ON MEASURING THE METALLICITY OF A TYPE IA SUPERNOVA'S PROGENITOR

We chose two realizations of a CO white dwarf with a central density of 2 × 10⁹ g cm⁻³. Krueger et al. (2012) created 5 CO white dwarf progenitors with central densities that ranged from 1 × 10⁹ to 5 × 10⁹ g cm⁻³. For each progenitor, 30 realizations were created by applying initial conditions in the form of random perturbations on the initial burned region. This was done to better understand potential systematic biases such as how the morphology of the initial conditions might influence the results. Many of their progenitors produced higher yields of ⁵⁶Ni than is typical in SNe Ia. In light of this, we selected realizations R01 and R10 due to their lower estimated ⁵⁶Ni yields, 0.8 ${M}_{\odot }$ and 0.7 ${M}_{\odot }$, respectively, which are closer to typical values for SN Ia (Howell et al. 2009a). In this paper, we refer to these two simulations as DDT-high and DDT-low.

We recalculated the simulations because the original simulations did not feature the tracer particles that are needed for this work (see below). For this work, we used a newer version of the FLASH code, version 4.0. The simulation results are nearly identical to those published in Krueger et al. (2012), with some small changes due to changes to the hydrodynamics method in the intervening FLASH releases. The overall software consists of the publically available version of the FLASH software instrument⁸ along with added components⁹ for efficient modeling of carbon–oxygen flames and detonations developed during our previous work (Calder et al. 2007; Townsley et al. 2007, 2009; Jackson et al. 2010; Krueger et al. 2012) as well as software for computing nucleosynthesis in post-processing (Townsley et al. 2015; see footnote 9). In addition to the hydrodynamics itself, the current public FLASH software release includes the advection–diffusion treatment used to model the propagation of the deflagration. The FLASH modules implementing the carbon–oxygen burning model (see footnote 9) will also be included in a future public release of the FLASH software instrument.

Our simulations begin at the end of the core convection phase of the runaway just after the ignition of a propagating thermonuclear flame (see Nonaka et al. 2012 for a discussion of this ignition process). We make the assumption that the flame is ignited in such a way that it spreads in the center of the star before plumes rise to lower densities. This is an analog of the "multi-point" ignition scenarios explored by other authors in three dimensions (e.g., Seitenzahl et al. 2013 for an example). Nonaka et al. (2012) suggest that multi-point ignitions are likely disfavored by nature. Nevertheless, as others have shown previously and we demonstrate again in this work, this does lead to an explosion that reproduces spectral observations of normal SNe Ia fairly well. Our simulations follow the nuclear and hydrodynamic evolution of the flame within the star until it reaches a density of ${10}^{7.1}$ g cm⁻³, at which point it is assumed that a detonation is ignited by a DDT mechanism (Khokhlov 1991; Poludnenko et al. 2011). This detonation is ignited with the introduction of a small heated region since all burning fronts, and therefore any physical DDT processes, are unresolved in our full-star simulations. The complete incineration of the star by the detonation then ensues. Active burning fronts cease due to the expansion of the star by about 1.8 s, and then we follow the expansion of the star to 4 s, at which point it is already close to force-free expansion.

Our model of carbon–oxygen burning used in hydrodynamics is a three-level burning model that is calibrated against steady-state deflagrations and detonations. The resulting temperature, T, and density, ρ, in the explosion simulation enable post-processing of Lagrangian tracer particles (see next section) to obtain abundances behind the reaction front with an accuracy of 5%–10% (Townsley et al. 2015). The global energy deposition based on the burning model is consistent with the yields obtained from post-processing to within 10%.

During the explosion simulations, the temperature–density histories of a set of 100,000 Lagrangian tracer particles are recorded. These tracer particles are inactive, meaning that they do not affect the hydrodynamics but are only advected with the flow, recording the history of local fluid variables. During the burning phase of the explosion, the histories are recorded every time step, giving the full time resolution data which is needed for accurate post-processing (Townsley et al. 2015). The equal-mass particles are distributed randomly. After the hydrodynamic simulation is complete, we use the TORCH¹⁰ software instrument to compute the evolution of a set of 225 nuclides subject to the conditions recorded for a given fluid element. This post-processing step allows the computation of detailed nucleosynthetic yields without having to compute the reaction or advection of all of these species in the main hydrodynamic simulation.

