Evidence for sub-Chandrasekhar Type Ia supernovae from the last major merger

We investigate the contribution of sub-Chandrasekhar mass Type Ia supernovae to the chemical enrichment of the Gaia Sausage galaxy, the progenitor of a significant merger event in the early life of the Milky Way. Using a combination of data from Nissen&Schuster (2010), the 3rd GALAH data release (with 1D NLTE abundance corrections) and APOGEE data release 16, we fit analytic chemical evolution models to a 9-dimensional chemical abundance space (Fe, Mg, Si, Ca, Cr, Mn, Ni, Cu, Zn) in particular focusing on the iron-peak elements, Mn and Ni. We find that low [Mn/Fe] $\sim-0.15\,\mathrm{dex}$ and low [Ni/Fe] $\sim-0.3\,\mathrm{dex}$ Type Ia yields are required to explain the observed trends beyond the [$\alpha$/Fe] knee of the Gaia Sausage (approximately at [Fe/H] $=-1.4\,\mathrm{dex}$). Comparison to theoretical yield calculations indicate a significant contribution from sub-Chandrasekhar mass Type Ia supernovae in this system (from $\sim60$% to $100$% depending on the theoretical model with an additional $\pm10$% systematic from NLTE corrections). We compare to results from other Local Group environments including dwarf spheroidal galaxies, the Magellanic Clouds and the Milky Way's bulge, finding the Type Ia [Mn/Fe] yield must be metallicity-dependent. Our results suggest that sub-Chandrasekhar mass channels are a significant, perhaps even dominant, contribution to Type Ia supernovae in metal-poor systems, whilst more metal-rich systems could be explained by metallicity-dependent sub-Chandrasekhar mass yields, possibly with additional progenitor mass variation related to star formation history, or an increased contribution from Chandrasekhar mass channels at higher metallicity.


INTRODUCTION
The progenitors of Type Ia supernovae (SNe Ia) are not well understood. Despite this, there has been great success in their use as standard candles for measuring the accelerating expansion of the Universe (Riess et al. 1998;Perlmutter et al. 1999). This has been possible due to empirical relations between luminosity and light-curve decay rate allowing for accurate calibration of their standard candle nature (e.g. Pskovskii 1977;Phillips 1993). However, the cause of such a relation is unclear. There is a growing zoo of different sub-classes of observed SNe Ia, indicating a range of formation channels. The commonality between most observed SNe Ia is they likely arise from the explosion of a carbon-oxygen (CO) white dwarf (WD) in a binary system (Whelan & Iben 1973;Iben & Tutukov 1984;Webbink 1984). Beyond this, there are many suggested scenarios for the nature of the progenitor systems and the explosion mechanisms (see the reviews of Hillebrandt & Niemeyer 2000;Maoz et al. 2014;Seitenzahl & Townsley 2017;Ruiter 2020).
The two leading theories for the progenitors of normal SNe Ia are the single-and double-degenerate scenarios. Single-degenerate progenitor systems are mostly anticipated to produce explosions near the Chandrasekhar mass ( Ch ), whilst double-degenerate systems can produce explosions well below the Chandrasekhar mass (sub-Ch ). ★ E-mail: jason.sanders@ucl.ac.uk (JLS) In the single-degenerate scenario, a CO WD typically accretes from a non-degenerate H-rich companion (non-degenerate He donors are also possible and at low accretion rates could resemble the doubledegenerate scenario described below) until the WD approaches the Chandrasekhar mass (Whelan & Iben 1973). At first, the WD enters a turbulent low-level carbon-burning phase lasting a few hundred to a thousand years, which eventually ignites the dynamical burning of carbon. Early models of this scenario (Arnett 1969) considered a super-sonic detonation which failed to produce the intermediatemass elements (e.g. Si) observed in SNe Ia remnants due to the high central density of the WD (Arnett et al. 1971). To remedy this, typically a deflagration flame is initialized which propagates subsonically and causes the WD to expand, producing regions of low density (Nomoto et al. 1984). Eventually, the deflagration flame transitions to a detonation wave leading to the production of intermediate mass elements in the new lower density regions, and crucially suppressing the production of neutron-rich species (e.g. 58 Ni) which are overproduced in pure deflagration models (Khokhlov 1991). This scenario is dubbed the deflagration-to-detonation transition (DDT, sometimes called delayed detonation).
It has further been hypothesised that in some systems the transition to detonation fails to occur so the WD undergoes a pure deflagration (e.g. Fink et al. 2014) leaving behind a remnant 'zombie' core. These systems could explain the observed class of under-luminous Type Iax supernovae (Foley et al. 2013). Despite the success the single-degenerate DDT models have in explaining many properties of the observed light-curve and remnant (e.g. Hoeflich & Khokhlov 1996;Badenes et al. 2005), a few issues remain: (i) the stable accretion rate of hydrogen onto the WD surface must be within a relatively narrow range such that it can successfully reach the Chandrasekhar mass (if the rate is too high a red giant envelope can form leading to common envelope evolution unless a strong wind is driven; if too low hydrogen flashes and nova explosions eject the mass, ), (ii) there is an observed lack of near-Chandrasekhar mass WDs from which these systems could originate (e.g. Kepler et al. 2007), (iii) the predicted rates from this channel fall short of explaining the observed SNe Ia rates (e.g. Ruiter et al. 2009 and see figure 8 of Maoz et al. 2014 for a comparison), (iv) low X-ray flux from nearby elliptical galaxies are consistent with only 5 per cent of all SNe Ia being singledegenerate (Gilfanov & Bogdán 2010), (v) the late-time spectrum of the Type Ia supernova SN2012cg lacks H emission as expected from the envelope of a non-degenerate companion (Shappee et al. 2018), (vi) and there are still some inconsistencies in matching the colours, velocities and light curves in all bands (Sim et al. 2013).
The double-degenerate scenario instead considers two white dwarfs in a binary system (either both CO WDs, or a CO WD and a lower-mass He WD) slowly merging due to the emission of gravitational waves (Iben & Tutukov 1984). Although early work presumed the disrupting WD would form an accretion disc around the primary (Benz et al. 1990) leading to steady growth up to Ch , it is now deemed more probable that one WD initialises a detonation in the other well before the Chandrasekhar mass is reached. Lower central densities in the sub-Ch WDs lead naturally to production of intermediate-mass elements without invoking a deflagration phase. If He is accreted through Roche-lobe overflow from a companion WD at a low enough rate to avoid helium flashes, a sufficiently massive He envelope can build up to initiate a detonation at the base of the He layer, triggering a detonation in the carbon core (Woosley & Weaver 1994). This scenario is called double-detonation (double-detonations may also occur in single-degenerate systems with a non-degenerate He-star although these are expected to have short delay times so cannot explain the bulk of SNe Ia with > 1 Gyr delays, and the initial detonation cannot be dynamically triggered, see below). The mass of the He shell required to initialise a detonation (∼ 0.2 ) produces high-velocity iron-peak elements from the burning of the He shell not seen in observations (Hoeflich & Khokhlov 1996) and an overabundance of lighter iron-peak elements (e.g. V and Cr, Kobayashi et al. 2020a). However, carbon detonation can occur with lower mass He shells (e.g. Fink et al. 2010) with the He detonation possibly being triggered dynamically through unstable mass transfer from a violent merger (Guillochon et al. 2010;Pakmor et al. 2012;Shen et al. 2018a). This scenario produces superior matches to observed multi-band light-curves Kromer et al. 2010;Townsley et al. 2019) and also predicts the ejection of the WD companion at high velocities (see Shen et al. 2018b, for potential candidates from Gaia).
One route for constraining the contributions of these different SNe Ia formation scenarios is through galaxy chemical evolution modelling. SNe Ia produce ∼ 0.6 of iron (formed from the radioactive decay of 56 Ni which causes SNe Ia to glow) and low (although as discussed not insignificant) quantities of intermediate-mass elements. The contribution of SNe Ia can then be inspected from the down-turning of a stellar population in the [ /Fe]-[Fe/H] diagram (Tinsley 1979). Although theoretical models for the different progenitor scenarios agree in the iron production per event, because of the variety of densities in the exploding media there is a range of different chemical abundances produced, particularly for the iron-peak elements. Most notably Mn (e.g. Seitenzahl et al. 2013b) and Ni (e.g. Kirby et al. 2019) are very sensitive to the central density of the exploding WD (see also McWilliam et al. 2018). Mn is mostly produced through the decay of 55 Co which is made in large quantities in normal freeze-out from nuclear statistical equilibrium achieved in Ch WDs, or in lower quantities in incomplete silicon burning at the lower densities typical in sub-Ch WDs, and is destroyed during alpha-rich freeze-out which occurs at lower densities (see Lach et al. 2020). Stable Ni is produced as 58 Ni which arises from the neutron-rich environments produced by electron-capture in the high density environments of Ch WDs (Seitenzahl & Townsley 2017). For these reasons, typically sub-Ch explosions lead to subsolar [Mn/Fe] and [Ni/Fe], whilst Ch events produce super-solar yields. Other elements (e.g. Cr, V) are also sensitive indicators of the explosion properties (Palla 2021). Studies of the [Mn/Fe] distribution in the Milky Way have thus concluded a significant fraction ( 75 per cent) of Chandrasekhar mass supernovae is required to explain the observations (Seitenzahl et al. 2013b;Kobayashi et al. 2020a) although comparative studies of different host environments (the bulge, thick disc and dSphs, see Cescutti et al. 2008;North et al. 2012) have appealed to significantly metallicity-dependent SNe Ia yields. A significant fraction of sub-Ch systems is in line with the ejecta mass measurements from Scalzo et al. (2014)  To build a consistent model of SNe Ia progenitors, we require a range of systems with different star formation histories and different metallicities which we can simultaneously fit.
With the arrival of data from the Gaia satellite (Gaia Collaboration et al. 2016), along with accompanying spectroscopic surveys, significant evidence has emerged that the Milky Way experienced a > 10 10 (total mass) merger approximately 8 − 10 Gyr ago. The progenitor system is known as the Gaia Sausage or Gaia-Enceladus. Evans (2020) outlines the history of this idea. Nissen & Schuster (2010) discovered two populations of local halo dwarf stars with distinct kinematics which follow distinct chemical abundance tracks, and attributed the lower [ /Fe] sequence to a remnant dwarf galaxy. Deason et al. (2013) put forward the hypothesis that the prominent break in the Galactic stellar halo's density profile is the consequence of an ancient accretion of a substantial satellite galaxy. This idea was tested by Belokurov et al. (2018) Belokurov et al. (2018) concluded that such a dominant, metal-rich and highly-eccentric halo component is naturally produced in a collision between the MW and an LMC/SMC-mass galaxy around redshift ∼ 2. Helmi et al. (2018) and Mackereth et al. (2019) presented a coherent chemo-dynamical picture of the merger using APOGEE data in combination with Gaia DR2, demonstrating that the progenitor's chemical evolution history can be reconstructed robustly using a set of simple orbital selection criteria. Further evidence for the merger has come from the properties of the globular cluster population (e.g. Myeong et al. 2018). The Gaia Sausage merger presents an ideal opportunity to study chemical evolution in a low metallicity, "high redshift" galaxy using a sample of bright, nearby stars. Although the galaxy is no longer intact, we can still study the distributions of its constituent stars in chemical abundance space and make inferences on the evolution within the progenitor system (e.g. Vincenzo et al. 2019;Aguado et al. 2021;Matsuno et al. 2021). The system nicely bridges the gap between the lower metallicity dSph systems studied by Kirby et al. (2019) and de los Reyes et al. (2020) and the higher metallicity environments within the Milky Way populations, whilst having abundant high-quality data. Furthermore, it is perhaps the best nearby example of a reasonably high-mass galaxy with a truncated early burst of star formation so presents a unique opportunity to study chemical evolution in this setting. With the current influx of highdimensional abundance data from large stellar spectroscopic surveys such as APOGEE (Ahumada et al. 2020) and GALAH (Buder et al. 2020), it is now possible to perform the necessary detailed studies of the chemical properties of this structure.
In this paper, we use a high-dimensional chemical abundance space to constrain simple chemical evolution models of the Gaia Sausage and place constraints on the properties of Type Ia supernovae in this system. We discuss the data used in Section 2. In Section 3 we describe the models from Weinberg et al. (2017) used to fit the data. Focussing specifically on the measured Mn and Ni yields, we discuss our results compared to a range of theoretical yields from Ch and sub-Ch Type Ia supernovae in Section 4, along with the impact of the assumption of local thermal equilibrium on our results. Finally, we compare the constraints on the Type Ia supernovae channel with results from other systems (the Milky Way and dwarf spheroidal galaxies) highlighting the necessary metallicity dependence of the yields, before we present our conclusions in Section 5.

