $\rm{}^{56}Ni$ and $\rm{}^{57}Ni$ are marked with
asterisk. SN 2011fe has a third estimate from very late-time light curve
marked as cross. We can see that only SN 2015F and SN 2013aa† are in the “only
sub-$\rm M_{Ch}$ region” i.e. in the region where 57/56 $\lessapprox$ 0.02,
while 2012cg∗ and 2014J∗ lie in the “near-$\rm M_{Ch}$” region, i.e. where
57/56 $\gtrapprox$ 0.05. SN 2012cg†, 2011fe∗, 2011fe†, 2011feX, 2014J† and
2013aa∗ lie in the overlap region.

To quantitatively compare the simulation models with the observations we
perform a $\chi^{2}$ test on each of the observations to find the set of
models which are consistent with them using:

$\begin{split}\chi^{2}=\\{\frac{(X^{57/56}_{i,obs}-X^{57/56}_{j,model})^{2}}{\sigma_{i,57/56}^{2}}+\frac{(X^{56}_{i,obs}-X^{56}_{j,model})^{2}}{\sigma_{i,56}^{2}}\\\
+\frac{(X^{55/56}_{i,obs}-X^{55/56}_{j,model})^{2}}{\sigma_{i,55/56}^{2}}\\}\end{split}$
(1)

and summarize the results in Table 3. We use the upper-tail one-sided
$\chi^{2}$ test with the null hypothesis defined as “the event being
consistent with a simulation model”, and select a significance value of 0.001.
An event being consistent with a model means that the model can explain the
observed late-time light curve of the SN event.

From the analysis we see SN 2012cg∗ is consistent with 4 near-$M_{\rm Ch}$ DDT
and GCD models of Dave et al. (2017). SN 2011feX is only consistent with
sub-$M_{\rm Ch}$ models of Shen et al. (2018a), Leung & Nomoto (2020), Perets
et al. (2019) and Papish & Perets (2016). SN 2015F can only we explained by
sub-$M_{\rm Ch}$ collision models of Papish & Perets (2016) and sub-$M_{\rm
Ch}$ models of Perets et al. (2019).

We find that the observational event SN 2011fe∗ is consistent with 11
near-$M_{\rm Ch}$ DDT and GCD models of Seitenzahl et al. (2013a), Dave et al.
(2017) and Leung & Nomoto (2018) and 6 sub-$M_{\rm Ch}$ double detonation
models of Leung & Nomoto (2020). SN 2011fe† is consistent with 20 sub-$M_{\rm
Ch}$ models from Leung & Nomoto (2020), Shen et al. (2018a), Perets et al.
(2019), Papish & Perets (2016) and Sim et al. (2010) and also consistent with
4 near-$M_{\rm Ch}$ models of Dave et al. (2017) and Seitenzahl et al.
(2013a).

For SN 2012cg†, 7 near-$M_{\rm Ch}$ DDT and GCD models of Seitenzahl et al.
(2013a), Dave et al. (2017) and Leung & Nomoto (2018) and 7 sub-$M_{\rm Ch}$
double detonation and pure detonation models of Leung & Nomoto (2020) and Sim
et al. (2010) can explain the late-time light curve.

We find SN 2014J† to be consistent with 25 near-$M_{\rm Ch}$ models of HESMA
(Seitenzahl et al., 2013a; Ohlmann et al., 2014; Seitenzahl et al., 2016),
Dave et al. (2017) and Leung & Nomoto (2018), and also to be consistent with
17 sub-$M_{\rm Ch}$ double detonation models from (Leung & Nomoto, 2018) and a
violent merger model of Pakmor et al. (2012). We note that SN 2014J∗ is not
consistent with any simulation model.

For SN 2013aa∗, 16 near-$M_{\rm Ch}$ DDT and GCD models of HESMA (Seitenzahl
et al., 2013a; Ohlmann et al., 2014; Seitenzahl et al., 2016), Dave et al.
(2017) and Leung & Nomoto (2018) and 18 sub-$M_{\rm Ch}$ double detonation and
pure detonation models of Leung & Nomoto (2020), Shen et al. (2018a) and Sim
et al. (2010) are consistent with the measurement. For SN 2013aa†, 23
sub-$M_{\rm Ch}$ progenitor models of Perets et al. (2019), Leung & Nomoto
(2020) and Shen et al. (2018a) and 9 near-$M_{\rm Ch}$ DDT models of
HESMA(Seitenzahl et al. (2013a), Ohlmann et al. (2014)), Dave et al. (2017)
and Leung & Nomoto (2018) are consistent with the measurement.

We can also explore constraints on the events using the very long-lived $A=55$
isobar chains, which have half-lives of $\sim$ 3 yrs. However, due to the
difficulty of observing SNe Ia at very late times, estimates of
$M(\rm{{}^{55}Co}$)/$M(\rm{{}^{56}Co}$) have been made for only a few events.
Shappee et al. (2017) put an upper limit on the mass ratio of 55/57 chain for
SN 2011fe† ($6.6\times 10^{-3}$) with 99% confidence and Jacobson-Galán et al.
(2018) estimated M(55Fe) using which we put an upper bound for 55/56 for SN
2013aa† $(1.42\times 10^{-3}$) and SN 2013aa∗ ($1.045\times 10^{-5}$). Tucker
et al. (2021) estimated log(55Fe/57Ni)=-0.61${}^{+0.13}_{-0.15}$. To use this
channel, we plot
($\rm{{}^{55}Ni}$+$\rm{{}^{55}Co}$+$\rm{{}^{55}Fe}$)/($\rm{{}^{56}Ni}$+$\rm{{}^{56}Co}$)
versus $M(\rm{{}^{56}Ni}$) for the simulation models in Figure 5. The black
and yellow dashed lines show the region where the yields of $\rm{}^{55}Mn$ are
in the range of 0.002 M⊙ to 0.005 M⊙, and we see that most of the
near-$M_{Ch}$ models fall in this range. We also show the upper bounds for SN
2011fe†, SN 2013aa† and SN 2013aa∗. We don’t use the 55/56 bound for SN
2013aa∗ in our analysis because the value and error bar on 55/56 are very
close to zero.

