SUBLUMINOUS TYPE Ia SUPERNOVAE AT HIGH REDSHIFT FROM THE SUPERNOVA LEGACY SURVEY^*

Modeling the colors of SNe Ia is fundamental for the LC fit and candidate selection of Section 3, as well as for the study of the recovery efficiency needed for the rate calculation of Section 4. The challenge in modeling SN Ia colors resides in the difficulty of disentangling intrinsic and extrinsic colors, the latter coming mainly from reddening of the host galaxy. As explained in Conley et al. (2008), the SiFTO method does not impose a single color model across all filters, which would probably not encompass the whole range of SN Ia variability; rather the SED is adjusted to match the observed colors in each filter.

We first remove highly reddened objects by the foreground Milky Way using the Schlegel et al. (1998) dust maps and a cut of E(B − V)_MW < 0.15. To get an idea of the intrinsic color–stretch relation of the objects, we take the transients that are hosted in less extincted passive galaxies, i.e., in galaxies catalogued as E/S0 for the nearby sample from the NED database¹⁵ and in galaxies with no star formation for the SNLS (Sullivan et al. 2006b). A color similar to B−V at B-band maximum brightness, c, as a function of stretch for these objects is shown in Figure 2 (filled circles and squares).

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We create a "low-extinction sample" by taking all SNe, regardless of their hosts, with a color below the upper limit from the maximum found for the SNe in only quiescent environments, c < 0.37. This is only done for normal-s SNe Ia, i.e., s>0.8, which have a relatively flat color–stretch relation. Low-s SNe Ia become redder at a steeper rate with decreasing stretch and are hosted in less dusty environments, so we do not use such a maximum color cut, although we remove known extremely extincted objects (e.g., SN1986G, SN2006br and SN2007ax). The fit to this larger sample (open and filled circles and squares) in Figure 2 is well represented by a piecewise linear model with a break at s = 0.77 (solid line):

$$\begin{equation} c(s)= {\left\lbrace {\begin{array}{@{}l@{\quad }l@{}}C_0+\gamma _1\,(s-1) & (s>0.77)\\[4pt] C_0+\gamma _2\,(s-1) + C_{\mathrm{match}} & (s\le 0.77). \\ \end{array}}\right.} \end{equation} \tag{ 1 }$$

The fit parameters are C₀ = 0.08 ± 0.02, γ₁ = −0.06 ± 0.02, γ₂ = −2.79 ± 0.07 and the transition stretch value s = 0.77 ± 0.01. C_match = −0.23(γ₁ − γ₂) is a correction factor to unite both lines at s = 0.77. The transition value for stretch (s = 0.77), together with the one found for the magnitude–color–stretch relation in next section (s = 0.82), argues for our chosen limiting value of s 0.8 between low-s and normal-s. The piecewise linear model has the best fit (reduced χ²_ν of 23.4) among other models (e.g., exponential with χ²_ν = 25.0 or power law with χ²_ν = 25.9). The figure shows that low-s SNe Ia are not only redder but their color–stretch relation is much steeper than for normal SNe Ia. Errors in stretch and color were propagated in the fits. Allowing a fit to all objects, also extincted ones, provides consistent results (with a transition stretch at s = 0.83) and is also shown in Figure 2. It is worth mentioning that if we force a piecewise linear model with a fixed lower transition value of s = 0.7, more similar to characteristic "subluminous" surveys (Taubenberger et al. 2008), the fit is slightly worse (χ²_ν = 24.6) and the slopes steeper. Due to the large scatter of colors at low-s, a range of consistent models (with χ² < χ²_min + 2) is obtained for transition values of 0.69–0.87.

We next derive a relation between the absolute peak magnitude as a function of stretch and color. To limit the effects of peculiar velocities, we require SNe in our training sample to have either z_CMB>0.01 (CMB frame redshift) or a known distance to the host taken from the NED database.

We simultaneously fit magnitude, stretch and color to different models. The best one is again bilinear in both, stretch and color. The absolute magnitude shows a steeper slope as a function of color for redder objects changing at c 0.07 (upper panel, Figure 3). Against stretch, the magnitude values (lower panel, Figure 3) are also well fit by a bilinear model with a shallower slope for the low-s stretch range. The effects in the differing slopes are not as strong as for the color–stretch relation but they will make the posterior analysis more precise. The errors in the absolute magnitude (from the apparent magnitude and redshift errors) and in the stretch and color are included in the fit. The magnitude–stretch–color relation can be summarized as follows:

$$\begin{equation} M_B=M+\alpha _{[1,2]}\,(s-1) + \beta _{[a,b]}\,c + M_{[2,b]}^{{\rm match}} \end{equation} \tag{ 2 }$$

with M = −19.11 ± 0.01, and where α₁ = −1.19 ± 0.03 and α₂ = −0.54 ± 0.09 are respectively used for stretch values greater and lower than the transition value of s = 0.82 ± 0.02. β_a = 3.00 ± 0.02 and β_b = 2.15 ± 0.05 are used for values of c greater and lower than 0.07 ± 0.01, respectively. M^match₂ = −0.18(α₁ − α₂) and M^match_b = 0.07(β_a − β_b) unite the fit at the changing values. This model results in a slightly better fit (χ²_ν = 19.6) than other models (e.g., exponential—in stretch and color—with χ²_ν = 19.9).

