# Underluminous Type Ia supernovae are standardizable candles

O. Graur1,2
1Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth,
PO1 3FX, UK
2Department of Astrophysics, American Museum of Natural History, Central Park
West and 79th Street, New York, NY 10024-5192, USA
E-mail<EMAIL_ADDRESS>0000-0002-4391-6137

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

###### Abstract

It is widely accepted that the width-luminosity relation used to standardize
normal Type Ia supernovae (SNe Ia) breaks down in underluminous, 1991bg-like
SNe Ia. This breakdown may be due to the choice of parameter used as a stand-
in for the width of the SN Ia light curve. Using the colour stretch parameter
$s_{\mathrm{BV}}$ instead of older parameters resolves this issue. Here, I
assemble a sample of 13 nearby 1991bg-like SNe Ia from the literature, all of
which have independent host-galaxy distance moduli and little to no reddening.
I use Gaussian process regression to fit the light curves of these SNe in
$U/u$, $g$, $B$, $V$, $R/r$, and $I/i$, and measure their peak absolute
magnitudes. I find statistically significant ($>5\sigma$ confidence level)
correlations between the peak absolute magnitudes of the underluminous SNe and
their $s_{\mathrm{BV}}$ values in the range $0.2<s_{\mathrm{BV}}<0.6$. These
correlations are broadly consistent with fits to $s_{\mathrm{BV}}<0.7$ SNe Ia
with preliminary $B$\- and $V$-band peak absolute magnitudes from the Carnegie
Supernova Project and significantly inconsistent with similar fits to normal
and transitional SNe Ia (with $0.7<s_{\mathrm{BV}}<1.1$). The underluminous
width-luminosity relation shown here needs to be properly calibrated with a
homogeneous sample of 1991bg-like SNe Ia, after which it could be used as a
rung on a new cosmological distance ladder. With surface-brightness
fluctuations (or another non-Cepheid method) used to calibrate distances to
nearby 1991bg-like SNe, such a ladder could produce an independent measurement
of the Hubble-Lemaître Constant, $H_{\mathrm{0}}$.

###### keywords:

methods: distance scale – supernovae: general

††pubyear: 2023††pagerange: Underluminous Type Ia supernovae are
standardizable candles–LABEL:lastpage

##  1 Introduction

Type Ia supernovae (SNe Ia) are famous for their use as standard candles, a
use that led to the discovery of the accelerating expansion of the Universe
and the existence of dark energy (Riess et al., 1998; Perlmutter et al.,
1999). Yet, SNe Ia are not truly standard; they are _standardizable_.
Independently, Rust (1974) and Pskovskii (1977) both showed that more luminous
SNe Ia had wider light curves, i.e., they took longer to rise to peak and then
decline. This correlation was corroborated and modernized by Phillips (1993)
and has been in use ever since (for a historical review of the width-
luminosity relation, see Phillips & Burns 2017; for a general review of the
history and physics of SNe, see Graur 2022).

Phillips (1993) found a tight correlation between the peak absolute
luminosities of ten SNe Ia and the widths of their light curves, parameterized
by $\Delta m_{\mathrm{15}}(B)$, the number of magnitudes by which a SN fades
15 days after it peaks in the $B$ band. Variations on this parameter are now
routinely used to fit SN Ia light curves and standardize them for use in
cosmology. Examples include $\Delta$, used in MLCS2k2 (Jha et al., 2007); $s$,
which is used in SALT and SiFTO (Guy et al., 2005; Conley et al., 2008); and
$x1$, used in SALT2 (Guy et al., 2007).

SNe are divided into several classes, each with its own subclasses. The same
is true for SNe Ia. The majority of SNe Ia fall into the so-called ‘normal’
subclass, but there are also overluminous 1991T-like SNe (Filippenko et al.,
1992b), transitional 1986G-like SNe (Phillips et al., 1987), underluminous
1991bg-like SNe (Filippenko et al., 1992a), peculiar SNe Iax (Jha et al.,
2006b; Foley et al., 2013), and more. The width-luminosity relation works
exceptionally well for normal SNe Ia and extends to overluminous SNe Ia as
well. However, this relation breaks down in underluminous, fast-evolving
1991bg-like SNe (Burns et al., 2014). Consequently, these SNe Ia are routinely
excised from cosmology samples, and the belief that ‘1991bg-like SNe are not
standard candles’ has become dogma.

Table 1: SN sample: basic parameters SN | Host | Classa | $\Delta m_{\mathrm{15}}(B)$ | $s_{\mathrm{BV}}$ | $\mu$ | $E(B-V)_{\mathrm{MW}}$ | $E(B-V)_{\mathrm{Host}}$
---|---|---|---|---|---|---|---
|  |  | (mag) |  | (mag) | (mag) | (mag)
Calibration sample
1991bg | NGC 4374 | E | 1.88(0.10) | 0.321(0.061) | 31.23(0.02) | 0.037 | 0.05(0.02)
1998de | NGC 0252 | S0 | 1.95(0.09) | 0.351(0.120) | 33.98(0.20) | 0.052 | 0.00(0.00)
1999by | NGC 2841 | Sab | 1.90(0.05) | 0.381(0.063) | 30.84(0.05) | 0.010 | 0.020(0.030)
2005bl | NGC 4059 | E | 2.00(0.06) | 0.387(0.061) | 35.21(0.11) | 0.025 | 0.17(0.08)
2006mr | NGC 1316 | Sab | 1.82(0.02) | 0.239(0.060) | 31.42(0.02) | 0.018 | 0.089(0.039)
2007ax | NGC 2577 | S0 | 1.87(0.06) | 0.355(0.061) | 33.20(0.14) | 0.045 | $<0.01$
2008R | NGC 1200 | S0 | 1.82(0.02) | 0.591(0.060) | 33.66(0.08) | 0.062 | 0.009(0.013)
2015bo | NGC 5490 | E | 1.91(0.01) | 0.480(0.010) | 34.27(0.08) | 0.023 | 0.00(0.00)
2017ejb | NGC 4696 | E | 2.04(0.04) | 0.470(0.010) | 32.88(0.06) | 0.097 | 0.05(0.02)
2019so | NGC 4622 | Sa | 1.96(0.04) | 0.410(0.010) | 34.27(0.48) | 0.128 | -0.01(0.02)
2021qvv | NGC 4442 | S0 | 2.05(0.03) | 0.280(0.050) | 30.85(0.07) | 0.019 | 0.00(0.00)
Test sample
1997cn | NGC 5490 | E | 1.90(0.05) | 0.350(0.060) | 34.27(0.08) | 0.023 | 0.00(0.00)
1999da | NGC 6411 | E | 1.94(0.10) | 0.442(0.160) | 33.12(0.27) | 0.046 | 0.00(0.00)
aGalaxy classifications sourced from de Vaucouleurs et al. (1991).