Further, we include results of the mass ratio of Mn/Fe of the remnant SNR 3C
397, which indicates a near-$\rm M_{Ch}$ progenitor. Yamaguchi et al. (2015)
and Ohshiro et al. (2021) determined the ratio of Mn/Fe for 3C 397 to be in
the range 0.018–0.033 and 0.042-0.060, respectively, which is shown as a
shaded yellow bar in Figure 5. Zhou et al. (2021) also determined the mass
ratio of Mn/Fe of SNR Sgr A East to be in the range of 0.023 - 0.032, being
similar to 3C 397, and concluded that the progenitor of Sgr A East is the
near-$\rm M_{Ch}$ white dwarf. Qualitatively looking at the Figure 5, we find
that SNR 3C 397 likely had a near-$\rm M_{Ch}$ progenitor, which is consistent
with the constraint put by Yamaguchi et al. (2015); Ohshiro et al. (2021);
Zhou et al. (2021). To further validate the potential of using the late-time
light curve channel, we plot the the estimates for Mn/Fe from De los Reyes et
al. (2020) for the dSph Sculptor galaxy, where they estimated the Mn/Fe
contributions from SNe Ia and used that to put constraints on nature of the
SNe Ia progenitor. The authors found that the major contribution to Mn/Fe in
the dSph Sculptor galaxy was from sub-$\rm M_{Ch}$ progenitors. We plot their
estimates in our Figure 5 as the green shaded region, and also find their
estimates to be consistent with sub-$\rm M_{Ch}$ progenitors. We also note
that De los Reyes et al. (2020) noted that some fraction (> 20 %) of near-$\rm
M_{Ch}$ progenitors are still needed to produce the observed yield. They
further suggested that near-$\rm M_{Ch}$ progenitors may become the dominant
channel of SNe Ia at later times in a chemical evolution of the dSph Sculptor
galaxy.

Table 3: Models Consistent with Events.

