11institutetext: CENTRA, Instituto Superior Técnico, Universidade de Lisboa,
Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
11email<EMAIL_ADDRESS>22institutetext: Institut d’Estudis Espacials de
Catalunya (IEEC), Gran Capità, 2-4, Edifici Nexus, Desp. 201, E-08034
Barcelona, Spain
22email<EMAIL_ADDRESS>33institutetext: Institute of Space Sciences
(ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, E-08193 Barcelona, Spain
44institutetext: European Southern Observatory, Alonso de Córdova 3107,
Casilla 19, Santiago, Chile 55institutetext: Millennium Institute of
Astrophysics MAS, Nuncio Monsenor Sotero Sanz 100, Off. 104, Providencia,
Santiago, Chile 66institutetext: Department of Applied Physics, School of
Engineering, University of Cádiz, Campus of Puerto Real, E-11519 Cádiz, Spain
77institutetext: Tuorla Observatory, Department of Physics and Astronomy,
University of Turku, FI-20014 Turku, Finland 88institutetext: School of
Sciences, European University Cyprus, Diogenes Street, Engomi, 1516 Nicosia,
Cyprus 99institutetext: School of Physics and Astronomy, University of
Southampton, Southampton, SO17 1BJ, UK

# Narrow absorption lines from intervening material in supernovae

I. Measurements and temporal evolution

Santiago González-Gaitán  11 Claudia P. Gutiérrez  2233 Joseph P. Anderson
4455 Antonia Morales-Garoffolo 66 Lluis Galbany  3322 Sabyasashi Goswami 1166
Ana M. Mourão  11 Seppo Mattila  7788 Mark Sullivan  99

Narrow absorption features in nearby supernova (SN) spectra are a powerful
diagnostic of the slow-moving material in the line of sight: they are
extensively used to infer dust extinction from the host galaxies, and they can
also serve in the detection of circumstellar material originating from the SN
progenitor and present in the vicinity of the explosion. Despite their wide
use, very few studies have examined the biases of the methods to characterize
narrow lines, and not many statistical analyses exist. This is the first paper
of a series in which we present a statistical analysis of narrow lines of SN
spectra of various resolutions. We develop a robust automated methodology to
measure the equivalent width (EW) and velocity of narrow absorption lines from
intervening material in the line of sight of SNe, including Na i D , Ca ii
H&K, K i and diffuse interstellar bands (DIBs). We carefully study systematic
biases in heterogeneous spectra from the literature by simulating different
signal-to-noise, spectral resolution, slit size and orientation and present
the real capabilities and limitations of using low- and mid-resolution spectra
to study these lines. In particular, we find that the measurement of the
equivalent width of the narrow lines in low-resolution spectra is highly
affected by the evolving broad P-Cygni profiles of the SN ejecta, both for
core-collapse and type Ia SNe, inducing a conspicuous apparent evolution. Such
pervading non-physical evolution of narrow lines might lead to wrong
conclusions on the line-of-sight material, e.g. concerning circumstellar
material ejected from the SN progenitors. We present thus an easy way to
detect and exclude those cases to obtain more robust and reliable
measurements. Finally, after considering all possible effects, we analyse the
temporal evolution of the narrow features in a large sample of nearby SNe to
detect any possible variation in their EWs over time. We find no time
evolution of the narrow line features in our large sample for all SN types.

###### Key Words.:

supernovae:general, ISM: lines and bands, ISM: dust and extinction

##  1 Introduction

The light emitted by astrophysical sources detected from Earth is subject to
interaction with matter that lies along the line of sight between the source
and the observer. Studying the effects of this line of sight material on the
background light source gives information on the nature of that material,
whether it is close to the light source, e.g. circumstellar material (CSM),
within galaxies, i.e. the interstellar medium (ISM), or between galaxies, i.e.
the intergalactic medium (IGM). It is important to understand these processes:
CSM for stellar mass loss and ISM and IGM for galaxy enrichment and evolution.
At the same time, this intervening material is also extremely important for
understanding the light sources themselves as it absorbs the light, especially
dimming it at optical wavelengths and re-emitting it at longer wavelengths. It
thus impacts on our understanding of the brightness and properties of the
emitters.

This material in the line of sight may be composed of dust and gas. Dust
grains are a product of stellar evolution and responsible for the effect of
absorption and reddening of optical starlight (e.g., Draine, 2011). Molecular
gas is known to exist, e.g. in the form of hydrogen (H2) and of carbon
monoxide (CO) (Dame et al., 2001) but also in the form of diffuse interstellar
bands (DIBs) which trace as-yet-unidentified molecules (e.g. Lan et al.,
2015). On the other hand, the atomic gas phase component of the intervening
material is mostly composed of hydrogen. Nevertheless, a fraction of metals is
also present and provides valuable information on the chemistry, ionization
state and gas temperature of the line-of-sight matter. These different species
will absorb the light at characteristic wavelengths, generating detectable
absorption lines in the spectroscopic observations.

Optical interstellar absorption lines in the form of sodium, calcium,
potassium and DIBs have long been observed in spectroscopic binaries (e.g.,
Heger, 1919a, b; Young, 1922), in early-type stars (Struve, 1928; Wilson &
Merrill, 1937; Merrill, 1937; Sanford, 1937; Evans, 1941), in novae (Merrill,
1935), in galaxies (e.g. Heckman & Lehnert, 2000), in active galactic nuclei
(e.g. Baron et al., 2016) and in supernovae (SN; e.g. Penston & Blades 1980).
In contrast to material moving at high velocity, as is the case for SN ejecta,
these intervening lines will appear much narrower than the characteristic
P-Cygni profiles of fast-expanding material. Today, narrow interstellar lines
are routinely detected in the optical spectra of a great number of
astrophysical transients (e.g., Galazutdinov, 2005; Megier et al., 2009; Chen
et al., 2010; Park et al., 2015). The relation of the strength of these lines
with colour excess and extinction has been studied in depth (e.g. Merrill &
Wilson, 1938; Spitzer, 1948; Buscombe & Kennedy, 1968; Hobbs, 1974) providing
empirical relations between the amount of dust extinction and strength of the
gas lines (e.g., Richmond et al., 1994; Munari & Zwitter, 1997; Turatto et
al., 2003; Poznanski et al., 2012; Murga et al., 2015), with caveats for the
sodium line since it saturates (Munari & Zwitter, 1997; Poznanski et al.,
2011). Alternative lines like potassium (e.g. Hobbs, 1974; Munari & Zwitter,
1997; Galazutdinov, 2005; Phillips et al., 2013) and DIBs (Kos & Zwitter,
2013; Phillips et al., 2013; Krełowski et al., 2019) have been claimed to be
superior extinction tracers.

SNe offer a particularly good laboratory to investigate the material in the
line of sight as they are bright and are found in many different environments.
On the other hand, understanding this material is also key to understanding SN
progenitor systems and their explosions, as it can give us information on
stellar evolution and mass loss. Indeed, the study of narrow line-of-sight
lines in transient objects has increased during the last few years, partly due
to the detection of time-variable absorption features in high-resolution
spectra. While intermediate-luminosity red transients embedded in dusty
cocoons show clear signs of sodium line strength evolution (Byrne et al.,
2023), the only SN coming from the core collapse of a massive star showing
this evolution is the broad-lined type Ic, SN 2012ap (Milisavljevic et al.,
2013), with changes in the strength of the DIBs but no variations in the
sodium lines. The authors conclude that the material producing these
variations is nearby, and the SN interacts with it. More surprisingly, a
handful of type Ia SNe – the thermonuclear runaway of white dwarfs in binary
systems – have shown evolving sodium lines in high-resolution spectra: SN
2006X (Patat et al., 2007), SN 1999cl (Blondin et al., 2009), SN 2007le (Simon
et al., 2009), SN 2013gh (Ferretti et al., 2016), and SN 2014J with time-
varying potassium instead of sodium (Graham et al., 2015). This has
significant implications for the debate surrounding the SN progenitor system:
it has been suggested that such variable lines could originate from the CSM
ejected by the progenitor system, in principle favouring the single degenerate
scenario of a binary system composed of a white dwarf and a main-sequence or
red giant (e.g. Moore & Bildsten, 2012). However, double degenerate models
composed of two white dwarfs could potentially also exhibit CSM
characteristics (Raskin & Kasen, 2013; Shen et al., 2013; Levanon & Soker,
2019). Such CSM claims are further emphasized with the observation of a
statistical excess of blueshifted (vs redshifted) sodium absorption lines in
high-resolution SN Ia spectra (Sternberg et al., 2013; Maguire et al., 2013;
Phillips et al., 2013; Clark et al., 2021), that may also be related to other
SN properties such as ejecta velocity and SN colour (Foley et al., 2012a).

On the other hand, measurements of narrow lines from intervening material in
SN Ia spectra of lower resolution have also shown intriguing correlations
between line strength and intrinsic SN properties such as the nebular velocity
shifts (Förster et al., 2012) and the late colour decline rate (Förster et
al., 2013), arguing again for a possible interaction of nearby material with a
sub-group of SNe Ia. This has been recently confirmed by Wang et al. (2019)
also with low/mid-resolution spectra who additionally find possible line
evolution in the interacting SNe Ia.

A significant caveat of the above studies is that lower resolution and signal-
to-noise ratio (S/N) spectra may affect or even prevent proper measurements.
Can low and mid-resolution spectra be used to infer narrow line equivalent
widths and velocities? Can weaker lines such as DIBs and potassium reliably be
detected in such spectra? Can the presence of the high-velocity, broad
features from the SN ejecta interfere with the narrow line measurements? Are
there any biases that could simulate an apparent evolution or blueshift? Given
the large amount of SN spectra gathered in the literature and the scarcity of
methods to investigate the material in the light of sight of the SN, the
capability to robustly use these spectra for narrow lines studies could
provide powerful, statistically meaningful results on the presence and
distribution of these species in the intervening matter.

In order to investigate narrow lines, we use here simulations and a large
sample of observed SNe from the literature that consists of more than 1000
objects of different types –including SNe Ia, SNe II (classical plateau and
linearly declining hydrogen-rich SNe), stripped-envelope SNe (SESNe) with SNe
Ic, SNe Ib and SNe IIb, and interacting SNe including SNe IIn, SNe Ibn and SNe
Icn. The spectra and simulations include various spectral resolutions that
allow us to test several different possible biases in the measurements of
equivalent widths and velocities of narrow line-of-sight lines. After defining
the limitations of our technique, we also look into the likelihood that these
lines in SN spectra evolve over time. In forthcoming papers, we will
investigate the differences in these lines for various SN types, SN properties
and environmental characteristics.

The paper is organised as follows. Section 2 introduces the lines from
intervening matter analysed in this study. We present the data sample in
Section 3 and the automated line measurement in Section 4. The systematic
biases are presented in Section 5, while the evolution analysis and discussion
are in Section 6. We conclude in Section 7.

##  2 Line-of-sight absorption lines

The properties of the ISM of a SN host galaxy and, more specifically, the
environment where the SN exploded – or of the CSM ejected by the progenitor
prior to explosion – can be studied with spectroscopy. The spectral
resolution, usually given by $R=\lambda/\Delta\lambda$ reveals the capacity of
the instrument to resolve narrower lines; we define here high-resolution
spectra as those with $R\gtrsim 10000$ (or $\Delta\mathrm{v}<30$km/s;
Appenzeller 1986) and mid-resolution as $5000<R<1000$ (or
30km/s$<\Delta\mathrm{v}<$60km/s). In nearby objects exploding in reddened
environments, it is possible to detect narrow lines close to the host galaxy’s
rest-frame line wavelength, but also due to the MW close to the observer-frame
line wavelength, at a variety of spectral resolutions and signal-to-noise
ratios, as will be shown later (sections 5.2 and 5.3).

