# Variations of Ramanujan’s Euler Products

Shin-ya Koyama111Department of Biomedical Engineering, Toyo University, 2100
Kujirai, Kawagoe, Saitama, 350-8585, Japan. & Nobushige Kurokawa222Department
of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo,
152-8551, Japan.

###### Abstract

We study the meromorphy of various Euler products of degree two attached to
Ramanujan’s $\tau$-function.

Key Words: Euler products; Zeta functions; Ramanujan’s $\tau$-function

AMS Subject Classifications: 11M06, 11M41,

## Introduction

In 1916 Ramanujan [R] studied the Euler product associated to Ramanujan’s
$\tau$-function defined as

$\Delta(z)=e^{2\pi iz}\prod_{n=1}^{\infty}(1-e^{2\pi
inz})^{24}=\sum_{n=1}^{\infty}\tau(n)e^{2\pi inz}$

for $\Im(z)>0$. This $\Delta(z)$ is a Hecke-eigen cusp form of weight 12 for
the elliptic modular group $SL(2,{\mathbb{Z}})$. Ramanujan’s Euler product is
written in the normalized form as

$L_{\Delta}(s)=\prod_{p:\,\text{prime}}(1-a(p)p^{-s}+p^{-2s})^{-1},$

where $a(n)=\tau(n)n^{-\frac{11}{2}}.$

Ramanujan conjectured that $|a(p)|\leq 2$ for all primes $p$ and it was proved
by Deligne [D] in 1974. Hence there exists a unique $\theta(p)\in[0,\pi]$
satisfying $a(p)=2\cos(\theta(p))$.

Thus

$\displaystyle L_{\Delta}(s)$
$\displaystyle=\prod_{p:\,\text{prime}}(1-2\cos(\theta(p))p^{-s}+p^{-2s})^{-1}$
$\displaystyle=\prod_{p:\,\text{prime}}\det(1-\alpha_{\Delta}^{0}(p)p^{-s})^{-1},$

where

$\alpha_{\Delta}^{0}(p)=\left[\begin{pmatrix}e^{i\theta(p)}&0\\\
0&e^{-i\theta(p)}\end{pmatrix}\right]\in\mathrm{Conj}(SU(2)).$

More general Euler products were constructed by Serre [Se], Langlands [L] and
Tate [T] as

$L_{\Delta}(s,\mathrm{Sym}^{m})=\prod_{p:\,\text{prime}}\det(1-\mathrm{Sym}^{m}(\alpha_{\Delta}^{0}(p))p^{-s})^{-1}$

for the irreducible representation $\mathrm{Sym}^{m}:\ SU(2)\to U(m+1)$ with
$m=0,1,2,3,...$. Quite recently, Newton-Thorne [NT1, NT2] proved that there
exists an automorphic representation $\pi_{m}$ of
$GL(m+1,\mathbb{A}_{\mathbb{Q}})$ satisfying

$L(s,\pi_{m})=L_{\Delta}(s,\mathrm{Sym}^{m})$

for each $m=0,1,2,3,...$ and that $\pi_{m}$ $(m\geq 1)$ are cuspidal
corresponding to the holomorphy of $L_{\Delta}(s,\mathrm{Sym}^{m})$.

The situation is quite similar for each holomorphic Hecke eigen cusp form of
general weight for the elliptic modular group, but we restrict to the case
$\Delta$ here for simplicity.

In this paper we first study the Euler product

$Z_{m}^{\pm}(s)=\prod_{p:\,\text{prime}}(1\pm
2\cos(m\theta(p))p^{-s}+p^{-2s})^{-1}$

for $m=0,1,2,3,...$ and show the following result.

######  Theorem A.

$Z_{m}^{\pm}(s)$ has an analytic continuation to all $s\in{\mathbb{C}}$ as a
meromorphic function.

In fact, this is easily seen from Newton-Thorne [NT1, NT2] since each
$Z_{m}^{\pm}(s)$ can be written explicitly via
$L_{\Delta}(s,\mathrm{Sym}^{n})$ for $n\geq 0$.

Next, we show the converse in some sense. Actually we determine all (non-
constant) monic polynomials $f(x)\in{\mathbb{Z}}[x]$ such that

$\displaystyle Z^{\pm}(s,f)$ $\displaystyle=\prod_{p:\,\text{prime}}(1\pm
f(a(p))p^{-s}+p^{-2s})^{-1}$ $\displaystyle=\prod_{p:\,\text{prime}}(1\pm
f(2\cos(\theta(p))p^{-s}+p^{-2s})^{-1}$

are meromorphic on ${\mathbb{C}}$.

######  Theorem B.

Let $f(x)\in{\mathbb{Z}}[x]$ be a monic polynomial of degree $m\geq 1$. Then
the following properties are equivalent.

  1. (1)

$Z^{\pm}(s,f)$ is meromorphic on ${\mathbb{C}}$.

  2. (2)

$|f(x)|\leq 2$ for $-2\leq x\leq 2$.

  3. (3)

$f(x)=2T_{m}(\frac{x}{2})$. Here, $T_{m}(x)$ is the Chebyshef polynomial
defined by $T_{m}(\cos\theta)=\cos(m\theta)$.

  4. (4)

$Z^{\pm}(s,f)=Z_{m}^{\pm}(s)$.

######  Example 1.

Let

$Z^{m}(s)=\prod_{p:\,\text{prime}}(1-(a(p^{2})-m)p^{-s}+p^{-2s})^{-1}$

for $m\in{\mathbb{Z}}$. Then,

$Z^{m}(s)$ is meromorphic on ${\mathbb{C}}$ $\Longleftrightarrow$ $m=1$.

Moreover,

$\displaystyle L_{\Delta}(s,\mathrm{Sym}^{2})$
$\displaystyle=\zeta(s)\prod_{p:\,\text{prime}}(1-(a(p^{2})-1)p^{-s}+p^{-2s})^{-1}$
$\displaystyle=\prod_{p:\,\text{prime}}(1-a(p^{2})p^{-s}+a(p^{2})p^{-2s}-p^{-3s})^{-1}.$

This expression was used by Shimura [S] for the proof of the holomorphy of
$L_{\Delta}(s,\mathrm{Sym}^{2})$.

######  Example 2.

For $m\in{\mathbb{Z}}$,

$\prod\limits_{p:\,\text{prime}}(1-(a(p)-m)p^{-s}+p^{-2s})^{-1}$ is
meromorphic on ${\mathbb{C}}$ $\Longleftrightarrow$ $m=0$.

Otherwise, each $Z^{\pm}(s,f)$ has the natural boundary
$\operatorname{Re}(s)=0$ as in the following result.

######  Theorem C.

Let $f(x)\in{\mathbb{Z}}[x]$ be a non-constant monic polynomial. Then the
following properties are equivalent.

  1. (1)

$Z^{\pm}(s,f)$ is not meromorphic on ${\mathbb{C}}$.

  2. (2)

$Z^{\pm}(s,f)$ has an analytic continuation in $\operatorname{Re}(s)>0$ with
the natural boundary $\operatorname{Re}(s)=0$.

Our results come from a general meromorphy theorem for Euler products shown in
[K1, K2] assuming [NT1, NT2]. We notice that the proof is similar to the case
of meromorphy of Dirichlet series $\sum\limits_{n=1}^{\infty}a(n)^{m}n^{-s}$
and $\sum\limits_{n=1}^{\infty}a(n^{m})n^{-s}$ treated in [K1, K2]. For
$m=1,2$ they are meromorphic on ${\mathbb{C}}$ and for $m\geq 3$ they are
meromorphic in $\operatorname{Re}(s)>0$ with the natural boundary
$\operatorname{Re}(s)=0$. We refer to [KK] for an application to rigidity of
Euler products.

##  1 Proof of Theorem A

###  (1) The case $m=0$

From

$Z_{0}^{\pm}(s)=\prod_{p:\,\text{prime}}(1\pm
2p^{-s}+p^{-2s})^{-1}=\prod_{p:\,\text{prime}}(1\pm p^{-s})^{-2}$

we see that

$Z_{0}^{-}(s)=\prod_{p:\,\text{prime}}(1-p^{-s})^{-2}=\zeta(s)^{2}$

and

$Z_{0}^{+}(s)=\prod_{p:\,\text{prime}}(1+p^{-s})^{-2}=\prod_{p:\,\text{prime}}\left(\frac{1-p^{-2s}}{1-p^{-s}}\right)^{-2}=\frac{\zeta(2s)^{2}}{\zeta(s)^{2}}$

are meromorphic on ${\mathbb{C}}$.

###  (2) The case $m=1$

By identifying

$Z_{1}^{-}(s)=\prod_{p:\,\text{prime}}(1-2\cos(\theta(p))p^{-s}+p^{-2s})^{-1}=L_{\Delta}(s)$

and

$\displaystyle Z_{1}^{+}(s)$
$\displaystyle=\prod_{p:\,\text{prime}}(1+2\cos(\theta(p))p^{-s}+p^{-2s})^{-1}$
$\displaystyle=\prod_{p:\,\text{prime}}[(1+e^{i\theta(p)}p^{-s})(1+e^{-i\theta(p)}p^{-s})]^{-1}$
$\displaystyle=\prod_{p:\,\text{prime}}\left[\frac{(1-e^{2i\theta(p)}p^{-2s})(1-e^{-2i\theta(p)}p^{-2s})}{(1-e^{i\theta(p)}p^{-s})(1-e^{-i\theta(p)}p^{-s})}\right]^{-1}$
$\displaystyle=\frac{L_{\Delta}(2s,\mathrm{Sym}^{2})}{L_{\Delta}(s,\mathrm{Sym}^{1})L_{\Delta}(2s,\mathrm{Sym}^{0})},$

where

$\displaystyle L_{\Delta}(s,\mathrm{Sym}^{1})$ $\displaystyle=L_{\Delta}(s),$
$\displaystyle L_{\Delta}(s,\mathrm{Sym}^{0})$ $\displaystyle=\zeta(s),$

we know that both are meromorphic on ${\mathbb{C}}$.

###  (3) The case $m\geq 2$

First

$\displaystyle Z_{m}^{-}(s)$
$\displaystyle=\prod_{p:\,\text{prime}}[(1-e^{im\theta(p)}p^{-s})(1-e^{-im\theta(p)}p^{-s})]^{-1}$
$\displaystyle=\prod_{p:\,\text{prime}}\left[\frac{(1-e^{im\theta(p)}p^{-s})(1-e^{i(m-2)\theta(p)}p^{-s})\cdots(1-e^{-im\theta(p)}p^{-s})}{(1-e^{i(m-2)\theta(p)}p^{-s})\cdots(1-e^{-i(m-2)\theta(p)}p^{-s})}\right]^{-1}$
$\displaystyle=\frac{L_{\Delta}(s,\mathrm{Sym}^{m})}{L_{\Delta}(s,\mathrm{Sym}^{m-2})}$

is a meromorphic function on ${\mathbb{C}}$ by means of [BLGHT, NT1, NT2].

Secondly

$\displaystyle Z_{m}^{+}(s)$
$\displaystyle=\prod_{p:\,\text{prime}}[(1+e^{im\theta(p)}p^{-s})(1+e^{-im\theta(p)}p^{-s})]^{-1}$
$\displaystyle=\prod_{p:\,\text{prime}}\left[\frac{(1-e^{2im\theta(p)}p^{-2s})(1-e^{-2im\theta(p)}p^{-2s})}{(1-e^{im\theta(p)}p^{-s})(1-e^{-im\theta(p)}p^{-s})}\right]^{-1}$
$\displaystyle=\frac{L_{\Delta}(2s,\mathrm{Sym}^{2m})}{L_{\Delta}(2s,\mathrm{Sym}^{2m-2})}\frac{L_{\Delta}(s,\mathrm{Sym}^{m-2})}{L_{\Delta}(s,\mathrm{Sym}^{m})}$

is meromorphic on ${\mathbb{C}}$ using [BLGHT, NT1, NT2]. ∎

##  2 Meromorphy of Euler products

We recall the meromorphy result of [K2, §3 Theorem 7] needed in this paper.
Take the triple

$E=(P({\mathbb{Z}}),\ SU(2),\ \alpha_{\Delta}^{0}),$

where $P({\mathbb{Z}})$ is the set of prime numbers, and the map

$\alpha_{\Delta}^{0}:\ P({\mathbb{Z}})\to\mathrm{Conj}(SU(2))$

defined as

$\alpha_{\Delta}^{0}(p)=\left[\begin{pmatrix}e^{i\theta(p)}&0\\\
0&e^{-i\theta(p)}\end{pmatrix}\right].$

We identify $[0,\pi]$ and $\mathrm{Conj}(SU(2))$ via the following map

$\begin{matrix}[0,\pi]&\longrightarrow&\mathrm{Conj}(SU(2)).\\\
\rotatebox{90.0}{$\in$}&&\rotatebox{90.0}{$\in$}\\\
\theta&\longmapsto&\left[\begin{pmatrix}e^{i\theta}&0\\\
0&e^{-i\theta}\end{pmatrix}\right]\end{matrix}$

Notice that the normalized measure on $\mathrm{Conj}(SU(2))=[0,\pi]$ coming
from the normalized Haar measure of $SU(2)$ is given by
$\frac{2}{\pi}\sin^{2}\theta d\theta$ used in the Sato-Tate conjecture proved
in [BLGHT].

