THE CARNEGIE SUPERNOVA PROJECT: ANALYSIS OF THE FIRST SAMPLE OF LOW-REDSHIFT TYPE-Ia SUPERNOVAE^*

The SNe Ia included in this analysis are listed in Table 1. Twenty-six of these SNe have optical observations beginning before B-band maximum light and with a cadence of a few days or less, which allows direct functional fits to their data. A subsample of nine SNe in this group also have enough YJH coverage to allow the same type of fits around maximum light. These fits provide direct measurements of light-curve parameters such as peak magnitude m^max, time of maximum t^max, and decline rate Δm₁₅(B), the latter measured from the B-band light curves (Phillips 1993).¹⁰ We therefore decided to directly measure the light-curve shape parameter, Δm₁₅(B), whenever possible rather than derive it from template fits in order to tie our luminosity corrections to a purely empirical parameter. These resulting parameters are given in Table 2, labeled as "Spline" in Column 7. The fits also serve to build a set of template ugriBVYJH light curves which may be used to fit more poorly observed SNe, as described in Section 2.2.

Spline functions are preferred for these direct fits since they are more readily adaptable to the shapes occurring in SN Ia light curves than are other functions such as polynomials. We use the FITPACK library (Dierckx 1993), which employs a variable number of knots depending on the sampling and precision of the data. These spline functions allow the whole time span of the observations to be fit without introducing undesirable oscillations where the coverage is less dense. The high quality of the photometry and the good time sampling of the observations ensures the accuracy of the fits. Errors in the parameters derived from the spline fits are modeled via a Monte Carlo calculation performed by randomly perturbing each data point according to its quoted photometric error and then measuring the scatter of the output measurements.

As mentioned by Contreras et al. (2010), one can expect a certain degree of correlation among light-curve points introduced by the host-galaxy subtraction process because the same template image is used to subtract all of the follow-up images. Not taking the correlation into account may lead to underestimating the derived uncertainties of the light-curve parameters. The effect is expected to be more noticeable when the host-galaxy subtraction is more uncertain, and the photometric precision and coverage are larger. We have estimated this effect by computing the covariance matrix for the data points and including this in the Monte Carlo calculations. The covariance matrix is estimated by examining the photometry of artificial stars which are added to the subtracted images at fixed locations around the SN and which are set to have the same flux as the SN at each epoch. In general, these experiments have yielded results in agreement with the case in which the correlations are neglected. Only in a few cases have the uncertainties in the peak magnitudes increased by more than a few thousandths of a magnitude. Therefore, we consider this effect to be negligible for the current analysis.

K-corrections are applied to the observed magnitudes to convert these to the rest frame. For this purpose, we utilize a library of SN Ia template spectra provided by Hsiao et al. (2007) in the optical, and E. Y. Hsiao et al. (2010, in preparation) in the NIR. The K-corrections are computed simultaneously with the light-curve fits. The template spectra corresponding to the epochs of the observations are first warped by a smooth function of wavelength in order to make the synthetic colors match the observed ones for all available bands. Since the template spectra cover a limited range of epochs from −15 to +85 days relative to B-band maximum, observations outside these limits are ignored.

The accuracy of the K-corrections can be tested by comparing them with values computed from the 194 follow-up spectra obtained by the CSP of the sample SNe. The spectra typically cover the optical griBV bands and a range of epochs from −12 to over +100 days with respect to B-band maximum light. In most cases, the agreement with the K-corrections derived from the template spectra is better than ±0.02 mag. Unfortunately, the lack of spectral coverage in the u and NIR bands prevents this test from being performed in those bands.

Figure 1 shows the spline fits for all SNe with pre-maximum data in each band. It should be noted that since normal SNe Ia peak earlier in u and i than they peak in B (see Section 2.4 and Figure 4), some SNe were not observed before maximum light in these two bands, reducing the sample from 26 in B, to 21 and 19 (respectively) in u and i.

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The value of Δm₁₅(B) obtained for each SN is indicated next to the corresponding curves in Figure 1. The range of decline rates covered by the sample is 0.7 <  Δm₁₅(B) <  1.9 for the BVgr bands, and 0.8 <  Δm₁₅(B) <  1.9 for the uiYJH bands, which corresponds well to the full range of observed values of Δm₁₅(B) for SNe Ia.

The optical data for these 26 SNe start between −12 and −1 days relative to B-band maximum light, and end between 17 and over 100 days past maximum, typically extending more than one month after maximum light. The YJH data for the nine SNe with pre-maximum coverage start between −10 and −1 days, and end between 5 and 90 days past maximum.

Following the work of Hamuy et al. (1996b) and Elias-Rosa et al. (2008), we searched for alternative characterizations of the SN Ia decline rates by comparing Δm₁₅(B) with values of Δm_t(X) for different bands X, and times t, since maximum. Figure 2 shows the cases for ugriBV which provide the strongest correlations with Δm₁₅(B); Figure 3 shows the same results for the NIR YJH bands. In general, at longer wavelengths, the best correlations with Δm₁₅(B) are found at later epochs. Even in the case of the NIR iYJH bands, where the double-peaked shape of the light curves complicates matters, reasonable correlations with Δm₁₅(B) are obtained at t ∼ 35–40 days—i.e., during the radiative decline.

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The behavior of the timing of maximum light for different bands with respect to B band was also studied. This is shown in Figure 4. For ugVr, the delay (or advance for u) is roughly constant for SNe of all decline rates. In the iYJH bands, there is a nearly constant advance of 2–4 days for SNe with Δm₁₅(B) < 1.5, whereas there is a delay of several days for SNe with Δm₁₅(B) ≈ 1.8. Note that the scatter in Figure 4 is generally greater than the error bars, implying that there is an intrinsic component.

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The availability of such a complete set of ugriBVYJH light curves with pre-maximum coverage allows template light curves to be built which can be used to derive light-curve parameters for SNe with inadequate coverage around maximum brightness. For this purpose, we used the SuperNovae in Object-Oriented Python (SNOOPy) package (C. Burns et al. 2010, in preparation), which is derived from the multi-filter light-curve fitting method developed by Prieto et al. (2006), with a few modifications. The technique for generating templates is briefly summarized below, and in more detail by C. Burns et al. 2010, in preparation.

The SNe that were fit with spline functions (see Section 2.1) form the training data set. The time of B-band maximum light t^max(B), the decline rate measured as Δm₁₅(B), and the peak magnitude m^max_X in each band X are used to place the photometric data points of the training set in a three-dimensional (3D) space for each band defined by the following variables: (1) the rest-frame epoch with respect to B-band maximum, Δt = [t − t^max(B)]/(1 + z), where t is the time of the observation and z is the heliocentric redshift of the SN; (2) the decline-rate parameter, Δm₁₅(B), considered as a single parameter to characterize all bands; and (3) the K-corrected magnitude relative to maximum light, Δm = m_X(t) − m^max_X − K_X(t), for each observed magnitude m_X(t).