[ht]  Observation Model Number of Models Z⊙ Parameters 2011fe∗ near-$M_{\rm
Ch}$ DDT - Leung+Nomoto, 2018 4 2.0-3.0 $\rho$: 3e9 - 5e9 g cm-3, c3 flame -
C/O:1 near-$M_{\rm Ch}$ DDT - HESMAa-Seitenzahl et al.(2013b) 4 1.0 $\rho$:
1e9 - 2.9e9 g cm-3 , N:100, 150, 200, 300 near-$M_{\rm Ch}$ DDT,GCD - Dave et
al, 2017 3 0.0-3.0 $\rho$: 6.0e9 g cm-3, bubble offset: 0km - 200km, bubble
radius: 8km - 100km, C/O:30/70 sub-$M_{\rm Ch}$ Double DET - Leung+Nomoto,
2020 6 1.0-3.0 $\rho$: 4.33e7 - 6.17e7 g cm-3, Helium Detonation: Spherical,
Bubble, Ring. 2011fe† sub-$M_{\rm Ch}$ Double DET - Leung+Nomoto, 2020 8
1.0-3.0 $\rho$: 2.26e7 - 6.17e7 g cm-3, Helium Detonation: Spherical, Bubble,
Ring. sub-$M_{\rm Ch}$ Pure DET - Shen et al, 2018 3 0.5-2.0 C/O = 50/50,
$WD_{\rm{Mass}}$: 1.0 sub-$M_{\rm Ch}$ Hybrid Double DET - Zenati+Perets, 2019
7 1.0 MHybrid: 0.53 M⊙ \- 0.68 M⊙ , MCO = 0.8 M⊙ \- 1.0 M⊙ sub-$M_{\rm Ch}$
Collision - Papish+Perets, 2015 1 0.0 MWD1: 0.6 M⊙ CO + 0.01 M⊙ He -
$\rho_{1}$: 3.4e6 g cm-3 \- MWD2: 0.6 M⊙ CO + 0.01 M⊙ He - $\rho_{2}$: 3.4e6
g/cm3 sub-$M_{\rm Ch}$ Pure DET - HESMAa-Sim et al. (2010) 1 3.0 $\rho$:
4.15e7 g cm-3, MWD: 1.06 M⊙ near-$M_{\rm Ch}$ DDT - HESMAa-Seitenzahl et
al.(2013b) 3 1.0 $\rho$: 1.0e9 - 2.0e9 g cm-3 , N:100, 150, 300 near-$M_{\rm
Ch}$ DDT - Dave et al, 2017 1 1.5 $\rho$: 6.0e9 g cm-3, bubble offset:0 km -
bubble radius:100km - C/O:30/70 2011feX sub-$M_{\rm Ch}$ Pure DET - Shen et
al, 2018 3 0.5-2.0 C/O = 50/50, $WD_{\rm{Mass}}$: 1.0 sub-$M_{\rm Ch}$ Double
DET - Leung+Nomoto, 2020 3 1.0 $\rho$: 2.26e7 - 4.33e7 g cm-3, Helium
Detonation: Spherical, Bubble, Ring. sub-$M_{\rm Ch}$ Hybrid Double DET -
Zenati+Perets, 2019 9 1.0 MHybrid: 0.41 M⊙ \- 0.73 M⊙ , MCO = 0.7 M⊙ \- 1.0 M⊙
sub-$M_{\rm Ch}$ Collision - Papish+Perets, 2015 2 0.0 MWD1: 0.6 M⊙ CO +
0.0-0.01 M⊙ He - $\rho_{1}$: 3.4e6 g cm-3 \- MWD2: 0.6 M⊙ CO + 0.0-0.01 M⊙ He
- $\rho_{2}$: 3.4e6 g/cm3 2012cg∗ near-$M_{\rm Ch}$ DDT,GCD - Dave et al, 2017
4 0.0-3.0 $\rho$: 6.0e9 g cm-3, bubble offset:200 km - bubble radius:8km -
C/O:30/70. 2012cg† near-$M_{\rm Ch}$ DDT - Leung+Nomoto, 2018 3 2.0-3.0
$\rho$: 5.0e9 g cm-3, c3 flame - C/O:1. near-$M_{\rm Ch}$ DDT - HESMAa \-
Seitenzahl et al.(2013b) 3 1.0 $\rho$: 2.9.0e9 g cm-3, N: 150,200,300.
near-$M_{\rm Ch}$ DDT - Dave et al, 2017 1 3.0 $\rho$: 6.0e9 g cm-3, bubble
offset:0km, bubble radius:100km, C/O:30/70. sub-$M_{\rm Ch}$ Double DET -
Leung+Nomoto, 2020 6 1.0-3.0 $\rho$: 2.26e7 - 6.17e7 g cm-3, Helium
Detonation: Spherical, Bubble, Ring. sub-$M_{\rm Ch}$ Pure DET - HESMAa-Sim et
al. (2010) 1 3.0 $\rho$: 4.15e7 g cm-3, MWD: 1.06 M⊙ 2014J∗ No Models are
Consistent with the Measurement 2014J† near-$M_{\rm Ch}$ DDT,GCD - HESMAa 7
0.01-1.0 $\rho$: 2.9e9 - 5.5e9 g cm-3, DDT Models N:40,100 , GCD model Offset:
200 km. near-$M_{\rm Ch}$ DDT - Leung+Nomoto, 2018 13 0.0-3.0 $\rho$: 1.0e9 -
5.0e9 g cm-3, c3 flame - C/O:1. near-$M_{\rm Ch}$ DDT - Dave et al, 2017 5
0.0-3.0 $\rho$: 2.2e9 - 6.0e9 g cm-3, bubble offset:0km -100 km, bubble
radius: 8km - 100km, C/O:30/70 , 50/50. sub-$M_{\rm Ch}$ Double DET -
Leung+Nomoto, 2020 16 0.0-3.0 $\rho$: 3.21e7 - 6.17e7 g cm-3, Helium
Detonation: Spherical, Bubble, Ring. sub-$M_{\rm Ch}$ Merger - Pakmor et al,
2012b 1 1.0 MWD1: 0.9 M⊙, MWD2: 1.1 M⊙ 2015F sub-$M_{\rm Ch}$ Collision -
Papish+Perets, 2015 2 0.0 MWD1: 0.6 M⊙ CO + 0.0-0.01 M⊙ He - $\rho_{1}$: 3.4e6
g cm-3 \- MWD2: 0.6 M⊙ CO + 0.0-0.01 M⊙ He - $\rho_{2}$: 3.4e6 g/cm3
sub-$M_{\rm Ch}$ Hybrid Double DET - Zenati+Perets, 2019 2 1.0 MHybrid: 0.41
M⊙ \- 0.48 M⊙ , MCO = 0.6 M⊙ \- 0.7 M⊙ 2013aa∗ near-$M_{\rm Ch}$ DDT -
Leung+Nomoto, 2018 7 0.0-1.0 $\rho$: 1.0e9 - 5.0e9 g cm-3, c3 flame - C/O:1.
near-$M_{\rm Ch}$ DDT,GCD - HESMAa 7 0.01-1.0 $\rho$: 2.9e9 - 5.5e9 g cm-3,
DDT Models N:20,40,100 , GCD model Offset: 200 km. near-$M_{\rm Ch}$ GCD -
Dave et al, 2017 2 0.0-1.5 $\rho$: 6.0e9 g cm-3, bubble offset: 100 km, bubble
radius: 16km, C/O:50/50. sub-$M_{\rm Ch}$ Double DET - Leung+Nomoto, 2020 14
0.0-1.0 $\rho$: 4.33e7 - 9.19e7 g cm-3, Helium Detonation: Spherical, Bubble,
Ring. sub-$M_{\rm Ch}$ Pure DET - Shen et al, 2018 3 0.5-2.0 C/O = 50/50, MWD:
1.1 M⊙. sub-$M_{\rm Ch}$ Pure DET - HESMAa \- Sim et al. (2010) 1 0.0 $\rho$:
7.9e7 g cm-3, MWD: 1.15 M⊙ 2013aa† sub-$M_{\rm Ch}$ DDT - Leung+Nomoto, 2020
11 0.0-1.0 $\rho$: 3.21e7 - 6.17e7 g cm-3, Spherical Det, Single He
detonation, Ring Detonation. sub-$M_{\rm Ch}$ Hybrid Double DET -
Zenati+Perets, 2019 9 1.0 MHybrid: 0.53 M⊙ \- 0.74 M⊙ , MCO = 0.8 M⊙ \- 1.0 M⊙
sub-$M_{\rm Ch}$ Pure DET - Shen et al, 2018 3 0.0-1.0 C/O = 50/50, MWD: 1.0
M⊙. near-$M_{\rm Ch}$ DDT - HESMAa 2 0.01-1.0 $\rho$: 2.9e9 g cm-3, DDT Models
N:100. near-$M_{\rm Ch}$ DDT - Leung+Nomoto, 2018 6 0.0-0.5 $\rho$: 3.0e9 -
5.0e9 g cm-3, c3 flame - C/O:1. near-$M_{\rm Ch}$ DDT - Dave et al, 2017 1 0.0
$\rho$: 6.0e9 g cm-3, bubble offset: 0 km, bubble radius: 100km, C/O:30/70.