The density and temperature histories for W7 were obtained from F. Thielemann (2001, private communication). These consist of about 500 time steps for 175 zones out to 4.15 s. These histories were treated in the same manner as Lagrangian tracks from our 2D simulations. Similar yields were used by De et al. (2014) to investigate the dependence of Si-group yields on metallicity in comparison to trends expected from quasi-equilibrium calculations.

The nucleosynthetic yields from the explosion simulations are determined by post-processing the histories of Lagrangian tracer particles. The reaction network calculation, described in Townsley et al. (2015), determines mass fractions for each particle. As described in Townsley et al. (2015), for fluid that is processed by the deflagration, in post-processing we reconstruct a portion of the Lagrangian history in order to obtain a more realistic $\rho (t),T(t)$ near the artificially thickened flame front. We do not do so for detonations or for the W7 element histories, instead using the $\rho (t),T(t)$ history directly. After post-processing, radioactive products with half-lives smaller than five days, according to beta decay rates in TORCH, have their calculated abundances converted into their decay products. Three important decay chains of three isotopes are calculated during the radiation transport simulations: ⁵⁶Ni, ⁵²Fe, and ⁴⁸Cr. The decay chain of ⁵⁶Ni to ⁵⁶Fe is well understood to be critical in the powering of SNe Ia light curves. ⁵⁶Ni has a half-life of 6.075 days¹¹ and transitions by beta decay to ⁵⁶Co. ⁵⁶Co has a half-life of 77.236 days (see footnote 11) and transitions by beta decay to ⁵⁶Fe. The decay of ⁵²Fe to ⁵²Cr is complex. ⁵²Fe has a half-life of 8.275 hr (see footnote 11) and transitions by beta decay to the 2+ excited state of ⁵²Mn. The excited state has a half-life of 21.1 minutes (see footnote 11), but has two possible decay branches. 98.25% will transition by beta decay to ⁵²Co, and the remaining 1.75% decays to the ground state of ⁵²Mn via internal transition. The ground state has a half-life of 5.591 days (see footnote 11; Huo et al. 2007; Dessart et al. 2014). However, we do not track this small amount of ⁵²Mn due to its negligible influence on spectral features. ⁴⁸Cr has a half-life of 21.56 hr (see footnote 11) and decays to ⁴⁸V. The ⁴⁸V has a much longer half-life at 15.9735 days (see footnote 11), and therefore its presence and changing abundance is important throughout the radiation transport calculations.

The particles are then mapped onto a 1D spherically symmetric velocity grid with 100 cells to match the mapping of the radiation transport calculations, using the final velocities of each particle at the end of the simulation. The mass fractions in each cell of the velocity grid follow from the average mass fractions of all particles in that cell. These mass fractions per cell, together with a spherically symmetrized density profile from the hydro simulation¹² and the geometrical volume of each cell, give the total yields (in gram) per species.

If a velocity grid cell contains many particles, then we limit the number of particles that are post-processed to 100, which are selected randomly. This reduces the amount of time needed for post-processing. An independently selected set leads to abundances that differ on the order of 10% in velocity grid cells that contain greater than 100 particles, and total yields that differ by less than 5%.

The progenitor white dwarf is broadly comprised of two regions: a convective core, that is, the inner 0.8 ${M}_{\odot }$ of the star, and a non-convective outer region. Our progenitor abundances are constructed to be similar to those in Domínguez et al. (2001), as well as to account for pre-explosion carbon burning (Piro & Chang 2008).

In this work, the progenitor metallicity is varied by changing the initial abundance state of the tracer particles, not by calculating separate hydro simulations for different initial abundances, because the energy release by carbon–oxygen burning is only marginally modified by the metallicity variations. The hydrodynamic initial model is at a single metallicity of Z/Z${}_{\odot }$ = 1.33. The three parameters that control the carbon–oxygen burning model used in the hydrocalculation stand for the abundances of ¹²C, ¹⁶O, and ²²Ne. We choose the convective core to be made up of 40% ¹²C, 3% ²²Ne, with the remaining 57% as ¹⁶O; the envelope is 50% ¹²C, 48% ¹⁶O, and 2% ²²Ne. The ²²Ne is a stand-in for the neutron excess contributed from the metallicity in both the core and the outer layers (Timmes et al. 2003). In the convective region, the additional 1% of ²²Ne stands in for the neutron excess from the products of pre-detonation simmering (Chamulak et al. 2008; Piro & Bildsten 2008).

