THE SPECTROSCOPIC DIVERSITY OF TYPE Ia SUPERNOVAE^*

As with the sample of 432 spectra published by Matheson et al. (2008), the majority of spectra presented here (2447 out of 2603 spectra or ∼94%) were obtained with the FAST spectrograph (Fabricant et al. 1998) mounted on the Tillinghast 1.5 m telescope at FLWO. The FAST spectrograph has been operational since 1994 January and the first SN Ia spectrum was one of SN 1994D taken on 1994 March 10 (all dates are given in UT). The observations were carried out in queue-scheduled mode, for the most part by two professional observers (P. Berlind and M. L. Calkins, who observed ∼72% of the FAST spectra presented here), as well as by CfA personnel. A total of 79 individual observers contributed to the FAST SN Ia sample presented here. During 1997–2008, typically 2–3 spectra were taken each night FAST was scheduled on the FLWO 1.5 m, namely, ∼20 nights per month, excluding August which corresponds to the annual shutdown at FLWO during the monsoon season. During 1994–1996, typically a single spectrum was taken in any given night. Observational details of the spectra are given in Table A1.

The usual setup for observations with FAST consisted of a ruled grating with 300 lines per mm and a 3'' slit, yielding a typical FWHM resolution of 6–7 Å over a wavelength range of ∼3700 to ∼7500 Å. From 2004 September onward (starting with observations of SN 2004dt), we changed the standard setup, extending it down to ∼3500 Å to cover the entire Ca ii H&K absorption profile. For a few SNe Ia (SN 1996ai, SN 1999by, SN 1999dq, SN 2001eh, SN 2001ep, and SN 2002bo) we requested additional observations with a different grating tilt to extend the wavelength range beyond 9000 Å. Other programs in the FAST queue would sometimes request different instrument setups, either with narrower slits (15 and 2'') or with a higher-resolution grating (cf. spectra of SN 1995ac, SN 1998aq, SN 1998bu, SN 1999by, and SN 1999cl). For most of the observations obtained during 1994–1998, the slit was oriented at a position angle of 90°. From late 1998 onward, the slit was generally positioned at the parallactic angle (unless the object was at airmass ≲ 1.1) so as to minimize the effects of atmospheric dispersion (Filippenko 1982). Table A1 gives the slit position angle for each spectrum as well as the absolute difference, |ΔΦ|, with the actual parallactic angle.

Additional spectra were obtained during classically scheduled nights at the MMT Observatory 6.5 m telescope with the Blue Channel (131 spectra) and Red Channel (3 spectra) spectrographs (Schmidt et al. 1989). A number of different spectrograph settings were used, yielding FWHM resolutions ranging between ∼3 and ∼13 Å, with a wavelength range typically extending below 3500 Å and beyond 8000 Å. A few spectra were taken with two settings of the Blue Channel spectrograph with non-overlapping wavelength ranges. This only concerns the five MMT spectra taken on 1994 June 12 (SN 1994D, SN 1994M, SN 1994Q, SN 1994S, and SN 1994T), all of which have ∼400 Å wide gaps (∼6150–6550 Å) between the two portions of the spectra. These spectra have been clearly marked in Table A1.

Last, a few spectra were obtained with the Magellan 6.5 m Clay (+LDSS-2/LDSS-3; 21 spectra) and Baade (+IMACS; one spectrum of SN 2003kf) telescopes. The FWHM resolution varies between ∼9 and ∼18 Å for LDSS-2, 9–12 Å for LDSS-3, and 5–6 Å for IMACS. The spectra taken with the Magellan 6.5 m telescopes typically do not reach bluer wavelengths than the FAST spectra, but they generally extend beyond 9000 Å.

The data reduction methods are the same as those presented by Matheson et al. (2008), and we refer the reader to that paper for complementary information. The FAST data were all reduced in the same consistent manner. This also applies to FAST data from 1994–1997 that were entirely re-reduced for the purposes of this paper (this enabled the recovery of two spectra of SN 1997do and SN 1997dt accidently omitted from the data set published by Matheson et al. 2008). The resulting FAST sample of 2447 spectra makes this by far the largest homogeneous SN Ia spectroscopic data set to date.

We used standard routines in IRAF¹⁶ to correct the CCD frames for overscan. The bias frames are sufficiently uniform so that we do not subtract them to avoid introducing additional noise. We did not correct for dark current as it is typically negligible with FAST. As noted by Matheson et al. (2008), however, a few spectra are affected by dark-current problems after UV flashing, resulting in a small emission feature at ∼7100 Å (observed) and increased noise in the red portion of the spectrum. For a handful of spectra taken in 1998 (SN 1998cs, SN 1998de, SN 1998ec, and SN 1998eg), the signal-to-noise ratio (S/N) degradation is large enough that we have simply trimmed off affected portions of the spectra (these have been clearly marked in Table A1). The CCD frames are then flat-fielded using a combined normalized flat-field image. One-dimensional spectra are then optimally extracted using the algorithm of Horne (1986) as implemented in the IRAF apall package. These spectra are then wavelength calibrated using HeNeAr lamps taken immediately after (occasionally before) each exposure. The same procedure was applied to spectra taken with the MMT and Magellan 6.5 m telescopes, although we subtracted a combined bias frame before the flat-fielding stage. Subsequent reduction steps described below are common to all telescope/instrument combinations.

We use our own set of routines in IDL to flux calibrate the extracted one-dimensional wavelength-calibrated spectra. Aside from the actual flux calibration, these routines also apply small adjustments to the wavelength calibration based on night-sky lines in the SN frames, and apply a heliocentric correction. The spectrophotometric standard stars observed on the same night (see Table A1) are also used to remove telluric absorption features from the spectra (see, e.g., Wade & Horne 1988; Matheson et al. 2000). We did not re-reduce the 37 MMT spectra from 1993 to 1997 that were flux calibrated using standard procedures in IRAF.

Approximately 50% of the spectra listed in Table A1 were cross-calibrated with two spectrophotometric standard stars of different colors to overcome the impact of second-order light contamination. Prior to the FAST refurbishment in 2003 August, the spectrograph blocked blue light such that the FAST spectra until SN 2003gq do not suffer from this effect (FAST spectra taken between 1997 September and 2003 July were usually calibrated with a single standard star). Starting in 2003 September, two standard stars were systematically used for the flux calibration. Our new reductions of the 1994–1997 data also uses two standard stars for consistency with the 2003–2008 data set, although the impact on the resulting flux calibration is minor.

Figure 1 illustrates our flux calibration technique with two standard stars on the FAST spectrum of SN 2008bf at 1 day past B-band maximum. On the same night as a given SN observation, we observed a relatively blue standard star (here Feige 34, sdO spectral type; top panel) and a relatively red standard star (here HD 84937, sdF spectral type; middle panel). The blue standard yields a more accurate flux calibration in the blue due to a greater number of counts and the absence of a Balmer jump, which complicates the calibration around 4000 Å, while the red standard yields a better calibration in the red where it does not suffer as much from second-order contamination. One clearly sees that the blue standard Feige 34 is poorly calibrated by the red standard blueward of ∼4000 Å (top panel; red line), while applying the blue standard to calibrate the red standard HD 84937 results in a flux deficit redward of ∼6000 Å (middle panel; blue line). The SN spectrum is calibrated using both standard stars (bottom panel), and the two resulting spectra are combined in a ∼100 Å wide region around ∼4500 Å (i.e., slightly redward of the Balmer jump). Blueward (redward) of this region, the spectrum calibrated using the blue (red) standard is used. The impact on the final spectrum (bottom panel; black line) may not seem spectacular, but we clearly recover flux in the Ca ii H&K absorption region and redward of the Si ii λ6355 line.

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We also illustrate the removal of atmospheric absorption features, here the O₂ B band around 6880 Å. Each standard star is used to derive a normalized telluric spectrum (top and middle panels; black line), which is then divided out from the SN spectrum after appropriate scaling (ratio of airmasses raised to some power α ≈ 0.6; see Wade & Horne 1988). The red standard star is more effective in removing these features, since the blue standard suffers from second-order contamination in these atmospheric bands. In this particular example the B band leaves a small imprint on the SN spectrum, but the inset in the lower panel of Figure 1 shows that this technique is successful in removing the unwanted absorption.

We can check the accuracy of the relative flux calibration of our spectra (no attempt was made to put the spectra on an absolute flux scale) by comparing the B − V color derived from photometry with that derived directly from the spectra (see Matheson et al. 2008, their Figure 4). We interpolated the corresponding B- and V-band light curves at the time each spectrum was taken, unless the difference was larger than 3 days (the interpolated measurements were visually cross-checked). FAST spectra taken before 2004 September do not extend to the blue edge of the B filter (3600 Å). We have run simulations based on the SN Ia spectral template of Hsiao et al. (2007) that show that even for spectra extending only down to 3750 Å the error on the inferred B − V color is less than 0.005 mag, so we include all the 1994–2008 FAST spectra in the comparison. The resulting 1128 B − V measurements are displayed in Figure 2. For SN spectra at phases less than 20 days past B-band maximum light taken within 10° of the parallactic angle (or at low airmass: $\sec {z}<1.1$; filled circles), the scatter around zero difference is ∼0.08 mag, slightly larger than the ∼0.06 mag scatter found by Matheson et al. (2008). The reason for this larger scatter is the inclusion of low-S/N spectra in our sample (S/N < 10 pixel⁻¹). Three significant outliers (∼1 mag difference in B − V color) correspond to spectra of the 2002cx-like SN 2005cc that are contaminated by host-galaxy light.

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The scatter is ∼2 times larger for spectra not observed at the parallactic angle (|ΔΦ ⩾ 10°|), regardless of phase (open circles and open stars). The scatter is also significantly larger for SN spectra at phases greater than 20 days past maximum observed at the parallactic angle (filled stars). Possible reasons for this were already noted by Matheson et al. (2008): the spectra become dominated by prominent emission features, giving rise to systematic errors when multiplied by a filter that is not precisely matched to the photometry. The spectra are also fainter, increasing the impact of host-galaxy contamination.