  * •

HESMA - Heidelberg Supernova Model Archive
2014J† \- Li et al. (2019), 2011fe† \- Shappee et al. (2017), 2012cg† \- Graur
et al. (2016),
2014J∗, 2011fe∗, 2012cg∗, 2015F - Graur et al. (2018a)
2013aa∗-Fit 3 , 2013aa†-Fit 2 Jacobson-Galán et al. (2018)
2011feX \- Tucker et al. (2021)

###  4.2 Stellar Progenitor Metallcities

As discussed in section 2.2, the yield of 56Ni drops with increased
metallicity. This effect can be seen in the sub-$M_{\rm{Ch}}$ models presented
by Shen et al. (2018a), and also in the near-$M_{\rm{Ch}}$ models presented by
Dave et al. (2017) and Leung & Nomoto (2018).

In section 2.2, we also discussed how the ratio 57/56 increases with an
increasing metallicity. This effect can be seen in the models presented by
Shen et al. (2018a), Dave et al. (2017) and Leung & Nomoto (2018) in Figure 1
– 4.

In our analysis, we find that the models consistent with the events SN
2011fe∗, 2011fe†, 2012cg∗ and 2014J† have metallcities ranging from 0.0 Z⊙ \-
3.0 Z⊙, while SN 2012cg† has metallicitiy in the range 1.0 Z⊙ \- 3.0 Z⊙. The
consistent models for SN 2011feX and 2013aa∗ have metallcities in the range
0.0 Z⊙ \- 2.0 Z⊙, while models consistent with SN 2013aa${\dagger}$ and 2015F
have metallcities in 0.0 Z⊙ \- 1.0 Z⊙.

We note that all the near-$M_{\rm Ch}$ simulation models that are consistent
with the observational events have metallicities in the range of 0.0 Z⊙ to 3.0
Z⊙. This is in contrast with the work presented by Martínez-Rodríguez et al.
(2017), who find the metallicity of the WD stellar progenitors of SNR 3C 397
and G337.2 to be 5.4 Z⊙. These very high metallicity values might be because
the models to which they compared their observations, only explored a range of
metallicities values, and the central densities of the WD stellar progenitors
were not explored (2 - 3 $\times 10^{9}$ g cm-3). A higher central density
stellar progenitor WD could suggest a high degree of neutronization for SNR 3C
397 as suggested by Dave et al. (2017). We note that the metallicity values
found by Martínez-Rodríguez et al. (2017) for SN remnants Tycho, N103B, and
Kepler were below 2Z⊙, and some of the models consistent with the events have
similar metallicities. These metallicity values between $\sim$ 1.5Z⊙ to $\sim$
3Z⊙ are on the higher end of the Galactic metallicity sprectrum (Rix & Bovy,
2013).

###  4.3 Near-$M_{\rm{Ch}}$ WD Progenitors: Central Densities

From the $\chi^{2}$ analysis, we find that the event SN 2012cg*, which has a
high 57/56 ratio ($>$ 0.05), points to a near-$M_{\rm{Ch}}$ WD progenitor,
with a very high central density. The models which are consistent with SN
2012cg* have central densities $\approx 6\times 10^{9}$ g cm-3. This result is
in contrast with the work presented by Iwamoto et al. (1999), which argued
that the central densities of an average progenitor WD in the near-$M_{\rm
Ch}$ channel should be $\lessapprox 2\times 10^{9}$ g cm-3, so as to avoid the
large ratios of 54Cr/56Fe and 50Ti/56Fe. Moreover, we note that the very high
fraction of 57/56 and 55/56 for the consistent models of the event SN 2012cg∗
suggests a near-$M_{\rm{Ch}}$ WD progenitor similar to that of SNR 3C 397.
Moreover, we find that SN 2015F is consistent with low central density models
of the order $\approx 10^{6}$ g cm-3 where the primary WD have masses in the
range from 0.6 M⊙ \- 0.7 M⊙.

##  5 Conclusions

We have looked at a set of five nearby SN Ia events (SN 2011fe, SN 2012cg, SN
2013aa, SN 2014J, 2015F) for which the ratio of 57Ni/56Ni was determined from
their late-time light curves. We perform a $\chi^{2}$ test to constraint the
parameters of these events with an extensive suite of simulation models in the
near-$M\rm_{ch}$ and sub-$M\rm_{ch}$ progenitors. We find the low and solar
metallicity sub-$M_{\rm Ch}$ collision models of Papish & Perets (2016) and
Perets et al. (2019) to be consistent with the event SN 2015F. Moreover, we
find the high central density DDT and GCD models of Dave et al. (2017) to be
consistent with the measurements made by Graur et al. (2018a) for SN 2012cg.
Models from Papish & Perets (2016), Shen et al. (2018a), Perets et al. (2019)
and Leung & Nomoto (2020) in the double detonation and pure detonation of a
sub-$M_{\rm Ch}$ were found to be consistent with the very late-time
measurements by Tucker et al. (2021) for the event SN 2011fe.

The measurements of SN 2011fe by Graur et al. (2016) and Shappee et al. (2017)
can be explained by both the near-$M_{\rm Ch}$ and sub-$M_{\rm Ch}$
progenitors. For SN 2014J, no model agrees with the measurement made by Graur
(2018), but both near-$M_{\rm Ch}$ and sub-$M_{\rm Ch}$ progenitor models are
consistent with the measurement made by Li et al. (2019). Finally, both the
single and double degenerate progenitor models are consistent with the
measurements of SN 2012cg made by Graur et al. (2016) and of SN 2013aa made by
Jacobson-Galán et al. (2018).