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We include an asymmetric error-snake on these relations, shown by the dotted lines in Figures 2 and 3. These delimit (twice) the region within which 95% of all SNe Ia in each bin of 0.1 in color and 0.07 in stretch are found around the best model. The width of the region has been doubled to allow for the larger photometric scatter of the high-z SNe, which will be important later in Section 3.2. This region also covers any uncertainties in the relations modeled in this section and allows for scatter in the color of the SNe. Requiring all objects to be inside those error snakes, i.e., color–stretch, magnitude–color and magnitude–stretch, we find that 90% of the normal and low-s SNe Ia pass the cut (see Table 3). However, only two contaminants (all SNe Ib/c) survive the same cull, an 6% of the total of contaminants. We find this tradeoff to be beneficial. Reducing the contamination is useful when limiting the systematic errors as discussed in Section 4.3.

In P11, accurate photometric redshifts from the SNe were obtained with the use of two normal SN Ia LC fitters. Their approach requires a good spectroscopic sample for testing the method and the precision of the final photometric redshifts. We do not possess any 91bg-like SN Ia spectra at high redshift with clear Ti ii features, only three low-s objects at the border of our definition, s 0.8, that have spectra of rather normal SNe Ia. The high degree of degeneracy in the LC fits, particularly with stretch and redshift, which compete to make an object appear faint, demand an unbiased redshift estimate that will lead to appropriate LC parameters like stretch and color.

Several hundred objects of the 4-year dataset possess spectroscopic redshifts (spec-z) from either the SN or from the host galaxy. For the SN-derived redshifts, the SN type has been determined from the spectra in the majority of cases (Howell et al. 2005; Bronder et al. 2008; Ellis et al. 2008; Balland et al. 2009; Walker et al. 2011). Most of these are normal SNe Ia (413), although a good sample of CC objects is also available (78). The host spec-z include data from DEEP/DEEP2 (Davis et al. 2003), VVDS (Le Fèvre et al. 2005), zCOSMOS (Lilly et al. 2007), VIMOS (J. Jönsson et al. 2011, in preparation), and additional FORS observations. Many candidates without spectroscopic redshifts have host photometric redshifts from the SNLS broadband photometry either using the spectral connectivity analysis method of Freeman et al. (2009), or, when no match is found, through fits to spectral energy distributions as in Sullivan et al. (2006b). The resulting host photo-z, z_hostphot, are very good to z 0.75 with $\frac{|\Delta z|}{(1+z_{{\rm spec}})}=0.042$ and a low (~5%) catastrophic failure rate ($\frac{|z_{\mathrm{spec}}-z_{\mathrm{hostphot}}|}{(1+z_{\mathrm{spec}})}>0.15$).

Fortunately, some of those SNe Ia with catastrophic redshifts can be identified with the help of the two LC fitters of P11 adjusted for low-s SN: estimate_sn (Sullivan et al. 2006a) and the aforementioned modified version of SiFTO (see Section 2). Their use in the multi-step process explained below provides good quality-of-fit parameters capable of distinguishing SNe Ia from CC-SNe, but also offer an independent mechanism for obtaining photometric redshifts using the SN, z_SNphot, in cases where host redshifts are lacking.

Once the redshift is known, SiFTO is used to obtain the stretch and color of each SN. Tests on the local sample (Section 2) show that it works well without imposing any stretch–color model. As shown in Figure 2 of P11, the derived stretch is largely independent of the redshift because the rise and fall of SNe Ia are faster at shorter wavelengths, which largely compensates for redshift uncertainties. In Figure 4, we demonstrate that this behavior extends to low-s objects. Shifting the redshift—either photometric or spectroscopic—for each SN in our final low-s sample (see Section 3.2) by as much as Δz = ±0.3, we obtain a stretch variation lower than ~s ± 0.1. While this shows more dependence on redshift than the normal-stretch population, it still allows us to obtain accurate stretch estimates even in the presence of moderate redshift uncertainties.

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The estimate_sn routine is described in more detail in Sullivan et al. (2006a). It performs a grid search on the χ² of a fit to SN photometry in flux space to a five-parameter model, including redshift. For this analysis, we modified it to include low-s objects using the color–stretch relations of Equation (1), and the magnitude relation of Equation (2). This fit depends on the assumed values of the cosmological parameters.

We calculated the SN photo-z for all objects in a similar fashion to P11 using the following multi-step process: First, we use estimate_sn to get a preliminary redshift estimate. We then use this redshift as a fixed input to a SiFTO fit to obtain an improved stretch measurement. Finally, we feed back this stretch to a final estimate_sn fit as a fixed parameter to obtain a more accurate z_SNphot. We use this SN photo-z for SNe that do not have any other redshift (spectroscopic or host photometric).

Objects with z_hostphot that have "bad fits," i.e., that do not pass all cuts of Section 3.2, are not discarded, but instead the cuts are re-applied using z_SNphot. If this also fails, then the object is rejected. This offers some protection against catastrophic host photo-z failures. This recovers 29 confirmed SNe Ia. All known contaminants fail with both photo-z values, so we are confident that this procedure still rejects non-Ias.