We outline here the main optical absorption lines that can be measured and
utilised to investigate the ISM and the extinction towards SNe.

  * •

Sodium (Na i D ) lines are very prominent and the typical tracers of the ISM.
They were discovered in the solar spectrum by Fraunhofer in 1814. They are the
doublet from the fine structure splitting of the excitation states of neutral
sodium at $\lambda=5891.58$ Å for D2 and $\lambda=5897.56$ Å for D1. Neutral
sodium has an ionization potential of 5.14 eV (Morton, 2003).

  * •

Calcium (Ca ii H&K) lines were also seen by Fraunhofer in sunlight and,
similarly, they correspond to the doublet of the fine structure splitting of
the excitation states of the singly ionized calcium at $\lambda=3934.8$ Å for
K and $\lambda=3969.6$ Å for H, with an ionization potential of 11.87 eV.

  * •

Potassium (K i ) lines were first detected by Dunham (1937). This resonant
doublet from the fine structure splitting of the neutral potassium excitation
states occurs at $\lambda=7664.90$ Å for K1 and $\lambda=7698.97$ Å for K2.
Neutral potassium has an ionization potential of 4.34 eV.

  * •

The diffuse interstellar bands (DIBs) were first discovered in stellar spectra
by Heger (1922). Today, hundreds of DIBs from unidentified molecules in the
ISM have been discovered (see Herbig 1995). Proposed carriers include
polycyclic aromatic hydrocarbons (PAHs), fullerenes and other hydrocarbons
ions and molecules (Salama et al., 1999; Kroto et al., 1985; Motylewski et
al., 2000). We choose here three of the strongest and more isolated DIBs at
$\lambda=4428.2,5780.5,6283.8$ Å.

Table 1 summarizes these narrow lines and their characteristics. We also
include the ”integration window”, a velocity interval around the central
wavelength in which the line will be measured (see section 4.1). It is related
to the strength of the line and therefore the wavelength range it may cover.

Table 1: Narrow spectral features from the intervening medium considered in this study. Line |  Wavelength (Å) |  Ionization potential (eV) |  Integration window (km/s)
---|---|---|---
Na i D |  5889.95 5895.92 |  5.14 |  $\pm$1000
Ca ii H |  3969.60 |  11.87 |  $\pm$900
Ca ii K |  3934.80 |  11.87 |  $\pm$900
K i 1 |  7664.90 |  4.34 |  $\pm$600
K i 2 |  7698.97 |  4.34 |  $\pm$600
DIB 4428 |  4428.20 |  – |  $\pm$750
DIB 5780 |  5780.50 |  – |  $\pm$600
DIB 6283 |  6283.80 |  – |  $\pm$750
Figure 1: Location of the narrow Na i D absorption lines in SNe II (left),
SESNe (middle) and SNe Ia (right panel). The names and phases with respect to
the $B$-band maximum in their observed light curves are marked in each inset.
Low-resolution spectra are presented in the top panels (pseudo-resolution –
see Section 3 – of $R_{p}=$ 1951,3087 and 3869 from left to right), while
higher-resolution spectra ($R_{p}=$ 11877, 9306 and 14879 from left to right)
are at the bottom. The vertical dashed lines indicate the position of the Na i
D for the MW (cyan) and the host galaxy (magenta), with the integration window
of $\pm$ 1000km/s shown in shaded areas.

SNe in low-reddening environments may only display the strongest line-of-sight
lines, which normally are the Na i D absorption features. If the strength of
the absorption line is very low, it indicates negligible material in the line
of sight and very low reddening. Figure 1 shows the location of the narrow Na
i D features for some low-redshift ($z<0.1$) SNe observed with a low (top
panels) and high (bottom panels) resolution instrument. In SNe II (left
panels), the narrow Na i D lines are on the top of the He i $\lambda 5876$
broad line from the ejecta at early phases, but after 30-40 days, they are
around the peak of the broad Na i D emission line (from the SN). In SESNe
(middle panels), they are between the He i/Na i D and the Si ii/He i or
H$\alpha$ (in SNe IIb), while in SNe Ia (right panels), the narrow Na i D
features are between the Si ii lines. The effect these broad lines have in the
narrow lines will be further discussed in Section 5.

In high-resolution spectra, the narrow Na i D lines can be resolved. It is
possible to detect four lines in the spectra (see, for example, bottom panel
of Figure 1), two from the MW ($\lambda 5890$ and $\lambda 5896$) and two from
the host galaxy ($\lambda 5890\times(1+z)$ and $\lambda 5896\times(1+z)$,
where $z$ is the SN redshift), as long as a single cloud component is present.
For low-resolution spectra, only two lines can be detected (one that
corresponds to the MW ($\lambda 5893$, on average) and another related to the
host galaxy ($\lambda 5893\times(1+z)$). When the SN is very nearby ($z<0.04$;
e.g. SN 1987A), the MW and the host galaxy lines can be blended, and just a
single narrow absorption line is visible. These cases are not included in the
current analysis (see the selection criteria in Sections 3 and 5).

The narrow line-of-sight lines play an important role in estimating the
reddening in the line of sight of SNe. Theoretically, this is somewhat
expected as the line equivalent width (EW) of a given gas species $X$ is given
by:

$\mathrm{EW_{X}}=\int\left(1-e^{-\tau_{X}(\lambda)}\right)\,d\lambda,$ (1)

where $\tau_{X}$ is the optical depth, which is a function of the column
density of the species, $N_{X}$, multiplied by the absorption coefficient,
$\kappa_{0}$. In turn, the gas column density depends on the gas-to-dust
ratio, $f_{\mathrm{gd}}$, the mass fraction of the gas, $f_{X}$, and the
ionization fraction, $f_{\mathrm{ion}}$. For low optical depths,
EW${}_{X}\propto\tau_{X}\propto N_{X}$, so that:

$\displaystyle\mathrm{EW_{X}}\propto\tau_{X}=\kappa_{0}N_{X}=\kappa_{0}(1-f_{\mathrm{ion}})f_{X}f_{\mathrm{gd}}N_{\mathrm{dust}},$
(2)

where the dust column density, $N_{\mathrm{dust}}$, is directly proportional
to the extinction $A_{V}$:

$\displaystyle\mathrm{EW_{X}}\propto\kappa_{0}(1-f_{\mathrm{ion}})f_{X}f_{\mathrm{gd}}A_{V}$
(3)

As the optical depth increases, the line profile becomes saturated, and the
equivalent width does not increase linearly with column density. Indeed, the
strength at the central wavelength stalls whereas the Doppler broadening
increases, so that EW${}_{X}\propto\sqrt{\ln{N_{X}}}$. At even larger optical
depths, the collisional broadening dominates and
EW${}_{X}\propto\sqrt{N_{X}}$.

The linear relation between EW and $A_{V}$ at low optical depths (eq. 3)
depends strongly on other factors that are quite uncertain and vary
substantially in our MW. For example, the low ionization potentials of the
gaseous metals (below the Hydrogen ionization potential of 13.6 eV, see Table
1) mean that they are easily photo-ionized, changing the ionization fraction.
Multiple gas clouds in the line of sight can also create several spectral
components whose varying velocities cannot be disentangled in mid- and low-
resolution spectra and thus contribute to the saturation of the line.

Despite these uncertainties in the theoretical relation, it has been shown
empirically, as mentioned above, that the strength of the narrow Na i D
intervening lines correlates with dust extinction, based either on nearby
stars (e.g., Munari & Zwitter, 1997), SNe (Turatto et al., 2003; Sollerman et
al., 2005) or large samples of extragalactic sources whose light is absorbed
in the MW gas and dust (Poznanski et al., 2012; Murga et al., 2015). The
trends are limited by the large dispersion and a distinct saturation for EW
values larger than 0.5 Å in low-resolution data (Munari & Zwitter, 1997;
Poznanski et al., 2011), and 1.8 Å in higher resolution spectra (1 Å for Na i
D 2 and 0.8 Å for Na i D 1; Poznanski et al. 2012) respectively. Poznanski et
al. (2011) suggested that the EW estimated from low-resolution spectra is a
bad estimator of $E(B-V)$, while Phillips et al. (2013) argued that the Na i D
gives an overabundance of $E(B-V)$ for several SNe Ia and the DIB at 5780 Å is
a better tracer of reddening than Na i D . Indeed, Baron et al. (2015)
demonstrate that eight different DIBs correlate well with extinction for a
large sample of extragalactic objects from the Sloan Digital Sky Survey.

In the context of SNe, the presence of the high-velocity, wide features from
the SN makes the measurement of the EW difficult (see section 5.4).
Additionally, time-variable Na i D absorption lines detected in a few SNe Ia
(Patat et al., 2007; Blondin et al., 2009; Simon et al., 2009) also complicate
their use as reddening tracers. However, considering their easy identification
and the scarcity of other methods to constrain dust extinction along the line
of sight, their use is still widespread. That is why a robust methodology and
a careful study of systematic biases, as the one presented here, are needed.

##  3 Sample

Figure 2: Left: Distribution of heliocentric redshifts for the 1719 SNe
included in our sample. Right: Histogram of the number of spectra per SN.

The data analysed in this paper consists of a large heterogeneous sample from
the literature obtained through large web compilations such as SUSPECT
(Richardson et al., 2001), WISEREP111https://www.wiserep.org/ (Yaron & Gal-
Yam, 2012), the Open Supernova Catalog222https://sne.space/ (Guillochon et
al., 2017), and from individual journal publications. It contains SNe Ia from
the thermonuclear runaway of white dwarfs in binary systems, including the
peculiar 91bg-like, 91T-like, 02es-like and Iax; it contains hydrogen-rich SNe
(SN II) from the core-collapse of massive stars that have retained their
hydrogen envelopes, including both old light-curve classifications, SNe IIP
(plateau) and SNe IIL (fast-decline); it contains stripped-envelope (SESNe)
from the core-collapse of massive stars that have lost their outer layers: IIb
with hydrogen shown only initially in their spectra but not later on, Ib
without hydrogen throughout their evolution, Ic without hydrogen nor helium in
their spectra and Ic-BL with broad lines from a central engine; and it
contains various transients related to stellar outbursts and death of massive
stars with clear signs of interaction with circumstellar material like SNe
IIn, SNe Ibn, SNe Icn, and SN impostors, collectively called here ”SNe-int”.

This dataset consists of 12257 spectra of 1788 SNe observed between 1937 and
2023. 2538 of these spectra correspond to 327 SNe II, 2053 spectra to 328
SESNe, 1269 spectra to 116 SNe-int and 6397 spectra to 1017 SNe Ia. The sample
also includes 174 spectra with mid- and high-resolution. Most of them are from
SNe Ia. Only 19 are from SESNe, and 60 are from SNe II. All spectra are
corrected for MW extinction with the maps of Schlafly & Finkbeiner (2011) and
brought to the rest frame. Due to the heterogeneous nature of the sources,
this sample is not complete, neither in volume nor in magnitude. The sample
details can be found in Table LABEL:table_SNe in the Appendix.