Burns et al. (2014) introduced a different stretch parameter,
$s_{\mathrm{BV}}$, defined as the time at which the $B-V$ colour curve reaches
peak, relative to the time of $B$-band peak, divided by 30 days. When used as
the basis for their own light-curve fitter, SNooPy (Burns et al., 2011), this
parameter made it possible to standardize not just the light curves of normal
and overluminous SNe Ia, but also underluminous SNe Ia as well. And yet,
underluminous 1991bg-like SNe Ia are still not treated as standardizable
candles. Some of this has to do with the continued use of light-curve fitters
such as SALT2 and MLCS2k2, which are optimized for normal SNe Ia, and some of
it has to do with the now deeply-ingrained belief that 1991bg-like SNe fall
off the width-luminosity relation. To wit, even Burns et al. (2018), who used
SNooPy to measure the Hubble-Lemaître Constant, $H_{\mathrm{0}}$, still
imposed an $s_{\mathrm{BV}}>0.5$ cut that excluded most of the 1991bg-like SNe
in their sample.

There are hints that the bias against 1991bg-like SNe may be starting to fade.
Burns et al. (2018) and Hoogendam et al. (2022) noted that distances derived
from the light curves of the 1991bg-like SNe 2006mr and 2015bo were consistent
with independent distance measurements of their host galaxies. Hoogendam et
al. (2022) also included a figure of SNooPy-derived $B$-band peak absolute
magnitudes vs. $s_{\mathrm{BV}}$ that extended all the way down to 1991bg-like
SNe ($0.2<s_{\mathrm{BV}}<0.6$). The apparent correlation between the peak
$B$-band luminosities and $s_{\mathrm{BV}}$ values of the 1991bg-like SNe is
striking but not elaborated on.

In this paper, I set out to test whether there truly is a correlation between
the luminosities of 1991bg-like SNe and the widths of their light curves, as
parameterized by $s_{\mathrm{BV}}$. In Section 2, I assemble a sample of 13
1991bg-like SNe that suffered little to no dust extinction and that exploded
in galaxies with independent distance measurements. Since all SN Ia light-
curve fitters are optimized towards normal SNe Ia, in Section 3 I fit the
light curves of the SNe myself with Gaussian process regression (GPR). The
results, shown in Section 4, are clear: there are statistically significant
correlations between the peak absolute magnitudes of 1991bg-like SNe and
$s_{\mathrm{BV}}$ in multiple filters ($g$, $B$, $V$, $R/r$, and $I/i$). This
leads to Section 5, in which I discuss the potential use of 1991bg-like SNe in
a new distance ladder, independent from the one currently dominating SN Ia
cosmology.

##  2 Sample

A review of the literature on 1991bg-like SNe revealed that only $\sim 30$
such objects had been methodically observed and published. Of these, only 13
had well-sampled multiwavelength light curves, exploded in galaxies with
independent distance measurements, and suffered relatively little host-galaxy
dust extinction ($E(B-V)<0.2$ mag). The latter is important because most
estimates of host-galaxy extinction are a byproduct of light-curve fitting,
which I avoid in this paper. The SN sample used in this work is shown in Table
1. Of the 13 SNe, 11 are used to test for the presence of a width-luminosity
relation, while the other two SNe are held back as test cases (Section 4.3).
Below, I describe the sources of the light curves used in this work. Basic
parameters of the SNe in the sample, including the names and types of galaxies
in which they exploded, are summarized in Table 1.

The eponymous SN 1991bg was extensively studied by Filippenko et al. (1992a),
Leibundgut et al. (1993), and Turatto et al. (1996). Here, I use the peak
magnitudes reported by Turatto et al. (1996), based on a combined analysis of
their data and those of Filippenko et al. (1992a) and Leibundgut et al.
(1993). $UBVRI$ observations of SN 1997cn were published by Turatto et al.
(1998) and Jha et al. (2006a). Modjaz et al. (2001) and Jha et al. (2006a)
published $BVRI$ light curves of SN 1998de, obtained as part of the Lick
Observatory Supernova Search (LOSS; Filippenko et al. 2001). $UBVRI$
observations of SN 1999by were published by Garnavich et al. (2004), who
presented an in-depth study of this object. Additional $BVRI$ observations,
obtained by LOSS, were published by Ganeshalingam et al. (2010). $BVI$ and
$BVRI$ observations of SN 1999da were published by Krisciunas et al. (2001)
and Ganeshalingam et al. (2010), respectively. Here, I use the latter as they
contain the additional $R$ filter.

The Carnegie Supernova Project (CSP; Contreras et al. 2010; Stritzinger et al.
2011; Krisciunas et al. 2017) have published $ugriBV$ observations of 16
underluminous, 1991bg-like SNe. Of these, only the host galaxies of SNe
2005bl, 2006mr, 2007ax, and 2008R had independent distance measurements and
host-galaxy reddening $<0.2$ mag (Burns et al., 2014). An independent analysis
of SN 2005bl (Taubenberger et al., 2008) estimated a host-galaxy reddening of
$0.17\pm 0.08$ mag and an $R_{V}$ value of 3.1, which I use here. Additional
observations of SN 2007ax were published by Kasliwal et al. (2008), who also
estimated a negligible host-galaxy reddening of $<0.01$ mag.

