Exploring the diversity of double detonation explosions for type Ia supernovae: Effects of the post-explosion helium shell composition

The detonation of a helium shell on top of a carbon-oxygen white dwarf has been argued as a potential explosion mechanism for type Ia supernovae (SNe~Ia). The ash produced during helium shell burning can lead to light curves and spectra that are inconsistent with normal SNe~Ia, but may be viable for some objects showing a light curve bump within the days following explosion. We present a series of radiative transfer models designed to mimic predictions from double detonation explosion models. We consider a range of core and shell masses, and systematically explore multiple post-explosion compositions for the helium shell. We find that a variety of luminosities and timescales for early light curve bumps result from those models with shells containing $^{56}$Ni, $^{52}$Fe, or $^{48}$Cr. Comparing our models to SNe~Ia with light curve bumps, we find that these models can reproduce the shapes of almost all of the bumps observed, but only those objects with red colours around maximum light ($B-V \gtrsim 1$) are well matched throughout their evolution. Consistent with previous works, we also show that those models in which the shell does not contain iron-group elements provide good agreement with normal SNe~Ia of different luminosities from shortly after explosion up to maximum light. While our models do not amount to positive evidence in favour of the double detonation scenario, we show that provided the helium shell ash does not contain iron-group elements, it may be viable for a wide range of normal SNe~Ia.


INTRODUCTION
One of the most debated aspects of research on type Ia supernovae (SNe Ia) is whether multiple progenitor systems are needed to explain the entire population (see Livio & Mazzali 2018;Wang 2018;Jha et al. 2019;Soker 2019 for recent reviews of SNe Ia). Despite significant work throughout the years, the question remains whether SNe Ia primarily result from Chandrasekhar or sub-Chandrasekhar mass white dwarfs.
To trigger the detonation of a sub-Chandrasekhar mass white dwarf, early models invoked scenarios in which a massive helium shell ( 0.2 ) accumulates on the surface of the white dwarf (e.g. Livne 1990; Livne & Glasner 1991;Woosley & Weaver 1994). As the mass of the helium shell increases through accretion, the density and temperature at the base of the shell also increase. Eventually convective nuclear burning may develop and potentially transition to a detonation. Following ignition of the shell, a secondary detonation may be triggered in the core. This secondary detonation can be triggered in multiple ways (converging shock, edge-lit, or scissors mechanism ;Livne 1990;Livne & Glasner 1991;Moll & Woosley 2013;Gronow et al. 2020), however the end result is the same -complete disruption of the white dwarf. This is the so-called double detonation scenario. Within these models, most studies find that burning in the ★ E-mail: mrmagee.astro@gmail.com helium shell proceeds mostly to nuclear statistical equilibrium (NSE) -producing a large amount of 56 Ni and other iron-group elements (IGEs). Such a large mass of IGEs in the outer ejecta leads to significant line blanketing that generally does not agree with observations of SNe Ia (Hoeflich & Khokhlov 1996;Nugent et al. 1997).
Given the adverse impact of the helium shell ash on the light curves and spectra, there has been significant interest in minimising its effects. Neglecting any helium shell altogether, models invoking pure detonations of isolated, bare sub-Chandrasekhar mass white dwarfs have been shown to broadly reproduce the light curves and spectra of normal SNe Ia (Sim et al. 2010b;Shen et al. 2018;Goldstein & Kasen 2018). Such white dwarfs however, will not spontaneously detonate and therefore these explosions do not occur naturally. Alternatively, models with thin helium shells may also be a viable pathway to explain normal SNe Ia. Bildsten et al. (2007) showed that ignition within the helium shell can be achieved for much lower masses of ∼0.02 , but they did not not consider the possibility of core ignition following the initial helium shell detonation. Subsequent core ignition was shown to be robustly achieved by Fink et al. (2007), Fink et al. (2010), and Shen & Bildsten (2014) for high-mass white dwarfs ( 0.8 ). In spite of these lower shell masses, models presented by Kromer et al. (2010) and Gronow et al. (2020) remain inconsistent with the observed light curves and spectra of normal SNe Ia. Recently, Polin et al. (2019) presented a suite of double detonation models covering a range of core and shell masses (from 0.6 -1.2 and 0.01 -0.1 , respectively) and find that some models with thin helium shells do produce spectra that resemble normal SNe Ia.
In addition to producing strong line blanketing, the presence of IGEs in the helium shell ash has an important consequence for the light curves predicted by double detonation explosions. Noebauer et al. (2017) and Jiang et al. (2017) have shown that the production of short-lived radioactive isotopes ( 56 Ni, 52 Fe, and 48 Cr) in the shell results in a distinct bump in the early light curve (within approximately three days of explosion). Studies of samples of SNe Ia (e.g. Bianco et al. 2011;Olling et al. 2015;Papadogiannakis et al. 2019;Miller et al. 2020b) have shown that the evidence for clear bumps is relatively rare, but a few candidate objects have been proposed (e.g. Jiang et al. 2017;Hosseinzadeh et al. 2017;Li et al. 2019;De et al. 2019;Miller et al. 2020a).
Qualitatively similar bumps in the early light curves of SNe Ia are also suggested to be produced via different mechanisms, such as the presence of a 56 Ni excess in the outer ejecta , interaction with a companion star (Kasen 2010), or interaction with circumstellar material (CSM; Piro & Morozova 2016). An excess of 56 Ni in the outer ejecta may result from plumes of burned ash rising to the surface of the white dwarf during explosion. As the 56 Ni decays to 56 Co, the radiation produced is able to quickly escape from the ejecta surface and results in a light curve bump. The luminosity and duration of the bump depends on both the mass and distribution of 56 Ni. In the interaction scenarios, a light curve bump may be produced due to cooling of the shocked ejecta following the interaction. For companion interaction, the bump is affected by the nature of the companion, with more evolved stars producing stronger interaction signatures. In both cases, the mass and extent of the interacting material will also determine the luminosity and duration of the bump. Maeda et al. (2018) specifically investigate the different early light curve signatures predicted by the double-detonation scenario and interaction. The models presented by Maeda et al. (2018) show significant overlap between these two scenarios, in terms of the duration and luminosity of the bump, but the double detonation in general produces somewhat redder colours. Maeda et al. (2018) show that this is at least partially due to the specific IGEs present in the shell.
Aside from the mass of the helium shell, it has also been suggested that its composition can play an important role during nuclear burning, and can dramatically affect the post-explosion observable properties. Kromer et al. (2010) presented a model in which the helium shell was polluted by carbon (34% by mass) and found that burning within the shell did not proceed to NSE, but instead stalled earlier in the -chain. In this case, the lack of IGE in the shell produced light curves and spectra that are generally consistent with normal SNe Ia. In addition, Townsley et al. (2019) recently showed that the inclusion of other isotopes, besides carbon, can also dramatically affect the post-explosion composition of the shell and produce observables comparable to normal SNe Ia. Therefore, there is considerable scope for variation in the burning products produced in the shell.
In this work, we present radiative transfer simulations exploring a range of ejecta models that are designed to parameterise and broadly mimic predictions from double-detonation explosion models. We perform the first large-scale exploration of various compositions for the helium shell following explosion, and determine the range of models that do and do not reproduce observations of SNe Ia. Although different helium shell compositions in parameterised models were previously studied by Maeda et al. (2018), here we explore a wider range of compositions in the helium shell, as well as multiple shell masses for a given core mass. In Sect. 2, we discuss the radiative transfer code used in this work, TURTLS (Magee et al. 2018  presents our approach to constructing parameterised double detonation models. In Sect. 4, we discuss the impact of the helium shell composition on the model observables, while in Sect. 5 we show the impact of the mass of burned material above the core. The rise times and early light curve bumps of our models are discussed in Sect. 6. In Sect. 7, we compare to existing models with varying 56 Ni distributions. Comparisons to observations of normal SNe Ia are presented in Sect. 8, while in Sect. 9 we compare to SNe Ia showing a bump in the early light curve. For all spectral comparisons, spectra have been corrected for Milky Way and host extinction, where appropriate, and were obtained from WISeREP (Yaron & Gal-Yam 2012). Finally, we present our conclusions in Sect. 10.