Main results of [K1, K2] give the criterion of the meromorphy of Euler product

$L(s,E,H)=\prod_{p:\,\text{prime}}H_{\alpha_{\Delta}^{0}(p)}(p^{-s})^{-1}$

for each polynomial

$H(T)\in 1+TR(SU(2))[T]$

when specialized to the present situation, where $R(SU(2))$ is the ring of
virtual characters of $SU(2)$ and

$H_{\alpha_{\Delta}^{0}(p)}(T)\in 1+T{\mathbb{C}}[T]$

denotes the polynomial obtained by taking values of coefficients.

We first report Theorems 1 and 2. It is remarkable that all the long-standing
assumptions made in [K1, K2] for
$E=(P({\mathbb{Z}}),SU(2),\alpha_{\Delta}^{0})$ which were necessary for
obtaining such theorems are now finally proved by Newton-Thorne [NT1, NT2].

######  Theorem 1.

Let $H(T)\in 1+TR(SU(2))[T]$ be a polynomial of degree $n$. Then the following
properties are equivalent.

  1. (1)

$H(T)$ is unitary in the sense that there exist functions $\varphi_{j}:\
{\mathbb{R}}\to{\mathbb{R}}$ $(j=1,2,...,n)$ such that

$H_{\theta}(T)=\prod_{j=1}^{n}(1-e^{i\varphi_{j}(\theta)}T)$

for $\theta\in[0,\pi]=\mathrm{Conj}(SU(2))$.

  2. (2)

$L(s,E,H)$ is meromorphic on ${\mathbb{C}}$.

######  Theorem 2.

Let $H(T)\in 1+TR(SU(2))[T]$. Then the following properties are equivalent.

  1. (1)

$H(T)$ is not unitary.

  2. (2)

$L(s,E,H)$ is meromorphic in $\operatorname{Re}(s)>0$ with the natural
boundary.

Here we prove a concrete example of degree two for applications in this paper.

######  Theorem 3.

Let $H(T)=1\pm hT+T^{2}\in 1+TR(SU(2))[T]$. Then the following properties are
equivalent.

  1. (1)

$H(T)$ is unitary.

  2. (2)

$|h(\theta)|\leq 2$ for all $\theta\in[0,\pi]$.

  3. (3)

$L(s,E,H)$ is meromorphic on ${\mathbb{C}}$.

Proof. Since $(1)\Longleftrightarrow(3)$ is implied in Theorem 1, it suffices
to show $(1)\Longleftrightarrow(2)$.

$(1)\Longrightarrow(2)$: From

$H_{\theta}(T)=(1-e^{i\varphi(\theta)}T)(1-e^{-i\varphi(\theta)}T)$

for a function $\varphi:\ {\mathbb{R}}\to{\mathbb{R}}$, we have

$|h(\theta)|=|2\cos(\varphi(\theta))|\leq 2$

for $\theta\in[0,\pi]$.

$(2)\Longrightarrow(1)$: From $|h(\theta)|\leq 2$ we have

$H_{\theta}(T)=\left(1\pm\frac{h(\theta)+i\sqrt{4-h(\theta)^{2}}}{2}T\right)\left(1\pm\frac{h(\theta)-i\sqrt{4-h(\theta)^{2}}}{2}T\right)$

and

$\left|\frac{h(\theta)\pm i\sqrt{4-h(\theta)^{2}}}{2}\right|=1.$

∎

##  3 Proofs of Theorems B and C

From Theorems 1, 2 and 3 we see that Theorems B and C are proved by the
following result.

######  Theorem 4.

For a monic polynomial $f(x)\in{\mathbb{Z}}[x]$ of degree $m\geq 1$, the
following conditions are equivalent.

  1. (1)

$|f(x)|\leq 2$ for $-2\leq x\leq 2$.

  2. (2)

$f(x)=2T_{m}(\frac{x}{2})$.

Actually, let

$H(T)=1\pm f(2\cos\theta)T+T^{2}\in 1+TR(SU(2))[T],$

then we see the following equivalence:

$\displaystyle L(s,E,H)\text{ is meromorphic on }{\mathbb{C}}$
$\displaystyle\overset{\text{Th }1}{\Longleftrightarrow}\quad H(T)\text{ is
unitary}$ $\displaystyle\overset{\text{Th
}3}{\Longleftrightarrow}\quad|f(2\cos\theta)|\leq 2\text{ for all
}\theta\in[0,\pi]$ $\displaystyle\Longleftrightarrow\quad|f(x)|\leq 2\text{
for }-1\leq x\leq 1$ $\displaystyle\overset{\text{Th
}4}{\Longleftrightarrow}\quad f(x)=2T_{m}(\frac{x}{2})$
$\displaystyle\Longleftrightarrow\quad L(s,E,H)=\prod_{p}(1\pm
2\cos(m\theta(p))p^{-s}+p^{-2s})^{-1}.$

We remark that this also gives a second proof of Theorem A.

Proof of Theorem 4. Let

$g(x)=\frac{1}{2^{m}}f(2x)\in{\mathbb{R}}[x].$

This is a monic polynomial of degree $m$. Then, (1) is equivalent to the
following condition:

$|g(x)|\leq\frac{1}{2^{m-1}}\text{ for }-1\leq x\leq 1.$

Such a monic polynomial $g(x)$ is uniquely determined as
$\frac{1}{2^{m-1}}T_{m}(x)$ by the famous theorem of Chebyshef. Hence
$f(x)=2T_{m}\left(\frac{x}{2}\right).$ ∎

## References


# A JWST Near- and Mid-Infrared Nebular Spectrum of the Type Ia Supernova
2021aefx

Lindsey A. Kwok Department of Physics and Astronomy, Rutgers, the State
University of New Jersey,
136 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA  Saurabh W. Jha
Department of Physics and Astronomy, Rutgers, the State University of New
Jersey,
136 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA  Tea Temim Department
of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ
08540-7219, USA  Ori D. Fox Space Telescope Science Institute, 3700 San Martin
Drive, Baltimore, MD 21218-2410, USA  Conor Larison Department of Physics and
Astronomy, Rutgers, the State University of New Jersey,
136 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA  Yssavo Camacho-Neves
Department of Physics and Astronomy, Rutgers, the State University of New
Jersey,
136 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA  Max J. Brenner Newman
Department of Physics and Astronomy, Rutgers, the State University of New
Jersey,
136 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA  Justin D. R. Pierel
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD
21218-2410, USA  Ryan J. Foley Department of Astronomy and Astrophysics,
University of California, Santa Cruz, CA 95064-1077, USA  Jennifer E. Andrews
Gemini Observatory/NSF’s NOIRLab, 670 North A‘ohoku Place, Hilo, HI
96720-2700, USA  Carles Badenes Department of Physics and Astronomy and
Pittsburgh Particle Physics, Astrophysics and Cosmology Center (PITT PACC),
University of Pittsburgh, 3941 O’Hara Street, Pittsburgh, PA 15260, USA
Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB),
Martí Franqués 1, E08028 Barcelona, Spain  Barnabas Barna Department of
Experimental Physics, Institute of Physics, University of Szeged, Dóm tér 9,
Szeged, 6720 Hungary  Astronomical Institute, Academy of Sciences, Bočnì II
1401, Prague, Czech Republic  K. Azalee Bostroem Steward Observatory,
University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA
Maxime Deckers School of Physics, Trinity College Dublin, The University of
Dublin, Dublin 2, Ireland  Andreas Flörs GSI Helmholtzzentrum für
Schwerionenforschung, Planckstraße 1, 64291 Darmstadt, Germany  Peter
Garnavich Department of Physics and Astronomy, University of Notre Dame, Notre
Dame, IN 46556, USA  Melissa L. Graham DiRAC Institute, Department of
Astronomy, University of Washington, Box 351580, U.W., Seattle, WA 98195, USA
Or Graur Institute of Cosmology & Gravitation, University of Portsmouth,
Dennis Sciama Building, Burnaby Road, Portsmouth PO1 3FX, UK  Department of
Astrophysics, American Museum of Natural History, Central Park West and 79th
Street, New York, NY 10024-5192, USA  Griffin Hosseinzadeh Steward
Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ
85721-0065, USA  D. Andrew Howell Las Cumbres Observatory, 6740 Cortona Drive,
Suite 102, Goleta, CA 93117-5575, USA  Department of Physics, University of
California, Santa Barbara, CA 93106-9530, USA  John P. Hughes Department of
Physics and Astronomy, Rutgers, the State University of New Jersey,
136 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA  Joel Johansson Oskar
Klein Centre, Department of Physics, Stockholm University, Albanova University
Center, SE-106 91, Stockholm, Sweden  Sarah Kendrew European Space Agency,
Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218,
USA  Wolfgang E. Kerzendorf Department of Physics and Astronomy, Michigan
State University, East Lansing, MI 48824, USA  Department of Computational
Mathematics, Science, and Engineering, Michigan State University, East
Lansing, MI 48824, USA  Keiichi Maeda Department of Astronomy, Kyoto
University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, 606-8502. Japan  Kate
Maguire School of Physics, Trinity College Dublin, The University of Dublin,
Dublin 2, Ireland  Curtis McCully Las Cumbres Observatory, 6740 Cortona Drive,
Suite 102, Goleta, CA 93117-5575, USA  Department of Physics, University of
California, Santa Barbara, CA 93106-9530, USA  John T. O’Brien Department of
Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
Armin Rest Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218-2410, USA  Department of Physics and Astronomy, The Johns
Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA  David
J. Sand Steward Observatory, University of Arizona, 933 North Cherry Avenue,
Tucson, AZ 85721-0065, USA  Melissa Shahbandeh Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD 21218-2410, USA  Louis-Gregory
Strolger Space Telescope Science Institute, 3700 San Martin Drive, Baltimore,
MD 21218-2410, USA  Tamás Szalai Department of Experimental Physics, Institute
of Physics, University of Szeged, Dóm tér 9, Szeged, 6720 Hungary  ELKH-SZTE
Stellar Astrophysics Research Group, H-6500 Baja, Szegedi út, Kt. 766, Hungary
Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Eötvös
Loránd Research Network (ELKH), Konkoly-Thege Miklós út 15-17, 1121 Budapest,
Hungary  Chris Ashall Department of Physics, Virginia Tech, Blacksburg, VA
24061, USA  E. Baron Homer L. Dodge Department of Physics and Astronomy,
University of Oklahoma, 440 W. Brooks, Rm 100, Norman, OK 73019-2061, USA
Hamburger Sternwarte, Gojenbergsweg 112, D-21029 Hamburg, Germany  Chris R.
Burns Observatories of the Carnegie Institute for Science, 813 Santa Barbara
Street, Pasadena, CA 91101-1232, USA  James M. DerKacy Department of Physics,
Virginia Tech, Blacksburg, VA 24061, USA  Tyco Mera Evans Department of
Physics, Florida State University, 77 Chieftan Way, Tallahassee, FL
32306-4350, USA  Alec Fisher Department of Physics, Florida State University,
77 Chieftan Way, Tallahassee, FL 32306-4350, USA  Lluís Galbany Institute of
Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, E-08193
Barcelona, Spain  Institut d’Estudis Espacials de Catalunya, Gran Capità, 2-4,
Edifici Nexus, Desp. 201, E-08034 Barcelona, Spain  Peter Hoeflich Department
of Physics, Florida State University, 77 Chieftan Way, Tallahassee, FL
32306-4350, USA  Eric Hsiao Department of Physics, Florida State University,
77 Chieftan Way, Tallahassee, FL 32306-4350, USA  Thomas de Jaeger Institute
for Astronomy, University of Hawai‘i, 2680 Woodlawn Drive, Honolulu, HI
96822-1839, USA  Emir Karamehmetoglu Department of Physics and Astronomy,
Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark  Kevin
Krisciunas Department of Physics and Astronomy, Texas A&M University, 4242
TAMU, College Station, TX 77843, USA  George P. and Cynthia Woods Mitchell
Institute for Fundamental Physics & Astronomy, College Station, TX 77843, USA
Sahana Kumar Department of Physics, Florida State University, 77 Chieftan Way,
Tallahassee, FL 32306-4350, USA  Jing Lu Department of Physics, Florida State
University, 77 Chieftan Way, Tallahassee, FL 32306-4350, USA  Justyn Maund
Department of Physics and Astronomy, University of Sheffield, Hicks Building,
Hounsfield Road, Sheffield S3 7RH, U.K.  Paolo A. Mazzali Astrophysics
Research Institute, Liverpool John Moores University, Liverpool, L3 5RF, UK
Max-Planck Institute for Astrophysics, Garching, Germany  Kyle Medler
Astrophysics Research Institute, Liverpool John Moores University, Liverpool,
L3 5RF, UK  Nidia Morrell Carnegie Observatories, Las Campanas Observatory,
Casilla 601, La Serena, Chile  Mark. M. Phillips Carnegie Observatories, Las
Campanas Observatory, Casilla 601, La Serena, Chile  Benjamin J. Shappee
Institute for Astronomy, University of Hawai‘i, 2680 Woodlawn Drive, Honolulu,
HI 96822-1839, USA  Maximilian Stritzinger Department of Physics and
Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark
Nicholas Suntzeff Department of Physics and Astronomy, Texas A&M University,
4242 TAMU, College Station, TX 77843, USA  George P. and Cynthia Woods
Mitchell Institute for Fundamental Physics & Astronomy, College Station, TX
77843, USA  Charles Telesco Department of Astronomy, University of Florida,
Gainesville, FL 32611 USA  Michael Tucker Center for Cosmology and
Astroparticle Physics, The Ohio State University, 191 West Woodruff Ave,
Columbus, OH 43215, USA  Lifan Wang Department of Physics and Astronomy, Texas
A&M University, 4242 TAMU, College Station, TX 77843, USA  George P. and
Cynthia Woods Mitchell Institute for Fundamental Physics & Astronomy, College
Station, TX 77843, USA