MULTI-DIMENSIONAL CHEMICAL ABUNDANCE DATA
We employ three samples of data: the local metal-poor 'halo' sample from Nissen & Schuster (2010), GALAH DR3 (Buder et al. 2020) and APOGEE DR16 (Ahumada et al. 2020). We describe the properties of each of the datasets in turn as well as the cuts we have employed to produce samples of Gaia Sausage member stars.

Nissen & Schuster (2010,11)
First, we take the nearby (distance < 335 pc) sample of low metallicity dwarf stars from Nissen & Schuster (2010, NS) and Nissen & Schuster (2011). The stars were observed with either the UVES spectrograph on the VLT ( ∼ 55000) or the FIES spectrograph on the NOT ( ∼ 40000) and LTE abundance measurements were obtained from equivalent widths for Fe, Na, Mg, Si, Ca, Ti, Cr, Mn, Cu, Zn, Ni, Y and Ba. For these stars we further utilise the non-LTE Cu measurements from Yan et al. (2016). We cross-match to the Gaia Early Data Release 3 (EDR3, Gaia Collaboration et al. 2016 to obtain proper motions and distances (utilizing a simple parallax inversion with a 0.017 mas offset), and find orbital properties (eccentricity , maximum Galactic height max ) using G (Bovy 2015) using the McMillan (2017)

GALAH DR3
Secondly, we utilise the 3rd data release (DR3) of the GALAH survey (Buder et al. 2020;Kos et al. 2017). GALAH (Galactic Archaeology with HERMES) is a large stellar spectroscopic survey performed with the HERMES spectrograph (Sheinis et al. 2015) on the Anglo-Australian Telescope. GALAH targets primarily stars with 9 < < 14 although the recent data release has processed similar surveys performed with HERMES, some of which are fainter. The majority of the targetted stars are within a few kpc of the Sun. In DR3, 30 elemental abundances are provided along with uncertainties, quality flags and bitmasks indicating the lines used for each star. 11 of these elements have 1D non-local thermal equilibrium (LTE) corrections (Amarsi et al. 2020). Furthermore, GALAH DR3 provides several value-added catalogues including the computation of spectro-photometric distances incorporating Gaia DR2 data and orbital parameters (computed in the Galactic potential of McMillan 2017). We restrict to stars with −2 dex < [Fe/H] < −0.7 dex (with flag_sp = 0, flag_fe_h = 0 and flag_alpha_fe = 0) and select a sample of high-probability Gaia Sausage members by requiring med( ) > 0. 85 The stricter eccentricity cut compared to that employed for NS is to reduce contamination from other accreted substructures (Myeong et al. 2019;Kim et al. 2021). The [Si/Fe] cut removes obvious contaminants not observed in the other datasets. We remove stars within 3 half-light radii (in projection) of known globular clusters (Harris 1996(Harris , 2010 and within known open clusters (Dias et al. 2002). Furthermore, we only use the abundances with uncertainties smaller than 0.1 dex and with no flags. This results in a sample of 1047 stars.
In Appendix A we inspect the abundance differences with respect to APOGEE DR16. We find significant offsets and trends with metallicity for many abundances. Part of this could arise from the non-LTE corrections applied in GALAH. For Ni (which does not have a non-LTE correction in GALAH), two lines are used in GALAH DR3. We find a significant difference in the residuals for stars analysed using both lines compared to when only one is used. We therefore choose to apply a correction to the GALAH DR3 Ni abundances to put them on the APOGEE scale.

APOGEE DR16
Thirdly, we use the APOGEE survey data as part of SDSS data release 16 (DR16, Ahumada et al. 2020). APOGEE (Apache Point Observatory Galactic Evolution Experiment, Majewski et al. 2017) is a large infra-red spectroscopic survey which has taken observations of ∼ 430, 000 stars in the band at resolution ∼ 22, 500.
The initial survey (APOGEE-1) used the 2.5m Sloan Foundation Telescope (Gunn et al. 2006) situated at Apache Point Observatory, but from DR16 the survey was extended (APOGEE-2) to include additional data taken with the 2.5m du Pont Telescope situated at Las Campanas Observatory. Both sub-surveys used similar spectrographs Wilson et al. (2019). Abundances for ∼ 20 elements are provided as measured by the ASPCAP pipeline (García Pérez et al. 2016;Jönsson et al. 2018Jönsson et al. , 2020 along with uncertainties and quality flags. We only use data with aspcapflag = 0 (and as with GALAH remove stars near known globular and open clusters). Unlike the GALAH survey, all abundances are derived under an LTE approximation, although non-LTE corrections are expected in the next data release (Osorio et al. 2020). Leung & Bovy (2019) have provided a valueadded catalogue with distances and orbital properties computed for all APOGEE stars via a convolutional neural network (astroNN). This catalogue assumes the MWPotential2014 from Bovy (2015) whilst for the other datasets we have assumed the McMillan (2017) potential. For stars matching between APOGEE and GALAH with eccentricity uncertainties less than 0.1, the two catalogues give a median eccentricity difference of 0.01 with a scatter of 0.06 so the difference between the Galactic potentials is not a concern. We select stars with