SN 2006X was removed from the training data set in ugriBV because of its peculiar behavior, especially in the B band, which showed an abnormally flat decline rate beginning one month after maximum light. This morphology has recently been explained as a light echo produced by dust surrounding this heavily reddened SN (Wang et al. 2008b). However, since there is no evidence that the NIR light curves of SN 2006X suffered such contamination, the data for this event were maintained in the training set for building YJH templates.

The data points of the training set define a surface in the three-dimensional space (Δt, Δm₁₅(B), Δm) for each band. To produce a smooth interpolation to any point in the three-dimensional space, we use a two-dimensional variation of the "gloess" algorithm, which is a Gaussian-windowed and error-weighted extension of data-smoothing methods outlined by Cleveland (1979) that was first implemented by Persson et al. (2004) for fitting Cepheid light curves. This is the main difference between the method of Prieto et al. (2006) and that of the SNOOPy package (C. Burns et al. 2010, in preparation). While the former method uses the fit spline functions as templates and interpolates among them, SNOOPy performs an interpolation among the data points  of the SNe. The gloess algorithm ensures a smooth interpolation even with heterogeneously sampled data.

The SNOOPy package generates templates for a given value of Δm₁₅(B) by making a slice through the (Δt, Δm₁₅(B), Δm) surface, interpolating along a constant Δm₁₅(B) line. The best-fit time of maximum light t^max, decline rate Δm₁₅(B), and peak magnitudes m^max are then determined by fitting these templates to the ugriBVYJH data of each SN via χ² minimization, with K-corrections computed in an iterative fashion, as described in Section 2.1. Uncertainties in the fit parameters are taken from the diagonals of the covariance matrix output from the Levenbert–Marquardt least-square routine used to fit the templates (C. Burns et al. 2010, in preparation). Additionally, we have tested the possible effect of correlated errors in the photometry due to the use of a single host-galaxy image in the subtraction process (see Contreras et al. 2010). Similarly to the results of the tests for spline fits that are given in Section 2.1, in the case of template fits we find that the effect is very small in all but a few examples. We have thus decided to use the uncertainties that are derived from uncorrelated errors.

This way, it is possible to derive the light-curve parameters for each SN, including the objects in our sample which were not observed until after maximum light. Table 2 lists the results of these template fits for the cases in which a spline function could not be fitted; these are labeled as "Templ." in Column 7. The template fits themselves are shown by C2010 together with the observed light curves for all SNe in the sample.

In order to test the accuracy of the method, template fits were performed for the subset of well-observed SNe and the resulting parameters were compared with those obtained directly from spline fits (Section 2.1). These comparisons are shown in Figure 5 for the peak magnitudes in ugriBVYJH, and for the decline-rate parameter Δm₁₅(B). The average difference in peak magnitudes is ±0.05 mag or less for all of the optical filters except the i band, with no obvious systematic differences except, perhaps, in u. The dispersion in the iYJH bands is ∼±0.1 mag, with some evidence for systematic differences in YJH. A similar situation holds for the differences in the time of maximum as measured with spline and template fits, where good consistency (±0.6 days or less) is found in ugrBV, but the iYJH bands yield worse agreement. The relatively poor precision of the YJH template fits is at least partly due to the small sample used to derive the templates. However, the large dispersion in i is the product of variations in the morphology of the secondary maximum (see Section 2.3). Figure 5 shows that, except at the fastest decline rates, there is reasonable consistency between the decline rate as measured by spline and template fits.

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The uncertainties associated with extrapolating the templates to obtain peak magnitudes and decline rates for those SNe caught after maximum light are more difficult to quantify. Clearly, the later the observations begin, the larger are the possible extrapolation errors. One way to approach this issue is to take the spline fits of the light curves of the SNe in the sample that were well observed before and after maximum, resample these to simulate an SN whose observations did not begin until after maximum, and then fit these data using the ugriBVYJH templates to recover the time of B maximum, Δm₁₅(B), and the maximum-light magnitudes. Calculations of this type will be presented in more detail in a future paper (C. Burns et al. 2010, in preparation), but for the representative case of an SN for which coverage began ∼1 week after maximum and continued until ∼70 days after maximum, we find a typical random uncertainty of ∼0.1 mag in the peak magnitude, averaged over all filters, and a much smaller systematic difference of ∼0.03 mag. Experience shows that for SNe with photometry beginning later than this, template fits are not reliable. Fortunately, only one event in the Contreras et al. (2010) sample, SN 2004dt, falls into this category; hence, it is excluded from the analysis in the present paper.

The lack of a sufficiently large set of SNe with good coverage around maximum light in the K_s band prevents the production of template light curves as was done for all other bands. Instead, we use the polynomial template light curve introduced by Krisciunas et al. (2004b). A stretch  factor (Perlmutter et al. 1997) based on the decline rate of each SN is applied to the time axis of the polynomial—along with the redshift-dependent time dilation—which is then fit to the data in order to obtain a peak magnitude. This kind of fit may be utilized for SNe with data in the range of validity of the template, which is between −12 and +10 days relative to t^max(B) for a SN of stretch factor S = 1. The specific formula employed to convert between Δm₁₅(B) and S is that given by Jha et al. (2006):

$$\begin{equation} S\,=\,\frac{3.06\,-\,\Delta m_{15}(B)}{2.04}. \end{equation} \tag{ 1 }$$

Höflich et al. (1995) and Pinto & Eastman (2000) were the first to attempt a theoretical understanding of the NIR secondary maximum in SNe Ia. More recently, Kasen (2006) modeled the NIR light curves of SNe Ia, arguing that the timing and strength of the secondary maximum is a consequence of the ionization evolution of the iron-peak elements in the ejecta. A specific prediction is that more luminous SNe should have a later and more prominent secondary maximum. In order to test this, we examined the delay and relative strength of the secondary maximum in the i-band light curves of the CSP SNe with the best coverage. Using the nomenclature of Kasen (2006), the following parameters were measured: (1) the phase of the peak of the secondary maximum with respect to B-band maximum, t₂ − t_B; (2) the difference in magnitudes between the secondary maximum and the local minimum between the primary and the secondary maxima, m₂ − m₀; and (3) the difference in magnitudes between the primary and secondary maxima, m₁ − m₂.

Figure 6 shows the results plotted as a function of Δm₁₅(B). If the decline-rate parameter Δm₁₅(B) (or peak luminosity) is associated with the amount of ⁵⁶Ni mass synthesized in the ejecta (Arnett 1982; Arnett et al. 1985; Nugent et al. 1995; Stritzinger et al. 2006), then Figure 6 can be compared directly with Figure 11 of Kasen (2006). Indeed, the upper panel of Figure 6 shows a strong correlation between Δm₁₅(B) and the timing of the secondary maximum, confirming previous results by Hamuy et al. (1996b) and Elias-Rosa et al. (2008). However, the middle and lower panels of Figure 6 show that there is little evidence for a dependence between Δm₁₅(B) and the strength of the secondary maximum, as measured with respect to either the primary maximum or the local minimum between the two maxima. Kasen (2006) points out that both the timing and strength of the secondary maximum can also be affected by the outward mixing of ⁵⁶Ni in the ejecta, the amount of stable iron group elements produced in the explosion, the progenitor metallicity, and the abundance of calcium in the ejecta. Of these, mixing appears to have the greatest effect on the strength of the secondary maximum, and therefore may be responsible for the lack of obvious correlations in the middle and bottom panels of Figure 6.

