© The Authors 2025

1. Introduction

Type Ia supernovae (SNe Ia) are fundamental tools in modern astrophysics and cosmology research. With their tremendous luminosities, SNe Ia act as standardisable candles (Phillips 1993; Hamuy et al. 1995; Riess et al. 1996; Tripp 1998), allowing us to probe vast distances in our Universe. Most famously, SNe Ia were used to discover the accelerated expansion of the Universe and the existence of a cosmological constant (Riess et al. 1998; Perlmutter et al. 1999a). In the decades since, they have been used to improve the measurement of the dark energy equation-of-state parameter, w, as well as the local Hubble constant, H₀, with ever-increasing precision (Garnavich et al. 1998; Perlmutter et al. 1999b; Brout et al. 2019; Riess et al. 2022). However, major issues with our understanding of cosmology and the nature of the Universe are still present, such as the fundamental origin of dark energy (Abbott et al. 2024) as well as the so-called Hubble tension, whereby the Hubble constant determined by SNe Ia in the local Universe is incompatible with the measurement derived from the early Universe at > 5σ (Planck Collaboration VI 2020; Di Valentino et al. 2021; Freedman 2021; Riess et al. 2022). If this disagreement is not due to (multiple) systematic uncertainties, this would be a sign of new unknown physics. It has become increasingly apparent that the physical nature of SNe Ia, how they are standardised, and any systematic uncertainties present must be better understood.

SNe Ia are widely believed to be the thermonuclear explosion of a carbon-oxygen white dwarf (C/O WD) in a binary system (Hoyle & Fowler 1960; Maoz et al. 2014). However, the exact nature of the companion star, the mechanism by which the white dwarf explodes, and how it relates to the observed diversity of SNe Ia, has not been solved (Liu et al. 2023). A popular model is the single degenerate model (Whelan & Iben 1973), consisting of a C/O WD with a non-degenerate companion star, such as a main-sequence or red giant star. The white dwarf accretes material from its companion star, increasing its mass. As the white dwarf’s mass approaches the theoretical maximum Chandrasekhar (1935) mass (∼1.4 M_⊙), central densities overcome the electron degeneracy, triggering a runaway thermonuclear reaction. The second main companion scenario (or progenitor channel) is the double degenerate model (Iben & Tutukov 1984), consisting of two white dwarfs. This system can trigger an SN Ia either by a direct merger (due to orbital decay via gravitational wave emission), or by accreting helium from one white dwarf onto the other, most likely resulting in a double-detonation (Nomoto 1982; Greggio 2005; Fink et al. 2010; Woosley & Kasen 2011; Shen et al. 2018). In this scenario, the companion star donates sufficient amounts of helium, resulting in a helium detonation on the surface of the white dwarf, providing the necessary increase in the white dwarf’s central density, and triggering a sub-Chandrasekhar mass explosion.

An important avenue to further our understanding of SNe Ia is studying their environments; that is, their host galaxies. In modern cosmology, there is a puzzling dependence of corrected Hubble residuals (remaining scatter after standardisation) on the SN Ia host galaxy stellar mass, whereby higher-mass galaxies contain SNe Ia that standardise to brighter luminosities (Sullivan et al. 2010; Kelly et al. 2010; Lampeitl et al. 2010; Kim et al. 2019; Smith et al. 2020; Rigault et al. 2020; Briday et al. 2022). This is a significant source of systematic uncertainty and is commonly corrected for in modern studies via a mass-step (Betoule et al. 2014; Scolnic et al. 2018; Brout et al. 2022). Several studies have shown that galaxy colour also contains a step, and may be an overall better tracer of the correlation between SNe Ia and their host galaxy (Rigault et al. 2013, 2020; Roman et al. 2018; Kelsey et al. 2021). Galactic properties such as mass and colour are known to correlate with the stellar population age, metallicity, star formation rate, dust content, et cetera. (Tremonti et al. 2004). However, it is unclear what the dominant physical driving factor behind the observed SN Ia mass-step is, necessitating a more physical understanding of the correlation between SN Ia properties and their host galaxies (Brout & Scolnic 2021; Briday et al. 2022).

The impressive resolution and observing depths of modern (all-sky) imaging surveys have allowed us to study the structures of galaxies at vast distances. Galaxy morphologies are very complex, with no two galaxies being identical. In classic galaxy formation, discs are young, dusty, and star-forming features, while bulges contain old populations of stars, with higher metallicity and very little dust and star formation. Visual classification is usually performed by recognising common structural features within galaxies such as discs, bulges, bars, and spiral arms (see Benson 2010; Conselice 2014, for a review on galaxy formation and structure). In order to extract structural information from these galaxy images, many ‘Bulge+Disc’ (+other component) decomposition codes have been created that fit two-dimensional (2D) surface brightness models to galaxy images, such as GALFIT (Peng et al. 2010) and GIM2D (Simard et al. 2002). These codes allow for easy extraction of useful structural information from galaxy images, while also benefiting from robust correctional methods for atmospheric and instrumental smearing effects.

In this paper, we make use of the second data release of SNe Ia from the Zwicky Transient Facility (ZTF; Bellm et al. 2019; Graham et al. 2019) between 2018 and 2020 (‘ZTF DR2’; Rigault et al. 2025) and perform automated image decomposition of their host galaxies into bulge, disc, and bar components. This allows us to link the SN position to a specific stellar population and provide insights into the possible SN Ia progenitor channels (different companion stars are more common in certain environmental types; for example, high-mass main-sequence stars are found in young discs), as well as the potential effects of metallicity, age and dust extinction on SN properties. This analysis uses a custom, gradient-descent based code¹, designed to perform fully automated 2D image decomposition for nearby (up to a redshift of z ∼ 0.15) ground-based galaxy images. The code takes as input g- and r-band images from the Dark Energy Spectroscopic Instrument Legacy Imaging Surveys (DESI-LS, Dey et al. 2019), and generates a 2D surface brightness model of the galaxy without any user input. The code handles external source masking (Sect. 3.1), PSF convolution during fitting (Sect. 4.3), initial condition estimation (Sect. 4.4), cosmological corrections (i.e. K-corrections; Sect. 4.5), and morphological classification (Sect. 4.6). Throughout this paper, the 2018 Planck Collaboration results (Planck Collaboration VI 2020) are used for cosmological calculations, made available by AstroPy (Astropy Collaboration 2022).

2. Data

In this section, we describe the SN Ia and host galaxy image data used in our analysis. In Sect. 2.1 we define the images used for analysing the host galaxies, while in Sect. 2.2 we detail the image preprocessing. In Sect. 2.3 we introduce the ZTF SN Ia sample, quality cuts that are applied to the sample, and detail the parameters measured from the light curves.

2.1. DESI Legacy Imaging Surveys

The galaxy images of the SN Ia sample used in this study come from DESI-LS (Dey et al. 2019). We make use of g- and r-band images, due to these bands having the deepest available observing depths (AB magnitude system). The DESI-LS consists of three separate surveys, with the joint goal of supplying optical and infrared imagery of 14 000 deg² of extra-galactic sky for the DESI project. The Beijing-Arizona Sky Survey (BASS) uses the 2.3m Bok telescope to observe in g and r bands, while the Dark Energy Camera (DECam) Legacy Survey uses the 4m Blanco telescope in Chile, to capture grz imagery. The third survey, the Mayall z-band Legacy Survey, only observes in the infrared and is not used in this paper. The latest two data releases DR9 and DR10 are used in this paper, which when combined, provide galaxy images for 83% of the SNe Ia in the ZTF DR2, as is seen in Fig. 1. The missing SNe Ia are due to the DESI sky coverage having a larger cut-out for the Milky Way, relative to the ZTF sky coverage. While the observing period for the ZTF DR2 overlaps with DESI-LS DR10, contamination of the galaxy images by SN light is not a concern, as this study uses fitted galaxy models to extract information, rather than using the actual pixels at the SN location.

Fig. 1. DESI-LS sky coverage shown using a Mollweide projection, with DR10 in light-grey and DR9 in dark-grey blue. The blue points represent covered ZTF Cosmo DR2 SNe Ia (83%), while the red points are SNe Ia outside the sky coverage. The green line marks the Galactic plane.

2.2. Preprocessing of DESI-LS images

Each galaxy image is obtained by querying DESI-LS image cutouts from their sky-viewer². The coadded images (image stacks) are used, which are fixed at a pixel scale of 0.262 arcsec/pixel. The sky level has already been subtracted from these images. The centre of the image is taken from the matched host galaxy information for each ZTF DR2 SN Ia (Smith et al. 2025), while the size of the image is set to be a 70 × 70 kpc² aperture, calculated using the ZTF SN Ia redshifts. The maximum image size is limited to 600 pixels, corresponding to a 70 kpc aperture at z ∼ 0.02. Consequently, galaxies closer than this redshift have apertures smaller than 70 kpc, and hence may have a larger angular diameter than the cutout image. The quality cuts detailed in Sect. 3.3 ensure that only galaxies smaller than their cutout image are fitted.

The DESI-LS cutout service returns an image in flux space (in units of nanomaggie) as well as a corresponding inverse-variance image. The header information contains the sky brick ID, which is used to obtain the full-width-half-maximum (FWHM, in units of arcsec) of the PSF and the 5σ detection depths for the g and r bands. Images that lie on brick edges are returned as multiple images and were stitched together (totalling approximately 4% of the sample). The PSF FWHM and the 5σ detection depths are estimated by averaging the values from each image. Images that lie on three or more bricks (and hence returned as three or more separate images) were discarded, as there were only nine such events (0.3%).

The flux image, F (in units of nanomaggie), is converted to apparent surface brightness, μ (in units of mag/arcsec²), using the equation

$$\begin{array}{c}\hfill \mu =\mathrm{ZP}-2.5{log}_{10}\left(\frac{F}{{\mathrm{scale}}^2}\right)-A_{\mathrm{MW}},\end{array}$$(1)

where the zero-point (ZP) = 22.5 mag and the scale = 0.262 arcsec/pixel are both defined by DESI-LS. A_MW is the Milky Way dust extinction, calculated using a total-to-selective extinction ratio, R_V, of 3.1 and the extinction model from Fitzpatrick & Massa (2007). Surface brightness is commonly used in galaxy studies. It is a distance independent quantity, related to the density of stars at any particular location within a galaxy. The 1/d² dependence of flux is cancelled out by dividing in the pixel scale. With increasing distance, the resolution of the galaxy image (i.e. how many pixels per galaxy) decreases, but the measured surface brightness remains constant.

A shortcoming of Eq. (1) is that it cannot represent non-detections; that is, flux values of less than zero. The image processing described in subsequent sections requires 2D images containing real numbers. A hyperbolic transformation is performed, as is described in Lupton et al. (1999), and this generates reasonable magnitudes for negative flux values, while leaving high signal-to-noise ratio magnitudes virtually unchanged. This transformation replaces the logarithm with

$$\begin{array}{c}\hfill -2.5{log}_{10}(F)=-\frac{2.5}{ln(10)}[\mathrm{asinh}\left(\frac{F}{2b_{\mathrm{s}}}\right)+ln(b_{\mathrm{s}})],\end{array}$$(2)

where b_s is a softening parameter, taken from Stoughton et al. (2002), as 0.9 × 10⁻¹⁰ in the g band and 1.2 × 10⁻¹⁰ in the r band. Using this transformation for high signal-to-noise values, the dispersion in magnitude from Eq. (1) is on the order of 10⁻⁶ mag.

2.3. ZTF Cosmo DR2

ZTF (Masci et al. 2018; Bellm et al. 2019; Graham et al. 2019; Dekany et al. 2020) is a state-of-the-art optical survey that scans the entire Northern sky every two-three nights (g, r, and i band). Since it began operation in 2018, it has detected a very large sample of SNe Ia. The ZTF SN Ia cosmology project (see Dhawan et al. 2022, for the first data release), aims to use these SNe Ia as distance indicators to further our understanding of cosmology. This paper makes use of the second data release (ZTF DR2), containing 3628 spectroscopically classified SNe Ia, from the first three years of operation (2018–2020). For a full overview of the dataset, including technical details on photometric and spectral data, redshift measurements, spectral classifications, and host galaxy matching, see Rigault et al. (2025) and Smith et al. (2025).

The redshifts used for this sample come from three sources. Public galaxy catalogues (such as the NASA/IPAC Extragalactic Database) are used for 60.6% of the sample, of which 71% comes from DESI through the Mosthost programme (Soumagnac et al. 2024). These redshifts, along with a further 8.8% coming from visible galaxy emission lines in the ZTF SN Ia spectra, are known as ‘galaxy redshifts’ and have a typical precision of Δz ≤ 10⁻⁴. The remaining 30% of the sample derive their redshift measurements from the SNID SN template matching programme (Blondin & Tonry 2007). A comparison between galaxy- and SNID-redshifts was carried out by Smith et al. (2025), who find a mean SNID-redshift precision of Δz ≈ 10⁻³.

This paper mainly uses originally derived galaxy parameters to investigate correlations with SNe Ia. However, we also make use of the computed host galaxy stellar masses, provided in the ZTF SN Ia DR2. As is described in Rigault et al. (2025), the host stellar masses are computed by spectral-energy-distribution fitting of global photometry using the PÈGASE2 galaxy spectral templates (Le Borgne & Rocca-Volmerange 2002). The global photometry is derived from the HostPhot package (Müller-Bravo & Galbany 2022), using public Pan-STARRS1 DR2 photometry.