We note that all simulation models have used different nucleosynthesis
networks (Timmes, 1999; Travaglio et al., 2004; Paxton et al., 2010) to
calculate the yields of the 55, 56, and 57 chain nuclides, with differing
nuclear reaction network sizes. The determination of formal uncertainties is a
challenging task even for simpler 1D spherically-symmetric stellar
evolutionary models (Fields et al., 2016). As such, the formal uncertainties
for multidimensional explosion models are generally left unreported.
Nonetheless, because widely varying methods produce similar outcomes for
broadly-similar initial conditions of sub-Mch versus near-Mch progenitors, the
basic conclusions arrived at here are likely robust.

In a recent paper, Polin et al. (2021) calculated the synthetic spectra of the
nebular phases for sub-$M_{ch}$ double detonation models and found it
difficult to explain the excessive flux corresponding to $[\rm
Ca\,\textsc{ii}]$ emissions in the models as compared to the spectra of SN
2011fe, a normal SN Ia. However, they found a good fit for SN 1999by, which is
a 1991bg-like sub-luminous SN Ia. These results are consistent with our, where
2015F, which is a sub-luminous SN Ia, is consistent with a sub-$M_{\rm{Ch}}$
progenitor.

We conclude that normal SNe Ia cannot be explained by a single progenitor or
explosion channel and both the single degenerate and the double degenerate
channels are at play here. This result is consistent with the work presented
by Flörs et al. (2020), where the authors looked at optical spectra of 58 SNe
Ia and concluded that 85% of them were in an agreement with a sub-$M_{\rm Ch}$
progenitor while 15% were in agreement with a near-$M_{\rm Ch}$ progenitor.
Our results are also in alignment with the work of Eitner et al. (2020), where
the authors compared the Galactic abundance of Mn/Fe and galactic chemical
evolution models from different sources of SNe Ia and SNe II and found a good
fit for the observations to the models with $\sim$ 75% of the SNe Ia coming
from the sub-$M_{\rm Ch}$ progenitors and $\sim$ 25% from the near-$M_{\rm
Ch}$ progenitors. On the other hand, Kobayashi et al. (2020) calculated
chemical evolution models, and from the evolutionary trends of the elemental
abundance ratios among Mn, Ni, Cr, and Fe, they concluded that the
contribution of sub-$M_{\rm Ch}$ progenitors is less than 25% in the solar
neighborhood, while it is higher in dSph. Moreover, Seitenzahl et al. (2019)
used supernova remnant tomography to put constraints on three young SNe Ia
remnants (SNR 0519-69.0, SNR 0509-67.5, and N103B) and found SNR 0519-69.0 to
be consistent with a near-$M_{\rm Ch}$ progenitor and SNR 0509-67.5 to be
consistent with a sub-$M_{\rm Ch}$ progenitor.

Finally, to put tighter constraints on the nature of the progenitors, future
observations of SNe Ia metallicity can be used as prior information to the
$\chi^{2}$ analysis. Moreover, as our knowledge of detonation physics
improves, SNe Ia simulation models with physics-informed detonation initiation
will help reduce the dimensionality of the explosion model parameter space.

## Acknowledgements

VT was supported by NASA through HST-GO-15693. OG was supported, in part, by an NSF Astronomy and Astrophysics Fellowship under award AST-1602595. RTF thanks the NASA ATP program for support under award 80NSSC18K1013. RTF also thanks the Institute for Theory and Computation at the Center for Astrophysics | Harvard & Smithsonian, and the Kavli Institute for Theoretical Physics, supported in part by the national Science Foundation under grant NSF PHY11-25915, for visiting support during which this work was undertaken. SCL acknowledges support from NASA grants HST-AR-15021.001-A and 80NSSC18K1017. KN is supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and JSPS KAKENHI Grant Numbers JP17K05382 and JP20K04024, and JP21H04499. KJS is supported by NASA through the Astrophysics Theory Program (NNX17AG28G).

## Data Availability

The data behind Figure 4 and Figure 5 is available here:
https://doi.org/10.5281/zenodo.5797233 .