When post-processing the particles with a large nuclear network, more complete and realistic progenitor abundances can be used. Figure 1 shows how the material affected by metallicity and the simmering ashes are distributed throughout the star for both the reduced abundance set and the abundances used for post-processing. A particle's initial location is used to determine whether or not the particle began in the convection region, and thus the initial ¹²C fraction and the relative contributions from metallicity and simmering. The initial abundances for post-processing are then set using four components.

• 1.  
  The mass fraction ${X}_{{}^{12}{\rm{C}}}$ is set to the value at the particle's initial position in the hydrodynamic progenitor.
• 2.  
  The abundance of all of the other nuclides are first set to scaled solar abundances based on $Z/{Z}_{\odot }$ and abundances from Anders & Grevesse (1989). In doing this, we assume that all metals lighter than ¹⁸O, i.e., ¹²C, ¹⁴N, and ¹⁶O, are converted into ²²Ne, and are thus added to that abundance instead.
• 3.  
  A simmering contribution is added to some abundances for particles that are initially in the convective region. We approximate that the simmering ashes, of a fraction ${X}_{{\rm{sim}}}$ determined below, are made up of an even mix of ¹³C, ¹⁶O, ²⁰Ne, and ²³Ne (Chamulak et al. 2008).
• 4.  
  Once the ¹²C, ¹³C, ²⁰Ne, ²²Ne, ²³Ne, and all heavier metal abundances are set in this fashion, the remainder of the material is assumed to be ¹⁶O.

For this study, the contribution from simmering ashes is held fixed as the metallicity is varied. Changing the metallicity thus affects the second and fourth steps.

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The amount of simmering ashes, ${X}_{{\rm{sim}}}$, is set to obtain consistency with the progenitor metallicity of $Z/{Z}_{\odot }=1.33$ used in the hydrodynamics, based on Y_e,

$$\begin{eqnarray}&&{Y}_{e}=\displaystyle \sum _{i}{X}_{i}\displaystyle \frac{{Z}_{i}}{{A}_{i}},\end{eqnarray} \tag{ 1 }$$

in the outer layers. Here Z_i and A_i are the number of protons and number of nucleons in species i. Matching Y_e between the progenitor and the initial abundances used for post-processing gives

$$\begin{eqnarray}&&{X}_{{}^{16}{\rm{O}},\mathrm{prog}}\displaystyle \frac{8}{16}+{X}_{{}^{12}{\rm{C}},\mathrm{prog}}\displaystyle \frac{6}{12}+{X}_{{}^{22}\mathrm{Ne},\mathrm{prog}}\displaystyle \frac{10}{22}\\ &&\qquad ={X}_{{}^{16}{\rm{O}}}\displaystyle \frac{8}{16}+{X}_{{}^{12}{\rm{C}}}\displaystyle \frac{6}{12}+{X}_{Z}{Y}_{e,Z}\\ &&\qquad \quad +{X}_{{\rm{sim}}}\displaystyle \frac{1}{4}\left(\displaystyle \frac{8}{16}+\displaystyle \frac{6}{13}+\displaystyle \frac{10}{20}+\displaystyle \frac{10}{23}\right),\end{eqnarray} \tag{ 2 }$$

where abundances with the "prog" subscript are those found at the initial particle position in the progenitor used in the hydrodynamics, X_Z is the sum of the scaled solar abundances of all material in the metallicity-scaled component, including ²²Ne, ${Y}_{e,Z}$ is the Y_e of the metallicity-scaled component, and equal abundances of the four simmering products have been assumed. For the hydrodynamic abundances given above, $Z/{Z}_{\odot }=1.33$, solar abundances from Anders & Grevesse (1989), and ${X}_{{}^{16}{\rm{O}}}=1-{X}_{{\rm{sim}}}-{X}_{Z}-{X}_{{}^{12}{\rm{C}}}$, Equation (2) gives ${X}_{{\rm{sim}}}=0.0175$ in the convection zone and ${X}_{{\rm{sim}}}=0$ outside it. This value of ${X}_{{\rm{sim}}}$ is then used to construct the initial abundances for post-processing at all metallicities.

We calculate light curves and spectra using Phoenix. Phoenix is a stellar atmosphere and radiation transport software instrument (Hauschildt & Baron 1999, 2004; van Rossum 2012a). Phoenix numerically solves the special relativistic radiative transfer equation using the efficient and highly accurate short characteristic and operator splitting methods. It samples millions of atomic lines individually. Phoenix solves for the time evolution using the radiation energy balance method. The code does not use the Sobolev approximation, diffusion approximations, or opacity binning approximations. We operate Phoenix in one dimension, assuming spherical symmetry.