###### Abstract

We present JWST near- and mid-infrared spectroscopic observations of the
nearby normal Type Ia supernova SN 2021aefx in the nebular phase at $+255$
days past maximum light. Our Near Infrared Spectrograph (NIRSpec) and Mid
Infrared Instrument (MIRI) observations, combined with ground-based optical
data from the South African Large Telescope (SALT), constitute the first
complete optical$+$NIR$+$MIR nebular SN Ia spectrum covering 0.3–14 $\mu$m.
This spectrum unveils the previously unobserved 2.5$-$5 $\mu$m region,
revealing strong nebular iron and stable nickel emission, indicative of high-
density burning that can constrain the progenitor mass. The data show a
significant improvement in sensitivity and resolution compared to previous
Spitzer MIR data. We identify numerous NIR and MIR nebular emission lines from
iron-group elements as well as lines from the intermediate-mass element argon.
The argon lines extend to higher velocities than the iron-group elements,
suggesting stratified ejecta that are a hallmark of delayed-detonation or
double-detonation SN Ia models. We present fits to simple geometric line
profiles to features beyond 1.2 $\mu$m and find that most lines are consistent
with Gaussian or spherical emission distributions, while the [Ar III] 8.99
$\mu$m line has a distinctively flat-topped profile indicating a thick
spherical shell of emission. Using our line profile fits, we investigate the
emissivity structure of SN 2021aefx and measure kinematic properties.
Continued observations of SN 2021aefx and other SNe Ia with JWST will be
transformative to the study of SN Ia composition, ionization structure,
density, and temperature, and will provide important constraints on SN Ia
progenitor and explosion models.

Supernovae (1668), Type Ia supernovae (1728), White dwarf stars (1799)

††facilities: JWST, SALT††software: Astropy (Astropy Collaboration et al.,
2018), Matplotlib (Hunter, 2007), NumPy (Oliphant, 2006), PyRAF (Science
Software Branch at STScI, 2012), PySALT (Crawford et al., 2010), dust
extinction (Gordon et al., 2022), jdaviz (Developers et al., 2022), jwst
(Bushouse et al., 2022), UltraNest (Buchner, 2021; Buchner et al., 2022)

published in ApJ Letters

††thanks: LSSTC Catalyst Fellow††thanks: CCAPP Fellow

##  1 Introduction

Type Ia supernovae (SN Ia) play an important role in astrophysics and
cosmology, yet we still lack a detailed understanding of their progenitor
systems and explosion physics. Nebular phase spectroscopy at late times
(beyond about 100 days past maximum light; Bowers et al., 1997; Branch et al.,
2008; Silverman et al., 2013; Friesen et al., 2014; Black et al., 2016)
reveals the SN ejecta when they have expanded, allowing us to see to the
innermost material. The observed flux is dominated by optically-thin
forbidden-line emission that directly probes the composition, density,
temperature, and ionization structure of the ejecta, constraining models of
thermonuclear explosions of a white dwarf (for a review, see Jerkstrand,
2017).

At early times, most SN Ia flux is at optical wavelengths, but as the ejecta
fade and cool to the nebular phase, the near- and mid-infrared (NIR and MIR)
comprise a large fraction of the emission (Axelrod, 1980; Fransson &
Jerkstrand, 2015). Nebular spectra have been obtained for hundreds of SNe Ia
in the optical but far fewer in the ground-accessible NIR windows. There are
only three SNe Ia to date with published nebular spectra in the MIR: one epoch
each of SN 2003hv and SN 2005df, observed with Spitzer ($\lambda=$ 5–15 $\mu$m
at spectral resolution R $\sim$ 90; Gerardy et al. 2007) and four epochs of SN
2014J, observed from the ground with Gran Telescopio Canarias (GTC)
($\lambda=$ 8–13 $\mu$m with R $\sim$ 60; Telesco et al. 2015) at phases of
$+$57, $+$81, $+$108, and $+$137 days. Atmospheric absorption and sky
background limit ground-based capabilities at these wavelengths. Spitzer was
pushed to its sensitivity limits and useful observations of these three SNe Ia
were only possible because they were nearby ($d\sim$ 3.5 Mpc for SN 2014J
(Dalcanton et al., 2009; Goobar et al., 2014), and $d\lesssim$ 20 Mpc for SN
2003hv and SN 2005df (Gerardy et al., 2007)).

Nebular phase observations in the NIR and MIR provide unique and powerful
constraints on models, including the density-dependent nucleosynthesis of
intermediate mass elements, radioactive iron-group elements, and stable iron-
group elements (Gerardy et al., 2007; Diamond et al., 2018; Dhawan et al.,
2018; Hoeflich et al., 2021). The JWST Near Infrared Spectrograph (NIRSpec;
Jakobsen et al., 2022) and Mid Infrared Instrument (MIRI; Rieke et al., 2015;
Wright et al., 2015) provide access to a wider range of elemental and ionic
species than the optical. Lines are also typically less blended in the
infrared, making it easier to derive line fluxes and abundance estimates, as
well as to infer the geometry of the emission region from the line profile
shape (e.g., Jerkstrand, 2017). Infrared spectra can also show evidence of
dust formation in the SN ejecta and polycyclic aromatic hydrocarbon (PAH) line
features that reveal the local and galactic environment (Tielens, 2005; Wang,
2005; Rho et al., 2008; Johansson et al., 2017).

JWST, with its wider NIR and MIR wavelength coverage, better spectral
resolution, and enormously increased sensitivity compared to previous
facilities, will be transformative to our understanding of SNe Ia. Here we
present the first NIR $+$ MIR SN Ia spectrum from JWST, of SN 2021aefx,
covering 0.6–14 $\mu$m. Our data include the previously unobserved 2.5–5
$\mu$m region and, combined with ground-based optical data, create a
continuous optical $+$ NIR $+$ MIR SN Ia spectrum. This observation, taken as
part of JWST Cycle 1 GO program 2072 “See Through Supernovae” (PI: S. W. Jha),
is the initial epoch of the first SN in a program to build a legacy, reference
sample of JWST NIR $+$ MIR nebular spectra of 9 white-dwarf (thermonuclear)
SNe over three cycles. SN 2021aefx will also be the target of two epochs of
data from the JWST Cycle 1 GO program 2114 (PI: C. Ashall). Combined, these
programs will observe SN 2021aefx in four epochs, providing the most
comprehensive time series of nebular IR spectra for a SN Ia.

SN 2021aefx is a “normal” (see e.g. Blondin et al., 2012) SN Ia that was
discovered within hours of explosion by the Distance Less Than 40 Mpc (DLT40)
survey (Tartaglia et al., 2018) on 2021 November 11.3 UT at
$\alpha=04\textsuperscript{h}19\textsuperscript{m}53\fs 400$ and
$\delta=-54\arcdeg 56^{\prime}53\farcs 09$ (J2000) (Hosseinzadeh et al.,
2022). It is located in the nearby galaxy NGC 1566 with a distance of 18.0
$\pm$ 2.0 Mpc ($\mu=$ 31.28 $\pm$ 0.23 mag; Sabbi et al., 2018) and a redshift
of $z=0.005017$ (Allison et al., 2014), making it a bright target for our
first observation with JWST.

SN 2021aefx peaked at an apparent magnitude of B $=$ 11.7 mag ($M_{B}=-19.6$
mag; Hosseinzadeh et al., 2022) and exhibited an exceptionally high silicon
velocity ($v\simeq$ 30,000 km s-1) in the earliest spectrum (Bostroem et al.,
2021). High-cadence intra-night observations of SN 2021aefx were carried out
by the Precision Observations of Infant Supernova Explosions (POISE; Burns et
al., 2021; Ashall et al., 2022) and the DLT40 surveys, and additional multi-
wavelength follow-up photometry revealed an early light-curve excess, perhaps
attributable to interaction between the ejecta and a nondegenerate companion
star, interaction with circumstellar material, or the effect of an unusual
nickel distribution (Hosseinzadeh et al., 2022). Part of the early light curve
evolution may also be explained by high and rapidly-evolving ejecta velocities
(Ashall et al., 2022). Aside from this peculiarity at the earliest epochs
(which have rarely been probed), SN 2021aefx subsequently evolved into a
normal SN Ia that would be included in cosmological samples. The evolution of
SN 2021aefx has been closely followed by ground-based observatories and has
generated significant interest in the SN community.

In Section 2 we detail our observations and data reduction; in Section 3 we
identify optical, NIR, and MIR nebular emission lines in SN 2021aefx; and in
Section 4 we present basic geometric line profile fits to the dominant
spectral features. We discuss the implications of our results and conclude in
Section 5.

Figure 1: Combined optical $+$ NIR $+$ MIR spectrum of Type Ia SN 2021aefx in
the nebular phase. The optical data are from SALT/RSS and the NIR and MIR data
were obtained with JWST/NIRSpec and JWST/MIRI, respectively. The optical flux
is calibrated to ground-based photometry and the MIRI flux is scaled to the
MIRI F1000W photometry. The NIRSpec flux is unscaled from the JWST pipeline
and matches up well to the optical and MIR. The spectrum has been dereddened
and corrected for the host-galaxy redshift. For presentation purposes, the
optical spectrum and the MIR spectrum past 12.5 $\mu$m have been rebinned to
lower resolution. The flux axis uses a non-linear (arcsinh) scale to better
show all the features across a wide range of wavelength and $F_{\nu}$. All
subsequent spectra presented in the paper use a linear flux scale.

##  2 Observations

We present the JWST spectrum of SN 2021aefx in Figure 1. We observed SN
2021aefx with both NIRSpec in the Fixed Slits (FS) Spectroscopy mode (Jakobsen
et al., 2022; Birkmann et al., 2022; Rigby et al., 2022) with the prism and
MIRI in the Low Resolution Spectroscopy (LRS) mode (Kendrew et al., 2015,
2016; Rigby et al., 2022) on 2022 August 11.7 UT at a rest-frame phase of
$+$254.9d, relative to B-band maximum (59546.54 $\pm$ 0.03 MJD; Hosseinzadeh
et al., 2022).