SN 2015bo was discovered by Howerton (2016) and studied by Hoogendam et al.
(2022). Here, I use the $BVugri$ light curves reported by the latter. $BVri$
observations of SNe 2017ejb and 2019so were obtained with the Las Cumbres
Observatory’s network of 1-m telescopes (Brown et al., 2013) and reported by
Chen et al. (2022). SN 2017ejb was discovered by the $D<40$ Mpc survey
(Tartaglia et al., 2017) and spectroscopically classified as a 1991bg-like SN
by Pan et al. (2017) and Valenti et al. (2017). SN 2019so was discovered by
the Asteroid Terrestrial-impact Last Alert System (ATLAS; Tonry et al. 2018)
and reported by Smith et al. (2019). Although a spectroscopic classification
of this SN has not yet been published, its light curves are consistent with
those of other 1991bg-like SNe.

The discovery of SN 2021qvv by ATLAS was reported by Tonry et al. (2021).
Spectra and $UBVgri$ imaging of SN 2021qvv were obtained as part of the Las
Cumbres Observatory’s Global Supernova Project and published by Graur et al.
(2023). The latter classified it as a 1991bg-like SN similar to SN 2006mr.

##  3 Analysis

Here, I describe the steps taken to measure the peak absolute magnitudes of
the SNe in my sample: measuring $s_{\mathrm{B}V}$ (Section 3.1), obtaining the
distance moduli of the SN host galaxies (Section 3.2) and the reddening
suffered by the SNe (Section 3.3), and measuring peak apparent and absolute
magnitudes (Section 3.4).

###  3.1 Colour stretch

SNe 1997cn, 2005bl, 2006mr, 2007ax, 2008R, 2015bo, 2017ejb, and 2019so have
been fit with SNooPy by other authors (Burns et al., 2014; Krisciunas et al.,
2017; Hoogendam et al., 2022; Chen et al., 2022). For these SNe, I use the
published $s_{\mathrm{BV}}$ values. For the rest of the sample (namely, SNe
1991bg, 1998de, 1999by, 1999da, and 2021qvv), I have derived $s_{\mathrm{BV}}$
by fitting their light curves with GPR fits, as done by Graur et al. (2023).
To ensure that the GPR-derived $s_{\mathrm{BV}}$ values are not systematically
offset from the SNooPy-derived values, I use GPR to fit the light curves of 16
1991bg-like SNe observed by CSP (SNe 2005bl, 2005ke, 2006gt, 2006mr, 2007N,
2007ax, 2007ba, 2008R, 2009F, 2006bd, 2006hb, 2007al, 2007jh, 2008bd, 2008bi,
and 2008bt) and compare my values to those published by Krisciunas et al.
(2017). As Fig. 1 shows, there is a tight correlation between the two
techniques, with $s_{\mathrm{BV}}(\mathrm{\texttt{SNooPy}})=(0.95\pm
0.14)s_{\mathrm{BV}}(\mathrm{GPR})-(0.06\pm 0.08)$ and a reduced
$\chi^{2}_{r}$ value of $15.2/14=1.1$. The GPR-derived $s_{\mathrm{BV}}$
values of the pre-2000 SNe have been corrected accordingly. The resultant
$s_{\mathrm{B}V}$ values, along with $\Delta m_{15}(B)$ measurements from the
literature, are presented in Table 1.

###  3.2 Distance moduli

No single work on nearby distance moduli encompasses all of the host galaxies
in my sample. Instead of picking and choosing between the various measurements
available for each galaxy, I take a weighted mean of all measurements
conducted from the year 2000 onwards. For some host galaxies, which have $>10$
measurements, this results in an underestimated uncertainty on the distance
modulus. I do not include distance modulus measurements derived from fitting
the light curves of the SNe in the sample, for two reasons. First, this would
be cyclical reasoning, as the goal of this paper is to ascertain whether
1991bg-like SNe can be used as standardizable candles. Second, as Graur et al.
(2023) noted, because current light-curve fitters are not optimized for
1991bg-like SNe, the distance moduli derived in this manner often span a range
of $\sim 1$ mag. The sources of the measurements used for each host galaxy are
gathered in Appendix A, and the resultant distance moduli and their
uncertainties are collected in Table 1.

###  3.3 Reddening

Galactic line-of-sight reddening towards the SNe in the sample,
$E(B-V)_{\mathrm{MW}}$, were extracted from the reddening maps produced by
Schlafly & Finkbeiner (2011), while host-galaxy reddening values,
$E(B-V)_{\mathrm{Host}}$, were extracted from the literature (see Section 2).
Unlike Galactic reddening, host-galaxy reddening is derived from light-curve
fits. Since most light-curve fitters are not optimized towards fitting 1991bg-
like light curves, these values are suspect. Thus, I chose to include only SNe
whose literature $E(B-V)_{\mathrm{Host}}$ values were $<0.1$ mag. The sole
exception is SN 2005bl, with $E(B-V)_{\mathrm{Host}}=0.17\pm 0.08$ mag
(Taubenberger et al., 2008). As it exploded in an elliptical galaxy, I chose
to keep it in the sample. To avoid any systematic uncertainties that have to
do with how host-galaxy reddening and $R_{V}$ values are derived, and since,
by design, the host-galaxy reddenings are low, I chose not take them into
account in the following analysis. A test in which these values were included
in the derivation of the peak absolute magnitudes showed that including them
did not impact the results in a significant manner.