The ZTF SN Ia DR2 contains detailed spectroscopic classification of the SN Ia sample. These can be grouped into five subclasses, ‘normal SNe Ia’, ‘91bg-like’ (Filippenko et al. 1992a), ‘91T-like’ (Filippenko et al. 1992b), ‘peculiar SNe Ia’ and ‘unclear SNe Ia’. The ‘91T-like’ subclass also contains ‘99aa-like’ events, as they are generally considered transitional SNe Ia between ‘91T-like’ and ‘normal SN Ia’, with very similar photometric properties (Phillips et al. 2022; Yang et al. 2022). The ‘peculiar’ subclass contains all identified subclasses that do not fall into these initial ones (e.g. SNe, Iax, ‘03fg-like’; see Burgaz et al. 2025). The last subclass of ‘unclear’ are the SNe Ia that are classified as SNe Ia, but their subclassification cannot be determined from their spectra. In this analysis, we remove these last two subclasses (‘peculiar’ and ‘unclear’), as they are not relevant for this analysis. The ‘peculiar’ subclass is removed, as it is composed of rare subtypes with too few numbers to do any meaningful analysis of their host galaxy properties. The ‘unclear’ subclass is removed, as it does not represent a physical subgroup of SNe Ia, but rather a sample of SNe Ia characterised by insufficient spectral information. This group can contain any of the SN Ia subclasses, with late-time spectra, or spectra with host contamination, where subclassification becomes difficult. These subgroups are removed after the galaxy fitting procedure and galaxy quality cuts.

The photometric parameters used in this study are those produced by the SALT2 light curve model (Guy et al. 2005, 2007; Betoule et al. 2014; Taylor et al. 2021), namely a parameter related to the width of the light curve x₁ and one related to the colour c, see Fig. 2. These are the commonly used parameters for the standardisation of SNe Ia used in modern precision cosmology. The ZTF SN Ia photometry and light-curve quality cuts are applied after the galaxy fitting procedure and galaxy quality cuts, as some figures do not require light-curve information. SNe Ia with satisfactory light-curve coverage are defined as having at least seven phase detections across two photometric bands, with at least two phase detections before and two after peak luminosity (same night detections in the same band, count as only one phase). We require the uncertainty in the SALT2 light curve width of Δx₁ < 1, the uncertainty in the colour of Δc < 0.1 and ‘fitprob’ > 10⁻⁷ (this is a SALT2 specific parameter quantifying the probability of the light-curve being that of a SN Ia). For further discussion on the applied cuts, see Rigault et al. (2025).

Fig. 2. Example SN Ia rest-frame light curve from ZTF DR2, with a fitted SALT2 model. The x1 parameter quantifies the stretch of the light curves, while c quantifies the colour. The black arrows are for illustrative purposes only, they do not describe how these parameters are calculated.

This paper, investigates the correlations between SN Ia light-curve properties and their host galaxies, with our main focus on non-peculiar SNe Ia that can be standardised for use in cosmology. For the ‘normal SN Ia’ and ‘91T-like’ SNe Ia, we apply the SALT2 parameter cuts of −3 < x₁ < 2 and −0.2 < c < 0.5. These are similar to the ‘Basic cuts’ from Rigault et al. (2025). We raised the upper bound of the colour parameter from the more standard cut of c < 0.3, to allow for an investigation into any potential effects of dust extinction. For the ‘91bg-like’ SNe Ia, we did not apply the ‘fitprob’ cut as the SALT2 model is not optimised for ‘91bg-like’ events, and we restricted the SALT2 parameter ranges to −5 < x₁ < −1 and −0.2 < c < 0.8.

This study does not concern itself with SN Ia rates or a volume limited sample, and hence we are lenient with the redshift cuts to maximise our sample size. No redshift cut is applied to the ‘normal SN Ia’, however due to the limitations of galaxy fitting at high redshift, the final sample only contains SNe Ia with z < 0.15. For our analysis in Sect. 5.1 concerning the SN Ia subgroups, we study the difference in host galaxy property distributions of the ‘91bg-like’ and ‘91T-like’ events, relative to each other and to the overall distribution of SNe Ia. As is discussed in Dimitriadis et al. (2025), and Burgaz et al. (2025), careful checking of the spectral subclasses was only performed up to a redshift of z = 0.06. It is important that we have the correct subclassifications for these rarer subtypes, hence we apply a redshift cut of z < 0.06 for these two subgroups. These redshift and light curve cuts are only applied after the galaxy fitting procedure, as the final sample of processed galaxies is studied first without SN dependent cuts.

3. Constructing an isophote map

This section details how isophote maps (contours of constant surface brightness) are created from DESI-LS images of SN Ia host galaxies. These maps are used in the galaxy fitting procedure, detailed in Sect. 4. In Sect. 3.1 we describe how galaxy pixels are isolated from neighbouring sources. In Sect. 3.2 we describe the isophote map generation procedure, while the applied quality cuts are described in Sect. 3.3.

3.1. Isolation of galaxy pixels

To isolate galaxy pixels from neighbouring sources, we developed a custom contour fitting algorithm (written in Python 3). After preprocessing of the DESI-LS image is complete (Sect. 2.2), a mean smoothing is applied to reduce noise. This consists of performing a numerical convolution to the image using a normalised 5 × 5 mean-kernel, in pixel space. The atmospheric PSF kernel size is nearly an order of magnitude larger, hence any information lost due to this smoothing is negligible (see Sect. 4.3). A two pixel strip is masked from all four edges to remove edge effects. To fit a contour for any given surface brightness value, the smoothed image is sampled for that value, within some range; for example, μ = 24 ± 0.1 mag/arcsec². Figure 3 shows the result of sampling the galaxy image before smoothing (top right panel) and after smoothing (bottom left panel).

Fig. 3. Schematic of the isophote map fitting procedure. Top left: Example g-band DESI-LS image of an elliptical host galaxy, with neighbouring field sources. Top right: The galaxy image sampled before mean-kernel convolution, at 24 ± 0.1 mag/arcsec2. Bottom left: The image sampled after mean-kernel convolution, with the target galaxy pixels highlighted in blue, external sources in red and a fitted 24 mag/arcsec2 isophote in black. The target galaxy pixels are isolated by running a connected-component labelling algorithm. Bottom right: The constructed g-band isophote map for this host galaxy, showing isophotes at every 0.2 mag/arcsec2.

This results in the individual sources within the image, forming isolated and connected regions. A connected-component labelling algorithm is applied from the Python image processing library, scikit-image (van der Walt et al. 2014). This algorithm labels each isolated region within the sampled pixels, allowing for easy differentiation between galaxy pixels with neighbouring sources, as is seen in the bottom left panel of Fig. 3. The host galaxy pixels are determined by the labelled connected region whose geometric centre is closest to the image centre (defined by Smith et al. 2025).

3.2. Isophote fitting

Having isolated the desired galaxy pixels, we fit a superellipse to the pixels, generating an isophote. A superellipse (or Lamé curve) is a generalised form of an ellipse described by

$$\begin{array}{c}\hfill {\left|\frac{x}{a}\right|}^{c_e}+{\left|\frac{y}{b}\right|}^{c_e}=1,\end{array}$$(3)

where a is the semi-major axis and b is the semi-minor axis. The deviation from an ellipse is controlled by the superellipse index, c_e (c_e = 2 for an ellipse). Increasing c_e above two increases the boxiness of the shape, while decreasing the value below two, flattens the edges towards a rhombus at c_e = 1. This approach is common in ‘bulge+bar+disc’ decomposition codes (Athanassoula et al. 1990), allowing for greater flexibility when fitting galaxy contours. The galaxy pixels are fitted for the semi-major axis (a), semi-minor axis (b), position angle (ϕ), superellipse index (c_e), and the central coordinates (x_c, y_c), totalling six free parameters (the position angle is introduced by fitting in polar coordinates).

This process is performed on a range of surface brightness values in both the g and r bands. The outermost galaxy isophote is defined at the 5σ depth for that particular image, with isophotes every 0.2 mag/arcsec², moving towards the centre. This process terminates when an isophote fit with fewer than 30 pixels is attempted; that is, at least five pixels are required for each of the six free parameters in a superellipse. The width of the pixel sampling (e.g. ±0.1 mag/arcsec² as shown in Fig. 3), is optimised for every isophote fit, by minimising the sum of the fractional uncertainty of the fitted superellipse parameters. This approach decreases the sampling width as far as possible without risking region fragmentation. This occurs when the sample width becomes too narrow and the galaxy pixels fragment into several different regions, preventing the connected-component labelling algorithm from extracting all the pixels associated with the galaxy.

During the superellipse fitting, each pixel is weighted by its associated inverse-variance σ⁻² (nanomaggie⁻²). These weights are converted to magnitude space using

$$\begin{array}{c}\hfill {\sigma }_{\mathrm{mag}}=\frac{2.5}{ln(10)}\left|\frac{\sigma }{\mathrm{Flux}}\right|·\end{array}$$(4)

The uncertainty of pixels is naturally lower for brighter surface brightness (easier to detect), and hence biases the fit towards pixels with brighter values; for example, fitting a superellipse to the range 24 ± 0.1 mag/arcsec², will result in an isophote closer to 23.9 mag/arcsec². To prevent this bias, the relation between pixel inverse-variance and surface brightness is divided out, resulting in a relative pixel uncertainty that can be used for isophote fitting.

This fitting procedure is used to generate g- and r-band isophote maps for all SN Ia host galaxies in the ZTF DR2 with DESI-LS images (2910 galaxies). An example isophote map is shown in the bottom right panel of Fig. 3.

If the algorithm reaches a surface brightness value brighter than the galaxy centre, the process terminates, as there are no more pixels to fit. However, if an external source exists within the image, and is brighter than the target galaxy (e.g. Milky Way star), then the algorithm will continue to detect pixels and start producing isophotes from the external source, rather than terminating. To ensure that only isophotes belonging to the target galaxy are included in the isophote map, we require the centre of each isophote to be within ten pixels of the image centre. This cut removes on average 12% of the generated isophotes for each isophote map. With this cut applied, the central coordinate of the galaxy is calculated as the median of all isophote centres. This remains fixed during the galaxy fitting procedure, detailed in Sect. 4.

3.3. Quality cuts on isophote map creation

Once an isophote map is generated, a number of quality cuts are applied to ensure sufficient coverage of the galaxy, before a galaxy decomposition is attempted. The breakdown of the quality cuts applied is shown in Table 1. As is seen in Fig. 1, 17% (602 SNe Ia) of our sample fall outside the DESI-LS coverage. From the host galaxies with DESI-LS image, 4% (116 SNe Ia) have no identified host galaxy. These are SNe Ia with no host galaxy within 100 kpc (see Smith et al. 2025) or due to these host galaxies covering fewer than 30 pixels on the image (galaxies with high redshift and/or low mass).

The first isophote map cut requires the isophote map to have at least ten isophotes in both the g and r bands, to ensure that the isophote map has enough coverage of the target galaxy, such that enough information (degrees of freedom) is available to fit all four of the galaxy models used in this paper (Table 2). Each isophote provides four degrees of freedom (centre not included), therefore, ten isophotes in both the g and r bands provide 80 degrees of freedom. The largest model (‘bulge+bar+disc’) has 17 free parameters, hence setting the cut at ten isophotes provides at least 80/17 ∼ 5 degrees of freedom per free parameter.

The second isophote map cut requires the difference in the number of isophotes between the g and r bands to be less than ten. This ensures that equal coverage is available in both g and r bands, otherwise the fitted galaxy colour (see Sect. 4.9) would be biased by whichever band had more coverage. For reference, the number of isophotes in either the g or r band, ranges from ten up to an observed maximum of 40 isophotes. For a galaxy to have a difference in isophote number higher than ten (same order of magnitude as number of isophotes), it likely means there is an issue with one of the images (e.g. image artefacts, large difference in atmospheric PSF/sky level, etc.).

The third isophote map cut requires the dimmest isophote in both the g and r bands to be dimmer than 23 mag/arcsec². This is applied to ensure that there are isophotes covering the outer regions of a galaxy, preventing the galaxy fitting routine from fitting only the bulge and not the disc. This cut will also remove galaxies which do not fit within the image aperture size. The value 23 mag/arcsec² is chosen as this surface brightness value is beyond the 5σ observational depth (g and r bands) in both DESI-LS DR9 and DR10.

A discussion of how these cuts affect low-mass and high-redshift galaxies is presented in Sect. 5.1. Following these quality cuts, 1995 successful isophote maps have been created with satisfactory coverage of the galaxy.

4. Fitting a galaxy model

This section details how we model the 2D surface brightness of galaxies, starting from the isophote maps generated in Sect. 3. In Sect. 4.1 we define the equations used to model the surface brightness of the various morphological components within galaxies. The fitting procedure is detailed in Sect. 4.2, which is visually summarised in Fig. 4. Section 4.3 describes how the effects of atmospheric distortion – that is, point spread function (PSF) corrections – are removed from the galaxy image. In Sect. 4.4, the initial conditions are described, which are designed to maximise the probability of converging on a correct solution. Section 4.5 details the required cosmological corrections. Section 4.6 describes how the galaxy morphology is identified. The results of the galaxy fitting and morphology identification is broken-down in Sect. 4.7. Section 4.8 describes how a SN is identified with a galaxy component, while Sect. 4.9 defines how galaxy colour is calculated.