Our NIRSpec observations used the S200A1 (0.2″ wide $\times$ 3.3″ long) slit
with the PRISM grating and CLEAR filter and our MIRI observations used the LRS
slit with the P750L disperser. The combined wavelength coverage spans 0.6–14
$\mu$m. Details of the observation settings are given in Table 1.

Figure 2: JWST MIRI F1000W verification image showing SN 2021aefx during target acquisition, before placement on the LRS slit. Part of the central region of the host galaxy, NGC 1566, can be seen in the bottom left corner; gas and dust features can be seen in the image on the right. Cosmic rays have been removed from the image. Table 1: JWST SN 2021aefx Observation Details Setting | NIR | MIR
---|---|---
Instrument | NIRSpec | MIRI
Mode | FS | LRS
Wavelength Range | 0.6 $-$ 5.3 $\mu$m | 5 $-$ 14 $\mu$m
Slit | S200A1 (0.2″ x 3.3″) | Slit
Grating/Filter | PRISM/CLEAR | P750L
R $=\lambda/\Delta\lambda$ |  $\sim$100 |  $\sim$40 – 250
Subarray | FULL | FULL
Readout Pattern | NRSIRS2RAPID | FASTR1
Groups per Integration | 5 | 134
Integrations per Exposure | 2 | 2
Exposures/Nods | 3 | 2
Total Exposure Time | 525 s | 1493 s
Target Acq. Exp. Time | 4 s | 89 s

###  2.1 JWST Data Reduction

The data were reduced using the publicly available
“jwst”111https://github.com/spacetelescope/jwst pipeline (version 1.8.0;
Bushouse et al., 2022) routines for bias and dark subtraction, background
subtraction, flat field correction, wavelength calibration, flux calibration,
rectification, outlier detection, resampling, and spectral extraction. The
final NIRSpec “stage 3” one-dimensional (1D) spectrum extracted by the
automated pipeline, available on the Mikulski Archive for Space Telescopes
(MAST)222https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html, was of
sufficiently good quality that we did not rerun any portion of the pipeline.
Unfortunately, the MIRI stage 3 1D spectrum extracted by the automated
pipeline, available on MAST, was noisy and unsuitable because the automated
extraction aperture was not properly centered on the SN trace. Thus, we re-
extracted the spectrum by manually running stage 3 of the pipeline
(calwebb_spec3) from the stage 2 (calwebb_spec2) data products, enforcing the
correct extraction trace and aperture.

We identified an issue with the original MIRI/LRS slit wavelength calibration
from the JWST pipeline that has been noted and confirmed by others and is
under investigation (S. Kendrew & G. Sloan, private communication). Spectra of
the candidate Herbig B[e] star VFTS 822, which exhibits hydrogen emission
lines (Kalari et al., 2014), were taken as part of calibration programs JWST
Cycle 0 COM/MIRI 1259333https://www.stsci.edu/jwst/phase2-public/1259.pdf (PI:
S. Kendrew) and JWST Cycle 1 CAL/MIRI
1530444https://www.stsci.edu/jwst/phase2-public/1530.pdf (PI: S. Kendrew).
From the data publicly available on the MAST archive, we measured wavelength
centroids and uncertainties for 12 H I emission line peaks in the data and
matched them with their known rest wavelengths. We found significant offsets
between the wavelengths from the pipeline calibration and the H I emission
line wavelengths, with larger deviation at shorter wavelengths ($\sim$0.1
$\mu$m at the longest wavelengths, rising to $\sim$0.5 $\mu$m at the shortest
wavelengths). We developed a custom wavelength solution correction that
informed updates to the JWST pipeline by the MIRI Team. All MIRI/LRS slit data
on MAST have been reprocessed by the new wavelength calibration, including the
MIRI data of SN 2021aefx presented in this work. The inaccuracy in the
wavelength calibration has been reduced to $\sim$0.02 – 0.05 $\mu$m (MIRI
Team, private communication). We caution that the H I emission line peaks in
the VFTS 822 data are weak and difficult to fit; full resolution of this
wavelength calibration issue may require additional observations of other
sources.

We measured MIRI F1000W photometry of SN 2021aefx from the LRS verification
image (see Figure 2) with $F_{\nu}=$ 0.309 $\pm$ 0.010 mJy, corresponding to
17.67 $\pm$ 0.04 mag AB. The photometry was done on the F1000W data from the
JWST pipeline using a 70% encircled energy aperture radius (4.3 pixels) and
inner and outer sky radii of 6.063 and 10.19 pixels (and a corresponding
aperture correction was also applied). Integrating the MIRI spectrum of the
supernova over the F1000W passband gave a flux that agreed with the measured
photometry to within 2%. We applied a rescaling of the spectrum to match the
photometry precisely. The NIRSpec spectrum does not have similarly measured
photometry, but the pipeline spectrum matches both the optical and MIR spectra
well, so we do not adjust its flux calibration.

To correct for extinction by dust, we use the Python dust-extinction package
(v. 1.1; Gordon et al., 2022). We deredden the NIRSpec spectrum out to 1.0
$\mu$m using the F19 model from Fitzpatrick et al. (2019) and the NIRSpec
spectrum past 1.0 $\mu$m as well as the MIRI spectrum with the G21_MWavg model
from Gordon et al. (2021). We correct for both the host-galaxy extinction of
E(B$-$V)${}_{\text{host}}=$ 0.097 mag (Hosseinzadeh et al., 2022) and the
Milky Way extinction of E(B$-$V)${}_{\text{MW}}=$ 0.008 mag (Schlafly &
Finkbeiner, 2011).

###  2.2 Optical Data

We obtained an optical nebular spectrum of SN 2021aefx with the Southern
African Large Telescope (SALT) Robert Stobie Spectrograph (RSS; Smith et al.,
2006) on 2022 August 7.1 UT (rest-frame phase of $+$250.3 d) using a
1$\farcs$5 longslit and the PG0900 grating in four tilt settings with a total
exposure time of 2294 s. Using a custom pipeline based on standard Pyraf
(Science Software Branch at STScI, 2012) spectral reduction routines and the
PySALT package (Crawford et al., 2010), we reduced the optical spectrum and
removed cosmic rays, host galaxy lines and continuum, and telluric absorption.
We scaled the optical spectrum to observed contemporaneous UBgVri photometry,
obtained with the Sinistro cameras on Las Cumbres Observatory’s 1 m telescopes
(Brown et al., 2013) and reduced automatically by the BANZAI (McCully et al.,
2018) and lcogtsnpipe (Valenti et al., 2016) pipelines, using the speccal
module in the Light Curve Fitting package (Hosseinzadeh & Gomez, 2020).
Lastly, we applied a redshift correction to the host-galaxy rest frame and
dereddened using the F19 model (Fitzpatrick et al., 2019) as above.

Figure 3: Optical line identifications for SN 2021aefx; only the most dominant
species for each feature are shown. The optical spectrum (black) from SALT/RSS
on 2022 August 7 at a rest-frame phase of $+$250d is dominated by forbidden-
line emission from iron-group elements. For comparison, we plot the
JWST/NIRSpec spectrum (grey) on 2022 August 11 at $+$255d, and the optical
spectra of SN 2011fe (blue; Mazzali et al., 2015) and SN 2014J (red; Childress
et al., 2015) at similar phase ($+$258d and $+$234d, respectively). Phases are
given with respect to B-band maximum and all spectra have been dereddened.

##  3 Line Identification

We identify nebular emission lines in the optical, NIR, and MIR spectra
presented in this work using line identifications from the Atomic Line
List555https://www.pa.uky.edu/~peter/newpage/; the Atomic-ISO line
list666https://www.mpe.mpg.de/ir/ISO/linelists/FSlines.html; previous optical
line identifications by Graham et al. (2022), Tucker et al. (2022), and Wilk
et al. (2020); previous NIR line identifications by Diamond et al. (2018),
Dhawan et al. (2018), Hoeflich et al. (2021), and Mazzali et al. (2015);
previous MIR line identifications by Gerardy et al. (2007) and Telesco et al.
(2015); and optical $+$ NIR $+$ MIR lines from models by Flörs et al. (2020).

In this work, we focus on line identifications for the 2.5–5 $\mu$m region in
the NIR and the full 5$-$14 $\mu$m MIR. Candidate lines in these wavelength
regions of interest were narrowed down by selecting atomic species consistent
with SN Ia abundance models (e.g. Nomoto et al., 1984; Thielemann et al.,
1986; Fink et al., 2010; Pakmor et al., 2012; Seitenzahl et al., 2013),
predicted strength of the forbidden-line transitions, and proximity to ground-
state transitions.

Selected prominent optical lines are marked in Figure 3. The optical lines
match closely with those presented by Graham et al. (2022), Tucker et al.
(2022), and Wilk et al. (2020), and are dominated by blended emission lines
from the iron-group elements: [Fe II], [Fe III], [Co III], and [Ni II]. We
compare to an optical spectrum of SN 2011fe from Mazzali et al. (2015) using
the William Herschel Telescope (WHT) with the Intermediate-dispersion
Spectrograph and Imaging System (ISIS) and dereddened by E(B$-$V) $=$ 0.023
mag. We also compare to a spectrum of SN 2014J from Childress et al. (2015)
taken with the Keck II telescope and the DEep Imaging Multi-Object
Spectrograph (DEIMOS), dereddened by $R_{V}$ = 1.4, E(B$-$V) $=$ 1.2 mag
(Foley et al., 2014; Amanullah et al., 2014; Mazzali et al., 2015). The
optical spectrum and line identification of SN 2021aefx closely matches that
of SN 2011fe and SN 2014J, indicating that SN 2021aefx is representative of a
typical SN Ia at about $+$250d. The spectrum of SN 2011fe is slightly
blueshifted compared to SN 2021aefx.

###  3.1 NIR Emission Lines

Figure 4: NIR line identifications for SN 2021aefx; only the most dominant
species for each feature are shown. We compare the JWST/NIRSpec spectrum of SN
2021aefx (black) at $+$255d with a scaled GN/GNIRS spectrum of SN 2014J (red)
at the closest available phase of $+$289d (Diamond et al., 2018; Graur et al.,
2020) and a scaled LBT/LUCIFER JHK${}_{\text{s}}$ spectrum of SN 2011fe (blue)
at a phase of $+$234d (Mazzali et al., 2015). Nearly all emission lines are
from significantly blended iron-group elements. The 2.5$-$5 $\mu$m region
shows two strong blended features of predominantly [Fe III] 2.874 & 2.905
$\mu$m, and [Ni I] 3.120 $\mu$m & [Fe III] 3.229 $\mu$m, and several weak
features. All spectra have been dereddened. Details of the NIR lines can be
found in Table 2.

In Figure 4, we mark prominent lines in the JWST/NIRSpec spectrum of SN
2021aefx. Most of the lines in the NIR are considerably blended, so for
clarity we only mark the most dominant species for each feature (Diamond et
al., 2018; Dhawan et al., 2018; Mazzali et al., 2015; Hoeflich et al., 2021).
A more comprehensive set of identifications and a detailed list of strong NIR
transitions from 0.8–2.4 $\mu$m can be found in Table 2 and associated figures
from Diamond et al. (2018) and additional model line transitions in this
region are given by Flörs et al. (2020) and Hoeflich et al. (2021).

Comparison to a dereddened ($R_{V}$ = 1.4, E(B$-$V) $=$ 1.2 mag; Foley et al.,
2014; Amanullah et al., 2014; Mazzali et al., 2015), scaled Gemini North GNIRS
(Elias et al., 2006a, b) spectrum of SN 2014J at the closest available phase
of $+$289d (Diamond et al., 2018; Graur et al., 2020) and a dereddened
(E(B$-$V)$=$0.023 mag; Mazzali et al., 2015), scaled Large Binocular Telescope
LUCIFER (Seifert et al., 2003) spectrum of SN 2011fe at similar phase of
$+$234d shows good agreement between the SNe, with nearly all of the same
lines present. The line strengths vary, though this may be attributed to the
differences in phase. The NIR spectral features are predominantly blends of
forbidden-line emission from the iron-group elements Fe, Co, and Ni. The [Ni
I], [Ni II], and [Ni III] transitions are of particular interest for
constraining models; however, none of the NIR nickel lines are isolated.