RADIATIVE TRANSFER MODELLING
We use the one dimensional radiative transfer code TURTLS (Magee et al. 2018) to perform our simulations. All model light curves and spectra presented in this work are freely available on GitHub 1 . TURTLS is described in detail by Magee et al. (2018). Here we provide a brief overview of the code and outline changes implemented for this study.
TURTLS is a Monte Carlo radiative transfer code following the methods of Lucy (2005) (see Noebauer & Sim 2019, and references therein, for a review of Monte Carlo radiative transfer methods). TURTLS is designed for modelling the early time evolution of thermonuclear supernovae. For each simulation, the density and composition of the model ejecta is defined in a series of discrete cells. Monte Carlo packets representing bundles of photons are injected into the model region, tracing the decay of radioactive isotopes. We have updated TURTLS to account for energy generated by the 52 Fe → 52 Mn → 52 Cr and 48 Cr → 48 V → 48 Ti decay chains, which can contribute significantly to the luminosity and overall evolution of the model in the double detonation scenario (Noebauer et al. 2017). Isotope lifetimes and decay energies are taken from Dessart et al. (2014).
For all simulations presented in this work, we use a start time of 0.5 d after explosion. In the appendix in Sect. A, we show the results of some of our convergence tests with earlier start times. These tests demonstrate that, despite the short half-lives of many of the included isotopes, a start time of 0.5 d after explosion does not significantly alter the synthetic observables and does not impact our conclusions. Once packets are injected into the model region, their propagation is followed until either they escape or the simulation ends. Due to the assumption of local thermodynamic equilibrium (LTE) within TURTLS, simulations are stopped at 30 days after explosion. We also note that as a consequence of this assumption, our models do not predict the presence of helium features, despite the potential for a large amount of unburned helium. Previous studies have shown that a non-LTE treatment of helium is required to produce spectral features for the conditions typical of SN ejecta (Hachinger et al. 2012;Dessart & Hillier 2015;Boyle et al. 2017). At the start of each simulation, packets are injected as -packets (representing -ray photons) and treated with a grey opacity of 0.03 cm 2 g −1 . Following an interaction with the model ejecta, -packets are converted to optical radiation packets ( -packets). For these packets, we use TARDIS (Kerzendorf & Sim 2014;Kerzendorf et al. 2018) to calculate the non-grey expansion opacities and electron-scattering opacities within each cell during the current time step. During each time step, we extract a 'virtual' spectrum using the so-called event-based technique (e.g Long & Knigge 2002;Sim et al. 2010a;Kerzendorf & Sim 2014;Bulla et al. 2015;. Light curves are calculated via the convolution of synthetic virtual spectra with the desired set of filter functions at each time step. In Fig. 1, we show a comparison of our model light curves including the new decay chains to those calculated by Noebauer et al. (2017), using the radiative transfer code, STELLA (Blinnikov et al. 1998(Blinnikov et al. , 2006, for the same model structure. This model involves the detonation of a 0.055 helium shell on a 1.025 carbonoxygen white dwarf. The resulting explosion leads to the production of 0.55 of 56 Ni in the white dwarf core. The helium shell ash following explosion is dominated by IGEs, which includes ∼0.002 of 56 Ni, 0.006 of 52 Fe, and 0.004 of 48 Cr. Figure 1 verifies that with our implementation of the additional decay chains, TURTLS can broadly match the light curves of Noebauer et al. (2017). The early light curve bump observed in our models is somewhat less pronounced than in the Noebauer et al. (2017) model, which is likely a result of differences in the treatment of opacities, for example.
We also show light curves in Fig. 1 calculated including either the 52 Fe → 52 Mn → 52 Cr chain, the 48 Cr → 48 V → 48 Ti chain, or neither, as a further demonstration of their contribution to the early luminosity. We note that in all cases, the 56 Ni → 56 Co → 56 Fe decay chain is included. Including these additional chains produces a ∼4 mag. increase in the brightness by approximately two days after explosion. Figure 1 shows that despite the short half-lives of both the parent and daughter isotopes ( 1/2 = 0.345 d and 0.015d, respectively), the 52 Fe → 52 Mn → 52 Cr chain contributes significantly to the early luminosity within the first few days of explosion. For this model, the early bump reaches a peak -band magnitude of −16.7 mag. approximately 1.8 d after explosion. At this time, the instantaneous energy deposition rate from material in the shell reaches ∼ 1.3 × 10 42 erg s −1 and dominates the luminosity output of the model, which is consistent with expectations from Arnett's law (Arnett 1982).

CONSTRUCTING THE DOUBLE DETONATION MODEL SET
In the following section, we discuss our approach to creating a parameterised description of the ejecta in double detonation explosions. Our strategy is based on capturing and exploring the variation present across a range of published models in a systematic way. Each of our models is controlled by the following parameters: the mass of the carbon-oxygen core, the mass of the helium shell, the fraction of the helium shell burned during the explosion, and the dominant -chain product produced in the shell burning. The range of input parameters used is shown in Table 1. The name of each model is also derived based on these parameters, for example WD1.00_He0.04_BF0.50_DP56Ni refers to a model with a core mass of 1.0 , a helium shell mass of 0.04 , of which 50% is burned, and the dominant product produced in the shell is 56 Ni. The range of parameters explored was chosen to broadly cover and bracket the values predicted by various explosion models, but we stress they are not exact reproductions of existing models.
For each model, we require a density profile for the ejecta. The density profiles presented in Magee et al. (2018) and  were designed to broadly mimic those from a variety of explosion scenarios. In particular, the exponential density profile with a kinetic energy of 1.4×10 51 erg from  bears a striking similarity to the models of Kromer et al. (2010) and Polin et al. (2019), although the density in the outer ejecta is slightly higher. We therefore take this model as our nominal profile shape and simply scale the density to the appropriate ejecta mass, which is given by the sum of the core and helium shell masses. A demonstrative comparison between model 3 of Kromer et al. (2010) and two of our models is shown Fig. 2(a). We note that the core mass of model 3 (1.025 ) is slightly higher than these models (1.0 ), and we show two shell masses (0.04 and 0.07 ) to bracket the 0.055 shell of model 3.

Composition of the core
Previous studies of double detonation explosions have shown that the amount of 56 Ni produced in the carbon-oxygen core during the explosion is directly related to the total mass of the white dwarf. In Fig. 3(a) we show the core 56 Ni mass produced as a function of total mass for a sample of models from the literature Shen et al. 2018;Polin et al. 2019;Gronow et al. 2020;Kushnir et al. 2020). As shown in Fig. 3 we use the former set of models as reference points throughout this study, allowing us to consistently select parameters for our model helium shells and cores masses. Between ∼0.9 -1.3 there is an approximately linear relation and broad agreement between these different model sets. As the Polin et al. (2019) sample covers a large range of total masses and different ignition conditions, we use a linear fit to this model set to determine the core 56 Ni mass of our models. The core 56 Ni mass is therefore given by: where core is the mass of the carbon-oxygen core and shell is the mass of the helium shell. All variables are in units of . This fit is shown as a dashed line in Fig. 3(a).
In Appendix. B, we present additional models exploring 56 Ni masses based on the Shen et al. (2018) and Kushnir et al. (2020) models. Given the uncertainty in the amount of 56 Ni produced, the total white dwarf mass should not be taken as a prediction from our models. Throughout this work, we give the values of core and shell masses simply as reference to identify each model. Instead, we consider the total luminosity (i.e. the 56 Ni mass) to be a robust prediction and expect that there may be a range of white dwarf properties that produce such a 56 Ni mass.
For the distribution of 56 Ni within the core, we follow the functional form used by Magee et al. (2018). The 56 Ni mass fraction at mass coordinate is given by: where Ni is the total 56 Ni mass in units of . The scaling parameter, , is used to control how quickly the ejecta transitions from a 56 Ni-rich to -poor composition. The models with = 21 presented by  Literature values are taken from their respective papers Shen et al. 2018;Polin et al. 2019;Gronow et al. 2020;Kushnir et al. 2020). For total masses between 0.90 -1.30 , we show a linear fit to the Polin et al. (2019) models, which is used to determine the core 56 Ni mass of our models. Panel b: Fraction of the helium shell that is burned (i.e. converted to elements heavier than helium following the explosion) as a function of total mass. Dashed horizontal lines show fractions of 0.2, 0.5, and 0.8. Black points show the specific models calculated in this work, which broadly extend the range predicted from a variety of explosion models. to those of Kromer et al. (2010) (Fig. 2(b)), therefore we fix = 21 for all models in this work. By adopting a similar method to  and , we also allow for a direct comparison to the models presented in both studies. Immediately below the base of the helium shell, we place a small amount (∼10 −3 -10 −1 ) of unburned carbon and oxygen assuming a Gaussian distribution (width ∼0.001 -0.01 ). The mass and distribution of this unburned material is comparable to that predicted by explosion models (e.g. Kromer et al. 2010;Polin et al. 2019), although in general a symmetric distribution is not predicted for all explosion parameters. We note that we have also tested narrower and broader distributions (width ∼0.0001 -0.1 ) and find the exact distribution does not have a significant impact on the model observables and does not affect our conclusions. The remaining material in the core is filled in with intermediate mass elements (IMEs).