## References

  * Baade & Zwicky (1934) Baade W., Zwicky F., 1934, Proceedings of the National Academy of Science, 20, 259
  * Benvenuto et al. (2015) Benvenuto O. G., Panei J. A., Nomoto K., Kitamura H., Hachisu I., 2015, The Astrophysical Journal Letters, 809, L6
  * Bravo (2019) Bravo E., 2019, Astronomy & Astrophysics, 624, A139
  * Bravo & Martínez-Pinedo (2012) Bravo E., Martínez-Pinedo G., 2012, Physical Review C, 85, 055805
  * Cappellaro et al. (2001) Cappellaro E., et al., 2001, The Astrophysical Journal, 549, L215
  * Chamulak et al. (2008) Chamulak D. A., Brown E. F., Timmes F. X., Dupczak K., 2008, The Astrophysical Journal, 677, 160
  * Childress et al. (2015) Childress M. J., et al., 2015, MNRAS, 454, 3816
  * Chiosi et al. (2015) Chiosi E., Chiosi C., Trevisan P., Piovan L., Orio M., 2015, MNRAS, 448, 2100
  * Churazov et al. (2014) Churazov E., et al., 2014, Nature, 512, 406
  * Crotts (2015) Crotts A. P., 2015, The Astrophysical Journal Letters, 804, L37
  * Dave et al. (2017) Dave P., Kashyap K., Fisher R., Timmes F., Townsley D. M., 2017, ApJ
  * De los Reyes et al. (2020) De los Reyes M. A., Kirby E. N., Seitenzahl I. R., Shen K. J., 2020, The Astrophysical Journal, 891, 85
  * Dimitriadis et al. (2017) Dimitriadis G., et al., 2017, Monthly Notices of the Royal Astronomical Society, 468, 3798
  * Do et al. (2021) Do A., et al., 2021, Monthly Notices of the Royal Astronomical Society
  * Drozdov et al. (2015) Drozdov D., Leising M., Milne P., Pearcy J., Riess A., Macri L., Bryngelson G., Garnavich P., 2015, The Astrophysical Journal, 805, 71
  * Drury (2012) Drury L. O. ., 2012, Astroparticle Physics, 39, 52
  * Dubey et al. (2009) Dubey A., Antypas K., Ganapathy M. K., Reid L. B., Riley K., Sheeler D., Siegel A., Weide K., 2009, Parallel Computing, 35, 512
  * Dubey et al. (2014) Dubey A., et al., 2014, International Journal of High Performance Computing Applications, 28, 225
  * Edwards et al. (2012) Edwards Z. I., Pagnotta A., Schaefer B. E., 2012, The Astrophysical Journal Letters, 747, L19
  * Eitner et al. (2020) Eitner P., Bergemann M., Hansen C. J., Cescutti G., Seitenzahl I., Larsen S., Plez B., 2020, Astronomy & Astrophysics, 635, A38
  * Elias-Rosa et al. (2021) Elias-Rosa N., et al., 2021, arXiv preprint arXiv:2106.15340
  * Elmegreen & Scalo (2004) Elmegreen B. G., Scalo J., 2004, ARA&A, 42, 211
  * Fields et al. (2016) Fields C., Farmer R., Petermann I., Iliadis C., Timmes F., 2016, The Astrophysical Journal, 823, 46
  * Fisher & Jumper (2015) Fisher R., Jumper K., 2015, The Astrophysical Journal, 805, 150
  * Flörs et al. (2020) Flörs A., et al., 2020, Monthly Notices of the Royal Astronomical Society, 491, 2902
  * Fransson & Jerkstrand (2015) Fransson C., Jerkstrand A., 2015, The Astrophysical Journal Letters, 814, L2
  * Fryxell et al. (1989) Fryxell B., Müller E., Arnett W., 1989, Max-Planck-Institut für Astrophysik, Garching
  * Fryxell et al. (2000) Fryxell B., et al., 2000, The Astrophysical Journal Supplement Series, 131, 273–334
  * Ginzburg & Syrovatskii (1964) Ginzburg V. L., Syrovatskii S. I., 1964, The Origin of Cosmic Rays
  * Graham et al. (2015) Graham M. L., Nugent P. E., Sullivan M., Filippenko A. V., Cenko S. B., Silverman J. M., Clubb K. I., Zheng W., 2015, Monthly Notices of the Royal Astronomical Society, 454, 1948
  * Graur (2018) Graur O., 2018, The Astrophysical Journal, 870, 14
  * Graur & Woods (2019) Graur O., Woods T. E., 2019, Monthly Notices of the Royal Astronomical Society: Letters, 484, L79
  * Graur et al. (2014) Graur O., Maoz D., Shara M. M., 2014, Monthly Notices of the Royal Astronomical Society: Letters, 442, L28
  * Graur et al. (2016) Graur O., Zurek D., Shara M. M., Riess A. G., Seitenzahl I. R., Rest A., 2016, The Astrophysical Journal, 819, 31
  * Graur et al. (2018a) Graur O., et al., 2018a, The Astrophysical Journal, 859, 79
  * Graur et al. (2018b) Graur O., Zurek D. R., Cara M., Rest A., Seitenzahl I. R., Shappee B. J., Shara M. M., Riess A. G., 2018b, The Astrophysical Journal, 866, 10
  * Graur et al. (2020) Graur O., et al., 2020, Nature Astronomy, 4, 188
  * Hernández et al. (2012) Hernández J. I. G., Ruiz-Lapuente P., Tabernero H. M., Montes D., Canal R., Méndez J., Bedin L. R., 2012, Nature, 489, 533
  * Iben & Tutukov (1984) Iben Jr. I., Tutukov A. V., 1984, ApJS, 54, 335
  * Iwamoto et al. (1999) Iwamoto K., Brachwitz F., Nomoto K., Kishimoto N., Umeda H., Hix W. R., Thielemann F.-K., 1999, The Astrophysical Journal Supplement Series, 125, 439
  * Jacobson-Galán et al. (2018) Jacobson-Galán W. V., Dimitriadis G., Foley R. J., Kilpatrick C. D., 2018, The Astrophysical Journal, 857, 88