We show example spectral series in Figure 3. Plots of all SNe Ia from the CfA SN Program will be made available alongside the actual data on the CfA SN Archive Web site.

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4.1.1. Branch et al. (2006) Classification

Branch et al. (2006) presented a new classification scheme for SNe Ia based on measurements of the pseudo equivalent widths (pEWs) of absorption features near 5750 Å and 6100 Å (generally attributed to Si ii λ5972 and Si ii λ6355, respectively) in their maximum-light spectra. The overall shape of the Si ii λ6355 absorption also enters this classification scheme. They divided SNe Ia into four subgroups based on their position in this two-dimensional parameter space, but noted the absence of strict boundaries between them, suggesting that the different subgroups were not physically distinct but instead corresponded to a continuous distribution of properties (with the possible exception of SN 2002cx-like events).

Figure 8 (left) shows the pEW of the Si ii λ5972 line versus that of the Si ii λ6355 line for 218 SNe Ia (see Figure 2 of Branch et al. 2009), based on spectra within 5 days from maximum light (see Table 4). The different symbols correspond to the various spectroscopic subclasses defined by Branch et al. (2006). Example maximum-light spectra within each subclass are shown in Figure 9. The exact locations of the boundaries between the different subclasses are ill-defined, but this does not affect our analysis.

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Table 4. SN Ia Classification using the Schemes of Branch et al. (2006) and Wang et al. (2009a)

SN	pEW(λ5972)a	pEW(λ6355)b	vabs(λ6355)c	Phased	Branche	Wangf
	(Å)	(Å)	(km s−1)	(days)	Class	Class
1981B	9.1	128.6	−12050	−1.5	BL	N
1984A	20.4	200.5	−16251	−2.8	BL	HV
1986G	44.4	122.7	−10128	−0.2	CL	91bg
1989B‡	20.9	120.6	−10634	−1.3	BL	N
1990N	10.9	87.1	−9352	1.6	CN	N
1990O	6.7	91.1	−12069	−0.9	CN	N
1991M	18.4	134.4	−12458	2.7	BL	HV
1991T	0.9	29.0	−9660	−1.5	SS	91T
1991bg	47.9	95.0	−10004	0.9	CL	91bg
1992A	19.1	108.9	−14192	−0.8	BL	HV

Notes. ^aPseudo-EW of the Si ii λ5972 absorption feature (formal uncertainty typically <1 Å). ^bPseudo-EW of the Si ii λ6355 absorption feature (formal uncertainty typically <1 Å). ^cAbsorption velocity the Si ii λ6355 absorption feature (formal uncertainty typically <100 km s⁻¹). ^dPhase of measurements in rest-frame days from B-band maximum. ^eBranch class based on pEW(λ5972) and pEW(λ6355) within 5 days from maximum light—CN: Core Normal; BL: Broad Line; SS: Shallow Silicon; CL: Cool. We also report a classification for SNe Ia with no spectra within 5 days from maximum (but within 7 days) based on the overall shape of the Si ii λ6355 absorption feature. ^fWang class based on v_abs(λ6355) within 7 days from maximum light (note that Figure 8 only displays measurements within 3 days from maximum)—N: Normal; HV: High-velocity; 91T: 1991T-like; 91bg: 1991bg-like. Peculiar SNe Ia (SN 2000cx, SN 2002cx, SN 2003fg, SN 2005hk, SN 2006gz, SN 2007if, SN 2008A, SN 2008ae, SN 2009dc) are not part of this classification scheme. ^gSN 2003fg is also known as SNLS-03D3bb (Howell et al. 2006). ^†Reddening and contamination by the host galaxy renders the Branch classification uncertain. ^‡Branch classification differs from that reported by Branch et al. (2009; indicated in parentheses)—SN 1989B: BL (CL); SN 1999ac: CN (SS); SN 1999ee: CN (SS); SN 1999gd: CN (BL); SN 2000E: CN (SS); SN 2002er: CN (BL); SN 2005cg: CN (SS).

Only a portion of this table is shown here to demonstrate its form and content. Machine-readable and Virtual Observatory (VO) versions of the full table are available.

Download table as:  Machine-readable (MRT)Virtual Observatory (VOT)Typeset image

The cluster of black dots forms the "Core Normal" (CN) subclass. SNe Ia with similar pEW(Si ii λ5972) but with broader Si ii λ6355 absorptions (characteristic of larger expansion velocities) are labeled "Broad Line" (BL). Together, these subclasses constitute the bulk of SNe Ia loosely referred to as "normal" in the literature. There is no strict boundary between the CN and BL subclasses, although the SNe Ia we classify as BL have pEW(Si ii λ6355) ≳ 105 Å. As noted by Branch et al. (2009), there is not a one-parameter sequence from the CN subclass to the more extreme BL SNe Ia (such as SN 1984A, SN 2006X, and SN 2006bq). Two SNe Ia classified as BL by Branch et al. (2009) are re-classified by us as CN (SN 1999gd, SN 2002er; see Table 4) based on the magnitude of their pEW(Si ii λ6355). We also classify SN 2009ig in the CN subclass, as opposed to the BL subclass (see Parrent et al. 2011).

The "Cool" (CL) subclass consists of SNe Ia with deeper Si ii λ5972 absorptions, often associated with low-luminosity events. They are referred to as "Cool" due to the notable absorption band around ∼4200 Å caused by lines of Ti ii, a signature of lower temperatures in the line-forming region (see, e.g., Hatano et al. 1999). The strength of this absorption band varies greatly within the CL subclass: it is strong in SN 2005ke, moderate in SN 1986G, and close to nonexistent in SN 2007au. The CL subclass contains SN 2006bt, which has a slowly declining light curve characteristic of luminous SNe Ia but with spectra displaying Ti ii absorption features characteristic of low-luminosity SNe Ia (Foley et al. 2010). It also includes SN 2004eo, labeled "transitional" by Pastorello et al. (2007) due to its intermediate properties between normal and low-luminosity SNe Ia. Based on its resemblance to SN 2004eo, Branch et al. (2009) classified SN 1989B as a CL SN Ia despite its shallower Si ii λ5972 absorption. This SN also displays narrower features in the 4600–5100 Å spectral region, indicative of lower ejecta expansion velocities. However, the width of its Si ii λ6355 absorption is comparable to other BL SNe Ia, and we choose to include SN 1989B in the BL subclass (the developer of this classification scheme concurs; D. Branch 2011, private communication).

As seen from the inset in Figure 8 (left), there appears to be a continuous distribution between CL SNe Ia at the low end of the pEW(Si ii λ5972) distribution and CN and BL SNe Ia at the high end of their pEW(Si ii λ5972) distributions. However, we identify one CL SN Ia (SN 2007al) with a remarkably shallow silicon absorption. We include this SN in the CL subclass based on the presence of a prominent Ti ii absorption feature in its maximum-light spectrum (see Figure 9).

The last subclass, "Shallow Silicon" (SS), consists of SNe Ia with weak Si ii λ5972 and Si  ii λ6355. It is a heterogeneous category, including luminous 1991T/1999aa-like events characterized by higher ionization lines (Fe iii), SNe Ia resulting from possible super-Chandrasekhar-mass progenitors (SNLS-03D3bb or SN 2003fg, SN 2006gz, SN 2007if, SN 2009dc), faint 2002cx-like SNe (SN 2005hk, SN 2008A), and otherwise spectroscopically "normal" events (e.g., SN 2006S). Several SNe Ia classified as SS by Branch et al. (2009) are re-classified by us as CN (SN 1999ac, SN 1999ee, SN 2000E, SN 2005cg; see Table 4) based on the magnitude of their pEW(Si ii λ6355). SN 1999ac was already noted as a borderline SS/CN by Branch et al. (2007), and its peculiar nature discussed at length by Garavini et al. (2005) and Phillips et al. (2006).

Of the 246 SNe Ia to which we assign a Branch et al. (2006) classification in Table 4, 94 (38.2%) are of the CN subclass, 74 (30.1%) are of the BL subclass, 43 (17.5%) are of the CL subclass, and 35 (14.2%) are of the SS subclass. Note that these numbers simply reflect the properties of the SN sample studied in this paper and not of the SN Ia class as a whole.

By ∼3 weeks past maximum light, spectra of SS SNe Ia are not distinguishable from those of the CN subclass at similar phases. By comparing the +19 day spectrum of the SN 1999aa to spectra of CN SNe Ia, Branch et al. (2009) found the best match to be SN 1994D at +14 days. The corresponding phase ratio (19/14 ≈ 1.36) closely matched the ratio of their B-band light curve stretch parameters (1.143/0.838 ≈ 1.36), prompting Branch et al. (2009) to suggest that SNe Ia from the CN and SS subclasses "age spectroscopically at the same rate that they decline photometrically." We have tested this hypothesis by finding the best-match CN SN Ia template to spectra of SS SNe Ia in the phase range [+19,+23] days using SNID, and plot the ratio of their phases against that of their stretch parameters (from SALT2) in Figure 10. While most SS SNe Ia indeed lie on or close to the 1:1 relation, there are a few significant outliers (including the peculiar SN 2000cx and the 1991T-like SN 1998ab), and the Pearson correlation coefficient between both ratios is only r = 0.36. We find similar results when considering SS SNe Ia at two and four weeks past maximum light, respectively. The hypothesis of Branch et al. (2009) therefore does not apply universally to SNe Ia of the SS subclass.