The complete final set of photo-z, after applying the cuts explained in the next section, contains 82% with host photo-z and 18% with SN photo-z. Figure 5 shows the comparison of photometric and spectroscopic redshifts for all SN Ia candidates with spectroscopic redshift either from the SN or the host. The median precision is $\frac{|\Delta z|}{(1+z_{{\rm spec}})}=0.030$. The photo-z are more accurate to z 0.6 (|Δz|/(1 + z_spec) = 0.025), where our rate study will concentrate.

We perform final SiFTO (and low-s-modified estimate_sn) fits to each object to obtain the parameters of each SN and quality-of-fit values. A series of cuts are then applied to screen out objects with poorly estimated parameters or those unlikely to be SNe Ia. A summary of the applied culls and the number of objects surviving each of them can be found in Table 4. The χ² culls are similar to those of P11, although we weaken the requirements slightly for s ≤ 0.8 objects based on the study of the low-z sample: reduced χ²s of $\chi ^2_{\mathrm{estimate\_sn}}<13$ for the overall fit, and for individual passbands $\chi ^2_{\mathrm{estimate\_sn}}(r_M)<11$ and $\chi ^2_{\mathrm{estimate\_sn}}(i_M)<16$, and a SiFTO cull of χ²_SiFTO < 11. Additionally, we require the objects to be inside the error snakes trained in Section 2 with low-z, low-s SNe in order to constrain their parameters (Figure 6). These cut values are derived empirically by trying to minimize the number of known contaminants that pass and minimize the number of rejected confirmed SNe Ia.

Table 4. Culls Applied to the SNLS Sample

Cull	Number
	All	Low-s	Confirmed	Confirmed
			SNe Ia	CC-SNe
None	5234	1497	407	86
Observation	2615	837	343	62
χ2SiFTO	1054	144	338	22
$\chi ^2_{\mathrm{estimate\_sn}}$	925	118	329	8
$\chi ^2_{f, \mathrm{estimate\_sn}}$	848	109	303	5
Snakes	710	70	295	1
FINAL	710	70	295	1
z < 0.6	174	18	118	0

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3.2.1. At z ≤ 0.6

Up to z = 0.6, the final sample contains 161 objects, of which 126 have spectroscopic redshifts. They span the stretch range 0.6 < s < 1.2, of which 18 have s ≤ 0.8. Six of these have known spectroscopic redshifts: three are confirmed SNe Ia with s ~ 0.8 with no outstanding 91bg-like feature in their spectra, and the other three have spectroscopic redshifts but have too low of a contrast with the underlying host galaxy to conclusively establish the spectroscopic SN type. The final parameter distributions are shown in Figures 7 and 8. The low-s photometric sample has median stretch and color of 〈s〉 = 0.74 and 〈c〉 = 0.24 (as compared to 〈s〉 = 1.00 and 〈c〉 = 0.00 for the normal-s population). The mean 〈Δ mag〉 = 0.76 ± 0.69 clearly shows that our low-s sample is not as extreme as the characteristic 91bg of Δmag ~ 2mag.

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The final errors in photo-z are obtained from Figure 5 by taking the standard deviation around the median as a function of z. The stretch and color errors are provided by SiFTO, which requires a known redshift input. Stretch and color variations due to redshift uncertainties will be taken into account with a Monte Carlo simulation in Section 4.2.

3.2.2. At z>0.6

The detection incompleteness at 0.6 < z < 1.0 increases rapidly for low-s SNe Ia, as will be shown in next section. We still find a significant population of 44 candidates (11 with spec-z from the host and 5 with low signal-to-noise spec-z from the SN, 2 of which are confirmed SNe Ia of s ~ 0.8 with no 91bg features) with median stretch, <s> = 0.76, and color, <c> = −0.02. The average color is extremely blue compared to the median color at z < 0.6 of 0.24 confirming that we are only observing the blue tail of the low-s distribution. This can be seen in Figure 9. They are not very faint objects, even for low-s, as shown in Figure 8. The lack of characteristic low-s red objects makes it difficult to reliably estimate the low-s rate at z>0.6 and proper corrections need to be made (see Section 4.2).

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Additionally, we do find a presence of blue low-s candidates at high-z which are absent at lower-z. Such blue objects with c < −0.05 are neither found in the nearby nor in the z < 0.6 SNLS samples, but represent ~30% (14) of all z>0.6 low-s objects. We do not expect all of these objects to be CC-SNe, which could potentially contaminate the low-s sample with their characteristic bluer colors. As will be shown later, besides going through our SN Ia pipeline, the environment of these candidates is not characteristic of CC-SNe.

A second possibility is that this group represents another class of transients that are fast and blue. The object found by Perets et al. (2010) is faint and fast-evolving but with clear different spectroscopic features and bluer colors than 91bg-like objects. It is hosted in an old stellar population as well as most of our candidates. However, our objects are not as faint (Figure 8). On the other hand, the transients found by Poznanski et al. (2010) and Kasliwal et al. (2010), suggested SN .Ia-like helium detonations of a WD, are blue and of similar brightness than ours, but evolve faster. When fit with our templates, we obtain very poor fits with s = 0.44 and s = 0.33, respectively, lower than our lower stretch limit for subluminous SNe Ia.