Common stars
In Appendix A we have analysed the differences in abundances for stars common to the three presented samples. When comparing APOGEE DR16 to NS we find only [Si/Fe] has a significant offset with APOGEE DR16 0.09 dex lower than NS. The only significant offset between GALAH DR3 and NS is in [Cr/Fe] which is 0.11 dex lower in GALAH DR3 although there are only a few stars in common. When comparing APOGEE DR16 and GALAH DR3 the differences arising from the non-LTE corrections of Mg, Si and Ca from Amarsi et al. (2020) are apparent. Mn has non-LTE corrections in GALAH but there is no significant systematic trend with respect to the APOGEE Mn abundances detected. As discussed in Section 2.2, the GALAH Ni abundances appear to be systematically biased at low metallicity compared to the APOGEE Ni abundances. Weinberg et al. (2017) demonstrate how galactic chemical evolution can be solved analytically under a number of simplifying assumptions: (i) the galaxy is represented by a single 'zone' so no radial flows or radial migration are included, (ii) the star formation efficiency (the star formation per unit gas mass ≡ 1/ ★ ) is a constant (instead of scaling non-linearly as per a Kennicutt-Schmidt law), (iii) supernovae (or other enrichment sources) either return their products immediately (e.g. for Type II) or, for Type Ia, according to an exponential delay-time distribution (DTD), e −( − D )/ Ia , with timescale Ia and minimum delay time D (or a linear combination of such distributions), (iv) stellar yields are independent of initial metallicity, (v) the rate of gas outflow is proportional to the star formation rate, via a constant , and finally, (vi) the star formation history, ★ ( ), is from a limited set of functions including exponential ★ ∝ e − / sfh or linear-exponential ★ ∝ e − / sfh for constant sfh . In these models, the gas reservoir is depleted on a timescale dep = ★ /(1 − + ) where ≈ 0.4 is the fraction of unprocessed material returned by stars (Weinberg et al. 2017). The advantage of these models is the speed with which they can be evaluated as they avoid a costly forward integration.

ANALYTIC CHEMICAL EVOLUTION MODELLING
For an assumed set of (IMF-integrated) net yields Y of element (i.e. mass of element produced per mass of stellar generation formed) and assuming a linear-exponential star formation law, the solution over time for the enrichment due to Type II supernovae (Y = II) is given by equation (56) of Weinberg et al. (2017), whilst the enrichment due to Type Ia supernovae (Y = Ia) is given by equation (58). By normalizing these equations relative to an assumed solar abundance (Asplund et al. 2009), we are able to predict the abundance tracks [X/H]( ) given a set of model parameters, = ( sfh , ★ , , Ia , D ).
The observed DTD for SNe Ia follows a power-law of the form ∼ −1 above some minimum delay time, D (Maoz & Mannucci 2012). As described in Weinberg et al. (2017), a linear combination of exponential DTDs, e =1 e −( − D )/ Ia,k , can be used to approximate a power-law distribution over the times of interest. For each choice of we have fitted a combination of e = 3 exponential DTDs to 1/ 1.1 normalized between and a Hubble time. We have then approximated the relationship between and the fitted timescales, Ia,k , and relative weights, , using cubic polynomials in the logarithms of these quantities. In this way, given a choice of we can simply find the combination of exponential DTDs which reproduces the 1/ 1.1 DTD. Our parameter set is then reduced to = ( sfh , ★ , , D ).
The key parameters for controlling the shape of the [X/Fe] vs. [Fe/H] tracks are sfh , ★ and dep . Increasing sfh (slower enrichment), increasing ★ (less enrichment per unit gas mass) or decreasing dep (increasing , more outflow) all cause the tracks in [ /Fe] vs. [Fe/H] to turn over at lower [Fe/H]. In the instantaneous recycling approximation, a metallicity-dependent Type II yield is equivalent to changing the depletion timescale, dep , (higher yields at higher metallicity increase the effective depletion timescale). Metallicitydependent Type Ia yields are not possible within the analytic framework but would have a similar effect. Over the range of considered metallicities, theoretical yields depend only very weakly on metallicity (see later discussion). We express this model in P , a P interface to the probabilistic programming language S (Carpenter et al. 2017) for performing Bayesian inference. For each star , we employ a birth time hyperparameter < max from which we compute {[X/Fe]( )} and [Fe/H]( ) to compare with the data. Assuming selection effects are minimal, the follow the star formation rate, ∼ ★ , up to the maximum time max (when the Gaia Sausage merged with the Milky Way and star formation ceased, Belokurov et al. 2020;Bonaca et al. 2020, , or more precisely in our analysis, the time when the interstellar gas metallicity reaches the maximum metallicity, [Fe/H] ∼ −0.7 dex, of our defined samples). We marginalize over the model parameters = ( sfh , ★ , , D , max ) adopting normal distributions 1 D ∼ N (0.15, 0.05) (with a minimum at 0.04 Gyr cor- responding to the lifetime of a 8 star assuming all more massive stars form Type II supernovae), ★ ∼ N (1, 100), ∼ N (10, 50), max ∼ N (4, 1) and sfh ∼ N (2, 1) (where all timescales are in Gyr). We place broad priors on [X/Fe] Y ∼ N (0, 2) except for [Mg/Fe] Ia which should be small (e.g. Kobayashi et al. 2020b) so we adopt a prior [Mg/Fe] Ia ∼ N (−2, 0.4) with a hard upper limit of [Mg/Fe] Ia = −0.5 (which approximately corresponds to the range of theoretical Type Ia models considered later). Early models (particularly those fitted to the GALAH data) produced [Mg/Fe] Ia ≈ 0 which is inconsistent with basic understanding of chemical evolution and essentially all SNe Ia models. For the mass of iron produced from Type II supernovae per total mass of stars formed, we use ln II Fe ∼ N (ln 0.00075, 0.45), which is consistent with a range of net IMF-integrated yields (evaluated at 10 −1.5 ) from the models of Chieffi & Limongi (2004), Kobayashi et al. (2006),  and NuGrid (Pignatari et al. 2016;Ritter et al. 2018) using IMFs from Salpeter (1955), Kroupa (2001), Kroupa et al. (1993) and Chabrier (2003) for stars between 0.1 and 100 . Similarly, for the range of Type Ia theoretical models considered later we find the mass of iron produced per event is (0.77 ± 0.15) . The total number of Type Ia supernovae over a Hubble time per stellar mass of star formation is approximately (2.2 ± 0.3) × 10 −3 −1 (Maoz & Mannucci 2012). Combining these numbers gives ln Ia Fe ∼ N (ln 0.0017, 0.24). In addition to the reported abundance uncertainties, we also allow for additional scatter in all [X/Fe] using a two-component mixture model with mixture weight mix ∼ N (1, 0.05) (where mix is the weight of narrower component restricted to 0 < mix < 1) and priors ln 1 ∼ N (−3, 1) and ln 2 ∼ N (−1, 1) for the additive additional scatters. We also use the mixture for [Fe/H] but set 1 = 0. Fig. 3 show the results of fitting these models to the NS, GALAH DR3 and APOGEE DR16 Gaia Sausage samples respectively. In Table 1 we give the model parameters governing the star formation, outflow and delay-time distribution of the Type Ia supernovae, and in Table 2 we give the abundances of the Type II and Type Ia channels. Here we discuss the results in general terms, mainly focusing on differences between the results from the different datasets. The next section concerns the meaning of the results in the context of nucleosynthesis in SNe Ia explosions.