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The spectroscopic properties of SNe Ia belonging to these different subclasses have been discussed at length by Branch et al. (2006, 2007, 2008) for maximum-light, pre-maximum, and post-maximum epochs, respectively, and by Branch et al. (2009) at all epochs based on a larger sample. Here, we focus on the spectroscopic variation at maximum-light. Within each subclass, we generate composite spectra using the same pre-processing as done by SNID: The individual spectra are "flattened" through division by a pseudo continuum. We then compute the mean flux in each wavelength bin, as well as the standard (and maximum) deviation from the mean. The result is a composite spectrum with error bands for each subclass, which we show in Figure 11. The large variation within the SS subclass is clearly visible throughout the optical range (3500–7500 Å). The CN subclass displays the smallest variation at any given wavelength, as expected, albeit with a noticeable variation in the strength of the Ca ii H&K absorption feature. The composite spectrum for the BL subclass reveals the presence of high-velocity components to the Si ii λ6355 line in the maximum-deviation spectrum (dark blue) in addition to the large variation blueward of ∼4000 Å as for the CN subclass (see, e.g., SN 2001bf and SN 2007jg in Figure 9). The CL composite spectrum shows the varying strength of the Ti ii absorption feature as well as the large range in depth and position of the Si ii λ6355 absorption (shallow in SN 2007al, less blueshifted in SN 2002es; see Figure 9).

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In a recent paper, Stritzinger et al. (2011) noted a similarity between the photometric properties of SN 2006ot and SN 2006bt (Foley et al. 2010): a broad light curve characteristic of luminous SNe Ia (Δm₁₅(B) = 0.84 mag), but a weak secondary i-band maximum characteristic of low-luminosity events. They also noted major differences between both SNe in their maximum-light spectra. SN 2006bt has a deep Si ii λ5972 absorption and evidence for absorption by Ti ii, both of which are characteristic of low-luminosity SNe Ia. SN 2006ot, on the other hand, is characterized by a shallower Si ii λ5972 absorption and a broad Si ii λ6355 absorption, with no evidence for features associated with Ti ii. In the classification scheme of Branch et al. (2006), SN 2006bt belongs to the CL subclass, while SN 2006ot is an extreme BL SN Ia, reflecting these spectroscopic differences. Nonetheless, Stritzinger et al. (2011) argue that at 3–4 weeks past maximum light, the spectra of SN 2006ot are more similar to those of SN 2006bt than to those of the normal SN 2006ax (part of the CN subclass), again stressing the similarity between both SNe. At such late phases, however, differences between the various subclasses are less apparent than at maximum light, as noted by Foley et al. (2010).¹⁸ We have run SNID on the +24 day spectrum of SN 2006ot and find good matches to spectra of SNe Ia from the CN, CL, and BL subclasses at similar phases. The best-match spectrum of SN 2006bt (at +33 days) is ranked only 21st overall. Similarly, the best-match template spectrum for the +33 day spectrum of the CL SN 2006bt (Foley et al. 2010) is the CN SN 2004S (Krisciunas et al. 2007), illustrating the difficulty in distinguishing between different SN Ia subclasses at late times. If anything, SN 2006bt and SN 2006ot demonstrate how two SNe Ia with similar light-curve shapes can have very different maximum-light spectra.

4.1.2. Wang et al. (2009a) Classification

Based on a restricted sample of nine SNe Ia from the CN subclass, Branch et al. (2009) inferred a mean absolute peak B-band magnitude of −19.48 mag with a scatter of only 0.14 mag, suggesting that SNe Ia belonging to this subclass could be true standard candles. They noted, however, that this should be tested on a larger sample of SNe Ia in the Hubble flow.

We use the BayeSN statistical models of Mandel et al. (2009, 2011) to infer the intrinsic peak absolute B-band magnitude (and hence the total extinction along the line of sight) of SNe Ia in a consistent manner (assuming a Hubble constant of H₀ = 72 km s⁻¹ Mpc⁻¹; see Table 1). These statistical models describe the apparent distribution of light curves as a convolution of intrinsic SN Ia variations and a dust distribution. Mandel et al. (2011) modeled the intrinsic covariance structure of the full multi-band light curves, capturing population correlations between the intrinsic absolute magnitudes, intrinsic colors, and light curve decline rates over multiple phases and wavelengths, as well as the distribution of host galaxy dust and an apparent correlation between the dust extinction A_V and its wavelength dependence, parameterized by R_V. The models fit individual optical and NIR SN Ia light-curve data to estimate the dust extinction, apparent and absolute light curves, and intrinsic colors for each SN. These models were trained on a nearby (z < 0.07) set of SNe Ia with optical (CfA3, Hicken et al. 2009; Carnegie SN Program, Contreras et al. 2010) and NIR (PAIRITEL; Wood-Vasey et al. 2008) data, plus light curves from the literature with joint optical and NIR observations. For the fits used in this paper, we have employed the distance-redshift constraint, except where alternative distance information was used (e.g., Cepheids; see Section 7.2).

Figure 12 shows the resulting intrinsic peak M_B distribution for SNe Ia at CMB-frame redshifts z_CMB > 0.01 with an inferred visual extinction A_V < 1 mag, for the various spectroscopic subclasses defined by Branch et al. (2006; top) and Wang et al. (2009a; bottom). The 41 SNe Ia in the CN subclass that satisfy these criteria have a mean intrinsic peak M_B = −19.40 mag with σ(M_B) = 0.16 mag. This is comparable to the values derived for the Normal subclass of Wang et al. (2009a) (M_B = −19.38 mag; σ(M_B) = 0.17 mag). The intrinsic peak M_B scatter is small for the CN (or Normal) subclass, but it is in fact larger than for the SS (σ(M_B) = 0.13 mag) and BL subclasses (σ(M_B) = 0.15 mag; the same is also true when comparing the Normal subclass to the 91T and HV subclasses of Wang et al. 2009a). We note, however, that the SS distribution does not include faint 2002cx-like SNe Ia (which typically have M_B ≳ −18 mag; e.g., M_B ≈ −17.5 mag for SN 2002cx and M_B ≈ −18.0 mag for SN 2005hk; see Phillips et al. 2007) nor luminous (possibly super-Chandrasekhar) events (e.g., SN 2006gz with M_B ≈ −19.9, Hicken et al. 2007; or SN 2009dc with M_B ≈ −20.2, Taubenberger et al. 2011) since BayeSN did not include such SNe Ia in its training set.

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The mean intrinsic peak M_B increases steadily along the SS→CN→BL sequence, with a corresponding increase in the mean Δm₁₅(B) (see Table 5). The same applies to the 91T→N→HV sequence. There is a hint of redder mean intrinsic B − V color for the BL (HV) subclass compared to the CN (N) subclass, as expected from the "brighter-bluer" relation of Tripp (1998), but the difference (0.02−0.03 mag) is small compared with the scatter (0.06–0.07 mag) within each subclass and only measurable with sample averages. Low-luminosity SNe Ia from the CL or 91bg subclasses are not included in this comparison since the BayeSN statistical model did not include SNe Ia with Δm₁₅(B) > 1.6 mag in its training set. SNe similar to SN 1991bg have intrinsic peak B-band magnitudes ∼2 mag fainter than normal SNe Ia, with Δm₁₅(B) ≳ 1.9 mag, and much redder intrinsic B − V colors at maximum light. A prime example is SN 2005bl, with M_B ≈ −17.2 mag, Δm₁₅(B) ≈ 1.9 mag, and (B − V)_max ≈ 0.5 mag (Taubenberger et al. 2008).

One might wonder whether SNe Ia in different subclasses obey different width–luminosity relations (WLRs). Figure 13 (left) shows the intrinsic peak M_B inferred from BayeSN fits versus Δm₁₅(B) for the SS, CN, and BL subclasses of Branch et al. (2006). Overplotted are best-fit relations of the form M_B = a[Δm₁₅(B) − 1.1] + b for the entire sample (solid line) and for the individual subclasses (dashed and dotted lines). The coefficients of the linear fits are given in each case. The slope of the WLR gets steeper along the SS→CN→BL sequence, while the intercept corresponds to a progressively fainter fiducial intrinsic peak M_B along this same sequence. The slopes and intercepts for the individual subclasses are, however, consistent within ∼1σ of one another, such that a single average WLR is adequate to describe all the subclasses (other than the CL subclass). As seen from the right panel of Figure 13, the same analysis holds when replacing (SS, CN, BL) with (91T, N, HV) for the Wang et al. (2009a) classification scheme, the only difference being the intercept for the HV subclass, which is ∼2σ larger than those for the Normal and 91T subclasses.

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Several authors have studied the correlation between various spectroscopic indicators and light-curve parameters with the aim to use such indicators to improve distance measurements to SNe Ia. Blondin et al. (2011b) showed that spectroscopic indicators alone could compete with the standard light-curve parameters for distance measurements, but that combining spectra with photometry yielded no statistically significant improvement (see also Silverman et al. 2012b). Their analysis, however, was based on a small number of SNe, so that larger samples may reveal spectroscopic measurements that provide independent information on the luminosity of SNe Ia. Such correlations are also potentially useful to constrain SN Ia models (see, e.g., Blondin et al. 2011a), and we investigate the relation of three spectroscopic indicators with Δm₁₅(B) in Figure 14.

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The upper left panel shows the $\mathcal {R}({\rm Si})$ ratio of Nugent et al. (1995; defined as the ratio of the relative absorption depth of the Si ii λ5972 line to that of Si ii λ6355), which is thought to define a temperature sequence from low-luminosity 1991bg-like SNe Ia to luminous 1991T-like events (see also Hachinger et al. 2008). The different symbols correspond to the spectroscopic subclasses defined by Branch et al. (2006), as in the left panel of Figure 8. There is a clear correlation of $\mathcal {R}({\rm Si})$ with Δm₁₅(B) (the Pearson correlation coefficient is r = 0.82), with some notable outliers including two SNe Ia from the CL subclass: SN 2002fb and SN 2006bt (whose peculiar nature has been discussed at length by Foley et al. 2010). At low Δm₁₅(B), the BL SN 2001ay (the most slowly declining SN Ia to date; Krisciunas et al. 2011) and the peculiar SS SN 2006gz (possibly resulting from a double WD merger; Hicken et al. 2007) are modest outliers.