Composition of the shell
The composition of the helium shell following the explosion remains one of the uncertain properties of double detonation explosions.   Kromer et al. (2010) models assuming a pure helium shell (model 3; black) and a shell that has been polluted with 34% carbon pre-explosion (model 3m; grey). Coloured lines show the mass fractions of all isotopes, assuming burning progresses to a specific point along the -chain -given by the colour. Solid lines show our standard isotope distributions, based on model 3m. The dashed line shows a broad distribution designed to mimic that of model 3 from Kromer et al. (2010), while the dotted line shows a narrow distribution in which more mass is burned to the dominant shell product.
goal of this work is to present models covering a large parameter space and systematically investigate differences in observables that result from various assumptions about the helium shell. In Fig. 3(b) we show the fraction of the helium shell that is burned following the explosion (i.e. converted from helium into heavier elements) for a selection of model sets from the literature. It is clear that there can be a large spread in how much of the shell is consumed, depending on different assumptions made within the models (such as when and how ignition is triggered). To investigate the impact of this on the observables, we choose fractions that bracket those predicted by the different explosion models. Specifically, for each total mass we calculate models for which 20%, 50%, and 80% of the helium shell is burned to other elements. These fractions are shown as black dashed lines in Fig. 3(b), while the individual models calculated in this work are shown as black points.
Aside from simply investigating how much of the shell is burned, we also aim to demonstrate the effects of elements that are produced during shell burning. This is strongly dependent on the initial composition of the shell. Kromer et al. (2010) present a model in which the helium shell is polluted and contains 34% 12 C (model 3m). This is shown in Fig. 4, along with the composition of the unpolluted model (model 3). The choice of 34% was specifically made to create a helium shell that is mostly burned to 36 Ar, which does not produce strong spectroscopic features. As discussed in other studies (e.g. Shen & Bildsten 2009;Waldman et al. 2011;Gronow et al. 2020) the presence of carbon has an important role to play in regulating helium burning and shaping the nucleosynthetic yields of the helium shell. Therefore, the final composition of the helium shell could be tuned by varying the level of pollution before explosion (Waldman et al. 2011). Piro (2015) has demonstrated that a wide range of carbon pollution fractions could indeed be achieved in the helium shell, depending on specifics of the binary system. This picture is complicated further however, by the presence of other isotopes besides 12 C. In particular, Townsley et al. (2019) have shown that -chain burning can stall at much lower pollution fractions (∼11%) when including 12 C, 14 N, and 16 O. It is clear that the nucleosynthetic yields of the helium shell could show significant variations following explosion. Linking these to specific compositions pre-explosion is a challenging prospect. Therefore, in this study we explore a wide variety of options and assume that burning in the helium shell could stall at any point along the -chain. We make no claims about specific pre-explosion compositions that could produce such yields. In the following, we refer to the point at which burning stalls as the dominant product in the shell. We calculate models for dominant shell products ranging from 32 S to 56 Ni. In our standard model distribution, the relative abundances of other isotopes along the -chain are taken following from the 3m model of Kromer et al. (2010). We chose model 3m for our standard isotope distribution as it represents an intermediate case to the other distributions explored in this work. In addition, Kromer et al. (2010) present abundances for each isotope produced in the helium shell. Although 36 Ar is the dominant shell product in this model, some amount amount of other isotopes are produced above and below 36 Ar in the chain. This is demonstrated in Fig. 4, which shows that 36 Ar is produced with a mass fraction of ∼60% while the previous -chain isotope ( 32 S) has a mass fraction of ∼30% and the next isotope ( 40 Ca) has a mass fraction of ∼8%. We use a skew normal distribution that approximates the Kromer et al. (2010) 3m distribution in order to determine the relative fraction of all -chain isotopes, for a given dominant shell product. Explosion models and detailed predicted yields covering a range pollution fractions within the helium shell are currently unavailable. Therefore, assuming that some amount of isotopes above and below the dominant shell product are also produced seems reasonable. Taking a functional form similar to an existing explosion model is a pragmatic choice, but we note that the exact quantities are unclear.
In Fig. 2(c), we verify that our parameterised approach produces spectra comparable to Kromer et al. (2010). We show a comparison between the maximum light spectrum of model 3 (core mass of 1.025 , shell mass of 0.055 ) and two of our models with similar parameters (core mass of 1.0 , shell masses of 0.04 and 0.07 ). In general, our models show similar results, however the velocities are typically too high. The purpose of Fig. 2(c) is to demonstrate that our parameterised description of the ejecta is not a limiting factor for the method used here. As discussed in Magee et al. (2018), differences in the radiative transfer code used here (TURTLS) and that of Kromer et al. (2010) (ARTIS;Kromer & Sim 2009) can lead to different observables. This is reflected in the spectra for our models, which are generally bluer than those of Kromer et al. (2010). In addition, the differences in the density profile will have some impact and a combination of these factors appears to result in a shift of features to higher velocities. We again stress that our models are not intended to be reproductions of existing model sets, but are designed to explore a large parameter space. As previously mentioned, by adopting a similar structure to the models of  and , we allow for a direct comparison with models from these works, which were all calculated with the same radiative transfer code.

Alternative abundances in the helium shell
In an effort to quantify the significance of our choice for the relative abundances of isotopes, we calculate two additional sets of models. Firstly, we consider a broad distribution similar to that found for model 3 of Kromer et al. (2010). This corresponds to a higher mass Together these two sets of models serve to bracket the distributions assumed throughout this work. The effects of these different compositions are discussed further in the appendix in Sect. C, but we note that in general the differences are relatively minor.

EFFECTS OF POST-EXPLOSION HELIUM SHELL COMPOSITION
In the following section we discuss the results of our radiative transfer modelling. We demonstrate the significant impact of the helium-shell composition on the model light curves and spectra. We compare models with the same core (1.0 ) and shell (0.07 ) masses, but different shell compositions for our standard isotope distribution (Sect. 3.2, Fig. 4). For this comparison of the effect of different dominant products in the helium shell, we focus on the models in which 50% of the helium shell is burned to heavier elements. Other models within our set show similar variations for different shell compositions. Figure 5 shows the effect of the shell composition on the light curve and colour evolution. Similar to previous studies, we find that those models with -chain burning progressing to IGEs ( 44 Ti -56 Ni), which therefore have relatively large amounts of short-lived radioactive isotopes in their shells ( 56 Ni, 52 Fe, and 48 Cr), display prominent bumps in their light curves within the days following explosion. Although these bumps are most pronounced at shorter wavelengths, they are also clearly seen in redder filters (e.g. -band). Aside from the shape of the light curve, models with IGE-dominated shells also show a distinct colour inversion. The colours are initially blue and quickly reach a peak red colour within a few days of explosion. At this point the colour evolution turns over and the models become somewhat bluer again, before again turning over and becoming progressively redder towards maximum light.

Light curves
For those models in which the shell is dominated by IMEs ( 32 S or 36 Ar), no early bump is observed due to the lack of the additional radioactive material. Instead, these models show a smooth rise to maximum light, as well as broader and brighter -band light curves than our IGE-dominated shell models. Our IME-dominated shell models also show a relatively flat colour evolution beginning approximately five days after explosion. We have also calculated models for which the assumed -chain burning stalls earlier than 32 S ( 20 Ne, 24 Mg, and 28 Si), however these models are very similar to each other and the 32 S-dominated model. Therefore, our models show that provided the initial composition of the helium shell is such that burning stops at IMEs, the relative abundances of this burned material are generally unimportant for shaping the evolution of the observables further.
Interestingly, our 40 Ca-dominated model represents an intermediate case between the IME-and IGE-dominated shells. No early light curve bump is observed and the -band in particular shows a longer dark phase (i.e. the time between explosion and the first light emerging from the supernova) than all other models. At the same time, the colour evolution does not show an inversion similar to the IGE-dominated shells, but is significantly redder at maximum light compared to the IME-dominated shells. Although 40 Ca is the dominant product in the shell (∼55% of the burned material), a small amount of 44 Ti is also present (∼5% of the burned material). The additional opacity contribution from 44 Ti will act to more effectively blanket the blue flux than in the other IME-dominated shell models, which do not contain 44 Ti. On the other hand, the 40 Ca-dominated model also lacks a contribution from any radioactive material in the shell, as in the case of the IGE-dominated shell models. Together, both of these properties will cause the lack of additional flux at early times and the redder colours at later times.

Spectra
In Fig. 6, we show the spectral evolution of our models with different shell compositions. Spectra are shown at 2.25 d, 7.25 d, and 18.25 d after explosion. At 2.25 d after explosion, our models dominated by 56 Ni and 52 Fe are substantially bluer than all other models and show relatively featureless spectra. Despite still containing shortlived isotopes near the surface of the ejecta, Fig. 6 shows that our 48 Cr-dominated model spectrum is much redder than either the 56 Nior 52 Fe-dominated model spectra. As shown in Fig. 1, 52 Fe is the dominant source of luminosity for the early light curve bump -due to its short half-life. Although some 52 Fe is present in the shell of our 48 Cr-dominated model, it has a much lower fraction than in the 56 Nior 52 Fe-dominated models -hence there is a lower luminosity and less heating, producing a fainter and redder spectrum during the early bump. Our 48 Cr-dominated spectrum also shows a strong absorption feature due to S at ∼4 800 Å. At 2.25 d, our IME-dominated models are still in the dark phase (i.e. very little luminosity has actually escaped). Despite their low luminosity, a weak Si 6 355 feature is still visible in our 36 Ar-and 32 S-dominated models, as well as the S feature around ∼4 800 Å that is also visible in our 48 Cr-dominated model.
One week after explosion, the 56 Ni-and 48 Cr-dominated shell models have become significantly redder. Much of the flux below 4 000 Å has been blanketed out in all models with IGE-dominated shells. These models also show a broad absorption feature due to Ti around ∼4 200 Å (with the exception of the 56 Ni-dominated model, which does not contain Ti in the shell). Our 48 Cr model shows remarkably little spectral evolution between the two epochs presented here relative to other models within our set. In the IGEdominated models, a few additional features are produced at longer wavelengths (most notably the Si 6 355 feature), but in general they are weaker and broader than in models that do not contain IGE in the shell. For our 36 Ar-and 32 S-dominated models, the spectra are now considerably bluer than the IGE-dominated models. Absorption profiles due to IME, such as Si 6 355 and the S -W feature can be observed. Both models also show strong Ca absorption at ∼3 600 Å.
Moving to maximum light, more of the blue flux in our IGEdominated models has been blanketed out. At this epoch, the spectra show very little flux below ∼4 300 Å and again show a broad, flat Ti profile between ∼3 900 -4 300 Å. Around ∼4 700 -4 900 Å, the IGE-dominated models show a similarly broad and flat feature due to Cr and Ti . For these models, features due to IME (Si 6 355, Si 5 972, and S -W) are again broader and weaker than in the IME-dominated shell models.
High-velocity features have been reported in a number of SNe Ia at early times (e.g. Childress et al. 2014;Maguire et al. 2014;Zhao et al. 2015). The origin of these features is unclear, but one proposed scenario is from an abundance or density enhancement that may be due to circumstellar interaction or intrinsic to SN itself (Mazzali et al. 2005;Tanaka et al. 2008). Double detonation models producing IMEs in the helium shell would be a natural method producing such an abundance enhancement. To investigate whether our IME-dominated shells produce similar features and if these can be attributed to high velocity material in the shell, we calculate the contribution of the shell material to the synthetic spectra. During the simulation, we track the location at which a Monte Carlo packet experiences its last interaction. In Fig. 7, we show separate spectra produced by binning Monte Carlo packets that last interacted with material in either the shell or the core. We note that we are only able to track the location of real packets (rather than virtual packets; see Magee & Maguire 2020), therefore the signal-to-noise ratio of these spectra is lower than others presented throughout this work. Nevertheless, Fig. 7 shows that within the first approximately one week since explosion, the shell material does contribute to the production of high velocity features. Specifically, the Si 6 355 feature is shifted to higher velocities and broadened when including contributions from the shell. Around maximum light however, there is a negligible impact from the shell material. While our models indicate that helium shell ash could provide one explanation for high velocity features seen in some SNe Ia at early times, further modelling work is required to fully constrain the abundance profiles required.