  * Kashi & Soker (2011) Kashi A., Soker N., 2011, Monthly Notices of the Royal Astronomical Society, 417, 1466
  * Katz & Dong (2012) Katz B., Dong S., 2012, arXiv preprint arXiv:1211.4584
  * Kelly et al. (2014) Kelly P. L., et al., 2014, The Astrophysical Journal, 790, 3
  * Kerzendorf et al. (2012) Kerzendorf W. E., Schmidt B. P., Laird J. B., Podsiadlowski P., Bessell M. S., 2012, The Astrophysical Journal, 759, 7
  * Kerzendorf et al. (2014) Kerzendorf W. E., Childress M., Scharwächter J., Do T., Schmidt B. P., 2014, The Astrophysical Journal, 782, 27
  * Kerzendorf et al. (2017) Kerzendorf W., et al., 2017, Monthly Notices of the Royal Astronomical Society, 472, 2534
  * Kobayashi et al. (2020) Kobayashi C., Leung S.-C., Nomoto K., 2020, The Astrophysical Journal, 895, 138
  * Kollmeier et al. (2019) Kollmeier J. A., et al., 2019, Monthly Notices of the Royal Astronomical Society, 486, 3041
  * Kozai (1962) Kozai Y., 1962, The Astronomical Journal, 67, 591
  * Kromer et al. (2013) Kromer M., et al., 2013, The Astrophysical Journal Letters, 778, L18
  * Kromer et al. (2016) Kromer M., et al., 2016, MNRAS,
  * Kushnir & Waxman (2020) Kushnir D., Waxman E., 2020, Monthly Notices of the Royal Astronomical Society, 493, 5617
  * Kushnir et al. (2013) Kushnir D., Katz B., Dong S., Livne E., Fernández R., 2013, The Astrophysical Journal Letters, 778, L37
  * Leonard (2007) Leonard D. C., 2007, The Astrophysical Journal, 670, 1275
  * Leung & Nomoto (2018) Leung S.-C., Nomoto K., 2018, The Astrophysical Journal, 861, 143
  * Leung & Nomoto (2020) Leung S.-C., Nomoto K., 2020, The Astrophysical Journal, 888, 80
  * Leung et al. (2015) Leung S.-C., Chu M.-C., Lin L.-M., 2015, Monthly Notices of the Royal Astronomical Society, 454, 1238
  * Li et al. (2011) Li W., et al., 2011, Nature, 480, 348
  * Li et al. (2019) Li W., et al., 2019, The Astrophysical Journal, 882, 30
  * Lidov (1962) Lidov M. L., 1962, Planetary and Space Science, 9, 719
  * Maoz & Mannucci (2008) Maoz D., Mannucci F., 2008, Monthly Notices of the Royal Astronomical Society, 388, 421
  * Maoz et al. (2014) Maoz D., Mannucci F., Nelemans G., 2014, Annual Review of Astronomy and Astrophysics, 52, 107
  * Marquardt et al. (2015) Marquardt K. S., Sim S. A., Ruiter A. J., Seitenzahl I. R., Ohlmann S. T., Kromer M., Pakmor R., Röpke F. K., 2015, Astronomy & Astrophysics, 580, A118
  * Martínez-Rodríguez et al. (2016) Martínez-Rodríguez H., Piro A. L., Schwab J., Badenes C., 2016, The Astrophysical Journal, 825, 57
  * Martínez-Rodríguez et al. (2017) Martínez-Rodríguez H., et al., 2017, The Astrophysical Journal, 843, 35
  * Nomoto (1982a) Nomoto K., 1982a, ApJ, 253, 798
  * Nomoto (1982b) Nomoto K., 1982b, ApJ, 257, 780
  * Nomoto & Leung (2018) Nomoto K., Leung S.-C., 2018, Space Science Reviews, 214, 1
  * Nomoto et al. (1984) Nomoto K., Thielemann F.-K., Yokoi K., 1984, The Astrophysical Journal, 286, 644
  * Ohlmann et al. (2014) Ohlmann S. T., Kromer M., Fink M., Pakmor R., Seitenzahl I. R., Sim S. A., Röpke F. K., 2014, Astronomy & Astrophysics, 572, A57
  * Ohshiro et al. (2021) Ohshiro Y., et al., 2021, The Astrophysical Journal Letters, 913, L34
  * Pakmor et al. (2010) Pakmor R., Kromer M., Röpke F. K., Sim S. A., Ruiter A. J., Hillebrandt W., 2010, Nature, 463, 61
  * Pakmor et al. (2012) Pakmor R., Kromer M., Taubenberger S., Sim S. A., Röpke F. K., Hillebrandt W., 2012, ApJ, 747, L10
  * Pankey (1962) Pankey Titus J., 1962, PhD thesis, HOWARD UNIVERSITY.
  * Papish & Perets (2016) Papish O., Perets H. B., 2016, The Astrophysical Journal, 822, 19
  * Patat (2005) Patat F., 2005, Monthly Notices of the Royal Astronomical Society, 357, 1161
  * Paxton et al. (2010) Paxton B., Bildsten L., Dotter A., Herwig F., Lesaffre P., Timmes F., 2010, The Astrophysical Journal Supplement Series, 192, 3
  * Paxton et al. (2013) Paxton B., et al., 2013, The Astrophysical Journal Supplement Series, 208, 4
  * Paxton et al. (2015) Paxton B., et al., 2015, The Astrophysical Journal Supplement Series, 220, 15
  * Perets et al. (2019) Perets H. B., Zenati Y., Toonen S., Bobrick A., 2019, arXiv e-prints, p. arXiv:1910.07532
  * Phillips (1993) Phillips M. M., 1993, ApJ, 413, L105
  * Piro & Bildsten (2008) Piro A. L., Bildsten L., 2008, The Astrophysical Journal, 673, 1009
  * Polin et al. (2021) Polin A., Nugent P., Kasen D., 2021, The Astrophysical Journal, 906, 65
  * Prieto et al. (2020) Prieto J. L., et al., 2020, The Astrophysical Journal, 889, 100
  * Pskovskii (1977) Pskovskii I. P., 1977, Soviet Astronomy, 21, 675
  * Quinn et al. (2006) Quinn J. L., Garnavich P. M., Li W., Panagia N., Riess A., Schmidt B. P., Della Valle M., 2006, The Astrophysical Journal, 652, 512