The $\mathcal {R}({\rm Si})$ ratio has a relatively large associated measurement error, since the relative absorption depth is measured at a specific wavelength. Using the ratio of pseudo-EW of both lines instead not only significantly reduces the error on individual measurements (since we are integrating over many wavelength bins), it also leads to a stronger and tighter relation with Δm₁₅(B) (r = 0.86; upper right). SN 2006bt remains an outlier using the pseudo-EW ratio, but SN 2002fb now follows the linear relation defined by the bulk of the sample. As noted empirically by Hachinger et al. (2006), and later confirmed theoretically by Hachinger et al. (2008), it is the Si ii λ5972 line that drives the correlation of $\mathcal {R}({\rm Si})$ and its pseudo-EW equivalent with Δm₁₅(B). Using the relative absorption depth or the pseudo-EW of the Si ii λ5972 line alone leads to equally strong correlations with Δm₁₅(B) (r = 0.79 and r = 0.85, respectively, not shown).

The lower left panel shows the pseudo-EW of the Si ii λ4130 line versus Δm₁₅(B) (see Bronder et al. 2008; Arsenijevic et al. 2008; Walker et al. 2011; Blondin et al. 2011b; Chotard et al. 2011). The correlation is very clear, although a nonlinear relation may need to be invoked to accommodate the CL SNe Ia at Δm₁₅(B) ≳ 1.5 mag. The peculiar SN 2006gz is again a modest outlier, but SN 2001ay is now a significant outlier in this relation. The largest outlier however is the BL SN 2006ot (which was not a significant outlier in the $\mathcal {R}({\rm Si})$ versus Δm₁₅(B) relation), whose similarity in terms of photometric properties with SN 2006bt was noted by Stritzinger et al. (2011). We argued in Section 4.1.1 that both objects are spectroscopically distinct, and this is clearly visible in this plot, where SN 2006bt follows the same relation as the bulk of the SN Ia sample.

Finally, the lower right panel shows the $\mathcal {R}({\rm Si,Fe})$ ratio of Hachinger et al. (2006), defined as the ratio of the pseudo-EW of the Si ii λ5972 line to that of the Fe ii λ4800 feature (see Blondin et al. 2011b, their Figure 14). The correlation with Δm₁₅(B) is again very strong (r = 0.80). In contrast to pEW(Si ii λ4130), SN 2006bt is this time the largest outlier, while SN 2006ot lies close to the mean relation. As was the case for the $\mathcal {R}({\rm Si})$ ratio, both SN 2001ay and SN 2006gz are modest outliers. Since $\mathcal {R}({\rm Si,Fe})$ and the pseudo-EW equivalent of $\mathcal {R}({\rm Si})$ are both positively correlated with Δm₁₅(B), one expects the pseudo-EWs of Fe ii λ4800 and Si ii λ6355 (at maximum light) to also be correlated. There is indeed a modest correlation (r = 0.60) between pEW(Si ii λ6355) and pEW(Fe ii λ4800), but the scatter is large and the relation flattens off for pEW(Fe ii λ4800) ≳ 150 Å.

Figure 15 shows the time evolution of the Si ii λ6355 absorption velocity for the different spectroscopic subclasses of Branch et al. (2006). Blondin et al. (2006) presented similar measurements (in different Δm₁₅(B) bins) along with a physical interpretation in terms of microscopic atomic-transition properties and macroscopic properties of the ejecta (density and velocity distributions). We refer the reader to that paper for more detail on the impact of these properties on the morphology of line profiles in SN Ia ejecta.

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For the CN subclass (top left), there is a large variation in v_abs(Si ii λ6355) at phases ≲ − 10 days (e.g., SN 2003du and SN 2006gr have v_abs ≈ −12,000 km s⁻¹ and −20,000 km s⁻¹ at −10 days, respectively). By maximum light, the spread in v_abs has decreased to ≲ 3000 km s⁻¹, SN 2009ig being a clear outlier (it is part of the High-velocity subclass of Wang et al. 2009a, but the pseudo-EW of its Si ii λ6355 line at maximum light places it unambiguously in the CN subclass of Branch et al. 2006; see Figure 8). The evolution is fast at early times and more gradual at post-maximum epochs, reflecting the recession of the line-forming region in the co-moving frame of the ejecta. There is also a large spread in v_abs at early times for SNe Ia in the Broad Line subclass (top right), but unlike the CN subclass it remains large at later phases (>5000 km s⁻¹ variation at maximum light). At phases later than −10 days, the absorption velocity is on average larger at any given time when compared with the CN subclass, illustrating the large overlap between the BL subclass and the High-velocity subclass defined by Wang et al. (2009a).

The bulk of the SS subclass (bottom left) has velocities comparable to the CN subclass, except at early times where there are no measurements |v_abs| > 15,000 km s⁻¹. A handful of SNe Ia display −5000 km s⁻¹ ≳ v_abs ≳ −10,000 km s⁻¹ at maximum light. These include 2002cx-like SNe Ia (SN 2005hk, v_{abs, max} ≈ −5500 km s⁻¹; SN 2008A, v_{abs, max} ≈ −7500 km s⁻¹), SNe Ia resulting from possible super-Chandrasekhar progenitors (SN 2009dc, v_{abs, max} ≈ −8000 km s⁻¹; SN 2007if, v_{abs, max} ≈ −7500 km s⁻¹), and the 1991T-like SN 2005M (v_{abs, max} ≈ −8000 km s⁻¹). This last SN stands out among other 1991T-like SNe Ia which all display ≳10,000 km s⁻¹ absorption blueshifts at maximum light.

The Cool subclass (bottom right) displays velocities similar to the CN and SS subclasses around maximum light, but with a more rapid evolution at post-maximum epochs. One significant outlier is SN 2002es (Ganeshalingam et al. 2012), with v_abs ≈ −6000 km s⁻¹ at +2 days (see also Figure 8, right), comparable to the SS SN 2005hk at a similar phase.

Benetti et al. (2005) noted the large variation in post-maximum evolution of the Si ii λ6355 absorption velocity for different SNe Ia. This is clearly visible in Figure 15, not only across the different subclasses but also within each subclass (the largest variation being apparent within the BL subclass). Benetti et al. (2005) introduced the velocity gradient, noted $\dot{v}$, representing the average |v_abs| decline rate past maximum light (units of $\rm {km\,s}^{-1}\,\rm {day}^{-1}$), and divided their SN Ia sample into three groups (FAINT for 1991bg-like SNe Ia, HVG and LVG for SNe Ia with high and low velocity gradients, respectively). They suggested that the LVG and HVG groups could consist of SNe Ia with different properties of their outer ejecta, resulting from different degrees of mixing or from interaction with circumstellar material. A recent study by Maeda et al. (2010a) associates the observed diversity in the velocity gradients with viewing-angle effects in off-center delayed-detonation models. Blondin et al. (2011a) showed that varying the criterion for DDT in explosions resulting from an isotropic distribution of ignition points (i.e., not off-center explosions) resulted in a range of velocity gradients comparable to that observed by Benetti et al. (2005).

The notion of velocity gradients has received much attention since the paper by Benetti et al. (2005), and there are different physical interpretations as to the origin of their variation. This however can only be properly addressed if the actual parameter is defined in a robust way, which we set out to do in the following section.

According to Benetti et al. (2005), the velocity gradient is derived "from a least-squares fit of the measurements taken between maximum and either the time the Si ii feature disappears or the last available spectrum, whichever comes first." This definition makes two implicit (and related) assumptions: (1) the evolution of v_abs is a linear function of time at post-maximum epochs and (2) the phase of the last v_abs measurement (or the range of available measurements) has little impact on the derived velocity gradient.

Figure 16 shows that both assumptions can have a significant impact on the derived gradient. In the left panel we show the time evolution of v_abs(Si ii λ6355) for the BL SN 2002bo (filled circles) and the CN SN 2005cf (open squares). While the v_abs evolution for SN 2005cf appears to be almost linear at post-maximum epochs, that of SN 2002bo is clearly not, with a more rapid change in the interval [+10,+15] days. Moreover, at phases ≳ + 25 days the Si ii λ6355 absorption profile is increasingly contaminated by the emergence of lines from different ions (see inset), such that the v_abs measurements at these times are not reliable. The right panel of Figure 16 shows the variation in the measured velocity gradient in the interval [+0,t_last] days as a function of the phase of the last available measurement, t_last. The velocity gradient for SN 2002bo is ∼115 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$ when measured over the interval [+0,+10] days, and ∼165 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$ when measured over the interval [+0,+20] days. Including the biased v_abs measurements at > + 25 days yields a velocity gradient ∼120 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$, comparable to that obtained for [+0,+10] days, but with a χ² for the linear fit that is two orders of magnitude larger. Benetti et al. (2005) report a velocity gradient of 110 ± 7 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$ for this SN, due to the inclusion of measurements out to +30 days (S. Benetti 2008, private communication). For SN 2005cf, the quasi-linear variation of v_abs with time results in a minor variation of the velocity gradient with t_last, from ∼65 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$ for t_last ≈ +10 days to ∼50 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$ for t_last ≈ +30 days. Wang et al. (2009b) report a velocity gradient of 38 ± 5 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$ for this SN using measurements out to +30 days.

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The classification of SN 2002bo and SN 2005cf, respectively, in the HVG and LVG groups still holds, regardless of t_last and whether or not the biased v_abs measurements of SN 2002bo are included. This is not always the case. For SN 2002er, Benetti et al. (2005) report a velocity gradient of 92 ± 5 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$, including v_abs measurements past +20 days when the Si ii λ6355 absorption profile is clearly contaminated by other lines. Using measurements in the phase interval [+0,+16] days we find 81 ± 3 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$, consistent with their value, and confirming the classification of SN 2002er as an HVG SN Ia. However, restricting the phase interval to [+0,+10] days, we infer a velocity gradient of 47 ± 7 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$, consistent with the LVG group. One can think of a variety of reasons why no spectra would have been obtained at later times (poor weather, telescope problems, conflict with other observations, etc.), none of which should affect the inferred intrinsic properties of SNe Ia.

