# Further evidence that galaxy age drives observed type Ia supernova
luminosity differences

P. Wiseman,1 M. Sullivan,1 M. Smith2 and B. Popovic3
1 School of Physics and Astronomy, University of Southampton, Southampton,
SO17 1BJ, UK
2 Univ Lyon, Univ Claude Bernard Lyon 1, CNRS, IP2I Lyon / IN2P3, IMR 5822,
F-69622 Villeurbanne, France
3 Department of Physics, Duke University, Durham, NC 27708, USA
E-mail<EMAIL_ADDRESS>(PW)

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

###### Abstract

Type Ia supernovae (SNe Ia) are explosions of white dwarf stars that
facilitate exquisite measurements of cosmological expansion history, but
improvements in accuracy and precision are hindered by observational biases.
Of particular concern is the apparent difference in the corrected brightnesses
of SNe Ia in different host galaxy environments. SNe Ia in more massive,
passive, older environments appear brighter after having been standardized by
their light-curve properties. The luminosity difference commonly takes the
form of a step function. Recent works imply that environmental characteristics
that trace the age of the stellar population in the vicinity of SNe show the
largest steps. Here we use simulations of SN Ia populations to test the impact
of using different tracers and investigate promising new models of the step.
We test models with a total-to-selective dust extinction ratio $R_{V}$ that
changes between young and old SN Ia host galaxies, as well as an intrinsic
luminosity difference between SNe from young and old progenitors. The data are
well replicated by a model driven by a galaxy-age varying $R_{V}$ and no
intrinsic SN luminosity difference, and we find that specific star-formation
rate measured locally to the SN is a relatively pure tracer of this galaxy age
difference. We cannot rule out an intrinsic difference causing part of the
observed step and show that if luminosity differences are caused by multiple
drivers then no single environmental measurement is able to accurately trace
them. We encourage the use of multiple tracers in luminosity corrections to
negate this issue.

###### keywords:

supernovae: general - cosmology: observations - dust, extinction - distance
scale

††pubyear: 2023††pagerange: Further evidence that galaxy age drives observed
type Ia supernova luminosity differences–Further evidence that galaxy age
drives observed type Ia supernova luminosity differences

##  1 Introduction

Type Ia supernovae (SNe Ia) are routinely used as standardizable candles to
directly probe the expansion history of the Universe, a technique that led to
the discovery of the accelerated cosmic expansion (Riess et al., 1998;
Perlmutter et al., 1999). These measurements are facilitated by empirical
relationships between the optical brightness of a SN Ia and its decline rate
(commonly known as ‘stretch’; Phillips 1993) and a SN Ia brightness and its
colour (Riess et al., 1996; Tripp, 1998). When combined, these two
relationships can reduce the dispersion in SN Ia distances to 7 per cent
($\simeq 0.15$ mag), enabling modern analyses to measure the dark energy
equation-of-state parameter to a precision of two per cent (Scolnic et al.,
2018; Abbott et al., 2019; Brout et al., 2022). Key to achieving an accuracy
that matches this precision is the comprehensive modelling of both SN Ia
selection biases in large surveys (e.g. Kessler et al., 2009; Perrett et al.,
2010; Betoule et al., 2014; Kessler et al., 2019; Vincenzi et al., 2021) and
astrophysical processes affecting the observed SN Ia properties.

In the latter case, significant effort has been dedicated to explaining and
accounting for the observed relationships between the standardized (i.e.,
stretch- and colour-corrected) SN Ia luminosities and their host galaxy
parameters, such as stellar mass (Kelly et al., 2010; Lampeitl et al., 2010;
Sullivan et al., 2010), star-formation rate (SFR; Sullivan et al. 2010),
morphology (Sullivan et al., 2003; Hakobyan et al., 2020), stellar age (Rose
et al., 2019, 2021; Millán-Irigoyen et al., 2022), metallicity (Gallagher et
al., 2008; Childress et al., 2013; Moreno-Raya et al., 2016; Millán-Irigoyen
et al., 2022), or galaxy rest-frame colour (Roman et al., 2018; Jones et al.,
2018; Kelsey et al., 2021, 2022). The relationships usually take the form of a
‘step’, where the mean standardized luminosities of SNe Ia on different sides
of a threshold in a given host galaxy parameter (e.g., a stellar mass of
$\log(M_{*}/\mathrm{M}_{\odot})=10$) differ by a fixed amount.

SN Ia stretch also correlates with various global (i.e., measurements of the
entire galaxy) host galaxy parameters (e.g., Hamuy et al., 1995; Sullivan et
al., 2006), and in particular those that trace stellar population age. Rigault
et al. (2013), Rigault et al. (2020) and Nicolas et al. (2021) show evidence
that there are at least two modes of SN stretch,one of which is dominant for
young white dwarfs and the other in older progenitor systems. There is also
evidence that the luminosity step is more significant when computed for a
‘locally’-measured111Measured in a small aperture centered at the SN location.
property (Rigault et al., 2020; Roman et al., 2018; Rose et al., 2019; Kelsey
et al., 2021, 2022), particularly properties linked to age, suggesting that
SNe Ia belonging to each stretch population may also have differing mean
intrinsic luminosities. Briday et al. (2022, hereafter B22) showed that the
magnitude of the step is directly related to the ability of a given host
galaxy measurement to trace the age of the stellar population near the SN:
locally measured specific SFR (the SFR normalised by stellar mass; Guzmán et
al. 1997) displays the largest step, while the step in global galaxy
morphology (which has a far less direct connection to the stellar population
age at the SN Ia position) is smallest. B22 propose that there should be a
linear relationship between the step size and how directly the host galaxy
parameter traces local sSFR, and their observed data agree with the
prediction.

An alternative explanation is that the stellar mass step is a result of
different average dust properties in low- and high-mass galaxies (Brout &
Scolnic, 2020). For example, a difference in the mean total-to-selective
extinction ratio ($R_{V}$) such that mean $R_{V}=2.0$ in low-mass galaxies and
$R_{V}=3.0$ in high-mass galaxies can reduce the mass step to $\sim 0.02$ mag
(Popovic et al., 2021a). Such a difference in $R_{V}$ can explain the observed
difference in the colour–luminosity coefficient $\beta$ between SNe Ia in low
and high mass host galaxies (Sullivan et al., 2011; González-Gaitán et al.,
2021; Chen et al., 2022; Kelsey et al., 2022), and also account for the
observed increase in the step size as a function of SN colour (Brout &
Scolnic, 2020; Popovic et al., 2021a; Kelsey et al., 2022) as well as the
increase in SN Ia luminosity dispersion as a function of SN colour (Brout &
Scolnic, 2020; Popovic et al., 2021a).