Summary
Our models clearly show the impact of the post-explosion helium shell composition on the observables. Those models in which the shell is burned mainly to IGE show an early bump in the light curve, a colour inversion, and significantly reddened spectra from approximately one week after explosion. Conversely, our models in which the shell mostly contains IMEs do not show an early bump and instead have a relatively flat colour evolution. In addition, we find that as long as burning within the shell does not progress to IGEs, the model observables show much smaller variations than those for which the shell is dominated by IGEs. This would indicate that meticulous fine-tuning is not necessary to avoid the impact of the helium shell ash on the observables -provided burning ceases at a certain point, the exact composition of the shell is mostly irrelevant. All models shown have a 1.0 core and we assume 80% of the shell is burned to elements heavier than helium. We show two representative cases in which the composition of the shell is dominated by either 32 S or 52 Fe.

EFFECTS OF THE BURNED MASS
In this section, we demonstrate how the amount of burned material above the core affects the model observables. To focus our discussion, we limit our comparisons to models with a core mass of 1.0 . Our models are controlled by both the mass of helium shell and the amount of the shell that is assumed to be burned during the explosion. As the mass of the helium shell also determines the amount of 56 Ni produced during the explosion, which will have a significant impact on the model observables, it is not possible to explore solely the effect of the total amount of material burned. Therefore, in Sect. 5.1 we discuss the effects of the helium shell mass and in Sect. 5.2 we discuss the role of the burned fraction.

Impact of helium shell mass on light curves and spectra
In Fig. 8 we show the light curve and colour evolution for models with varying shell masses, while the spectral evolution is shown in Fig. 9. We limit our comparison to the 52 Fe-and 32 S-dominated shells, which are representative of trends observed for IGE-and IME-dominated shells, respectively (see Sect. 4).
As discussed in Sect. 4, our 32 S-dominated shell models do not produce an early bump in the light curve, but there is still some variation among the different shell masses. This is not primarily driven by material in the shell, but rather the different 56 Ni masses in the core of the white dwarf (Sect. 3.1, Fig. 3). Therefore, models with more massive shells are brighter and somewhat bluer simply due to the increased ejecta mass and hence 56 Ni mass. Figure 9 shows that these models also produce similar spectra, with the primary differences being the luminosity and colour. Differences between the spectra of models with different shell masses are most pronounced at early times, where lower mass shells show stronger Si and S features, likely due to their lower temperatures. For our IME-dominated models we also note there is also a degeneracy between the core and shell masses. For the models presented here, provided the total ejecta mass is the same, the distinction between the core and shell is unimportant. For example our 32 S-dominated model with a 1.0 core and 0.1 shell and model with a 1.1 core and 0.01 shell produce similar light curves and spectra.
For our 52 Fe-dominated shells, Fig. 8 shows that all models produce a light curve bump. The timescale of the bump varies significantly (∼1 -5 days) for the broad-band light curves, with lower mass shells producing shorter-lived and more rapidly evolving bumps. As demonstrated by Fig. 8, this is primarily due to temperature evolution for the different models, as there is significantly smaller spread in the bump timescales in bolometric light. Unlike the 32 S-dominated shell models, the shell mass has a considerable impact on the colour evolution for the 52 Fe-dominated models. Smaller shell masses produce a more rapid and extreme change in colour within the first five days after explosion. In addition, beginning approximately 10 days after explosion, the lowest mass shell model (0.01 ) shows a relatively flat colour evolution towards maximum light. In contrast, models with more massive shells become significantly redder over this same period. The 0.10 shell model remains redder than both the 0.07 and 0.04 shell models for all times presented here, however the overall difference between their respective colours decreases with time. This likely points to two competing effects -the influence of line blanketing from the shell and the different 56 Ni masses causing different temperatures. More massive shells will produce more line blanketing and hence one may expect redder colours, but this is not observed for the models presented here. In this case, as the core mass is the same, the increase in the shell mass results in an increased 56 Ni mass that keeps the ejecta hotter and bluer. Figure 9(d) shows that, at 2.25 d after explosion, the temperature is the primary difference between the models and few features are present. At later epochs, our models show that larger shell masses produce broader Si 6 355 features and weaker IME features overall.

Impact of the burned fraction percentage on light curves and spectra
The amount of material in the shell converted from helium to heavier elements is also a free parameter of our models. We have investigated burned fractions of 20%, 50%, and 80%, which approximately span the range predicted by different explosion models (see Fig. 3(b)). The differences between these models are fairly straightforward and follow the trends one may expect. For IME-dominated shell models, the burned fraction has no effect on the resultant observables. For IGE-dominated shell models, a higher burned fraction will result in a brighter bump at early times and redder colours at later times.

MODEL RISE TIMES AND BUMP TIMESCALES
Here we discuss the rise times and peak magnitudes of the models presented in this work and demonstrate the range of magnitudes and timescales for early light curve bumps. In Fig. 10(a) we show the -band rise times and peak absolute -band magnitudes for our models with the standard isotope distribution. The difference between our IGE-and IME-dominated shell models is readily apparent. We find that, in general, those models with IGE-dominated shells show shorter rises, with a median rise time of 13.8±2.3 d, compared to those with IME-dominated shells, which have a median rise time 18.25d Figure 9. Spectra for models with different shell masses. All models shown have a 1.0 core and we assume 80% of the shell is burned to elements heavier than helium. We show two representative cases in which the composition of the shell is dominated by either 32 S or 52 Fe. Spectra are shown at three epochs: 2.25 d, 7.25 d, 18.25 d after explosion. All spectra are normalised to the flux between 7 000 -7 500 Å. Figure 10. Panel a: Peak absolute -band magnitudes against rise time to peak -band magnitude. Panel b: Peak absolute -band magnitudes of the early light curve bump against time to reach the peak of the bump. We note that all models with IME dominated shells and some models with high core masses do not show early bumps and therefore are neglected. For models in which the light curve is already declining at the start of our simulation (0.5 d), we consider these as upper limits and show them as black arrows. In both panels, each model is coloured based on the dominant element produced in the shell. The size of each point is proportional to the burned mass of the helium shell (i.e. the product of the shell mass and burned fraction), while the shape of each point denotes the mass of the core. of 17.6±0.7 d. The longer rise time of the IME-dominated models is more typical of normal SNe Ia (e.g. Ganeshalingam et al. 2011;Firth et al. 2015;Miller et al. 2020a). Although in general we find that models with IGE-dominated shells show shorter rise times, there are some notable exceptions. For a 0.9 core, some models with low-mass shells (0.01 and 0.04 ) can show longer rise times than similar models with higher mass cores. In these cases, the longer rise times result from a combination of the compact 56 Ni distribution and less extreme line blanketing of the low-mass shell. For our IMEdominated models, the scatter in the peak absolute -band magnitude is driven simply by differences in 56 Ni mass due to the various core and shell masses explored here. Models with IGE-dominated shells however, show a significantly larger scatter ( 4 mag compared to ∼2 mag for IME-dominated shells) due to line blanketing from the material in the shell. Indeed at longer wavelengths that are less sensitive to line blanketing (e.g. -band), both the IGE-and IMEdominated shell models show a similar scatter in peak magnitudes (again, due to the differences in the 56 Ni mass), although the IGEdominated models are systematically brighter as much of the blue flux has been reprocessed to longer wavelengths by the shell. Figure 10(b) shows the properties of the early light curve bumps. We calculate the time since explosion to reach the peak of the bump and the magnitude at this point. For some models, the light curve is already declining at the beginning of our simulations (0.5 d after explosion). We therefore consider these points as limits and show them as black arrows in Fig. 10(b). We do not include models with IMEdominated shells as they do not show a bump at early times. In addition some models, such as those with large total masses ( 1.2 ), do not show pronounced bumps in their light curves due to their high 56 Ni masses and extended distributions. In other words, there is no clear decline in the light curve within the first few days of explosion. These models are also not included in Fig. 10(b). Among the models shown in Fig. 10(b), there is a general trend that brighter bumps are also typically longer lasting. Models with 44 Ti-dominated shells however, deviate from this trend. Following from Fig. 4, in our 44 Ti dominated model only a small amount of 48 Cr is contained within the shell. Therefore this set of models contain a significantly smaller mass of radioactive isotopes in the shell compared to our other IGE dominated models. We also note that the models shown as limits in Fig. 10(b) hint at the possibility of bright and very short lived bumps -less than 0.5 d. It is highly likely that such bumps could be missed in most current surveys.