We also note that the velocity gradient for SN 2006bt reported by Foley et al. (2010) is in error. We measure a velocity gradient of 93 ± 16 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$ over the time interval [+0,+16] days, while Foley et al. (2010) report 51 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$ over the same interval. R. J. Foley (2011, private communication) has revised this measurement to 96 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$. According to Figure 7 of Foley et al. (2010), this new measurement places SN 2006bt well into the HVG group, whereas Foley et al. (2010) originally included this SN in the LVG group (albeit close to the border with HVG SNe Ia).

A more robust definition of the velocity gradient should consider v_abs measurements over a fixed phase range, as done for instance by Foley et al. (2011) over the interval −6 days ⩽ t ⩽ +10 days using linear functions to describe the v_abs(t) curves. This still does not account for nonlinear variations in absorption velocity. As an alternative, we suggest the measurement of the mean velocity decline rate Δv_abs/Δt over some fixed time interval [t₀, t₁]:

$$\begin{equation} \left.\frac{\Delta v_{\rm abs}}{\Delta t}\right|_{[t_0,t_1]} = \frac{v_{\rm abs}(t_1)-v_{\rm abs}(t_0)}{t_1-t_0}. \end{equation} \tag{ 1 }$$

In practice there are not always measurements at precisely t₀ and t₁, so one may choose to either select the v_abs measurement closest in time to t₀ and t₁ (within a small time interval, e.g., ±2 days) or to interpolate the v_abs measurements at the desired phases, again making sure the v_abs(t) data are well sampled around t₀ and t₁. Figure 17 (left) shows the mean velocity decline rate for the Si ii λ6355 line for the time interval [+0,+10] days versus Δm₁₅(B), where we choose the v_abs measurement closest in time and within 2 days from these time boundaries. The clustering into the three groups (FAINT, LVG, HVG) is less obvious than in Figure 3 of Benetti et al. (2005), but the points occupy similar regions of parameter space, with a smaller scatter in Δv_abs/Δt_{[ + 0, +10]} for Δm₁₅(B) ≳ 1.5 mag. The Broad Line subclass displays the largest variation in mean velocity decline rate (confirming what was visually apparent in Figure 15), ranging from ∼ − 20 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$ for SN 2006cj to +215 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$ for SN 2006X. Two other SNe have Δv_abs/Δt_{[ + 0, +10]} < 0, SN 1991T and SN 2005M (both part of the SS subclass). A combination of optical-depth effects (e.g., an Si iii→Si ii recombination wave) and measurement error (the Si ii absorption is relatively weak and occurs on a steep pseudo-continuum) could explain why the absorption blueshift apparently increases over this phase range. Several objects have Δv_abs/Δt_{[ + 0, +10]} consistent with zero, including the CN SN 2006gr. This was also the case for SN 2005hj, for which Quimby et al. (2007) invoke a shell-like density structure (formed within the context of a pulsating delayed detonation or WD–WD merger scenario) to explain the ∼10 day long plateau in absorption velocity of the Si ii λ6355 line.

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The CN and CL subclasses appear to form a sequence of linearly increasing Δv_abs/Δt_{[ + 0, +10]} with Δm₁₅(B). Both quantities are indeed strongly correlated (r = 0.80) for these two subclasses. Within the BL subclass, there is a strong correlation (r = −0.80) between the Si ii λ6355 absorption velocity at maximum light and its subsequent mean decline over the interval [+0,+10] days. In other words, SNe Ia with the largest absorption blueshifts at maximum light also display steeper velocity gradients (see also Benetti et al. 2005; Wang et al. 2009a; Foley et al. 2011).

In the limit (t₁ − t₀) → 0, Equation (1) yields an instantaneous velocity decline rate at a given time, $dv_{\rm abs}/dt|_{t=t_0}$. We determine this using second- or third-order polynomial fits to the v_abs(t) data, as shown in Figure 16 (left; solid curves). The instantaneous velocity decline rates at t = +0 days for Si ii λ6355 in SN 2002bo and SN 2005cf are shown as the error bars in the right panel of Figure 16. Figure 17 (right) shows the same quantity for a larger sample versus Δm₁₅(B). The different subclasses occupy similar regions of parameter space as when considering the mean velocity decline rate (not surprisingly, since the mean and instantaneous decline rates are strongly correlated), but reveals an even larger variation within the BL subclass, with instantaneous velocity decline rates at maximum light as large as ∼300 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$ for SN 2001ay and ∼400 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$ for SN 2003W. SN 2005hj has the lowest instantaneous velocity decline rate at maximum light (14 ± 5 $\rm {km\,s}^{-1}\,\rm {day}^{-1}$), in line with the results of Quimby et al. (2007).

Based on the results by Wang et al. (2009a), Foley & Kasen (2011) noted that SNe Ia in the High-velocity subclass were intrinsically redder (in B^max − V^max pseudo-color¹⁹) than SNe Ia in the Normal subclass. They provided a simple physical explanation using radiative transfer calculations by Kasen & Plewa (2007) based on the detonating failed deflagration (DFD) model of Plewa (2007), although Blondin et al. (2011a) showed that this relation did not hold in the two-dimensional delayed-detonation models of Kasen et al. (2009). Using the Si ii λ6355 absorption velocity measurements presented in this paper, Foley et al. (2011) quantified the relation between Si ii velocity and intrinsic B^max − V^max pseudo-color at maximum light for Normal and High-velocity SNe Ia with 1.0 mag ⩽ Δm₁₅(B) ⩽ 1.5 mag. They find that the slope of the relation between both quantities is non-zero at the ∼9σ level using a linear weighted least-squares fit (but only at the ∼3.5σ level when using a Bayesian approach to linear regression; see Kelly 2007), with a Pearson correlation coefficient r = −0.39.

We show the relation between the intrinsic B − V color at B-band maximum light, as derived from BayeSN fits to SN Ia light curves (see Table 1), and the absorption velocity of the Si ii λ6355 line in Figure 18 (left) for the Normal and High-velocity subclasses. We restrict the Δm₁₅(B) range to 1.0 mag ⩽ Δm₁₅(B) ⩽ 1.5 mag, as done by Foley et al. (2011). Moreover we use their interpolated velocity measurements at maximum light (their Table 1). The right panel shows histograms of intrinsic B − V color at maximum light for the Normal (N_SN = 51; open) and High-velocity (N_SN = 22; filled) subsamples. Of the 65 SNe Ia used by Foley et al. (2011) in their analysis, 42 (27 Normal and 15 High-velocity) are part of our sample.

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A linear weighted least-squares fit to the data in Figure 18 yields the following relation between intrinsic B − V color and Si ii velocity at maximum light:

$$\begin{eqnarray} B-V &=& -0.26(0.05) - 0.013(0.005)\nonumber\\ && \times [v_{\rm Si}/10^3\,\rm {km\,s}^{-1}]\mathrm{\ mag}, \end{eqnarray} \tag{ 2 }$$

with a Pearson correlation coefficient r = −0.35 (see Table 6), consistent with that found by Foley et al. (2011). Taking into account the uncertainty in the intrinsic color estimates via a Monte Carlo simulation results in almost no correlation (r ≈ −0.20). The slope of the best-fit linear relation is shallower than that found in their paper, and with a larger associated error that reflects the larger uncertainty we derive on intrinsic color estimates (typically 0.05 mag using BayeSN cf. 0.03 mag for the estimates reported by Foley et al. 2011). The statistical significance of the linear relation that we find is also significantly lower than that reported by Foley et al. (2011). Our fit rejects a non-zero slope at the ∼3σ level²⁰, compared with ∼9σ in their paper. The scatter about the best-fit linear relation (dashed line) is the same to within ∼10% for both samples, in conflict with the results of Foley et al. (2011), who infer a ∼50% larger scatter for the HV sample.

The robustness of the relation between Si ii velocity and intrinsic color remains questionable given the large uncertainty associated with the determination of the latter quantity. Foley et al. (2011) fit for a relation between light-curve shape corrected peak absolute M_V (not corrected for host-galaxy reddening) and observed B^max − V^max pseudo-color (their Figure 15), based on Equation (1) of Wang et al. (2009a). They assume that the horizontal scatter about this relation is entirely due to intrinsic variations in B^max − V^max pseudo-color, where it in fact also includes some intrinsic scatter about the luminosity–Δm₁₅(B) relation. It is thus unlikely that their "intrinsic color estimates" are simply offset from the true intrinsic color, as postulated in their paper. Moreover, Foley et al. (2011) do not propagate the errors associated with their nuisance parameters (α, M_ZP, R_V, and C in their Equation (15)) in the intrinsic color error bars, which artificially enhances the statistical significance of their correlation. The same limitations apply to the recent study of this relation for high-redshift SNe Ia by Foley (2012), with the additional caveat that their analysis makes use of an indirect measure of the observed B^max − V^max pseudo-color.

BayeSN uses a more elaborate statistical treatment of the population correlations between the intrinsic absolute magnitudes (BVRI, including JH when available), intrinsic colors, and light-curve shape, as well as the distribution and properties of host galaxy dust. BayeSN therefore provides an estimate of the true intrinsic color, with realistic error bars reflecting the uncertainty in the various population correlations. It is then not surprising that our estimates for the intrinsic B^max − V^max pseudo-color are only weakly correlated (r = 0.21) with those of Foley et al. (2011). Likewise, the mean intrinsic colors that we derive for both Normal and High-velocity samples are significantly bluer (by ∼0.08 mag and ∼0.12 mag, respectively, when using the intrinsic B^max − V^max pseudo-color from BayeSN) than those using the color estimates of Foley et al. (2011). The larger uncertainties derived from BayeSN fits reflect the difficulty in obtaining accurate estimates of intrinsic color, properly accounting for the population distributions (or variance) of intrinsic color and dust extinction.

The difference in the weighted mean intrinsic B − V color between the High-velocity and Normal samples is 0.030 ± 0.013 mag, corresponding to a ∼2σ difference (cf. ∼7σ in Foley et al. 2011). A Kolmogorov–Smirnov (K-S) test shows that the two samples are discrepant at the p = 0.023 significance level, which is less than the canonical cutoff of 0.05 for statistical significance,²¹ with the caveat that this statistic does not consider the (large) uncertainty in intrinsic color. This is still significantly larger (i.e., less significant) than the K-S probability inferred for the Foley et al. (2011) estimates of intrinsic B^max − V^max pseudo-color (p = 0.003).