The deterministic radiation transport method employed in Phoenix calculates spectra without the noise that is inherent to Monte Carlo methods. This allows the study of small effects of individual atomic transition lines on the calculated spectrum through knockout spectra (van Rossum 2012b). Knockout spectra are calculated by post-processing a normal calculation of light curves and spectra. For the calculation of knockout spectra, the opacity of individual transition lines is artificially set to zero while keeping the temperatures, electron densities, and atomic occupation numbers fixed to the values determined in the original calculation. The difference between the original spectrum and a knockout spectrum is attributed to the same line opacity that was artificially set to zero. Here, as a diagnostic, we use knockout spectra to identify which chemical elements are responsible for the differences that metallicity changes cause in the calculated spectra.

We note that a spectrum is not merely a linear combination of individual lines. When multiple knockout spectra are shown together in one plot, each of the knockout spectra is interpreted as described above, but there is no useful interpretation to a linear combination of knockout spectra.

The total yields for six different progenitor metallicities, Z/Z${}_{\odot }$ ∈ [0.1, 0.5, 1.33, 2.0, 3.0, 4.0], of each of the three explosion simulations are listed in Table 1.

Table 1.  Nucleosynthetic Yields Grouped by Atomic Mass in ${M}_{\odot }$ for Three Simulations and Six Progenitor Metallicities

Z/${Z}_{\odot }$	0.1	0.5	1.33	2	3	4
DDT-high
COa	0.053	0.053	0.053	0.053	0.052	0.052
IMEb	0.322	0.320	0.314	0.310	0.302	0.295
IGEc	0.992	0.994	1.003	1.006	1.012	1.025
56Ni	0.848	0.831	0.800	0.775	0.738	0.703
DDT-low
CO	0.076	0.077	0.077	0.076	0.076	0.075
IME	0.385	0.382	0.375	0.369	0.360	0.351
IGE	0.906	0.909	0.916	0.922	0.932	0.941
56Ni	0.771	0.756	0.726	0.703	0.669	0.637
W7
CO	0.166	0.166	0.165	0.163	0.161	0.159
IME	0.306	0.304	0.300	0.297	0.292	0.288
IGE	0.889	0.890	0.896	0.900	0.907	0.913
56Ni	0.698	0.684	0.658	0.638	0.607	0.577

Notes.

^aCO: A = 12 and A = 16. ^bIME: $16\lt A\leqslant 40$. ^cIGE: $A\gt 40$.

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The yields are grouped into Carbon and Oxygen (CO, A = 12 and A = 16), IMEs ($16\lt A\leqslant 40$), and iron-group elements (IGEs, $A\gt 40$). ⁵⁶Ni is also included in the table because of its importance as an energy source for light curves and spectra. Table 1 shows that the amount of radioactive Nickel decreases with increasing progenitor metallicity while the total yields of IGEs increases due to the increased neutronization.

The relative change in abundances with progenitor metallicity is plotted in Figure 2. The effect of progenitor metallicity on the nucleosynthetic yields of the explosion are fairly similar among the three models for most elements, especially for the DDT-high and DDT-low models. The most marked difference between W7 and the 2D DDT models is that the Ne and Na yields increase more with metallicity in the W7 model.

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Figure 3 shows the integrated yields of ²⁸Si, ⁴⁰Ca, and ⁵⁴Fe for the six progenitor metallicities. We find that with increasing metallicity, the overall yields of ²⁸Si stay relatively constant. Conversely, the total yields of ⁴⁰Ca decrease and ⁵⁴Fe increase with increasing metallicity similar to the trends seen in De et al. (2014) with decreasing Y_e. The relatively flat trend in the ²⁸Si yields makes this species a useful reference for comparing the trends in the yields of other species between the three models. The ratios of ⁴⁰Ca and ⁵⁴Fe relative to ²⁸Si are shown in Figure 4. As expected from Figure 2, the trends are similar among the models, although the absolute ratio of ⁴⁰Ca to ²⁸Si is different in the W7 model due to Ca being produced in higher amounts at a wider extent in the DDT models (Figure 5). The trends are also smooth over the metallicity range that we are studying.