Wiseman et al. (2022, hereafter W22) explored variations of the Brout &
Scolnic (2020) model by simulating the stellar age distributions of galaxies
and the delay-time distributions (DTDs) of SNe Ia. The DTD is the probability
distribution describing the ‘delay’ between the star formation event in which
the SN progenitor was formed and the epoch when the SN progenitor explodes.
This is commonly modelled as a power law of form $t^{\lambda}$ (e.g., Maoz et
al., 2014)222Usually the power-law index is represented by $\beta$. Here we
use $\lambda$ to avoid confusion with the SN colour – luminosity relation.,
with $\lambda\sim-1$ favoured by observations. By incorporating the DTD with
the evolution of stellar ages within simulated galaxies, W22 trace the
connection between the intrinsic properties of stellar age and SN Ia
progenitor age, with measurable galaxy properties such as stellar mass, SFR,
and rest-frame colour, and how these link to SN Ia distance measurements.

W22 find that model with a mean $R_{V}$ of a SN Ia population
($\overline{R_{V}}$) that changes with galaxy age (rather than stellar mass)
best reproduces the observed step and its evolution with SN colours – in
particular when this model is compared to observations of both host galaxy
stellar mass and host galaxy colour. This is a complex, multi-dimensional
parameter space and several permutations of the model can explain the broad
trends in the data equally well, although no model is able to reproduce the
data in both host parameters simultaneously. In particular, models with and
without an intrinsic SN Ia luminosity step perform equally well, with
different observed luminosity step sizes accounted for by varying differences
in the $R_{V}$ population means. A model where $R_{V}$ changes with respect to
the age of the SN Ia progenitor itself (e.g., the extinction is caused by
circumstellar material that is affected by the astrophysics of the white dwarf
and its companion) does not provide good fits to the data.

Other analyses have found conflicting results. By measuring $R_{V}$ for
individual SNe Ia using a hierarchical Bayesian model, Thorp et al. (2021) and
Thorp & Mandel (2022) found no significant difference in the $R_{V}$
population means between SNe in low and high mass galaxies. There is also some
evidence that the step remains when measuring distances using near infra-red
(NIR) light curves, where the effects of dust should have a smaller impact
(Ponder et al., 2021; Uddin et al., 2020; Jones et al., 2022), although, by
contrast, Johansson et al. (2021) found the step was removed in both the NIR
and when fitting each SN Ia for its $R_{V}$.

Evidence for a large difference between the dust laws in the general
population of old/massive galaxies and young/low-mass galaxies is also
limited, hindered by the difficulty in measuring dust laws for large samples
of galaxies typically only observed with optical and sometimes NIR photometry.
Salim et al. (2018) showed that the $R_{V}$ values for star-forming galaxies
increase with stellar mass, opposite to the inference of the Brout & Scolnic
(2020) model. This increase corresponds to an increase in the dispersion of
the $R_{V}$ parameter and an increase in the optical depth (which correlates
with stellar mass). However, Salim et al. (2018) also show that the $R_{V}$ of
passive galaxies is $\sim 0.5-1$ smaller than star-forming galaxies,
indicating that the dust is dependent on age/star-formation history in the
direction matching the W22 model. A similar effect has been observed in SN Ia
host galaxies by Meldorf et al. (2022) and Duarte et al. (2022). The
increasing fraction of passive galaxies (and also of passive SN Ia hosts) with
stellar mass may thus explain the inferred decrease in $R_{V}$ with stellar
mass inferred by Brout & Scolnic (2020), Popovic et al. (2021b) and W22.

Overall, dust-based models can explain the optical SN Ia data, but it is
unclear which host galaxy parameter is the key driver of a changing dust law
and whether dust can account for the full step. The tests conducted by B22
provide a framework to disentangle the problem: a successful model must
reproduce both the trend of luminosity step as a function of SN Ia colour as
well as the relationship between step magnitude and tracer contamination.

In this work, we test the dust-based models explored in W22 in the framework
of B22, which we outline in Section 2. Using a simulated sample of SNe Ia, we
assume each potential astrophysical driver of a changing dust law as a ‘truth’
and assess the resulting relationship between step size and contamination
between the two SN Ia populations traced by each host galaxy property, which
we present in Section 3. In Sections 4 and 5 we asses and review the
implications of our findings. Where appropriate, we assume a reference
cosmology described by a spatially flat $\Lambda$CDM model with
$H_{0}=70~{}\mathrm{km}~{}\mathrm{s}^{-1}~{}\mathrm{Mpc}^{-1}$ and
$\Omega_{M}=0.3$.

Table 1: SN Ia luminosity models tested. $Y$ is the environmental property on which $R_{V}$ varies between $\overline{R_{V,1}}$ and $\overline{R_{V,2}}$. $X$ is the astrophysical driver of additional luminosity steps that have magnitude $\gamma_{X}$. Model Name | $Y$ | $\overline{R_{V,1}}$ | $\overline{R_{V,2}}$ | $X$ | $\gamma_{X}$
---|---|---|---|---|---
SN step only | - | 2.5 | 2.5 | SN progenitor age | 0.2
Age $R_{V}$ | Host stellar age | 1.5 | 3.0 | - | 0.0
Age $R_{V}~{}+$ SN step | Host stellar age | 1.75 | 2.5 | SN progenitor age | 0.15
Figure 1: The measured SN Ia luminosity step $\gamma$ (Section 2.3) as a
function of the contamination of environmental tracers, using two different
reference tracers. The simulations have a constant $R_{V}$ and a $0.2$ mag
luminosity step on SN progenitor age. The solid line (the ‘model’) illustrates
a direct relationship between the true step at zero contamination and a zero
step at 100 per cent contamination. Circles correspond to the simulation from
this work, diamonds are observational measurements taken from B22. Open
circles are intrinsic galaxy properties taken from the simulation, while
filled markers are galaxy properties estimated from spectral energy
distribution fitting (either of data or of simulations outputs). Simulated
properties are shown without uncertainties since in the simulation they are
known exactly. The dashed line and shaded region show a least-squares fit to
the simulated steps (forced through the point [100%, 0]) and $1$ $\sigma$
uncertainties respectively. Left: contamination computed assuming SN
progenitor age as the reference tracer; Right: contamination computed
incorrectly assuming galaxy age as the reference tracer.