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Krisciunas et al. (2001) suggested a different way of measuring the strength of the I-band second maximum which consists of converting the light-curve observations to fluxes normalized to the flux at the primary maximum, fitting a high-order polynomial to these, and then determining the mean flux (based on an integration of the polynomial) from 20 to 40 days after the time of B maximum. They found a good correlation between this parameter, 〈I〉₂₀₋₄₀, and the decline rate. Similar measurements were carried out for the i-band light curves of the best-observed subsample of CSP SNe, and plotted versus Δm₁₅(B) in Figure 7. Illustrated as a solid line are the same measurements for the family of SNOOPy templates. There is a clear trend in this diagram in the sense that luminous, slowly declining SNe generally display stronger secondary maxima. However, there is also a significant and real dispersion which is illustrated by the four labeled SNe—2006D, 2006bh, 2004ef, and 2004eo. The left half of Figure 8 shows that all four have very similar B and V light curves covering a narrow range of decline rates (Δm₁₅(B) = 1.37–1.42). Nevertheless, there are significant differences in the strength and morphology of the i-band secondary maximum which, again, may reflect varying amounts of mixing of the ⁵⁶Ni into the ejecta (Kasen 2006).

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These variations in the strength of the secondary maximum represent a significant problem for template fitting in the iYJHK bands, where the strength of the secondary maximum is assumed to be a smoothly varying function of Δm₁₅(B). This is illustrated in the right half of Figure 8, where the difference in the maximum-light magnitudes in BVi as derived from template and spline fits is plotted for the same four SNe (2006D, 2006bh, 2004ef, and 2004eo) with nearly identical decline rates. In the B band, the template fits give results that are in excellent agreement (±0.02 mag) with the direct measurements from the spline fits. The agreement in V is nearly as good, but in the i band, the template fits give errors as large as ∼±0.15 mag. Figure 8 shows that these errors are a function of the strength of the secondary maximum: template fits to SNe with a weak secondary maximum will yield maximum-light magnitudes that are too faint, whereas fits to SNe with a strong secondary maximum will give maximum-light magnitudes that are too bright. This effect most likely explains the relatively large dispersion in the difference in the peak i magnitudes measured from template and spline fits observed in Figure 5. Until an observational parameter can be found which makes these differences predictable, they will remain a significant impediment to the fitting of template light curves in the iYJHK bands. Quantities such as 〈I〉₂₀₋₄₀ or m₂ − m₀ may prove to be such a parameter, but a larger sample of SNe is needed to test their utility.

The results of the light-curve fits for the SNe in our sample are summarized in Table 2, where the times of maximum (t^max) and apparent peak magnitudes (m^max) are listed for all bands. Also given are decline-rate parameters (Δm_t) for each filter expressed as the amount in magnitudes that the light curve declines between maximum light and an epoch t days after maximum. Finally, the number of data points and range of epochs covered with respect to the time of maximum, as well as the fitting method (spline functions, Section 2.1, or templates, Section 2.2), are specified for each band.

Based on the quality of the data and fits, we define a subsample of SNe with the best-measured light-curve parameters, mostly via direct fits but also through template fitting. Twenty-nine SNe belong to this group of "best-observed" events. These objects are identified in Table 1; in Table 2, their names are set in bold face. Special use will be made of this subsample when studying the relation between peak luminosity and decline rates in Section 4.

In order to use color information to determine reddening of the SN light produced in the host galaxy, we follow the procedure of Phillips et al. (1999) of selecting a subsample of SNe which are suspected to have suffered little or no dust extinction. This subsample will be used to derive intrinsic colors for SNe Ia as a function of decline rate which, in turn, will serve as a reference for computing color excesses for the whole SN sample.

The criteria employed to select the low-reddening subsample of SNe are (1) SNe which occurred in E/S0-type galaxies, or SNe located away from the arms or nuclei of spiral galaxies, and (2) absence of detectable interstellar Na i D lines in early-time spectra.

Ten SNe from the sample fulfill both criteria: SNe 2004eo, 2005M, 2005al, 2005am, 2005el, 2005hc, 2005iq, 2005ke, 2005ki, and 2006bh. In what follows, these objects are used to derive intrinsic color relations as a function of the decline-rate parameter Δm₁₅(B). We use colors corrected for Galactic reddening since such correction is, in most cases, small (see Table 1) and known to higher accuracy than the host-galaxy reddening. We thus do not perform any cut based on Galactic reddening. We assume a value of R_V = 3.1, which may not be exactly correct for each SN. However, since the maximum Galactic reddening for this sample of ten SNe is E(B − V) = 0.14 mag, uncertainties in this correction will be small.

The subsample of low-reddening SNe can be used to study the behavior of the B−V color approximately one month after maximum light. It was first found by Lira (1995) that unreddened SNe Ia follow a very similar linear regime in B−V between one and three months after maximum. As shown in Figure 9, we confirm this result and derive the following intrinsic color law for the "tail" of the B−V color evolution:

$$\begin{equation} (B-V)_0 = 0.732 (0.006) - 0.0095 (0.0005) \, (t_V - 55). \end{equation} \tag{ 2 }$$

For consistency with Phillips et al. (1999), rest-frame days since maximum V light, t_V, is used for the time axis. Equation (2) is valid over the range 30 < t_V < 80 days and yields a dispersion of 0.077 mag. The SNe employed in the fit cover a broad range of decline rates, 0.85 < Δm₁₅(B) < 1.76 mag. This relation is similar to the one of Phillips et al. (1999), who found, for a sample of six SNe and a different photometric system, (B − V)₀ = 0.784 − 0.0118(t_V − 55) with a dispersion of 0.06 mag. Figure 9 shows that the residuals for individual SNe are strongly correlated, which implies that some of the 10 SNe in this subsample actually do suffer measurable host-galaxy reddening, or that there is an intrinsic dispersion in the Lira relationship, or both. We also note that the three SNe in our sample which occurred in early-type galaxies (E/S0), namely SNe 2005al, 2005el, and 2005ki, show remarkably similar B−V color evolution—and also in other optical colors—during the tail, with an average deviation of −0.05 ± 0.003 mag from Equation (2), and a dispersion of ∼0.035 mag.