Figure 2 (left panel) shows the redshift distribution of our sample. The mean
redshift value is 0.026, while the median is 0.019. The nearest object is SN
1987A with a redshift of 0.00001, while the farthest object is iPTF-13ajg with
a redshift of 0.74. In the right panel of Figure 2, we present the
distribution of spectra per object. 720 SNe have at least two spectra, 873 SNe
have at least three spectra, and 937 SNe have between 3 and 10 spectra. The
objects with the most data are the type Ia SN 2005cf (115), SN 2006X (85) and
SN 2001V (82). On average, we have 6.7 spectra per SN. For the sample of SNe
II, the average is 7.6, 6.4 for SESNe, and 6.4 for SNe Ia.

To give a better idea of the spectral quality of this large and diverse
sample, we show in the top panel of Figure 3 the average resolution of all
spectra, where we define this resolution as the average sampling around 10,000
km s-1of the narrow line (in this case Na i D ):
$\Delta\mathrm{v}_{\mathrm{res}}=\Delta\lambda_{p}c/\lambda_{\mathrm{rest}}$
and we also show the corresponding pseudo-resolution,
$R_{p}=\lambda_{\mathrm{rest}}/\Delta\lambda_{P}$, in the upper axis.
$\Delta\lambda_{p}$ is the wavelength sampling provided in the spectral files
instead of the full-width half-maximum of unresolved lines, which
characterizes an instrument resolution. This is why we call $R_{p}$ a pseudo-
resolution, which is only a proxy for the real spectral resolution of the
instrument. In particular, some spectra have higher wavelength spacing without
meaning the spectral resolution is better. This is especially true for many
old SN spectra from years before 1995 that have a narrower spacing data format
but are not of particularly high resolution. These represent most of the first
peak shown in the distribution, although some real high-resolution spectra are
also present. Other peaks in the distribution are mostly but not exclusively
populated by the spectra of large surveys with its instruments and their
characteristic wavelength spacing: the supernova group at the Harvard-
Smithsonian Center for
Astrophysics333https://lweb.cfa.harvard.edu/supernova/SNarchive.html (CfA;
e.g. Blondin et al., 2012; Modjaz et al., 2014; Hicken et al., 2017), the UC
Berkeley SN group444https://heracles.astro.berkeley.edu/sndb/ (Berkeley; e.g.
Silverman et al., 2012a; Shivvers et al., 2019; Stahl et al., 2020) and
Carnegie Supernova Project555https://csp.obs.carnegiescience.edu/data (CSP;
e.g. Folatelli et al., 2013; Gutiérrez et al., 2017; Stritzinger et al., 2023)
are the most characteristic ones.

On the bottom panel of Figure 3, we also show the average S/N in the same
wavelength regime calculated with the inverse of the root mean square (RMS) of
the difference between the original spectrum and a smoothed spectrum with a
cosine kernel. The distribution is exponentially falling with a mode around 30
and a median of 150.

With the aim of doing a systematic analysis and decreasing the bias in our
sample, we will define several criteria for our data selection (see Table 2)
that will be discussed in the next sections.

Figure 3: Distribution of average velocity resolution (top) around the sodium
doublet and signal-to-noise (bottom) for all spectra in our sample. Vertical
blue dashed lines show the median values of the distributions
($\Delta\mathrm{v}_{\mathrm{res}}=99.2$ km/s and $S/N=155$).

##  4 Automated line measurement

In this section, we outline the automated methodology we have developed to
measure EWs and velocities in a diverse set of spectra.

###  4.1 Equivalent width measurement

Figure 4: SN 2009au (SN II) spectrum around the Na i D lines. The black line
is the spectrum, the blue lines are the estimated flux errors, the red stars
are the nodes, and the red dashed line is the continuum crossing the nodes.
Red dotted lines represent the continua at $\pm 25\%$ of the node separation
being considered. The grey-shaded area is the integrated line within $\pm$1000
km/s. The black dashed vertical line is the central wavelength of the line.
The other vertical blue dotted lines indicate other wavelengths where we
measured EW and whose dispersion provides a systematic uncertainty.

To measure the equivalent width of the various line-of-sight narrow lines from
the host galaxy and/or the MW, we use a fully automated code inspired by
Förster et al. (2012). First, a continuum is found by defining several nodes
around the centre wavelength of the narrow line with separations between 50
and 2000 km s-1. The nodes are evenly spaced (in velocity space), and the flux
at each node is obtained by smoothing locally the spectrum with a cosine
kernel.

Once the continuum is defined, we measure the EW:

$\displaystyle\mathrm{EW}=\sum_{i}\frac{c_{i}-f_{i}}{c_{i}}\Delta\lambda=\sum_{i}1-\frac{f_{i}}{c_{i}}\Delta\lambda,$
(4)

where the flux at each pixel, $f_{i}$, is compared to the continuum at that
pixel $c_{i}$, and $\Delta\lambda$ is the wavelength separation between pixels
in Å.

We sum the area under/over the continuum in a fixed velocity range around the
central wavelength, similar to Förster et al. (2012). The velocity interval
changes for each line that we consider and is given in Table 1. These
intervals were estimated by a visual inspection of the spectra with the most
pronounced narrow features, e.g. SN 2013fc (Kangas et al., 2016) or SN 2003cg
(Elias-Rosa et al., 2006); it is largest for the stronger Na i D ($\pm$1000 km
s-1) which includes two lines. We note that the EW measured in this way sums
all contributions in the given range, meaning that, in principle, more than
one feature could be present. Additionally, it is important to remember that
emission lines are taken into account giving possible negative values of EWs.
The net sum should be close to zero when there is only noise.

Figure 5: Measured equivalent width, EW, over measured uncertainty,
$\sigma_{\mathrm{EW}}$, (top) and equivalent width (bottom) for a spectrum of
SN 2014J around the K i 7665Å (see Figure 27) as a function of the node
separation defining the continuum. The maximum EW is shown with a blue dashed
vertical line, whereas the maximum —EW—/$\sigma_{\mathrm{EW}}$ is shown with a
red line.

The error associated with the EW of a single spectrum has four components: i)
the flux error, $\sigma_{f}$, which is calculated from the RMS between the
flux and the continuum outside the range of the line (but within $\pm$10000
km/s), and an average of this outside error is assumed within the range of the
line (see solid blue lines in e.g. Figures 4 and 27); ii) the error in the
continuum, $\sigma_{c}$, which comes from the dispersion obtained when
assuming a difference of $\pm 25\%$ in the node separation that determines the
continuum (see red dotted lines in Figures 4 and 27); iii) the error in the
integration window for which we re-calculate the EW at $\pm 25\%$ of the range
given in Table 1 for each line; and iv) a systematic error,
$\sigma_{\mathrm{sys}}$, from the dispersion obtained when measuring the line
at eight different positions where no line is expected (see vertical blue
dotted lines in Figures 4 and 27) and the EW should be zero. The first two
sources of error are propagated in Eq. 4 to obtain the error on the EW, and
together with the latter, they are added in quadrature.

The node separation for the continuum plays a crucial role in this technique
(see Figures 4 and 27), so the optimal separation is found by maximizing the
absolute value of the signal-to-noise of the EW (see Figure 5). This is done
by calculating the EW for various node separations in bins of 50 km s-1 and
through a secondary loop in bins of 10 km s-1 centred at the highest
—EW—/$\sigma_{\mathrm{EW}}$ found in the first iteration. The absolute value
is used to make sure that we do not necessarily force positive values on the
EW. We also require that the profile of the narrow line intersects with the
continuum within the velocity window considered.

To verify the accuracy of the automated method, we compare the results
obtained by measuring the EW of Na i D manually using iraf (Tody, 1986, 1993)
and with our automatic software. Figure 6 shows the comparison for a sub-
sample of 1145 low-resolution spectra and 23 high-resolution spectra. The
methods give consistent results with a median absolute deviation of
0.01$\pm$0.22 Å.

Figure 6: Comparison of the EW of Na i D doublet measured manually and with
our automatic software for a sub-sample of 1145 low-resolution spectra (empty
circles) and 32 high-resolution spectra (purple circles). The red line shows a
$x=y$ match. The blue shaded area represents the median scatter between the
two measurements per EW bin. The median and deviation of the full sample are
indicated at the top of the plot. The median error from the automatic method
is 0.11Å. Below zero in the manual measurement, we show in blue the median and
absolute median deviation, $-0.04\pm 0.47$Å, of the automated measurements for
spectra where no line was found visually.

###  4.2 Velocity measurement

Figure 7: Example of a double Gaussian fit (purple) for the host galaxy sodium
doublet within a spectrum of SN 2002bo (black). The wavelength separation
between both Gaussians is kept fixed. In this case, the fitted velocity is
$-37.1\pm 1.5$ km/s compared to $-48.9\pm 16.7$ km/s (without the uncertainty
floor of 35 km/s) obtained with the flux-weighted wavelength average. The
vertical dashed line is the mean rest-frame centre of both lines, and the red
horizontal dashed line marks the continuum.

The velocity measurement is trickier compared to the EW: for weak or absent
lines, it cannot be done, while for stronger lines, there might be multiple
components that are only resolvable with high-resolution spectroscopy.
Obtaining an unbiased estimate of the velocity similar to the EW is perhaps
impossible, but we attempt two approaches. The first is model independent and
consists simply of a line-weighted average wavelength of the area under/over
the continuum measured above (grey shaded areas in e.g. Figure 4) compared to
the rest wavelength of the line:

$\displaystyle\mathrm{v}(\mathrm{km/s})=\frac{c}{\lambda_{\mathrm{rest}}}\left[\frac{\sum_{i}(1-f_{i}/c_{i})\lambda_{i}}{\sum_{i}(1-f_{i}/c_{i})}-\lambda_{\mathrm{rest}}\right],$
(5)

where $c$ is the speed of light in vacuum and $\lambda_{\mathrm{rest}}$ the
restfame wavelength of the line.

The first source of uncertainty stems from the wavelength calibration, which
may vary substantially across our heterogeneous sample and is impractical to
check individually given the lack of, e.g. sky spectra. Instead, we measure
the emission lines from the gas phase of the environment, sometimes present in
SN spectra, and calculate the average velocity dispersion between those lines.
We obtain a characteristic dispersion of 35 km/s (see Appendix A), which we
take as an uncertainty floor from the wavelength calibration. Additional
sources of uncertainty come from re-measuring the velocities when i) the flux
error is added/subtracted to the original flux, ii) when the continuum is
changed by $\pm$25%, and iii) when the integration window is changed by
$\pm$25%.

This simple technique works well when the narrow line is relatively symmetric
but otherwise suffers from several biases. For example, in the case of the Na
i D , if one of the two doublet components is stronger, the flux-weighted
wavelength will be affected, as seen for SN 2002bo (see Figure 7). The bias
may be even stronger in the case of multiple components with different
velocities.