##  2 Method

In W22 a framework was introduced for tracing the effects of galaxy-scale
astrophysical effects through to SN Ia distance measurements in a simulation.
The simulation comprises two core components described in detail in previous
works and summarised in this section: firstly, we generate a library of
potential host galaxies (Wiseman et al., 2021), and then we generate samples
of SN Ia light-curve parameters associated with those galaxies (W22). The
simulations are optimised to best reproduce SN samples from the Dark Energy
Survey supernova program (DES-SN; Bernstein et al. 2012), but in principle
they can be adapted to mimic any SN survey. Unless otherwise stated, we use
identical model parameters to W22, shown in Table 1.

###  2.1 Galaxy models

To track stellar populations through the evolution of galaxies, we use the
galaxy mass assembly prescriptions of Wiseman et al. (2021) based upon the
work of Childress et al. (2014). These simulations begin with seed galaxies at
a range of look-back times and follow them as they evolve along the star-
forming main sequence according to an empirical relation between stellar mass,
SFR, and redshift. The simulations also include the quenching of star
formation. Here we follow the updated prescription of W22, where each seed
galaxy is replicated 100 times, with the time of quenching onset drawn from a
probability distribution. Small bursts of star formation happening at random
times are drawn in a similar fashion. The output of the simulation is a
representative library of galaxies across a large range of stellar mass and
redshift, for which the full stellar age distribution is known at a resolution
of 0.5 Myr. This star-formation history for each galaxy is convolved with a SN
Ia DTD to produce a probability distribution for the age of SN Ia progenitors
at the time of explosion. We use the power-law DTD from Wiseman et al. (2021)
with $\lambda=-1.13$.

###  2.2 Supernova models

SNe Ia in the simulation are represented by their light-curve parameters and
corresponding uncertainties. We parametrise light curves in the SALT framework
(Guy et al., 2007), with SN colour represented by the parameter $c$ and SN
stretch by the parameter $x_{1}$. We assign intrinsic $c$ and $x_{1}$ to each
SN by drawing from probability distributions that relate to their host galaxy
or their progenitor age. In this work we use the two-component colour
distribution of Brout & Scolnic (2020), with an intrinsic component following
a Gaussian distribution and an external reddening component drawn from an
exponential distribution. We draw $x_{1}$ from a Gaussian mixture introduced
by Nicolas et al. (2021), with the component weights determined by SN
progenitor age, using identical parameters to W22.

In cosmological SN Ia samples, SN absolute magnitudes show scatter around
their mean. Various models have been introduced in order to account for this
scatter, which are summarised in Brout & Scolnic (2020) and Popovic et al.
(2021b). In the Brout & Scolnic (2020) framework used in this work, the
intrinsic scatter is caused by SN to SN variation in the intrinsic colour-
luminosity coefficient $\beta_{\mathrm{SN}}$ and dust extinction $R_{V}$. In
our simulations, the SN Ia intrinsic luminosity $M_{B}$ is fixed to $-19.365$
mag. This intrinsic brightness is modified according to the intrinsic stretch
using a linear relationship with a coefficient fixed to
$\alpha_{\mathrm{SN}}=0.15$, and by the intrinsic colour–luminosity
relationship drawing the coefficient $\beta_{\mathrm{SN}}$ from a Gaussian
distribution with mean 2.0 and standard deviation 0.35, which were the best-
fitting values for the scatter model in Brout & Scolnic (2020). This adjusted
absolute magnitude is then converted to an apparent magnitude $m_{B}$ using a
distance modulus computed from a fixed reference cosmology. The brightness is
reduced by $\Delta m_{B}$, which is determined according to the extinction via

$\Delta m_{B}=(R_{V}+1)\times E(B-V)\,,$ (1)

where $R_{V}$ is the total-to-selective extinction ratio and reddening
$E(B-V)$ is drawn from an exponential distribution.

The W22 models draw the $R_{V}$ value for a given SN Ia from one of two
Gaussian distributions with different means but identical standard deviations,
with the means depending on a given host galaxy parameter $Y$: those SNe Ia
hosted in galaxies above a specific value of $Y$ are drawn from a distribution
with a different mean than those SNe Ia hosted in galaxies below that value of
$Y$. In this work, we focus only on using the host galaxy stellar-mass-
weighted mean stellar age (which we refer to simply as stellar age) for $Y$ as
that was the best matching parameter in W22. Additionally, we allow for an
intrinsic luminosity step driven by different host properties/progenitor ages
that is not related to $R_{V}$ differences: this luminosity step
$\gamma_{X,\mathrm{sim}}$ is added or subtracted to the distance modulus of
the SNe Ia directly, rather than as a modification to the peak brightness
$m_{B}$. Here, we use the SN progenitor age for $X$. A summary of the models
we test is presented in Table 1.

We simulate 2000 SNe Ia in the redshift range $0\leq z\leq 1.2$. To mimic the
effect of observational survey selection effects, we apply efficiency
corrections following the DES host galaxy redshift detection efficiency as
modelled by Vincenzi et al. (2021), such that SNe in fainter host galaxies are
less likely to be included in the final sample. We add Gaussian noise to each
of $m_{B}$, $c$ and $x_{1}$ according to the SN Ia brightness and redshift
using empirical relations measured from DES-SN data (W22).

###  2.3 Supernova distances and residual steps

Using the ‘observed’ parameters modelled in the previous section, distance
moduli $\mu_{\mathrm{obs}}$ of the simulated SNe are computed using a form of
the Tripp estimator:

$\mu_{\mathrm{obs}}=m_{B}-M_{B}+\alpha x_{1}-\beta
c+\gamma_{X,\mathrm{obs}}\Gamma_{X}\,,$ (2)

where $\alpha$, $\beta$333Note that $\beta$ here accounts for all
colour–luminosity dependence, different from $\beta_{\mathrm{SN}}$ in the
simulations which only corresponds to intrinsic variation and not dust.,
$\gamma_{X,\mathrm{obs}}$, $M_{B}$ are nuisance parameters, and $\Gamma_{X}$
represents the step for some environmental property $X$:

$\Gamma_{X}=\left\\{\begin{array}[]{@{}rc@{}}0.5,&X<X_{\mathrm{split}}\\\
-0.5,&X\geq X_{\mathrm{split}}\\\ \end{array}\right.\,,$ (3)

where $X_{\mathrm{split}}$ is the location of the step for that property. To
find the values of the nuisance parameters $\alpha$, $\beta$ and $\gamma_{X}$,
we minimize the $\chi^{2}$ statistic for the observed distance moduli
$\mu_{\mathrm{obs}}$ compared to the fixed cosmological model
$\mu_{\mathrm{mod}}$, given the distance uncertainties and their covariance.