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Equation (2) can be used to estimate color excesses, E(B − V)_tail, for the SNe in our sample with observations in the tail.¹¹ The results are listed in Column (2) of Table 4. The uncertainties in these color excesses reflect only the photometric errors in the data points. In considering the color excess for any particular SN, the observed dispersion of 0.077 mag in the Lira law should also be added in quadrature to these errors. However, in the analysis presented in Section 4.2 of the precision to which SNe Ia may be used as standardizable candles, this dispersion is included in the σ_SN values that are yielded by the fits, and the errors used for the color excesses are those given in Table 4. Note that since Equation (2) was derived by fitting to the average color evolution of the low-reddening sample, it is guaranteed to generate some small negative color excesses. Phillips et al. (1999) dealt with this by applying a Bayesian prior consisting of a one-sided Gaussian distribution of A_B values with a maximum at zero and σ = 0.3 mag. More recently, Jha et al. (2007) used a prior consisting of a Gaussian with σ_(B−V) = 0.068 mag, representing the intrinsic dispersion in color, convolved with an exponential reddening distribution with a scale length τ_E(B−V) = 0.138 mag which they derived from fits to nearby SN Ia light curves by assuming a particular intrinsic color distribution. However, until it is possible to confidently separate dust reddening from intrinsic color variation, we choose not to apply a prior to the color-excess measurements.

For each SN, we tested the agreement of the B−V tail slope with that of Equation (2). Good consistency was generally found, with the notable exception of SN 2006X, the most highly reddened SN in the sample. As mentioned in Section 2.1, this SN showed a peculiar behavior in the B band at the epochs considered here, probably due to the appearance of a light echo (Wang et al. 2008b).

Figure 10 displays a plot of E(B − V)_tail versus the equivalent width of the absorption in the Na i D lines produced in the host galaxies as measured from spectra we obtained of the SNe. Generally speaking, these spectra are of low wavelength resolution, and the Na i D lines are only visible when the equivalent width reaches ∼1 Å or more. The red symbols with dashed error bars in Figure 10 indicate the low-reddening sample. Four of the eight SNe with the largest color excesses show detectable Na i D absorption, and the two with the largest color excesses, 2004gu and 2005lu, correspond to the two with the highest Na i D equivalent widths. However, two SNe with slightly negative values of E(B − V)_tail, SNe 2005bg and 2006mr, also show strong Na i D absorption. In the case of SN 2005bg, the estimate of E(B − V)_tail is based on a single B−V color measurement in the tail, and therefore may be unreliable. SN 2006mr was a very rapidly declining event which occurred in the dusty central regions of Fornax A (NGC 1316). Garnavich et al. (2004) studied a small sample of fast-declining SNe Ia and found that they followed the Lira relationship fairly well, but SN 2006mr suggests that there may be exceptions. In any case, the correlation between color excess and the Na i D equivalent width is known to be poor (Blondin et al. 2009).

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Figure 10 shows that five additional SNe (2004ey, 2005ag, 2006D, 2006ax, and 2006gt) have color excesses E(B − V)_tail < 0.08 mag and undetectable Na i D absorption. In the following analysis, these five SNe are added to the original subsample of ten assumed to have suffered little or no host-galaxy reddening.

Since the CSP data—and generally most SN Ia observations—have better coverage around maximum light than during the declining tail, it is important to develop methods to determine color excesses from observations around the time of maximum. Using the peak magnitudes (corrected for Galactic reddening and the K-correction) derived in Section 2, pseudocolors¹² can be computed as the differences between the maximum-light magnitudes in two given bands. The dependence of these pseudocolors on the decline rate, parameterized by Δm₁₅(B), can then be investigated for a subsample of SNe with low reddening.

Figure 11 shows the pseudocolors as a function of Δm₁₅(B) for several band combinations which will be used in the analysis of Section 4. The solid points in the figure correspond to the 15 SNe with low reddening. As expected, in all colors these SNe lie at the lowest values, following a roughly linear trend with decline rate over the range 0.8 < Δm₁₅(B) < 1.7 mag. At the fastest decline rates (Δm₁₅(B)>1.7 mag), however, the pseudocolors at maximum light are significantly redder than these linear trends would predict. Table 3 summarizes straight-line fits to the colors of the low-reddening SNe with decline rates 0.8 < Δm₁₅(B) < 1.7 mag; these fits are plotted as dashed lines in Figure 11. Note that the slopes of the relations are largest when the uYJH bands are involved. In general, there is a correlation between the dispersion in color and the difference in the effective wavelengths, Δλ, of the two filters that form the color. Within this trend, the colors involving the u filter have a relatively higher dispersion for their Δλ values, whereas the colors involving the J filter have a lower dispersion.

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In the case of the (B^max − V^max) pseudocolor, the following linear law for the 14 SNe with low reddening and 0.8 < Δm₁₅(B) < 1.7 mag is found:

$$\begin{equation} (B^{\mathrm{max}}-V^{\mathrm{max}})_0 = -0.016 (0.014) + 0.12 (0.05) \, [\Delta m_{15}(B)- 1.1].\vspace*{3pt} \end{equation} \tag{ 3 }$$

The root mean square (rms) of the fit is 0.06 mag. This relation is slightly redder  than that given by Phillips et al. (1999) and Altavilla et al. (2004). We use this intrinsic-color law to derive color excesses E(B − V)_max for all SNe in the range 0.7 < Δm₁₅(B) < 1.7 mag, computed as (B^max − V^max) − (B^max − V^max)₀. The resulting values are listed in Column 3 of Table 4. The uncertainties in these color excesses are calculated from the photometric errors in the peak magnitudes.

The top-left panel of Figure 12 shows a comparison of the E(B − V)_max measurements with the E(B − V)_tail values from Section 3.2. The sample of 21 objects consists of the best-observed SNe for which both color excesses could be measured. In addition, the highly reddened SN 2006X has been excluded since it clearly deviates from the behavior of the rest of the SNe in the B−V tail, as mentioned in Section 3.2. One can see that there is a small but systematic difference between these two color-excess measurements which, unexpectedly, appears to be correlated with the magnitude of the color excess. A formal fit to the data (plotted as a solid line in Figure 12) gives E(B − V)_tail − E(B − V)_max = 0.032 − 0.496 E(B − V)_max, with a dispersion of 0.06 mag. The top-right panel of Figure 12 shows that this difference also seems to correlate with the color of the SN. Since E(B − V)_tail and E(B − V)_max are "observed" color excesses, as opposed to "true" color excesses (see Phillips et al. 1999), and are measured at different epochs in the color evolution of the SN, one would expect there to be a small systematic difference between the two that would correlate with the SN color. This effect is produced primarily because the effective wavelengths of the filters evolve due to the significant color evolution of the SN spectral energy distribution, but there is also a smaller dependence on the total amount of dust reddening itself. The Hsiao et al. (2007) spectral template may be used to calculate the expected magnitude of these combined effects, which is plotted as a dotted line in both of the top panels of Figure 12. The observed systematic differences are greater than can be explained in this way, although the significance is only at the ∼2σ level.

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Since the method of estimating the intrinsic colors at maximum by bootstrapping from the Lira relation was first utilized by Phillips et al. (1999), we have examined the color excesses derived by these authors. Interestingly, the same trends observed in the top two panels of Figure 12 are clearly present in those measurements. Not only does this appear to confirm the reality of the effect, but it also argues that it is not an artifact of the CSP data set.