Fig. 4. Schematic showing the main steps of the galaxy fitting procedure. Top left: DESI-LS image of a barred host galaxy, containing a ZTF SN Ia. Top right: The green superellipses are the isophotes of the constructed g-band isophote map (every second isophote has been removed for visual clarity). The blue dots mark the sampled points used during the galaxy fitting. Middle panels: Two example fitted surface brightness profiles are shown. The first panel is located at 120°, close to the semi-major axis of the bar, while the second panel is located at 240°, close to the semi-minor axis of the bar. The black points correspond to the sampled blue dots from the top right panel, with 3σ error bars. The coloured lines (orange, green and red) represent the fitted intrinsic profiles for each component in the ‘Bulge+Bar+Disc’ model. The shaded red region shows the 3σ confidence interval for the combined model, convolved with the PSF. Bottom left: The reconstructed 2D surface brightness model of the galaxy, convolved with the PSF. The three superellipses represent the intrinsic effective radius for each component in the model. The shaded regions are computed by linearly mapping the surface brightness profile to the opacity. The central brightness has an opacity of one, while zero opacity is at the 25 mag/arcsec2 isophote. Bottom right: Subtraction of model from original image in surface brightness space.

4.1. Galaxy models

The galaxy profiles used in this study are the commonly used Sérsic profiles. These are 1D functions that relate surface brightness to the radial distance from the galaxy centre, R. For a summary, see Graham & Driver (2005). The profile used for elliptical galaxies, bulges and bars is the general Sérsic profile:

$$\begin{array}{c}\hfill \mu (R)={\mu }_e+\frac{2.5b_n}{ln(10)}[{(R/R_e)}^{1/n}-1],\end{array}$$(5)

where R_e is the effective radius (radius containing half of the galaxy light), μ_e is the effective surface brightness (the surface brightness at R_e) and n is the Sérsic index, which controls the shape of the profile (higher n means more light concentrated near the centre). The geometric term b_n is computed by numerically solving, Γ(2n) = 2γ(2n, b_n), where Γ and γ are the complete and incomplete gamma functions, respectively. Using Sterling’s approximation, b_n can be expressed as an asymptotic expansion (Cappellari et al. 2000), avoiding the need for expensive numerical solving,

$$\begin{array}{c}\hfill b_n=2n-\frac{1}{3}+\frac{4}{405n}+\frac{46}{25515n^2}+\mathrm{O}(n^{-3}).\end{array}$$(6)

An exponential profile is used for galaxy discs; that is, setting the Sérsic index to one,

$$\begin{array}{c}\hfill \mu (R)={\mu }_0+\frac{2.5}{ln(10)}\left(\frac{R}{h}\right)·\end{array}$$(7)

where μ₀ is the central surface brightness and h is the scale length. These can be transformed to effective surface brightness or radius using

$$\begin{array}{cc}\hfill {\mu }_e& ={\mu }_0+\frac{2.5b_n}{ln(10)},\hfill \end{array}$$(8)

$$\begin{array}{cc}\hfill R_e& ={b_n}^{n}h.\hfill \end{array}$$(9)

This paper uses four main galaxy models, summarised in Table 2. A single Sérsic profile is used for elliptical galaxies (early-type galaxies), labelled as the ‘Elliptical’ model. For late-type galaxies, a single disc profile is used for a ‘Disc-only’ galaxy model, a Sérsic and a disc profile are used for the ‘Bulge+Disc’ model, and a disc with two Sérsic profiles are used for the ‘Bulge+Bar+Disc’ model.

4.2. Galaxy fitting procedure

Galaxy fitting is the procedure that takes the 1D Sérsic profiles (Eqs. 5 and 7) and constructs a 2D surface brightness profile of the galaxy, μ(R, θ). In this paper, the effective radius, R_e, along some angle, θ, is given by the superellipse polar equation:

$$\begin{array}{c}\hfill R_e(\theta )={({\left|\frac{cos(\theta -{\varphi }_e)}{a_e}\right|}^{c_e}+{\left|\frac{sin(\theta -{\varphi }_e)}{b_e}\right|}^{c_e})}^{(-1/c_e)},\end{array}$$(10)

where a_e and b_e are the effective radii along the semi-major and semi-minor axis, respectively, ϕ_e is the position angle of the semi-major axis, and c_e is the superellipse index. The same equation also applies to the disc scale length, h. This superellipse is taken to be common between both the g and r bands, along with the Sérsic index, while the effective surface brightness, μ_e, is allowed to vary between the bands. This assumes that the structure of the galaxy – the geometric distribution of stars – is the same when observing in either band, but the brightness coming from any particular region will be different in both bands.

The fitting procedure begins with sampling the isophote map along radial lines spaced at 30°, as is seen in the top right of Fig. 4. The g-band isophote map is sampled starting at 0°, while the r-band map is sampled starting at 15°. This results in the galaxy being sampled along twenty-four radial directions across the g and r bands. The angles are measured anti-clockwise from the positive x axis (negative right ascension), with the point of rotation being the galaxy centre. Since isophotes have varying centres, a correction is required for the sample angle, θ, so that the isophotes are sampled along straight lines. This corrected sample angle, ϕ, is found by numerically solving

$$\begin{array}{c}\hfill d_{c}sin(\theta -\beta )-R(\varphi )sin(\varphi -\theta )=0,\end{array}$$(11)

where d_c and β are the distance and relative angle between the galaxy origin and the target isophote centre, respectively, and R(ϕ) is the sampled superellipse point of the target isophote.

The fitting was performed by the ODRPACK implementation found in SciPy (Boggs & Rogers 1990; Virtanen et al. 2020). This package performs orthogonal distance regression (ODR); that is, it weighs the fit based on variance in both the x and y axes. The uncertainties in surface brightness were calculated by taking the mean variance of the pixel values used in the corresponding isophote fit, while the uncertainty in the radial direction was found by error propagation through Eq. (10). The 1D galaxy profiles were simultaneously fitted to each radial direction across both the g and r bands (middle panels of Fig. 4), which subsequently constructed a 2D superellipse describing the effective radii or scale lengths. These superellipses, coupled with the fitted effective surface brightness and Sérsic index, created a 2D surface brightness profile (bottom left, Fig. 4).

4.3. Point spread function

The effects of atmospheric seeing on image decomposition are significant and require careful consideration. The PSF spreads flux away from the galaxy centre, artificially diffuses light into the halo, and tends to make elliptical sources more circular (the latter effect requires the PSF correction to be modelled in 2D). In this paper, the PSF was modelled using a 2D circular Moffat function,

$$\begin{array}{c}\hfill \mathrm{PSF}(R)=\frac{\beta -1}{\pi {\alpha }^2}{[1+{\left(\frac{R}{\alpha }\right)}^2]}^{-\beta },\end{array}$$(12)

where the power index was chosen to match the turbulence theory of our atmosphere, β = 4.765 (Trujillo et al. 2001). The core-width, α, is related to the full width at half maximum (FWHM) by $\mathrm{FWHM}=2\alpha \sqrt{2^{1/\beta }-1}$, where the g- and r-band PSF FWHM for each sky brick is supplied by DESI-LS. The mean FWHM is 1.51 arcsec in the g band and 1.36 arcsec in the r band. The PSF was corrected for by numerically convolving Eq. (12) with the galaxy model (in flux space) during each iteration of the ODR fitting routine.

4.4. Initial conditions and bulge truncation

Continuing the approach used in this paper, the galaxy fitting procedure has been designed to require zero input from the user. Therefore, each galaxy fit requires robust initial conditions that maximise the probability of the ODR fitting routine of converging on the correct solution for any type of galaxy. In addition, the ODR fitting routine is not bounded, so initial conditions must be close to the true solution, to minimise the time it takes to search for a solution.

The single-component models (‘Elliptical’ and ‘Disc-only’) were run first. Having only a single component makes the fitting easy, as there is only one solution. The central surface brightness for both models is initialised to the mean surface brightness of the nine closest pixels to the galaxy centre (defined as the median of all fitted isophotes). The semi-major axis of the effective radius and scale length is initialised to the mean semi-major axis of all isophotes used for the fit. The same is repeated to initialise the semi-minor axis and the position angle, while the superellipse index is initialised to an ellipse (c_e = 2). The Sérsic index for the elliptical galaxy fit is initialised to n = 4 (de Vaucouleurs’ profile).

For the two component ‘bulge+disc’ fit, the disc was initialised to the results of the ‘disc-only’ fit, except for the superellipse index, which was reset to two. The bulge central brightness was initialised to the mean surface brightness of the nine closest pixels to the galaxy centre, and the bulge Sérsic index was initialised to one. The superellipse describing the effective radius of the bulge was initialised to the isophote whose corresponding surface brightness is closest to the initialised surface brightness.

For the three component fit, the disc is initialised to the results from the two component ‘bulge+disc’ fit (except for the superellispe index, which is reset to 2). The bulge effective surface brightness and Sérsic index are also initialised from the ‘bulge+disc’ fit. The ellipticity of the bulge effective radius is degenerate with the bar ellipticity, hence for the three component fit, the bulge is defined by a circle rather than a superellipse (reduces the number of free parameters from four to one). The bulge effective radius is initialised to the two-component bulge semi-minor axis. The bar’s effective radius is initialised to the bulge effective radius from the two-component fit, with the superellipse and Sérsic indices being reset to 3 and 0.5, respectively. The bar’s central surface brightness is set to be halfway between that of the bulge and the disc.

An effective approach to further preventing non-physical solutions is to truncate the bulge profile during the fitting procedure. In this paper, the bulge is truncated at the point where the slope of the bulge profile equals that of the disc. Through differentiation of Eqs. (5) and (7), the truncation radius is given by

$$\begin{array}{c}\hfill R_{\mathrm{trunc}}=R_e{\left(\frac{n{R}_e}{b_{n}h}\right)}^{(n/(1-n))}.\end{array}$$(13)

Due to the singularity at n = 1, this truncation only occurs if the bulge Sérsic index is greater than 1.1 (since the bar Sérsic index is always less than one, this truncation will not apply). The result of this truncation is that it prevents the ODR fitting routine from using the bulge profile to fit the outer regions of the galaxy, as can be seen in Fig. 5. Using the same initial conditions for both panels, the bulge truncation prevents the fit from converging on the non-physical solution.

Fig. 5. Results of a two-component ‘bulge+disc’ fit to a galaxy. The surface brightness in the g band is shown as a function of radial distance from the centre (along an arbitrary angle). The black points are the radially sampled points from the corresponding isophote map for this galaxy, with 3σ error bars. The coloured lines (green and red) represent the fitted intrinsic profiles for each component in the ‘Bulge+Disc’ model. The shaded red region shows the 3σ confidence interval for the combined model, convolved with the PSF. The top panel has no bulge truncation and converges on a non-physical solution. The bottom panel shows a fit using the same initial conditions, but with bulge truncation active during the fitting procedure. The dotted black line in the bottom panel marks the defined boundary between the bulge and the disc.

4.5. Cosmological corrections

So far in our analysis, the apparent surface brightness (μ) has been used. To convert to absolute surface brightness (μ_abs), two cosmological corrections are required,

$$\begin{array}{c}\hfill {\mu }_{\mathrm{abs}}=\mu -10{log}_{10}(1+z)-\mathrm{K}(z),\end{array}$$(14)

where z is the cosmological redshift. The first correction is known as cosmological dimming (Tolman 1930, 1934), and is due to the break-down of the inverse-square law for propagating light in an expanding Universe.

The second correction is known as a K-correction, which corrects for the cosmological redshifting of a galaxy’s spectrum due to an expanding Universe; in other words, it corrects the g and r photometric bands to the observed galaxy’s rest frame. The underlying spectrum of the galaxy impacts the size of the K-correction, so we use different spectral templates for the different galaxy types. The galaxy templates used are the ‘elliptical’ and three of the disc templates (S0, Sb, Sc) from the Superfit SN spectroscopic typing tool (Goldwasser et al. 2022), with an extra bulge template coming from Kinney et al. (1996). The bulge template is used for both the bulge and the bar (bars primarily consist of an older population of stars, similar to a galaxy bulge).

For the ‘disc-only’ galaxy model, the Sc template is used. To choose the most suitable K-correction template for the disc component of the ‘bulge+disc’ and ‘bulge+bar+disc’ models, we make use of the colour difference between the central bulge and the disc, calculated from the fitted galaxy models. The bulge and three disc templates are integrated in the g and r bands to estimate a reference colour difference between the bulge and the disc, c_b − d, which is defined as the g − r colour of the bulge minus the g − r colour of the disc for each galaxy disc type (S0, Sb, Sc). The fitted colour difference, derived from the galaxy models, is then matched to the closest template colour difference for use in K-corrections. Based on our templates, we define the following ranges for each disc type; S0: 0 < c_b − d < 0.17 mag, Sb: 0.17 ≤ c_b − d < 0.5 mag and Sc: 0.5 ≤ c_b − d < 2 mag. It is important to note that these colour ranges are not used to classify the morphology of a galaxy, but are only used to find the closest matching template to be used for K-corrections.