###  2.4 Step drivers, tracers, and contamination

To measure the accuracy of environmental tracers of SN Ia luminosity
variations we follow the method of B22. This prescription is based on the
assumption that SNe Ia are divided into two populations ($a$ and $b$) that
have a dependency on some physical property that we denote the driver.
Different environmental measurements, tracers, map to those populations with
differing levels of accuracy: a tracer that is closely related to the true
driver of the population split will accurately divide the populations, while a
tracer less closely linked to the SN populations will not split the
populations well. The degree of accuracy with which an environmental tracer
recovers the true driver of the population split is parametrised by
contamination ($C$), which is defined as the fraction of SNe Ia that are
classified as belonging to population $a$, but truly belong to $b$, plus the
opposite: SNe classified as $b$ but truly $a$, which we define briefly below.

For a given tracer, $N_{a}$ represents objects that truly belong to the class
$a$ while $N^{a}$ is the number of objects classified as $a$. Thus the
contamination can be defined as the fraction of objects classified as
belonging to $a$ that truly belong to $b$: $C^{a}\equiv N^{a}_{b}/N^{a}$.
Similarly there is contamination $C_{b}$: objects classified as $b$ but truly
$a$: $C^{b}\equiv N^{b}_{a}/N^{b}$. Total contamination $C$ is then defined as
$C\equiv C^{a}+C^{b}$. A tracer that perfectly separates the two SN
populations has $C=0$, while a random tracer carrying no information about the
SN populations has $C=100$ per cent.

The framework presented in B22 predicts that there is a direct relationship
between the measured step size and contamination: a tracer with no
contamination has the largest step, while that with 100 per cent contamination
has no step. Observations should lie directly on the line that connects these
points but are dispersed by observational noise. We call this line the B22
step – contamination relation. While the true driver is not known
astrophysically, B22 define a reference tracer to split the SNe, for which the
contamination is defined to be zero. Hypothesising that the SN Ia populations
are separated by their progenitor age, they use spectroscopic local sSFR as a
reference tracer, against which all other tracers display some contamination.
Local sSFR itself may be – and probably is – an imperfect tracer: therefore
against the perfect reference tracer, local sSFR will also show some
contamination.

In this analysis, there are two key differences to the work of B22. Firstly,
for each simulation, we know the driver(s) of the SN Ia luminosity steps in
that simulation. For simulations where these steps are caused by a single
driver, we can use that driver as a reference tracer knowing it has zero
contamination, but also test other reference tracers to simulate the real-
world effects of choosing an incorrect reference tracer.

Secondly, not only can we use observable properties such as stellar mass,
galaxy colours, and sSFR as tracers, but we can also use intrinsic properties
such as SN Ia progenitor age and galaxy age, which cannot be observed
directly. Thus, by simulating SN Ia samples with luminosity steps driven by SN
Ia progenitor age and galaxy age, measuring their steps and contamination from
observable properties, and comparing to the model of B22, we can break the
degeneracies between the models of W22. In our analysis we investigate the
following tracers: global host-galaxy rest-frame $U-R$ colour, global host-
galaxy sSFR (as would be measured by a photometric spectral energy
distribution fitting method), and the global host-galaxy stellar mass. Note
that our simulations are not resolved, i.e., our simulations cannot estimate
local tracers (e.g., local sSFR).

To compare step sizes measured in data to those in our simulation, we include
the data used by B22. Our simulations do not provide the measurements of local
sSFR that B22 use as a reference tracer in the data, so we calibrate the
contamination measurements by fixing the measured contamination of global
stellar mass from B22 to be the same as in our simulation, and use the
relative differences between the contamination of stellar mass and the other
tracers used by B22 to include in our analysis. The B22 contaminations and
steps were measured on data, and so the relative differences are only valid
given the true underlying model. However, we show them against all of our
simulations: the agreement between our simulated and the measured
contamination and steps for each tracer and in each model provides additional
information on which model is most realistic.

Figure 2: The SN Ia luminosity step as a function of the contamination of
environmental tracers for the model where galaxy age is the astrophysical
driver: an $R_{V}$ difference of 1.5 between old and young galaxies and no
intrinsic luminosity step on SN Ia progenitor age. Left: contamination
computed assuming, incorrectly, SN Ia progenitor age as the reference tracer;
Right: contamination computed correctly assuming galaxy age as the reference
tracer. Figure 3: The same as Fig. 2, but where galaxy age and SN progenitor
age are both astrophysical drivers: an $R_{V}$ difference of 0.75 between old
and young galaxies and a 0.15 mag intrinsic luminosity step on SN Ia
progenitor age. Left: contamination computed assuming SN Ia progenitor age as
the reference tracer; Right: contamination computed assuming galaxy age as the
reference tracer.

##  3 Results

B22 demonstrated that, under the assumption that SNe Ia can be split into two
populations with different mean luminosities, there is a linear relationship
between the ability of an environmental tracer to divide the SNe between those
two populations and the size of the recovered luminosity step. In that
analysis, low-redshift samples of SNe Ia followed the step–contamination
relationship when using local sSFR as a reference tracer. Any model of SN Ia
populations that seeks to explain the luminosity step should display a similar
trend to that observed in the data. We validate this method by testing a
simulation with a simple luminosity step on SN Ia age (Section 3.1). We then
construct step–contamination relationships for the two best-matching models to
the step versus SN Ia colour relationships from W22 in Section 3.2.

###  3.1 SN Ia progenitor age step only

We begin with a simulation with fixed $R_{V}$ and a $0.2$ mag step in the
intrinsic SN Ia luminosity at a progenitor age of $0.75$ Gyr. We then use the
progenitor age as a reference tracer – in the simulation this is the true
driver, so it has a true contamination of zero. The contamination and step
measurements measured for the other tracers are shown in the left-hand panel
of Fig. 1. The $0.2$ mag step is almost fully recovered when fitting for
$\gamma_{\mathrm{SNage}}$ – a slight reduction in the step is caused by SN Ia
progenitor age affecting both $\gamma$ and $x_{1}$ simultaneously.