A tempting explanation is that the input photometric redshift for the LC fit is consistently wrong and affects the obtained color and stretch. This hypothesis is ruled out because seven of these objects have spectroscopic redshift and the rest (except one) have photometric redshift from the host (with mean photo-z error of ~0.08), which do not show any significant systematic. We have seen the small effect in the measured stretch when the redshift is varied (Figure 4). We now show the effect in the magnitude and color of photometric candidates in Figure 10. Their variation due to changes in photo-z cannot account for all the bluer colors but for some of them—only if the redshifts were all consistently overestimated by +0.05, the colors would be redder by +0.08 (and magnitudes fainter by +0.17, whereas stretches would only change by −0.004).

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Photometric errors in stretch and color get larger as we probe higher redshifts. Most of the candidates found at these redshifts have stretch close to the limiting definition of s = 0.8, so that they are consistent with being fast normal-s SNe Ia. Only three candidates have s + s_error < 0.8, one of them with s < 0.6. Of these, two also have c +  c_error < −0.05, inconsistent with typical low-s colors, and will be further discussed in 4.3. The rest can therefore be explained as the blue tail of s ~ 0.8 objects while the redder counterpart remains unseen due to selection effects. In the remainder of this work, we treat this group as such and correct for this selection bias when calculating the rates.

The detection efficiencies _i account for observing biases measuring the rate. They depend on numerous factors and are specified as a function of the SN characteristics such as redshift, stretch and color (along with field and year of observation). The efficiencies used here are described in P11 and Perrett et al. (2010), and are based on a Monte Carlo study of 2.4 million artificial SNe Ia injected into the actual SNLS images. This approach avoids the need to make assumptions about temporal and spatial data coverage.

The artificial SNe are generated using uniform distributions of redshift and stretch (down to s = 0.5), and the color is determined using a relation like Equation (1), including an intrinsic scatter around the stretch–color relation (a Gaussian noise with dispersion σ = 0.04 added independently to U and V). The peak magnitudes are generated from the luminosity–stretch–color relation, as in Equation (2), and the redshift and assumed cosmology, plus an additional unmodeled magnitude dispersion from a Gaussian distribution with an intrinsic scatter of σ_int = 0.15. The LC is generated from integrating, at a given epoch and through the filter functions, the SN Ia spectral templates scaled for the B-magnitude and corrected for U − B and B − V colors, and Milky Way and host extinction. The artificial SN is then inserted into each i_M image obtained by the SNLS. The images are then processed through the same pipeline used in the real-time SN discovery, and the recovered fraction is used to derive the efficiency as a function of various parameters, including redshift, stretch, color, observing season and field (see a more detailed description in Perrett et al. 2010). The low-s candidates, which are faint, red and have a rapid rise and fall, are more difficult to detect and therefore have a lower efficiency, so are accordingly given a higher weight when calculating the rates.

The efficiencies are calculated in bins of width Δz = Δs = Δc = 0.1. The measured efficiencies, linearly interpolated according to each SN, are shown in Figure 11. It can be seen that the mean efficiencies are lower for the low-s sample up to z ~ 0.6. Above z = 0.6, the mean efficiencies seem to approach the normal-s sample. At these redshifts they become low for all but the bluest low-s objects. This leads to a higher fraction of low-s blue objects with very few red ones recovered, as found in Section 3.2. We show the expected efficiency distribution for the missing red, low-s SNe Ia of an unevolving color distribution as green squares in Figure 11 based on the observed low-s color distribution at low redshift from Section 7. The rates at z>0.6 will need to be corrected for these missing objects (see the next section). We note that the efficiencies shown in Figure 11 are the raw efficiencies—the recovery fraction from the simulated SNe—without the observational constraints of section Section 3 nor the sampling time correction that depends on each field and season. These corrections can make the actual efficiencies much lower (see P11). For instance, one of our low-s candidates at high-z with z = 0.80, s = 0.63, and c = 0.24 has a raw efficiency of 0.54 but with observational constraints and sampling time this efficiency is only 0.18, which explains why these objects are hard to observe.

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We calculate a rate evolution, r_meas, shown in Column 5 of Table 5 solely based on Equation (3) and with the efficiencies from Section 4.1. We use bins of Δz = 0.2 due to the low sample statistics (the last z-bin is only 0.1 wide as no low-s candidate exists beyond z = 1). The measured errors in stretch and color from the LC fit, and especially the errors in the photometric redshift of each candidate, can lead to uncertainties in the rate measurement. We account for these via a Monte Carlo (MC) simulation that randomly shifts the photometric redshift according to its error (for those SNe Ia with photo-z). For each redshift, a corresponding LC fit provides new parameters and its covariance matrix. This latter is used to generate a new random set of stretch and color using standard techniques (James & Roos 1975). We recalculate the rates from Equation (3) for each iteration based on the randomized values, and use the median value over the 1000 realizations to determine the final rate, r_MC. We estimate the error from the variance of the rates in each bin (and call them "MC" errors shown in Column 6 of Table 5 and Figure 12) and add it in quadrature to the weighted 1σ Poisson statistical errors in Figure 12 (confidence levels calculated using the Clopper–Pearson method used for example in Gehrels 1986). The error simulation differs from P11, as the SNLS does not possess a known spectroscopic sample of low-s SNe Ia from which we could obtain parameter distributions. Instead, we treat each SN individually by randomly drawing parameters according to its errors and covariances. Although the precise details of our simulation differ from P11, an SN Ia rate evolution within the errors is obtained.