Fig. 1, Fig. 2 and
In the NS data we see little evidence for a plateau or knee in the [ /Fe] abundances for the Gaia Sausage stars suggesting the Type Ia supernovae began contributing below [Fe/H] = −1.5 dex. This is somewhat corroborated by the APOGEE DR16 data where the knee occurs around [Fe/H] ≈ −1.5 dex. However, the picture is less clear from the GALAH DR3 data where there is seemingly a plateau until a metallicity of [Fe/H] ≈ −1 dex and the subsequent turnover due  Table 1 that all models have a large ★ indicating inefficient star formation (high gas fraction) and high indicating a large outflow. These features are necessary to explain the down-turning [ /Fe] at low metallicity. In Fig. 4 we show the star formation histories and iron enrichment over time are very similar despite the differences in parameters. The NS data are equally well fitted using the exponential star formation history which matches better the constraints from the GALAH DR3 and APOGEE DR16 data. Therefore, we conclude that the model parameter differences in Table 1 lead to moderate changes in the shapes of the tracks. More important is the variation in the abundances ( Y ) which alters the slope of the tracks. From Table 2 we see that all models produce ∼ 0.0015 per solar mass of star formation, consistent with the prior of 0.0017 . On the other hand, the results for the iron contribution of Type II are slightly more varied with, in general, a larger contribution of ∼ 0.0025 per solar mass of star formation compared to the prior of around ∼ 0.00075 . This is driven by the shape of the metallicity distributions which may be subject to some selection effects, and is also sensitive to the specific choice of star formation history. However, it could also point towards the initial mass function of the Gaia Sausage producing more supernovae per generation. The production of the Type II supernovae corresponds to the observed plateaus in each dataset, and the differences between different datasets can largely be attributed to the observed discrepancies between common stars (due in part to NLTE effects). The Type Ia production of elements is low for all datasets and elements. Mg is consistent with the prior, demonstrating our models are sufficiently flexible to reproduce the Mg trends without invoking significant Mg production. For all datasets we see Ca production is greater than Si production (relative to solar) but the different dataset constraints are not consistent within the errors. This demonstrates the challenge of measuring the significant depletion of these elemental ratios.
We now focus on the iron-peak elemental abundances produced by the two channels. For all three datasets the three iron peak elements       Cu and Zn are also produced predominantly in supernovae but with very weak contributions from AGB stars (e.g. Kobayashi et al. 2020b). Additionally, as an odd-element, Cu production (particularly from Type II supernovae) appears sensitive to metallicity (Kobayashi et al. 2020b). The model fits do not completely capture the trends in the GALAH data, possibly for this reason, although for Zn in particular the scatter is large. Despite these caveats, our results indicate a typical [Cu/Fe] from Type Ia of ∼ −0.5 dex. The downwards trend in [Zn/Fe] from the NS data gives [Zn/Fe] ∼ −1 dex, whilst the GALAH DR3 produces a result around solar (although the results are marginally consistent within the errors).
As a check of the robustness of the results, we also considered the exponential star formation law solution from Weinberg et al. (2017) and found the model fits and the conclusions regarding the inferred yields very similar (see Table B1 for a comparison for the NS dataset).

DISCUSSION
We now discuss the constraints derived from the chemical evolution models of the Gaia Sausage with comparison to theoretical calculations of Type Ia supernovae yields and similar constraints for dwarf spheroidal galaxy data.
[Cr/Fe] Ia = + 0.03 +0.03    Kobayashi et al. (2020a), whilst the black pdf shows the combined result. The grey pdf shows the combined result assuming the models of  which adopt a different initial composition. The arrow shows the 95 per cent lower bound using the models of Seitenzahl et al. (2013a) and Gronow et al. (2021a,b).

Theoretical models
As discussed in the Introduction, we consider three main categories of theoretical Type Ia supernovae explosion calculation: deflagrationto-detonation transition, pure deflagration and double-detonation models. All considered models (bar one) follow a single white dwarf system which is triggered to explode in a variety of ways. We evaluate the theoretical yields for a SN Ia with metallicity = 10 −1.2 by linearly interpolating in log 10 ( ) or if = 10 −1.2 is not covered by the grids we use the lowest available metallicity model. Only the pure deflagration WDD2 model of , the pure deflagration model of Fink et al. (2014) and the violent merger model of Pakmor et al. (2012) are only available at solar metallicity.

Deflagration-to-detonation transition models
Deflagration-to-detonation transition (DDT) models are the favoured types of models for Ch SNe Ia. A sub-sonic deflagration phase is introduced in a carbon-oxygen white dwarf which initially burns at high density producing iron-group elements. This causes the star to expand producing lower density regions containing unburnt fuel.
After ∼ 1 s the transition occurs to a super-sonic detonation phase which burns the remaining fuel predominantly producing intermediate mass elements (such as Si) due to the now lower densities. The deflagration flame front is highly textured in this scenario so multidimensional models are essential. We consider three different sets of models. Seitenzahl et al. (2013a, S13) present yields from 3D DDT models using a varying number of ignition sites and four metallicities ( = , 0.5 , 0.1 , 0.01 ). We use the benchmark models with 100 ignition sites and a central density of 2.9 × 10 9 g cm −3 which produce ∼ 0.6 of 56 Ni. The strength of the deflagration phase increases with the number of ignition sites, which decreases the production of Fe, Cu and Zn but increases the production of intermediate mass elements due to increased energy injection and expansion and increases the production of Cr, Mn and Ni. Leung & Nomoto (2018, LN18) ran a series of 2D DDT models with a range of central densities, metallicities, C/O ratios and ignition kernels. We consider their benchmark models with central density 3×10 9 g cm −3 at = 0, 0.1, 0.5, 1, 2, 3, 5 which produce 0.63 of 56 Ni, and their updated WDD2 model (Iwamoto et al. 1999) which we denote LN18WDD2. Kobayashi et al. (2020a) updated the models of  using a more realistic solar-scaled initial composition (elements lighter than Ne are presumed to be converted into Ne during normal H and He burning), again varying both the initial mass and metallicity ( = 0 to = 0.04). As in , the Ch model with a central density of 3 × 10 9 g cm −3 produced 0.63 of 56 Ni. Lower masses produce more Fe and less Si, Ca, Cr, Mn, Ni and Zn. We consider all four mass models (1.30, 1.33, 1.37, 1.38 ). We denote the models as K20DDT_M*. The lower of these masses can be considered as sub-Ch models. However, the difference with the double-detonation scenario (see below) is that central carbon ignition does not initiate a detonation due to the high electron-degeneracy pressure, but rather a deflagration. Also, in these cases, the physical trigger for the explosion isn't clear. Prior to ignition, a near-Ch WD undergoes a simmering period where low-level carbon burning slowly increases the neutron fraction. This is equivalent to raising the metallicity of the WD, such that below a metallicity floor of / ≈ 2/3 (Piro & Bildsten 2008). The models of Kobayashi et al. (2020a) and Seitenzahl et al. (2013a) have relatively weak metallicity dependence on the Mn production so we ignore this detail.

Pure deflagration models
Pure deflagrations (Def) have been proposed as scenarios to explain Type Iax supernovae (spectroscopically similar to Type Ia but with lower peak brightness, Foley et al. 2013). The deflagration does not completely destroy the star and they primarily produce intermediate-mass elements with lower iron yields and an overabundance of neutron-rich species (e.g. 58 Ni). Their contribution to galactic chemical evolution is believed to be quite small (Kobayashi et al. 2020b). Fink et al. (2014, F14) ran 3D pure deflagration models using a similar setup to Seitenzahl et al. (2013a). We select the model with 100 detonation sites. Mn production has a weak dependence (∼ 0.2 dex variation) on the number of ignition sites as shown in de los Reyes et al. (2020). We also take the 2D pure deflagration models from Leung & Nomoto (2018, LN18Def).  have rerun the classic W7 model of Iwamoto et al. (1999) using updated electron capture rates and run a series of new models with varying central density. We take the model with a central density of 10 9 g cm −3 (LN18W7). Higher central densities lead to higher Mn yields.

Double-detonation models
Double-detonation (DD) models are the favoured scenario for sub-Ch SNe Ia. In this scenario, He is accreted onto the surface of the CO WD from a non-degenerate He-star, a companion He WD or even another CO WD as they are expected to have a small surface He layer. The He shell ignites and detonates at the base of the layer when it becomes sufficiently massive or if driven dynamically through the violent merger of two white dwarfs (Pakmor et al. 2012). This initial detonation triggers a subsequent detonation in the C/O core. There are two general classes of these controlled detonation models: those with and without a He shell. The He shell is required to initiate the first detonation but as observations suggest only a low mass He shell is required , several groups have run pure detonation simulations without a He shell (e.g. Sim et al. 2010 Gronow et al. (2021a) for example, although of low mass, the He shell is important to the explosion mechanism and subsequent yields of the model. Leung & Nomoto (2020) ran a series of 2D double-detonation simulations varying the core mass, the He shell mass, metallicity and initial He detonation mechanism. We consider their benchmark model with core mass = 1.1 , a 0.1 He shell and detonated with a bubble 50km above the coreenvelope interface (110-100-x-50) which produces 0.62 of 56 Ni. We also consider the benchmark models with a spherical detonation which have = 1.0 and a 0.05 He shell (100-050-x-S50). Differences in the detonation geometry are largely characterised by differences in the abundances of light iron-peak elements (e.g V and Cr) with aspherical detonations producing higher abundances of these elements (Leung & Nomoto 2020;Palla 2021, although note that the two inspected models differ in their He shell mass which also affects these elements). As with the DDT models, the models of Leung & Nomoto (2020) were updated with a more realistic initial mix of heavy elements by Kobayashi et al. (2020a) who ran models at a series of core masses and metallicities ( = 0 to = 0.04) with 0.05 He shell and a spherical detonation pattern. We use the 1, 1.1 and 1.2 total mass models, ignoring the 0.9 model as it produces a low quantity of Fe. The 1 model produces 0.63 of 56 Ni. Very recently, Gronow et al. (2021a,b, G21) have presented a series of 3D double-detonation models investigating a range of core masses and He shell masses, and post-processed for a range of metallicities ( = 0.01 , 0.1 , , 3 ). We use the models with 0.9 and 1 core mass and 0.05 He shell which produce around 0.6 of 56 Ni, and ignore the 0.8 model as the iron production is low. Despite similar core and shell masses, the Gronow et al. (2021a,b) model yields differ significantly from those of Kobayashi et al. (2020a) possibly related to the core-shell mixing in Gronow et al. (2021a,b), the difference in the dimensionality of the models, or the difference in the nuclear network size (7 isotopes in Kobayashi et al. 2020acompared to 35 in Gronow et al. 2021a as for the low density regions found in sub-Ch WDs some burning pathways are neglected by smaller networks (Shen et al. 2018a).
The described bare and double-detonation models all follow the explosion of a single white dwarf presuming some initial configuration from the result of an ongoing merger event. Simulations of the violent mergers of two white dwarfs (Pakmor et al. 2012) have been considered but there are more limited available nucleosynthetic yields for this case due to the increased complexity of the simulations. A set of four violent merger models from Pakmor et al. (2010Pakmor et al. ( , 2012 and Kromer et al. (2013Kromer et al. ( , 2016