The other environmental tracers are found at higher contaminations and lower
step sizes, as expected from the B22 step – contamination relationship (shown
as a solid line). We show a least-squares regression line between the points
(the dashed line and shaded uncertainty), forcing the line through the point
of [100% contamination, zero step]. The small uncertainty in this best-fitting
line illustrates the validity of using progenitor age as the reference tracer
given this SN model, and the B22 measurements lying close to the predicted
step – contamination line provides reassurance that our ad-hoc connection
between the simulated and observed measurements is reasonable. The exception
is stellar mass, which shows a large discrepancy between the observed and
simulated step size. For the correct model, the recovered contamination and
step size should be consistent between observations and the simulation. The
implication is thus that an SN-progenitor-age – luminosity step does not
explain the full observed effect. Indications that this model is not a full
description of the effect are also apparent in step size versus SN colour
relationships analysed Kelsey et al. (2022) and W22.

Our second test is to incorrectly use galaxy age as a reference tracer for the
simulation that is driven by an SN Ia progenitor-age step (Fig. 1, right). As
expected, the agreement is worse than when using SN Ia progenitor age as the
reference tracer. SN progenitor age has the largest contamination (60 per
cent) but (as expected) the largest step. The points all lie above the B22
step – contamination relationship, indicating that the chosen reference tracer
is not the true driver: the actual contamination of that tracer should be non-
zero, which would increase the gradient of the step – contamination line and
allow it to be more consistent with the other points. The observed local
measurements of spectroscopic sSFR and $u-r$ colour show less contamination
here than when SN progenitor age is used as a reference tracer. These lower
contamination measurements, along with the high contamination for SN
progenitor age, imply that local spectroscopic sSFR and local $u-r$ are more
closely related to galaxy age than they are to SN progenitor age. This is a
consequence of the SN Ia DTD which decouples the observed stellar population
age with that of SN progenitors: roughly half of SNe Ia explode over 1 Gyr
since the star formation event that formed the progenitor star.

There are some notable discrepancies between the observed and simulated
contaminations, whichever reference tracer is used. A clear example is that
the contamination for galaxy colour is much larger in the simulations than in
the observations. Part of this difference can be attributed to the simulations
only tracing global galaxy colour while the observations are measured locally
to the SN: the inability of our simulations to predict locally-measured
properties is a limitation that will be rectified with future developments.

A second discrepancy is the relative lack of dynamic range in the
contamination of simulated tracers, whereas the contaminations of observed
tracers show greater dispersion. For example, using SN progenitor age as the
reference tracer, the other simulated tracers show a range of $55$–$70$ per
cent contamination; using galaxy age as the reference tracer the other
simulated tracers cover a $40$–$60$ per cent range. A specific example is that
both stellar mass and $U-R$ colour (both routinely used in the literature) are
$\sim 60$ per cent contaminated with SN progenitor age as the reference
tracer, and around $40$ per cent contaminated with galaxy age as the reference
tracer. This results from the strong covariance between SNe in calculating the
contamination of these tracers. For example, when using galaxy age as a
reference tracer, 89 per cent of the SNe misclassified by stellar mass are
also misclassified by $U-R$ colour. This strong correlation between the
simulated tracers may indicate oversimplifications in the galaxy evolution
model, such as the nature of starbursts, the quenching of star formation,
ultraviolet ionising continuum, dust attenuation, or active galactic nucleus
activity.

###  3.2 Galaxy age, dust, and SN progenitor age steps

Having introduced the method using a single driver (a fixed luminosity step
driven by SN Ia progenitor age), we next investigate models that have
previously been shown to best reproduce the variation of SN distance moduli as
a function of host and SN properties. W22 showed that a SN progenitor-age
–luminosity step alone cannot explain the observed trend of step size
increasing with SN colour. We therefore introduce the two successful models
from that paper that are built upon a dust law that varies with galaxy age, as
introduced in Section 2.2. First, we use only that varying dust law (Section
3.2.1), and then we combine it with the intrinsic SN progenitor-age –
luminosity step (Section 3.2.2).

####  3.2.1 Galaxy age $R_{V}$ step only

Fig. 2 shows the relationships between step size and contamination for a model
with a large $\overline{R_{V}}$ difference of 1.5 between old and young host
galaxies, and no intrinsic SN luminosity step. As expected, using SN
progenitor age as a reference tracer (left panel) does not result in a good
agreement between prediction and either simulations or data: all of the points
lie above the B22 step – contamination relationship. Assuming galaxy age as
the reference tracer performs much better (right panel): the dispersion on the
best-fitting line is small and in agreement with the model prediction.

As also expected, local spectroscopic sSFR displays the largest step, as found
by Rigault et al. (2020) and B22. Interestingly, SN progenitor age is the most
contaminated and has the smallest step of all tracers tested. As with the
model with the SN progenitor-age step only, the simulated host stellar mass
step is smaller than the observed one, meaning that the model does not account
for the full luminosity difference observed in galaxies of different masses,
despite adequately reproducing the step sizes for the other tracers
considered.

####  3.2.2 Galaxy age $R_{V}$ step plus SN progenitor-age – luminosity step

In Fig. 3 we show a model where the difference between $R_{V}$ in young and
old galaxies is smaller, and an additional intrinsic luminosity step as a
function of SN progenitor age is added. With this model, neither using SN
progenitor age nor galaxy age as reference tracers results in good agreement
between the step – contamination prediction (solid line) and the
data/simulations. This is because neither is the sole driver of the step in
the model and so assuming their contamination to be zero is not valid. In
fact, given that there are multiple drivers that are related but distinct from
each other, there is no one-dimensional space in which the SN populations can
be perfectly separated. It follows that for such a model, a single
$\gamma_{X}$ for any tracer $X$ can never remove the full step.

##  4 Discussion

Identifying the underlying driver of the SN Ia luminosity step is important
for precision cosmological measurements, as well as in the physical
understanding of SN Ia explosions. This driver is unlikely to be a direct
observable (e.g., a magnitude or colour) or a simple physical property that
can be directly inferred from observations (e.g., global stellar mass), hence
the search for the best indirect tracer has seen increasing focus in recent
work. Here we discuss the implications of our simulations in two aspects:
first, we discuss the contamination of tracers and their relation to physical
parameters, and then we discuss our work in the context of cosmological
distance measurements.