Figure 1: $s_{\mathrm{BV}}$ values derived with GPR are $\approx 5$ per cent
larger than the values derived by the SNooPy light-curve fitter. The
correlation shown here is $s_{\mathrm{BV}}(\mathrm{\texttt{SNooPy}})=(0.95\pm
0.14)s_{\mathrm{BV}}(\mathrm{GPR})-(0.06\pm 0.08)$ with a reduced
$\chi^{2}_{r}$ value of $1.1$.

###  3.4 Peak magnitudes

Here, I follow Graur et al. (2023) and fit the light curves of the SNe in my
sample using the Matlab 2023a GPR-fitting routine fitrgp with the default
parameters. These fits provide the date and magnitude of maximum light in each
of six filters: $U/u$, $g$, $B$, $V$, $R/r$, and $I/i$. To estimate the
uncertainties on the derived parameters, I repeat the fit in each filter 100
times, each time varying the photometry of the SN randomly according to the
measurement uncertainties. The mean and standard deviation of the results are
then taken as the estimated peak date and its $1\sigma$ uncertainty. The same
method is applied to estimating the peak apparent magnitude in each filter.
Magnitude uncertainties smaller than $0.01$ mag are rounded up. The resultant
peak apparent magnitudes are presented in Table 2.

In Fig. 2, I compare the peak apparent magnitudes derived using this method
with published values from the literature, derived using various fitting
techniques. In all filters except $U/u$, the mean difference between the GPR
and literature values is $<0.1$ mag. In $U/u$, this value rises to $0.2$ mag,
but this is due to a single outlier, SN 2005bl.

With the fitted peak apparent magnitudes, Galactic reddening values, and
distance moduli described above, the peak absolute magnitude of the $i$-th SN
in my sample in filter $\lambda$ is simply:

$M(\lambda)_{\mathrm{max},i}=m(\lambda)_{\mathrm{max},i}-A(\lambda)_{i}-\mu_{i},$
(1)

where $\mu_{i}$ is the distance modulus of the $i$-th SN, and $A(\lambda)_{i}$
is derived from the Galactic line-of-sight reddening,
$E(B-V)_{\mathrm{MW},i}$, the Galaxy’s $R_{V}=3.1$, and the Cardelli et al.
(1989) extinction law. Due to the nature of the SN sample, which includes SNe
observed by different surveys in different filters (e.g., $U/u$, $R/r$, and
$I/i$), I do not attempt to perform $K$ or $S$ corrections. The resultant peak
absolute magnitudes are presented in Table 3 and Fig. 3.

##  4 Results

Here, I fit the peak absolute magnitudes measured in the previous section and
show that correlations between them and the colour stretch of the SNe are
statistically significant in all filters except $U/u$ (Section 4.1); compare
these correlations to fits applied to independent CSP data (Section 4.2); and
demonstrate the effect of these correlations on our understanding of SNe
1997cn and 1999da (Section 4.3).

###  4.1 Width-luminosity relations

I perform two independent tests to determine whether the peak absolute
magnitudes in a specific filter are correlated with colour stretch,
$s_{\mathrm{BV}}$, the results of which are collected in Table 4. First, all
filters display a strong, negative Pearson correlation coefficient, $r$, with
$r\lesssim-0.9$ in $g$, $B$, $V$, and $R/r$, and $r\sim 0.7$ in $U/u$ and
$I/i$. The same test shows that these correlations have statistically
significant $p$-values of $<0.05$ in all filters except $U/u$ (and a more
stringent $p<0.01$ in all filters except $U/u$ and $I/i$). Second, A
likelihood-ratio test (see, e.g., Graur et al. 2017a) prefers a 1st-order
polynomial over a 0th-order polynomial fit to the data in every filter with a
statistical significance of $>5\sigma$. The same test shows that a 2nd-order
polynomial is not preferred over a 1st-order polynomial.

Based on the likelihood-ratio test, I fit 1st-order polynomials to the peak
absolute magnitudes in each filter, the results of which are presented in
Table 4 and Fig. 3. An analysis of the residuals produced by subtracting the
best-fitting polynomials from the data reveals a scatter, $S$, of $<0.3$ mag
in all filters except $U/u$, and $<0.2$ mag in $B$ and $V$. I attribute the
high scatter in $U/u$ to the difficulty in estimating the time of peak from
photometry that, only in this band, did not always cover the rise and peak of
the light curves (SNe 1999by, 2007ax, 2008R, and 2021qvv). Note that, even
without $K$ and $S$ corrections, and with distance moduli derived using a
variety of techniques, the scatter found here is on par with, or better than,
the scatter found in the initial width-luminosity relations measured by
Phillips (1993) for normal SNe Ia.