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Additionally, we have seen in the previous section that we do not find red objects in the higher-z bins. This selection bias results in underestimated rate values at high-z, which are not corrected by the efficiencies of the bluer objects found. There is a lack of objects in certain red color bins, so that no efficiency can account for them. To measure this effect, we take the low-s distribution of color in an underestimated high-z bin (for each MC iteration), we normalize this distribution to the known color distribution at z < 0.6 (right Figure 7), which has already been corrected for the efficiencies of each SN, and then generate fake SNe in the missing color bins. The normalization is done with the two bluest matching bins. Figure 13 shows an example for 0.7 < z < 0.9. The green open histogram is the final corrected histogram from the initial observed diagonal-shaded distribution. The correcting factor for this case is $f_{b,\mathrm{corr}}=1+\frac{N_b(\mathrm{fake})}{N_b}=1.9$, where N_b(fake) is the number of generated SNe in the bin and N_b is the observed number, so that the corrected rate is r_b,corr = r_b × f_b,corr. The final correcting factors from the median of all MC iterations, summed Poisson (from the number of objects used in the bins used to normalize both distributions) and MC errors, as well as the resulting corrected rates are shown in the last two columns of Table 5. In Figure 12, the corrected rates are shown as stars.

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Finally, errors in the efficiencies can affect the rate values. As explained in P11, those come mainly from systematic errors in the underlying assumed Δmag distribution of SNe Ia, the distribution of residuals between the absolute magnitude of a fiducial s = 1 SN Ia and the magnitudes of observed SNe Ia after correction for stretch and color. To check for this systematic error, we allow for random efficiencies in the MC simulation calculated from a range of σ_int = 0.12–0.15 in the Δmag distribution, obtaining results that agree very well with the rates of Figure 12.

An important systematic error comes from core-collapse SNe mimicking the LC of low-s SNe Ia. In P11, a misclassification fraction of less than ≤3% was found for the normal-s population. We expect this fraction to be higher for low-s SNe Ia due to two effects. First, they are dimmer, and more easily confused with the intrinsically fainter CC-SNe at a given redshift. Second, as we probe higher redshift ranges, CC-SNe at low-z might masquerade as low-s SNe Ia at higher-z since we do not have spectroscopic redshifts for all candidates. SNe II have characteristic LCs that are very different from SNe Ia and present less danger of contamination. SNe Ib/c, on the other hand, represent a smaller fraction of CC-SNe, ~30% according to Cappellaro et al. (1999); Smartt et al. (2009) and 25% according to Li et al. (2011a), but are more similar to SNe Ia and will be the most probable contaminants.

The first error—where the photo-z is correct, yet the fit of an SN Ia template manages to pass the quality-of-fit cuts—can be examined using the training sample (as in Section 2). We found a 6% CC-SN misclassification rate, all SNe Ib/c. This fraction can be propagated to higher redshift using the SN Ia rate evolution of P11, $R_{\rm Ia}(z)\propto (1+z)^{\alpha _{\rm Ia}}$ with α_Ia 1.91, and a CC-SN rate evolution reflecting the star formation rate (SFR) from Hopkins & Beacom (2006), $R_{\rm CC}\propto (1+z)^{\alpha _{\rm CC}}$ with α_CC 3.6. We assume a ratio of SNe Ia to CC-SNe of ~0.22 at z = 0.3 from Bazin et al. (2009), which translates to a local ratio of ~0.39—as compared to ~0.48 found by Cappellaro et al. (1999) and 0.24 from Li et al. (2011a). The percentage fraction of contamination as a function of z, f_cont(z), can then be described as

$$\begin{equation} f_{\mathrm{cont}}(z)=\frac{N_{\mathrm{passcont}}(z)}{N_{\mathrm{pass}}(z)}=\frac{1}{1+0.39\, r\,(1+z)^{\alpha _{\mathrm{Ia}}-\alpha _{\mathrm{CC}}}}, \end{equation} \tag{ 4 }$$

where r = 0.88/0.06 is the cull passing efficiency (the probability of an SN passing all selection criteria) ratio of Ia with respect to CC, assuming it does not evolve with z. This equation allows us to get a crude estimate of the number of contaminants that would pass all cuts, N_passcont(z). By dividing the expected fraction of contaminants per z-bin by the mean weight of low-s SNe found therein (per MC simulation), we translate this into a rate error lower than 20% for z < 0.6 up to as high as 30% at z 0.9. The estimates are sensitive to the power law of the rate evolution, the initial fraction of SNe Ia to CC-SNe and the local misclassification fraction of 6%. Increasing this latter fraction to 10% or instead using SN Ia to CC-SN local ratio of 0.241^+3.7_−3.8 by Li et al. (2011a), we obtain contamination percentages up to 30% for z < 0.6 and 40% at z 0.9. This latter rate error from contamination is added in Table 5 and we believe it to be an upper limit. The contamination error is smaller than the combined statistical and MC errors at z < 0.6 but becomes comparable at higher-z. One should note that the misclassification error increases with redshift and would only make the rate higher than the actual one.

The second source of contamination comes from uncertainties in the photo-z. Incorrect redshift estimates can cause low-z CC-SNe to simulate low-s SNe Ia at high-z. This effect can be tested by running all spectroscopic objects through our selection criteria with their calculated photo-z instead of the spec-z. By doing this, we retain most of our candidates with a similar fit stretch and, most importantly, we still reject all the CC-SNe as contaminants (even though their photo-z was calculated with SN Ia templates and can occasionally be very different from the spec-z).