###  4.1 What drives the step and what is the best tracer of it?

B22 find that spectroscopic local sSFR is the best environmental tracer of the
SN Ia luminosity step: using it as a reference tracer minimises the dispersion
around the best-fitting step versus contamination relationship. The local sSFR
measurement itself is unlikely to relate directly to SN Ia luminosities.
Instead, local sSFR presumably acts as a tracer of some fundamental underlying
properties of the local stellar population or SN Ia progenitor. These might
include their respective ages, which trace relationships between galaxy
evolution and feedback, dust creation and destruction, metallicity, and white
dwarf mass, which all have differing levels of importance. By testing purely
SN Ia progenitor-age-driven models (Section 3.1), galaxy-age-driven models
(Section 3.2.1), and a combination of the two (Section 3.2.2), we have been
able to test both the underlying drivers and how well they are traced by
measurable parameters.

####  4.1.1 Contamination

We first consider tracer contamination. The contamination measurements
presented in B22 are derived directly from observations but are only relative
and depend upon tracer choices, while those measured in our simulations are
connected directly to step drivers but depend upon the relationships built
into the galaxy evolution model of Wiseman et al. (2021). By comparing both
observed and simulated contamination for given observables against our
synthetic reference tracers, no tracer particularly closely follows the SN
progenitor age (see left-hand panels of Figs. 1–3). Galaxy age is 55 per cent
contaminated with reference to SN progenitor age. This is because of the
declining power-law nature of the DTD: roughly half of SNe Ia explode within 1
Gyr of progenitor system formation and half beyond 1 Gyr. This smooths out the
relationship between the age of a stellar population and the SNe Ia that
explode within it. The best tracer (of all observed and simulated) of SN Ia
progenitor age is spectroscopic local sSFR, as identified in Rigault et al.
(2020) and B22. Interestingly it is a better tracer of SN progenitor age than
the overall galaxy age: the fact that the measurement is local carries more
information about the progenitor than is lost by using a tracer (sSFR) instead
of knowing the property directly.

When choosing galaxy age as a reference instead of SN progenitor age, the
measurable properties of spectroscopic local sSFR, (local) galaxy colours, and
stellar mass are much less contaminated. This is because they are more closely
related to the age of the stellar population than they are to an individual SN
progenitor age. Spectroscopic local sSFR can still be used to broadly trace SN
progenitor age – spectroscopic local sSFR separates the SN Ia stretch
population well (Nicolas et al., 2021), and SN stretch is much more likely to
be linked to progenitor age than galaxy age. Nevertheless, our results show
that neither spectroscopic local sSFR nor any other tracer should be expected
to separate progenitor populations based on their age any better than with
$\geq 30$ per cent contamination. On the other hand, our results indicate that
spectroscopic local sSFR can separate stellar population ages with $\simeq 10$
per cent contamination. At low redshift, where galaxies can be more easily
resolved and this tracer is therefore measurable, we encourage observations to
obtain spectroscopic local sSFR in SN host galaxies in order to trace the step
driver accurately.

At higher redshifts where it is not currently possible to obtain spectroscopy
with 1 kpc resolution, local photometric and global measurements are the only
available tracers. In such cases, if it is possible to measure local galaxy
colour then this property traces either SN progenitor age or galaxy age best.
At high redshift (roughly $z\gtrsim 0.6$ in DES, although this limit will be
higher with better seeing in the Legacy Survey of Space and Time; Ivezić et
al. 2019), where the imaging cannot resolve local regions smaller than a few
kpc, local measurements of colour and sSFR are not possible. In this case, the
specific choice of global tracer makes little difference when tracing SN
progenitor age.

Photometric sSFR is marginally disfavoured when tracing galaxy age. This
result itself is slightly counter-intuitive because sSFR is usually assumed to
be a strong discriminator of galaxy SFHs and thus ages. We find the large
contamination is due to the relatively broad range of stellar mass at which
galaxy quenching can occur in our simulations, which was chosen to best
reproduce observations of galaxy populations. This range translates to a range
of ages at which quenching occurs, hence blurring out the correlation.

The lack of range between the contamination of the simulated tracers compared
to the much larger respective range for the observed data is likely a
combination of two factors. Firstly, the observations are local measurements,
while the simulation does not carry spatial information so only global
measurements are possible. Future work will improve the model to include
spatial dimensions in the galaxy evolution model. A second reason for the lack
of range is that the simulations suffer from oversimplifications that lead to
missing or washed-out relationships, or a lack of diversity in the galaxy
population. Again, these issues will be addressed in future versions of the
simulations.

####  4.1.2 The true driver of the step

So far we have shown that measurable galaxy properties are better tracers of
galaxy age than of SN progenitor age, but that doesn’t necessarily mean that
galaxy age is the true step driver. Instead, a model representing the true
driver should result in a tight correlation between step magnitudes and
contamination for each tracer. In Section 3.2 we find a good match between the
simulated and measured steps and tracer contamination assuming a model with a
large $R_{V}$ difference and no intrinsic luminosity step, using galaxy age as
the reference tracer. Such a result indicates that a galaxy-age varying
$R_{V}$ is consistent with causing the full step independent of the choice of
tracer and that there is no need for an additional difference in the
luminosity of SNe as a function of their age. The model with a small $R_{V}$
difference plus an intrinsic luminosity step shows a worse agreement than the
no-intrinsic step model, using either SN progenitor age or galaxy age as
reference tracers. Of these, SN progenitor age provides a better match but
with all other tracers displaying steps larger than predicted.

Taken at face value this would add further weight to the argument for a single
galaxy-age $R_{V}$ step as the sole driver. The reality is less clear cut: in
the model with both $R_{V}$ and intrinsic steps, we know that neither SN
progenitor age nor galaxy age are the sole true driver since the luminosity
varies as a combination of the two. Thus, neither has a true contamination of
zero rendering the step–contamination diagram inaccurate. We thus do not rule
out that model as being the true driver of the observed steps, but that a one-
dimensional step (or two-dimensional contamination–step parameter space) is an
inadequate discriminator for a model with multiple drivers of luminosity
variations. It has been shown in previous work that a combination of
environmental tracers can provide a better standardization than a single
tracer (e.g., Rose et al., 2021; Kelsey et al., 2022). The development of a
statistical discriminator between multi-driver models is left for future work.