Figure 2: Peak apparent magnitudes derived with GPR are in good agreement with similar values derived by previous works with various light-curve fitters. The solid curve represents the 1:1 line, and the various symbols and colors represent the six filters tested here (see legend). In five of the six filters, the mean difference between the GPR and literature values is $<0.1$ mag. Only in $U/u$, where the light curves are sparsest, does the mean difference between the values rise to $0.2$ mag, due to a single outlier, SN 2005bl. Figure 3: Peak absolute magnitude vs. colour stretch, $s_{\mathrm{BV}}$. Magnitudes have been corrected for Galactic line-of-sight reddening. The correlations, calculated using the calibration sample (filled squares) are statistically significant in all filters except $U/u$. SNe 1997cn and 1999da (described in detail in Section 4.3) are shown as an open circle and triangle, respectively. SN 1999da is shown twice, once when using a host-based distance modulus ($\mu_{\mathrm{Host}}$) and once when using a distance modulus derived by SN light-curve fitters ($\mu_{\mathrm{SN}}$). For clarity, the second measurement of SN 1999da is shown without its horizontal error bar. Table 2: Peak apparent magnitudes SN | $m(U/u)_{\mathrm{max}}$ | $m(g)_{\mathrm{max}}$ | $m(B)_{\mathrm{max}}$ | $m(V)_{\mathrm{max}}$ | $m(R/r)_{\mathrm{max}}$ | $m(I/i)_{\mathrm{max}}$
---|---|---|---|---|---|---
| (mag) | (mag) | (mag) | (mag) | (mag) | (mag)
Calibration sample
1991bg | $\cdots$ | $\cdots$ | 14.75(0.10) | 13.96(0.05) | 13.63(0.05) | 13.50(0.05)
1998de | $\cdots$ | $\cdots$ | 17.56(0.04) | 16.82(0.04) | 16.61(0.04) | 16.61(0.05)
1999by | 13.77(0.19) | $\cdots$ | 13.60(0.01) | 13.14(0.01) | 12.90(0.01) | 12.89(0.01)
2005bl | 19.85(0.07) | 18.28(0.02) | 18.70(0.02) | 17.90(0.02) | 17.66(0.03) | 17.71(0.03)
2006mr | 16.72(0.03) | 15.03(0.01) | 15.43(0.04) | 14.64(0.01) | 14.47(0.05) | 14.48(0.01)
2007ax | 18.10(0.09) | 16.02(0.03) | 16.44(0.02) | 15.69(0.02) | 15.45(0.01) | 15.59(0.01)
2008R | 16.01(0.13) | 15.37(0.01) | 15.51(0.01) | 15.37(0.01) | 15.32(0.01) | 15.58(0.02)
2015bo | 18.16(0.04) | 16.62(0.01) | 16.85(0.01) | 16.45(0.01) | 16.29(0.01) | 16.65(0.02)
2017ejb | $\cdots$ | $\cdots$ | 15.81(0.02) | 15.38(0.02) | 15.26(0.02) | 15.54(0.02)
2019so | $\cdots$ | 16.81(0.08) | 17.16(0.03) | 16.66(0.02) | 16.55(0.02) | 16.81(0.04)
2021qvv | 14.33(0.19) | 14.31(0.03) | 14.51(0.01) | 13.85(0.01) | 13.78(0.01) | 13.91(0.02)
Test sample
1997cn (GPR) | 18.20(0.20) | $\cdots$ | 17.20(0.10) | 16.55(0.10) | 16.30(0.10) | 16.35(0.10)
1997cn (2007ax) | 18.23(0.20) | 16.74(0.03) | 17.16(0.10) | 16.41(0.10) | 16.17(0.10) | 16.31(0.10)
1999da ($\mu_{\mathrm{Host}}$) | $\cdots$ | $\cdots$ | 16.81(0.03) | 16.22(0.01) | 15.94(0.02) | 15.93(0.02)
1999da ($\mu_{\mathrm{SN}}$) | $\cdots$ | $\cdots$ | 15.98(0.03) | 15.39(0.01) | 15.11(0.02) | 15.10(0.02)
Table 3: Peak absolute magnitudes SN | $M(U/u)_{\mathrm{max}}$ | $M(g)_{\mathrm{max}}$ | $M(B)_{\mathrm{max}}$ | $M(V)_{\mathrm{max}}$ | $M(R/r)_{\mathrm{max}}$ | $M(I/i)_{\mathrm{max}}$
---|---|---|---|---|---|---
| (mag) | (mag) | (mag) | (mag) | (mag) | (mag)
Calibration sample
1991bg | $\cdots$ | $\cdots$ | -16.64(0.10) | -17.38(0.05) | -17.70(0.05) | -17.81(0.05)
1998de | $\cdots$ | $\cdots$ | -16.64(0.20) | -17.32(0.20) | -17.51(0.20) | -17.48(0.21)
1999by | -17.12(0.20) | $\cdots$ | -17.28(0.05) | -17.73(0.05) | -17.97(0.05) | -17.97(0.05)
2005bl | -16.31(0.13) | -17.67(0.11) | -17.33(0.11) | -17.91(0.11) | -18.07(0.11) | -17.91(0.11)
2006mr | -14.79(0.04) | -16.46(0.02) | -16.07(0.04) | -16.84(0.02) | -17.00(0.06) | -16.98(0.02)
2007ax | -15.32(0.17) | -17.35(0.14) | -16.95(0.14) | -17.65(0.14) | -17.87(0.14) | -17.71(0.14)
2008R | -17.95(0.13) | -18.53(0.08) | -18.41(0.08) | -18.49(0.08) | -18.51(0.08) | -18.21(0.08)
2015bo | -17.52(0.08) | -17.89(0.08) | -16.22(0.09) | -17.74(0.08) | -18.04(0.08) | -17.67(0.08)
2017ejb | $\cdots$ | $\cdots$ | -17.48(0.06) | -17.80(0.06) | -17.88(0.06) | -17.54(0.06)
2019so | $\cdots$ | -17.94(0.49) | -17.65(0.48) | -18.01(0.48) | -18.07(0.48) | -17.73(0.48)
2021qvv | -16.61(0.20) | -16.61(0.07) | -16.42(0.07) | -17.05(0.07) | -17.12(0.07) | -16.98(0.07)
Test sample
1997cn (GPR) | -16.18(0.22) | $\cdots$ | -17.17(0.13) | -17.79(0.13) | -18.03(0.13) | -17.97(0.13)
1997cn (2007ax) | -16.16(0.22) | -17.62(0.09) | -17.21(0.13) | -17.93(0.13) | -18.16(0.13) | -18.01(0.13)
1999da ($\mu_{\mathrm{Host}}$) | $\cdots$ | $\cdots$ | -16.51(0.27) | -17.05(0.27) | -17.31(0.27) | -17.28(0.27)
1999da ($\mu_{\mathrm{SN}}$) | $\cdots$ | $\cdots$ | -17.34(0.08) | -17.88(0.08) | -18.14(0.08) | -18.12(0.08)
Table 4: $M$–$s_{\mathrm{BV}}$ correlation parameters Filter | $a$ | $b$ | $\chi^{2}_{r}$ | $L$ | $r$ | $p$ | $S$
---|---|---|---|---|---|---|---
| (mag) | (mag) |  | ($\sigma$) |  |  | (mag)
$U/u$ | -7.5(0.3) | -13.0(0.1) | 26 | 5 | -0.72 | $6.6\times 10^{-2}$ | 0.74
$g$ | -5.8(0.2) | -15.1(0.1) | 1.6 | 5 | -0.95 | $1.1\times 10^{-3}$ | 0.23
$B$ | -6.4(0.2) | -14.6(0.1) | 3.9 | 5 | -0.96 | $4.1\times 10^{-6}$ | 0.19
$V$ | -4.7(0.2) | -15.8(0.1) | 3.6 | 5 | -0.93 | $4.4\times 10^{-5}$ | 0.18
$R/r$ | -4.0(0.2) | -16.2(0.1) | 7.9 | 5 | -0.88 | $3.1\times 10^{-4}$ | 0.21
$I/i$ | -3.7(0.2) | -16.2(0.1) | 23 | 5 | -0.72 | $1.2\times 10^{-2}$ | 0.28
Columns are: (1) filter; ($2+3$) best-fitting parameters for a first-order
polynomial
$M=as_{\mathrm{BV}}+b$ (uncertainties appear in parentheses); (4) reduced
$\chi^{2}$ of fit;
(5) significance of likelihood-ratio test; ($6+7$) Pearson $r$ coefficient and
its $p$-value;
and (8) scatter after subtracting the best-fitting polynomial from the data.
|
---|---
Figure 4: Absolute $B$\- (left) and $V$-band (right) magnitudes vs. colour
stretch, $s_{\mathrm{BV}}$ for the 1991bg-like SNe from this work (blue and
green squares, respectively) and CSP SNe (gray triangles; preliminary
measurements shown in Hoogendam et al. 2022, with finalised measurements
expected in Uddin et al. 2023). The dashed, black curves have been fit to
normal and transitional SNe Ia ($0.7<s_{\mathrm{BV}}<1.1$), while the dot-
dashed, black curves were fit to CSP SNe with $s_{\mathrm{BV}}<0.7$. As
detailed in the text, the normal and underluminous SN curves have
significantly different slopes. On the other hand, the slopes of the fits to
the peak absolute magnitudes measured here (blue and green solid curves,
respectively) are broadly consistent with those of the low-stretch CSP SNe.