4.6. Galaxy morphology identification

Each galaxy is fitted with all four models listed in Table 2. A series of cuts are then applied to determine which galaxy model fits have produced a physically plausible result (see Table 3). The first four cuts are applied to all the models.

The first general cut is on the residuals of the model and requires that the mean residual is < 0.02 mag/arcsec². This cut was determined by manually inspecting several model fits, to find the approximate cut-off that is strict enough to only include visually very well fit galaxies, but not too strict as to remove galaxies with spiral arms. The second is a cut on the model disc ellipticity (1 − b/a) to remove fits with ellipticities of ≥0.6. This is because the surface brightness profiles used in this paper describe the face-on profile, and would be inappropriate to use for edge-on systems. This cut-off value of the ellipticity is estimated from Pastrav et al. (2013), who showed that the effects of dust on the derived disc axis-ratio (i.e. inclination) becomes prominent at an ellipticity of > 0.65. The bulge and disc would also overlap for more edge-on systems, making it difficult to tell where a SN originates from. The ‘Elliptical’ models are also cut at the same ellipticity, to prevent them from fitting edge-on disc systems.

The third cut on all models is based on the value of the superellipse index. The model superellipse cuts are loose bounds, to prevent the inclusion of non-physical shapes, the majority of galaxy fits have values that do not come close to these boundaries.

The surface brightness colour, μ_g − r, was calculated by subtracting surface brightness in the r band from the g band. While each model has its own colour bounds, the overall lower and upper colour boundaries of 0 and 2 mag are chosen to exclude non-physical models, which primarily arise when a model component fails to fit to one of the bands. As will be seen in Sect. 5.3, the colour values range from a minimum of 0.2 mag to a maximum of 0.9 mag.

The single-component models are given additional colour cuts to prevent the inclusion of non-physical models. The ‘Elliptical’ model has a lower colour cut of μ_g − r > 0.5 mag, while the ‘Disc-only’ has an upper colour cut of μ_g − r < 0.6 mag. These values were determined by manually checking the tail-end of the respective colour distributions. Beyond these two colour cut-offs, well-fit ‘Elliptical’ and ‘Disc-only’ models were found to be non-physical. The ‘Elliptical’ colour cut effectively removes the rare case of star-forming elliptical galaxies, which we choose to remove to keep our final sample cleaner. The ‘Disc-only’ galaxies naturally have blue colours, as redder discs are much more likely to have a significant bulge and hence be fit by one of the multi-component models. These cuts removed seven blue ‘Elliptical’ galaxies and five red ‘Disc-only’ galaxies. The overall results and conclusions of this paper are unaffected by the inclusion or exclusion of these galaxies.

In the overlap region between these two models, 0.5 < μ_g − r < 0.6 mag, a degeneracy is present as the ‘Elliptical’ model is able to mimic a disc by setting its Sérsic index close to unity (Cameron et al. 2009). This is not a concern for identifying elliptical galaxies as the ‘Disc-only’ model cannot fit the profile shapes of elliptical galaxies, however ‘Disc-only’ galaxies will be well fit by both the ‘Disc-only’ and ‘Elliptical’ models in this colour region. To resolve these cases, we simply choose the ‘Disc-only’ model. This relies on the assumption that elliptical galaxies generally do not have exponential surface brightness profiles, which is well supported by the literature (e.g. Lange et al. 2015).

For the multi-component models, the central bulge has the same colour cuts as the elliptical galaxies (similar stellar populations), the bulge must be redder than the disc, and the bulge must have a greater r-band central brightness than the disc. The r-band bulge-disc luminosity ratio is limited to < 0.5, to remove any potential ambiguity between the ‘Elliptical’ and ‘Bulge+Disc’ models; that is, it prevents the ‘Bulge+Disc’ model from setting the disc brightness very low and using only its bulge component to mistakenly fit an elliptical galaxy. This phenomenon is known as a false disc, which is a common pitfall for automated galaxy image decomposition (Allen et al. 2006; Cameron et al. 2009). The cut-off value was chosen as the largest bulge-disc ratio found in Table 7 of Graham & Worley (2008).

The bar in the ‘Bulge+Bar+Disc’ model requires extra cuts to ensure that it is real and bar shaped. The bar ellipticity is set to be greater than 0.3, so that it is not used to fit ring- or disc-like objects. The bar has the same colour cuts as the bulge, and the semi-major axis of the bar’s effective radius must extend beyond the bulge effective radius. The upper bound of the bar superellispe index is set to 10, to enable the bar to take on a more box-like shape, a common feature of bars (Athanassoula et al. 1990, the value of 10 is a conservative upper bound). The bar’s Sérsic index must be below unity to generate the correct surface brightness profile. However, the cut is lowered to 0.9, to prevent the model from erroneously switching the bar and disc components.

To further aid in the classification of galaxies, recent results from the Galaxy Zoo project (Lintott et al. 2008) were used. Walmsley et al. (2023) have performed 8.67 million morphology measurements of nearby galaxies using the DESI-LS. These were achieved using deep learning models trained on Galaxy Zoo volunteer data. These results overlap with 41% of the galaxies in the ZTF DR2 sample, providing useful information for distinguishing between the different galaxy models. Four parameters from this study are used: probability of a disc, probability of a bulge, probability of a bar, and probability of the galaxy merging. If the probability of the galaxy merging is > 0.5, the galaxy is discarded.

If following the cuts from Table 3 only one model remained, it was used as long as it was consistent with the Galaxy Zoo measurement (if any exist). For example, if our ‘Elliptical’ model was the only model to pass all the cuts and the disc probability from Galaxy Zoo was less than 0.5, then we selected this model. If multiple models passed our parameter cuts, the Galaxy Zoo measurements were used to distinguish between them; for example, if the ‘Bulge+Disc’ and ‘Bulge+Bar+Disc’ models both passed the cuts, then the bar probability was used to choose the appropriate model (the bar was chosen if bar probability was greater than 0.5). In the case in which no Galaxy Zoo measurement was available, and both ‘Bulge+Disc’ and ‘Bulge+Bar+Disc’ models passed, the ‘Bulge+Bar+Disc’ model was chosen. The most likely scenario for both of these models to pass is when a bar does exist, and the ‘Bulge+Disc’ model has absorbed the bar’s profile into that of the bulge. If the ‘Elliptical’ model and any of the disc models both passed the cuts, and no Galaxy Zoo measurement was available, then the galaxy was marked as an ambiguous ‘E-S0’.

4.7. Galaxy model fitting results

The results of the galaxy fitting procedure are summarised in Table 4. From the initial 1995 SN Ia host galaxies, 728 (37%) galaxies have been successfully fitted, and have an identified galaxy type. The Galaxy Zoo measurements, described in Sect. 4.6, flagged 111 galaxies as merging, and hence these were removed before the galaxy fitting began. From the initial sample of 1884 host galaxies, 25% (466) failed the residual cut, whereby all four galaxy models were unable to fit the isophote map with satisfactory residuals. A further 19% were removed, as they have ellipticities greater than 0.6; that is, they are edge-on systems that cannot be fit using the models employed in this paper, as is discussed in Sect. 4.6.

Of the remaining 1145 galaxies, a further 353 were removed due to the fitted model parameters being non-physical; that is, the fitted models did not pass all the cuts detailed in Table 3. These cuts are applied simultaneously to the fitted models, and generally when a galaxy model fails these cuts, it usually fails a combination of them. Therefore, it is not possible to give a breakdown of how many galaxies fail each of the cuts in Table 3. From manually checking the failed models, the main reasons for failure include: highly asymmetric galaxies, merging galaxies (that were not flagged by Galaxy Zoo), very prominent spiral arms, globular clusters, or dust lanes, the fitting routine converging on non-physical solutions, a high atmospheric PSF FWHM relative to the galaxy size, and intervening Milky Way stars.

Following the procedure detailed in Sect. 4.6, the galaxy type (Table 2) was identified for 728 galaxies, while 64 galaxies were removed due to multiple models passing the cuts, and having no Galaxy Zoo measurement to resolve them. Figure 6 shows four example fitted galaxy models, one for each model type.

Fig. 6. Example images of the four galaxy models used in this paper. The left panels are the DESI-LS images in surface brightness space with the location of the SN marked as the green cross, the centre panels are the fitted galaxy models, where the superellipses are the effective radius for each galaxy component. The shaded regions are computed by linearly mapping the surface brightness profile to the opacity. The central brightness has an opacity of one, while zero opacity is at the 25 mag/arcsec2 isophote. The right panels are the residuals, calculated by subtracting the model from the image. The four galaxy models (from top to bottom) are ‘Elliptical’, ‘Bulge+Disc’, ‘Bulge+Bar+Disc’ and ‘Disc-only’.

From the final sample of 728 galaxies, 448 galaxies have a Galaxy Zoo measurement that can be used to benchmark the custom morphology identification used in this paper. The agreement for elliptical galaxies is 92%, while the agreement for galaxies containing discs is 90%. The reasons for disagreement between our morphologies and those from Galaxy Zoo are predominantly due to the difficulty in distinguishing between elliptical and lenticular galaxies.

4.8. Association of a supernova with a galaxy model component

In the multi-component galaxy models, to determine which model component each SN Ia is associated with, the intrinsic flux ratio at the SN position is used. If, at the location of the SN, the flux coming from the bulge or bar is greater than 20% of the disc flux, then it is classified as a ‘bulge/bar SN’. The SN is classified as a ‘disc SN’ if the bulge and bar components are < 20% of the disc flux, and the surface brightness at its location is brighter than 25 mag/arcsec² in the r band (conventional isophote for the galaxy edge). If the SN is outside this isophote, it is marked as a ‘halo SN’. This boundary between the bulge or bar and the disc is shown in the bottom panel of Fig. 5. This value was chosen as it is the approximate flux ratio where the combined surface brightness profile begins to diverge from the pure exponential disc profile.

The galaxy models used in this paper are fitted to projected images, giving rise to the possibility that the SN lies in front of the associated component and should be marked as a ‘halo SN’. However, as will be seen in our results, the number of halo SNs is small, and hence this potential misclassification is very unlikely to affect our results. The possibility of misclassification between bulge and disc due to projection effects is also unlikely to be significant. Our final sample does not include any highly inclined systems (Sect. 4.6) that would make this distinction difficult.

4.9. SN-component-colour

The ‘SN-component-colour’ (SCC_g − r) is defined as the intrinsic g − r colour (i.e. Milky Way dust, cosmological and PSF effects removed) of the galaxy component that contains the SN (e.g. bulge, bar, disc, or elliptical). This was found by subtracting the r-band from the g-band surface brightness of the model, at the location of the SN. The area dependence cancels out; hence, the units are given in magnitudes. Since the galaxy shape (effective radius and Sérsic index) is common between both bands during the fitting procedure, each component of the galaxy only has a single colour. Therefore, the SCC_g − r captures the average colour of the galaxy component that contains the SN, rather than the colour at the actual SN location, commonly referred to as the local colour.

This colour definition has potential benefits over the local colour for correlating SN Ia properties with their host galaxies. The conditions of the explosion site of a SN Ia cannot fully explain its intrinsic properties, as the progenitor system may have orbited several times around its host galaxy before it exploded. Therefore, if a SN is observed to explode within a spiral arm for example (which has a much bluer colour than the disc average), the very blue local galaxy colour of the SN Ia may not better explain its intrinsic properties, as they should be driven more by the environment where the progenitor was born and evolved in. The SCC_g − r used in this paper, is an average colour of the entire area that the progenitor system was born and evolved in, and hence could correlate stronger with SNe Ia properties. This approach also removes the issue of having SN light within the galaxy image, as it does not rely on the pixels at the location of the SN. In the case of the ‘Elliptical’ and ‘Disc-only’ models, this colour parameter refers to the global intrinsic colour of the galaxy, as there is only one component in these models. However, for the multi-component galaxies, this colour definition is preferred to the global colour as, for example, it is able to determine the colour of a disc without being biased by the redder flux coming from a central bulge.

5. Results

In Sect. 5.1, the results in terms of selection biases against high-redshift and low-mass galaxies are provided. Section 5.2 provides a breakdown of the final sample of host galaxies used to correlate with SN Ia properties. Section 5.3 presents the correlations between the SALT2 light curves parameters (defined in Sect. 2.3) and the host galaxy properties of SN component-colour (defined in Sect. 4.9) and host stellar mass. In Sect. 5.4, the SALT2 parameters are presented in terms of local intrinsic surface brightness, derived from the galaxy models.