Perhaps the most concerning signature of contamination is the low-s SN Ia sample at z>0.6. These have rather blue colors and could be CC-SNe instead. As an additional test, we analyzed the host galaxy properties (Figure 14). In general these objects are found in massive and passive galaxies, not characteristic of CC-SNe. This provides evidence that most of these candidates, including the z>0.6 population, are likely the actual blue tail of the low-s SN Ia distribution, as mentioned earlier.

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Nevertheless, two of these objects, 05D3hp (with s = 0.59 ± 0.04 and c = −0.16 ± 0.09) and 07D2bc (with s = 0.72 ± 0.04 and c = −0.13 ± 0.03), have unusually blue colors at very low-s. The host of 05D3hp is small (M = 10^7.9M_☉) and star-forming (SFR/mass = 10^−9.8M_☉ yr⁻¹) and points rather toward a CC event imitating a low-s SN Ia. 07D2bc happened also in a star-forming (SFR/mass = 10^−10.0M_☉ yr⁻¹) but rather massive galaxy (M = 10^11.0M_☉). If these two objects are CC-events, they would constitute an ~5% contamination fraction. Although it is dangerous to assume the local star formation from general galaxy properties, if we were to assume that all events in "burst" hosts with SFR/mass < −9.5M_☉ yr⁻¹ are contaminants, then ~30% of the z>0.6 objects are misclassified (~25% at z ≤ 0.6), in agreement with our generous estimates.

Recently, Li et al. (2011b) have calculated a low-redshift SN Ia rate and a corresponding 17.9^+7.2_−6.2% fraction of 91bg-like objects (Li et al. 2011a). To obtain a rough local rate estimate of low-s SNe Ia defined according to our study, i.e., s ≤ 0.8, we use their luminosity-function sample (Table 3 of Li et al. 2011a) and directly fit objects for which we have photometric data (48 of 74) with SiFTO. For the rest we fit instead their closest R-band LC template match (Column 11). Using the completeness-corrected numbers (Column 9), we obtain 16.4 objects with s ≤ 0.8, or a 21.9^+6.9_−5.4% fraction. This fraction is complete within a volume of 80 Mpc and translates to a low-s rate of (6.3 ± 1.8) × 10⁻⁶ yr⁻¹ Mpc⁻³h³₇₀ based on the local rate of Li et al. (2011b), adjusted for h₇₀ = 1, and including the uncertainty from the Poisson error. We note that by using the whole low-z training sample of Section 2, the 41 low-s objects correspond to a 22% fraction of the total, in good agreement.

This rate estimate is based on small sample statistics and suffers from the biases that arise from a distance-limited host-targeted survey, where the rate of SNe is converted to a volumetric rate via the galaxy luminosity function. This can lead to systematic errors, as brighter galaxies may contain more SNe Ia of a certain type (as is expected for low-s). However, a local low-s rate estimate is a valuable tool for comparison to the high-z rates.

The low-s SN Ia rate, as opposed to the normal-s population, does not seem to increase with redshift and follow the SFH trend found in P11 (Figure 12). If we fit a simple power law, r ∝ (1 + z)^α, to the SNLS rate to z = 0.6 and the low-z estimate, we obtain α = −0.90 ± 1.38, smaller than the value of the normal-s population found by P11 (α_normal = 1.91 ± 0.24). This means that an evolution similar to the normal-s population is rejected with ~94% probability. If we include the corrected z>0.6 SNLS points, we find a higher slope more consistent with zero: α = −0.34 ± 1.41.

It is known that SNe Ia occur in both star-forming and non-star-forming regions, whereas low-s objects are usually found in passive ones. In the model of Mannucci et al. (2005); Scannapieco & Bildsten (2005), two populations are considered: a "prompt" component for SNe exploding in 0.1Gyr or less that is proportional to the SFH, $\dot{M}(z)$ and a "delayed" component for events with higher delay times that is proportional to the host mass, M(z). It is described as

$$\begin{equation} \mbox{SNR}_{\rm {Ia}}(z)=A M(z)+B\dot{M}(z), \end{equation} \tag{ 5 }$$

where A and B are scale factors for both components. The delayed model for low-s SNe Ia is supported when one looks at the host properties of the low-s sample. CFHT Legacy Survey host galaxy broadband photometry can be fitted to several template galaxy SEDs using PEGASE/ZPEG (Le Borgne & Rocca-Volmerange 2002) to determine average galaxy SFRs, stellar masses and photometric redshifts (Sullivan et al. 2006b). The properties of SNe Ia are compared to their host type (see Figure 14) to investigate the behavior of low-s objects in contrast to the normal-s population.

We use the specific star formation rate (sSFR), the star formation per unit stellar mass, as the indicator of star formation activity, and separate the host galaxy sample in passive galaxies, log(sSFR(yr⁻¹)) < −11.5, weak star formers, −11.5 ≤ log(sSFR(yr⁻¹)) ≤ −9.5, and strong star formers log(sSFR(yr⁻¹))> − 9.5, where SFR refers to the average star formation over the last 0.5 Gyr. This approach ignores important internal variations of SF in the galaxy, as shown in Raskin et al. (2009a), and provides only a general estimate of the SF. The sSFR distribution for two different stretch ranges (Figure 14, left), normal-s and low-s, provides clear evidence that the low-s population is less dependent on star formation. Neill et al. (2009) find that no local low-s SN Ia is found in a host with log(sSFR(yr⁻¹))> − 9.5. Here, we corroborate their results for high-z objects finding that only 3 (17%) at z < 0.6, and 11 (25%) at higher redshift, have higher mass-weighted SFRs.