Another approach, commonly used in the literature, consists of a parametric
profile fitting with, e.g. one or multiple Gaussians. For this, one has to
assume the number of components (i.e. number of Gaussians), which one may not
be able to distinguish in the case of low-resolution spectra. In Figure 7, we
show a fit with two Gaussians with a fixed wavelength separation of 6 Å
between the two sodium lines and a common Gaussian width. We directly fit the
ratio of the flux-to-continuum using the continuum obtained in the previous
section. For this particular case, the line-weighted average estimates a
slightly bluer velocity because of the stronger line of the doublet. A
comparison of the automated EW and flux-weighted velocity measurements of the
sodium doublet with the values obtained from a double Gaussian fit is shown in
Figure 8. The match of the EWs is remarkable, while there is a larger
dispersion for the velocity – still consistent within the large uncertainties.
This dispersion is mostly driven by cases where the EW is very weak, and the
measurement of the velocity becomes very uncertain for both methods. Spectra
with stronger lines (see green points) agree better and present lower
uncertainties. The two approaches are overall quite consistent, and we use our
automated technique as our primary method but compare, when necessary, with
the Gaussian approach.

Figure 8: Comparison of EWs (top) and velocities (bottom) measured with the
automated technique and a double Gaussian fit for Na i D with the wavelength
separation fixed. Each plot shows the corresponding best linear MCMC fit
(linmix_err; Kelly 2007) as a solid orange line and the variance with respect
to the fit line in yellow. The $x=y$ line is indicated in dashed green lines.
In the bottom plot, the green points show that cases for which the
EW/$\sigma_{\mathrm{EW}}>1.3$ and EW$>0.5$ have lower uncertainties and agree
better within the two methods. The uncertainties here do not include the
uncertainty floor of 35 km/s.

###  4.3 SN average of multiple spectra

In the case where multiple spectra of a SN are available, and no evolution is
expected or found (see section 6), one can get a better estimate of the EW and
velocity of the intervening line by averaging the individual values found in
the previous sections (sections 4.1 and 4.2). Nonetheless, given the
heterogeneous set of spectra taken with different instruments and with
different underlying SN ejecta throughout the SN evolution, we find that a
better approach consists of taking the median of the individual flux-to-
continuum ratios to generate a stacked ratio with higher signal-to-noise. This
approach is similar to the work done by Poznanski et al. (2012) for Na i D
measured in galaxy spectra. We show an example of 12 stacked flux-to-continuum
spectra of SN 2003cg around the DIB-5780 Å line in Figure 9. After the
stacking, the EW and velocity are calculated in the same way as for individual
spectra. To measure the corresponding error of this stacked value, we do a
bootstrap with 100 iterations in which the same number of flux-to-continuum
spectra are randomly taken from the original sample (some spectra may be
repeated or missing in each iteration), and the EW and velocity are measured
each time. The final asymmetric uncertainties are given by 16% and 84% of the
distribution. The blue lines in Figure 9 contain the 1$\sigma$ bootstrap
realizations for illustration purposes. As for individual spectra, we also
include the systematic errors from the continuum, the integration window and
the dispersion of measuring the line at other wavelengths. A comparison of the
EW measured from stacked spectra with the median or weighted averages of
individual EW measurements is shown in Figure 26.

Figure 9: Stacked flux-to-continuum ratios of 12 spectra of the SN Ia SN
2003cg around the DIB-5780Å line (black) with the 1$\sigma$ uncertainty from
100 bootstrap realizations (blue). The EW and velocity values measured with
the automated technique (grey shading) are shown in the plot. The vertical
dotted line indicates the wavelength of the line.

##  5 Spurious temporal evolution of narrow lines and systematic biases

Figure 10: Na i D EW measured at different epochs since explosion for the MW
narrow line-of-sight line. Each panel shows the SN name and its type. Open
symbols show all available measurements whereas filled symbols are the ones
passing the cuts (see section 5).

Although the methodology laid out in the previous section correctly measures
the EW of narrow lines of spectra of various resolutions, there are some
general caveats inherent to the use of low-resolution spectra that we
investigate further. Some of these biases become evident when studying the
behaviour of the narrow lines over time: we see a conspicuous evolution of the
EW for some SNe. In particular, in Figure 10, we show the evolution of the Na
i D EW for the MW line for different SN types measured with the automated
technique. All objects show variations in their EWs; however, in SN 1999em and
SN 2007gr, a constant decrease is striking. Narrow intervening lines from the
MW are expected to be generally constant as a function of time since they come
from material in our own Galaxy that does not interact with the SN; therefore,
variations in their EWs suggest the interference of external factors. A
similar, albeit more extreme, behaviour was observed for the narrow lines from
the host galaxy (see section 5.4). This is worrying since it can lead to false
physical interpretations like interaction with circumstellar material. These
results motivate us to explore the possible biases that affect the strength of
the narrow line-of-sight lines. They are described in detail in the following
subsections.

Table 2: Number of SNe and spectra after various cuts.

Cut | SNe Ia | SNe II | SE SNe |  SNe-int∗ | Total | Comment
---|---|---|---|---|---|---
| $N_{\rm SN}$ | $N_{\rm spec}$ | $N_{\rm SN}$ | $N_{\rm spec}$ | $N_{\rm SN}$ | $N_{\rm spec}$ | $N_{\rm SN}$ | $N_{\rm spec}$ | $N_{\rm SN}$ | $N_{\rm spec}$ |
None | 1017 | 6397 | 327 | 2538 | 328 | 2053 | 116 | 1269 | 1788 | 12257 | –
$z>0.004$ | 988 | 5759 | 285 | 1890 | 303 | 1694 | 110 | 1177 | 1686 | 10520 |  Section 5.1
S/N$>15$ | 988 | 4171 | 285 | 1515 | 303 | 1099 | 110 | 834 | 1686 | 7619 |  Section 5.3
$m_{c}<0.002\leavevmode\nobreak\ $Å-1 | 701 | 2466 | 214 | 809 | 233 | 856 | 98 | 439 | 1246 | 4570 |  Section 5.4†
$N_{\rm sp}>1$ | 400 | 2165 | 140 | 735 | 142 | 765 | 71 | 412 | 753 | 4077 |  Section 5.6

  * •

∗ These include interacting SNe: SNe IIn, Ibn and Icn.

  * •

† Values for Na i D . The exact cuts and number of SNe/spectra changes
according to line species.

###  5.1 Redshift

In order to ensure that the narrow Na i D absorption line from the MW and the
host galaxy are separately resolved (i.e., to ensure they are not blended), we
set a lower limit cut at 0.004 on the redshift of our sample (independently of
the spectral resolution). This eliminates a small number of very nearby SNe
(see Table 2). As will be shown later in section 5.4, the redshift also plays
an important role in the location of the broad lines of SNe with respect to
the narrow intervening lines affecting their measurement.

###  5.2 Spectral resolution

In this section, we explore the impact of varying spectral resolutions in
measuring the EW and velocity of the lines from intervening matter. We pick
high-resolution spectra ($\Delta v_{\mathrm{res}}<50$ km s-1) and smooth them
with a cosine kernel (see Section 4) with increasing velocity windows to
simulate lower resolutions (Figure 28). We try velocity windows spanning
50-500 km s-1, which conservatively cover our sample’s average velocity
resolutions (see Figure 3). As long as the continuum is relatively flat
(Section 5.4), we find excellent agreement in the measured EW and velocity of
the line-of-sight line for all resolution windows with the original high-
resolution values. This is shown in Figure 11 for several SNe of different Na
i D line strengths. We highlight that the uncertainties in the EW become
larger, as expected, for worse resolutions. Therefore, no cut on the
resolution is applied to our sample.

Figure 11: Measured velocity (top) and equivalent width (bottom) of Na i D as
a function of the decreasing simulated resolution,
$\Delta\mathrm{v}_{\mathrm{res}}$ for the SN 2010ev spectra shown in Figure 28
(black) and also for smoothed high-resolution spectra of SN 2008bc (purple),
SN 2014ao (green) and SN 2014J (red). The squares show the original high-
resolution values of each SN.

###  5.3 Signal-to-noise

We now evaluate the performance of the automated technique when subject to
varying S/N ratios in the spectral range of the narrow line. The S/N, as
explained in section 3, is estimated by calculating the RMS of the observed
spectrum with respect to a smoothed version. We perform a simulation by taking
spectra with high signal-to-noise and randomly perturbing them according to
various decreasing S/N ratios (see Figure 29). For each given S/N, we do 10
simulations and take the mean and standard deviation to show the effect in
Figure 12. The measured EWs are remarkably consistent with decreasing S/N, and
the uncertainties measured by the automated code (dashed error bars) are
always larger than the standard deviation (solid error bars) of the
simulations. Nevertheless, we do see that for weak lines (e.g. those in the
spectra of SN 2002an and SN 2007oc), the average EW starts to increase at very
low S/N, albeit always within the errors. Because of this, we add a
conservative additional minimal S/N cut of 15 that rejects around 1500 spectra
(see Table 2).

The recovered velocities are also consistent, but we note that, as expected,
there is a large dispersion for weak lines as there is not much line flux to
obtain the flux-weighted average wavelength. However, the uncertainties on the
velocities are always much larger than the standard deviations of the
simulated spectra (measured uncertainties are larger than 300 km s-1 and are
not shown in the Figure).

Figure 12: Measured equivalent width (bottom) and velocity (top) as a function
of the S/N for the perturbed SN 2002an spectra shown in Figure 29 (black) and
also for perturbed spectra of SN 2013K (purple), SN 2013ai (green) and SN
2007oc (red). The points are the mean of 10 different simulations for a given
S/N, and its associated error bars are the standard deviations (SN 2007oc has
too large errors on the velocities and are thus not shown). The dashed error
bars of the EW are the mean of the errors calculated from the automated
technique. Upper right ticks show the S/N of the original unperturbed spectra.

###  5.4 Broad line interference

Figure 13: Optical spectral evolution of SN II 2004fc (top) and SN Ia 2006X
(bottom) around the Na i D doublet. An apparent evolution of the narrow line
is seen for both cases that coincide with the P-Cygni profile evolution of Na
i D at the left of the narrow line and of Si II at the right of the narrow
line, respectively.

When measuring the narrow line-of-sight lines in individual spectra of SNe and
investigating their evolution, it becomes evident that several objects have a
seemingly temporal change in the strength of Na i D . Particularly for SNe II,
there is often a strong apparent decrease in the EW as a function of time, but
variations can also be seen for SNe Ia (see, e.g. Figure 13). Upon closer
examination of the SN spectra, these changes appear to be strongly linked to
the appearance and evolution of the emission and absorption components in the
P-Cygni profile of the broad Na i D (for SNe II) and Si ii 6355 Å (for SNe
Ia). If the variations are connected to the development of broad lines (see
Gutiérrez et al. 2017 for the appearance and evolution of P-Cygni lines in SNe
II), they do not constitute real intrinsic variations. In that case, one will
infer a wrong evolution and, more generally, EW measurements that depend on
the underlying profile and are thus not a reliable reflection of the column
density of the intervening material.

To investigate such bias in more detail, we create a simple simulation in
which we input fake narrow lines into various differing P-Cygni profiles and
calculate the recovered EW. For these simulations, we take both observed
optical spectra of seven SNe II, three SNe Ia, and five SESNe, together with
two SN radiative transfer models: the delayed-detonation model of SNe Ia
(Blondin et al., 2015), and the m15mlt3 for SNe II (Dessart & Hillier, 2011;
Dessart et al., 2014). Most of the selected observed SNe do not show evident
narrow Na i D lines; however, when the lines are clearly detected, they are
removed from the spectra by interpolating the continuum to the line
wavelengths so that we can then add simulated lines of our choice.