###  4.2 Comparison with CSP

In Fig. 4, I compare my $B$\- and $V$-band measurements to similar CSP peak
absolute magnitudes from Hoogendam et al. (2022). These measurements are
preliminary (the finalized CSP measurements will appear in the published
version of Uddin et al. 2023; W. B. Hoogendam, private communication) but
provide a good sanity test and the option to compare the correlations found
above with the well-established width-luminosity relation of normal SNe Ia.

The $B$-band CSP data were obtained by fitting the light curves with SNooPy
using the EBV2 model, while the $V$-band peak absolute magnitudes were
obtained via GPR fits, which explains the larger scatter in this filter
(Hoogendam et al., 2022). To recreate figure 9 from Hoogendam et al. (2022), I
have removed all $V$-band measurements with uncertainties $\sigma[M(V)]>0.3$
mag and $\sigma(s_{\mathrm{BV}})>0.06$ (Hoogendam et al. 2022 used a stricter
$\sigma(s_{\mathrm{BV}})>0.05$, but my sample has an average
$\langle\sigma(s_{\mathrm{BV}})\rangle\sim 0.06$). No cuts were applied to the
$B$-band data.

A likelihood-ratio test applied to the entire range of data in the $B$ band
prefers a 2nd-order polynomial over a 1st-order polynomial at a $>5\sigma$
confidence level, indicating that the underluminous SNe Ia follow a different
trend than the normal ones. The scatter in this filter is small enough that
this preference can be seen by eye as well. The same likelihood-ratio test
conducted on the $V$-band CSP measurements finds no statistically significant
preference for a 2nd-order polynomial over a 1st-order polynomial, or even for
a 1st-order polynomial over a 0th-order polynomial.

Based on the results of the likelihood-ratio test on the $B$-band CSP data, I
fit the CSP measurements in both filters with 1st-order polynomials in two
$s_{\mathrm{BV}}$ ranges: $0.7<s_{\mathrm{BV}}<1.1$ to capture all normal SNe
Ia (as well as transitional SNe Ia), and $s_{\mathrm{BV}}<0.7$ to include only
underluminous SNe Ia. In the $B$ band, I find that the normal SNe Ia follow
$M(B)_{\mathrm{CSP}}=(-1.5\pm 0.1)s_{\mathrm{BV}}-(17.9\pm
0.1)~{}(\chi^{2}=5.7/42=0.14)$ and underluminous SNe Ia follow
$M(B)_{\mathrm{CSP}}=(-5.3\pm 0.2)s_{\mathrm{BV}}-(15.5\pm
0.1)~{}(\chi^{2}=22/9=2.4)$. In the $V$ band, the correlations are
$M(V)_{\mathrm{CSP}}=(-1.8\pm 0.1)s_{\mathrm{BV}}-(17.2\pm
0.1)~{}(\chi^{2}=1010/55=18)$ and $M(V)_{\mathrm{CSP}}=(-4.0\pm
0.2)s_{\mathrm{BV}}-(15.7\pm 0.1)~{}(\chi^{2}=498/15=33)$, respectively. The
higher $\chi^{2}$ values, divided by the number of degrees of freedom, are due
to the larger scatter of the measurements in this filter.

Overplotting the $B$\- and $V$-band peak absolute magnitudes from Fig. 3 shows
that the width-luminosity relations from my sample are in broadly good
agreement with the fits to the CSP data. This is somewhat surprising, given
the heterogeneity of my sample, the variety of methods used to derive distance
moduli to the SN host galaxies, and the lack of $K$ corrections. It is too
much to assume that all of these factors would conspire to produce
correlations that coincidentally agreed with the homogeneous, well-calibrated
CSP data. Instead, we must conclude that there are, indeed, clear correlations
between the peak absolute magnitudes of underluminous SNe Ia and the widths of
their light curves, as parameterized by $s_{\mathrm{BV}}$.