Using the intrinsic B − V color at B-band maximum as opposed to the B^max − V^max pseudo-color does not affect our results nor their interpretation, as both quantities are strongly correlated (r = 0.72; see Table 1). The weighted mean difference between intrinsic B^max − V^max pseudo-color and intrinsic B − V color at B-band maximum is only 0.02 mag with an rms of 0.04 mag. Pseudo-colors are difficult to interpret since, for an individual SN, the V band may reach its maximum brightness between ∼0 days and ∼4 days later than the B band (see, e.g., Blondin et al. 2011a, their Figure 9). We prefer to use the true color at a given time. The resulting correlation between Si ii absorption velocity and intrinsic B^max − V^max pseudo-color is slightly weaker than when using the B − V color at B-band maximum light (r = −0.26). While the High-velocity sample is found to have a redder average intrinsic B^max − V^max pseudo-color than the Normal sample at the ∼2σ level (i.e., the same significance than when using the B − V color), a K-S test shows that the two samples are consistent at the p = 0.14 > 0.05 significance level.

A larger sample of SNe Ia with combined optical and NIR photometry should enable a more precise determination of the intrinsic colors of SNe Ia (see Mandel et al. 2011) and a more reliable study of the relation with Si ii velocity.

In Section 5.1, we investigated the time evolution of the Si ii λ6355 absorption velocity as a function of spectroscopic subclass (see Figure 15). This velocity corresponds to the location of maximum absorption. Here, we are interested in the maximum velocity at which Si ii absorbs, since this value gives an estimate of the extent of incomplete C/O nuclear burning in the progenitor WD star.

In Figure 19, we show the Si ii λ6355 line profile for SNe Ia in the CfA sample for which we have spectra prior to −10 days from B-band maximum light (excluding 1991T/1999aa-like objects and SN 2006cc; see the caption). There is a large variation in the maximum Si velocity (roughly corresponding to the blue edge of the Si ii λ6355 absorption profile), from ∼ −28,000 km s⁻¹ for SN 2003W to ∼ −18,000 km s⁻¹ for SN 1998aq. These values are consistent with the lower limit of ∼12,000 km s⁻¹ found by Mazzali et al. (2007) for the outer extent of the Si-rich layer in SN Ia ejecta, although as noted by Branch et al. (2007) this layer generally extends to much higher velocities, in conflict with pure deflagration models (e.g., Maeda et al. 2010b).

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Mazzali et al. (2007) assume that pre-maximum Si ii λ6355 profiles are affected by HVFs (see the next section) that are not representative of the bulk of IMEs. In Figure 19, we clearly see variations in the line-profile morphology of the Si ii line, from single absorptions (both broad as in SN 2003W and narrow as in SN 1998aq) to evidence for multiple components (e.g., SN 2006cp, SN2007le, SN 2005cf). In the latter case the absorption profile is interpreted as consisting of a photospheric and a "detached" high-velocity component, as in the analysis of SN 2005cf by Wang et al. (2009b). In this interpretation both components would then be blended together in SNe Ia displaying a single broad absorption profile (see, e.g., Tanaka et al. 2008).

The emission component of the P-cygni profiles in Figure 19 appears blueshifted with respect to the rest-frame wavelength of the Si ii transition (6355 Å). In SN 2003W, the peak emission occurs at a Doppler velocity of ∼ − 6000 km s⁻¹. Such peak-emission blueshifts are routinely observed in Type II SNe, and result from the steep density gradient in the ejecta, confining the absorbing and emitting regions (as seen by an external observer) to a narrow range of impact parameters (Dessart & Hillier 2005). Sizeable peak-emission blueshifts are also expected theoretically in SNe Ia (Blondin et al. 2006), although line overlap renders their identification problematic (see discussion by Branch et al. 2007). In several cases we note a small absorption notch in the Si ii λ6355 emission profile (downward-pointing arrows), usually attributed to C ii λ6580 (see Section 6.3), which obviously biases the measurement of the velocity at peak emission to higher apparent blueshifts.

The Ca ii near-infrared triplet (IRT; 8498 Å, 8542 Å, and 8662 Å) is less subject to overlap with lines from other ions than Ca ii λ3945 and is hence more appropriate to search for high-velocity absorption features. Mazzali et al. (2005a) found HVFs to be present in all SN Ia spectra with early-time data covering the Ca ii IRT, with evidence of similar features in the Si ii λ6355 line for some. They suggested that an abundance enhancement in Ca and Si was unlikely to be the sole cause, as extreme enhancements were required in the outer ejecta (e.g., factor of 20 increase in the Ca abundance above 20,000 km s⁻¹ for SN 1999ee; Mazzali et al. 2005b). A density enhancement was found to be a more likely explanation, either intrinsic to the explosion or through an interaction with circumstellar material. In the former case one expects a large diversity in HVFs due to line-of-sight effects dependent on the geometry of the high-velocity material (e.g., thick torus or clumpy structure; see Tanaka et al. 2006). There is direct evidence for the multi-dimensional nature of HVFs in one SN Ia, SN 2001el, where spectro-polarimetric observations revealed the high-velocity component of the Ca ii IRT to be polarized at the ∼1% level (Wang et al. 2003; Kasen et al. 2003).

Figure 20 shows Ca ii λ3945 (black) and IRT (red) line profiles in pre-maximum SN Ia spectra, ordered by phase. The diversity in line-profile morphology noted by other authors (e.g., Tanaka et al. 2008) is clearly seen, ranging from single to multiple detached absorption profiles. The correspondence between the Ca ii λ3945 and IRT absorption profiles is obvious in some cases (e.g., SN 1997bq), more questionable in others (e.g., SN 2001da), and nonexistent for a few cases (e.g., SN 2002he, SN 1996X). One possible source of confusion comes from the overlap of the Si ii λ3858 line with Ca ii λ3945, which can lead to an apparent HVF in the H&K line where none are visible in the IRT (e.g., SN 1996X, SN 1989B). The dotted lines in Figure 20 show the location of maximum absorption by Si ii λ3858, assuming it has the same absorption velocity as Si ii λ6355. In several cases it coincides with a blue absorption in the Ca ii λ3945 absorption profile, sometimes resulting in a flat-bottomed absorption (e.g., SN 2002he). Another difficulty when interpreting HVFs associated with Ca ii is that a mass fraction as low as ∼10⁻⁵ is sufficient to yield a strong Ca ii absorption at high velocities (see, e.g., Blondin et al. 2011a).

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Tanaka et al. (2006) noted that if HVFs result from a density enhancement (regardless of its origin), other lines (e.g., Si ii λ6355) should be affected, and the relative strengths and velocities of these HVFs should be correlated in some way. Stanishev et al. (2007) suggest that the relative strengths of HVFs in Ca ii λ3945, Si ii λ6355, and Ca ii IRT are indeed correlated in SN 2003du, but Tanaka et al. (2008) find that the velocities of the Si ii λ6355 and Ca ii IRT HVFs in several SNe Ia are at odds with this hypothesis. We note based on Figures 19 and 20 that five of the SNe Ia with the largest extent of Ca ii IRT absorption (SN 2002bo, SN 2005cf, SN 1997bq, SN 2002dj, SN 2003W) also have some of the largest Si ii λ6355 absorption velocities at early times (the same is also true for SN 2009ig, shown in Figure 20 but not in Figure 19). Interestingly, Tanaka et al. (2006) also suggest that a combination of density and abundance enhancements might provide a better explanation for the origin of HVFs, on the account that dense blobs covering the entire photosphere would result in Si ii λ6355 absorption velocities in excess of 20,000 km s⁻¹, "never observed in any SN." In fact, such high absorption blueshifts are seen in one SN from the CfA sample, SN 2003W (see Figure 15), and more recently in SN 2009ig (Foley et al. 2012).

Inferring the presence of unburnt carbon in SN Ia spectra can yield important clues on the properties of the explosion. Pure deflagration models predict that large amounts of unburnt C/O fuel are left over, sometimes mixed to the innermost ejecta regions (see, e.g., Röpke 2005), although one then predicts strong forbidden lines of C i and O i in late-time spectra that are not observed (Kozma et al. 2005). Delayed-detonation models, on the other hand, consume all but the outermost regions of the SN ejecta, leaving varying amounts of unburnt C/O at high expansion velocities (e.g., Maeda et al. 2010b), which can affect early-time SN Ia spectra.

As noted by several authors (e.g., Tanaka et al. 2008), carbon is expected to be ionized once in the ejecta of SNe Ia at early times, such that the "best bet" at constraining the abundance of carbon, let alone inferring its presence, is through its strongest optical line C ii λ6580 (although see the analysis by Marion et al. 2006 using C i lines in NIR spectra). Until recently, firm detections of this line had been reported in only a handful of SNe Ia, the most convincing cases being those of SN 2006D (Thomas et al. 2007) and SN 2006gz (Hicken et al. 2007), both of which show gradually fading C ii λ6580 lines with respect to the earliest detection. The scarcity of C ii detections led to the suggestion of a clumpy carbon distribution in the outer ejecta of SNe Ia combined with line-of-sight effects (e.g., Branch et al. 2007).

In a recent survey of C ii features in SN Ia spectra, Parrent et al. (2011) find evidence for C ii λ6580 absorption in ∼30% of SNe Ia in their sample, a significantly larger fraction than previously suspected (e.g., Thomas et al. 2007). Moreover, they show that these features are most common in SNe in the low-velocity gradient group, in line with the idea that LVG SNe Ia correspond to off-center delayed detonations seen from the deflagration side (Maeda et al. 2010a; although see Blondin et al. 2011a for an interpretation of velocity gradients in the context of symmetric explosions) or at least to WDs that underwent less intense burning (Tanaka et al. 2008).