Once the spectra are clean from the Na i D lines, we smooth them using a
moving average box by convolution with a cosine kernel. Although most spectra
have a good S/N, the smoothing process provides easier identification of the
continuum and the narrow lines. Following the smoothing, the spectra are
normalised at the average position of the narrow lines ($\lambda 5893$). In
parallel, we generate narrow Na i D absorption features with similar
characteristics to those observed in SN spectra. These features are created
for two cases: i) for low-resolution spectra that assume the Na i D doublet is
blended and just one single feature is detected, and ii) for high-resolution
spectra that produce the Na i D doublet (two resolved lines). For the two
cases, we test different strengths of the narrow line.

Figure 14: Narrow Na i D lines artificially placed at various locations for SN
2003bn. Here, we added the narrow Na i D lines with low resolution (single
line). The location of the narrow line is presented in the different panels.

To see the effect of the narrow lines in the SN spectra, we add the narrow Na
i D feature at different locations offset from the central wavelength in all
SN types to simulate different ejecta velocities and redshifts. Figure 14
shows the type II SN 2003bn spectra at 100 days from explosion centred on the
broad Na i D P-Cygni profile with the added narrow Na i D feature at different
locations. Here, the narrow Na i D features were added for low (single line)
resolution spectra. As one can see, the strength of the narrow Na i D lines at
low-resolution visually changes depending on the location in the spectrum; for
example, at $l_{2}$ and $l_{3}$, the line looks weaker. This same procedure of
adding the narrow lines at different positions was also done for other
observed SNe and for the aforementioned SN Ia and SN II models.

Figure 15: EW measurements for simulated intervening Na i D lines located at
rest-frame ($z_{0}=5893$ Å) on top of real sodium P-Cygni profiles from
observed spectra of SN 1999em (left) and on top of simulated spectra of
Dessart & Hillier (2011) (right). Horizontal lines show the EW expected from
the simulated narrow Na i D line: $EW1=1.1$ Å, $EW2=2.2$ Å and $EW3=4.4$ Å.
The measured continuum slope at around 5893 Å is presented in the bottom
panel. The horizontal line shows the cut applied in this study: a maximum
continuum slope value of 0.002 Å-1.

When we then calculate the EW of these simulated narrow lines, we find
significant changes and a non-intrinsic evolution, as can be seen for observed
and modelled spectra of SNe II in Figure 15. It is evident that as the P-Cygni
profile of Na i D develops with time during the SN photospheric evolution, the
narrow line measurement is deeply affected: the EW we measure is lower and
drops as a function of the steepness of the underlying continuum. Not only
does this affect the EW measured for the narrow line, but it may also affect
the photospheric or nebular line from the SN, e.g. producing an apparent
redshift in the broad Na i D emission line at late times in CC-SNe ($>100$
days from explosion; see Figure 30). This effect depends on the SN
cosmological redshift and ejecta expansion velocity, which determine the
location of the narrow line with respect to the P-Cygni profile. At different
locations, the inferred behaviour changes accordingly. However, if we do the
same simulations for high-resolution narrow lines, we do perfectly recover the
EW, no matter the underlying P-Cygni profile.

Figure 16: Same as Figure 15, but for SESNe. Simulated narrow Na i D lines on
top of observed spectra of iPTF13bvn (left) and on top of modeled spectra of
Dessart et al. (2020) (right).

Similarly, we show in Figure 16 the results for a simulation of SESN models
and observed spectra. We also find that the Na i D P-Cygni profile of the SN
ejecta may swallow portions of the narrow line at later epochs in low-
resolution spectra and consequently induce smaller measurements of the EW.
Regarding SNe-int, there are no available models in the literature, and
simulating narrow lines on top of a few objects will misrepresent a very
heterogeneous class. Nonetheless, we expect –and visually confirm– the effect
to be smaller, since strongly interacting SNe generally do not show prominent
P-Cygni profiles.

Figure 17: Same as Figure 15, but for SNe Ia. Simulated narrow Na i D lines on
top of observed spectra of SN 2006X (left) and on top of modeled spectra of
Blondin et al. (2015) (right).

Although less pronounced than SNe II, SNe Ia can also exhibit an apparent
evolution from contamination by the broad ejecta features. This is seen for
observed and simulated spectra (see Figure 17). In this case, the P-Cygni
profile of Si ii 6355 Å at early times, has very high expansion velocities,
and part of it may reach the rest wavelength of the narrow Na i D; later as
the ejecta slows down, the narrow line is no longer engulfed by the absorption
of the P-Cygni profile, and the EW is well recovered. The exact behaviour
naturally depends strongly on the expansion velocity that moves different
parts of the broad line into the narrow Na i D, and the effect will be
stronger for higher-velocity objects (bottom panel of Figure 13). Again, this
is not seen for high-resolution spectra.

It is thus no surprise that using low-dispersion spectra, Wang et al. (2019)
find the evolution of Na i D lines in high-velocity SNe Ia when not properly
considering this effect. Their evolution is strikingly similar to our
simulated case shown in Figure 17. Therefore, variations in the strength of Na
i D narrow lines in SNe Ia at early times are not always indications of
circumstellar interaction. In most cases, when low-resolution spectra are
used, variations are produced by the interference of SN broad lines; in this
case, by the Si ii line.

Figure 18: Fractional residual of measured EW to simulated EW (—EWauto-
EW${}_{\mathrm{sim}}|/$EWsim) as a function of the continuum slope around the
Na i D line for simulated spectra of SNe II (top) and SNe Ia (bottom). EW
measurements include simulated line-of-sight Na i D lines at 6 different
positions around the central wavelength and for three different simulated
narrow EW strengths. The vertical dashed line is our adopted cut to eliminate
spectra with too large a continuum slope, i.e., a big underlying broad line
profile, eliminating cases with more than 30% residuals (horizontal dashed
lines).

Having seen the clear important bias of the interference of the broad line in
the measurement of the narrow lines, is there a way to identify these dubious
cases? Since the bias stems basically from a steep continuum from the P-Cygni
profile, we can measure for a given spectrum the average absolute slope of the
inferred continuum, $m_{c}$, from the automatic method (see Section 4.1) by
doing:

$m_{c}=\frac{1}{<f_{c}>}\frac{1}{N}\sum_{i}^{N}\frac{|f_{c}^{i+1}-f_{c}^{i}|}{\lambda_{c}^{i+1}-\lambda_{c}^{i}},$
(6)

where the sum is over all N nodes of the continuum within $\pm$50Å of the
line, and $<f_{c}>$ is a normalization average of the continuum flux in the
same range. The normalized slope continuum $m_{c}$[Å-1] is a measure of the
steepness of the line in the immediate vicinity of the narrow line. Such
slopes clearly reflect the presence of underlying broad profiles, as can be
seen in Figures 15 and 17. The discrepancy with the real simulated EW
increases for larger continuum slopes, as can also be seen in Figure 18. We
find that a cut of continuum slopes of $m_{c}<0.002$ Å-1 ensures deviations of
less than 30% in the EW measurement for all simulated cases. We strongly
encourage such a cut in $m_{c}$ for the EW measurement of narrow lines in mid
and low-resolution spectra.

###  5.5 Slit size and orientation

Figure 19: MUSE image of the galaxy NGC 988. This synthetic image is created
by collapsing the MUSE data cube across the full wavelength range of the
observation. Orientation is North-up, East-left. The white rectangles mark
four slit angles (0, 45, 90 and 135 degrees) at six different positions,
labelled with numbers from 1 to 6. The width of the slit is 1″ in this case,
but the tests are also performed for a 1.5″ slit.

The stability of the strength of narrow lines could be affected by different
external instrumental factors (besides the resolution and S/N), such as the
size and orientation of the slit used for spectroscopy or the amount of host-
galaxy light entering into the slit. These factors could introduce changes in
the EWs that could be easily confused with intrinsic variations, given the
diversity of instruments and observing conditions. To estimate the effect of
some of these potential biases in the strength of the narrow lines, we measure
the EW of the Na i D lines in different environments and with different
simulated slits. For this aim, we use data obtained with MUSE at the Very
Large Telescope (VLT) as a part of the All-weather MUse Supernova Integral
field Nearby Galaxies (AMUSING, Galbany et al. 2016a) collaboration.

To evaluate the effect of the amount of host-galaxy light entering the slit,
the impact of the slit orientation and its size, we define six different
positions, with four distinct slit angles for each location (0, 45, 90 and 135
degrees with respect to the horizontal West-East axis) and two widths for the
slit, one of 1″ and another one of 1.5″. The length of the slit is taken to be
15″. Figure 19 shows an example of the procedure for NGC 988, the host galaxy
of the type II SN 2017gmr. As one can see, the chosen positions are
distributed along the galaxy, from outer regions to the centre. We note that
position 1 actually includes the SN 2017gmr in the slit, making this a
particularly good study case. We then integrate all spectra within a
rectangular aperture of the slit dimension. The resulting profiles centred on
the narrow Na i D lines are presented in Figure 20 for all different slit
orientations and positions. We show the profiles obtained using the slit of 1″
width on the left panels and the slit of 1.5″ on the right panels. The average
and standard deviations of the Na i D EWs for every position, displayed in
each panel, show small differences (the largest deviation for the slit size is
$\sim 3\%$ and for orientation is $\sim 13\%$ at position 3). Although
positions 2-6 do not enclose a supernova, we expect the effect of just
integrated host light to be an upper limit on the EW since the light
contribution of the SN will dominate. In fact, position 1 with the SN does
present only minor changes of less than 3% with slit size and orientation.
Similar results were also found using several other galaxies targeted by
AMUSING, particularly the largest error associated with these components is
about 0.20 Å, which is generally always lower than the uncertainty estimate of
our automated technique and thus not a major concern henceforth.

Figure 20: Spectra around the Na i D line-of-sight line for six slit positions
of the MUSE cube shown in Figure 19 and with four orientations with respect to
the horizontal W-E orientation shown in colours: blue ($0^{\circ}$), red
($45^{\circ}$), green ($90^{\circ}$) and yellow ($135^{\circ}$). The left
panels are for the 1″ and the right panels for the 1.5″ slits. The measured
EWs and velocities are always consistent within the uncertainties for the
different slit orientations and apertures. We note that the emission line to
the left of Na i D absorption lines is the nebular He I at restframe 5875 Å.

###  5.6 Minimum number of spectra

Figure 21: Top: Median and standard deviation of equivalent width measurements
from multiple spectra randomly taken from the numerous sets of spectra of five
SNe: SN 1999aa, SN 2001V, SN 2002bo, SN 2006X and SN 2012ec, as a function of
the number of spectra used for the measurement. The solid error bars represent
the standard deviation, while the dashed error bars are the median error
calculated by the automated technique. Bottom: Average ratio of the standard
deviation to the calculated uncertainty from the automated technique for
nearly 30 SNe with more than ten spectra. The error bars show the dispersion.

In this section, we evaluate how the automated measurement of the EW changes
with the number of spectra used in the stacking procedure of section 4.3. We
take non-evolving SNe with more than 20 spectra, passing the previous cuts (in
particular, the continuum slope). We then explore the effect of using 1 to 15
spectra for calculating the EW. For each number of spectra used, we draw 50
random combinations of spectra and calculate the median and standard
deviation, as shown in Figure 21. Besides the dispersion of all draws, we also
show the typical median uncertainty obtained from our method with dashed error
bars. The bottom plot shows that the uncertainties are conservative when using
more than 2-3 spectra. The standard deviation indicates a high dispersion in
the measured values for one and sometimes two spectra, even beyond the
measured uncertainties. The bottom panel of Figure 21 shows that this
underestimation can reach a factor of 2-10 when using only one spectrum. We
thus prefer to discard SNe with only one spectrum because of the large
dispersion of $1-2$ Å of some cases.