Between 2.5$-$5.0 $\mu$m, the NIR spectrum shows weak features, apart from two
prominent features at around 2.9 $\mu$m and 3.2 $\mu$m, shown in Figure 4 and
Figure 5. The peak flux of the feature at 2.9 $\mu$m is about half that of the
3.2 $\mu$m feature and is dominated by two strong [Fe III] lines at 2.874 and
2.905 $\mu$m. The red side of the peak may be blended with weaker [Ni II]
2.911 $\mu$m and [Fe III] 3.044 $\mu$m (Flörs et al., 2020).

The broad, somewhat-boxy feature centered near 3.2 $\mu$m is attributable to
the [Ni I] 3.120 $\mu$m line on the blue side and the [Fe III] 3.229 $\mu$m
line on the red side. Other potential blended contributors include [Fe III]
3.044 $\mu$m, [Co II] 3.286 $\mu$m and [Co II] 3.151 $\mu$m lines. Models by
Hoeflich et al. (2021) predict the [Ni I] 3.120 $\mu$m line to be strong and
narrow, with the [Fe III] 3.229 $\mu$m line being weaker but broader. Our data
suggest that the [Fe III] 3.229 $\mu$m line is indeed broader, but more
detailed modeling of all potential lines in this region is needed to
unambiguously determine line strengths in this blended feature. We further
discuss the line profile shapes and measurements of the 3.2 $\mu$m feature in
Section 4.1.1.

The remainder of the 2.5$-$5.0 $\mu$m region shows only unidentifiable weak
lines and strong, isolated lines do not reappear until the MIR around 6.5
$\mu$m. Small bumps in the spectrum at $\sim$3.4 $\mu$m (S/N $\simeq$ 9) and
$\sim$3.8 $\mu$m (S/N $\simeq$ 8) might be attributable to [Fe II] 3.393
$\mu$m and [Ni III] 3.802 $\mu$m, respectively, or a blend of unidentifiable
features. Interestingly, several lines that are predicted to be strong do not
appear in our JWST/NIRSpec observations of SN 2021aefx. Models from Hoeflich
et al. (2021) predict strong [Fe II] 4.076 and 4.115 $\mu$m lines, and while
there may be a small bump in the data in this region, it is weak (S/N $\simeq$
7). These models also predict weaker lines of [Co I] 2.526 $\mu$m and [Co I]
3.633 $\mu$m that we do not see strongly in our data. However, the models from
Hoeflich et al. (2021) were made for a subluminous SN, so it may be expected
that the ionization state and relative line strengths are different between
the models and our data. Additionally, the [Co III] 3.492 $\mu$m line has been
expected to contribute to the flux in the Spitzer CH1 (3.6 $\mu$m) photometric
band (e.g., see Gerardy et al., 2007; Johansson et al., 2017); however, it
does not appear in our NIR spectrum.

Figure 5: JWST/NIRSpec line identifications in the 3 $\mu$m region, previously
unobserved for SN Ia. The main line features arise from [Fe III] 2.874 & 2.905
$\mu$m, and [Ni I] 3.120 $\mu$m & [Fe III] 3.229 $\mu$m emission (grey).
Several possible weaker lines that may be blended with the dominant lines are
shown in light grey. Details of the dominant lines can be found in Table 2.

###  3.2 MIR Emission Lines

Figure 6: MIR line identifications for our JWST/MIRI spectrum of SN 2021aefx
(black) at $+$255d. We compare to scaled Spitzer/IRS MIR spectra of SN 2005df
(red) at $+$117d and SN 2003hv (green) at $+$359d (Gerardy et al., 2007), SN
2006ce (blue) $+$127d, and a MIR spectrum from GTC/CanariCam of SN 2014J
(orange) at $+$121d (Telesco et al., 2015). All phases are given in rest-frame
days relative to B${}_{\text{max}}$. The JWST/MIRI spectrum shows impressive
improvement in S/N, allowing for additional identification of weaker lines.
The dominant emission lines are marked in grey, with uncertain line
identifications marked in light grey. Details of the MIR lines can be found in
Table 2.

Starting from the MIR line identifications by Gerardy et al. (2007) for SN
2005df, we mark the dominant emission lines in the JWST/MIRI spectrum of SN
2021aefx in Figure 6. Compared with the Spitzer/IRS MIR spectra of SN 2005df
and SN 2003hv ($+$117 and $+$359 rest-frame days from B${}_{\text{max}}$,
respectively) (Gerardy et al., 2007), as well as an unpublished Spitzer
spectrum of SN 2006ce ($+$127 rest-frame days from B${}_{\text{max}}$;
Blackman et al. 2006) (PID 30292, PI: P. Meikle) downloaded from the CASSIS
database777https://cassis.sirtf.com/ (Lebouteiller et al., 2011), and the Gran
Telescopio Canarias (GTC)/CanariCam MIR spectrum of SN 2014J ($+$121 rest-
frame days from B${}_{\text{max}}$)(Telesco et al., 2015), the improvement in
signal-to-noise ratio of the JWST spectrum is striking. Despite differences in
phase, the MIR spectra all show fairly close agreement, with the same major
emission lines present but varying in strength and shape. The JWST/MIRI
spectrum reveals additional, weaker emission lines and helps to confirm and
clarify noisy lines in the Spitzer/IRS and GTC/CanariCam spectra. Details of
our MIR line identifications can be found in Table 2.

The optical and NIR nebular spectra are dominated by iron-group elements with
most of the strongest lines attributed to Fe emission, while the most dominant
emission lines in the MIR instead come from Co and Ni. Prominent emission
lines from Ar, an intermediate-mass element, also emerge in the MIR, providing
new physical insight into the SN emission structure. These MIR emission lines
are significantly less blended than in the NIR, making their identification
and subsequent fitting easier.

####  3.2.1 Iron-Group Elements: Cobalt and Nickel

The brightest MIR feature is the isolated [Co III] 11.888 $\mu$m line, which
shows close agreement with the SN 2005df and SN 2003hv spectra. The [Co II]
10.521 $\mu$m line is also fairly strong and its peak is clearly resolved,
although its base is blended with other lines on both sides. This line is
potentially indistinguishably blended with the weaker and very nearby [S IV]
10.510 $\mu$m line (Gerardy et al., 2007). [Co II] 10.521 $\mu$m has roughly
25% the strength of the [Co III] 11.888 $\mu$m line, suggesting that there is
comparatively little [Co II] emission.

On the blue side, the [Co II] 10.521 $\mu$m line is blended with a previously
unidentified line which creates a shoulder feature. We tentatively suggest
that this unidentified line near 10.2 $\mu$m is [Fe II] 10.189 $\mu$m, [Fe
III] 10.201 $\mu$m, or a combination of both. This shoulder feature is also
visible in SN 2003hv and potentially as a single pixel excess in SN 2005df,
though it is not present in SN 2006ce or SN 2014J. It was not previously
identified by Gerardy et al. (2007) due to high spectral noise.

The broad curve redward of the [Co II] 10.521 $\mu$m line exhibits two
distinct bumps, indicating blended emission. We identify the expected strong
[Ni III] 11.002 $\mu$m line and tentatively identify the weaker [Ni II] 10.682
$\mu$m line as the main contributing species. Weak [Co II] 11.167 $\mu$m
emission may also broaden the redder of the two bumps, which is predominantly
[Ni III] 11.002 $\mu$m. Inspection of the SN 2005df spectrum reveals a single
pixel excess at the location of the bluer bump, the SN 2003hv spectrum also
shows a peak (though without the JWST/MIRI spectrum for comparison, these very
faint signals look like, and could be, noise). SN 2014J clearly exhibits two
distinct peaks in this region while the SN 2006ce spectrum looks smoothly
blended into one peak. All of the supernovae exhibit emission consistent with
the [Ni III] 11.002 $\mu$m line.

The second brightest MIR feature, at roughly 50% of the [Co III] 11.888 $\mu$m
line strength, is the relatively isolated [Ni III] 7.349 $\mu$m line. This
feature in SN 2021aefx is slightly weaker in relative strength than in SN
2006ce and SN 2003hv, but is stronger than in SN 2005df.

The [Ni IV] 8.405 $\mu$m line in SN 2021aefx has a roughly equal peak flux as
the neighboring [Ar III] 8.991 $\mu$m line and is fairly well isolated, with
minimal blending. This line was identified for SN 2005df, though its strength
was only about 30% the strength of the [Ar III] 8.991 $\mu$m line. SN 2006ce
and SN 2014J also exhibit the [Ni IV] 8.405 $\mu$m line, and like SN 2021aefx,
it has roughly equal peak flux to the [Ar III] 8.991 $\mu$m line. The [Ni IV]
feature in SN 2003hv was only speculatively identified by Gerardy et al.
(2007) due to noise, but it is confirmed more clearly when compared to the
MIRI spectrum of SN 2021aefx.

While Gerardy et al. (2007) found no convincing detection of the [Ni II] 6.636
$\mu$m line (partially due to excess noise in the 7.4 $\mu$m region from the
overlapping edges of spectral orders), our JWST/MIRI spectrum displays a clear
“shoulder” feature on the blue side of the 7$-$8 $\mu$m region explained by
[Ni II] 6.636 $\mu$m blending with the nearby [Ar II] 6.985 $\mu$m line.

Further analysis of these iron-group element features in the MIR spectrum is
given in Section 4.2 and Section 4.3, where we fit simple geometric line
profiles and estimate kinematic properties for each identified line.

####  3.2.2 Intermediate-Mass Elements: Sulfur and Argon

[S IV] 10.510 $\mu$m line emission theoretically contributes to the [Co II]
10.521 $\mu$m peak (Gerardy et al., 2007); however, it is too closely blended
and our spectral resolution is too low for direct identification or further
analysis, especially given the additional blending with Ni on both sides.
Detailed theoretical models will be needed to disentangle the contribution of
[S IV] 10.510 $\mu$m.

Two important argon emission lines appear in the MIR. In the MIRI spectrum,
the [Ar III] 8.991 $\mu$m line is only slightly blended with the [Ni IV] 8.405
$\mu$m line. We see a broad, well-resolved flat-topped profile indicative of a
lack of emission at low projected velocities implying a spherical shell of
emission. This boxy [Ar III] shape differs significantly from the forked [Ar
III] and [Ar II] profiles that Gerardy et al. (2007) found for both SN 2005df
and SN 2003hv, attributed to an asymmetric ring of emission. SN 2006ce and SN
2014J appear to have an asymmetrically sloped [Ar III] profile, highlighting
interesting differences in the distribution of argon between these supernovae.

Visible as a small bump on top of the broad blue wing of the [Ni III] 7.349
$\mu$m feature, the [Ar II] 6.985 $\mu$m line in SN 2021aefx is nearly
completely blended into the Ni emission lines that surround it. With such
strong blending, it is difficult to conclusively determine the shape of this
line; we further analyze the shape of the [Ar III] 8.991 $\mu$m and [Ar II]
8.405 $\mu$m lines and discuss their implications in Section 4.4.

####  3.2.3 Unidentified and Speculative Features

Gerardy et al. (2007) suggest possible detection of silicon monoxide (SiO)
molecular emission in the $\sim$7.5-8 $\mu$m region of SN 2005df,
corresponding to the fundamental ($\Delta_{v}=1$) rovibrational band. Hoeflich
et al. (1995) concluded that CO and SiO might form in subluminous SNe Ia with
very low 56Ni yield, and SN 2005df was a subluminous SN Ia, making detection
of SiO an interesting possibility. However, SN 2021aefx is a normal SN Ia;
furthermore, our full NIR $+$ MIR spectrum does not show convincing evidence
for CO or SiO fundamental emission elsewhere in the spectrum. Thus, we favor
an explanation for the weak emission feature at $\sim$7.8 $\mu$m by [Fe III]
7.791 $\mu$m, which is predicted by models from Flörs et al. (2020).

We speculate that the faint emission feature at $\sim$6.2 $\mu$m is [Co II]
6.214 $\mu$m. [Ni II] 12.729 $\mu$m may be detected redward of the strong [Co
III] 11.888 $\mu$m line, but an increase in noise toward the end of the
spectrum prevents conclusive detection. Finally, a small spike on top of the
red side of the [Co III] 11.888 $\mu$m peak is tentative evidence for [Ni I]
12.001 $\mu$m.