COMPARISON WITH A CHANDRASEKHAR-MASS MODEL CONTAINING A 56 Ni EXCESS
Magee & Maguire (2020) present light curves and spectra of Chandrasekhar mass models that contain an excess of 56 Ni (a 56 Ni shell) in the outer ejecta. Qualitatively, these models show similar behaviour (light curve bumps at early times and line blanketing closer to maximum light) to double detonations in which a significant fraction of IGEs is produced in the shell. Here we perform a comparison between these two cases and investigate ways in which they may be distinguished from each other, based on the early light curve bump and spectra at maximum light. In Fig. 11(a), we show the 56 Ni distributions of Chandrasekharmass model with and without a 56 Ni excess compared to the sub-Chandrasekhar double detonation models explored in this work. We show one of the Chandrasekhar mass models from Magee & Maguire (2020) that does not contain a 56 Ni excess (black in Fig. 11 with a model in which a 0.03 56 Ni shell has been added to the outer ejecta (Fig. 11, green). We present an additional Chandrasekhar mass model without a 56 Ni excess, but in which the 56 Ni distribution has been extended, such that 56 Ni is present throughout the ejecta with a mass fraction that decreases monotonically towards the outer ejecta (Fig. 11, grey). For our sub-Chandrasekhar mass double detonation model with a 1.0 core and a 0.07 shell dominated by 56 Ni (WD1.00_He0.07_BF0.50_DP56Ni; Fig. 11, red), the total 56 Ni mass and 56 Ni distribution in the outer ejecta is similar to the 56 Ni excess model. Finally, we also show the same model with a 32 S-dominated shell (WD1.00_He0.07_BF0.50_DP32S; Fig. 11, blue) as representative of a sub-Chandrasekhar mass model without an excess of 56 Ni in the outer ejecta. We note that the density profile and ejecta mass differs slightly between the double detonation and Chandrasekhar mass models shown here. Figure 11(b) demonstrates that the sub-Chandrasekhar mass double detonation 56 Ni-dominated shell model shows a more pronounced bump that rises and declines within a few days compared to the more plateau-like shape of the Chandrasekhar mass 56 Ni excess model. Even for double detonation models with a lower 56 Ni mass fraction in the outer ejecta (i.e. different burned fractions), the 56 Ni-dominated shells do not reproduce the shape of the 56 Ni excess models. Such a difference in light curve shape, despite similar 56 Ni distributions, serves to further highlight the importance of the additional radioactive isotopes produced in the double detonation models. For the Chandrasekhar-mass 56 Ni excess model, 56 Ni is the only radioactive isotope considered in the ejecta, while the double detonation model also contains 52 Fe and a small amount of 48 Cr. Hence, the bump produced in the light curve of the 56 Ni-dominated shell model is more pronounced due to the presence of 52 Fe, 52 Mn, and 48 Cr, which have considerably shorter half-lives compared to 56 Ni. At later epochs, the double detonation model with a 56 Ni-dominated shell becomes significantly redder and fainter than the 56 Ni excess model. Again this points to important differences in the ejecta composition -the presence of additional IGEs in the double detonation model will more effectively blanket blue flux than just the 56 Ni decay chain as in the 56 Ni excess model. For our 32 S-dominated shell model, the light curve shows a sharper rise and slightly longer dark phase than the model without a clump. In this case, the 56 Ni distribution is somewhat less extended and shows a more rapid change from 56 Ni-rich to -poor ejecta than the Chandrasekhar mass model without a 56 Ni excess.
In Fig. 11(c) we show the spectra of all models at 18.25 d after explosion. At this epoch, our Chandrasekhar mass model without a 56 Ni excess and 32 S-dominated shell model show extremely similar spectra (black and purple lines), with the most noticeable difference being that the double detonation model is marginally bluer. Conversely, the 56 Ni-dominated shell model (red line) is significantly different from all other models, including the Chandrasekhar mass 56 Ni excess model (green line). The flux below ∼4 000 Å is essentially completely removed from the spectrum and redistributed to wavelengths 5 000 Å, which show a significantly higher continuum flux. The Cr and Ti features present in the double detonation model at ∼4 000 -5 000 Å easily distinguish it from the 56 Ni excess model.
Comparing Chandrasekhar mass models with a 56 Ni excess in the outer ejecta and sub-Chandrasekhar mass double detonation models in which a 56 Ni-dominated shell is produced as a result of helium shell burning, we find that the two are easily distinguished despite qualitatively similar behaviour. Although thought to be created via a different mechanism, models with 56 Ni excess can also produce an early light curve bump. The shape of the bump however, more closely resembles a plateau compared to the clear peak in the double detonation models. The significant amount of IGEs produced during helium shell burning leads to extremely red colours -even more so than the 56 Ni excess models, which also show red colours at maximum. Finally, our IGE-dominated shell models also show shorter rise times than the 56 Ni excess models of .

COMPARISONS WITH NORMAL SNE IA
In the following section, we discuss whether our double detonation models are consistent with observations of SNe Ia. We compare to light curves and spectra of two well-observed and prototypical SNe Ia, SNe 2011fe (Nugent et al. 2011;Richmond & Smith 2012;Vinkó et al. 2012) and 2005cf (Pastorello et al. 2007;Garavini et al. 2007). Figure 12 shows the light curves of both objects compared to our models, while spectra are shown in Fig. 13 ; Nugent et al. 2011). For SN 2005cf, we find that a larger total mass is required to reproduce the higher core 56 Ni mass (0.7 ; Pastorello et al. 2007). Our models with either a 1.0 core and 0.10 shell or 1.1 core and 0.01 shell both produce similar light curves and spectra for a 32 S-dominated shell and may be considered interchangeable. Here we present the model with a 1.0 core and 0.10 shell for SN 2005cf. As previously mentioned (Sect. 3.1), there is disagreement between various studies over the amount of 56 Ni produced for a given white dwarf mass during explosion. For this reason, the core and shell masses presented here should not be taken as predictions for the objects discussed here, but are simply given as reference to identify the comparison models shown. SN 2011fe has been corrected for a total extinction of ( − ) = 0.01 mag. (Nugent et al. 2011), while SN 2005cf has been corrected for ( − ) = 0.1 mag. (Pastorello et al. 2007). Figure 12 demonstrates that our double detonation models with 32 S-dominated shells provide good agreement with the light curve shapes of both objects beginning a few days after explosion and extending to approximately maximum light. The largest discrepancies between models and observations are observed in the -band, but we note this is likely related to the simplified composition and ejecta structure used (see , and will be explored in future work. Townsley et al. (2019) have also previously shown that a double detonation model with a 1.0 core and 0.02 helium shell dominated by IMEs can reproduce the light curve of SN 2011fe. For epochs 4 days after explosion however, both models clearly show a rise that is too sharp and a dark phase that is too long to match the observed flux. By comparing to several Chandrasekhar mass models of different 56 Ni distributions,  found that the early light curve points of SN 2011fe can be reproduced by a relatively extended 56 Ni distribution with a mass fraction in the outer ejecta of ∼0.03. In contrast, the double detonation models shown here, which have 32 S-dominated shells, do not contain any 56 Ni in the outer ejecta. As shown in Fig. 2, the functional form used for the models presented here produces a 56 Ni distribution that is somewhat more compact than that predicted by Kromer et al. (2010). A slightly more extended core 56 Ni distribution for the double detonation models with 32 S-dominated shells could likely reproduce the earliest detections of SNe 2011fe and 2005cf, without adversely affecting the light curve at later times. Figure 13 shows a comparison between the spectra of SNe 2011fe and 2005cf and their corresponding double detonation models with 32 S-dominated shells. Previous comparisons to double detonation and bare sub-Chandrasekhar mass models have focused only on spectra around maximum light. Here we show spectra for both objects at multiple epochs, beginning ∼4 days after explosion and extending to maximum light. For our double detonation models, we find that spectra at 4.25 d after explosion are consistent with those of both objects approximately two weeks before maximum. Although many of the features are reproduced with approximately the correct strength and shape, such as Si 6 355 & 5 972, S -W, Mg 4 481, O 7 774, and the Ca NIR triplet, they are all noticeably offset to higher velocities in the models compared to the observed spectra. Therefore, in Fig. 13, we also show our model spectra with a velocity shift of ∼6 000 km s −1 applied and find improved agreement. As discussed in Sect. 3.2, the systematic shift to high velocities in our spectra could be due to simplifications made in our model density profiles, particularly in the outer regions. Closer to maximum light, velocities of many features (such as the S -W feature) in our model spectra show good agreement with both SNe, although Si velocities are still somewhat higher than those observed. In Sect. 4, we show how the material in the helium shell can impact the spectroscopic features within the first week of explosion. Qualitatively, this is similar to the broad Si 6 355 feature in SN 2005cf, which has been argued to have a high velocity component (Garavini et al. 2007). Again we note that there is a systematic shift of all features to higher velocities at this time. Nevertheless, our models provide tentative evidence that the high velocity components in some SNe Ia may be due to interactions with a helium shell containing IMEs.
Our models verify the claims of Kromer et al. (2010)   with different helium shell masses are also capable of reproducing multiple normal SNe Ia and at various epochs up to maximum light. While our models generally show good agreement from a few days after explosion, the initial rise of the light curve is sharper than observed, supporting the claims of  that a more extended distribution for 56 Ni in the core may be required.