Our final set of spectra and SNe is considerably reduced after the four cuts
(see Table 2): redshift, signal-to-noise in the spectra, continuum slope,
signal-to-noise in the EW and minimum number of spectra. However, we believe
these cuts provide a clean sample suitable for the automated measurement of
narrow lines.

##  6 Na i D EW evolution analysis and discussion

The evolution of narrow lines from intervening material has been largely
studied in the literature, and variations in the EW have been reported for
various transients (e.g. Byrne et al., 2023; de Jaeger et al., 2015). Such
observations have important implications for the presence and possible
interaction with circumstellar matter or nearby patchy interstellar clouds. In
high-resolution spectra of SNe Ia, SN 2006X (Patat et al., 2007), SN 2007le
(Simon et al., 2009), SN 2013gh (Ferretti et al., 2016), and SN 2014J (Graham
et al., 2015) showed changes in the EWs and profiles of the narrow intervening
features. In the previous section, we showed with simulations that the EW
measurement of narrow lines in high-resolution spectra is not affected by
systematic biases; therefore, changes in profiles and intensities presented in
these works probably represent true changes in the absorption of material in
the line of sight of those SNe. On the other hand, we have shown that
variations in low-resolution spectra should be analyzed with care. In
particular, the effect of the varying morphology of the underlying ejecta
P-Cygni profile must be considered. Wang et al. (2019) found variable Na i D
absorption lines in low-resolution spectra of high-velocity SNe Ia at early
phases. We found a similar behaviour at similar phases in our low-resolution
simulation: a decrease in the EW, followed by an increase, finally settling
onto a constant evolution. As shown in Figure 17, the changes in the EW are
closely related to the slope of the continuum. When a continuum cut is
considered, the variability in most objects disappears. This does not mean
that the variability of the narrow absorption lines in low-resolution spectra
cannot be real, but we suggest that careful cuts must be considered before
performing such analysis and drawing conclusions.

After including the cuts described in previous sections (see summary in Table
2), we evaluate the behaviour of the strength of the narrow Na i D absorption
lines as a function of time. Since EW measurements from an individual spectrum
can be unreliable (Section 5.6), we require at least two spectra per epoch
range for a given SN; we then stack them and recalculate the EW (see Section
4.3). For this analysis, we explore various possibilities, using at least two
or at least three spectra per SN in intervals of 10 to 20 days. While stacking
SN spectra in time intervals will wash out shorter timescale variations, it
ensures that we can study the time evolution of the EW with our methodology
and low-resolution spectra. Once the spectra are stacked, we look at the
evolution of the narrow lines both statistically and individually.
Particularly, we focus on the EW of the line because it is more robust than
the velocity (see section 4.2).

###  6.1 Statistical evolution of EW

We study the main SN types, SNe Ia, SNe II, SESNe and SNe-int, for statistical
signs of evolution. Finding population-wide signs of evolution could give us
important insight into the progenitor systems and nearby material of a given
population. For instance, a large number of SNe II are known to be subject to
very early interaction with CSM (e.g. Bostroem et al., 2023; Martinez et al.,
2023) that could perhaps be revealed in the narrow absorption line evolution.
SNe-int are by definition subject to interaction with nearby material and thus
good candidates to observe evolution in the narrow absorption lines, as is the
case for SN 2011A(de Jaeger et al., 2015); however, the line is often absent
despite strong CSM interaction(e.g. Ofek et al., 2010; Pastorello et al.,
2018). To investigate this, we require a common reference epoch for each
group. We use the maximum $B$-band dates obtained through multi-wavelength
light-curve fits with spectrophotometric templates of each SN type (see
Appendix B). The epochs are normalized to this reference epoch and put in
rest-frame. All spectra of a given SN in time intervals of, e.g. 10 days, are
stacked, and the EW is measured. We investigate the evolution of the
$\sigma$-deviations per epoch interval $i$ with respect to the all-epoch
stacked EW measurement of each SN, i.e. $\overline{\mathrm{EW}}$, so that:

$\delta\sigma_{i}=\frac{\mathrm{EW}_{i}-\overline{\mathrm{EW}}}{\sqrt{\sigma_{\mathrm{EW}_{i}}^{2}+\sigma_{\overline{\mathrm{EW}}}^{2}}}.$
(7)

Figure 22 shows the evolution of the distribution of these deviations for the
Na i D lines of the four SN types considered. In general, we can see that
there is no large evolution of these groups within the uncertainties. For
SESNe, there seems to be a decrease after 50 days post maximum, although this
is likely due to low-number statistics. Similarly, SNe Ia appear to increase
within 30 and 50 days post maximum, but with a small number of SNe at these
epochs, this is mainly driven by one SN (SN 2012dn), as we will see in the
next section. SNe-int also have very low statistics making any conclusion
rather difficult. When investigating the evolution with different epoch
intervals between 10-15 days and also requiring a minimum of three spectra per
SN per interval instead of two, we obtain similar results.

Additionally, we also investigate narrow lines from other species, such as Ca
ii and K i finding no evidence of evolution. However, it is possible that
smaller sub-groups might show signs of changes in time. Given the small
statistics after all our cuts, we leave this for a future investigation. In
the next section, we analyse the evolution of individual well-sampled SNe.

Figure 22: Na i D Violin plots of Na i D EW evolution for each SN type (from
top to bottom: SNe II, SNe Ia, SESNe and SNe-int) showing the mean and extreme
of the distribution (white point in the box and edges of central vertical
lines) and a kernel density probability estimation (coloured area). The number
of SNe within each distribution at each epoch is indicated above each bin. We
require at least two spectra per SN in intervals of 15 days for stacked EW
measurements.

###  6.2 Individual evolution of EW

To analyze the evolution of the EW of narrow absorption lines for individual
objects, we take the stacked EW of a minimum of two spectra in intervals of 10
days, as in the previous section. Besides a visual inspection, we apply two
methods to identify possible candidates for evolution automatically:

  * •

Linear slope: we fit the EW of all intervals of each SN to a simple line with
a Bayesian fit from which the posterior distribution gives us the probability
for the slope to be different from zero (Kelly, 2007). A significantly non-
zero slope ($p>95\%$) indicates that there is a general monotonic trend in the
evolution of the SN.

  * •

Cumulative sum: to try to capture more complicated evolution, we also do a
cumulative sum analysis (CUSUM, Page, 1954) that monitors deviations from a
target, in this case, the EW value obtained from the stack of all the spectra
(at all epochs) of a given SN, $\overline{\mathrm{EW}}$. Then the algorithm
measures accumulated deviations, chosen here as a drift of half the
uncertainty on the stacked EW, $0.5\sigma_{\overline{\mathrm{EW}}}$, to both
positive and negative directions, so that:

$\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle L_{0}$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle H_{i}$ $\displaystyle=$
$\displaystyle
max\left[0,H_{i-1}+(\mathrm{EW}_{i}-\sigma_{\mathrm{EW}_{i}})-(\overline{\mathrm{EW}}+0.5\sigma_{\overline{\mathrm{EW}}})\right]$
$\displaystyle L_{i}$ $\displaystyle=$ $\displaystyle
min\left[0,L_{i-1}+(\mathrm{EW}_{i}+\sigma_{\mathrm{EW}_{i}})-(\overline{\mathrm{EW}}-0.5\sigma_{\overline{\mathrm{EW}}})\right]\,$

where $H_{i}$ and $L_{i}$ indicate the positive and negative cumulative
deviations updated at each time step $i$. We also take into account the
uncertainty in each individual EWi. This measures the cumulative deviations
starting from the earliest epoch interval of the SN and increasing in time. In
order to not give higher weight to the initial time bins, we also start the
algorithm from the latest epoch interval going in decreasing time steps. The
final cumulative $H$ and $L$ are the average of both sequential cumulative
deviations. SNe with more than two cumulative deviations above the error on
the stacked EW are considered viable candidates.

Figure 23: Top: Na i D EW evolution for SN 2006X (left) and SN 2014G (right)
before/after cuts (empty/filled black points) and stacked measurements of
points after culls in intervals of 15 days requiring minimum 2 spectra (red
points). Horizontal lines indicate the stacked EW and error from all spectra
that passed the cuts. A Bayesian linear fit to the red points is shown with
light red lines (1000 realizations) and a corresponding median with a dotted
red line. The resulting slope is negative with 68.7% and 93.7% probability,
respectively for both SNe. Bottom: Cumulative deviation analysis above/below
(upper/lower triangles) the stacked EW (corresponding to the zero horizontal
line) and its associated errors (dashed lines). Significant points are shown
in purple.

Figure 23 shows these two methods applied to the measurements of SN 2006X (SN
Ia) and SN 2014G (SN II) after the cuts (filled black points). Doing a
Bayesian linear fit to the stacked EWs per epoch interval (red points) results
in a slope consistent with zero for SN 2006X and no significant points in the
cumulative deviation analysis. SN 2014G, on the other hand, has a slope with a
probability of 93.7% to be non-zero and one significant cumulative deviation
point. Both of these SNe are considered to be not evolving in time according
to our diagnostics. However, it is important to observe that if we were to do
our analysis on all data points without cuts (empty black points), these SNe
would have non-zero slope probabilities of 88.4% and 100%, with 4 and 12
significant cumulative deviation points, respectively. Both SNe would be
considered to have an evolution. This shows the importance of our cuts,
particularly the continuum slope that prevents strong blending with the broad
P-Cygni profile (see Section 5.4).

When analysing various narrow lines for all SNe after cuts and stacking at
least two spectra per 10 or 15-day intervals, we find that very few SNe are
tagged as possibly evolving according to our two diagnostics. For every case,
we ensure that no equivalent evolution is seen for the same lines of the MW
and inspect them visually. The few SNe tagged (e.g. SN 2001N, SN 2012dn) have
few points in the nebular phase with stronger EW. However, these are a very
small number and, in some cases, still noisy after stacking per interval. In
the case of SN 2012dn, a 03fg-like ”super-Chandra” SN Ia (Chakradhari et al.,
2014; Taubenberger et al., 2019), the narrow Na i D at $\sim 40$ days after
the maximum gets mixed with features due to iron-group elements and it is
difficult to disentangle the two.

SNe known to show variation in their lines with high-resolution spectra, e.g.
SN 2006X, SN 2007le, SN 2013gh or SN 2014J, are not picked up by our culls,
but we have predominantly low-resolution spectra for which such small
variations remain within the uncertainties. We have no access to high-
resolution spectra for SN 2006X, SN 2013gh and only one for SN 2007le. For SN
2014J, if we restrict our analysis to the 10 high-resolution spectra we have
and do not make any cuts on the continuum slope, we find no evolution of the
EW for Na i D and no evolution of the three spectra that cover the wavelengths
of K i . Graham et al. 2015 found evolution in K i lines in other spectra of
even higher resolution (R$>$100000). In the case of the type IIn SN 2011A (de
Jaeger et al., 2015), we do see the increase in EW with time that they report.
Nonetheless, with only 5 spectra after cuts (and only two stacked points per
15-day interval), the linear slope is not significantly positive, and the
stacked EW of all spectra is too influenced by the late spectra with
increasing EW, such that no significant deviation points are found.