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Figure 5 shows the chemical abundance profiles in velocity space from the DDT-high, DDT-low, and W7 models for two metallicities: Z = 0.5 and 2.0 ${Z}_{\odot }$. Differences in explosion model produce differences in the profiles for a single metallicity. While all three models feature a ⁵⁶Ni core that extends to approximately $1.0\times {10}^{4}\;\mathrm{km}\;{{\rm{s}}}^{-1}$, W7 is the only model that has a "nickel hole," which is a known characteristic of 1D models. In the higher-velocity portion of the ejecta, from ≈$1.6\times {10}^{4}$ km s⁻¹ to $2.5\times {10}^{4}$ km s⁻¹, the W7 model's ejecta consist of unburnt C, O, and Ne, which is much simpler than that produced in the DDT-high and DDT-low models. In this region of the material, the DDT-high and DDT-low models' profiles consist of predominantly C and O, with $\lesssim 10 \%$ of Si, Mg, Ne, S, Ni, Ar, and Ca. At the highest velocities, $\gtrsim 2.7\times {10}^{4}$ km s⁻¹, the DDT-high and DDT-low ejecta take on a configuration similar to that of the outer layers of W7, unburnt C, O, and Ne. At intermediate velocities, between $1.0\times {10}^{4}$ km s⁻¹ and $1.6\times {10}^{9}$, Si peaks in all three models. This is also the location of other IME peaks such as S, Ar, and Ca.

Examining the same explosion models at different metallicities presents additional features. At Z = 2 ${Z}_{\odot }$, ⁵⁶Ni makes up less of the total Ni yield in the three explosion models, although there is little change in the overall mass fraction of Ni. Overall, the Fe mass fraction increases with the increased metallicity. However, not only is the mass fraction of Fe higher, the Fe-rich layers extend further outward into the higher-velocity ejecta. In all three modes, the Fe-rich layer reaches into the material that is mostly IMEs. The increase in metallicity also causes the Ca peak to be both lower and narrower. The Si peak is largely unaffected by the increase in metallicity. This is also reflected in how little the overall Si abundance varies in Figure 3, and the small change in the Si ii spectral feature at 6150 Å with metallcity as shown in Section 3.3.

Figure 6 shows the abundance profiles of Si and Ti in the DDT and W7 models 30 days post-explosion. It demonstrates how the Ti production is sensitive to the progenitor metallicity while the Si production is largely unaffected. The variation of Ti with metallicity is strongest in the region of incomplete Si burning and the unburned regions. A shift in the quasi-equilibrium abundances due to metallicity is expected in the Si-burning region (De et al. 2014). The direct Ti production is mainly in the form of ⁴⁴Ti, which has a half-life of 60 years (see footnote 11), while the progenitor metallicity contributes mainly ⁴⁸Ti. Any ⁴⁸Cr produced during the explosion will decay with a half-life of 21.56 hr (see footnote 11) to ⁴⁸V, which will decay further with a half-life of 15.97 days (see footnote 11) to stable ⁴⁸Ti. We show the profile at day 30 post-explosion to facilitate discussion of Ti spectral features at this epoch in Section 3.3. At this time, 71% of the ⁴⁸Cr produced during the explosion has decayed to ⁴⁸Ti so that the Ti curves in Figure 6 in some regions are higher than the "direct Ti" curves.

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The bolometric light curves for the three explosion models with two extreme metallicities are shown in Figure 7. Higher progenitor metallicities systematically make the bolometric light curve rise more slowly and peak lower. The slower rise is caused by the higher abundance of metals in the outer layers of the ejecta, which raise the opacity and make radiation diffuse out more slowly from the hot ⁵⁶Ni-rich core. The lower peak brightness is caused by the lower total ⁵⁶Ni mass in the models with higher progenitor metallicities, as shown in Table 1.

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Figure 8 shows the UBVRI-band model light curves compared to the light curves from the data-driven model MLCS (Jha et al. 2007). The systematic trend with progenitor metallicity in most light curves is small but significant, especially in the UBV bands. This is apparent in Figure 9 where the light curve shape parameters, peak brightness, and decline rate ΔM₁₅(B) are plotted, along with the Phillips et al. (1999) relation (see also Mandel et al. 2011). The model light curves have slightly brighter peak magnitudes than the bulk of observed SNe Ia at the same ${\rm{\Delta }}{m}_{15}(B)$. We find for the two 2D DDT cases used here, which have above-average yields, that the changes in ignition conditions that lead to different ⁵⁶Ni yields cause the shape parameters to change perpendicular to the Phillips et al. (1999) relation. This confirms a similar result found by Seitenzahl et al. (2013) using three-dimensional (3D) DDT models. This failure of multi-dimensional DDT models to reproduce the Phillips relation in the expected way makes it difficult to make comparisons with empirical metallicity–luminosity relations. In contrast, the progenitor metallicity introduces a change in the B-band light curve shapes that roughly follows the observed width–luminosity relation.