The lower-left panel of Figure 12 shows that E(B − V)_tail − E(B − V)_max does not depend significantly on the decline-rate parameter Δm₁₅(B). The measurements suggest that there may be a weak correlation with the absolute B magnitude (corrected for decline rate), but this requires confirmation by more data. We do not currently understand the underlying causes of these correlations, but the fact that they exist implies that there is a component of one or the other of these color-excess measurements that is not due to interstellar dust.

Keeping this caveat in mind, the fits corresponding to (V^max − X^max_λ) for X_λ = ugriBYJHK_s can be used to derive color excesses E(V − X_λ)_max for all SNe with decline rates in the range 0.8 < Δm₁₅(B) < 1.7 mag. Columns 4 through 11 of Table 4 list these color excesses.

The availability of photometric data spanning the u through K_s bands enables the properties of the host-galaxy reddening to be examined. Assuming that the dust extinction in the host galaxies obeys the law introduced by Cardelli et al. (1989) and modified by O'Donnell (1994) (hereafter referred to as the "CCM+O law"), we can investigate which value of the total-to-selective-absorption coefficient R_V is favored by SN Ia data. Extinction in the Galaxy is well approximated by the CCM+O law with an average value of R_V ≈ 3 (e.g., Fitzpatrick & Massa 2007).

3.4.1. Optical–NIR Color Excesses

An average value of R_V for the subsample of best-observed SNe can be derived by comparing the color excesses between the optical and NIR bands presented in Section 3.3 with the values of E(B − V)_max. The use of optical–NIR color excesses ensures the necessary leverage in wavelength to provide a more precise measurement of the reddening law, even for SNe with low reddening. In Figure 13, comparisons of E(V − X)_max with X ≡ iYJH are shown.

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Fits to the slopes E(V − X)_max/E(B − V)_max were carried out for all four bands considered here. The slope were then converted to values of R_V, using Equations (A5) and (A6) in Krisciunas et al. (2006) and the CCM+O law coefficients a_X and b_X derived as explained in Appendix B. Table 5 summarizes the results of these fits. The data favor a low value of R_V, with an average of R_V = 1.69 ± 0.05 from the four fits. The corresponding slope for this average is indicated by the solid red lines in Figure 13.

Table 5. R_V Values from Fits to E(V − X)_max Versus E(B − V)_max

Sample	E(V − i)	E(V − Y)	E(V − J)	E(V − H)	Weighted
	Max	Max	Max	Max	Average
Whole	1.1 ± 0.2	1.6 ± 0.2	1.8 ± 0.1	1.7 ± 0.1	1.7 ± 0.1
Excl. 2005A and 2006X	3.8 ± 1.5	3.5 ± 1.4	3.1 ± 0.7	3.1 ± 0.7	3.2 ± 0.4

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If, however, the two highly reddened SNe, SN 2005A and SN 2006X, are excluded from these fits, the uncertainties increase but the favored values of R_V become larger and compatible with the standard R_V ≈ 3. An average of R_V = 3.2 ± 0.4 is derived from the four fits, and is indicated by the dashed blue lines in Figure 13.

This is a very interesting result which may indicate that SNe with low or moderate reddening are affected by standard dust extinction, while very highly reddened SNe suffer relatively more reddening than extinction, as inferred from a low value of R_V. We will return to this finding after analyzing the calibration of SNe Ia as standardizable  candles in Section 4.

3.4.2. Highly Reddened Supernovae

Highly reddened objects provide the most precise information about the behavior of extinction as a function of wavelength. Our sample includes two objects, SNe 2005A and 2006X, for which E(B − V) ≳ 1.0 mag. Although the optical light curves of SN 2006X showed evidence of a light echo, this did not appear until ∼1 month after maximum light (Wang et al. 2008b), and so the peak magnitudes should be uncontaminated. Figure 14 shows the values of E(V − X_λ) derived for both SNe (see Section 3.3) plotted as a function of the effective wavelength of each filter X. Note that at these extreme values of the color excesses, it is necessary to convert the observed values to true color excesses, and to also correct Δm₁₅(B) for reddening effects (see Phillips et al. 1999). Table 6 gives the linear relations for converting from observed to true color excesses that are derived using the Hsiao et al. (2007) template spectra. Using the same spectra, the following relation for correcting the observed decline rate for reddening is found:

$$\begin{equation} \Delta m_{15}(B)_{\rm true} = \Delta m_{15}(B)_{\rm observed} + 0.065 E(B-V)_{\rm observed}. \end{equation} \tag{ 4 }$$

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Table 6. Conversion of Observed to True Color Excesses

Color Excess	Correction Factor CX
E(V − u)	1.0576
E(V − B)	1.1023
E(V − g)	1.1855
E(V − r)	1.2028
E(V − i)	1.0218
E(V − Y)	1.0370
E(V − J)	1.0142
E(V − H)	0.9951
E(V − Ks)	0.9818

Note. E(V − X)_true = C_X*E(V − X)_observed.

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Assuming that the reddening of SNe 2005A and 2006X is produced by dust that is described by the CCM+O reddening law, fits to the color excesses in Figure 14 give values of R_V of 1.68 ± 0.10 and 1.55 ± 0.07, respectively. These fits are plotted as solid lines in Figure 14, and are to be compared with the dotted lines which show the best-fitting CCM+O reddening law where R_V is fixed at its canonical Galactic value of 3.1. Although the CCM+O law does a good job in the optical and NIR, it fails miserably in fitting the (V−u) color excesses. Recently, Goobar (2008) has considered a model where multiple scattering of photons by circumstellar dust steepens the effective extinction law, converting it to a power law. When this model is fit to the color excesses of SNe 2005A and 2006X, excellent agreement at all wavelengths is obtained with Large Magellanic Cloud (LMC) dust and a power-law index p ≈ −2.4. These fits are plotted as dashed lines in Figure 14. Thus, the data would appear to favor a model where the reddening of these two objects is produced by dust that is local to the SN. In this context, it is interesting to note that short-term variations in the interstellar Na i D absorption lines in the spectrum of SN 2006X were observed by Patat et al. (2007) and interpreted as evidence for the existence of significant circumstellar material. Wang et al. (2009) have recently pointed out that both SNe 2005A and 2006X belong to the group of high-velocity SNe Ia, and that this group shows systematically different reddening properties (with lower R_V values) than normal SNe Ia, probably due to a circumstellar dust component or to intrinsic properties of this subgroup of SNe.