COMPARISONS WITH SNE IA SHOWING AN EARLY LIGHT CURVE BUMP
A handful of objects with early light curve bumps have now been discovered. Among these objects, two distinct groups are clearly apparent based on their optical colour close to maximum lightthose that are extremely red, with − 1 (SNe 2016jhr, 2018byg) and those that are relatively normal with a blue colour, − 1 (SNe 2017cbv, 2018oh, 2019yvq, and iPTF14atg). In Sect. 9.3, we discuss the case of PTF10ops separately, which has been suggested to show excess flux at early times ), but its true nature is unclear due to a limited data set. In the following sections, we compare our models to observations of SNe Ia with early bumps split into 'blue' and 'red' SN sub-classes, and discuss our models in the context of other scenarios that have been proposed for these objects.
Again, we stress that exact values for the core mass should not be taken literally due to the uncertainty in the amount of 56 Ni produced (Sect. 3.1), but are given for reference. The shell masses presented here are likely more robust predictions as our models cover a wide range of burned masses and products, and the observed shape of the bump will be highly sensitive to the mass of the shell. Figure 14 shows a comparison between the light curves of SNe 2017cbv, 2018oh, 2019yvq, iPTF14atg, and four of our double detonation models with 56 Ni-dominated shells. These models are broadly able to reproduce the shape of the early light curve bump. In Fig. 15, we compare spectra of each SN and model around maximum light.

Blue SNe Ia with an early bump
The light curve of SN 2017cbv was previously compared to double detonation models by Maeda et al. (2018), however their models were somewhat too faint (SN 2017cbv was a bright SN Ia and showed a peak absolute magnitude of B = −20.4; Hosseinzadeh et al. 2017). In Fig. 14(a), we show that our model with a 1.1 core and massive helium shell of 0.10 dominated by 56 Ni can generally reproduce the shape of the early light curve of SN 2017cbv. The 56 Ni mass of this model is 0.94 . For the model shown in Fig. 14(a), the -band shows a more pronounced bump than observed; we speculate that minor changes to the composition within the helium shell could be made to find improved agreement in this band. Regardless of this discrepancy, beginning approximately three weeks after explosion the model light curves show a much faster decline in the bluer bands ( , , and ) than SN 2017cbv. This is further demonstrated by Fig. 15, which shows that the maximum light spectrum of our double detonation model with a 56 Ni-dominated shell exhibits significant line blanketing that is inconsistent with SN 2017cbv. In addition, the spectral features are also inconsistent with SN 2017cbv. Our model shows a significantly broadened Si 6 355 feature that has blended with Si 5 972. In contrast, the observed spectra of SN 2017cbv show a well defined Si 6 355 feature and lacks Si 5 972. Hosseinzadeh et al. (2017) compare observations of SN 2017cbv to models of the interaction between the SN ejecta and a non-degenerate companion star. While they find that interaction with a 56 subgiant is able to reproduce the bump in the optical bands, the model over-predicts the flux in the UV bands ( 1, 2, and 2).  Hosseinzadeh et al. (2017) argue that this could indicate the flux resulting from the interaction is not a black-body, or an alternative explanation is required. Based on nebular spectra hundreds of days after explosion, Sand et al. (2018) rule out the presence of any significant H signatures, which are expected if material is stripped from a non-degenerate companion. Finally, the presence of CSM was also ruled out by Ferretti et al. (2017). Our models indicate that SN 2017cbv likely did not result from a double detonation explosion. Taken together, the nature of SN 2017cbv is still uncertain. Dimitriadis et al. (2019) compare SN 2018oh to a double detonation explosion model with a 0.98 core and 0.05 helium shell, which produces 0.45 of 56 Ni. To match the early light curve bump of SN 2018oh, Dimitriadis et al. (2019) invoke mixing of the SN ejecta after explosion. Therefore, the model does not produce a well defined early bump in the light curve, but rather shows an extended 'shoulder' that more closely resembles SN 2018oh. It is not clear however, whether such mixing could be achieved in double detonation explosions. Figure 14(b) shows that our model with a 1.0 core and thin helium shell of 0.04 (i.e. a 56 Ni mass of 0.49 ) with a narrow isotope distribution dominated by 56 Ni provides reasonable agreement to the light curve shape of SN 2018oh. A narrow isotope distribution is required to reduce the mass of 52 Fe in the shell and avoid the peak produced by its short lifetime. Even for this distribution however, the bump in the model Kepler band light curve is more pronounced than observed in SN 2018oh. Reducing the 52 Fe mass further may again provide improved agreement as  have shown that the light curve bump in SN 2018oh can be reproduced by a model containing a clump of pure 56 Ni in the outer ejecta.
Although this model is less affected by line blanketing than the model for SN 2017cbv, due to the lower mass of the helium shell, Fig. 15 shows that the spectral features of SN 2018oh are also inconsistent with a double-detonation scenario. Dimitriadis et al. (2019) did not consider the spectra of their double-detonation models compared to SN 2018oh. In summary, our models indicate that SN 2018oh did not result from a double-detonation explosion as we are unable to simultaneously match the spectroscopic features and early light curve bump, regardless of the composition of the helium shell.
The favoured interpretation for SN 2018oh by Dimitriadis et al. (2019) is that of interaction with a non-degenerate companion. Again similar to SN 2017cbv, no evidence for material stripped from a non-degenerate companion has been found in late-time spectra of SN 2018oh (Tucker et al. 2019). The case of interaction with CSM was investigated by Shappee et al. (2019), who found that none of their interaction models could satisfactorily reproduce the initial early light curve shape of SN 2018oh. An alternative case of interaction for SN 2018oh was suggested by Levanon & Soker (2019). In this scenario, the explosion follows from the merger of two white dwarfs. An accretion disk forms around the primary (Raskin et al. 2012;Zhu et al. 2013), which serves as the source of the CSM. After explosion, the SN ejecta shocks material in the disk, producing a flash of UV radiation that may be similar to that of SN 2018oh (Levanon et al. 2015). This scenario warrants further investigation with radiative transfer simulations, as alternatives appear to be ruled out for SN 2018oh.
In Fig. 14(c), we compare a double-detonation model to SN 2019yvq (Miller et al. 2020a). SN 2019yvq was a somewhat peculiar SN -it was slightly under-luminous, but showed high ve-  Fig. 14(c) we show our model with comparable values -a 0.9 core and 0.04 shell dominated by 56 Ni. The similarity of these values is unsurprising, given that Miller et al. (2020a) use the same modelling treatment as Polin et al. (2019), upon which our models are at least partially based. As in Miller et al. (2020a), our model generally matches the early light curve bump in the redder bands ( and ), but the -band shows a larger decrease in magnitude than is observed immediately following the peak of the bump. Even for models with different dominant products (e.g. 52 Fe and 48 Cr) in the shell, we are not able to simultaneously match the bump in both the -and -bands. As with SNe 2017cbv and 2018oh, Fig. 15 shows that the maximum light spectrum of SN 2019yvq is significantly bluer than the model and does not exhibit strong line blanketing.
In addition to double detonation explosions, Miller et al. (2020a) investigate other scenarios to explain SN 2019yvq, including an excess of 56 Ni in the outer ejecta, a violent merger, and companion interaction. None of the proposed scenarios fully explain all of the observed features. One possible exception is the violent merger sce- nario. While it is likely that this scenario does produce some CSM, models including this material are currently unavailable. Based on the identification of calcium in nebular spectra and favourable comparisons with model nebular spectra, Siebert et al. (2020) argue that SN 2019yvq was indeed the result of a helium shell detonation. The light curve curve of the model favoured by Siebert et al. (2020) however, does not reproduce what is observed in SN 2019yvq. The model is simultaneously too bright and does not show a pronounced bump at early times. Whether it is possible to simultaneously match the early-and late-time observations of SN 2019yvq requires further investigation.
Finally, Fig. 14(d) shows iPTF14atg compared to a doubledetonation model with a 0.9 core and thin helium shell of 0.01 . This model contains only 0.13 of 56 Ni. Unlike the other objects discussed here, the early bump observed in iPTF14atg was less pronounced in the optical bands, but clearly apparent at UV wavelengths. (Cao et al. 2015). The origin of this early excess was discussed by Cao et al. (2015), who argued that it is consistent with theoretical predictions of the collision between the SN ejecta and companion star. They also discuss the possibility of this excess arising from 56 Ni at the surface of the ejecta, such as in double detonation models, and estimate that this would require a 56 Ni mass of ∼0.01 at the surface. Figure 14(d) shows that even for our model with a 0.01 shell, the early light curve bump produced is inconsistent with iPTF14atg. Figure 15 also shows the maximum light spectrum of iPTF14atg. Our model predicts a spectrum at maximum light that is substantially redder than iPTF14atg, and shows strong flux suppression for wavelengths 4 200 Å. The origin of iPTF14atg was also considered by Kromer et al. (2016), who favour the violent merger of two white dwarfs. This particular realisation of the violent merger scenario did not predict a light curve bump, but interaction due to the presence of some CSM for similar models may be consistent with the observations.