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The bottom panel of Figure 9 shows the bolometric peak magnitude and decline rate. While higher metallicity leads to lower bolometric peak magnitudes, just as it leads to a lower B-band peak, the effect on ${\rm{\Delta }}{M}_{15}(\mathrm{bol})$ is very small. This indicates that the range of ${\rm{\Delta }}{M}_{15}(B)$ (top panel of Figure 9) spanned by metallicity is due entirely to a color effect, in which the B-band light decreases more quickly at higher metallicity while the decline rate of the total energy output is mostly unchanged. ${\rm{\Delta }}{M}_{15}(\mathrm{bol})$ is even slightly slower at higher metallicity, which is the opposite of ${\rm{\Delta }}{M}_{15}(B)$. In reality, any color effects caused by progenitor metallicity would be convolved with environmental factors such as extinction from dust. Light curve modeling techniques such as MLCS (Riess et al. 1996; Jha et al. 2007) and BayeSN (Mandel et al. 2011) are able to separate the two, allowing them to have the possibility of properly capturing the effects of the progenitor metallicity.

Model spectra, calculated for W7 and the two DDT simulations, with different progenitor metallicities, and at four different epochs post-explosion, are shown in Figure 10 through 13.

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The model spectra are compared against the Hsiao et al. (2007) spectral templates. These templates represent the typical spectral shape of normal SN Ia, as they are constructed as an average over many observed spectra. We set day 0 of the Hsiao07 templates (which is the time of maximum brightness) to the time that the B-band luminosity peaks in the model spectra, averaged over the set of progenitor metallicities, and fit a "distance" to the imaginary source of the Hsiao07 templates to simultaneously fit the spectra of all epochs. We do not apply any other corrections, such as stretch or color corrections, to the Hsiao07 templates.

The model spectra generally agree reasonably well with the templates, but there are a few significant discrepancies. First, none of the model spectra reproduce the far-UV flux below 2500 Å as it is observed on days 10 and 20 post-explosion (pe). Second, although the Si ii feature at 6150 Å is present in the model spectra, none of them accurately reproduce the observed shape of this feature. It is unclear whether this is a consequence of an under- or over-produced nucleosynthetic yield, or a missing line, or a combination of lines from the radiative transfer calculation. Third, the W7 model spectra do not reproduce the observed strong absorption feature around 3300 Å at day 10 pe.

Different progenitor metallicities give rise to variations in the model spectra. These variations change with time. Generally, the variations appear to affect the flux more on the blue side of the spectrum than on the red side and, over time, the range in which variations are strong gradually extends further to longer wavelengths. At day 10 pe, the DDT and W7 spectra are most effected in the 2600–3600 Å range. At days 20 and 30 pe. that window extends to 2600–4500 Å and 2600–5200 Å, respectively. In the latest spectra, at day 40 pe, the effect of progenitor metallicity presents only a small amount of variation.

The systematic decrease of blue (short-wavelength) flux with progenitor metallicity is correlated in part to the decline in total ⁵⁶Ni yield with metallicity. This makes it more difficult to use this region as an indicator of metallicity that is independent of overall ⁵⁶Ni yield. The total ⁵⁶Ni yield affects the brightness of the light curve as well as the temperature of the radiating ejecta, and thus the spectral color. At the same time, the progenitor metallicity affects the composition of the SN Ia ejecta, which may affect the strength of individual spectral features.

In order to help demonstrate the difference between the ⁵⁶Ni-mass–temperature–color effect and the composition-feature-strength effect, Figure 14 compares spectra at similar ⁵⁶Ni yields. From the top panel, for spectra at days 10 and 20 pe, we see that the region of the spectrum around 3000 Å does not show as strong a metallicity variation in this comparison as it does in Figure 10. The case in the bottom panel, with slightly higher yield and lower metallicity, shows more metallicity variation in this region. Together, these comparisons show that it will be necessary to carefully control for total ⁵⁶Ni yield in order to use a metallicity indicator from this spectral region.

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Features that vary consistently in both Figures 10 and 14 are good candidates for robust metallicity indicators that may not require careful control for ⁵⁶Ni yield, which renders the feature useless if the ⁵⁶Ni yield is not known. Two features that clearly vary with progenitor metallicity at day 30 pe in Figure 14 as well as in all spectra in Figure 10 are (1) an absorption feature around 4200 Å and (2) an absorption feature around 5200 Å. These two features are potential spectral indicators of progenitor metallicity. We will examine these features more below.