Comparison with observations
Fig . 5 shows the Type Ia abundance constraints from the Gaia Sausage datasets with the theoretical yields from the different models. We show the five iron-peak elements in our modelling (Cr, Mn, Ni, Cu and Zn). Focusing on Mn and Ni, we see that all models lie along an anti-correlation in [Mn/Fe] Ia vs. [Ni/Fe] Ia as highlighted by Kobayashi et al. (2020a). As discussed in the Introduction, these elements are particularly sensitive to the density and hence mass of the model. Pure deflagration models sit at the high abundance end of the trend along with the DDT models. The lower mass DDT models of Kobayashi et al. (2020a) sit further down the trend with the sub-Ch models as they too can be considered as sub-Ch models. The violent merger model of Pakmor et al. (2012) sits part way between the two groups of models. It is clear our derived constraints are able to distinguish between the different scenarios. Based on this panel alone, all datasets are consistent with either (i) the 1 G21 model, (ii) a mixture of 'normal' DDT (40%) and sub-Ch (60%) Kobayashi et al. (2020a) models (as shown in the plot), (iii) the violent merger model of Pakmor et al. (2012) or (iv) a low mass 1.33 DDT model (which is perhaps not well physically motivated). The distribution of the models in Cr abundances is not so easy to understand and is perhaps more reflective of the ignition mechanism rather than its structure (Palla 2021). There are DDT models and sub-Ch models consistent with the solar/sub-solar measured abundances.
The mixture model matches the NS well and as discussed these are systematically higher than the corresponding GALAH DR3 and APOGEE DR16 models. Finally, Cu and Zn are heavier iron-peak elements largely produced in the detonation phase (e.g. Kobayashi et al. 2020a). We observe that deflagration models are inconsistent with the constraints. Also, despite its success in reproducing [Mn/Fe] and [Ni/Fe], the violent merger model underproduces both of these elements. sub-Ch models tend to produce more Cu than DDT models producing better consistency with the data. The Zn measurements of essentially all DDT and sub-Ch models are consistent with the data.
In Fig. 6 Fig. 5 with only the two benchmark Ch and sub-Ch models of Kobayashi et al. (2020a) shown. We also display the sequence of linear combinations of these two models. The APOGEE DR16 constraints agree very well with a mixture of 60 per cent sub-Ch SNe Ia. Both the NS and GALAH DR3 are less conclusive in this 2D projection although they are consistent at the 2 level with the APOGEE result.  Fig. 6 we display the inferred sub-Ch percentage based on the Mn measurements alone. GALAH DR3 yields slightly higher percentages of ∼ 65 per cent whilst NS and APOGEE DR16 are more consistent with ∼ 55 per cent. This is despite there being minimal differences in the Mn abundances from common stars so the difference must be driven by how the distribution of all the abundances constrains the SNe Ia contribution. Combining the constraints gives (57±2) per cent sub-Ch . Adopting instead the Kobayashi et al. (2020a) 1.1 or 1.2 total mass sub-Ch models yields essentially identical constraints of 56 and 57 per cent respectively so any mass distribution of sub-Ch WDs would produce a similar constraint.
If instead we adopt the models of  (which include only 22 Ne for the metallicity rather than a scaled-solar composition and so are perhaps less realistic) we yield ∼ 52 per cent sub-Ch using the 1.1 core mass sub-Ch model and ∼ 40 per cent using the spherical detonation 1.0 model. The more recent metallicity-dependent 3D sub-Ch models from Gronow et al. (2021a,b) in general produce significantly higher [Mn/Fe] than the Kobayashi et al. (2020a) models. The 1 core mass model implies essentially 100 per cent sub-Ch supernovae ( 87 per cent sub-Ch SNe Ia at 95 per cent confidence), whilst the 1.1 core mass model requires (67 ± 3)per cent sub-Ch (if combined with the Seitenzahl et al. 2013a, models) and the 0.9 produces too high [Mn/Fe] to be consistent with the data. Although our inference is still quite uncertain as it depends on the specific models we adopt, the general conclusion of a significant fraction of sub-Ch appears robust.

Non-LTE effects
Mn abundance measurements are known to be sensitive to the assumption of local thermal equilibrium (LTE, Bergemann & Gehren 2008;Bergemann et al. 2019). As our conclusions regarding the SNe Ia enrichment channel depend on Mn, it is worth discussing the status of non-LTE (NLTE) corrections for Mn. In general, the difference between 1D NLTE Mn abundances and 1D LTE Mn abundances (Δ NLTE ) is positive, increases for lower metallicity stars and is typically larger for giant stars than dwarf stars, although the differences are line dependent with resonance lines more sensitive to NLTE effects. The metallicity dependence has the effect of steepening correlations in [ Chieffi & Limongi (2004) or the fast rotating models of . The GALAH DR3 abundances already include the 1D NLTE corrections for Mn yet yield a higher [Mn/Fe] Ia than the corrected NS data. It should also be noted that similar trends in [Mn/Fe] vs. [Fe/H] for the NS dataset are seen in both the Gaia Sausage and the higher-in-situ or Splash (Belokurov et al. 2020) stars. As noted in Nissen & Schuster (2011), this does suggest some part of the trend is due to NLTE effects although clearly the two groups are still distinct in this projection.
The 1D NLTE corrections do not tell the full story, as 3D corrections are also important for Mn. However, 3D calculations are expensive so much more limited information is available. Bergemann et al. (2019) compute the difference between 1D and 3D NLTE calculations for a set of four models (giant and dwarf at [Fe/H] = −1, −2 dex) finding, in general, 3D NLTE Mn abundances are larger than 1D NLTE. From Bergemann et al. (2019) the lines around 4800Å and 6015Å in dwarf stars have 3D NLTE larger than 1D by 0.06 dex and 0.08 dex respectively. Applying this correction to our dataset will shift the measurement of [Mn/Fe] Ia up by ∼ 0.05 dex which would lower the contribution of sub-Ch to around 50 per cent in line with the Milky Way result from Seitenzahl et al. (2013b).
GALAH uses the lines at 4754.04Å and 4761.51Å. which have 3D NLTE larger than 1D NLTE by approximately 0.02 − 0.05 dex and 0.08−0.09 dex for the −2 and −1 dex stars respectively. Interestingly, for Mn I in the infra-red, Bergemann et al. (2019) report that 3D NLTE should be consistently higher than 1D NLTE by 0.15 dex for all analysed stars. However, as shown in Appendix A, no difference between the 1D LTE APOGEE and the 1D NLTE GALAH Mn abundances is observed, whilst the results in Bergemann et al. (2019) suggest a possible discrepancy of ∼ 0.15 dex at −1 dex and ∼ 0.2 dex at −2 dex (APOGEE being lower).
Although there is clear evidence that 1D LTE Mn abundances are too low at low metallicity, NLTE corrections depend sensitively on the details of the modelling and input data. Their dependence on effective temperature and surface gravity can be tested by requiring globular cluster stars have similar abundances. For example, Kirby et al. (2018) applied Cr and Co NLTE corrections to LTE measurements for globular cluster members but found the corrections introduced systematic trends which were not present in the LTE calculations. However, Kovalev et al. (2019) show how NLTE measurements of Fe, Mg, and Ti globular cluster stars do not increase the intra-cluster scatter. These studies suggest that it is far from trivial to apply corrections to the LTE abundances.