###  4.3 Two test cases

When constructing the SN sample in Section 2, the underluminous SNe 1997cn and
1999da were set aside. Below, I describe the circumstances that led me to
isolate each of these SNe and how they help validate the width-luminosity
relations from Section 4.1.

####  4.3.1 SN 1997cn

SN 1997cn was discovered by Li et al. (1997) and observed by both Turatto et
al. (1998) and Jha et al. (2006a), but only after it had already peaked and
begun to decline. Turatto et al. (1998) estimated that the SN reached $B$-band
maximum light on JD 2450587.5, while Hoogendam et al. (2022) used SNooPy to
fit the Jha et al. (2006a) light curves and derive a peak date of
$2450580.7\pm 0.7$, 6.8 d earlier than the Turatto et al. (1998) date and
$\sim 2$ d prior to its discovery. A spectroscopic fit with SNID (Blondin &
Tonry, 2007) yielded a third date: 2450588.3, 0.8 d after the Turatto et al.
(1998) estimate. As the light curves of this SN did not cover the date of
maximum light, I did not include it in my sample.

The SNooPy fit conducted by Hoogendam et al. (2022) provided an
$s_{\mathrm{BV}}$ value of $0.35\pm 0.06$. As this value is consistent with
the $s_{\mathrm{BV}}$ value of SN 2007ax, $0.355\pm 0.061$ (Krisciunas et al.,
2017), the light curves of the two SNe can be compared to each other directly.
In Fig. 5, I fit the $V$-band measurements of SN 2007ax to those of SN 1997cn
taken by both Turatto et al. (1998) and Jha et al. (2006a) by varying the date
of maximum light and the offset between the light curves. I find
$t(B)_{\mathrm{max}}=2450586.1\pm 0.2$ d, with SN 1997cn $0.72\pm 0.01$ mag
fainter than SN 2007ax, with a reduced $\chi^{2}_{r}=2.1$.

Fig. 3 includes peak absolute magnitudes of SN 1997cn estimated by adding an
offset of $0.72$ mag to the peak apparent magnitudes of SN 2007ax. The
resultant values are consistent with peak apparent magnitudes estimated by GPR
fits to the data from Jha et al. (2006a), with the exception of the $g$ band,
in which there were no previous measurements. In all filters, the resultant
peak absolute magnitudes of SN 1997cn are consistent with the width-luminosity
relations within the scatter of the calibration sample. This indicates that
the width-luminosity relations measured in Section 4.1 can be used to estimate
the peak absolute magnitudes of individual underluminous SNe Ia.

####  4.3.2 SN 1999da

The host galaxy of SN 1999da only had two independent, post-2000 distance
measurements (both from Blakeslee et al. 2001), which averaged to
$\mu=33.12\pm 0.27$ mag. However, distance moduli derived by fitting the light
curves of this SN by various groups (Riess et al., 2004; Wang et al., 2006;
Jha et al., 2007; Kowalski et al., 2008; Hicken et al., 2009), using different
light-curve fitters, produced an average value of $\mu_{\mathrm{SN}}=33.95\pm
0.08$, $\sim 0.8$ mag fainter. Fig. 3 shows SN 1999da twice, using each of the
distance moduli above. In all four filters in which this SN was observed, the
peak absolute magnitudes derived using the host-based distance modulus are
systematically dimmer than the width-luminosity relations. The peak absolute
magnitudes derived with the light-curve-based distance modulus, on the other
hand, are consistent with the correlations. This indicates that, at least for
this SN, existing light-curve fitters already did a good job of deriving its
luminosity distance.

Figure 5: A comparison between SN 1997cn (filled circles and triangles from
Turatto et al. 1998 and Jha et al. 2006a, respectively) and SN 2007ax (empty
squares; Krisciunas et al. 2017). Derived dates of maximum light are shown as
dashed (Turatto et al., 1998), dot-dashed (SNooPy; Hoogendam et al. 2022),
dotted (SNID; Hoogendam et al. 2022), and solid (this work) curves. According
to the fit between the two SNe, SN 1997cn was $\approx 0.7$ mag dimmer than SN
2007ax.

##  5 Conclusions

In this work, I have assembled a sample of 13 underluminous, 1991bg-like SNe
Ia from the literature to test whether this type of SN Ia can be used as a
standardizable candle. To minimize systematic uncertainties, I have chosen SNe
that exploded in host galaxies with low to no reddening and well-sampled light
curves. Even so, the sample still suffers from systematic uncertainties that
stem from the variety of filters in which the SNe were observed as well as a
lack of $K$ and $S$ corrections. This sample still exhibits statistically
significant ($>5\sigma$) correlations between the peak absolute magnitudes of
the underluminous SNe and the widths of their light curves, as parameterized
by the colour stretch, $s_{\mathrm{BV}}$.

The underluminous width-luminosity relations measured here are similar in
nature to those of normal SNe Ia but with significantly steeper slopes. The
$B$\- and $V$-band width-luminosity relations are consistent with preliminary
CSP data, which both strengthens the existence of the correlations and shows
that SNooPy, the light-curve fitter used by CSP, is able to standardize
underluminous as well as normal SNe Ia. The correlations shown here are
further strengthened by two test cases, SNe 1997cn and 1999da, which were not
included in the calibration sample.

The work done here shows that underluminous, 1991bg-like SNe Ia can be
standardized and hence used to measure extragalactic distances. However, these
width-luminosity relations must first be properly calibrated by compiling a
homogeneous sample of 1991bg-like SNe Ia observed by a single survey in a
single set of filters. Even then, because 1991bg-like SNe are rarer than
normal SNe Ia (15 vs. 70 per cent of all SNe Ia, respectively; Graur et al.
2017b), their inclusion in SN Ia cosmology samples will not make much of a
difference. Moreover, their inherent dimness makes them more susceptible to
Malmquist bias, which limits their use as probes of dark energy.