SN 2006X has also been studied in detail by Wang et al. (2008a), who found a best fit to the data with R_V = 1.48 ±  0.06, in excellent agreement with our results. However, these authors concluded that the data, including their U-band photometry, were matched quite well by the CCM reddening law. The Wang et al. (2008a) color-excess measurements for SN 2006X were derived through comparison with a few individual SNe as well as via the method we employ here of looking at the maximum-light colors as a function of the decline rate. In general, there is good agreement between their measurements and our own except for the critical U band, where Wang et al. (2008a) find E(U − V)_true = 2.65 ± 0.14 but we derive E(u − V)_true = 3.11 ± 0.14. Some of this discrepancy may be due to the fact that the sample of SNe Ia with well-observed light curves in the U or u bands is still fairly small, although in looking at Figure 11, it is difficult to see how the value of E(u − V) could be overestimated by 0.4 mag. Although the u filter response function is the least well determined of the CSP bands, the color excesses for SNe 2005A and 2006X were derived with respect to other SNe Ia observed with exactly the same filter, and in the same photometric system. Thus, the color excess is well determined. The largest uncertainty is the effective wavelength that is associated with the CSP u band, which we estimate to be <200 Å. This maximum possible error is indicated by the horizontal error bars on the u points in Figure 14. The conclusion is that the uncertainty in the u-band response function cannot explain the discrepancy.

Only two other objects in our sample, SNe 2005kc and 2006eq, have significant host-galaxy reddenings and observations in the YJH bands. Fits to the color excesses of these two SNe with both the CCM+O and Goobar (2008) reddening laws are displayed in Figure 15. The observations of SN 2005kc are consistent with either the CCM+O law with R_V = 4.4 ± 0.6, or a Goobar power law with p = −0.7 ± 0.2. In the case of SN 2006eq, the measurements may be fit with either a CCM+O law with R_V = 1.1 ± 0.8, or a Goobar power law with p = −3.5 ± 1.1. Thus, the results for SN 2006eq are similar to those obtained for SNe 2005A and 2006X, but the reddening law for SN 2005kc appears to be quite different, perhaps even consistent with normal Galactic reddening. These results confirm those of Figure 13, where SNe 2005kc and 2006eq are the two points with highest E(B − V) values among the moderately reddened SNe. These two SNe illustrate the power of optical–NIR color excesses in differentiating the reddening law. More well-observed, moderately reddened events are clearly needed to advance further in the study of the reddening laws of SNe Ia, and to look for possible correlations with other properties of the SN or its environment.

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The two-parameter model of Tripp (1998) assumes that the distance modulus of an SN has the following dependence on decline rate and color:

$$\begin{equation} \tilde{\mu }_X=m_X-M_X(0)-b_X\,[\Delta m_{15}(B)-1.1]-\beta _X^{YZ}\,(m_Y-m_Z), \end{equation} \tag{ 6 }$$

where m_X is the observed peak magnitude in a given band X, and (m_Y − m_Z) is a pseudocolor at maximum brightness from any choice of bands Y and Z. The fit parameters are the peak absolute magnitude of SNe Ia with Δm₁₅(B) = 1.1 and zero color, M_X(0); the slope of the luminosity versus decline-rate relation, b_X; and the slope of the luminosity–color relation β^YZ_X. Using measurements of peak magnitudes, pseudocolors, and decline rates {m_Xi; Δm₁₅(B)_i; (m_Y − m_Z)_i} (i = 1, ..., N) for a sample of N SNe, the best-fit parameters are solved for via χ² minimization, as explained in Appendix A. This is equivalent to minimizing the dispersion in the Hubble diagram. Note that a constant term has been added to the measurement uncertainties that appear in the denominator of the χ² in Equation (A4) to account for possible intrinsic dispersion of the SN Ia data about the model given by Equation (6).

The Tripp (1998) model has traditionally been applied to fits of B versus Δm₁₅(B) and B−V. Such an analysis is presented here, along with a fit of J versus Δm₁₅(B) and V−J.

Table 8 summarizes the results of the fits. Fit 1 was done using the entire sample of 32 SNe with reliable distances and the combination of B versus Δm₁₅(B) and B−V. This fit is shown in panel (a) of Figure 16, and yields a total dispersion of 0.17 mag and an intrinsic dispersion σ_SN = 0.12 mag. Performing the fit with the subsample of 26 best-observed SNe (Fit 2) reduces the scatter to 0.15 mag and the derived intrinsic dispersion to just 0.09 mag without modifying substantially the fit parameters. This is shown in panel (b) of Figure 16.

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Note that the fits described above include SNe with all  the available decline rates; specifically, they include the fast-declining SNe with Δm₁₅(B)>1.7 (SN 2005bl, SN 2005ke, and SN 2006mr) which are marked with red circles in Figure 16. These SNe follow the behavior of the rest of the sample. This is substantiated by the results of Fits 3 and 4, done excluding these three SNe; the resulting fit parameters are in agreement within the uncertainties with those of Fits 1 and 2, and the dispersions found are nearly the same. This is a remarkable result since the red colors of the fast-declining SNe Ia—which occur preferentially in E and S0 galaxies—are clearly largely intrinsic in origin, whereas the reddening of slower-declining SNe—which are most common in spiral galaxies—is presumably due at least in part to dust.

If it is assumed for the moment that the pseudocolors (after correction for Galactic reddening) vary due to reddening by dust in the host galaxy, the color term, β, can be converted to a total-to-selective absorption coefficient, R_V, adopting the CCM+O reddening law and following the prescriptions of Appendix B. The value of R_V obtained from each fit is given in Column 7 of Table 8. As is seen, all the fits thus far considered yield values of R_V ≈ 1.5, which is significantly different from the standard Galactic value of 3.1. In the right-hand panels of Figure 16, the fits are labeled with the corresponding R_V values. For comparison, the reddening vector predicted by R_V = 3.1 is also plotted as a dotted line.

The Tripp (1998) model was also fit to the subset of best-observed SNe that meet the condition that (B^max − V^max) < 0.4 mag. This excludes both the fast-declining objects, which are intrinsically red, and SNe 2005A and 2006X, which suffered heavy reddening due to dust. Fit 5 in Table 8 gives the results of such a fit. Once again, the zero point and slope of the luminosity versus decline-decline rate relation (M_B(0) and b_B) do not change significantly in comparison with the previous fits. The luminosity-color slope, β^BV_B, does change slightly (by ∼1σ), resulting in an even lower value of R_V. These results are relevant to the study of high-redshift SNe Ia for which red (and therefore faint) objects (with (B^max − V^max)>0.4 mag) are rarely observed (e.g., Kowalski et al. 2008).

The results of applying the Tripp (1998) model to J versus Δm₁₅(B) and V−J for the sample of best-observed SNe with J-band coverage (21 objects) are given in Fit 6 of Table 8, and are plotted in the lower part of Figure 16. Two additional fits are presented in Table 8 for the J versus Δm₁₅(B) and V−J combination. In Fit 7, the two fast-declining SNe with coverage in J are excluded. A slightly lower value of the slope β^VJ_V is obtained, with an almost negligible intrinsic dispersion of σ_SN = 0.02 mag. Fit 8 employs a color cut of (V^max − J^max) < 0.0 mag, which is equivalent to limiting the sample to (B^max − V^max) < 0.4 mag. Again, the best-fit parameters are in good agreement with those of Fit 7, and the dispersions are likewise very similar.