  * Reinecke et al. (1998) Reinecke M., Hillebrandt W., Niemeyer J. C., Klein R., Gröbl A., 1998, arXiv preprint astro-ph/9812119
  * Reinecke et al. (2002) Reinecke M., Hillebrandt W., Niemeyer J., 2002, Astronomy & Astrophysics, 386, 936
  * Rest et al. (2012) Rest A., Sinnott B., Welch D., 2012, Publications of the Astronomical Society of Australia, 29, 466
  * Rix & Bovy (2013) Rix H.-W., Bovy J., 2013, The Astronomy and Astrophysics Review, 21, 61
  * Rust (1974) Rust B. W., 1974, Technical report, Use of supernovae light curves for testing the expansion hypothesis and other cosmological relations. Oak Ridge National Lab.(ORNL), Oak Ridge, TN (United States)
  * Schaefer & Pagnotta (2012) Schaefer B. E., Pagnotta A., 2012, Nature, 481, 164
  * Schmidt et al. (1994) Schmidt B. P., Kirshner R. P., Leibundgut B., Wells L. A., Porter A. C., Ruiz-Lapuente P., Challis P., Filippenko A. V., 1994, arXiv preprint astro-ph/9407097
  * Seitenzahl et al. (2013a) Seitenzahl I. R., et al., 2013a, MNRAS, 429, 1156
  * Seitenzahl et al. (2013b) Seitenzahl I. R., Cescutti G., Röpke F. K., Ruiter A. J., Pakmor R., 2013b, A&A, 559, L5
  * Seitenzahl et al. (2014) Seitenzahl I. R., Timmes F., Magkotsios G., 2014, The Astrophysical Journal, 792, 10
  * Seitenzahl et al. (2016) Seitenzahl I. R., et al., 2016, Astronomy & Astrophysics, 592, A57
  * Seitenzahl et al. (2019) Seitenzahl I. R., Ghavamian P., Laming J. M., Vogt F. P., 2019, Physical review letters, 123, 041101
  * Shappee et al. (2017) Shappee B. J., Stanek K., Kochanek C. S., Garnavich P., 2017, The Astrophysical Journal, 841, 48
  * Shen & Schwab (2017) Shen K. J., Schwab J., 2017, The Astrophysical Journal, 834, 180
  * Shen et al. (2018a) Shen K. J., Kasen D., Miles B. J., Townsley D. M., 2018a, The Astrophysical Journal, 854, 52
  * Shen et al. (2018b) Shen K. J., et al., 2018b, The Astrophysical Journal, 865, 15
  * Sim et al. (2010) Sim S., Röpke F., Hillebrandt W., Kromer M., Pakmor R., Fink M., Ruiter A., Seitenzahl I., 2010, The Astrophysical Journal Letters, 714, L52
  * Sim et al. (2012) Sim S. A., Fink M., Kromer M., Röpke F., Ruiter A., Hillebrandt W., 2012, Monthly Notices of the Royal Astronomical Society, 420, 3003
  * Soker (2019) Soker N., 2019, New Astronomy Reviews, 87, 101535
  * Sparks et al. (1999) Sparks W., Macchetto F., Panagia N., Boffi F., Branch D., Hazen M., Della Valle M., 1999, The Astrophysical Journal, 523, 585
  * Springel (2005) Springel V., 2005, Monthly notices of the royal astronomical society, 364, 1105
  * Srivastav et al. (2016) Srivastav S., Ninan J., Kumar B., Anupama G., Sahu D., Ojha D., Prabhu T., 2016, Monthly Notices of the Royal Astronomical Society, 457, 1000
  * Telesco et al. (2014) Telesco C. M., et al., 2014, The Astrophysical Journal, 798, 93
  * Thielemann et al. (1986) Thielemann F.-K., Nomoto K., Yokoi K., 1986, in , Nucleosynthesis and Its Implications on Nuclear and Particle Physics. Springer, pp 131–142
  * Timmes (1999) Timmes F. X., 1999, ApJS, 124, 241
  * Timmes et al. (2003) Timmes F. X., Brown E. F., Truran J. W., 2003, ApJ, 590, L83
  * Travaglio et al. (2004) Travaglio C., Hillebrandt W., Reinecke M., Thielemann F.-K., 2004, Astronomy & Astrophysics, 425, 1029
  * Tucker et al. (2021) Tucker M., Shappee B., Kochanek C., Stanek K., Ashall C., Anand G., Garnavich P., 2021, arXiv preprint arXiv:2111.01144
  * Vallely et al. (2019) Vallely P., et al., 2019, Monthly Notices of the Royal Astronomical Society, 487, 2372
  * Wang et al. (2008) Wang X., Li W., Filippenko A. V., Foley R. J., Smith N., Wang L., 2008, The Astrophysical Journal, 677, 1060
  * Webbink (1984) Webbink R. F., 1984, ApJ, 277, 355
  * Whelan & Iben (1973) Whelan J., Iben Jr. I., 1973, ApJ, 186, 1007
  * Woosley et al. (1973) Woosley S. E., Arnett W. D., Clayton D. D., 1973, Astrophysical Journal Supplement
  * Yamaguchi et al. (2015) Yamaguchi H., et al., 2015, The Astrophysical Journal Letters, 801, L31
  * Yang et al. (2017) Yang Y., et al., 2017, The Astrophysical Journal, 834, 60
  * Yang et al. (2018a) Yang Y., et al., 2018a, The Astrophysical Journal, 852, 89
  * Yang et al. (2018b) Yang Y., et al., 2018b, The Astrophysical Journal, 854, 55
  * Zenati et al. (2018) Zenati Y., Toonen S., Perets H. B., 2018, Monthly Notices of the Royal Astronomical Society, 482, 1135
  * Zhou et al. (2021) Zhou P., Leung S.-C., Li Z., Nomoto K., Vink J., Chen Y., 2021, The Astrophysical Journal, 908, 31
  * Zingale et al. (2011) Zingale M., Nonaka A., Almgren A., Bell J., Malone C., Woosley S., 2011, The Astrophysical Journal, 740, 8