Comparison with Milky Way dwarf spheroidal galaxies
In a similar vein to our work, Kirby et al. (2019) and de los Reyes et al. (2020) have estimated the contributions of different Type Ia supernovae to the five dwarf spheroidal (dSph) galaxies: Sculptor, Leo II, Draco, Sextans and Ursa Minor (UMi). These isolated lowmetallicity systems provide an alternative way to inspect chemical evolution of entire galaxies. Kirby et al. (2019) fitted linear models with a break to [X/Fe] vs. [Fe/H] data with X=(Mg, Si, Ca, Cr, Fe, Co, Ni) from which the yields of the supernovae could be computed. The advantage of this procedure is that the metallicity dependence of the supernovae yields can be more simply captured. However, unlike our procedure where there is a common set of evolutionary parameters (e.g. star formation, outflow), the fits for each abundance are independent. Based on the sub-solar [Ni/Fe] of Type Ia supernovae, Kirby et al. (2019) concluded there must be a significant contribution from sub-Chandrasekhar mass Type Ia supernovae in these systems. de los Reyes et al. (2020) measured Mn abundances for Sculptor and fitted the same models as Kirby et al. (2019) to find [Mn/Fe] Ia = (−0.30±0.03) dex. These authors also suggest that systems with early concentrated star formation (such as Sculptor) have lower [Mn/Fe] Ia , and hence higher fractions of sub-Chandrasekhar mass Type Ia supernovae, than systems with more extended star formation. This suggests that perhaps the dominant channel of Type Ia enrichment depends on delay time with Chandrasekhar mass systems dominating at late times.
In Fig. 7 we show our constraints for the Gaia Sausage alongside the constraints from the dSph models of Kirby et al. (2019) Kobayashi et al. (2020a, K20 Ch DDT, light green), and the sub-Ch double-detonation metallicity-dependent yields from Gronow et al. (2021a,b, G21 1 sub-Ch, thick light blue, and other masses in thinner lines) and Kobayashi et al. (2020a, K21 1 sub-Ch, dark blue). We also display in grey the Type II yields from Kobayashi et al. (2006) used in the Kobayashi et al. (2020b) models. The purple dashed line is the proposed metallicity dependence of the Type Ia Mn yields from Cescutti et al. (2008). to the details of the explosion initialization and geometry, and so is more an indicator of different simulation setups. The elements (Si and Ca) are sub-solar for all dwarf spheroidals as well as for the Gaia Sausage (again except for UMi which has a high [Si/Fe]). As with Cr, this is largely consistent with any set of theoretical SNe Ia yields.

Type Ia yields across a range of systems
The comparison with the dwarf spheroidal data in the previous section demonstrates that both the Gaia Sausage and dSphs have significant contribution from sub-Ch SNe Ia. However, there is also the suggestion of metallicity dependence in the Type Ia yields with Sculptor having a ∼ 0.1 dex lower [Mn/Fe] yield than Gaia Sausage, despite both systems having similar early bursts of star formation. To investigate the potential metallicity dependence of the Type Ia yields, we fit the simple analytic models to data from a wider range of systems. We consider APOGEE DR16 data for the Milky Way bulge (selected as giants within 5 deg of the Galactic centre), the Large Magellanic Cloud (LMC, fainter than = 12 within 15 deg of the centre of the LMC and with a line-of-sight velocity cut), the Small Magellanic Cloud (SMC, fainter than = 12 within 10 deg of the centre of the SMC and with a line-of-sight velocity cut) and the Sagittarius dSph (Sgr, fainter than = 12 within 5 deg of the centre of the Sgr, with a line-of-sight velocity cut and removing highin-situ stars). We use similar quality cuts on the abundances to those in Section 2.3. Furthermore we take the dSph data from Kirby et al. (2018) combined with the Mn measurements from de los Reyes et al. (2020) for Ursa Minor, Canes Venatici I, Sculptor, Leo I and Fornax.
We fit the analytic chemical evolution models to Mg, Si, Ca, Cr, Mn and Ni adopting identical priors to those in Section 3 except for max and sfh which are chosen on the basis of star formation history measurements (de Boer et al. 2012a(de Boer et al. ,b, 2015Barbuy et al. 2018;Harris & Zaritsky 2009Gallart et al. 1999;Carrera et al. 2002). We display the results of the model fits in Fig. 8 along with the theoretical yields from the Ch models of Seitenzahl et al. (2013a) and Kobayashi et al. (2020a), and the sub-Ch 1 models from Kobayashi et al. (2020a) and Gronow et al. (2021a,b). The full set of model constraints is given in Table B2 and the model fits for the most massive systems are given in Fig. B1. We see that for all systems of lower metallicity than the SMC there is a weak gradient of [Mn/Fe] with metallicity. This gradient may be consistent with NLTE effects. Based on the discussion of Section 4.2, we have rerun the modelling with a [Mn/Fe] NLTE correction of −0.1[Fe/H], although this is possibly an overestimate of the effect. The shift in the results are shown by the small arrows in Fig. 8. It is clear NLTE effects could flatten the gradient at the metal-poor end. However, more robust is the difference in the yield required to explain the higher metallicity systems (Milky Way bulge, LMC, Sgr) compared to the lower metallicity systems (e.g. Gaia Sausage). The lower metallicity systems cover a range of star formation histories (e.g. the early star formation of Gaia Sausage and Sculptor vs. the more extended star formation history from Fornax) as do the high metallicity systems (early star formation in the Milky Way bulge vs. extended, albeit bursty, star formation in the LMC). This suggests that although evolutionary differences may give rise to some variation at fixed metallicity (see later and de los Reyes et al. 2020), the main factor governing the Type Ia [Mn/Fe] yields is metallicity dependence.