Note that the interpretation of σ_SN as the intrinsic dispersion of the SN data about the model relies on the correct estimation of the measurement uncertainties and their covariances. An error in the treatment of the uncertainties would lead to different σ_SN. Consequently, the fit parameters themselves could potentially change. In order to test this, the dependence of the fit parameters on the adopted uncertainties in the peak magnitudes, colors, and decline rates was examined. This was done by alternatively adding an arbitrary constant uncertainty on each quantity, up to 0.06 mag. While the values of σ_SN obtained were found to decrease with increasing uncertainties as expected, the best-fit values of the parameters remained well within the uncertainties.

In order to test the dependence of our results on the method used to derive light-curve parameters, we repeated the fits using peak magnitudes, decline rates and colors from template light-curve fits (see Section 2.2). We used the sample of best-observed SNe for which we performed both spline and template fits. The results are equivalent within the quoted errors to the ones presented above.

In this section, the color excesses derived in Section 3 are used to produce fits of absolute peak magnitude versus decline rate and reddening. It is explicitly assumed that the color excesses, E(Y − Z), are due to dust and can be converted into an absorption in the X band via A_X = R^YZ_X E(Y − Z). This absorption is used to correct the observed peak magnitudes according to the model

$$\begin{equation} \tilde{\mu }_X\;=\;m_X\;-\;M_X(0)\;-\;b_X\,[\Delta m_{15}(B)-1.1]\;-\;R_X^{YZ}\,E(Y-Z). \end{equation} \tag{ 7 }$$

Note the similarity of this model to that of Equation (6) of Section 4.1, especially if one considers that the color excesses derived in Section 3.3 are based on linear relationships between the observed pseudocolors and Δm₁₅(B).¹³ Since the color excesses were obtained for SNe in the range 0.7 < Δm₁₅(B) < 1.7 mag, the fast-declining SNe are necessarily excluded from the analysis of this section.

We first focus on fits done with B and E(B − V) data. To provide a reference, the sample was initially limited to the twelve SNe which were previously identified as having suffered little or no dust extinction (see Section 3.2). These low-reddening objects were used to fit the model of Equation (7) with the last term on the right-hand side set to zero—that is, without applying any reddening correction. The resulting parameters are given in Fit 1 of Table 9. These twelve SNe with low reddening yield a total dispersion of 0.19 mag, with an intrinsic dispersion σ_SN of the same amount.

Next, the entire SN sample was fit by first fixing R_V to the standard Galactic value of 3.1. The results are given in Fit 2 of Table 9, where we solve only for M_B(0) and b_B. This fit implies substantially higher peak luminosities and, most notably, a very large dispersion of ∼0.5 mag. Thus, correcting for the observed reddening as if it were produced by typical Galactic interstellar dust yields a dispersion which is a factor of ∼2.5 greater than that obtained for the subsample of low-reddening SNe.

Beginning with Fit 3 of Table 9, R^BV_B was allowed to be a free parameter. Here the full sample of SNe with 0.7 < Δm₁₅(B) < 1.7 mag was employed, while in Fit 4 only the best-observed subsample was considered. The resulting parameters for these two fits are indistinguishable within the errors. The best-fit value of R^BV_B in both cases is ∼2.8, corresponding to R_V ≈ 1.5. These two fits are shown in panels (a) and (b) of Figure 17.

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The values of R^BV_B derived in Fits 3 and 4 are strongly influenced by the two highly reddened SNe 2005A and 2006X. Fits 5 and 6 in Table 9 give solutions where these two SNe are excluded from the sample, with Fit 5 corresponding to the whole sample and Fit 6 to the best-observed SNe. As is seen, these fits still prefer a value of R_V ≈ 1.5.

Bearing in mind the slight differences between photometric systems, the slope of the luminosity versus decline-rate relation found for Fits 1–6 is in reasonable agreement with previously published results (Hamuy et al. 1996a; Altavilla et al. 2004; Prieto et al. 2006)¹⁴. As shown in the left panels of Figure 17, the quadratic relationship found by Phillips et al. (1999) is also consistent with the data, although the CSP observations themselves are fitted perfectly well by a simple linear relation.

Next, we considered fits using the peak magnitudes in ugriVYJHK_s, combining each band with a suitable color excess. The choice of this color excess is somewhat arbitrary. For the ugr and JHK_s bands, color excesses involving the filter itself were used. However, when this is done for the i and Y filters, the resulting fits yield significantly higher dispersions than are obtained when using E(B − V). We attribute this to the poor quality of the template fits in the i and Y bands (see Section 2.3), which not only introduces an error in the peak magnitude, but also in the color excess if the latter involves either of these bands.

Fits 7–15 in Table 9 give the results for all bands other than B using the sample of best-observed SNe. The best-fit values of R_V range from 1.1 to 1.8 for ugriYJ. For H and K_s, somewhat larger values in the range 2.4–2.7 are obtained, but with such large uncertainties that they are compatible with both low and standard values of R_V. A weighted average of the results for Fits 7–15 yields R_V = 1.38 ± 0.04, which is consistent with the results obtained for B versus E(B − V) (see Fit 4). The scatter in the corrected magnitudes for Fits 7–15 is in the range 0.12–0.16 mag, with the intrinsic components of the dispersion amounting to as small as 0.04 mag and as large as 0.13 mag.

The set of NIR light curves presented here constitutes a significant contribution to the available data in the literature. The advantages of using NIR observations of SNe Ia to determine distances were demonstrated by Krisciunas et al. (2004a) and, more recently, by Wood-Vasey et al. (2008). Apart from being less affected by dust extinction and scattering—which considerably reduces the effect of uncertainties in the color excesses and reddening law—the peak luminosities of SNe Ia in the NIR present a shallower dependence on decline rates than their optical counterparts.

Indeed, previous studies have shown no clear evidence for a correlation between luminosity and decline rate in the JHK_s bands except for the very fastest declining SNe (Krisciunas et al. 2004c). The improved quality and coverage of the CSP NIR light curves allow a weak dependence of the luminosity on the decline rate to be discerned in the J band. Fit 13 of Table 9 yields a slope of b_J = 0.58 ± 0.09 over 0.7 < Δm₁₅(B) < 1.7 mag, and is shown in panel (c) of Figure 17. In the H band, the evidence is also suggestive for a weak correlation between luminosity and decline rate, although the significance of the measured slope is only ∼2σ (see Fit 14 of Table 9). In a future paper (S. Kattner et al. 2010, in preparation), we will use a larger sample of CSP SNe to test the reality of these relations.

To compare with previous studies, we compute the dispersion in absolute peak magnitudes without any correction for decline rate. Table 10 shows the results of weighted averages of the absolute peak magnitudes in YJHK_s. These averages correspond to the sample of best-observed SNe, correcting for extinction using the values of E(V − X) and R^YZ_X listed in Fits 12–15 of Table 9. For comparison, averages were also taken limiting the sample to those SNe with low/moderate reddening (i.e., excluding SN 2005A and SN 2006X), and assuming two different values of R^YZ_X: zero, which is equivalent to ignoring host-galaxy extinction corrections, and the values of R^YZ_X that correspond to R_V = 3.1.