Red SNe Ia with an early bump
Observations of SN 2016jhr were presented by Jiang et al. (2017), who found reasonable agreement with models of double detonations and either sub-or near-Chandrasekhar mass white dwarfs. In Fig. 16, we show a comparison between some of our double-detonation models and SN 2016jhr. We note that unlike Jiang et al. (2017), we do not apply K-corrections to the observed light curve as these may be uncertain due to the peculiar nature of SN 2016jhr. Instead, we transform our model spectra into the observer frame (redshift of 0.117) and calculate observer frame light curves. For all comparisons, we assume an explosion date of MJD = 57482.0, which is a few hours before the first detection. We also assume a distance modulus of = 38.99 mag, which is 0.3 mag higher than that used by Jiang et al. (2017). (2019) (1.0 core and 0.05 helium shell). Our model contains 0.49 of 56 Ni and is able to broadly reproduce the light curve shape of SN 2016jhr during the first few days after explosion, but shows a much faster decline in the -band than observed. Assuming a shell dominated by 56 Ni (Fig. 16(b)), we again find that our model can reproduce the early light curve bump. We also find improved agreement in the -band close to maximum light, however the model still declines somewhat faster than observed. In Fig. 16(c), we show a model assuming our narrow isotope distribution. In this model, a much larger fraction of 56 Ni is present in the shell relative to 52 Fe than in our standard distribution. In this case, the -band light curve does not display a pronounced bump and instead shows a shoulder to the light curve that is still generally consistent with SN 2016jhr. For the -band, this model is clearly brighter during the bump than the observations, however a lower burned fraction may produce more favourable agreement (the models shown in Fig. 16(a) & (b) both have burned fractions of 0.2, while the model in Fig. 16(c) has a burned fraction of 0.5). Around maximum light, this model also produces a broader -band light curve that more closely resembles SN 2016jhr. As a further point of comparison, we also show a model with a 32 Sdominated shell (Fig. 16(d)). In this case, the model clearly does not reproduce the early light curve bump, but provides good agreement around maximum light.
In Fig. 17, we show spectra for the models presented in Fig. 16  at 18.25 d after explosion and compare them to SN 2016jhr approximately two days before maximum light. As expected from the light curve comparison, Fig. 17 shows that our standard isotope distribution models with 52 Fe-and 56 Ni-dominated shells do a reasonable job of reproducing the maximum light spectrum. The 56 Ni-dominated shell produces better agreement at shorter wavelengths, while the 52 Fe-dominated shell model shows more extreme flux suppression. In both cases, the continuum flux around ∼5 000 -5 550 Å is higher than observed. The narrow distribution model shows a Si 6 355 velocity closer to SN 2016jhr, but also does not manage to reproduce the spectrum at shorter wavelengths. Again, we speculate that a smaller mass of burned material would produce improved agreement in this case. For our 32 S-dominated shell model, we find that the Taken together, our models corroborate the claims of Jiang et al. (2017) that SN 2016jhr is consistent with a double detonation explosion. The exact composition of the shell required is unclear, although it must include at least some amount of short-lived radioactive isotopes. Minor changes to the models presented here could provide improved agreement. Assuming a helium shell dominated by IMEs, we also find good agreement with the light curves and spectrum close to maximum light, although the model spectrum is too blue, which could indicate an alternative explanation for the early light cure bump is also possible. Indeed, a small 56 Ni excess in the outer ejecta may also provide good agreement with the early light curve shape and produce a redder spectrum consistent with SN 2016jhr.
In addition to SN 2016jhr, SN 2018byg also shows a peculiar early light curve and extremely red colours close to maximum light. The observations of SN 2018byg were presented by De et al. (2019), who show that SN 2018byg displays a shoulder to the early rise of the -band light curve. De et al. (2019) argue that SN 2018byg is consistent with the double detonation of a low mass white dwarf core (∼0.75 ) and massive helium shell (∼0.15 ). De et al. (2019) were unable to reproduce the early light curve shape of SN 2018byg within the standard double detonation scenario and find all models produce a significant light curve bump that is not observed. Instead, De et al. (2019) artificially performed mixing of the ejecta to match the light curve shape, as in Dimitriadis et al. (2019) for the case of SN 2018oh. Again, it is not clear how such mixing could be achieved.
As the parameter space of our model set does not cover the appropriate 56 Ni mass range predicted for SN 2018byg, our models are all much brighter than the observations. Therefore, in Fig. 18, we show spectra for one of our models with a low mass white dwarf (0.9 ) and thick helium shell (0.1 ) dominated by 56 Ni that is scaled to match the flux of SN 2018byg. Figure 18 shows that approximately 10 d after explosion, our model generally reproduces the spectrum of SN 2018byg at −10 d relative to maximum light. SN 2018byg shows a relatively flat continuum between ∼5 500 -7 000 Å, while our model shows high-velocity Si . Closer to maximum light, our model provides excellent agreement with SN 2018byg and is able to reproduce the extreme flux suppression at wavelengths 5 000 Å as well as the Si features around ∼6 000 Å.

PTF10ops
PTF10ops was a peculiar SN Ia that showed a light curve significantly broader than expected for its low luminosity ( B = −17.77 ± 0.04; Maguire et al. 2011). The early light curve of PTF10ops is not well sampled and therefore it is unclear whether it belongs to the group of SNe showing bumps at early times. For this reason, we opt to discuss it separately. Jiang et al. (2018) argued that there was evidence for an early flux excess, however they stress this is based on a single point. First detection occurred approximately 17 d beforeband maximum, while the next detection was six days later. Making a definitive statement on the origin of PTF10ops is therefore a challenging prospect. Here we discuss whether PTF10ops is consistent with models of double detonation explosions.
In Fig. 19, we show a comparison between the light curve of PTF10ops and models with either a 56 Ni-or 32 S-dominated shell. These models contain a core mass of 0.9 and shell mass of 0.01 . The mass of 56 Ni produced in the core is 0.13 . For the 56 Ni-dominated shell, our model shows a short-lived bump approximately one day after explosion that is consistent with the earliest detection of PTF10ops. The lack of detections in the following days and in other bands means that the decline from this initial bump could have simply been missed. This model also provides a good match to the later light curve evolution, but is slightly too faint in the -band. Conversely, the 32 S-dominated model does not match the earliest detection. Again, this model is able to match the light curve towards maximum light, but remains too faint in the -band. Figure 20 shows spectra for both models compared to PTF10ops approximately one week before maximum light. While both models Comparison between the spectrum of PTF10ops approximately one week before -band maximum (black) and our models with a 0.9 core, 0.01 shell and either a 56 Ni-dominated (red) or 32 S-dominated (purple) composition for the shell. Phases for the model spectra are given as days since explosion. All spectra are shown in an absolute flux scale.
provide good agreement for wavelengths 4 500 Å, it is clear that the 56 Ni-dominated shell model shows significant line blanketing that is inconsistent with PTF10ops. The 32 S-dominated shell model provides improved agreement, although it shows much stronger Ca features than those observed.
Our models show that if PTF10ops did indeed have an excess of flux at early times, this was not due to a helium shell detonation as its spectra do not show significant line blanketing. Double detonation models in which the helium shell is dominated by IMEs provide better agreement overall, but they are not able to match the first detection in the light curve. As with normal SNe Ia (Sect. 8), this could indicate that a somewhat extended 56 Ni distribution may be required for these sub-Chandrasekhar mass models. Alternatively, any early excess emission could be due to interaction, but the lack of nebular spectra and indeed the poorly sampled early light curve makes a definitive conclusion about the nature of PTF10ops a challenging prospect.

Summary
By comparing our models to observations of SNe Ia with early bumps, we show that a variety of shell masses and compositions is necessary to reproduce the diversity observed. While double detonation models can match the shapes of the early bumps for all objects (with the exception of iPTF14atg), only those with red colours at maximum light are well matched both photometrically and spectroscopically throughout their evolution following the bump. An investigation of the extent to which double detonations can explain these blue objects requires further observations for a larger sample of objects. We note however, that all of the currently proposed mechanisms for producing early light curve bumps appear inconsistent with these blue objects in at least some way. These discrepancies may be due to incorrect colours or lacking features predicted from models of companion and CSM interaction in nebular spectra.

CONCLUSIONS
We have presented a large-scale parameter study of the double detonation explosion scenario. Using the Monte Carlo radiative transfer code presented by Magee et al. (2018), we calculated light curves and spectra for parameterised ejecta structures that were designed to broadly mimic predictions from theoretical explosion models (e.g. Kromer et al. 2010;Polin et al. 2019;Gronow et al. 2020). We considered a range of white dwarf core masses (0.9 -1.2 ) and helium shell masses (0.01 -0.10 ), which effectively amounts to a range of 56 Ni masses. We also considered, for the first time, a large range of possible compositions for the burned material produced in the helium shell, which may result from different levels of pollution in the shell pre-explosion (Shen & Bildsten 2009;Waldman et al. 2011;Kromer et al. 2010;Gronow et al. 2020).
Broadly, our model set may be separated into two categories: those that contain iron-group elements (IGE) in the shell and those that do not. Consistent with previous studies (e.g. Noebauer et al. 2017;Jiang et al. 2017;Polin et al. 2019), we find that those models containing IGE in the shell produce a bump in their respective light curves within the days following explosion. The luminosity and timescale of the bump can show considerable variation, reaching up to B ∼ −18 and lasting a few days for massive shells. Although the bump is most pronounced for bluer bands (e.g. ), it is also visible at longer wavelengths. At later times, light curves and spectra show extremely red colours and much of the flux below ∼4 500 Å has been suppressed. This also leads to fast declining light curves with rise times to -band maximum typically around two weeks. Conversely, models that do not contain IGE in the shell show a relatively flat and blue colour evolution, and longer rise times that are more typical of normal SNe Ia.
As shown previously Townsley et al. 2019), models that do not contain IGE in the shell provide good agreement with observations of normal SNe Ia around maximum light. Here, we have extended this and shown that the double detonation scenario is consistent with normal SNe Ia beginning a few days after explosion. Our models do not provide evidence that the helium shell must contain specific elements (e.g. 32 S), but rather show that it cannot contain IGE and beyond this requirement the composition has little effect. Therefore, provided the helium shell does not produce IGE during the explosion (which could be due to some amount of pollution), the double detonation scenario may be considered viable for a range of normal SNe Ia and cannot be excluded. Future explosion models should investigate this further by exploring a range of core and helium shell masses, as well as initial helium shell compositions.
We also compared our models to SNe Ia that show early bumps in their light curves. While we find that the bumps of all objects (with the exception of iPTF14atg) can be reproduced, only those objects with red colours at maximum light ( − 1) are matched throughout their evolution. For blue objects, the model spectra at maximum light typically show broader features than observed, in addition to strong flux suppression. Regardless of the composition of the shell, we are unable to simultaneously match the early light curve and maximum light spectra of these blue events. The discovery of additional objects with early light curve bumps will help to determine the limit of the double-detonation scenario in reproducing observed light curve bumps.
Given that our double detonation models are unable to reproduce the complete evolution of blue SNe Ia showing bumps at early times, this would indicate that an alternative source for the light curve bumps of these blue objects is necessary. Previous studies have also considered alternative scenarios and generally there is at least some disagreement between these scenarios and the observations. This may be due to either an over-or under-prediction of UV flux or the lack of features predicted by companion and CSM interaction scenarios in nebular spectra. It is therefore clear that there is much that remains unknown about the origin of the light curve bumps in SNe Ia. As current and future facilities, such as the Zwicky Transient Facility and the Vera C. Rubin observatory (LSST), discover more SNe Ia within hours of explosion, an investigation of general trends among the class will become possible. This should provide further insights into the nature of these enigmatic bumps.