5.1. Redshift and host mass biases

As is seen in Table 1, no successful isophote map was generated for 915 host galaxies; that is, the isophotes for these galaxies did not pass the quality cuts detailed in Sect. 3.3. The majority of these galaxies (693) failed due to the second quality cut that requires at least ten isophotes in both g and r bands. This is a natural consequence of having a fixed image resolution and observational depth. The redshift and host stellar mass distributions of SN Ia host galaxies in the full DR2, those for which a successful isophote map is produced, and those in the final sample for which a galaxy morphology is identified, are shown in the top panels of Fig. 7. The biases in generating isophote maps in terms of redshift and host stellar masses are quantified in the middle row of Fig. 7, which show the rate of generating a successful isophote map given a galaxy image, as a function of redshift and host stellar mass. For redshift, the rate is lower for nearby objects due to the galaxy being too large to fit on a DESI-LS image. The galaxy images are limited to 600 pixels (computational cost is too high for larger images), hence large galaxies at low redshift do not fully fit on the image, and fail the quality cuts. This has the benefit of only including galaxies that are within the Hubble flow, which ensures that the redshift gives an accurate measure of the galaxy distance. The success rate peaks at a redshift of around z = 0.06 and begins to drop at a redshift of z = 0.1. No galaxy with a redshift greater than z = 0.18 possessed a usable isophote map; however, these are also very rare events in the DR2 sample. For a fixed pixel scale, the number of pixels per galaxy drops with increasing redshift, and hence small galaxies begin to fail to produce isophote maps at high redshifts. For the host mass, a similar trend is apparent. The success rate is around 0.9 for higher mass galaxies but begins to drop below a host mass of ∼10⁹ M_⊙. The lower the host mass, the dimmer it will be, as well as the smaller it will be in angular size, and hence the pixel scale and observing depth limitation will create a bias against low-mass galaxies from being included in the final sample. This bias is unavoidable due to the limitations of galaxy image decomposition methods such as this one.

Fig. 7. Fit success rate as a function of redshift and host mass. Top panels: Blue histogram shows the full ZTF Cosmo DR2 binned in redshift (left) and host mass (right). Orange histogram shows galaxies with isophote maps that pass all quality cuts. Green histogram shows final sample (successful galaxy fit + galaxy morphology identified). Centre panels: Success rate for generating a successful isophote map, given a galaxy image, as a function of redshift (left) and host mass (right). Bottom panels: Success rate for generating a galaxy model (successful galaxy fit + galaxy morphology identified), given an isophote map, as a function of redshift (left) and host mass (right).

The success rate of generating a galaxy model fit that passes the required cuts, given an isophote map, is shown on the bottom panels of Fig. 7. The success rate of generating a galaxy fit as a function of redshift is reasonably constant at ∼40%. There is larger scatter in the highest redshift bins due to small number statistics. The average success rate for generating a galaxy fit as a function of host stellar mass is ∼70% for the most massive galaxies (> 10¹¹ M_⊙) and ∼40% for galaxies with masses of 10^7.5–10¹¹ M_⊙. More massive galaxies have more pixels covering them and are brighter, increasing the successful galaxy model fit rate. Elliptical galaxies are the easiest type of galaxy to fit as they only have a single component and are in general very massive or luminous, contributing to the increased success rate for massive galaxies. No galaxy with a mass of less than 10^7.5 M_⊙ is in the final sample. Such galaxies are likely too small and young to have formed pronounced discs; hence, none of the galaxy models used in this paper can properly describe these galaxies (e.g. faint blue galaxies, dwarf irregular galaxies, etc.). This is a notable limitation of our study in that our sample is significantly biased, with an over-representation, compared to the intrinsic rate, of SNe Ia occurring in the most massive galaxies. However, we can still investigate trends and correlations that do not depend on the absolute or relative rates of host galaxies.

5.2. Final sample statistics

A breakdown of the final sample of host galaxies is presented in Table 5. Out of our final sample, 51% are classified as elliptical galaxies. To quantify how biased this value is, we compared this value with the expected rate of SNe Ia in elliptical versus disc galaxies. We used the results from Li et al. (2011) who have calculated the rates of SNe Ia as a function of Hubble type, per unit solar mass. We also made use of Kelvin et al. (2014), who estimated the distribution of mass across each of the galaxy Hubble types. By weighting the SNe Ia rates by this mass distribution, we find that the expected rate of SNe Ia in elliptical galaxies should be approximately ∼37%. Our ratio of elliptical galaxies to disc galaxies of 51% is higher than the expected rate, indicating a bias towards ellipticals in our final sample. This can be understood by the fact that elliptical galaxies are smooth and featureless, and hence are much easier to fit. Furthermore, as is discussed in Sect. 4.6, a fraction of the disc galaxies that are edge-on were excluded from the final sample, decreasing the total number of disc galaxies.

Within the disc sample, 62% have no detected bulge or bar. This number is higher than the true rate due to the physically small nature of bulge/bar systems relative to the disc. Very small bulges and/or bulges at high redshift can become too small and have an insufficient number of pixels to fit a bulge model, resulting in the galaxy being classified as disc only. A very large PSF FWHM can also have this effect, as it can spread a significant amount of bulge flux away from the centre.

In the top panel of Fig. 8, we show the SALT2 x₁ distribution of the full ZTF DR2 and our final SN Ia sample with a successful galaxy morphological classification (our light curve quality cuts are applied to both samples). In the bottom panel of Fig. 8, we show the rate of a SN Ia with any particular x₁ value, entering our final galaxy sample. There is a clear trend, with an excess of SNe Ia with negative x₁ values (with respect to the average ∼25%), while there is a deficit of SNe Ia with positive x₁ values in the final sample. As will be seen in Sect. 5.3, low-stretch (negative x₁) SNe Ia are predominately found in passive environments – elliptical galaxies – while high-stretch (positive x₁) SNe Ia are predominately found in star-forming regions; that is, disc galaxies. Therefore, Fig. 8 reiterates the result that there is an excess of elliptical galaxies relative to disc galaxies in our final sample. However, there is still broad coverage of the x₁ distribution to investigate the correlations between host environment and x₁. No apparent bias is seen in the SALT2 c distribution.

Fig. 8. SALT2 x1 completeness of the final galaxy sample. Top panel: Blue histogram shows the ZTF DR2 binned in SALT2 x1, with the light curve quality cuts applied, see Sect. 2.3. The orange histogram shows the final sample containing SNe Ia with successfully fitted galaxy models. Bottom panel: Rate of finding a SN Ia with any particular x1 value in the final sample. The dashed grey line shows the average rate.

Table 6 shows the number of SNe Ia in each morphological component (elliptical, disc, bulge or bar, and halo), in our final sample, split based on their spectral properties, and after additionally applying the light-curve quality cuts described in Sect. 2.3. The ‘halo’ column is split into SNe Ia found in the haloes of elliptical and disc-containing galaxies; that is, they have local r-band surface brightness μ_r > 25 mag/arcsec². Host galaxies with large outlier SN-component-colour values were manually inspected. This resulted in the removal of nine host galaxies (1.2%), reducing the final sample size from 728 to 719. The host galaxies removed are those associated with the SNe Ia; ZTF18abugthp, ZTF19abhzewi, ZTF20abffaxl, ZTF18abuykiu, ZTF19aatgznl, ZTF19abquwvx, ZTF19abzkiuv, ZTF20abxidyb and ZTF20abdxuew. By inspection, these galaxies have all converged on a non-physical ‘Bulge+Disc’ or ‘Bulge+Bar+Disc’ decomposition solution.

5.3. The link between SN-component-colour and host stellar mass

In this section, the relation between SN-component-colour (defined in Sect. 4.9) and the host stellar mass is investigated. These galaxy properties are plotted in Fig. 9 for the final sample with only the subclassification cuts applied. The top panel is divided into the three main SN Ia subtypes based on their spectral classifications (normal SNe Ia, ‘91bg-like’ and ‘91T-like’). The middle panel shows the density functions of host mass for the final sample of SNe Ia with successful host galaxy modelling, along with the SNe Ia from the ZTF DR2 that were cut from the final sample. As was seen in Fig. 7, we are missing more SNe Ia at lower stellar mass than at higher stellar mass due to biases in our fitting method. The bottom plot is divided into three regions, demonstrating the classical picture of galaxy evolution. This was achieved by using an agglomerative clustering algorithm from scikit-learn (Pedregosa et al. 2011). The number of clusters was set to three, with the remaining hyperparameters at their default values. The low-mass and blue-coloured galaxies (μ_g − r ≲ 0.55 mag) make up the Blue Cloud (Schawinski et al. 2009), representing the young galaxies with active star formation. This population is dominated by disc-containing galaxies. The high-mass and red (μ_g − r ≳ 0.65 mag) elliptical galaxies make up the old and non-star-forming red sequence (Eales et al. 2018). In between these populations lies the Green Valley (Smethurst et al. 2015), populated by both elliptical and disc galaxies, representing transitional galaxies with recent star formation.

Fig. 9. SN-component-colour (g − r) versus host stellar mass. Top panel: SNe Ia separated into the main spectral subtypes, normal SNe Ia (blue), ‘91bg-like’ SNe Ia (red), and ‘91T-like’ SNe Ia (orange). Centre panel: Kernel density plots showing host mass density function of the final sample (dashed line) and the ZTF DR2 SNe Ia that were cut from the final sample (solid line). Bottom panel: Normal and ‘91T-like’ SNe Ia grouped into three main clusters: red sequence, green valley, and blue cloud, using a clustering algorithm. The elliptical ‘91bg-like’ (purple diamonds) and elliptical ‘91T-like’ (orange triangles) subpopulations are highlighted. The dashed black and solid grey lines are linear regressions for the normal and ‘91bg-like’ subpopulations in the red sequence, respectively.

A dashed black line in the bottom panel of Fig. 9 shows a linear regression to the normal SN population in the red sequence. It can be seen that ‘91bg-like’ events reside above this line. For fixed mass, ‘91bg-like’ events reside in elliptical galaxies that are 0.035 ± 0.01 mag redder (3.5σ). The slopes of the two (uncertainty-weighted) linear regressions seen in the bottom panel of Fig. 9 for normal (5.7σ) and ‘91bg-like’ (2.9σ) SNe Ia in the red sequence are consistent. The ‘91T-like’ SNe Ia which reside in the disc, are predominantly found in the Blue Cloud, while ‘91T-like’ events in elliptical galaxies are predominantly contained within the Green Valley. From Table 6 it can be seen that within ellipticals galaxies (z < 0.06), 87% are normal SNe Ia, 2% are ‘91T-like’ and 11% are ‘91bg-like’. In disc-containing galaxies, 92.5% are normal SNe Ia, 6.1% are ‘91T-like’ and 1.4% are ‘91bg-like’. The ‘91bg-like’ events in disc-containing galaxies reside entirely in either the central bulge/bar or in the halo.

In Fig. 10, the relationship between SN Ia light-curve stretch (SALT2 x₁) and host galaxy SN-component-colour and stellar mass is investigated. The left panels show these colour coded by the SN Ia spectral classification (normal, 91bg-like, 91T-like) and the right panels, show the normal SNe Ia only, separated by the associated galaxy component. This plot, as well as subsequent plots, have the subclassification and light-curve cuts applied. We identify a high significance linear relation (16.8σ) between SN-component-colour and SALT2 x₁ for the normal SNe Ia occurring in the elliptical galaxy population (top right panel of Fig. 10). SNe Ia with narrower light curves (more negative x₁) occur in redder elliptical galaxies than those with broader light curves (more positive x₁). The Pearson correlation coefficient for this relation is −0.61, giving a R² value of 0.37, implying that up to 37% of the variation seen in the x₁ values of SNe Ia located in elliptical galaxies, is explained by the intrinsic colour of their host environment. The bulge/bar subpopulation has too few SNe Ia for meaningful statistics, but they can be seen to follow a similar trend as the elliptical galaxies. This is not surprising, since they are also expected to be passive environments with older stars than those seen in galaxy discs. Using a Kolmogorov-Smirnov test (two-sided), the probability that the bulge/bar SN Ia x₁ distribution is different from the distribution seen in elliptical galaxies is only 1.2σ, while the probability that they are different from the disc SNe Ia is 4.6σ.

Fig. 10. Galaxy colour and host mass versus SALT2 x1. The top panels show SN-component-colour (Sect. 4.9) versus x1, while the bottom panels show host stellar mass versus x1. The left panels show the distribution of normal SNe Ia (blue circles), ‘91bg-like’ SNe Ia (red diamonds), and ‘91T-like’ SNe Ia (orange triangles). The right panels contain only the normal SNe Ia, with the associated galaxy component highlighted (Sect. 4.8). Weighted linear regressions are shown for statistically significant relationships of SN-component-colour/stellar mass and x1 in elliptical galaxies (dashed red lines) and SN-component-colour/stellar mass and x1 in disc galaxies (dashed blue line).

A relation between SN-component-colour and SALT2 x₁ for the normal SNe Ia occurring in galaxy discs is also identified. This correlation is much steeper than the one seen in elliptical galaxies, as well as much weaker, however it is still significant (5.1σ). With a Pearson correlation coefficient of −0.39, the disc colour explains approximately 15% of the variation seen in x₁ values of SNe Ia occurring in galaxy discs. The 91T-like SNe Ia that occur in disc galaxies appear to follow the same SN-component-colour and x₁ relation as the normal SN Ia population. In the bottom right panel of Fig. 10, a trend is also seen between host stellar mass and x₁ in the elliptical population. However, this trend is much less significant (3.3σ, R² = 0.13) than that between SN-component-colour and x₁ (16.8σ, R² = 0.37).