We conclude that the current dataset, after all our cuts and with our
methodology, is consistent with no variation in the EW of narrow lines over
time within the uncertainties. More subtle variations are naturally possible
but necessitate better sampling and, foremost, higher spectral resolution.

##  7 Conclusions

In this paper, we have presented a statistical analysis of the evolution of
narrow line-of-sight lines from intervening material (Na i D , Ca ii H&K, K i
, DIBs) visible in nearby SNe. We have analyzed over 11000 spectra of more
than 1600 SNe with various resolutions. To quantify the properties of these
narrow lines, we developed a robust automated methodology allowing for the
estimation of their EWs and velocities. When searching for possible EW
evolution, we found systematic variations in the Na i D line from the MW and
the host galaxies. Changes in the Na i D depth from the MW narrow interstellar
lines are in principle not expected because the material in our own Galaxy is
at sufficiently large distances from the SN. The strength of the narrow
absorption lines in low and mid-resolution spectra may thus be affected by
different external factors that we explore: the signal-to-noise of the
spectra, the spectral resolution, the slit size and orientation, the number of
spectra used and most importantly, the interference of the broad P-Cygni lines
from the SN. We found that these biases, particularly those associated with
presence of the broad lines, induce a conspicuous apparent evolution
(variability) over time in many SNe. For instance, in SNe II, the narrow
sodium line overlaps with the broad sodium from the SN ejecta, and in SNe Ia
the narrow sodium overlaps with the P-Cygni profile of Si ii . Indeed, we
confirm this with simulations in which we input fake narrow lines into
different locations of the broad SN P-Cygni lines and calculate the recovered
EW. In our simulations of low-resolution spectra, we found large significant
changes in the EW depending on the underlying profile. However, in our high-
resolution simulations, the strength of the narrow lines was not affected by
the interference of the broad SN P-Cygni profiles.

Based on these findings, we presented an easy way to detect and exclude cases
of high contamination of the SN profile in the narrow line: when the
normalized slope of the continuum is too high, i.e. $m_{c}>0.002$ Å-1, there
is a broad component underneath blending with the narrow line. We show that
excluding these cases with a steep continuum is essential in the study of
narrow lines from intervening material in the line of sight. Not taking this
into account may lead to important biases in the frequently used estimations
of dust extinction through the EW$\propto A_{V}$ relation (Eq. 3) in low-
resolution spectra, and it may lead to wrong inferences of the evolution of
the lines and the presence of nearby material. Previous claims of the
evolution of narrow lines in low-resolution spectra should be revisited in
light of this study.

We also strongly encourage the use of more than one low-resolution spectrum in
the estimation of the EW. Fluctuations of the S/N, the spectral resolution,
and the morphology of the underlying P-Cygni profile over time can generate
changes in these lines that are washed out with a larger number of spectra
stacked to obtain a more accurate EW. We show that at least 2-3 spectra are
needed to measure robust EWs with conservative uncertainties.

Finally, after considering all these effects, we analysed the temporal
evolution of line-of-sight features in a large sample of nearby SNe to detect
any possible statistical variation in their EWs. We found no overall evolution
for three principal SN groups: SNe II, SNe Ia and SESNe, in all narrow lines
considered. Significant individual SN variations were also ruled out based on
our methodology and low-resolution spectra.

In this work, we have paved the way for robust measurement of narrow
absorption lines in a variety of spectra from different instruments and
resolutions. Armed with this methodology, we can now use a large sample to
study the variations of the SN-averaged narrow lines according to SN type, to
SN properties, and to host galaxy characteristics. A wealth of information on
the pervading intervening material is contained in those lines providing
essential clues to the immediate and farther vicinity of SNe and their
progenitors, with far-reaching consequences for our understanding of the ISM
cycle of galaxies and the persistent question of extinction in supernova
cosmology.

###### Acknowledgements.

We sincerely thank the referee for very useful comments that improved the
quality of the paper. We heartily thank all the research groups and observers
for making their data public. This research has made use of the Berkeley SNDB
database as well as the CfA Supernova Archive, which is funded in part by the
National Science Foundation through grant AST 0907903. S.G.G. acknowledges the
financial support from the visitor and mobility program of the Finnish Centre
for Astronomy with ESO (FINCA), funded by the Academy of Finland grant nr
306531. S.G.G. and A.M.M. acknowledge support by FCT under Project CRISP
PTDC/FIS-AST-31546/2017 and Project No. UIDB/00099/2020. C.P.G. acknowledges
financial support from the Secretary of Universities and Research (Government
of Catalonia) and by the Horizon 2020 Research and Innovation Programme of the
European Union under the Marie Skłodowska-Curie and the Beatriu de Pinós 2021
BP 00168 programme. C.P.G. and L.G. recognize the support from the Spanish
Ministerio de Ciencia e Innovación (MCIN) and the Agencia Estatal de
Investigación (AEI) 10.13039/501100011033 under 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. L.G. also thanks the financial support from the
European Social Fund (ESF) ”Investing in your future” under the 2019 Ramón y
Cajal program RYC2019-027683-I. J.P.A. acknowledges support by ANID,
Millennium Science Initiative, ICN12_009. A.M.G. acknowledges financial
support by the European Union under the 2014-2020 ERDF Operational Programme
and by the Department of Economic Transformation, Industry, Knowledge, and
Universities of the Regional Government of Andalusia through the FEDER-
UCA18-107404 grant. S.M. acknowledges support from the Research Council of
Finland project 350458.

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##  Appendix A Wavelength calibration uncertainty

The heterogeneous dataset used in our analysis has been wavelength-calibrated
by different groups for different instruments and observing conditions. Here
we estimate an average uncertainty of the wavelength calibration for the
entire sample. For this, we randomly take 1556 spectra of 180 core-collapse
SNe from our sample and use our automated technique (see §4.2) around three
narrow emission lines characteristic of gas-rich environments: H$\alpha$
(6563Å), H$\beta$ (4681Å) and [OIII] (5007Å). We use the following integration
windows: $\pm$ 750 km/s, $\pm$ 750 km/s and $\pm$ 900 km/s, respectively. An
example measurement of H$\alpha$ for a spectrum of SN 2009bz is shown in
Figure 24. The dispersion of the three line velocities for this spectrum gives
33.2 km/s. Of the 1556 spectra, we select 696 that clearly show at least two
of the three lines, i.e. requiring that EW/$\sigma_{\mathrm{EW}}>1.3$ and EW
$>0.5$Å; and we calculate the median dispersion for the sample finding 35.0
km/s. Doing the same analysis using Gaussian fits instead, we obtain proper
fits with EW$>0.5$ in at least two lines for 767 spectra for which the average
of the velocity dispersion gives 36.6 km/s. This is in remarkable agreement
with the automated value confirming that this dispersion is not dependent on
the discussed methodologies. This value could reveal calibration uncertainties
or real physical differences in the elements of the gas clouds, or a
combination of both. We conservatively take it here as an intrinsic
calibration uncertainty that is added in quadrature to other uncertainties of
the method.

Figure 24: Spectrum of SN 2009bz around the emission line H$\alpha$ and
automated measurement of EW and velocity. The quoted uncertainty in velocity
does not include the wavelength calibration. The dispersion of the velocities
of the three lines (H$\alpha$, H$\beta$ and [OIII]) for this spectrum is 33.2
km/s.

##  Appendix B $B$-band maximum dates

Figure 25: Example of multi-band light-curve fits with SiFTO (Conley et al.
2008) and different spectrophotometric templates to SN2004du (SN II, up),
SN2011hs (SN IIb, middle) and SN2012hr (SN Ia, bottom).

To obtain the $B$-band maximum reference epochs used in section 6, we perform
multi-band light-curve fits to every SN. We use a light-curve fitter commonly
used for SNe Ia (Conley et al. 2008) but with different spectrophotometric
templates for each type: the template by Hsiao et al. (2007) for SNe Ia, the
template of Nugent et al. (2002) for SESNe and a model of SNe II used in the
PLAsTiCC challenge (Kessler et al. 2019) based on empirical spectral energy
distributions that are a linear combination of three ‘eigenvector’ components.
For SNe-int, we try both, the template for SNe II and for SESNe, choosing
visually the best obtained fit. Example fits are shown in Figure 25. All fits
are visually inspected, and SNe with poor data or bad fits are discarded.
Although core-collapse SNe are not as homogeneous as SNe Ia, and this fitter
was devised for SNe Ia, this simple approach gets rough estimates that allow
us to have epochs in the same reference system. We also attempted to use the
explosion epochs as a reference for SNe II (Gutiérrez et al. 2017), confirming
no general evolution of EW with time in all three SN types.

##  Appendix C Figures

We present here additional figures: Figure 26 shows the comparison of the EW
measured from the stacked flux-to-continuum ratios of several spectra of a
given SN (see §4.3) versus the median and the weighted average of individual
spectra for a given SN (the uncertainties come from the median absolute
deviation and the weighted error, respectively); Figure 27 displays the impact
of the continuum node separation on the EW measurement (see §4.1); Figures 28
and 29 shows how degrading the spectral resolution (§5.2) and the signal-to-
noise (§5.3) affect the narrow line and its EW measurement; and Figure 30
demonstrates that the narrow line at nebular times can emulate a blueshift in
the line from the SN ejecta (see §5.4).

Figure 26: Comparison of Na i D EW measured from the stack of multiple spectra
of each SN vs the median of individual EW (left) and the weighted average
(right). The dashed lines indicate a $x=y$ relation. Figure 27: SN 2014J
spectrum around the K i 7665Å line showing different measured continua by
increasing the node separation from top left to bottom right: 250, 400, 600,
750, 950 and 1150 km/s. The black line is the spectrum, the blue lines are the
estimated flux errors, the red stars are the nodes and the red dashed line is
the continuum crossing the nodes. Red dotted lines represent the continua at
$\pm 25\%$ of the node separation being considered. The grey-shaded area is
the integrated line within $\pm$600 km/s. The dashed vertical line is the
central wavelength of the line.  Figure 28: SN 2010ev high-resolution spectrum
(top left) smoothed with decreasing kernel resolutions (increasing
$\Delta\mathrm{v}_{\mathrm{res}}=100-500$km/s) around Na i D from left to
right and top to bottom. The estimated error on the flux is shown in blue. The
automated methodology outlined in section 4 calculates a continuum (shown in
red) and integrates through a fixed spectral window (grey shading). Figure 29:
SN 2002an high-signal spectrum (top left) perturbed to various decreasing S/N
around Na i D from left to right and top to bottom. The estimated error on the
flux is shown in blue. The automated methodology outlined in section 4
calculates a continuum (shown in red) and integrates through a fixed spectral
window (grey shading). The vertical dashed line shows the rest-frame central
line of Na i D .

Figure 30: Observed spectra (black) of SN 1999em (left) and 2007aa (right) at
165 and 105 days from the explosion. The vertical dashed lines show the rest-
frame position of the strongest lines. The simulated spectrum of SN 2007aa,
where a fake Na i D narrow line was added at $\lambda=5893$ Å (EW$=1.1$ Å), is
presented on top of the observed spectrum (cyan). The peak of the Na i D seems
redshifted in SN 1999em, compared to H$\alpha$, which is at the rest-position.
A comparable shift can be obtained when a narrow line is located on the top of
the broad SN P-Cygni line of SN 2007aa.