Thomas et al. (2011a) investigated the presence of C ii λ6580 in early-time SN Ia spectra from the SNfactory collaboration and found definite detections in five SNe Ia. All five SNe are part of the CN/LVG subclass and are otherwise unremarkable, apart from SN 2005cf, which displays clear HVFs in the Si ii λ6355 and Ca ii IRT lines. Of the remaining four SNe, three (SN 2005el, SN 2005ki, and SNF20080514-002) are characterized by blue colors and narrow light-curve widths, suggesting that carbon-positive SNe Ia may be preferentially associated with a specific photometric behavior, and one (SN 2005di) is highly reddened by host-galaxy dust. Thomas et al. (2011a) estimate that ∼20% of "normal" SNe Ia display a C ii λ6580 absorption as late as −5 days from B-band maximum light, in line with the results of Parrent et al. (2011; see also the paper recently submitted by Silverman & Filippenko 2012). They also illustrate using a toy model that a spherically symmetric distribution of carbon can account for the C ii λ6580 line-profile morphology, although they cannot rule out the large-scale asymmetries inferred by Maeda et al. (2010a) to explain the spectroscopic diversity of SNe Ia.

In Figure 21 we show the Si ii λ6355 region in previously unpublished spectra from the CfA SN Program where we tentatively identify an absorption due to C ii λ6580. For each of the 21 SNe Ia in this figure, we only show the earliest spectrum where the C ii λ6580 absorption is seen, except for a few cases where the earliest spectra had too low S/N (SN 2004at, SN 2004fu, SN 2006ax, SN 2007cq, and SN 2008A). Furthermore, we do not show spectra of SNe Ia already included in Figure 19. In all this represents a total of 24 SNe Ia not included in the study by Parrent et al. (2011).²² A C ii λ6580 line in SN 2005el was already noted by Thomas et al. (2011a) in their earliest spectrum.²³ We do not attempt to confirm these detections with parameterized spectrum synthesis codes as done by Parrent et al. (2011) using SYNOW (e.g., Branch et al. 2005) or by Thomas et al. (2011a) using SYNAPPS (Thomas et al. 2011b), but most of the tentative C ii identifications (17 out of 24) correspond to firm detections in spectra from different nights. Of the remaining seven SNe Ia, three only have a single pre-maximum spectrum (SN 2004bg, SN 2004fz, and SN 2005mz) and two have possible detections in multiple spectra (SN 2000dn and SN 2003U). For SN 2001ep, we have a definite detection in the −7 day spectrum but none in the −3 day spectrum. The small notch seen in the −2 day spectrum of SN 2007ax is no longer visible one day later.

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The majority of SNe Ia with C ii λ6580 detections in Figures 19 and 21 are part of the CN subclass (19 out of 30). Moreover, 7/9 CN SNe Ia with spectra prior to −10 days from B-band maximum (see Figure 19) show C ii absorptions, suggesting that this feature is a generic property of this subclass, and highlighting the need for very early-time spectroscopy. The other 11 SNe Ia are almost equally divided among the SS (four objects), CL (three), and BL (four) subclasses. The carbon-positive SS SNe Ia include the 1991T-like SN 2007cq, the 2002cx-like SN 2008A, and the possibly super-Chandrasekhar SN 2006gz. One of the carbon-positive CL (SN 2007ax) has an obvious 1991bg-like spectrum, while the remaining two (SN 2002fb and SN 2005mz; both similar to SN 2006D) show evidence for more moderate Ti ii absorption. Of the four carbon-positive BL SNe Ia, only one (SN 2004fu) is part of the high-velocity subclass of Wang et al. (2009a), with a high-velocity gradient in the Si ii λ6355 line ($\dot{v}\approx 120$ $\rm {km\,s}^{-1}\,\rm {day}^{-1}$ measured between +2 days and +22 days). This also confirms the results of Parrent et al. (2011) who find only one carbon-positive SN Ia of the HVG group in their sample (SN 2009ig; see also Foley et al. 2012). Parrent et al. (2011) suggest line blending as a possible explanation for the apparent paucity of high-velocity carbon-positive SNe Ia.

The C ii λ6580 blueshifts at maximum absorption in this sample of 30 SNe Ia are typically in the range −10,000 ≳ v_abs ≳ −14,000 km s⁻¹, with the notable exceptions of the 2002cx-like SN 2008A (v_abs ≈ −8000 km s⁻¹ at −6 days, similar to the Si ii λ6355 absorption velocity at the same phase; see Figure 15) and the peculiar SN 2006gz (v_abs ≈ −15,500 km s⁻¹ between −13 days and −11 days), both part of the SS subclass. The absorption velocities of C ii λ6580 and Si ii λ6355 are similar for most objects at any given pre-maximum epoch, their ratio being within 10% of unity (see also Figure 8 of Parrent et al. 2011). Two SNe Ia (SN 1990N at −14 days and SN 2005cf between −13 days and −11 days) have v_abs(C ii)/v_abs(Si ii) < 0.8, but both display HVFs in the Si ii λ6355 line (see Figure 19), which biases this ratio to lower values. Conversely, SN 2006gz, SN 2008Z (both SS SNe Ia), and the 1991bg-like SN 2007ax display v_abs ratios closer to 1.2.

The maximum velocity at which C ii λ6580 absorbs (v_max) is in the range −12,000 km s⁻¹ ≳ v_max ≳ −16,000 km s⁻¹ for most SNe Ia. Since v_max and v_abs are strongly correlated, we find the same two outliers as above (SN 2008A, v_max ≈ −10,000 km s⁻¹; SN 2006gz, |v_max| ≳ 18,000 km s⁻¹). Within the same sample, the maximum velocity at which Si ii λ6355 absorbs is typically significantly higher (−20,000 km s⁻¹ ≳ v_max ≳ −28,000 km s⁻¹). This suggests that some silicon is ejected at larger velocities than the bulk of unburnt carbon, incompatible with current delayed-detonation models (see, e.g., Maeda et al. 2010b). As noted by Branch et al. (2007), the pulsating reverse detonation models of Bravo & García-Senz (2006) could provide a viable explanation for the low carbon velocities compared to the bulk of IME.

Thomas et al. (2011a) suggest that the presence of C ii lines in normal SNe Ia could be preferentially associated with blue colors and narrow light-curve widths, based on the SALT2 parameters (x₁, c). They note, however, that this trend does not appear to hold for the sample analyzed by Parrent et al. (2011). We have reliable SALT2 fits for 24 out of 30 SNe Ia in our carbon-positive sample, and do not find evidence for bluer colors with respect to the carbon-negative sample (171 SNe Ia with reliable SALT2 fits). The median SALT2 x₁ is approximately −0.6 for both samples. Since the SALT2 color parameter does not distinguish between intrinsic color and extinction by dust, we re-ran this analysis using the intrinsic B − V color at B-band maximum and Δm₁₅(B) values from BayeSN. Within the CN sample, the carbon-positive SNe Ia appear to occupy a narrower Δm₁₅(B) range (1 mag ≲ Δm₁₅(B) ≲ 1.5 mag) than the carbon-negative sample (0.8 mag ≲ Δm₁₅(B) ≲ 1.6 mag), but a K-S test shows that the two distributions are consistent at the p = 0.37 > 0.05 significance level. The distribution of intrinsic B − V color is the same for both subsamples (p = 0.34). These conclusions hold when considering only SNe Ia with persistent C ii features (i.e., present later than −5 days), although the numbers become then too small for a robust analysis.

We have 21 spectra of 12 SNe Ia with spectra taken later than 150 days past maximum in the CfA sample presented in this paper (see Table 2). At these epochs, the spectra are dominated by forbidden lines of Fe ii and Fe iii, reflecting the NSE composition of the inner ejecta regions. Based on the observed correlation of the FWHM of the iron emission feature centered at ∼4700 Å with Δm₁₅(B) in nebular SN Ia spectra from the Asiago SN archive, Mazzali et al. (1998) found a relation between the kinetic energy of the explosion and the mass of synthesized ⁵⁶Ni. In particular, the low-luminosity SN 1991bg was associated with a low mass of ⁵⁶Ni but also a low explosion energy, suggesting that the fast-evolving light curve was the result of a small (sub-Chandrasekhar) exploding mass.

We revisit the relation between the FWHM of the nebular iron feature at ∼4700 Å and Δm₁₅(B) in Figure 22 (see Figure 2 of Mazzali et al. 1998). We have used our own FWHM measurements (see Table 7) on the publicly available data from Mazzali et al. (1998; i.e., with the exception of SN 1989M, SN 1991M, and SN 1993L), supplemented with 10 of the 12 SNe Ia from the CfA SN Program with nebular spectra (we exclude SN 2002cx due to low S/N and SN 2003cg due to the large associated error on FWHM), and additional data from the literature (including a nebular spectrum of SN 1991T from Schmidt et al. 1994, publicly available in the CfA SN Archive; see Table A2). When several measurements are available for a given SN, we plot the error-weighted mean FWHM. We also show the original relation published by Mazzali et al. (1998; long-dashed line).

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We have performed multiple least-squares fits of the relation Δm₁₅(B) = a + b(FWHM − 15) (FWHM in units of 10³ km s⁻¹) of Mazzali et al. (1998) to the data, the results of which are reported in Table 8, including the reduced χ²_ν and Pearson correlation coefficient (r). We also include the results of fits to the original data of Mazzali et al. (1998, see their Table 1). The entire CfA+Literature sample yields a slope (b = −0.069) that is somewhat shallower than the relation published by Mazzali et al. (1998; b = −0.103), although the correlation is strong in both cases (r = −0.71 and −0.80, respectively). The large χ²_ν indicates that the scatter about the best-fit linear relation is real and not the result of measurement uncertainties. Mazzali et al. (1998) suggested that this intrinsic scatter could be taken as evidence for a second parameter affecting the light-curve widths (in addition to the ⁵⁶Ni mass) or the velocities of the iron-dominated nebula (in addition to the kinetic energy of the explosion).