Table 2: Line Identification and Fitting for SN 2021aefxaaLine information from the Atomic Line List version 3.00b4 $\lambda_{\textrm{rest}}$ | Species | Fit Profile | $\lambda_{\text{peak}}$ | $v_{\text{peak}}$ | FWHM | Transition | $A_{ki}$ | $E_{lu}$
---|---|---|---|---|---|---|---|---
($\mu$m) |  |  | ($\mu$m) | (km s-1) | (km s-1) |  |  | (eV)
Optical + NIR Lines |  |  |
0.589 | [Co III] | Gaussian | 0.591 | 1000 $\pm$ 1500 | 10,800 $\pm$ 2500 | a4F${}_{9/2}-$a2G9/2 | $4.20\text{\times}{10}^{-1}$ | 0.000$-$2.105
0.716 | [Fe II] | Gaussian | 0.717 | 600 $\pm$ 1100 | 8900 $\pm$ 2200 | a4F${}_{9/2}-$a2G9/2 | $1.46\text{\times}{10}^{-1}$ | 0.232$-$1.964
0.738 | [Ni II] | Gaussian | 0.740 | 900 $\pm$ 1200 | 9700 $\pm$ 3600 |  2D${}_{5/2}-^{2}$F7/2 | $2.30\text{\times}{10}^{-1}$ | 0.000$-$1.680
1.257 | [Fe II] | Gaussian | 1.263 | 1300 $\pm$ 1500 | 9400 $\pm$ 1500 | a6D${}_{9/2}-$a4D7/2 | $4.74\text{\times}{10}^{-03}$ | 0.000$-$0.986
1.547 | [Co II] | Gaussian | 1.545 |  $-$400 $\pm$ 1300 | jussi6100 $\pm$ 2200 | a5F${}_{5}-$b3F4 | $2.81\text{\times}{10}^{-02}$ | 0.415$-$1.217
1.644 | [Fe II] | Gaussian | 1.651 | 1300 $\pm$ 1200 | 11,100 $\pm$ 2400 | a4F${}_{9/2}-$a4D7/2 | $6.00\text{\times}{10}^{-03}$ | 0.232$-$0.986
1.939 | [Ni II] | Gaussian | 1.949 | 300 $\pm$ 1000 | 11,300 $\pm$ 1000 |  4F${}_{9/2}-^{2}$F7/2 | $8.70\text{\times}{10}^{-02}$ | 1.041$-$1.680
2.219 | [Fe III] | Gaussian | 2.226 | 1100 $\pm$ 1800 | 9900 $\pm$ 1800 |  3H${}_{6}-^{3}$G5 | $3.40\text{\times}{10}^{-02}$ | 2.486$-$3.045
2.874 | [Fe III] | Gaussian | 2.867 | $-$100 $\pm$ 1500 | 13,900 $\pm$ 4300 |  3F4${}_{4}-^{3}$G4 | $2.70\text{\times}{10}^{-02}$ | 2.661$-$3.092
2.905 | [Fe III] | 2.906 |  3F4${}_{3}-^{3}$G3 | $2.90\text{\times}{10}^{-02}$ | 2.690$-$3.117
3.120 | [Ni I] | Gaussian | 3.118 |  $-$200 $\pm$ 1400 | 6300 $\pm$ 1900 |  3D${}_{3}-^{1}$D2 | $7.80\text{\times}{10}^{-02}$ | 0.025$-$0.423
3.229 | [Fe III] | Gaussian | 3.209 |  $-$1900 $\pm$ 1400 | 11,300 $\pm$ 3000 |  3F4${}_{4}-^{3}$G5 | $1.70\text{\times}{10}^{-02}$ | 2.661$-$3.045
MIR Lines |  |  |
6.636 | [Ni II] | Gaussian | 6.725 | 4000 $\pm$ 1500 | 12,000 $\pm$ 3500 |  2D${}_{5/2}-^{2}$D3/2 | $5.54\text{\times}{10}^{-02}$ | 0.000$-$0.187
6.985 | [Ar II] | Gaussian | 7.084 | 4000 $\pm$ 1500 | 20,500 $\pm$ 4100 |  2P${}^{o}_{3/2}-^{2}$P${}^{o}_{1/2}$ | $4.23\text{\times}{10}^{-02}$ | 0.000$-$0.177
7.349 | [Ni III] | Gaussian | 7.422 | 3000 $\pm$ 1400 | 11,200 $\pm$ 1300 |  3F${}_{4}-^{3}$F3 | $6.50\text{\times}{10}^{-02}$ | 0.000$-$0.169
8.405 | [Ni IV] | Sphere | 8.447 | 1300 $\pm$ 1200 | 13,600 $\pm$ 1600 |  4F${}_{9/2}-^{4}$F7/2 | $5.70\text{\times}{10}^{-02}$ | 0.000$-$ 0.148
8.991 | [Ar III] | Shell | 9.012 | 700 $\pm$ 1100 | 23,700 $\pm$ 1600 |  3P${}_{2}-^{3}$P1 | $3.10\text{\times}{10}^{-02}$ | 0.000$-$0.138
10.521 | [Co II] | Gaussian | 10.562 | 1200 $\pm$ 1000 | 8300 $\pm$ 1300 | a3F${}_{4}-$a3F3 | $2.24\text{\times}{10}^{-02}$ | 0.000$-$0.118
11.002 | [Ni III] | Gaussian | 11.051 | 1300 $\pm$ 1900 | 10,700 $\pm$ 2400 |  3F${}_{3}-^{3}$F2 | $2.70\text{\times}{10}^{-02}$ | 0.169$-$ 0.281
11.888 | [Co III] | Gaussian | 11.911 | 500 $\pm$ 1900 | 10,200 $\pm$ 1300 | a4F${}_{9/2}-$a4F7/2 | $2.01\text{\times}{10}^{-02}$ | 0.000$-$0.104
Tentative Lines |  |  |
2.911 | [Ni II]? | Gaussian | 2.913 | 4800 $\pm$ 2000 |  $-$400 $\pm$ 2600 |  4F${}_{5/2}-^{2}$F7/2 | $1.40\text{\times}{10}^{-02}$ | 1.254$-$1.680
3.044 | [Fe III]? | – | – | –abcde | –abcde |  3F4${}_{2}-^{3}$G3 | $1.80\text{\times}{10}^{-02}$ | 2.710$-$3.117
3.286 | [Co II]? | Gaussian | 3.287 |  $-$300 $\pm$ 1400 | 5900 $\pm$ 1200 | c3F${}_{4}-$a1F3 | $1.86\text{\times}{10}^{-03}$ | 5.089$-$5.467
6.214 | [Co II]? | – | – | –abcde | –abcde | a1D${}_{2}-$a3P2 | $3.08\text{\times}{10}^{-02}$ | 1.445$-$1.644
6.920 | [Ni II]? | Gaussian | 7.055 | 5800 $\pm$ 2000 | 12,000 $\pm$ 4000 |  2F${}_{7/2}-^{2}$F5/2 | $4.71\text{\times}{10}^{-02}$ | 1.680$-$1.859
6.985 | [Ar II]? | Shell | 7.039 | 2300 $\pm$ 1500 | 23,700 $\pm$ 1600 |  2P${}^{o}_{3/2}-^{2}$P${}^{o}_{1/2}$ | $4.23\text{\times}{10}^{-02}$ | 0.000$-$0.177
7.791 | [Fe III]? | – | – | –abcde | –abcde |  3P4${}_{2}-^{3}$P41 | $4.70\text{\times}{10}^{-02}$ | 2.406$-$2.565
10.189 | [Fe II]? | Gaussian | 10.235 | 1000 $\pm$ 1000 | 6700 $\pm$ 1700 | b4P${}_{5/2}-$b4P3/2 | $2.30\text{\times}{10}^{-02}$ | 2.583$-$2.704
10.201 | [Fe III]? |  3H${}_{4}-^{3}$F44 | $1.60\text{\times}{10}^{-03}$ | 2.539$-$2.661
10.510 | [S IV]? | – | – | –abcde | –abcde |  2P${}^{o}_{1/2}-^{2}$P${}^{o}_{3/2}$ | $7.30\text{\times}{10}^{-03}$ | 0.000$-$0.118
10.682 | [Ni II]? | Gaussian | 10.842 | 4500 $\pm$ 1000 | 4200 $\pm$ 1500 |  4F${}_{9/2}-^{4}$F7/2 | $2.71\text{\times}{10}^{-02}$ | 1.041$-$1.157
11.167 | [Co II]? | Gaussian | 11.214 | 1300 $\pm$ 1900 | 3100 $\pm$ 1400 | b3F${}_{4}-$b3F3 | $1.88\text{\times}{10}^{-02}$ | 1.217$-$1.328
12.002 | [Ni I]? | – | – |  $\sim$ 1500abc | –abcde |  3D${}_{2}-^{3}$D1 | $2.10\text{\times}{10}^{-02}$ | 0.109$-$0.212
12.729 | [Ni II]? | – | – | –abcde | –abcde |  4F${}_{7/2}-^{4}$F5/2 | $2.76\text{\times}{10}^{-02}$ | 1.157$-$1.254

##  4 Line Velocities and Profiles

At late times in the nebular phase, the supernova ejecta opacity drops and
emission streams freely from all regions, revealing important properties of
the ejecta composition and ionization structure. The shape of the nebular
emission lines is determined by the ejecta emissivity, which depends on both
the density and excitation (for a review, see Jerkstrand, 2017). Several
simple ejecta geometries that produce common line profiles include a uniform
sphere resulting in a parabolic shape, a uniform spherical thick shell
resulting in a boxy, flat-topped shape with parabolic wings, and a Gaussian
density sphere resulting in a Gaussian line profile (Jerkstrand, 2017). The
MIR, where the features are comparatively isolated, is particularly useful for
inferring the kinematic distribution of the emission.

Most of the lines that we identify can be well modelled by a superposition of
Gaussian line profiles. Using the $+$266d emission line model of SN 2015F from
Flörs et al. (2020), and following the approach of Maguire et al. (2018) and
Flörs et al. (2018, 2020), we model the superposition of all [Fe II], [Fe
III], [Ni II], [Co II], and [Co III] lines contributing to selected optical
and NIR features. The relative line strengths of each line are fixed by the
model, and all lines of the same species are restricted to have the same
Gaussian width and kinematic offset from the central wavelength. Because this
model was computed for temperatures and densities specific to SN 2015F, not SN
2021aefx, we do not attempt to fit the entire optical or NIR spectrum, but
rather fit the model to selected regions containing features of interest.

Past 3 $\mu$m, the emergence of lines from species not included in the SN
2015F model from Flörs et al. (2020) prevents us from modeling each feature in
such a thorough and self-consistent way. Furthermore, the model relative line
strengths begin to deviate significantly from the data for the NIR 3.2 $\mu$m
feature and the MIR. Thus, we fit each feature redward of 3 $\mu$m as a
superposition of basic geometric line-emission profiles for each
distinguishable contributing line, allowing the amplitude, kinematic offset,
and width of each individual line to be free parameters. Fit profile shapes
for each line are chosen based upon visual inspection of best overall feature
fit. A proper, bespoke model for SN 2021aefx is beyond the scope of this
paper, but will be the focus of future effort.

We use UltraNest (Buchner, 2021), a Bayesian inference package for parameter
estimation using nested sampling, to fit our line profiles and recover
uncertainties on our measurements of kinematic offset ($v_{\text{peak}}$) and
full-width half-maximum (FWHM). The likelihood function optimized in the
UltraNest fitting is given by:

$\ln(\text{likelihood})=-\frac{1}{2}\sum\left[\frac{\left(\text{data}-\text{model}\right)^{2}}{s^{2}}+\ln(2\pi
s^{2})\right]$

where

$s^{2}=\text{data\\_uncertainty}^{2}+f^{2}~{}\text{model}^{2}$

and the uncertainties are underestimated by some fractional amount $f$ that is
marginalized over in the fit. For the MIR lines, we include a systematic
uncertainty in the wavelength calibration of 0.034 $\mu$m, derived from the
root-mean-square of the wavelength calibration residuals in the region
encompassing the SN features (6.5–12.5 $\mu$m), and add this in quadrature to
the kinematic offset uncertainties. The typically low resolution of our data
results in instrumental broadening ($c\Delta\lambda/\lambda$) that is
significant: $\sim$4500 km s-1 at 1.25 $\mu$m and $\sim$1000 km s-1 at 3.3
$\mu$m for the NIR, and $\sim$1900 km s-1 at 6.5 $\mu$m and $\sim$450 km s-1
at 12 $\mu$m for the MIR. We remove this in quadrature from our FWHM
measurements and impose a floor on our line centroid uncertainty equal to one-
third the instrument resolution. For a sense of scale, the instrumental
resolution is roughly 50% of the FWHM of the [Co III] 1.257 $\mu$m line, 10%
of the FWHM of the [Fe III] 3.229 $\mu$m line, 15% of the FWHM of the [Ni III]
7.349 $\mu$m line, and 5% of the FWHM of the [Co III] 11.889 $\mu$m line.