Modelling variants
Before discussing the implications of our results, we will detail some caveats and model variants we have explored to test the robustness of our derived constraints.
Complex star-formation histories: Several of these systems have more complicated star formation histories than can perhaps be captured by the linear-exponential star formation law of Weinberg et al. (2017). For example, the Magellanic Clouds are known to have relatively extended, bursty star formation histories, possibly linked to their mutual interaction (Harris & Zaritsky 2009;Nidever et al. 2020), or the Milky Way bulge may have gone through bursts and quenching episodes (Lian et al. 2020). The models of Weinberg et al. (2017, see section 5.6) permit more complex star formation histories as linear sums of the analytic solutions (linear-exponential and exponential). Due to the linearity of the chemical evolution equations, the expressions for the mass of each element are just linear sums of the solutions for each component. We have considered more complex models of two linear-exponential terms with relative weight , with one star formation law beginning at = 0 and the other offset to begin at = 0 , a free parameter. Each component has its own separate sfh . Despite the added complexity, the constraints on the yields, particularly Mn and Ni, are very similar to those obtained with the single component model. However, in general these models are much harder to sample from. Furthermore, removing the constraint that follows the star-formation law in the modelling produces significantly poorer matches to the metallicity distributions but the conclusions regarding [Mn/Fe] Ia and [Ni/Fe] Ia are unchanged. This suggests, despite the simplicity of the adopted star formation prescription and despite the sampling of the systems possibly being subject to selection effects that reshape the metallicity distributions, our results regarding the Mn and Ni abundances are robust to reasonable variations in the model star formation history parametrization. Note we are not arguing that the average Type Ia yields for each system do not correlate with the star formation history (see later discussion), rather our recovery of the average yields for each system is insensitive to modelling variations in the star formation history.
Metallicity-independent yields: A further caveat is that several of the studied systems (in particular the Milky Way bulge) span a large range in metallicity such that the metallicity-independent yields prescription of the analytic models is possibly questionable. For example, in the case of the Milky Way bulge, the simplistic models fail to perfectly capture the morphology of the nearly linearly rising [Mn/Fe] vs. [Fe/H]. This is indicative of the need for metallicitydependent yields for this system (Gronow et al. 2021b). For both of these reasons, our measurements should be considered as average SNe Ia yields corresponding approximately to the metallicity at which the bulk of the stars were formed.
Metallicity-dependent Type II yields may also play a role and, as discussed, these are not well incorporated in our modelling framework. In Fig. 8, we show the weak metallicity dependence of Type II yields from the models of Kobayashi et al. (2006Kobayashi et al. ( , 2020b where the [Mn/Fe] difference between [Fe/H] = −1 and 0 dex is ∼ 0.1 dex. Other Type II supernovae yields give ∼ 0.13 dex (Chieffi & Limongi 2004), ∼ 0.27 dex (Woosley & Weaver 1995;Pignatari et al. 2016;Ritter et al. 2018) or as large as ∼ 0.6 dex Prantzos et al. 2018) for this quantity. Such a large variation could in principle explain the [Mn/Fe] trend for the Milky Way bulge. However, the maximum [Mn/Fe] II yield at solar metallicity for any model is ∼ 0.1 dex Prantzos et al. 2018). Assuming the same relative contribution to stars of solar metallicity from Type Ia and Type II yields as found in the default model fit, [Mn/Fe] Ia would be lowered from 0.28 dex to ∼ 0.07 dex in this case. Even in this extreme case, super-solar Type Ia yields are required for high metallicity systems. As a further test, we incorporated a form of metallicity-dependent Type II yields in our models by including a different parameter for each element. As referenced in Section 3, a Type II yield, II , which depends linearly on the current mass fraction of element relative to solar, , is equivalent to changing the depletion timescale dep for this element. We included a unit normal prior on the gradient of II / II ( = 0) with respect to (consistent with the spread from the theoretical yields). Furthermore, we enforced the yield for a mass fraction 10 times solar to be positive. This produced no difference in [Mn/Fe] Ia for systems with [Fe/H] < −1 dex (including the Gaia Sausage). For the LMC, Sgr and the Milky Way bulge, [Mn/Fe] Ia was significantly lower (−0.09, −0.06 and −0.18 dex respectively) but, in the cases of the Milky Way bulge and the LMC, only by invoking unphysically steep Type II metallicity gradients such that [Mn/Fe] II ∼ 0.3 − 0.5 dex at solar metallicity. Thus it seems that although metallicity-dependent Type II yields could contribute to the interpretation of the data, they will only weakly flatten the gradient seen in Fig. 8 and the observation that the Milky Way bulge data require SNe Ia Mn yields inconsistent with the lower metallicity systems (e.g. Gaia Sausage) is robust.
Tensions with priors: From inspecting Table B2, there are a few parameter constraints which are in tension with our adopted priors. In particular, log 10 Fe for the Type II and Type Ia channels are in some instances much higher than the priors (e.g. the Milky Way bulge and the LMC). Running models with much tighter priors on these quantities (0.004 dex) produces very similar results for [X/Fe] Ia,II and by eye a similar quality of fit to the abundance trends. This is typically achieved through lowering D such that it pushes against the 40 Myr lower bound and is consistent with very prompt SNe Ia exploding soon after their progenitor star becomes a WD. Another quirk of our modelling is the low constraint for the Milky Way Bulge (consistent with zero) indicating very low outflow. Allowing to be negative produces a constraint of = (−0.15 ± 0.12), which, as discussed previously, we could physically interpret as a Type II iron yield increasing with metallicity. Relaxing the positivity constraint on does not affect the inferred abundances.
We conclude that, despite some deficiencies in our modelling approach, the main results that we focus on are robust to reasonable variations of the model. This is further corroborated by the results of our models for Sculptor and Ursa Minor compared to the models of Kirby et al. (2019) andde los Reyes et al. (2020) as shown in Fig. 7, demonstrating the good agreement between the two modelling frameworks. We now discuss the meaning of these results.  Cescutti et al. 2008;Weinberg et al. 2019). There are two possible routes for how such a metallicity dependence could arise. If we simply consider two different Type Ia channels, it could either be that the individual [Mn/Fe] yields from the two channels are metallicity dependent, or the relative contribution from the two channels is metallicity dependent. Both the models of Kobayashi et al. (2020a) and Gronow et al. (2021b) demonstrate that metallicity dependence of [Mn/Fe] Ia can be significant for sub-Ch models above [Fe/H] −1 dex which is driven by the neutron excess for more metal-rich models altering the relative abundance of iron-group elements from incomplete silicon burning (typically during He detonation). On the other hand, the [Mn/Fe] Ia for the Ch DDT models (Seitenzahl et al. 2013a;Kobayashi et al. 2020a) are much less metallicity dependent as in this case 55 Co is typically produced in nuclear statistical equilibrium so is sensitive more to the neutron fraction from electron capture and less to the initial composition. Additionally, pre-explosion simmering (Piro & Bildsten 2008) should make the pre-explosion neutron abundance more insensitive to the initial metallicity. As highlighted by Gronow et al. (2021b), the rising trend of [Mn/Fe] vs. [Fe/H] for the Milky Way then requires significant metallicity-dependent sub-Ch contributions. On the other hand, there are theoretical arguments that the Ch channel becomes more significant for higher metallicity systems as winds are required to grow WDs stably to Ch which are more effective at higher metallicity (Kobayashi & Nomoto 2009). In the recent Milky Way models of Kobayashi et al. (2020a, Figure 14), Ch SNe Ia only begin contributing after = 4 Gyr when the Galaxy is enriched to around [Fe/H] = −1 dex. However, we must also consider the birth and evolutionary properties of binary systems with metallicity. Recently, it has become clear that binarity is more common at low metallicities (Moe et al. 2019) possibly due to increased fragmentation at lower metallicity caused by lower dust opacities and enhanced gas cooling through dust coupling. Whether such an effect could differentially affect the two channels is not clear. Additionally, as highlighted by Ruiter (2020), mass-loss is more effective at higher metallicities. At very low metallicity ([Fe/H] ≈ −2 dex) low mass-loss can produce fewer single-degenerate channel SNe Ia as ONe WDs are formed preferentially. At higher metallicities, increased mass loss over the stellar lifetime can change the evolutionary pathway of a binary, make the Chandrasekhar mass harder to reach or prevent contact and subsequent stable mass transfer (as highlighted above). These three effects can significantly alter the predicted rates of different channels.

Star-formation-history-dependent Type Ia yields
Finally, we consider the possibility that some of the variation we see is due to different star formation histories in the systems. There is a well-studied trend for more massive galaxies to host intrinsically fainter SNe Ia events (e.g. Howell et al. 2009), which could be related to either age or metallicity effects. One plausible explanation is that the more massive systems host longer delay time SNe Ia, which could arise from lower mass progenitor systems in the sub-Ch scenario (Howell 2001) although disentangling age and metallicity effects is non-trivial. This observation would appear at odds with the theory of increased dominance of Ch systems in higher metallicity environments, although crucially the host galaxy correlations are against gas-phase metallicity which could be quite different from the progenitor metallicity depending on the star formation history. In the sub-Ch scenario there is the additional parameter of the progenitor mass. As noted previously, the models of Kobayashi et al. (2020a) show weak mass dependence at fixed metallicity, whilst there is more significant variation with mass in the Gronow et al. (2021a,b) models. In Fig. 8 we show the 0.8, 0.9 and 1.1 core mass models from Gronow et al. (2021a,b) as thin lines. At fixed metallicity, the [Mn/Fe] yield increases with decreasing core mass. This is mostly driven by the correlation of 56 Ni mass with core mass, as the mass of Mn is linked to the He detonation at the lower density core/shell interface and is more constant with core mass. We observe that systems such as the LMC or Sgr with perhaps more extended star formation histories may be entirely consistent with 100 per cent sub-Ch models but of lower mass. Such a picture would be in line with the supernovae population studies. It appears from [Mn/Fe] alone that breaking the degeneracy between core mass and metallicity is not trivial. Flörs et al. (2020) have used spectroscopic observations of the ratio 58 Ni to 56 Ni to infer the mass of individual SNe Ia. As a neutronrich species, the 58 Ni abundance depends on neutron fraction so is higher for higher mass, higher density WDs (Timmes et al. 2003). This difference in behaviour with mass for [Ni/Fe] compared to [Mn/Fe] allows breaking of the degeneracy when using only [Mn/Fe]. In Fig. 9 we display the joint constraints on Type Ia [Mn/Fe] and [Ni/Fe] yields for the six best constrained systems we have analysed alongside the sub-Ch models from Gronow et al. (2021a,b). We observe that, despite higher Type Ia [Mn/Fe] abundance in the LMC and Sgr compared to the Gaia Sausage, the [Ni/Fe] abundance in these systems is lower than that for the Gaia Sausage and appears more consistent with 0.8 or 0.9 core mass models, in line with our expectation from their more extended star formation history. We have chosen the 0.05 He shell mass models from Gronow et al. (2021a,b) but similar conclusions hold for other He shell mass models although there is more variance in the abundances for lower (0.8, 0.9 ) core mass models with varying shell mass. To draw firmer conclusions requires a more detailed modelling of the star formation histories of the systems and the delay-time distributions with WD mass, but the combined constraints from [Mn/Fe] and [Ni/Fe] appear to provide a route to more strongly constrain the SNe Ia progenitor scenario and could even point towards sub-Ch dominance in all systems.
Based on the currently available theoretical models, the combination of results from the different systems suggest an increasing significance of Ch explosions with increasing metallicity, although the differences between different theoretical models hint there may be a model where the metallicity dependence is explained entirely by sub-Ch explosions, possibly with a distribution of masses related to the star formation history of the system. The plane of [Mn/Fe] vs.
[Ni/Fe] seems ideal to distinguish these scenarios.