In Fig. 11, it is apparent that SN Ia light-curve colour (SALT2 c) does not correlate as strongly with host galaxy properties as x₁ does. As has been seen before, the SALT2 colours of 91bg-like SNe are redder, on average, than normal and 91T-like events, and as was seen in Fig. 9, they occur among the reddest elliptical and bulge/bar environments. In the right panels, the only apparent trend (2.96σ, R² = 0.07) is between SN-component-colour and SALT2 c, for normal SNe Ia in discs (with the caveat that outlier points with galaxy colours of > 0.65 mag are excluded. Including these points drops the significance to 2.5σ). Using a Kolmogorov-Smirnov test (two-sided), the probability that the SN Ia colour distribution seen in the elliptical galaxies is different from those in the disc, is only 1.9σ (with elliptical SNe Ia having slightly redder colours). No significant trends between host stellar mass and colour are seen in the bottom row of Fig. 11.

Fig. 11. Galaxy colour and host mass versus SALT2 c. The top panels show SN-component-colour (Sect. 4.9) versus c, while the bottom panels show host stellar mass versus c. The left panels show the distribution of normal SNe Ia (blue circles), ‘91bg-like’ SNe Ia (red diamonds), and ‘91T-like’ SNe Ia (orange triangles). The right panels contain only the normal SNe Ia, with the associated galaxy component highlighted (Sect. 4.8). A Weighted linear regression (dashed blue line) is shown for SN-component-colour and c in disc galaxies.

5.4. Light curve properties and local surface brightness

In Fig. 12, the SALT2 light-curve parameters of x₁ (left panels) and c (right panels) are plotted against the local r-band surface brightness (intrinsic surface brightness of the galaxy model at the position of the SN). The top panels show SNe Ia occurring in ‘disc-only’, ‘bulge+disc’, ‘bulge+bar+disc’ galaxies and the bottom panels show the SNe Ia that occur in elliptical galaxies. Their membership of a disc, bulge/bar, elliptical, or halo is indicated by the symbols and colours. In the top panels, as expected, the SNe Ia in bulges or bars have, on average, the brightest local surface brightness, followed by SNe Ia in discs, and the SNe Ia in the halo are the regions of lowest surface brightness. In the top left panel, a significant linear correlation (6.1σ, R² = 0.09) is seen between x₁ and local r-band surface brightness, for the normal SN Ia population originating from galaxies containing discs (disc, bulge or bar). The normal SNe Ia residing in the halo do not appear to follow the same trend and have a mean stretch of x₁ = 0.23 ± 0.16. This stretch value is more consistent with the mean stretch of disc SNe Ia x₁ = 0.02 ± 0.07, than with SNe Ia found in the bulge/bar x₁ = −0.81 ± 0.17.

Fig. 12. Local r-band surface brightness (derived from model) versus SALT2 parameters (x1 on the left and c on the right), highlighted by SN classification and the associated galaxy component. The top panels contain SNe Ia located in galaxies containing a disc, while the bottom panels contain SNe Ia located in elliptical galaxies. Dashed black lines indicated significant linear correlations. A horizontal red line is plotted at c = 0.035. See Sect. 6.4 for discussion.

In the bottom left panel of Fig. 12, no correlation is seen between SALT2 x₁ and local r-band surface brightness for the elliptical galaxy population of SNe Ia. However, the interpretation of surface brightness for elliptical galaxies is more difficult than for disc galaxies due to the difficulty in determining where SNe Ia occur within the 3D elliptical galaxies from a projected image. SNe Ia located near the centre of elliptical galaxies (and hence have a bright surface brightness), may in fact lie in front of the galaxy, making the true surface brightness much dimmer. This projected surface brightness problem does not affect disc SNe Ia to the same extent, as the face-on discs used in this study can be considered as 2D objects, and hence the measured surface brightness relates to the true stellar density at the explosion site. The 91bg-like population shown in the bottom left panel of Fig. 12 does not appear to reside in brighter or fainter surface brightness components than the normal SNe Ia populations.

In the top right panel of Fig. 12, local r-band surface brightness is plotted against SALT2 c, for the SN Ia population located in galaxies containing discs, highlighted by the SN classification and the associated galaxy component (disc, bulge/bar or halo). A horizontal red line has been plotted at SALT2 c = 0.035 (the choice of this value is arbitrary). Below this line, normal SNe Ia are present across all surface brightness values, while above this line, the normal SNe Ia are localised between 19 < μ_r < 24.5 mag/arcsec², and have a significant linear trend (3.3σ), whereby redder SNe Ia reside at brighter surface brightness. The normal SNe Ia residing in the halo, have a mean SN colour of c = −0.052 ± 0.015, with all events having SALT2 c < 0.035. In the brightest regions around the galaxy centres (where bulge/bars are located), the apparent cluster of normal SNe Ia (−0.05 < c < 0.035 and μ_r < 20.8 mag/arcsec²), group tightly at c = −0.011 ± 0.004.

In the bottom right panel of Fig. 12, no significant correlation between SALT2 c and local r-band surface brightness is seen for normal SNe Ia in elliptical galaxies. However, SNe Ia with c > 0.035 do populate the bright surface brightness values, which are missing from the top right panel. The ‘91bg-like’ SNe Ia (red diamonds) appear to be absent from the brightest regions (μ_r ≲ 19 mag/arcsec²) of elliptical galaxies. This is likely an observational bias, as ‘91bg-like’ SNe Ia are generally less luminous, and therefore are more difficult to detect near the bright centres of elliptical galaxies.

6. Discussion

6.1. Galaxy morphology and supernova properties

Many studies have shown empirical correlations between SN Ia stretch and host galaxy properties, whereby higher-stretch SNe Ia (and hence more luminous SNe Ia) are preferentially found in less massive star-forming environments, while low-stretch SNe Ia are predominately found in more massive passive galaxies (Hamuy et al. 1996; Sullivan et al. 2003; Graur et al. 2017; Kim et al. 2019; Pruzhinskaya et al. 2020; Nicolas et al. 2021; Garnavich et al. 2023). We identify a similar trend between x₁ and host stellar mass for the much larger sample of ZTF DR2 SNe Ia studied here (bottom panels of Fig. 10), which is also seen in the ZTF companion paper, Ginolin et al. (2024).

We have also investigated in a large sample for the first time, the relation between the distribution of SNe Ia in different galaxy components (elliptical, bulge/bar, disc) and SALT2 parameters. It can be clearly seen in the top panels of Fig. 10 that SNe Ia in elliptical galaxies have a different x₁ distribution than those in disc galaxies. SNe Ia located in the disc have a mean x₁ of 0.12 ± 0.06, while the ellipticals have lower x₁ SNe Ia, with a mean x₁ of −1.04 ± 0.8. The SNe Ia in elliptical galaxies overlap with the x₁ values of discs in the region of x₁ > −1, however, the elliptical SNe Ia account for ∼75% of the SNe Ia with x₁ < −1. The few SNe Ia in discs (17%) with low stretch values (x₁ < −1) may originate above or below the disc (e.g. in globular clusters), where the stellar population is much older (Barkhudaryan 2023).

The right panels of Fig. 10 show that the SNe Ia identified to reside in a bulge/bar have x₁ and SN-component-colour values that match the elliptical population. Using a two-sided Kolmogorov-Smirnov (KS) test, there is only a 1.2σ level probability of the bulge/bar x₁ distribution being different from that of the elliptical x₁ distribution. This supports the idea that the bulge/bar of disc-containing galaxies have an older population of stars, similar to those found in elliptical galaxies.

In a ZTF companion paper, Ginolin et al. (2024) investigate the standardisation procedure used in cosmology, focusing on the environmental dependencies of SN Ia stretch and luminosity of the ZTF DR2 sample. They find a significant (14.3σ) non-linearity between standardised Hubble residuals and x₁, quantified by a ‘broken-α’ model (where α refers to the fitted variable used to standardise x₁). They report that the best-fit value to separate the low- and high-stretch modes is at x₁ of −0.49 ± 0.06. In another ZTF companion paper, Deckers et al. (2025) investigate the secondary maximum seen in the light-curves of SNe Ia in the ZTF DR2 sample. They report a non-linearity between the r-band integrated flux under the secondary maximum and x₁, with their best fit ‘broken-line’ splitting the low and high-stretch modes at x₁⁰ = −0.6 ± 0.2, consistent within the uncertainties with that of Ginolin et al. (2024). Different versions of this ‘broken-α’ model have been previously seen (Burns et al. 2018; Garnavich et al. 2023).

It is difficult to estimate the point between potential high- and low-stretch dominated regimes for our sample. However, we can attempt this using the intersection point of the two linear regressions between SN component colour and x₁ seen in the top right panel of Fig. 10. This intersection point is at a value of x₁ = −0.39 ± 0.06. This x₁ value is less negative but consistent with both Ginolin et al. (2024) and Deckers et al. (2025). However, we reiterate that it is difficult to determine whether this intersection is an accurate measure.

6.2. Type Ia supernovae in elliptical galaxies

This is the first study that demonstrates that significant linear correlations exist in both elliptical and disc SNe Ia between SN-component-colour/host stellar mass and the SALT2 parameter x₁. We have identified a 16.8σ correlation between SN-component-colour and x₁ for SNe Ia occurring in elliptical galaxies. The correlation has a R² of 0.37, which points to a fundamental link between SNe Ia x₁ and the colour of the galaxy component (bulge, bar, or disc) that the SN resides in. A lower significance correlation (3.3σ, R² = 0.13) between host stellar mass and x₁ is found for SNe Ia in elliptical galaxies, suggesting the SN-component colour is the preferred environmental tracer. A similar result was seen by Kelsey et al. (2021), who found a larger galaxy colour step compared to the host mass step in the Hubble Residuals.

These trends that are seen when SNe Ia in elliptical galaxies are separated from SNe Ia in discs, as well as the fact that there appears to be a distinct population of SNe Ia in the reddest elliptical galaxies that have significantly smaller x₁ values than SNe Ia in the discs of galaxies, requires a physical explanation. To attempt to reach such an explanation, the driving factors behind these correlated parameters must be first identified. The driving force in the colour of elliptical galaxies is commonly believed to be a function of both stellar age and metallicity. While the relative strength of these driving factors depend on the colour being investigated, Chang et al. (2006) argue that metallicity is the primary driver of galaxy colours (relative to luminosity-weighted age), but age has the most significant effect on optical colours (such as the g and r bands).

There is a well known relation between galaxy colour and magnitude. The driving force in the colour of elliptical galaxies is commonly believed to be a function of both stellar age and metallicity (Thomas et al. 2010; Maiolino & Mannucci 2019), with age and metallicity both increasing for higher mass galaxies. Thomas et al. (2010) provided scaling relation between age/metallicity and dynamical mass for early-type galaxies. However, these scaling relations cannot be directly applied since we have host stellar masses and not velocity dispersion measurements that could be used to estimate the dynamical mass of the galaxies.

Studies of the delay-time distributions (DTD) of SNe Ia have suggested that they are best fit by two-component models (‘prompt’ and ‘delayed’), such as Mannucci et al. (2006). A comparison between SN Ia DTD and age and α-element abundances in early-type or elliptical galaxies has suggested these distributions are best fit by a single power-law DTD model (Walcher et al. 2016), which could imply a single progenitor origin for SNe Ia in elliptical galaxies. We have identified a strong trend between SN component colour of the galaxy and SALT2 x₁, for ‘normal’ SNe Ia in galaxies we have classified as elliptical. In our SN Ia sample, nearly all ‘91bg-like’ events are found in elliptical galaxies (there is just one in a bulge/bar location) and within these elliptical galaxies, they occur in the reddest galaxies. The ‘91bg-like’ SNe Ia follow the correlation between SN component colour and SALT2 x₁ seen for the ‘normal’ events in elliptical galaxies, suggesting that, from this aspect at least, their progenitor origin may be similar. For a sample of SNe Ia in 29 early-type galaxies, a strong correlation between SN peak V-band magnitude and host age (measured by the strength of typical absorption lines in old stellar populations) was identified (Gallagher et al. 2008). ‘91bg-like’ SNe Ia have also been suggested to come from progenitors with delay times of > 6 Gyr (Panther et al. 2019).

Shen et al. (2017) investigated the relation between SN Ia decline rate and stellar age using binary population synthesis calculations for sub-Chandrasekhar mass double-detonation systems. They found that the fastest declining SNe Ia (equivalent to the most negative SALT2 x₁) are associated with the oldest populations. This relation is driven by the fact that binaries with higher mass primaries interact at younger ages compared to systems with less massive white dwarf primaries. Higher metallicity SN Ia progenitors can result in more neutron capture, resulting in more stable Fe-group isotopes at the expense of radioactive Fe-group elements resulting in a fainter SN Ia, but this effect is expected to be small. Therefore, it is not expected that metallicity plays a major role in x₁ diversity. This is also seen in studies of galaxy-based forward models investigating the impact of age on SN Ia luminosities, where variations in age are found to be consistent with the SN luminosity variations observed (Wiseman et al. 2022, 2023). Kim et al. (2024) investigated a population of SNe Ia in elliptical galaxies to attempt to break the degeneracy between age and metallicity in their birth environments, but no clear trends were seen. However, Hayden et al. (2013) investigated the fundamental metallicity relation for a sample of star-forming hosts and concluded that the correlation between Hubble residuals and host stellar mass is most likely driven by metallicity, highlighting the somewhat conflicting results in the literature.