###  4.2 Impact on cosmological measurements

Since their discovery (Kelly et al., 2010; Lampeitl et al., 2010; Sullivan et
al., 2010), host galaxy–SN luminosity correlations have usually been removed
from SN Ia distance measurements via a single standardization parameter,
$\gamma$ (Eq. 2). In parallel, the last decade has seen vast improvements in
the methodology for correcting Malmquist-like biases, or selection biases, in
SN Ia samples. From simple redshift-only corrections (e.g., Betoule et al.,
2014) the frameworks are now in place to perform bias corrections in many
dimensions, including based upon parameters of the SN light curves and host
galaxies (Scolnic & Kessler, 2016; Popovic et al., 2021a; Popovic et al.,
2021b). At this point, the standardization parameters and bias correction
routines become intricately connected. For example, correcting for a bias
involving the light-curve stretch is also host galaxy dependent (because of
stretch–host galaxy relationships) and thus affects the measured value of
$\gamma$ (Smith et al., 2020).

A recent bias correction model, ‘BBC4D’ (Popovic et al., 2021b), is based
directly on the $R_{V}$ models used in this work. BBC4D corrects for dust-
related luminosity bias by assuming different values of $R_{V}$ in hosts below
and above a threshold in some tracer, nominally stellar mass. The results of
this work demonstrate that if galaxy age is the true driver of the $R_{V}$
difference, populations split by stellar mass are nearly 40 per cent
contaminated, meaning using stellar mass as the host parameter in BBC4D may
not fully account for the bias/step. On the other hand, we show that other
global measurements, particularly host colour, are no less contaminated. We
thus predict, assuming galaxy age as the driver, that SN distances inferred
using BBC4D may still display a residual luminosity step or may not have their
intrinsic scatter reduced as much as would be achievable with a less
contaminated measurement such as spectroscopic local sSFR. Similar, if not
worse, biases are expected for an analysis not implementing $R_{V}$-based bias
corrections but using simple $\gamma$ nuisance parameters. Future work will
focus on estimating and reducing the bias on cosmological measurements
introduced by tracer contamination.

##  5 Conclusion

In this work we have investigated the astrophysical driver of the SN Ia
environmental luminosity step, by comparing the size of steps inferred by
different environmental tracers in a simulated sample of SNe Ia, and how
contaminated those tracers are with respect to the driver of the step. We
first validated the method by inputting a simple SN-progenitor-age –
luminosity step and testing the resulting relationship between the measured
step and contamination of each environmental tracer. We then tested a model
where the entire step is caused by a large difference in dust extinction slope
$R_{V}$ between young and old galaxies, and one where the $R_{V}$ difference
is smaller and there is an additional, intrinsic step in SN luminosity as a
function of the SN progenitor age. We find:

  1. 1.

in our simulation of a simple SN progenitor age step, we recover the expected
linear relationship between step size and tracer contamination which matches
the trend seen in observations.

  2. 2.

modelling SNe Ia with an $R_{V}$ that changes with host galaxy age as the sole
driver of luminosity differences, and fixing galaxy age as the reference
tracer, results in the best correlation between step size and contamination.

  3. 3.

local sSFR is less contaminated when assuming galaxy age as a reference tracer
than when SN progenitor age is the reference.

  4. 4.

that the step versus contamination relationship is looser for a model with
both a galaxy-age – $R_{V}$ step and an intrinsic SN luminosity step,
regardless of whether the reference is SN progenitor age or galaxy age.

  5. 5.

that the simple tracer contamination versus step parameter space is not
adequate to constrain models with multiple parameters driving the step.

  6. 6.

that the implication of point (v) is that using a single $\gamma$ or a single
tracer in a host-dependent bias correction will not remove the full effect of
the step.

The choice of tracer is deeply woven into the measurement of cosmological
parameters. At face value, we confirm the result of B22 that spectroscopic
local sSFR is the closest representation of the driver and should be used to
correct distances where available. At high redshift, where that measurement is
not possible, both stellar mass and $U-R$ colour are the best, non-ideal,
tracers. We caution against performing cosmological measurements using a
simplistic $\gamma$ correction based on a single tracer and encourage the
modelling of the step alongside selection effects.

## Acknowledgements

PW acknowledges support from the Science and Technology Facilities Council
(STFC) grant ST/R000506/1.

## Data Availability

This work is based on simulations from publicly available code
(https://github.com/wisemanp/des_sn_hosts) and data from a published paper
(B22).

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Cloud-native 5G experimental platform section:platform

The 5G architecture consists of two parts that have remarkably changed from
previous generations: the new radio access network (NG-RAN) supporting the new
radio (NR) and the 5G Core Network (5GC). In this section, we describe the
main components of our experimental platform and propose a MEC-enabled testbed
to demonstrate the end-to-end monitoring system.

[width=.86]img/sba.png

Figure : 5G System Architecture in reference point representation (based on
[3gpp_sba]). Some NFs and interfaces are not included for the sake of
representation.

##  Open-source 5G core

The movement toward softwarization of telecommunication networks has deeply
influenced the creation of the 5G core (5GC) [5gamericas]. Rather than relying
on monolithic elements, 5G adopts a service-based architecture (SBA) composed
of NFs that modularize the tasks of the core. As shown in Fig. , these NFs
interact through Service-Based Interfaces (SBIs), which employ
Representational State Transfer (REST) interfaces. A key feature of such SBA
modularization is network slicing, which benefits from softwarization and
cloudification. In essence, slices represent logical instances of the network
that can be tailored to optimize services and thus cope with different service
level agreements (SLAs) according to the use case. Further, within the SBA, we
may find the NWDAF (Network Data Analytics Function) and the MDAF (Management
Data Analytics Function) for generating insights from NFs data and taking
actions to enhance performance, including slice selection and control
[pateromichelakis2019end]. This new architecture allows 5G stakeholders much
more flexibility and openness, paving the way for an environment where open-
source perfectly suits. In this regard, some alternatives of open-source 5GC
are available, such as OpenAirInterface [nikaein2014openairinterface] CN,
Free5GC [free5gc], and Open5GS [open5gs]. In our experimental framework, we
use Open5GS v2.4.0, since it includes most of the 5GC NFs defined in 3GPP and
also allows deploying more than one UPF instance, thus supporting MEC-enabled
networks. Nevertheless, our experimental framework is designed to also work
with other open-source 5G cores under acceptable adjustments.111Preliminary
deployments of our containerized framework with OpenAirInterface CN and
Free5GC have been already successfully validated.