Moreover, the host mass distribution for these stretch ranges (Figure 14, right) shows clearly different distributions, in which low-s SNe Ia occur almost solely in massive hosts. We extend the results of Neill et al. (2009) to the distant universe: low-s SNe Ia are found mostly in hosts with masses above ~10^9.5M_☉. For all cases, the distribution of host-galaxy mass for the two populations, normal-s and low-s, has a low two-sided Kolmogorov–Smirnov (KS) statistic, indicating a small probability that they are randomly drawn from the same distribution.

The above relations of SN Ia properties with host characteristics demonstrate that previous findings extend to the extreme of the SN Ia population at z>0.1, and at the same time are a confirmation that the observed photometric low-s sample obtained is real and not strongly contaminated from core-collapse events that are only found in late-type galaxies.

They also provide additional support to model the low-s SN Ia rate as just an A-component. Fitting the latter to the local and SNLS and rates up to z = 0.6 provides A = (1.05 ± 0.28) × 10⁻¹⁴ SNe yr⁻¹M⁻¹_☉ and is shown in Figure 12. If we fit both components, we obtain A = (1.29 ±  0.66) ×  10⁻¹⁴ SNe yr⁻¹M⁻¹_☉ and B = (−0.27 ± 0.64) × 10⁻⁴ SNe yr⁻¹(M_☉ yr⁻¹)⁻¹. When we include the corrected SNLS z>0.6 points, we obtain A = (1.14 ± 0.61) × 10⁻¹⁴ SNe yr⁻¹M⁻¹_☉ and B = (0.03 ± 0.62) × 10⁻⁴ SNe yr⁻¹(M_☉ yr⁻¹)⁻¹. In the latter two cases, B is close to zero (B < 0 is an unphysical result) suggesting that the mass component dominates over the component proportional to the SFH.

Comparing our delayed component (for z < 0.6) with the normal SN Ia population, the ratio of this value to the one in P11 is $\frac{A_{\mathrm{low}-s}}{A_{\mathrm{P11}}}\simeq 30\%$, which is particularly important at low-z, where the delayed component becomes dominant. This ratio sets a constraint on the fraction of low-s progenitors.

Ideally, the rate should be able to help constrain the SN Ia delay-time distribution as follows:

$$\begin{equation} \mbox{SNR}_{\rm {Ia}}(t)=\nu \int _{t_F}^t \mbox{SFR}(t^{\prime })\mbox{DTD}(t-t^{\prime })dt^{\prime }, \end{equation} \tag{ 6 }$$

where t' is the age of the universe, t_F is the time at which the first stars formed (effectively zero for the SNLS), DTD(t_d) is the delay-time (t_d) distribution and ν is the number of SNe Ia that explode per unit stellar mass formed. This number is related to the progenitor mass range and IMF, Ψ(M), as well as the efficiency η or the fraction of WDs that explode as SNe Ia, and the range of WD progenitor masses, M^low_WD and M^up_WD:

$$\begin{equation} \nu =\eta \frac{\int _{M_{\mathrm{WD}}^{\mathrm{low}}}^{M_{\mathrm{WD}}^{\mathrm{up}}}\Psi (M)dM}{\int _{0.1\,M_{\odot }}^{125\,M_{\odot }}M\Psi (M)dM}. \end{equation} \tag{ 7 }$$

Equation (6) can set constraints on the progenitors through the delay-time distribution if the SFH is known. We assume here the parameterization of Hopkins & Beacom (2006) for an SalA IMF (h = 0.7):

$$\begin{equation} \mathrm{SFH}\propto \frac{(0.017+0.13z)h}{1+(z/3.3)^{5.3}}. \end{equation} \tag{ 8 }$$

A simple parameterization power-law DTD ∝tⁿ has been used by several authors to model the SN Ia rate. n −1 is obtained in the SN Ia rate studies made by Totani et al. (2008); Maoz et al. (2010, 2011; Maoz & Badenes (2010) and is consistent with the DD scenario. Pritchet et al. (2008) obtain a different power law of n −0.5 investigating the SN Ia rate as a function of sSFR in the SNLS. It is interesting to see how the sub-sample of low-s behaves, in particular as the inferred DTDs tend to have a low tail at higher delay times (after a sharp initial peak) where the subluminous SNe Ia are expected to lie.

By fitting this model to the local and SNLS data up to z = 0.6, we obtain n = −0.93 ± 2.13 reflecting the large uncertainties in the rates. With the corrected z>0.6 rates, the power law becomes n = −2.67 ± 6.98 (see Figure 15). This is accomplished by increasing the other free parameter, the scale ν, by an order of magnitude such that many more SNe Ia occur at early times, in order to compensate for the steady rates at higher-z.

We also use the simple empirical models employed in Strolger et al. (2004): an exponential function motivated by the double-degenerate scenario (Tutukov & Yungelson 1994) and a Gaussian function motivated by a DTD with a specific delay time (Dahlen & Fransson 1999). The Gaussian distribution can be a proper approach for sub-groups of SNe Ia, such as the low-s one, but not for the entire population, which has been shown to be either bimodal or slowly declining in time. We use the formalism of the unimodal skew-normal DTD of Equation (6) of Strolger et al. (2010) as a generalization of these models encompassing a wide variety of shapes and distributions.