There are a couple of other features that are promising, but their variation with metallicity is less consistent for both varying (Figure 10) and fixed (Figure 14) ⁵⁶Ni. The variation with metallicity around 4400 Å at day 20 pe in Figure 10 is much less clear in Figure 14. The feature around 5500 Å at day 20 pe that appears to vary in Figure 14 does not vary in the same way in Figure 10. For this latter feature, it appears that the two DDT models produced different amounts of the chemical elements that are responsible for these features, and so this is not a simple effect of progenitor metallicity.

The Si ii P-Cygni feature at 6150 Å is present in the DDT and W7 model spectra at peak brightness and before peak brightness. Even though, as noted before, the observed shape of this feature is not very accurately reproduced in the model spectra, the change in metallicity has very little effect on both the depth of the absorption wing and the height of the emission wing. This is in agreement with the observation in Figures 2 and 3, namely, that the Si abundance remains relatively static for changes in the progenitor metallicity. This allows for its use as a possible calibration when examining the strengths of other features.

Two spectral features were identified above as potential spectral indicators of progenitor metallicity in all three models: (1) an absorption feature around 4200 Å and (2) an absorption feature around 5200 Å; both at day 30 pe. Both of these features become systematically stronger with increasing progenitor metallicity. Figure 15 shows the trend of the feature strength with progenitor metallicity, with the pseudo-equivalent width pEW as measure for feature strength:

$$\begin{eqnarray}&&\mathrm{pEW}={\int }_{{\lambda }_{a}}^{{\lambda }_{b}}\displaystyle \frac{{F}_{{\rm{pCont}}}(\lambda )-F(\lambda )}{F(\lambda )}d\lambda ,\end{eqnarray} \tag{ 3 }$$

where ${F}_{\mathrm{pCont}}$ is the pseudo continuum flux level as approximated by a linear interpolation between the flux levels F at the beginning and end points of integration ${\lambda }_{a}$ and ${\lambda }_{b}$. The interpolation range for the two features is chosen as [4000, 4400] and [4900, 5400] in units of Å.

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While the dependence of the pEW on metallicity is promising, the variation across metallicities in the $Z/{Z}_{\odot }=0.5$ to 2 range is similar to that among the three explosion models considered here. This means that in order to use a plot like Figure 15 to determine the metallicity, it is necessary to obtain a control variable that distinguishes among the models. Unfortunately, the most accesible control variable ${\rm{\Delta }}{m}_{15}(B)$ does not appear to fill this role well. The DDT-high model at $Z/{Z}_{\odot }=0.5$ and the DDT-low model at $Z/{Z}_{\odot }=2$ both have very similar ${\rm{\Delta }}{m}_{15}(B)$ and very similar 4000–4400 Å pEWs. Our limited sample of explosions available here does not allow us to robustly test control parameters. However, we are hopeful that a larger sample, either theoretical or observational, may be able to exploit the trends in pEW shown here to calibrate a measure of metallicity.

In order to investigate which chemical species are responsible for these two specific characteristic features, we use knockout spectra (see Section 2.5). Figures 16 and 17 displays a series of knockout spectra for the DDT-low and the W7 model at day 30 pe for Z/${Z}_{\odot }$ = 0.1, 1.33 and 4.

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These knockout spectra are created by removing the opacity from all of the transition lines of a single chemical element from the radiative transfer calculation and repeating the simulation with fixed temperatures, electron densities, and population numbers as in the original calculation. The recalculated spectra show the effect of the absence of opacity. This allows us to identify what and how strongly different elements are involved in forming specific spectral features.

In the knockout spectra, Figures 16 and 17, the full spectra are represented by the solid black line. The color patches represent the knockout effect that each element has on the full spectrum. Patches below the spectra indicate emission features: the flux level is that much lower in the absence of opacity from that species. In the same way, patches above the full spectrum represent absorption features.

Using the knockout spectra, we see that the two identified spectral indicators of progenitor metallicity, around 4200 and 5200 Å at day 30 pe, can be associated with Titanium and Iron production, respectively. The higher the progenitor metallicity, the stronger the Titanium and Iron abosption features appear in the spectra. Iron also shows a shift from emission to absorption in this feature as metallicity increases. Looking back at Figures 5 and 6, we see that, indeed, the Ti and Fe abundances increase with the progrenitor metallicity in the outer regions of the ejecta, i.e., at velocities $\gt {10}^{4}\;\mathrm{km}\;{{\rm{s}}}^{-1}$.