##  Cloud-native deployment of the 5G core

5G is expected to support use cases that go beyond raw throughput performance,
where the focus is to be put on service flexibility and agility. In this
regard, the procedure to deploy 5G NFs has a critical impact. In essence,
these functions can be instantiated as physical NFs (PNFs), virtual NFs
(VNFs), or containerized NFs (CNFs). Naturally, VNFs gained momentum against
siloed PNFs since the conceptualization of the modularized 5G SBA because of
the virtualization benefits in terms of efficiency, scalability, or cost.
Recently, it is the turn of Containerized Network Function (CNFs) to gain
momentum among operators [chun2019kubernetes] against conventional Virtualized
Network Function (VNFs) due to their higher degree of scalability, efficiency
for operation and management, energy-saving, and suitability for resource-
constrained edge applications. According to Docker [docker], a container is a
standard unit of software that packages up code and all its dependencies, so
the application runs quickly and reliably from one computing environment to
another. This makes a container image a lightweight, standalone, executable
package of software that includes everything needed to run an application.
Container deployments, which may span multiple hosts, are managed with an
orchestrator responsible for automating container creation, deletion, and
modification without service disruption. Notice that such tasks on containers
match the NFV lifecycle management. In this work, we adopt Kubernetes
[kubernetes] as a container orchestrator, since it is the de facto solution in
multiple industries for high-demand services with complex configurations. As
explained in the following subsections, we deploy both the 5GC NFs and
monitoring system within the same Kubernetes cluster using Helm charts [helm],
a collection of files that describe a related set of Kubernetes resources (see
Fig. ).222Notice that all the NFs in Open5GS can be compiled and deployed
separately, making it a suitable candidate for evaluating the performance of
distributed and cloud-native deployments of the 5GC. This way, the whole
framework is deployed in just two commands, one for installing the monitoring
system and the other for installing Open5GS. Once the deployments are
instantiated, connectivity among containers and toward external services must
be provided. In particular, our Kubernetes cluster relies on Calico [calico],
a well-known container network interfaces (CNIs) plugin to implement such
networking capabilities.

[width=.8]img/helm.png

Figure : Helm and Kubernetes flowchart. The Open5GS chart is composed of the
NF templates in blue (e.g., AMF), whereas the monitoring chart includes
templates for kube-prometheus in red and the Amarisoft sampling function in
yellow.

##  Experimental RAN integration

The radio access network (RAN) is another critical component of 5G networks,
since it provides individual users with wireless connectivity to the core and
external data networks. There are different alternatives when it comes to
experimenting with 5G RAN, which can be classified in simulated/emulated (e.g,
UERANSIM [ueransim]) and physical/real (e.g., Amarisoft). In this work, in
order to provide actual over-the-air-transmissions, we rely on Amarisoft’s
AMARI Callbox Ultimate [amari] acting as a gNB with high-performing NR
capabilities. Nevertheless, other RAN alternatives like UERANSIM have been
also successfully integrated within the testbed without requiring any
configuration changes in the Open5GS Helm chart. Notice that the RAN in our
framework is essentially a physical NF (PNF), whereas the 5GC and monitoring
system are built on CNFs. As for how to integrate the AMARI gNB with the
Open5GS core, we shall indicate the AMF’s IP address in the gNB configuration
file to establish the NG Application Protocol (NGAP) connection. In our case,
since the AMF runs as a CNF, a custom Kubernetes service exposes the AMF
functionality to let the gNB point to the master node IP rather than to the
AMF’s pod IP. This is a common practice, since pod IPs may change after
deletion, while services remain fixed.

##  A MEC-enabled 5G testbed

Testbeds are essential in telco research, as new architectures, techniques,
and features can be conveniently assessed and validated in the lab before
going into field trial campaigns. In this work, we implement and integrate the
testbed shown in Fig  to showcase our cloud-native end-to-end 5G experimental
platform. In particular, the 5G network is composed (from left to right) of
the following elements: two UEs emulated with Amarisoft AMARI UE Simbox Series
[amari_simbox], with UE1 always targeting best-effort services while UE2 may
target both best-effort and time-critical MEC applications (e.g., AR/VR); an
Amarisoft Callbox acting as a stand-alone 5G gNB; an edge node running the MEC
UPF and an iperf [iperf] server; a core node running the Open5GS CNFs and
another iperf server; a monitoring node hosting the monitoring containers; and
a master node managing the Kubernetes cluster. In the current testbed
implementation, since the focus is on the monitoring of the RAN and core
domain components, the access and transport networks have been simplified to
Ethernet links in a local area network. As shown in the Kubernetes
representation in Fig. LABEL:fig:k8s, the cluster is composed of the
aforementioned nodes (master, core, edge, and monitoring). Therefore, the
master is responsible for deploying, controlling, deleting, and updating the
containers of each of the nodes.

[width=]img/marsal-csndsp-testbed.png

Figure : MEC-enabled 5G testbed. Data planes are shown in green (dashed arrow
for the MEC data plane). Monitoring connections are represented with red
dashed arrows.

Remarkably, all the 5GC NFs are deployed in one click (through Helm) in the
core and edge host. In particular, the core runs all the Open5GS NFs available
in v2.4.0, including both critical (e.g., AMF or SMF) and secondary NFs (BSF).
Besides, for the sake of enabling MEC platforms, the presented cloud-native
deployment provides two UPFs: one located at the core and the other, at the
edge node. This way, each UPF serves as a PDU session anchor and provides a
connection point to different access networks, one being the conventional
Internet, and the other any data network that can benefit from MEC processing
capabilities, like AR/VR application components. Finally, the monitoring node
is in charge of running all the containers related to the end-to-end
monitoring of the components. To that aim, and as explained in
§LABEL:sec:monitoring, kube-prometheus [kube-prom] is deployed for monitoring
Kubernetes elements, while a custom sampling function is developed to pull
metrics from the Amarisoft Callbox API. So, these monitoring CNFs can be
viewed as 5G’s application functions (AFs) within 5G’s SBA.