##  Appendix D Tables

In table LABEL:table_SNe we present all SNe used in this analysis with their
type, redshift, number of spectra and the references for their spectra.

Table A1: Sample of SNe initially included in our sample. SN | Type | Redshift | # of spectra | References
---|---|---|---|---
|  | SNe Ia |  |
SN 1937C | Ia | 0.001 | 2 | Casebeer et al. 2000
SN 1937D | Ia | 0.002 | 1 | Casebeer et al. 2000
SN 1939A | Ia | 0.003 | 2 | Casebeer et al. 2000
SN 1954B | Ia | 0.005 | 10 | Casebeer et al. 2000
SN 1956A | Ia | 0.004 | 8 | Casebeer et al. 2000
SN 1957A | Ia | 0.002 | 2 | Casebeer et al. 2000
SN 1957B | Ia | 0.004 | 7 | Casebeer et al. 2000
SN 1959C | Ia | 0.010 | 1 | Casebeer et al. 2000
SN 1960F | Ia | 0.006 | 8 | Casebeer et al. 2000
SN 1960H | Ia | 0.002 | 3 | Casebeer et al. 2000
SN 1960R | Ia | 0.002 | 2 | Casebeer et al. 2000
SN 1962J | Ia | 0.005 | 3 | Casebeer et al. 2000
SN 1963I | Ia | 0.001 | 1 | Casebeer et al. 2000
SN 1964E | Ia | 0.004 | 2 | Casebeer et al. 2000
SN 1966J | Ia | 0.002 | 1 | Casebeer et al. 2000
SN 1968E | Ia | 0.013 | 1 | Casebeer et al. 2000
SN 1968I | Ia | 0.006 | 1 | Casebeer et al. 2000
SN 1971G | Ia | 0.006 | 1 | Casebeer et al. 2000
SN 1981B | Ia | 0.006 | 13 | Branch & Venkatakrishna 1986
SN 1983G | Ia | 0.004 | 8 | Foley et al. 2008
SN 1986G | Ia | 0.002 | 39 | Cristiani et al. 1992; Foley et al. 2008
SN 1989A | Ia | 0.008 | 1 | [2]
SN 1989B | Ia | 0.002 | 12 | [2]
SN 1989M | Ia | 0.005 | 4 | [2]
SN 1990G | Ia | 0.036 | 2 |  [2]; Gomez et al. 1996
SN 1990M | Ia | 0.009 | 6 |  [2]; Gomez et al. 1996
SN 1990N | Ia | 0.003 | 11 |  [2]; Gomez et al. 1996
SN 1990O | Ia | 0.031 | 4 |  [2]; Gomez et al. 1996
SN 1990R | Ia | 0.016 | 6 |  [2]; Gomez et al. 1996
SN 1990Y | Ia | 0.039 | 1 | [2]
SN 1991ad | Ia | 0.070 | 7 | Gomez et al. 1996
SN 1991ak | Ia | 0.010 | 3 | [2]
SN 1991am | Ia | 0.061 | 1 | [2]
SN 1991as | Ia | 0.011 | 1 | [2]
SN 1991at | Ia | 0.041 | 1 | [2]
SN 1991ay | Ia | 0.050 | 1 | [2]
SN 1991B | Ia | 0.009 | 5 |  [2]; Gomez et al. 1996
SN 1991bb | Ia | 0.027 | 2 | [2]
SN 1991bc | Ia | 0.021 | 3 |  [2]; Gomez et al. 1996
SN 1991bd | Ia | 0.013 | 2 |  [2]; Gomez et al. 1996
SN 1991bf | Ia | 0.030 | 1 | [2]
SN 1991bg | Ia | 0.004 | 34 |  [2]; Filippenko et al. 1992a; Gomez et al. 1996; Turatto et al. 1996
SN 1991bh | Ia | 0.028 | 1 | [2]
SN 1991bj | Ia | 0.018 | 2 |  [2]; Gomez et al. 1996
SN 1991F | Ia | 0.006 | 3 | Gomez & Lopez 1995
SN 1991K | Ia | 0.017 | 6 | Gomez et al. 1996; Matheson et al. 2001
SN 1991M | Ia | 0.007 | 7 |  [2]; Gomez et al. 1996
SN 1991O | Ia | 0.049 | 1 | [2]
SN 1991S | Ia | 0.055 | 2 |  [2]; Gomez et al. 1996
SN 1991T | Ia | 0.006 | 30 |  [2]; Gomez et al. 1996; Filippenko et al. 1992b; Mazzali et al. 1995
SN 1991X | Ia | 0.009 | 1 | Gomez et al. 1996
SN 1992ac | Ia | 0.052 | 3 | Gomez et al. 1996; Gómez & López 1998
SN 1992ah | Ia | 0.028 | 2 |  [2]; Gomez et al. 1996
SN 1992ap | Ia | 0.030 | 1 | [2]
SN 1992G | Ia | 0.005 | 12 |  [2]; Gomez et al. 1996
SN 1992M | Ia | 0.047 | 2 | [2]
SN 1993aa | Ia | 0.025 | 1 | [2]
SN 1993ab | Ia | 0.014 | 1 | [2]
SN 1993ac | Ia | 0.049 | 4 | [2]; [3]
SN 1993ae | Ia | 0.020 | 3 | [2]; [3]
SN 1993ai | Ia | 0.035 | 1 | [2]
SN 1993aj | Ia | 0.078 | 3 | [2]
SN 1993C | Ia | 0.013 | 3 | [2]
SN 1993Y | Ia | 0.020 | 1 | [2]
SN 1993Z | Ia | 0.004 | 9 | [2]
SN 1994ab | Ia | 0.034 | 1 | [2]
SN 1994ae | Ia | 0.004 | 21 | [2]; [3]
SN 1994B | Ia | 0.089 | 2 | [2]
SN 1994D | Ia | 0.001 | 77 |  [2]; [3]; Patat et al. 1996; Gomez et al. 1996
SN 1994E | Ia | 0.064 | 1 | [2]
SN 1994J | Ia | 0.056 | 1 | [2]
SN 1994M | Ia | 0.023 | 10 |  [3]; Gomez et al. 1996
SN 1994Q | Ia | 0.030 | 9 |  [2]; [3]; Gomez et al. 1996
SN 1994S | Ia | 0.015 | 6 |  [2]; [3]; Gomez et al. 1996
SN 1994T | Ia | 0.035 | 6 | [2]; [3]
SN 1994U | Ia | 0.004 | 2 |  [2]; Gomez et al. 1996
SN 1994X | Ia | 0.056 | 1 | [2]
SN 1995A | Ia | 0.032 | 1 | [2]
SN 1995ac | Ia | 0.050 | 6 | [2]; [3]
SN 1995ak | Ia | 0.023 | 7 | [2]; [3]
SN 1995al | Ia | 0.005 | 15 | [2]; [3]
SN 1995bd | Ia | 0.015 | 9 | [3]
SN 1995C | Ia | 0.027 | 1 | [2]
SN 1995D | Ia | 0.007 | 18 | [2]; [3]
SN 1995E | Ia | 0.012 | 9 | [2]; [3]
SN 1995L | Ia | 0.024 | 1 | [2]
SN 1995T | Ia | 0.056 | 1 | [2]
SN 1995Y | Ia | 0.018 | 4 | [3]
SN 1996ab | Ia | 0.130 | 1 | [3]
SN 1996ac | Ia | 0.026 | 1 | [3]
SN 1996ai | Ia | 0.003 | 12 | [2]; [3]
SN 1996bk | Ia | 0.007 | 1 | [3]
SN 1996bl | Ia | 0.024 | 4 | [3]
SN 1996bo | Ia | 0.017 | 5 | [3]
SN 1996bt | Ia | 0.009 | 1 | [3]
SN 1996bv | Ia | 0.017 | 2 | [2]; [3]
SN 1996by | Ia | 0.014 | 6 | [3]
SN 1996C | Ia | 0.027 | 3 | [3]
SN 1996ca | Ia | 0.016 | 4 | [3]
SN 1996O | Ia | 0.037 | 1 | [2]
SN 1996P | Ia | 0.016 | 1 | [2]
SN 1996V | Ia | 0.024 | 1 | [3]
SN 1996X | Ia | 0.007 | 32 | [3]
SN 1996Z | Ia | 0.008 | 6 | [3]
SN 1997bp | Ia | 0.008 | 14 | [2]; [3]
SN 1997bq | Ia | 0.009 | 13 | [3]
SN 1997br | Ia | 0.007 | 27 | [2]; [3]
SN 1997by | Ia | 0.045 | 3 | [3]
SN 1997bz | Ia | 0.030 | 2 | [3]
SN 1997C | Ia | 0.023 | 1 | Matheson et al. 2001
SN 1997cn | Ia | 0.016 | 16 | [2]; [3]
SN 1997ct | Ia | 0.030 | 1 | [3]
SN 1997cw | Ia | 0.018 | 4 | [2]; [3]
SN 1997do | Ia | 0.010 | 16 | [2]; [3]
SN 1997dt | Ia | 0.007 | 8 | [3]
SN 1997E | Ia | 0.014 | 5 | [2]; [3]
SN 1997fb | Ia | 0.054 | 1 | [2]
SN 1997fc | Ia | 0.054 | 1 | [2]
SN 1997T | Ia | 0.042 | 1 | [2]
SN 1997Y | Ia | 0.016 | 5 | [2]; [3]
SN 1998ab | Ia | 0.027 | 13 | [3]
SN 1998an | Ia | 0.019 | 1 | [3]
SN 1998aq | Ia | 0.004 | 61 | [2]; [3]
SN 1998bn | Ia | 0.006 | 9 | [2]; [3]
SN 1998bp | Ia | 0.010 | 17 | [2]; [3]
SN 1998bu | Ia | 0.003 | 81 | [2]; [3]
SN 1998cd | Ia | 0.025 | 1 | [2]
SN 1998cl | Ia | 0.030 | 1 | [2]
SN 1998cm | Ia | 0.080 | 1 | [2]
SN 1998cn | Ia | 0.009 | 1 | [3]
SN 1998co | Ia | 0.018 | 8 | [3]
SN 1998cs | Ia | 0.033 | 7 | [2]; [3]
SN 1998D | Ia | 0.012 | 4 | [3]
SN 1998de | Ia | 0.016 | 10 | [2]; [3]
SN 1998dh | Ia | 0.009 | 14 | [2]; [3]
SN 1998dj | Ia | 0.014 | 1 | [2]
SN 1998dk | Ia | 0.013 | 14 | [2]; [3]
SN 1998dm | Ia | 0.007 | 15 | [2]; [3]
SN 1998dw | Ia | 0.049 | 1 | [2]
SN 1998dx | Ia | 0.054 | 9 | [2]; [3]
SN 1998eb | Ia | 0.013 | 2 | [2]; [3]
SN 1998ec | Ia | 0.020 | 7 | [2]; [3]
SN 1998ef | Ia | 0.018 | 9 | [2]; [3]
SN 1998eg | Ia | 0.025 | 7 | [2]; [3]
SN 1998en | Ia | 0.021 | 1 | [2]
SN 1998es | Ia | 0.011 | 33 | [2]; [3]
SN 1998ex | Ia | 0.037 | 1 | [3]
SN 1998fc | Ia | 0.107 | 1 | [2]