By linking these studies with our identified trend between x₁ and SN component colour, where the SNe Ia with more negative x₁ values occur in redder environments, it suggests that if the SN Ia population in elliptical galaxies is due to double degenerate white-dwarf systems (and it appears hard to explain in the context of a Chandrasekhar-mass explosion in these very old systems), then it suggests that the stellar age of the environment is more important than metallicity for the elliptical galaxies hosting SNe Ia.

Briday et al. (2022) investigated the best environmental tracers to explain the ‘mass step’ correction applied in SN Ia cosmology. They concluded that local specific star formation rate from Hα emission and global stellar mass are better tracers of this galaxy environment step function than host morphology and other tracers such as local mass or local colour. For their morphologies, they used two different methods, for the SN Factory data, the ratio of the radii containing 50 and 90% of the Petrosian flux in the r band (Kauffmann et al. 2003) supplemented with literature events with a visual morphological classification of Pruzhinskaya et al. (2020).

6.3. Type Ia supernovae in galaxies containing discs

A 5.1σ correlation exists between SN component colour and x₁ for SNe Ia occurring in discs (top right of Fig. 10). However, with a R² of only 0.15, this trend is not as strong as that seen in the elliptical population. It is clear that for the actively star-forming disc galaxies, with a SN-component-colour below ∼0.55 mag, there is no trend. Normal SNe Ia originating from actively star-forming environments (e.g. very blue discs) typically have −1 < x₁ < 2. Disc environments, with g − r SN-component-colour of > 0.55 mag, start to overlap with the colours of elliptical galaxies (i.e. in the ‘Green Valley’). These discs may have quenched star formation (hence redder colours), and therefore, resemble older environments that typically contain lower-stretch SNe Ia. Therefore, these ‘Green Valley’ SNe Ia are likely driving the observed correlation between SN component colour and x₁.

From Fig. 11, it can be seen that SN Ia colour shows little correlation with galaxy colour or host mass in galaxies containing discs, which is in agreement with the literature (e.g. Sullivan et al. 2010; Uddin et al. 2017). From the top right panel, it can be seen that there is a weak trend (2.96σ) in the SNe Ia located in discs, where redder discs tend to host redder SNe Ia. However, with such a large scatter, this result is better interpreted as a lack of red SNe Ia in very blue discs. A similar result was also reported in Sullivan et al. (2010), Kelsey et al. (2023). The colours of star-forming disc galaxies are generally much harder to relate to metallicity or age, as many star formation bursts can occur during a disc’s lifetime, complicating the age and chemical composition each time. Colours of elliptical galaxies likely correlate better than the colours of discs, as the period of time post-star-formation is much longer relative to the timescales between star formation bursts.

In the top left panel of Fig. 12, a significant linear correlation (6.1σ) is seen between x₁ and the local r-band surface brightness in galaxies with discs (‘disc-only’, ‘bulge+disc’, ‘bulge+bar+disc’ models). We can deduce that this must be an environmental dependence with the SN Ia progenitors, as a purely observational explanation should also produce a correlation for the SNe Ia in elliptical galaxies, which is not seen (bottom left panel of Fig. 12). In addition, we can rule out any observational bias on the SN Ia magnitude as an explanation, as the trend is in the opposite direction. Lower-stretch SNe Ia are generally less luminous (Phillips 1993), and hence should be more difficult to observe in brighter regions. However, the trend shows the opposite, with lower-stretch SNe Ia being more abundant near the galaxy centre.

This surface brightness correlation is apparent for disc systems but not for elliptical galaxies and can be explained by considering the distinction between SNe Ia explosion location and progenitor birthplace, as the local host environment at the birthplace should correlate stronger with the intrinsic properties of the SN Ia, rather than the explosion site. For disc galaxies, stars generally orbit in circular orbits within the plane of the disc, resulting in the stellar density (i.e. surface brightness) following the gravitational potential of the galaxy. Therefore, the SN Ia progenitors in disc systems, will likely be born, evolve and explode all at approximately the same surface brightness value; in other words, on the same isophote. In contrast, the stellar orbits within elliptical galaxies are much more chaotic and are generally not circular. Therefore, the measured surface brightness at the SN Ia explosion site in elliptical galaxies, has no correlation with the local surface brightness at the time of progenitor birth or during the progenitor evolution.

Disc galaxies are generally believed to grow inside-out, whereby the oldest population are closest to the centre and active star formation of young stars occurs further out in the plane of the disc (Delgado et al. 2015). Hence, surface brightness, which is directly related to the galactic separation and stellar density, is correlated with both age and metallicity in galaxy discs, where age and metallicity increase towards the centre (MacArthur et al. 2004; Sánchez et al. 2014). The observed trend of lower-stretch SNe Ia being more abundant closer to the galaxy centre, implies that lower-stretch SNe Ia are associated with older and more metal-rich environments. This is the same result as that from the SN component colour in Fig. 10. Lower-stretch SNe Ia reside in redder colours, which is known to also correlate with older and more metal-rich environments.

6.4. The impact of dust extinction

The top right panel of Fig. 12 shows an interesting correlation between SALT2 c- and r-band surface brightness for SNe Ia located in disc-containing galaxies. The 3.3σ linear trend, seen in the normal population with SALT2 c > 0.035, could be due to dust extinction, as dust content is known to increase towards the centre of disc galaxies (i.e. follows surface brightness; Muñoz-Mateos et al. 2009). As dust is predominantly located in the plane of the disc, the bulk of the SNe Ia with blue colours likely sit in front of the central plane (from our point-of-view), experiencing minimal dust extinction, while SNe Ia within/behind the central plane will suffer more dust extinction, leading to redder SNe Ia. This picture is consistent with the observed relative number of red/blue SNe Ia, as the SNe Ia residing behind the plane will be harder to detect and hence have lower numbers. Curiously, SNe Ia located very close to the centre (μ < 19.5 mag/arcsec²), appear to break away from the reddening trend and cluster closely to SALT2 c ∼ 0. Central bulges are old 3D stellar objects, with relatively little dust. However, dust is still present in the disc that cuts through the bulge. Consequently, any SNe Ia reddened by dust must reside relatively close to the centre of the plane, where the SNe Ia will almost certainly be obscured by the very bright surrounding bulge. Hence, only SNe Ia that lie sufficiently in front of the plane (away from the majority of the dust) will be visible at the bright central regions of galaxy bulges.

If this colour trend were due to age or metallicity, we would expect the innermost SNe Ia to follow the overall trend and be the reddest, which is not seen. The lack of red SNe Ia in the bright centres of disc galaxies disfavours age or metallicity as an explanation for the apparent reddening of SN Ia colours. This is further supported by the right panels in Fig. 11, which shows almost no correlation between SALT2 c and galaxy colour (which was demonstrated to be a good tracer of age and metallicity). In fact, the only apparent observation in Fig. 11; that red SNe Ia are missing from the bluest disc galaxies, can be explained by dust extinction, as the bluest disc galaxies are in general very young, and hence have less dust content. Additionally, from the bottom right panel of Fig. 12, at the same surface brightness values in elliptical galaxies (μ < 19.5 mag/arcsec²), many red normal SNe Ia are observed. For SNe with c > 0.035 in disc-containing galaxies, there are zero counts with μ < 19.5 mag/arcsec² and 54 counts with μ > 19.5 mag/arcsec². Whereas for SNe with c > 0.035 in elliptical galaxies, there are 21 counts with μ < 19.5 mag/arcsec² and 35 counts with μ > 19.5 mag/arcsec². This strongly disfavours any observational argument based on redder normal SNe Ia being intrinsically less luminous, and hence harder to observe in bright galaxy regions.

These arguments result in the conclusion that the majority of normal SNe Ia in disc galaxies, have intrinsic SALT2 c ≲ 0.035, with anything redder being due to dust. However, this cannot be stated with significant certainty, as it is not known how the different progenitor channels affect this picture. Elliptical galaxies, similarly to bulges, generally have relatively little dust and hence we would not expect any significant dust extinction. The significant presence of red (SALT2 c > 0.035) normal SNe Ia in elliptical galaxies (at all surface brightness values), could point to differing progenitor channels between Elliptical and Disc galaxies, which is also supported by the significantly different x₁ distributions.

6.5. Host galaxies of ‘91bg-like’ and ‘91T-like’ Type Ia supernovae

The preferred galaxy locations of ‘91T-like’ and ‘91bg-like’ SNe Ia in terms of their g − r SN-component-colour and host stellar mass (shown in Fig. 9) is similar to the u − r galaxy colour (estimated from 25 mag arcsec² elliptical apertures) against stellar mass of Hakobyan et al. (2020). They found the same tendency for ‘91bg-like’ SNe Ia to reside in the red sequence, while ‘91T-like’ SNe Ia are much less likely to occur in the reddest and highest mass hosts. Barkhudaryan et al. (2019) showed that within the red sequence, the ‘91bg-like’ SNe Ia have consistent host masses with the normal SNe Ia but, on average, have redder host galaxies (p-value of 0.047). This result is statistically significant (3.5σ) in our analysis, as is seen in the bottom panel of Fig. 9. The ‘91bg-like’ events have g − r SN-component-colour that is 0.035 ± 0.01 mag redder than the normal SNe Ia in elliptical galaxies. This colour offset, the consistency (0.5σ) of the slopes of the linear regressions for the normal and ‘91bg-like’ SNe Ia in the red sequence as well as the ‘91bg-like’ distributions in the left-hand plots of Fig. 10, show that the difference in host-environments between normal and ‘91bg-like’ SNe Ia is much better captured by the galaxy colour rather than the host mass. However, it should be noted that these results are dependent on the agglomerative clustering that was used to find the red sequence.

6.6. Benefits of the disc, bulge, or bar decomposition method

In contemporary galaxy studies, the u − r galaxy colour is usually preferred to g − r colour due to its ability to distinguish between young and old stellar populations (e.g. Lintott et al. 2008). However, as is seen in Fig. 9 using the g − r SN-component-colour (defined in Sect. 4.9), recovers similar galaxy correlations without relying on UV photometry. This may be of importance, as all-sky optical photometry, such as the g and r bands, are much more readily available (and to greater observing depths) than the corresponding UV bands. The benefit of the SN component colour over a standard global colour, is that it removes the red bias of central bulges from the estimated global colours of discs, allowing for greater contrast between young and old stellar populations (or star-forming and non-star-forming regions).

A shortcoming of some SN Ia galaxy studies is that they separate young from old stellar populations by global properties (e.g. global colour, host stellar mass, global star formation rate, Hubble morphology). This separation is problematic as SNe Ia originating from the bulges and bars of disc galaxies are grouped with SNe Ia from young star-forming populations, even though they contain an older population of stars, much closer to those found in elliptical galaxies (Kauffmann 1996). We overcome this issue by using a decomposition of the disc, bulge and bar (where required), allowing for easy separation between disc and bulge SNe Ia.

The results from this paper that were previously not seen in the literature likely highlight the importance of galaxy modelling as well as having a clean sample of host galaxies, when studying host properties. This study removes merging galaxies, edge-on disc systems, galaxies with intervening Milky Way stars, etc. These types of systems are difficult to accurately measure host stellar mass for and likely only introduce scatter in the relations between host stellar mass and x₁ for example. Therefore, it can be argued that SNe Ia with these types of host systems, should be removed when host environmental properties (e.g. mass-step) are being used to standardise SNe Ia for use in cosmology.

While it is clear that there is great benefit from galaxy decomposition, the technique used in this paper does not fully reproduce a representative sample of host galaxies. In particular, low-mass galaxies, asymmetric irregular galaxies, very prominent spiral arms and edge-on disc systems are underrepresented in our final sample due to being very difficult to model. Future studies could make use of more advanced decomposition codes that provide more freedom when fitting galaxies, however the more advanced the decomposition code is, the more difficult it will become to automate the procedure for a large sample of host galaxies. This is of importance as ZTF and other future sky surveys are expected to detected tens of thousands of new SNe Ia in the coming decade.

7. Conclusions

In this work, we have performed 2D host image decomposition for SN Ia host galaxies in the ZTF DR2. Using four surface brightness models: ‘Elliptical’, ‘Bulge+Disc’, ‘Bulge+Bar+Disc’, and ‘Disc-only’, we successfully modelled 719 galaxies, which were used to investigate the correlation between SN Ia photometric properties (using the SALT2 light-curve fitter) and their host environment. Our main results are as follows:

1. We demonstrate that the previously observed bimodal SALT2 x₁ distribution seen in the literature can be seen in the morphological galaxy type. Normal SNe Ia residing in galaxy discs are predominantly contained within the high-stretch mode, while SNe Ia residing in elliptical galaxies exist across all stretch values but constitute the majority of the low-stretch mode. We also show that SNe Ia originating from the central bulges of disc galaxies have lower x₁ values than SNe Ia in discs (4.8σ), and are consistent with the elliptical population of SNe Ia. This provides further evidence that the low-stretch SNe Ia originate from older environments.

2. We identify for the first time that SNe Ia in elliptical galaxies exhibit a strong linear correlation between x₁ and the model-derived (g − r) galaxy colour (‘SN-component-colour’) used in this study, whereby SNe Ia with lower x₁ values are found in redder elliptical galaxies (16.8σ, R² = 0.37). If a double degenerate white-dwarf progenitor model is assumed, then this relation may be driven by age rather than metallicity and normal SNe Ia in elliptical galaxies may originate from a single progenitor channel.