CONCLUSIONS
The range of different SNe Ia scenarios are distinguished by different nucleosynthetic yields, which are in turn imprinted in the chemical abundances of stars. SNe Ia channels potentially vary with the mass, metallicity, initial-mass-function and star-formation-history of the host systems which could have profound consequences on the use of SNe Ia as cosmological probes. However, high-quality resolved stel-  Gronow et al. (2021a,b, black) and the Ch DDT models of Seitenzahl et al. (2013a, green). The yields are functions of both core mass and metallicity of the sub-Ch progenitor. Increased mass increases 56 Ni production (and hence Fe) whilst Mn remains constant (as it is formed in lower density regions at the core-shell interface), and increases the neutronization elevating the abundance of neutron-rich 58 Ni. lar spectroscopic abundances are available only for the Milky Way and several Local Group dwarf galaxies. Recently, it has been discovered that a substantial fraction of the Milky Way's halo consists of a population of kinematically and chemically distinct stars consistent with originating from a significant merger, dubbed the Gaia Sausage, early in the life of the Milky Way. This gives us a unique probe into a low metallicity, early-Universe galaxy using sets of bright local tracers.
We have taken three samples of Gaia Sausage stars from Nissen & Schuster (2010), GALAH DR3 (Buder et al. 2020) and APOGEE DR16 (Ahumada et al. 2020) for which high-dimensional chemical data are available (8+ abundances), and fitted their chemical-space distributions with the simple analytic chemical evolution models of Weinberg et al. (2017) to measure the abundance signatures of SNe Ia. We find the down-turning [ /Fe] sequence indicative of the onset of SNe Ia is accompanied by a relatively flat [Mn/Fe] and weakly declining [Ni/Fe] trend indicating low production in SNe Ia (∼ −0.15 dex and ∼ −0.3 dex respectively). Comparison with theoretical nucleosynthetic yields for SNe Ia exploded as Ch pure deflagrations, Ch deflagration-to-detonation transition and doubledetonation sub-Ch demonstrate such behaviour is only possible with a significant ( 60 per cent) contribution of sub-Ch systems, with some theoretical calculations implying near 100 per cent sub-Ch systems. This corroborates findings from dwarf galaxies of the Local Group.
Finally, we concluded our study with a combined analysis of several systems with varying star formation histories and metallicity distributions (Milky Way bulge, Magellanic clouds, Gaia Sausage, dwarf spheroidal galaxies). We found to reproduce the chemical enrichment of all of these systems requires a metallicity-dependent Type Ia [Mn/Fe] yield. Only sub-Ch models have significant metallicitydependent yields. This then indicates either (i) a significant contribution from sub-Ch SNe Ia in all systems, possibly with variation in the WD mass distribution with star formation history which could be elucidated using a combination of [Ni/Fe] and [Mn/Fe] measurements, or (ii) an enhancement of the Ch channel at higher metal-licity, possibly due to stabilising winds allowing white dwarf growth to the Chandrasekhar mass.
We close by reflecting on the implications of our work on using SNe Ia as cosmological probes. The understanding of the progenitors of SNe Ia is an important part of using these objects as standard candles. However, the success in their use has arisen from the ability to standardise the variety of observed light curves, even despite an understanding of what drives the variation. From the Ch perspective, the free parameter is perhaps the time a deflagration flame transitions into a detonation wave (Blondin et al. 2013), whilst in the sub-Ch picture, it is the mass of the WD. In both cases, however, the primary driver in light curve variation is then the 56 Ni production which gives rise to the width-luminosity relation (Scalzo et al. 2014). Beyond this, there is an intrinsic scatter of ∼ 0.15 mag in SNe Ia luminosities, the cause of which is not understood. If the residual depends on the progenitor scenario, and possibly more crucially the host galaxy properties, there is scope to improve the calibrations and remove potential systematics (e.g. Sullivan et al. 2010). For instance, several studies (e.g. Howell et al. 2009;Kelly et al. 2010;Scalzo et al. 2014) have demonstrated SNe Ia in less massive galaxies tend to be brighter (even after calibrating the light curves), reflecting possible age (Childress et al. 2014) or metallicity effects (Timmes et al. 2003) on the progenitors. Such correlations favour the sub-Ch channel where lower mass older WDs could explode at longer delay times producing lower luminosity SNe Ia (Howell 2001). Such studies are concerned with the properties of observed supernovae, whilst here we are only sensitive to the SNe Ia which enrich future star-forming gas, and furthermore, supernovae observations are unable to probe the gas metallicity when the progenitor was born. For these reasons, our results are complementary to SNe Ia population studies and taken together give a route into further understanding the SNe Ia progenitor properties with both metallicity and star formation history of the host.

APPENDIX A: COMMON STARS
For the analysis in this paper, it is important that the absolute abundance scale of each dataset is well determined, and that the stellar samples are free from systematic trends with stellar parameters (in particular, metallicity). As we have access to multiple datasets we are able to compare abundances for stars observed by two surveys. We initially compare the APOGEE DR16 and GALAH DR3 datasets as they have the largest overlap, and then we go on to compare these datasets with NS.

A1 APOGEE vs. GALAH
We take the same sample of GALAH DR3 as detailed in Section 2.2 without restricting to just Gaia Sausage stars, and crossmatch to APOGEE DR16 stars with no ASPCAP flags. This results in 479 stars. We fit a series of linear models to the difference in the Mg, Si, Ca, Cr, Mn, Ni and Cu abundances (only using unflagged abundances) as a function of the stellar parameters ( eff , log and [Fe/H]). Note that APOGEE Cu is known to be unreliable particularly at low metallicity (Jönsson et al. 2020) so we don't use it in the main body of the paper. We account for the uncertainties in the abundances but not in the stellar parameters (which on the whole are small). Importantly we account for the floor in the APOGEE [X/H] abundances below which abundances aren't reported. Excluding this effect biases the trends.
In Table A1 we show the results of fitting linear models to the abundance differences as a function of eff , log and [Fe/H] separately, and using all three together. The results of the fits against [Fe/H] are shown in Fig. A1. The results at the metal-rich end are mostly consistent with 0.1 dex difference. Jönsson et al. (2020) show how the majority of APOGEE abundances agree (within ∼ 0.05 dex) in the median with those obtained from optical spectra (using LTE). For Mg and Si we find significant trends with metallicity with APOGEE abundances larger than GALAH at the low metallicity end. Both of these elements have non-LTE corrections in GALAH (Amarsi et al. 2020). Ca also has non-LTE corrections in GALAH but we don't observe a trend, just an offset of −0.06 dex. Osorio et al. (2020) found that non-LTE corrections for Ca and Mg in the Sun, Arcturus and Procyon are smaller in the H-band than in the optical, and always 0.1 dex. For Cr and Cu, we find a significant trend with APOGEE overestimating relative to GALAH at the metal-poor end. However, Cu in APOGEE is known to be unreliable (Jönsson et al. 2020). For Mn we find essentially no bias between APOGEE and GALAH across the full metallicity range. This is despite the non-LTE corrections in GALAH. This gives confidence to the conclusions in the main body of the paper which use the Mn abundances. Finally, with Ni the GALAH analysis used different combinations of the 5847Å and 6586Å lines. We found different trends with respect to the APOGEE Ni abundances when restricting to subsets of GALAH abundances derived from the same combinations. In particular, we found a significant trend when using only 5847Å whilst no trend for only 6586Å. This suggests there is a systematic variation in the GALAH Ni abundances.

A2 APOGEE and GALAH vs. Nissen & Schuster
We find there are three stars in the NS sample also observed as part of GALAH DR3, and nine also observed in APOGEE DR16. In Fig. A2 we plot the difference in the reported abundances and report the inverse-variance-weighted mean difference. Despite the small number of datapoints, we find good correspondence between the two datasets with only [Cr/Fe] showing a significant offset (GALAH DR3 smaller than NS). Six of the inspected abundances have 1D non-LTE corrections in GALAH DR3 (Amarsi et al. 2020) whilst NS use an LTE analysis (except in the case of [Cu/Fe] where we have used the results from Yan et al. 2016).

APPENDIX B: AUXILIARY DATA TABLES
Here we provide additional data tables from out study. The results using the exponential star formation rate of Weinberg et al. (2017) applied to the NS data are shown in Table B1. The full set of constrained parameters for the Local Group systems are shown in Table B2. We show the corresponding models for the most massive five systems (Gaia Sausage, Sgr, SMC, LMC and the Milky Way bulge) in Fig. B1. This paper has been typeset from a T E X/L A T E X file prepared by the author. Table A1. Offset parameters between common stars observed by APOGEE DR16 and GALAH DR3. Models of the difference between APOGEE and GALAH (i.e. positive numbers mean APOGEE is larger than GALAH) have been fitted with the linear models of the form + eff ( eff /1000 K − 4.5) + log (log − 2.5) + [Fe/H] ( [Fe/H] + 1). The top set of fits correspond to a constant model 0 ( eff = log = [Fe/H] = 0) and three models where the gradient in only one parameter is fitted for (e.g. eff , eff correspond to the case where ( log = [Fe/H] = 0). The second section shows the results for a full multivariate fit. The final section shows the results for fitting a metallicity-dependent model to the Ni differences using the three different line combinations. The spectroscopic parameters ( eff , log , [Fe/H]) used are those of GALAH DR3 although they differ from APOGEE DR16 parameters by 3.6 K, −0.002 dex and −0.05 dex in the median.    Table B2. Type Ia and Type II supernovae abundances and evolutionary parameters for chemical evolution models fitted to a range of Local Group systems. We give the median metallicity of the stars used to model each system.