Our line fits do not include any radiative transfer and we only attempt basic
accounting of line blending by superposition. Table 2 gives the chosen fit
profile shape, measured peak wavelength, kinematic offset, FWHM, and the
estimated uncertainties of each fitted feature.

###  4.1 Optical $+$ NIR: Iron and Nickel

Figure 7: Optical $+$ NIR emission-line profiles of SN 2021aefx at $+$255d
(black) for complex spectral features dominated by iron. The model profile for
the central line is shown in dashed red, with contributions from other lines
shown in dashed light red and dashed light blue, and the full modeled
superposition of lines shown in solid blue.

Using the SN 2015F model from Flörs et al. (2020) as described above, we fit
the NIR [Fe II] 1.257 $\mu$m, [Fe II] 1.644 $\mu$m, and [Fe III] 2.218 $\mu$m
features, shown in Figure 7. We find agreement within the uncertainties, in
both $v_{\text{peak}}$ and FWHM, between all of these lines, which exhibit a
moderately redshifted $v_{\text{peak}}$.

Like the models by Maguire et al. (2018), the SN 2015F model from Flörs et al.
(2020) only contains significant line contributions in the 1.3 $\mu$m region
from [Fe II], as shown in Figure 7. The model fits the data well in this
region and the measured FWHM and $v_{\text{peak}}$ agree closely with those
measured from the optical [Fe II] 0.716 $\mu$m and NIR [Fe II] 1.644 $\mu$m
lines. This supports the conclusion of Maguire et al. (2018) that the 1.3
$\mu$m feature is a relatively contaminant free way to measure line velocities
and widths for [Fe II].

We also fit the optical 7300 Å line complex of [Fe II] 0.716 $\mu$m and [Ni
II] 0.738 $\mu$m with the SN 2015F model, as shown in Figure 7. The measured
FWHM and $v_{\text{peak}}$ for the [Fe II] 0.716 $\mu$m line are within the
uncertainties of the NIR Fe lines. The [Ni II] 0.738 $\mu$m line is consistent
in FWHM with the other [Ni II] and [Ni III] lines in the NIR and MIR. Its
$v_{\text{peak}}$ is comparable to [Ni II] 1.939 $\mu$m, [Ni III] 11.002
$\mu$m, and [Ni IV] 8.405 $\mu$m, but lower than [Ni II] 6.644 $\mu$m and [Ni
III] 7.349 $\mu$m, which may be partly attributed to the imperfect JWST
MIRI/LRS wavelength calibration that is worse at the shorter wavelengths.
Overall, from the optical 7300 Å line complex, the [Fe II] 0.716 $\mu$m and
[Ni II] 0.738 $\mu$m kinematics are roughly consistent with the Fe and Ni
kinematics from the NIR and MIR.

We detect the [Ni II] 1.939 $\mu$m line feature in our NIR spectrum, shown in
Figure 10, which has also been seen in other ground-based studies of the NIR
(Dhawan et al., 2018; Flörs et al., 2020; Blondin et al., 2022). Fitting this
feature with the 2015F model, we find a FWHM that is comparable to the FWHM of
Ni II in the MIR. The $v_{\text{peak}}$ is within the uncertainties of the [Ni
II] 0.738 $\mu$m, [Ni III] 7.349 $\mu$m, [Ni III] 11.00 $\mu$m, and [Ni IV]
8.41 $\mu$m lines and slightly smaller than the [Ni II] 6.64 $\mu$m
$v_{\text{peak}}$, again possibly affected by the MIR wavelength calibration.

Following Flörs et al. (2020), we find that while we cannot rule out weak [Ca
II] line contamination in the 7300 Å line complex, strong [Ca II]
contamination in the optical would require a weaker [Ni II] 0.716 $\mu$m line,
thus predicting a weaker [Ni II] 1.939 $\mu$m line since no [Ca II] is present
there. Our fits to [Ni II] 0.738 $\mu$m and [Ni II] 1.939 $\mu$m have similar
amplitudes and we do not see a weaker [Ni II] 1.939 $\mu$m line than expected
from [Ni II] 0.738 $\mu$m. Thus, in agreement with the findings by Maguire et
al. (2018) and Flörs et al. (2020), we do not find a compelling reason to
invoke contamination from [Ca II] to reconcile the differences in kinematic
properties between the optical and NIR $+$ MIR.

Figure 8: NIR emission-line profiles of SN 2021aefx at $+$255d (black) for the
3 $\mu$m region features. The model profile for the central line is shown in
dashed red, with contributions from other lines shown in dashed blue and the
full modeled superposition of lines shown in solid blue.

####  4.1.1 NIR: 2.9 and 3.2 $\mu$m Features

The [Fe III] 2.874 $\mu$m $+$ [Fe III] 2.905 $\mu$m line feature, fit to the
SN 2015F model from Flörs et al. (2020), has contributions from [Ni II] 2.911
$\mu$m and [Fe III] 3.044 $\mu$m, shown in Figure 8. This feature exhibits a
broad FWHM consistent with [Fe III] 2.218 $\mu$m and [Fe III] 3.229 $\mu$m,
and a blueshifted $v_{\text{peak}}$ between the other two NIR [Fe III] lines.

We fit the feature at 3.2 $\mu$m with a superposition of three Gaussian
emission distribution profiles, shown in Figure 8. The strongest lines
expected in this feature are [Ni I] 3.120 $\mu$m and [Fe III] 3.229 $\mu$m,
but to improve the fit of the full feature, we include a third [Co II] 3.286
$\mu$m line. This fit produces a strong, broad [Fe III] 3.229 $\mu$m line
consistent in FWHM with [Fe III] 2.874 $\mu$m and [Fe III] 2.218 $\mu$m, but
with a large blueshifted $v_{\text{peak}}$. This may indicate that a more
comprehensive model is needed to explain the 3.2 $\mu$m feature. The fits to
the [Ni I] 3.120 $\mu$m and [Co II] 3.286 $\mu$m lines show weaker, narrower
profiles of similar strength, FWHM, and low kinematic offset.

Alternatively, we can fit the 3.2 $\mu$m feature nearly equally well without
[Co II] 3.286 $\mu$m, but it requires boxy, flat-topped profiles for both [Ni
I] 3.120 $\mu$m and [Fe III] 3.229 $\mu$m. In this alternative fit, both lines
are of similar strengths, nearly equally broad, and display large redshifts.
However, these profiles would imply a large central hole of both [Fe III] and
[Ni I] emission that would be difficult to explain physically. The Fe comes
from 56Ni decay and is expected to be broad, whereas the observed Ni must be
stable 58Ni, as all radioactive 56Ni has decayed away at this phase. The
stable Ni is expected to be centrally concentrated and thus with a narrower,
peaked profile. Furthermore, flat-topped profiles are not seen in other Ni and
Fe features throughout the optical, NIR, and MIR spectra. Therefore, we favor
the Gaussian profile fits with inclusion of an unexpectedly strong [Co II]
3.286 $\mu$m line. More detailed future modeling of this feature may reveal
additional contributing lines.

Figure 9: Cobalt emission-line profiles of SN 2021aefx at $+$255d (black)
across the optical, NIR, and MIR compared with model line emission from
Gaussian distributions of emission. The model profile for the central line is
shown in dashed red, with contributions from other lines shown in dashed blue
and the full modeled superposition of lines shown in solid blue.

###  4.2 MIR: Cobalt

Shown in Figure 9, the [Co III] 11.888 $\mu$m line is well fit by a fairly
broad Gaussian profile and low kinematic offset from the host-galaxy rest
frame. Gerardy et al. (2007) find evidence for a parabolic, slightly flat-
topped [Co III] 11.888 $\mu$m profile for SN 2005df resulting from a spherical
distribution with a hollow inner region, and a significantly blueshifted,
fairly flat and weak [Co III] 11.888 $\mu$m line for SN 2003hv. SN 2021aefx
shows a Gaussian [Co III] distribution with clear wings not expected from a
uniform spherical distribution. However, close inspection of the Gaussian fit
shows that the peak may be marginally flat topped, potentially indicating a
Gaussian distribution with a small central hole corresponding to the electron
capture zone where little Co is produced (Gerardy et al., 2007).

The [Co II] 10.521 $\mu$m line is blended so closely with the predicted weaker
[S IV] 10.510 $\mu$m line that they are indistinguishable and we model them as
one line. We model the full blended $\sim$10-11.3 $\mu$m feature as a linear
combination of five Gaussians: [Fe II] 10.189 $\mu$m/[Fe III] 10.201 $\mu$m,
[Co II] 10.521 $\mu$m, [Ni II] 10.682 $\mu$m, [Ni III] 11.002 $\mu$m, and [Co
II] 11.167 $\mu$m. The FWHM, $v_{\text{peak}}$, and Gaussian profile shape
agree nicely between the [Co III] 11.888 $\mu$m and [Co II] 10.521 $\mu$m
lines.

For comparison to the optical and NIR, we also fit the SN 2015F model from
Flörs et al. (2020) to the [Co III] 0.589 $\mu$m and [Co III] 1.547 $\mu$m
lines in Figure 9. The [Co II] and [Co III] measurements agree quite well
across the optical, NIR, and MIR and we conclude that the emission structure
between Co ionization states is similar, though the MIR line strengths suggest
there is less emission from [Co II]. The tentative identifications of [Co II]
11.167 $\mu$m and [Co II] 3.286 $\mu$m show decent agreement as well, though
they are significantly less broad and slightly blueshifted, respectively.

###  4.3 MIR: Nickel

Figure 10: Observed NIR and MIR nickel emission-line profiles for SN 2021aefx
at $+$255d (black) compared to model line emission from Gaussian
distributions, except for [Ni IV] 8.405 $\mu$m which is better fit by emission
from a uniform spherical emission distribution. The model profile for the
central line is shown in dashed red, with contributions from other lines shown
in dashed blue and the full modeled superposition of lines shown in solid
blue.

The SN 2021aefx MIR nickel emission-line profiles, shown in Figure 10, are all
blended with other emission-line profiles, with the [Ni IV] 8.405 $\mu$m and
[Ni III] 7.349 $\mu$m lines being the most isolated. Despite blending, the
JWST/MIRI Ni lines in SN 2021aefx are significantly better resolved than the
Spitzer/IRS and GTC/CanariCam spectra and we can fit the blended Ni lines well
by Gaussian profiles, except for [Ni IV] 8.405 $\mu$m which prefers a
parabolic profile. The wings of a Gaussian profile for [Ni IV] contribute too
much to the boxy [Ar III] 8.991 $\mu$m feature, implying the [Ni IV] emission
arises from a uniform spherical, rather than Gaussian, geometry.

The Ni lines show significantly redshifted kinematic offsets, except for
neutral [Ni I] 3.120 $\mu$m, which is also the only NIR Ni line we fit. [Ni
II] 6.636 $\mu$m and [Ni III] 7.349 $\mu$m exhibit the highest
$v_{\text{peak}}$ values, which may be partially due to the MIRI wavelength
solution being more uncertain at shorter wavelengths. Taking the wavelength
uncertainties into account, these lines are within the uncertainty range of
the [Ni IV] 8.405 $\mu$m and [Ni III] 11.002 $\mu$m measurements. These MIR
[Ni II] and [Ni III] lines are consistent in FWHM, while the NIR [Ni I] 3.120
$\mu$m line is narrower. The tentatively identified [Ni II] 10.682 $\mu$m line
shows a highly redshifted $v_{\text{peak}}$, consistent with [Ni II] 6.636
$\mu$m and [Ni III] 7.349 $\mu$m, but with an inconsistently narrower FWHM.