3. We identify a weaker but still significant linear correlation between x₁ and the model-derived (g − r) galaxy colour for SNe Ia in discs (5.1σ, R² = 0.15). While this trend also has lower x₁ values in the redder disc, the overall slope of the relation is fundamentally different from that seen in the elliptical population of SNe Ia. This trend is likely due to a combination of age and metallicity, but is complicated by the potential diverse progenitor channels that can exist for high-stretch SN Ia.

4. By taking the intersection point between the elliptical and disc linear trends, we estimate an approximate cross-over point between low- and high-stretch SNe Ia at x₁ = −0.39 ± 0.06. This value is less negative but still consistent with the split point in the ZTF companion papers: Ginolin et al. (2024) and Deckers et al. (2025). However, the exact position to choose is uncertain.

5. We identify for the first time that normal SNe Ia in disc-containing galaxies have a significant correlation with the local r-band surface brightness, whereby SNe Ia at higher surface brightness (that are hence closer to the centre) have lower x₁ values (6.1σ, R² = 0.09). This relation is consistent with the known metallicity and age gradients that exist in disc galaxies, and provides further evidence that SNe Ia in older and more metal-rich environments have lower x₁ values. No surface brightness trend is seen in the elliptical galaxy population, further highlighting the importance of separating disc SNe Ia from those in elliptical galaxies.

6. As has been seen previously in the literature, SN Ia colour (SALT2 c) shows much less correlation with the host environment. No significant relation was found between galaxy colour and SALT2 c. When analysing c versus local surface brightness, a potential dust effect may have been identified. For SNe Ia in disc-containing galaxies, there is a linear correlation (3.3σ) for SNe Ia with c > 0.035, where redder SN Ia are found in brighter regions (more dust closer to the centre). This suggests that normal SNe Ia in disc-containing galaxies have intrinsic colours of c < 0.035. This result cannot be concluded confidently, however, as it is still unclear how different progenitor channels play a role in this picture.

7. We show that the 91bg-like SNe Ia reside entirely in older populations of stars; that is, in elliptical galaxies and in the central bulge of disc galaxies. Within the classical red sequence, for fixed host stellar mass, the host galaxies of 91bg-like SNe Ia were found to be 0.035 ± 0.01 mag redder than the normal SNe Ia, suggesting that the underlying progenitor system of 91bg-like SNe Ia are on average older than those of normal SNe Ia in the red sequence.

This work has shown that by 2D modelling the host galaxies of SNe Ia and separating out different environmental types, many strong environmental correlations are recovered. In the current era of precision cosmology, we conclude that image decomposition techniques such as the one used in this study could better uncover the dependence of host environment on SN Ia properties, allowing for better standardisation of SN Ia luminosities. While this study has developed an image decomposition method that is fully automated, and that hence can be scaled to larger sample sizes, the primary shortcoming is the loss of a large fraction of the starting sample as well as a bias against low-mass, irregular, and merging galaxies and the exclusion of edge-on systems. This should be addressed by future works using more sophisticated decomposition programs.

Acknowledgments

RS, KM, UB, GD, MD, and JHT are supported by the H2020 European Research Council (ERC) grant no. 758638. LG acknowledges financial support from AGAUR, CSIC, MCIN and AEI 10.13039/501100011033 under projects PID2020-115253GA-I00, PIE 20215AT016, CEX2020-001058-M, and 2021-SGR-01270. This project has received funding from ERC grant no. 759194 – USNAC. LH is funded by the Irish Research Council under grant number GOIPG/2020/1387. This work has been supported by the research project grant “Understanding the Dynamic Universe” funded by the Knut and Alice Wallenberg Foundation under Dnr KAW 2018.0067 and the Vetenskapsrådet, the Swedish Research Council, project 2020-03444. Y-LK has received funding from the Science and Technology Facilities Council [grant number ST/V000713/1]. Based on observations obtained with the Samuel Oschin Telescope 48-inch and the 60-inch Telescope at the Palomar Observatory as part of the Zwicky Transient Facility project. ZTF is supported by the National Science Foundation under Grant No. AST-1440341 and a collaboration including Caltech, IPAC, the Weizmann Institute of Science, the Oskar Klein Center at Stockholm University, the University of Maryland, the University of Washington, Deutsches Elektronen-Synchrotron and Humboldt University, Los Alamos National Laboratories, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, and Lawrence Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW. The ZTF forced-photometry service was funded under the Heising-Simons Foundation grant #12540303 (PI: Graham). The Gordon and Betty Moore Foundation, through both the Data-Driven Investigator Program and a dedicated grant, provided critical funding for SkyPortal. The Legacy Surveys consist of three individual and complementary projects: the Dark Energy Camera Legacy Survey (DECaLS; Proposal ID #2014B-0404; PIs: David Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID #2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Survey (MzLS; Prop. ID #2016A-0453; PI: Arjun Dey). DECaLS, BASS and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF’s NOIRLab; the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOIRLab. Pipeline processing and analyses of the data were supported by NOIRLab and the Lawrence Berkeley National Laboratory (LBNL). The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the Tohono O’odham Nation. NOIRLab is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. LBNL is managed by the Regents of the University of California under contract to the U.S. Department of Energy. This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energeticas, Medioambientales y Tecnologicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciencies de l’Espai (IEEC/CSIC), the Institut de Fisica d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig Maximilians Universitat Munchen and the associated Excellence Cluster Universe, the University of Michigan, NSF’s NOIRLab, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University. BASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Priority Research Program “The Emergence of Cosmological Structures” Grant # XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance. The BASS is also supported by the External Cooperation Program of Chinese Academy of Sciences (Grant # 114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant # 12120101003, # 11433005). The Legacy Survey team makes use of data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), which is a project of the Jet Propulsion Laboratory/California Institute of Technology. NEOWISE is funded by the National Aeronautics and Space Administration. The Legacy Surveys imaging of the DESI footprint is supported by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract; and by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to NOAO.

References

All Tables

All Figures

	Fig. 1. DESI-LS sky coverage shown using a Mollweide projection, with DR10 in light-grey and DR9 in dark-grey blue. The blue points represent covered ZTF Cosmo DR2 SNe Ia (83%), while the red points are SNe Ia outside the sky coverage. The green line marks the Galactic plane.
In the text

	Fig. 2. Example SN Ia rest-frame light curve from ZTF DR2, with a fitted SALT2 model. The x1 parameter quantifies the stretch of the light curves, while c quantifies the colour. The black arrows are for illustrative purposes only, they do not describe how these parameters are calculated.
In the text

Fig. 3. Schematic of the isophote map fitting procedure. Top left: Example g-band DESI-LS image of an elliptical host galaxy, with neighbouring field sources. Top right: The galaxy image sampled before mean-kernel convolution, at 24 ± 0.1 mag/arcsec2. Bottom left: The image sampled after mean-kernel convolution, with the target galaxy pixels highlighted in blue, external sources in red and a fitted 24 mag/arcsec2 isophote in black. The target galaxy pixels are isolated by running a connected-component labelling algorithm. Bottom right: The constructed g-band isophote map for this host galaxy, showing isophotes at every 0.2 mag/arcsec2.

In the text

Fig. 4. Schematic showing the main steps of the galaxy fitting procedure. Top left: DESI-LS image of a barred host galaxy, containing a ZTF SN Ia. Top right: The green superellipses are the isophotes of the constructed g-band isophote map (every second isophote has been removed for visual clarity). The blue dots mark the sampled points used during the galaxy fitting. Middle panels: Two example fitted surface brightness profiles are shown. The first panel is located at 120°, close to the semi-major axis of the bar, while the second panel is located at 240°, close to the semi-minor axis of the bar. The black points correspond to the sampled blue dots from the top right panel, with 3σ error bars. The coloured lines (orange, green and red) represent the fitted intrinsic profiles for each component in the ‘Bulge+Bar+Disc’ model. The shaded red region shows the 3σ confidence interval for the combined model, convolved with the PSF. Bottom left: The reconstructed 2D surface brightness model of the galaxy, convolved with the PSF. The three superellipses represent the intrinsic effective radius for each component in the model. The shaded regions are computed by linearly mapping the surface brightness profile to the opacity. The central brightness has an opacity of one, while zero opacity is at the 25 mag/arcsec2 isophote. Bottom right: Subtraction of model from original image in surface brightness space.

In the text

Fig. 5. Results of a two-component ‘bulge+disc’ fit to a galaxy. The surface brightness in the g band is shown as a function of radial distance from the centre (along an arbitrary angle). The black points are the radially sampled points from the corresponding isophote map for this galaxy, with 3σ error bars. The coloured lines (green and red) represent the fitted intrinsic profiles for each component in the ‘Bulge+Disc’ model. The shaded red region shows the 3σ confidence interval for the combined model, convolved with the PSF. The top panel has no bulge truncation and converges on a non-physical solution. The bottom panel shows a fit using the same initial conditions, but with bulge truncation active during the fitting procedure. The dotted black line in the bottom panel marks the defined boundary between the bulge and the disc.

In the text

Fig. 6. Example images of the four galaxy models used in this paper. The left panels are the DESI-LS images in surface brightness space with the location of the SN marked as the green cross, the centre panels are the fitted galaxy models, where the superellipses are the effective radius for each galaxy component. The shaded regions are computed by linearly mapping the surface brightness profile to the opacity. The central brightness has an opacity of one, while zero opacity is at the 25 mag/arcsec2 isophote. The right panels are the residuals, calculated by subtracting the model from the image. The four galaxy models (from top to bottom) are ‘Elliptical’, ‘Bulge+Disc’, ‘Bulge+Bar+Disc’ and ‘Disc-only’.

In the text

Fig. 7. Fit success rate as a function of redshift and host mass. Top panels: Blue histogram shows the full ZTF Cosmo DR2 binned in redshift (left) and host mass (right). Orange histogram shows galaxies with isophote maps that pass all quality cuts. Green histogram shows final sample (successful galaxy fit + galaxy morphology identified). Centre panels: Success rate for generating a successful isophote map, given a galaxy image, as a function of redshift (left) and host mass (right). Bottom panels: Success rate for generating a galaxy model (successful galaxy fit + galaxy morphology identified), given an isophote map, as a function of redshift (left) and host mass (right).

In the text

Fig. 8. SALT2 x1 completeness of the final galaxy sample. Top panel: Blue histogram shows the ZTF DR2 binned in SALT2 x1, with the light curve quality cuts applied, see Sect. 2.3. The orange histogram shows the final sample containing SNe Ia with successfully fitted galaxy models. Bottom panel: Rate of finding a SN Ia with any particular x1 value in the final sample. The dashed grey line shows the average rate.

In the text

Fig. 9. SN-component-colour (g − r) versus host stellar mass. Top panel: SNe Ia separated into the main spectral subtypes, normal SNe Ia (blue), ‘91bg-like’ SNe Ia (red), and ‘91T-like’ SNe Ia (orange). Centre panel: Kernel density plots showing host mass density function of the final sample (dashed line) and the ZTF DR2 SNe Ia that were cut from the final sample (solid line). Bottom panel: Normal and ‘91T-like’ SNe Ia grouped into three main clusters: red sequence, green valley, and blue cloud, using a clustering algorithm. The elliptical ‘91bg-like’ (purple diamonds) and elliptical ‘91T-like’ (orange triangles) subpopulations are highlighted. The dashed black and solid grey lines are linear regressions for the normal and ‘91bg-like’ subpopulations in the red sequence, respectively.

In the text

Fig. 10. Galaxy colour and host mass versus SALT2 x1. The top panels show SN-component-colour (Sect. 4.9) versus x1, while the bottom panels show host stellar mass versus x1. The left panels show the distribution of normal SNe Ia (blue circles), ‘91bg-like’ SNe Ia (red diamonds), and ‘91T-like’ SNe Ia (orange triangles). The right panels contain only the normal SNe Ia, with the associated galaxy component highlighted (Sect. 4.8). Weighted linear regressions are shown for statistically significant relationships of SN-component-colour/stellar mass and x1 in elliptical galaxies (dashed red lines) and SN-component-colour/stellar mass and x1 in disc galaxies (dashed blue line).

In the text

Fig. 11. Galaxy colour and host mass versus SALT2 c. The top panels show SN-component-colour (Sect. 4.9) versus c, while the bottom panels show host stellar mass versus c. The left panels show the distribution of normal SNe Ia (blue circles), ‘91bg-like’ SNe Ia (red diamonds), and ‘91T-like’ SNe Ia (orange triangles). The right panels contain only the normal SNe Ia, with the associated galaxy component highlighted (Sect. 4.8). A Weighted linear regression (dashed blue line) is shown for SN-component-colour and c in disc galaxies.

In the text

Fig. 12. Local r-band surface brightness (derived from model) versus SALT2 parameters (x1 on the left and c on the right), highlighted by SN classification and the associated galaxy component. The top panels contain SNe Ia located in galaxies containing a disc, while the bottom panels contain SNe Ia located in elliptical galaxies. Dashed black lines indicated significant linear correlations. A horizontal red line is plotted at c = 0.035. See Sect. 6.4 for discussion.

In the text