Foley & Kirshner (2013), and more recently, Graham et al. (2015) investigated the two SNe Ia SN2011fe, and SN2011by. These two objects exhibited remarkably similar optical spectra and decline times, and even peak brightnesses after accounting for possible errors in the measured distance to 2011by. Thus, the two objects become nearly identical. However, the two objects display differences in the near-UV regions of their spectra. Graham et al. (2015) propose that one possible source of these differences could be a difference in progenitor metallicity. To investigate this proposal, they first normalize the two spectra across 4000–5500 Å and then take the ratio of the two spectra, 11fe:11by. This ratio is then compared to the ratios of model spectra at difference metallicites produced by Lentz et al. (2000). By comparison, they find the best match between the observed spectral ratio and the model spectra corresponds to a factor of 30 difference in progenitor metallicity (Graham et al. 2015, Figure 8). In this section, we discuss how well our models reproduce the observed UV spectra in these cases, and challenges in making this type of comparison when metallicity and explosion strength (⁵⁶Ni mass) are allowed to vary independently.

In Figure 18, we take 4 pairs of spectra from our models at day 20 pe and follow a similar procedure as that of Graham et al. (2015). Two pairs, DDT-high Z = 0.5 ${Z}_{\odot }$ and DDT-low Z = 2.0 ${Z}_{\odot }$ and DDT-high Z = 2.0 ${Z}_{\odot }$ and DDT-low Z = 4.0 ${Z}_{\odot }$, have similar decline times as shown in Figure 9. The other two pairs, DDT-high Z = 2.0 ${Z}_{\odot }$ and DDT-low Z = 0.1 ${Z}_{\odot }$ and DDT-high Z = 4.0 ${Z}_{\odot }$ and DDT-low Z = 4.0 ${Z}_{\odot }$, have similar optical spectral features at day 20 pe. Each of the spectra are normalized across the 4000–5500 Å region. The top and middle panels of the left column of Figure 18 show the normalized spectra in log space of each pair selected for similar decline times. The top and middle panels of the right column of Figure 18 show the normalized spectra in log space of each pair selected for similar optical features. Also included in the top and middle panels of both columns of Figure 18 are the spectra of SN2011fe at -2 days from peak brightness (Mazzali et al. 2014) and SN2011by at -1 day (both M. Graham 2015, private communication). These spectra were first binned in increments of 5 Å and then normalized in the same way as our calculated spectra. Our calculated spectra share a similar sequence of features, especially blueward of about 3000 Å, with these two observed spectra. However, the features in the model spectra are both stronger and located at slightly shorter wavelengths. Also, the overall relative flux level blueward of about 2700 Å is about a factor of 10 lower in the model spectra.

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The bottom panels of both columns in Figure 18 show the ratios taken between each pair as well as the ratio of SN2011fe to SN2011by. One of the most prevalent differences in the model spectra is in the 3000 Å region, which is one of the regions most strongly effected by varying metallicity. The model pairs selected to have the same decline time show a factor of 4–6 ratio in this region, while the observed pair shows no significant difference. Comparing the ratios of pairs, DDT-high Z = 0.5 ${Z}_{\odot }$ and DDT-low Z = 2.0 ${Z}_{\odot }$ (Figure 18, bottom left panel, red curve) and DDT-high Z = 4.0 ${Z}_{\odot }$ and DDT-low Z = 4.0 ${Z}_{\odot }$ (Figure 18, bottom right panel, green curve), demonstrates the difficulty in separating metallicity from ⁵⁶Ni yield variations. The location of the peaks in these two pairwise comparisons is similar, having ratio peaks at about 1900, 2350, 2500, and 3600 Å. However, while the first pair are at different metallicities, the second pair are at the same metallicity but with different ⁵⁶Ni yields. A possible factor in the similarity can be seen in the ⁵⁶Ni yields. Looking at Table 1, it is clear that none of the four cases produced equal amounts of ⁵⁶Ni, although the ratios of ⁵⁶Ni produced in each pair are very similar at ≈1.1. Another contribution comes from the apparent blueshifting of features in the DDT-low spectra compared to the same features in the DDT-high spectra. In Section 3.3, we have discussed that many of the changes in the bluer side of the NUV region are caused by temperature effects from the changing ⁵⁶Ni yield. Since changes in spectral features from this temperature effect occur alongside changes from the varying ejecta composition, isolating the two effects by taking spectral ratios appears challenging. A more conclusive comparison will require model spectra that are more similar to the observations in this spectral range.