The model of Strolger et al. (2010) has three independent variables. We obtain for the local and z < 0.6 rates ξ = 6.4 ± 3.8, ω = 1.6 ± 5.2 and α = −81.2 ± 264 00, which translate to a Gaussian behavior of characteristic delay time τ = (5.11 ± 5.65) Gyr and width σ = 0.94 ± 3.16. This fit is shown in Figure 15. It indicates that low-s SNe Ia come from progenitors 3–10 Gyr old following a single Gaussian-shaped distribution. Such delay times correspond to masses of 1.5–2.5 M_☉ using t ~ m^−2.5. This sustains the argument that low-s SNe Ia come exclusively from a delayed channel of SNe Ia. Such estimates are higher than the lower delay-time limit obtained by Schawinski (2009) based on individual SN Ia studies in early-type galaxies with optical and UV data. On the other hand, they agree with Brandt et al. (2010) who find that SDSS SNe Ia with s < 0.92 have delay times of 2.4 Gyr. Fitting all points, corrected SNLS and low-z, we obtain τ = (5.88 ± 4.84) Gyr and σ = 0.15 ± 2.41 (ξ = 5.8 ± 3.6, ω = 0.2 ± 4.0 and α = 0.9 ± 17300) instead. This approximates to a much narrower Gaussian spanning 5.3–6.3 Gyr (1.95–2.10 M_☉), time at which all low-s SNe Ia are created. This explains the sharp drop in the modeled rate at z ~ 0.9 seen in Figure 15. Nevertheless, due to the large error uncertainties, all these results are poorly constrained.

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The explosion fraction is around 0.0001–0.0003 low-s SNe Ia per M_☉ for all models except for the power-law fit to all corrected data (t^−2.7) which is 0.001 low-s SNe Ia per M_☉. These fractions are lower than for the normal-s population—generally higher than ~10⁻³ (e.g., Dahlen et al. 2004; Maoz et al. 2011). Both populations differ by an order of magnitude which can be due either to (1) a lower conversion efficiency η of WD into SN Ia, or (2) a lower progenitor mass range, taking into account that low mass stars are more frequent, or a combination of both. If one assumes that the conversion efficiency is constant over mass—this can be the case for certain considerations of the SD and DD scenarios (Pritchet et al. 2008)—the allowed mass range of low-s SNe Ia would need to be extremely narrow. With an SalA IMF, and assuming that all main-sequence stars with 0.8 M_☉ < M < 8 M_☉ become WDs, the cutoff value between low-s and normal-s progenitors (from Equation (7)) would lie at ~1 M_☉, an extremely narrow range of stars with too long delay times (>10 Gyr) to be seen at z 0.3. Changing the range of progenitor masses to 3 M_☉ < M < 8 M_☉ (Nomoto et al. 1994), the cutoff would need to be at ~3.2 M_☉. On the other hand, using the lower WD mass limit of 1.95 M_☉ and cutoff at 2.10 M_☉, from the Gaussian distribution previously obtained, we also get a consistent explosion fraction ratio of low-s to normal-s.

We note that our sample does not consist of extremely subluminous SNe Ia. Looking at the local sample of typical subluminous objects, such as SN1991bg or SN1999by, the stretch is on the low end, i.e., s = 0.5–0.65. In the SNLS 0.1 < z < 0.6 range, we only obtain one such object (see Figure 16). This SN has a spectroscopic redshift of z = 0.57 with LC parameters s = 0.63 ± 0.08 and c = 0.02 ± 0.06. The candidate is hosted in a galaxy of mass M = 10^10.47 M_☉ and sSFR = 10^−11.93 M_☉ yr⁻¹, typical of very low-s objects.

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This single SN corresponds to a single z = 0.35 rate of (0.25^+0.56_−0.21(stat)^+0.06_−0.03(MC)) × 10⁻⁶ yr⁻¹Mpc⁻³ (see Figure 17). Despite the obvious uncertainty in this number, it is tempting to argue that perhaps this "extremely low-s" population, i.e., s < 0.65, actually behaves differently than our s ≤ 0.8 group. If we maintain the extremely low-s definition for the local luminosity-function sample of Li et al. (2011a) as well, we obtain five objects with s ≤ 0.65 (5.16 corrected for completeness), which corresponds to 6.9^+4.5_−2.9% of their rate (if we were to use the entire local sample, 15 objects have s < 0.65, i.e., an ~9% fraction), that is a rate of (2.0 ± 1.1) × 10⁻⁶ yr⁻¹Mpc⁻³. We note that this is much smaller than the 91bg fractions generally quoted, such as the 17.9^+7.2_−6.2% by Li et al. (2011a), indicating that this photometric definition is more restrictive than their definition (which is also based on spectra). Contrarily, if we were to use their fraction of "subluminous" SNe Ia to estimate the stretch limit, we would obtain s ~ 0.75. We emphasize therefore the importance of having a clear and consistent definition of "subluminous" SNe Ia across the redshift range. As shown in Figure 17, the rate evolution for this population also reveals a decline, perhaps even more rapid than for our standard low-s sample.

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