###  4.4 MIR: Argon

Figure 11: Observed argon emission-line profiles for SN 2021aefx at $+$255d
(black) compared to model line emission from a Gaussian emission distribution
for [Ar II] 6.985 $\mu$m (top); a thick uniform shell/hollow uniform sphere
emission distribution for [Ar III] 8.991 $\mu$m with an inner radius
corresponding to $v_{\text{min}}=$ 8700 $\pm$ 200 km s-1 and an outer radius
corresponding to $v_{\text{max}}=$ 13,500 $\pm$ 300 km s-1 (bottom); and a
thick shell distribution for [Ar II] 6.985 $\mu$m which has been forced to
have the same inner and outer radius as the [Ar III] fit (middle). The model
profile for the central line is shown in dotted red, with contributions from
other lines shown in dotted blue and the full modeled superposition of lines
shown in solid blue. Figure 12: FWHM versus kinematic offset for species with
fitted line profiles in Section 4. MIR lines (red) are compared against NIR
lines (blue) and optical lines (grey). Error bars show fit uncertainties, in
combination with wavelength calibration uncertainties for the MIR lines, and
take into account instrument resolution. Low opacity points represent
uncertain line identifications. Values and errors for these measurements are
given in Table 2.

Appearing only in the MIR, the argon lines reveal new structure in the SN
emission. Gerardy et al. (2007) find that the [Ar II] 6.985 $\mu$m and [Ar
III] 8.991 $\mu$m lines have distinctive forked profiles in both SN 2005df and
SN 2003hv, indicating an asymmetric, ring-shaped argon distribution. In
contrast, for SN 2021aefx, the [Ar III] 8.991 $\mu$m feature in SN 2021aefx
has a very distinctive boxy shape, indicative of a hollow uniform sphere, or
thick shell, of emission.

Shown in Figure 11, [Ar III] 8.991 $\mu$m has a very broad FWHM $\approx$
23,700 km s-1with an inner shell radius corresponding to a minimum velocity of
$v_{\text{min}}=$ 8700 $\pm$ 200 km s-1 and an outer radius corresponding to
$v_{\text{max}}=$ 13,500 $\pm$ 300 km s-1. The flat-top of the [Ar III] 8.991
$\mu$m line is only slightly sloped and is more symmetric than in the other
SNe observed by Spitzer and GTC/CanariCam. Indeed, the [Ar III] 8.991 $\mu$m
line shape exhibits considerable variation across the SNe in Figure 6.

The [Ar II] 6.985 $\mu$m feature in SN 2021aefx is heavily blended on both
sides by [Ni II] 6.636 $\mu$m and [Ni III] 7.349 $\mu$m, making it difficult
to conclusively determine its geometry. We find that the full $\sim$6.5–7.5
$\mu$m feature is well fit by a sum of three Gaussian profiles, with the
Gaussian representing [Ar II] 6.985 $\mu$m having a very broad FWHM $\approx$
20,600 km s-1 and high redshifted $v_{\text{peak}}\approx$ 4000 km s-1.
However, most lines from the same element share similar shapes, and we can fit
the feature equally well with a boxy, shell profile for [Ar II] 6.985 $\mu$m
if we include an additional line which could be [Ni II] 6.920 $\mu$m. In
Figure 11 we show a fit to the [Ar II] 6.895 $\mu$m line where we force it to
have the same inner and outer shell radii as the [Ar III] 8.991 $\mu$m fit.
This forced fit reduces $v_{\text{peak}}$ of [Ar II] but yields a very high
redshifted $v_{\text{peak}}\approx$ 5800 km s-1 for [Ni II] 6.920 $\mu$m.
Without the boxy [Ar III] profile, we would not consider a boxy profile for
[Ar II], and the fit is not unique since we could allow the [Ar II] shell to
have different radial boundaries than [Ar III]. More detailed modeling of this
feature is needed to fully disentangle the profile of [Ar II]. We further
discuss the shapes of the Ar lines and their implications in Section 5.

##  5 Summary and Conclusions

We present nebular SN Ia NIR and MIR spectra of SN 2021aefx from JWST,
including the first observation of the 2.5$-$5 $\mu$m region and the highest
S/N MIR SN Ia spectrum to date.

At the phase of our observations ($+$255d), all of the radioactive 56Ni has
decayed away and the Ni emission lines that we see come from stable 58Ni. We
unambiguously detect stable Ni in the ejecta of SN 2021aefx via the strong [Ni
I] 3.120 $\mu$m and [Ni II] 7.349 $\mu$m lines, a clearly resolved [Ni IV]
8.405 $\mu$m line, and several other weaker, blended Ni lines. Electron
capture reactions producing stable iron-group elements require high density
burning above $\sim$108 g cm-3 that is found in carbon-oxygen white dwarfs
with $M\gtrsim$ 1.2 $M_{\odot}$ (Hoeflich et al., 1996; Iwamoto et al., 1999;
Seitenzahl et al., 2013; Seitenzahl & Townsley, 2017; Hoeflich et al., 2017).
The strong detection of stable Ni advocates for a massive, perhaps near-
Chandrasekhar mass, progenitor for SN 2021aefx. However, our relatively low
instrumental resolution does not allow us to probe the ejecta structure at the
lowest velocities (the central region), and a more detailed quantitative
analysis is needed to determine whether the stable nickel mass could be
consistent with a lower-mass progenitor in a double detonation scenario (Flörs
et al., 2020; Blondin et al., 2022).

We fit emission-line profiles to prominent NIR and MIR lines to investigate
their kinematic properties and geometric emission distributions. As shown in
Figure 12, the iron-group elements (Fe, Co, and Ni) largely cluster around a
redshifted kinematic offset of $\sim$1000 $\pm$ 1000 km s-1 and a FWHM of
$\sim$ 11,000 $\pm$ 4000 km s-1. We find consistent kinematic offsets between
the Co lines, with [Co III] having a slightly higher FWHM than [Co II],
hinting that it may extend out to larger radii. This could indicate
recombination in the higher density central region. McClelland et al. (2013)
suggested that the more rapid decline seen in Spitzer IRAC photometry of SN
2011fe at 3.6 $\mu$m compared to 4.5 $\mu$m could also be a result of
recombination of doubly ionized species in the 3.6 $\mu$m band; indeed, our
data show [Fe III] emission should dominate that region.

The Ni lines have a slightly higher overall redshifted kinematic offset than
Co and Fe. However, we caution that the largest redshifted kinematic offsets
are found at the shortest MIRI wavelengths where the calibration is most
uncertain.

The reliably identified Fe lines are all consistent in FWHM. The [Fe III]
3.229 $\mu$m line, which was not modeled with the SN 2015F model from Flörs et
al. (2020) like the other NIR Fe lines, is significantly blueshifted in
kinematic offset compared to the other Fe lines; more detailed future modeling
of this feature may bring it into closer agreement with the other Fe line
measurements. Comparing the width of the [Fe II] 1.644 $\mu$m line
($\sim$12,000 km s-1) to the models for SN 2014J from Diamond et al. (2018,
e.g. see Figure 9), we find that SN 2021aefx has a higher central density than
SN 2014J. Though the uncertainties are large, there is a slight hint that a
redshifted kinematic offset is seen for [Fe II] compared to [Fe III]; as
described by Maeda et al. (2010), the low-ionization lines trace the
deflagration ash and this may be a signature of a delayed-detonation explosion
where the initial deflagration produces an offset innermost ejecta while the
subsequent detonation creates a spherically symmetric outer ejecta.

The argon lines are significantly broader than those of the iron-group
elements, implying that argon extends to higher velocities, and
correspondingly, radii. This result may support detonation models which
produce stratified ejecta from nucleosynthesis in the low density outer layers
leading to intermediate mass elements, and nucleosynthesis in the high density
interior producing the iron group elements (Khokhlov, 1991; Woosley & Weaver,
1994; Wiggins et al., 1998; Fink et al., 2007; Sim et al., 2010).

The flat-topped shape of the [Ar III] 8.991 $\mu$m line indicates a thick
shell geometry for the [Ar III] emission, which could either be produced by a
physical lack of Ar in the central regions, or a lack of doubly ionized Ar in
the interior. The [Ni IV] 8.405 $\mu$m line, which has an ionization energy of
35.2 eV, has a smooth parabolic profile suggesting that [Ni IV] is present in
the central region. Based just on ionization energies, it should then be
possible to doubly ionize argon to [Ar III], which requires 27.6 eV, in the
center. This would argue for a physical absence of Ar in the innermost ejecta,
a flat-topped profile for [Ar II], and a stratified ejecta from a detonation.
Comparison of ionization energies is likely too simplistic; indeed, Fransson &
Jerkstrand (2015) show that at late times in the high density regions, a large
fraction of the energy comes from ionizations and excitations, rather than
just thermal heating. A detailed future analysis should explore whether [Ni
IV] can be formed in the central region without central emission of [Ar III].

Overall, the Gaussian, parabolic, and flat-topped shapes of the fit profiles
for SN 2021aefx point to spherically symmetric distributions of emission for
all species. This is consistent with the low level of continuum polarization
of SNe Ia during the photospheric phase, as strong asymmetry of the
radioactive distribution will lead to directional dependence of polarization
(Wang & Wheeler, 2008; Yang et al., 2022). More JWST MIR data of SNe Ia will
improve our understanding of asymmetry in the explosions.

More detailed analyses, such as the derivation of elemental abundances and
inferred masses, and modeling of this and future JWST data of SN 2021aefx will
be the subject of future work. Continuing observations of SN 2021aefx with
JWST will build a time series of four epochs of JWST spectra and will be the
most comprehensive SN Ia nebular IR dataset available. As part of JWST Cycle 1
GO program 2072 (PI: S. W. Jha), another NIRSpec/Prism $+$ MIRI/LRS spectrum
will be obtained roughly 100 days after the first, and two additional MIRI
spectra of SN 2021aefx will be obtained through JWST Cycle 1 GO program 2114
(PI: C. Ashall), including planned MIRI/MRS spectroscopy doubling the
wavelength range out to 28 $\mu$m, with higher spectral resolution. Analysis
of SN 2021aefx over time will reveal the evolution of radioactivity,
ionization, and density structure of the ejecta.

SN 2021aefx is an excellent first reference SN Ia for JWST observations and
the initial analysis that we present here highlights the promise of JWST to be
transformative for the study of nebular phase SN Ia.

We thank Shelly Meyett for her consistently excellent work scheduling these
JWST observations, Greg Sloan for assistance with the MIRI LRS wavelength
solution correction, and Glenn Wahlgren for help with the NIRSpec
observations.

This work is based on observations made with the NASA/ESA/CSA JWST. The data
were obtained from the Mikulski Archive for Space Telescopes at the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127
for JWST. These observations are associated with JWST program #02072. Support
for this program at Rutgers University was provided by NASA through grant
JWST-GO-02072.001.

The SALT data presented here were obtained with Rutgers University program
2022-1-MLT-004 (PI: Jha). We are grateful to SALT Astronomer Rosalind Skelton
for taking these observations.

L.A.K. acknowledges support by NASA FINESST fellowship 80NSSC22K1599.

T.S. is supported by the NKFIH/OTKA FK-134432 grant of the National Research,
Development and Innovation (NRDI) Office of Hungary, the János Bolyai Research
Scholarship of the Hungarian Academy of Sciences and by the New National
Excellence Program (ÚNKP-22-5) of the Ministry for Innovation and Technology
of Hungary from the source of NRDI Fund.

Time-domain research by the University of Arizona team and D.J.S. is supported
by NSF grants AST-1821987, 1813466, 1908972, and 2108032, and by the Heising-
Simons Foundation under grant #2020-1864.

K.M. acknowledges support from the Japan Society for the Promotion of Science
(JSPS) KAKENHI grant JP18H05223 and JP20H00174.

L.G. acknowledges financial support from the Spanish Ministerio de Ciencia e
Innovación (MCIN), the Agencia Estatal de Investigación (AEI)
10.13039/501100011033, and the European Social Fund (ESF) “Investing in your
future” under the 2019 Ramón y Cajal program RYC2019-027683-I and the
PID2020-115253GA-I00 HOSTFLOWS project, from Centro Superior de
Investigaciones Científicas (CSIC) under the PIE project 20215AT016, and the
program Unidad de Excelencia María de Maeztu CEX2020-001058-M.

K.M. and M.D. are funded by the EU H2020 ERC grant #758638.

J.P.H. acknowledges support from the George A. and Margaret M. Downsbrough
bequest.

The Aarhus supernova group is funded in part by an Experiment grant (#28021)
from the Villum FONDEN, and by a project 1 grant (#8021-00170B) from the
Independent Research Fund Denmark (IRFD).

C.L. acknowledges support from the National Science Foundation Graduate
Research Fellowship under Grant No. DGE-2233066

A.F. acknowledges support by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (ERC Advanced
Grant KILONOVA No. 885281).

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