APPENDIX A: START TIME CONVERGENCE TEST
All models presented in this work were calculated assuming a start time of 0.5 d after explosion, which is comparable to the decay timescale of some of the radioactive isotopes included in the model. For packets injected before the start of the simulation, diffusion relative to the matter in the ejecta is assumed to be negligible. The energy of these packets is also reduced to account for work done on the ejecta (see e.g. Lucy 2005).
In Fig. A1 we show 52 Fe-dominated shell models calculated with earlier start times of 0.2 d and 0.4 d after explosion, as well as later times of 0.6 d, 0.8 d, and 1.0 d after explosion. As demonstrated by Fig. A1, the choice of 0.5 d after explosion does not significantly impact the light curve. From ∼1 d after explosion, all models produce comparable light curves to each other and show only minor variations ( 0.05 mag.) consistent with Monte Carlo noise. Assuming a later start time of 0.8 d after explosion does not reproduce the first light curve point of our nominal 0.5 d case. For earlier start times however, the light curves show good agreement with our nominal case. Therefore, although the half-life of 52 Fe, for example, is shorter than our 0.5 d start time, the light curves in this case do not show significant variation when assuming earlier start times.  Figure B1. Light curves and colours for models with different core 56 Ni masses. All models shown have a 1.0 core and a 0.10 shell, of which 50% is burned to elements heavier than helium. We show models with dominant shell products of 52 Fe and 32 S, as representative of models with IGE-and IME-dominated shells. et al. (2019). Kushnir et al. (2020) present a detailed study of bare, sub-Chandrasekhar mass detonation models and show that the 56 Ni masses presented in these works may be systematically lower than those determined by other studies. Predicted 56 Ni masses calculated by Kushnir et al. (2020) are shown in Fig. 3(a) for their default setup. Also shown are 56 Ni masses presented by Shen et al. (2018) for white dwarfs in the range ∼0.9 -1.1 , which show only minor variations for different metallicities and agree with those of Kushnir et al. (2020). Figure 3(a) shows that differences in predicted 56 Ni masses may be relatively large -particularly for lower mass white dwarfs. To account for this uncertainty in the 56 Ni mass produced, we present an additional set of models in which the core 56 Ni mass is based on the Kushnir et al. (2020) models. Using a linear fit to these models, the core 56 Ni mass is given by:

APPENDIX B: EFFECTS OF CORE
( 56 Ni) = 2.6 × ( core + shell ) − 2.1, where core and shell are again the mass of the carbon-oxygen core and helium shell, respectively, in units of . In Fig. B1, we show a comparison between models with 56 Ni masses based on Polin et al. (2019) (Eqn. 1) and those based on Kushnir et al. (2020) (Eqn. B1). As expected, those models with increased 56 Ni masses show systematically brighter peak luminosities, earlier rises, and overall bluer colours compared to their lower mass counterparts. Figure B1 shows that the increased 56 Ni mass results in brighter -and -band peaks by ∼0.35 mag for our 32 S-dominated shell model. This model also begins to rise slightly earlier (by ∼0.5 d), although there is not a significant shift in the time of peak brightness (i.e. 0.5 d). For our 56 Ni-and 52 Fe-dominated shell models, the changes from an increased 56 Ni mass are more dramatic, particularly in the -band. In these cases, the -band peaks are brighter by ∼0.6 -0.7 mag., while the -band light curves experience slightly more modest increases of ∼0.4 -0.5 mag. Again, these models are brighter at early times, but there is no shift in the time of peak brightness. Although the shape of the main rising light curves differ due to the increased 56 Ni masses, the shapes of the early light curve bumps are unaffected as these are primarily driven by the material in the . Light curves and colours for models with different core 56 Ni masses. All models shown have a 1.0 core and a 0.10 shell, of which 50% is burned to elements heavier than helium. We show models with dominant shell products of 52 Fe and 32 S, as representative of models with IGE-and IME-dominated shells.
shell. The rise time and magnitude of the bump peak is unaffected, however the decline after the peak of the bump is less pronounced. As the increased 56 Ni mass models begin to rise earlier, the difference between the peak of the bump and the minimum after the bump is reduced. This may make bumps less distinguishable in some cases as the earlier rise of the light curve produces more of a 'shoulder' in the light curve than a well-defined rise and decline (such as those shown in the -band in Fig. B1. In Fig. B1, we also show the colour evolution for these models. This is further reflected in Fig. B2, which shows spectra for each model at 3.25 d, 7.25 d, and 18.25 d after explosion. For our 32 Sdominated shell model, the − colour is bluer by 0.15 mag. throughout its evolution. This model also exhibits slightly higher velocities for the spectral features produced at all epochs. Again, the 56 Ni-and 52 Fe-dominated shell models show larger changes. At their reddest points (∼4 d after explosion), the higher 56 Ni mass models show a shift to bluer − colours between ∼0.6 -0.8 mag, while at their bluest points this shift is ∼0.3 -0.4 mag. Therefore the 'red bump' (i.e. the transition a few days after explosion from blue colours to red, and back to blue again) because less pronounced and the Δ − for this transition decreases from ∼1.6 -1.7 mag. to ∼1.0 -1.4 mag. Interestingly, our 56 Ni-and 52 Fe-dominated shell models with increased 56 Ni masses show a bluer − colour than the 32 Sdominated shell model between approximately one and two weeks after explosion. This is somewhat misleading, as Fig. B2 shows that the spectra at these epochs for our 32 S-dominated shell model are bluer and this is indeed reflected in the − and − colours. The appearance of the bluer − colours for the 56 Ni-and 52 Fedominated shells is likely due to increased fluorescence emission at these wavelengths. Similar to the 32 S-dominated shell model, the increased 56 Ni mass also causes a slight shift to higher velocities for spectral features in the case of our 56 Ni-and 52 Fe-dominated shell models.  Figure C1. Light curves and colours for models with different shell isotope distributions. All models shown have a 1.0 core and a 0.04 shell, of which 50% is burned to elements heavier than helium. We show models with dominant shell products of 52 Fe and 32 S, as representative of models with IGE-and IME-dominated shells.

APPENDIX C: EFFECTS OF RELATIVE ISOTOPE ABUNDANCES
In Fig. 4, we show the assumed relative abundances of isotopes in the helium shell for our models. In addition, we also show distributions in which the relative mass fraction of the dominant shell product decreases (broad distribution) or increases (narrow distribution). Here we discuss how these different distributions affect the light curves and spectra for our models.
As demonstrated in Fig. C1, changes in the relative abundances of isotopes in the helium shell have only minor effects on the light curves. For our 32 S-dominated shell models, only a slight change in colour is observed at early times. For the 52 Fe-dominated shell models, the effect is most pronounced in the -band at early times. Relative to our standard case, the broad distribution shows a somewhat fainter early bump, which is not surprising given the decrease in mass fraction of radioactive isotopes. At later times, the narrow distribution is brighter than both our standard and broad distributions in the -band. In the narrow distribution, as the mass fraction of 52 Fe increases, the relative fractions of all other isotopes decrease. Therefore, that the narrow distribution is brighter at later times likely points to the decreased contribution to line blanketing from having fewer different elements present in the ejecta.
In Fig. C2, we show how the spectra are affected by changes in the relative abundances of the shell isotopes. Again, it is clear that our 32 S dominated shell model shows only minor changes throughout its spectral evolution. At early times, the effect of a decreased 52 Fe fraction is clearly apparent from the fainter and redder spectrum. This paper has been typeset from a T E X/L A T E X file prepared by the author. . Spectra for models with different shell isotope distributions. All models shown have a 1.0 core and a 0.04 shell, of which 50% is burned to elements heavier than helium. We show models with dominant shell products of 52 Fe and 32 S, as representative of models with IGE-and IMEdominated shells. Spectra are shown at three epochs relative to explosion: 3.25 d, 7.25 d, and 18.25 d.