As noted by Mazzali et al. (1998), the low-luminosity SN 1991bg appears to be a clear outlier, being the only SN Ia with FWHM < 10,000 km s⁻¹ (FWHM ≈ 2500 km s⁻¹). Excluding this single SN from our sample results in a similar slope (b = −0.079) but a significantly degraded correlation (r = −0.37). This is somewhat different to the result obtained using the original measurements of Mazzali et al. (1998), where the exclusion of SN 1991bg leads to a steeper slope (b = −0.169) and a minor impact on the correlation strength (r = −0.69). The new CfA+Literature sample indicates that the scatter about the best-fit linear relation (excluding SN 1991bg) is ∼30% larger than for the original Mazzali et al. (1998) sample (∼50% larger when including SN 1991bg). Further excluding the low-luminosity SN 1986G (i.e., only considering SNe Ia with Δm₁₅(B) < 1.6 mag), there is essentially no correlation between Δm₁₅(B) and the FWHM of the 4700 Å Fe line (r = −0.17). If the latter quantity is indeed a proxy for the kinetic energy of the explosion, as suggested by Mazzali et al. (1998), our measurements imply that the kinetic energy of the explosion does not strongly depend on Δm₁₅(B), and hence on the SN Ia luminosity, for all but the least luminous events. This is expected in delayed-detonation models where the WD star is almost completely incinerated, as burning of C/O to Ni or Si-group elements releases similar amounts of energy (see, e.g., Hoeflich et al. 1995). One will need to see whether additional data (in particular for low-luminosity SNe Ia) continue to disfavor a strong relation between both quantities.

A recent study of nebular-phase SN Ia spectra by Maeda et al. (2010c) revealed that some forbidden lines associated with Ni ii and Fe ii displayed up to ±3000 km s⁻¹ velocity shifts about the central wavelength, while others did not (this was first noted in mid-IR spectra of SN 2003hv by Gerardy et al. 2007). This result was interpreted by Maeda et al. (2010c) in the context of a simple kinematic model crafted to reproduce basic characteristics of an off-center delayed-detonation model. These include an inner high-density region of stable iron-group elements and ⁵⁶Ni, and an outer low-density ⁵⁶Ni-rich region. The former results from the deflagration phase, and hence presents a global velocity offset in the case of an off-center ignition, while the latter results from the detonation phase, resulting in a spherically symmetric distribution of ⁵⁶Ni in the outer region. Different spectral lines in nebular SN Ia spectra thus trace different zones of the inner ejecta, and the diversity in velocity shifts for lines tracing the deflagration ash (in particular [Fe ii] λ7155 and [Ni ii] λ7378) then stems from viewing-angle effects: the lines are blueshifted when the SN is viewed from the ignition side and redshifted when it is viewed from the opposite direction.

Subsequent investigations by Maeda and coworkers showed that these nebular line shifts were related to the velocity evolution of the Si ii λ6355 line around maximum light (Maeda et al. 2010a) and to the intrinsic color at maximum light (Maeda et al. 2011), suggesting that both quantities were also related to viewing angle effects in off-center delayed-detonation models.

We investigate the relation between nebular line shifts and intrinsic color in Figure 23. We use the intrinsic B − V color at B-band maximum light derived from BayeSN fits as opposed to the intrinsic B^max − V^max pseudo-color used by Maeda et al. (2010a, 2010c, 2011; see also discussion in Section 5.3). We were not able to determine reliable intrinsic colors for SN 1986G, SN 2000cx, or SN 2009dc, since BayeSN does not include such objects in its training sample. We also highlight two objects with an inferred visual extinction A_V > 1 mag, and note that objects at redshifts <0.01 are subject to a larger uncertainty given the impact of peculiar velocities on the redshift-based distance. We have checked, however, that using external distance priors based on Cepheids or Surface Brightness Fluctuations results in inferred intrinsic colors that are consistent within <1σ with those reported here. We computed nebular line shifts based on a two-Gaussian fit to the emission profiles of [Fe ii] λ7155 and [Ni ii] λ7378. The nebular velocity, v_neb, is simply computed as the mean velocity of both lines, while the error is set to the standard deviation of both measurements (see Table 9). For cases where only one of two lines has a well-defined emission profile (usually [Fe ii] λ7155), we set the error in v_neb to 600 km s⁻¹, as done by Maeda et al. (2011). Our measurements are consistent within the errors with those reported by Maeda et al. (2010a, 2011), with a scatter about zero difference of ∼150 km s⁻¹ when the same spectrum was used (marked with a "†" in Table 9) and of ∼700 km s⁻¹ when using a different spectrum of the same object. One notable exception is SN 2006X for which we infer a ∼1500 km s⁻¹ larger v_neb (2832 ± 343 km s⁻¹ cf. 1331 ± 164 km s⁻¹ reported by Maeda et al. 2011). This difference is suspiciously close to the host-galaxy redshift (cz = 1550 km s⁻¹ from Falco et al. 1999) and results from Maeda et al.'s accidental use of spectra of SN 2006X available in the SUSPECT²⁴ archive that were already de-redshifted by 1587 km s⁻¹ (K. Maeda 2011, private communication; this error does not affect their conclusions, however). We include four SNe Ia (SN 1994ae, SN 1995D, SN 2003kf, SN 2007af) that are not part of the sample studied by Maeda et al. (2010a, 2010c, 2011) and for which we have spectra past +150 days from B-band maximum light (see Table 2).

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The data points in Figure 23 are color coded according to their spectroscopic subclass ("Normal" and "High-velocity" in the classification scheme of Wang et al. 2009a). Our spectroscopic coverage for SN 2007sr only starts ∼2 weeks past B-band maximum (see Table A1), so we infer its spectroscopic subclass (HV) based on the Si ii λ6355 absorption velocity of −12,300 km s⁻¹ at +5 days reported in CBET 1173 by Naito et al. (2007). We could not determine a spectroscopic subclass for SN 2006dd, since there are no available spectra around maximum light (Stritzinger et al. 2010).

The distribution of nebular line shifts (Figure 23, top) shows a clear separation at v_neb ≈ 1000 km s⁻¹ between the N/HV spectroscopic subclasses. One notable exception is the HV SN 2004dt, whose nebular spectrum was shown by Maeda et al. (2010a) to resemble most that of the faint SN 1991bg than other HV SNe Ia. We thus reproduce the main result of Maeda et al. (2010a), although they base their spectroscopic classification on the velocity gradient of the Si ii λ6355 line defined by Benetti et al. (2005) as opposed to its absorption velocity at maximum (all the HV SNe Ia in Figure 23 are classified as HVG by Maeda et al. 2010a). In the context of the simple kinematic model of Maeda et al. (2010c), the HV subclass consists of SNe Ia viewed from the direction opposite to the initial off-center ignition (hence the pronounced redshift of nebular lines), where IMEs synthesized during the detonation phase are ejected at higher velocities.

The overall correlation between intrinsic B − V color at B-band maximum and nebular line shift is rather modest (r = 0.36; r = 0.58 when excluding the peculiar HV SN 2004dt). Considering only the 10 SNe Ia part of our low-extinction sample (by which we mean SNe Ia with E/S0 hosts or in the outskirts of spiral galaxies, following Maeda et al. 2011; see SNe Ia marked with an asterisk in Figure 23 and Table 9), the correlation is even lower than for the entire sample (r = 0.30). The bulk of SNe Ia in our sample with v_neb > −2000 km s⁻¹ (18 out of 21 with intrinsic color estimates) do nonetheless appear to follow a clear trend of redder intrinsic color for larger v_neb (r = 0.82). It is unclear whether the two Normal SN 1994D and SN 2003hv at v_neb < −2000 km s⁻¹ simply reflect a larger scatter at large nebular line blueshifts or reveal a non-monotonic relation between nebular line shifts and intrinsic color (which would then also accommodate SN 2004dt).

Our results appear to differ slightly from those of Maeda et al. (2011), who find a strong correlation and an obvious linear relation between intrinsic color and nebular line shift for their low-extinction sample (which includes SN 1994D and SN 2003hv, both modest outliers in the relation defined by the bulk of our sample), and which also applies to all other SNe Ia in their sample (including SN 2004dt) after excluding three highly reddened objects (SN 1998bu, SN 2002bo, and SN 2006X; their Figure 4). In fact, Maeda et al. (2011) use the relation between "intrinsic" B^max − V^max pseudo-color²⁵ and Δm₁₅(B) found by Folatelli et al. (2010) to correct their own pseudo-color measurement. Using this same approach we too find a strong correlation between this color residual and v_neb for our low-extinction sample (r = 0.72). We note, however, that the Δm₁₅(B)–color relation of Folatelli et al. (2010) has a low statistical significance (∼2σ), and that there is no correlation between the B^max − V^max pseudo-color and Δm₁₅(B) for the low-extinction sample of Maeda et al. (2011; r = 0.05; their Figure 3). Whether or not we apply this correction makes little difference to the strength of the correlation between B^max − V^max pseudo-color and v_neb for our low-extinction sample (r = 0.68 with no correction).

In any case these nebular line shifts do seem to provide information on the intrinsic properties of SNe Ia independently from Δm₁₅(B). We confirm the lack of correlation between v_neb and Δm₁₅(B) (|r| < 0.1 whether or not we exclude peculiar SNe Ia) noted by Maeda et al. (2010a), although Maeda et al. (2011) argue that such a correlation is expected theoretically given the correlation they find between Δm₁₅(B) and viewing angle (for their model with 0.3 M_☉ of ⁵⁶Ni; their Figure 12). The recent study of two-dimensional delayed-detonation models of Kasen et al. (2009) by Blondin et al. (2011a) shows, however, that the relation between Δm₁₅(B) and viewing angle is a complex one, both non-monotonic and nonlinear in character regardless of the ⁵⁶Ni mass. While we agree with Maeda et al. (2011) that more data are needed to study the relation between nebular line shifts and Δm₁₅(B), we also note the limiting predictive power of simple models crafted to reproduce the observed trends.

Table A1 presents the observational details of the spectra and Table A2 presents the references for spectroscopic data from the literature used in this paper.