Instead, I suggest using 1991bg-like SNe Ia as a rung on a new cosmological
distance ladder, one that would provide an independent check on the distance
ladder based on Cepheids and normal SNe Ia. While extremely successful (e.g.,
Riess et al. 2022), the latter ladder is biased towards star-forming galaxies,
which host young Cepheid variables. Underluminous SNe Ia, on the other hand,
are mostly found in massive, passive galaxies (Graur et al., 2017b). This
makes them less prone to host-galaxy reddening and removes the systematic
uncertainty produced by the mass step found in Hubble residuals (e.g., Kelly
et al. 2010; Brout & Scolnic 2021; Kelsey et al. 2021, 2023; Meldorf et al.
2023).

Bereft of Cepheids, the galaxies that host underluminous SNe Ia will have to
be calibrated by other means, such as surface-brightness fluctuations. Several
groups have already attempted to use this method to either measure
$H_{\mathrm{0}}$ directly or to calibrate the host galaxies of normal SNe Ia
(e.g., Blakeslee et al. 2021; Jensen et al. 2021; Khetan et al. 2021;
Garnavich et al. 2023; Uddin et al. 2023; Wood et al. 2023). A new distance
ladder, based on surface-brightness fluctuations (or some other non-Cepheid
method) and 1991bg-like SNe Ia could provide an independent measurement of
$H_{\mathrm{0}}$, a necessary step towards resolving the current ‘Hubble
tension’ (e.g., Di Valentino et al. 2021; Kamionkowski & Riess 2023).

## Acknowledgments

I thank Chris R. Burns and Saurabh W. Jha for helpful discussions, and Willem
B. Hoogendam and the CSP for sharing their data with me. This research has
made use of NASA’s Astrophysics Data System and the NASA/IPAC Extragalactic
Database (NED), which are funded by the National Aeronautics and Space
Administration and operated by the California Institute of Technology.

## Data availability

All data used in this work, except for the CSP data shown in Fig. 4, are
public and have been published elsewhere. The measurements and fits produced
by this work are included in the tables throughout this paper. Any remaining
data or measurements will be shared on request to the corresponding author.

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##  Appendix A Distance modulus references

In this section, I summarize the sources of the various distance moduli
measurements used to calculate the weighted means shown in Table 1. The
acronyms used in Table 5, below, are: Cepheids (Cep), cosmic microwave
background (CMB), dwarf galaxy diameter (DGD), fundamental plane (FP),
globular cluster luminosity function (GCLF), globular cluster radius (GCR),
planetary nebula luminosity function (PNLF), surface brightness fluctuations
(SBF), and the Tully-Fisher relation (TF).

Table 5: Distance modulus references SN | Host | Method | References
---|---|---|---
1991bg | NGC 4374 | FP | Blakeslee et al. (2001)
GCLF | Gómez & Richtler (2004); Jordán et al. (2007); Villegas et al. (2010)
GCR | Jordán et al. (2005)
PNLF | Ferrarese et al. (2000); Ciardullo et al. (2002)
SBF | Ferrarese et al. (2000); Blakeslee et al. (2001); Blakeslee et al. (2009); Mei et al. (2007); Cantiello et al. (2011)
TF | Courtois & Tully (2012)
1998de | NGC 0252 | CMB | Modjaz et al. (2001)
1999by | NGC 2841 | Cep | Macri et al. (2001); Saha et al. (2006)
TF | Russell (2002); Tully et al. (2009); Springob et al. (2009); Courtois & Tully (2012); Sorce et al. (2012); Sorce et al. (2014)
1999da | NGC 6411 | FP | Blakeslee et al. (2001)
SBF | Blakeslee et al. (2001)
2005bl | NGC 4059 | FP | Saulder et al. (2016)
TF | Theureau et al. (2007)
2006mr | NGC 1316 |  $D$–$\sigma$ | Bernardi et al. (2002)
DGD | Misgeld & Hilker (2011)
FP | Graham (2002)
GCLF | Villegas et al. (2010)
GCR | Masters et al. (2010)
PNLF | Ciardullo et al. (2002); Feldmeier et al. (2007)
TF | Sakai et al. (2000); Tully & Pierce (2000); Theureau et al. (2007); Tully & Courtois (2012)
Courtois & Tully (2012); Sorce et al. (2013); Sorce et al. (2014); Neill et
al. (2014); Tully et al. (2016)
|  | SBF | Tonry et al. (2001); Ajhar et al. (2001); Jensen et al. (2003); Cantiello et al. (2007)
|  | Blakeslee et al. (2009, 2010); Cantiello et al. (2013)
2007ax | NGC 2577 | TF | Theureau et al. (2007)
2008R | NGC 1200 | FP | Springob et al. (2014)
2015bo | NGC 5490 | TF | Theureau et al. (2007)
SBF | Jensen et al. (2021)
2017ejb | NGC 4696 | DGD | Misgeld & Hilker (2011)
FP | Blakeslee et al. (2001)
FP+SBF | Tully et al. (2013)
GCLF | Mieske et al. (2005)
TF | Theureau et al. (2007); Tully et al. (2016)
SBF | Tonry et al. (2001); Blakeslee et al. (2001); Mieske & Hilker (2003); Cantiello et al. (2005)
2019so | NGC 4622 | FP | Springob et al. (2014)
2021qvv | NGC 4442 | GCLF | Jordán et al. (2007); Villegas et al. (2010)
|  | GCR | Jordán et al. (2005)
|  | TF | Theureau et al. (2007)
|  | SBF | Mei et al. (2007); Blakeslee et al. (2009)
Note. SN 1997cn exploded in the same host galaxy as SN 2015bo, NGC 5490.