The most striking implication of Table 10 is that, when the highly reddened SNe are excluded from the sample, the absolute magnitudes change only very slightly depending on the value of R_V. This illustrates the great advantage of working in the NIR, where lack of knowledge of the exact value of R_V has little influence, and where dust-extinction corrections can essentially be ignored for all but the most heavily reddened SNe.

Table 10 shows that the dispersions in the absolute magnitudes are 0.18–0.21 mag in the J and H bands, depending only slightly on the assumed value of R_V. These are to be compared with the rms dispersions of 0.12 mag and 0.16 mag, respectively, obtained when the absolute magnitudes are corrected for the decline-rate dependence (cf. Fits 13 and 14 of Table 9). For reference, Krisciunas et al. (2004c) obtained rms values (uncorrected for decline rate) of 0.13 mag in J and 0.15 mag in H for a sample of 22 SNe Ia, and Wood-Vasey et al. (2008) found dispersions of 0.33 mag in J and 0.15 mag in H for 21 SNe.

In Table 11, the average absolute magnitudes at maximum of the best-observed subsample of CSP SNe are compared with those obtained by Krisciunas et al. (2004c). The values in J and H are consistent at the ∼2σ level. In this same table, a comparison is also attempted with the absolute magnitudes at the epoch of B maximum given by Wood-Vasey et al. (2008). For J and H, these were derived using all the measurements available for the CSP SNe inside a bin of [−2.5,2.5] rest-frame days with respect to B maximum, converting these to absolute magnitudes as explained below. A total of 33 and 28 data points entered into the calculations for J and H, respectively. For the K_s band, the relative scarcity of observations forces the use of a slightly larger bin size of [−4,4] days in order to include 14 data points in the average. Again, the results in J and H are consistent with those of Wood-Vasey et al. (2008) at only the ∼1–2σ level. In both cases, we suspect that the relatively poor agreement is due more to differences in the methods used to fit and combine the data rather than differences in the data themselves.

Figure 18 shows the absolute iYJH light curves for the sample of best-observed CSP SNe. The time axis is plotted as rest-frame days since maximum light in B. The Hubble-flow distances of Equation (5) along with the Cepheid and SBF distances given in Table 7 are used to convert the observed magnitudes to absolute values. K-corrections (see Section 2), Galactic-extinction corrections with R^Gal_V = 3.1, and host-galaxy extinction corrections using the measurements of color excesses E(V − X) and the R^YZ_X parameters obtained from Fits 11–14 for iYJH have also been applied. For the fast-declining SNe 2005ke and 2006mr, zero host-galaxy extinction is assumed.

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Ignoring for the moment the two fast-declining SNe 2005ke and 2006mr, Figure 18 indicates that the smallest dispersion in absolute magnitude occurs around the time of the primary maximum in each filter. By the time of the minimum between the primary and secondary maximum, the dispersion increases in all bands. This observation is at odds with the suggestion by Kasen (2006) that the minimum of the J-band light curves of SNe Ia should show a low dispersion in luminosity. At epochs later than the second maximum, the data in iYJH show increasing spread due to differences in the decline rates among the SNe (see also Section 2 and Figure 2).

The two fast-declining events (SN 2005ke and SN 2006mr) do not reach maximum brightness until ∼5 days after the epoch of B maximum. Interestingly, Figure 18 shows that at this epoch, their J-band luminosities are similar to those of the rest of the SNe which are already declining from the first maximum.

The fits presented in Section 4.2 can be used to place the SNe with 0.7 < Δm₁₅(B) < 1.7 mag in a Hubble diagram. Distance moduli were computed for each SN in ugriBVYJH using the parameters given in Table 9 for the fits to the best-observed SNe (Fits 4 and Fits 8–14). Note that Fit 7 involving V and E(B − V) was not included because it is mathematically equivalent to that involving B and E(B − V) (Fit 4). The distance moduli for all available bands were then averaged for each SN. The full correlation matrix among distance moduli from all bands was estimated in order to weight the average. Correlation might be expected to arise from the fact that all fits involve the same decline-rate parameter Δm₁₅(B), and in several cases, the same bands in the color-excess term. These averaged values are plotted versus the luminosity distance, d_L, in the top half of Figure 19. A combined scatter of 0.11 mag, or ∼5% in distance, is found for the 23 best-observed SNe that are included in the diagram.

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In the bottom panel where the residuals from the fit are plotted, it may be seen that the dispersion decreases with distance, implying that it is due predominantly to the peculiar velocities of the individual host galaxies. Plotted for reference as dotted lines is the spread predicted by the 1σ dispersion of 382 km s⁻¹, as derived by Wang et al. (2006a) from an independent sample of 56 SNe Ia. In general, the distance modulus residuals are consistent with the latter value, although it is interesting that the four most-distant SNe all have negative residuals (meaning that they are overluminous). The most discrepant of these objects is SN 2004gu which, as discussed in C2010, appears to have been spectroscopically similar to the peculiar SN Ia 2006gz, a slow-declining, overluminous SN Ia that may have resulted from the merger of two white dwarfs (Hicken et al. 2007). Both SNe displayed nearly identical decline rates (Δm₁₅(B) ≈ 0.7), and very similar luminosities (M_B ≈ −19.6 mag). The other three SNe (2005ag, 2005ir, and 2006py) are all slow-decliners (0.86 ⩽ Δm₁₅(B) ⩽ 1.02 mag) and, therefore, luminous events. In principle, the luminosity versus decline-rate relationship should correct for this. Nevertheless, SN 2005ag was discovered by the LOSS, and must have been at the limit of detection. Hence, Malmquist bias could explain its overluminous nature. However, SNe 2005ir and 2006py were discovered by the much deeper SDSS II survey which must have negligible bias at these redshifts. SN 2004gu had a moderately large color excess of E(B − V)_max = 0.20, but the other three events had color excesses very close to the median value of E(B − V)_max = 0.05 for the full subset of best-observed SNe. Hence, it is unlikely that the negative residuals in Figure 19 can be ascribed to inaccurate reddening corrections. Since the points for SNe 2005ag, 2005ir, and 2006py all lie within 1σ of the expected velocity spread due to peculiar velocities, we consider small-number statistics to be the most likely explanation. Clearly it will be interesting to re-examine the Hubble diagram residuals once the full CSP sample has been produced.

Interestingly, although the Hubble diagram in Figure 19 was produced by averaging the distance moduli derived in each filter, the resulting dispersion is not much better than the dispersions obtained in the individual filters (cf. Table 9). The explanation for this is found in Figure 20, where the distance-modulus residuals in ugriYJHK are plotted versus the residual in B. A significant correlation is observed between the distance-modulus residuals in one band versus those in another, particularly in the ugri bands. For this reason, the combined residuals do not reduce significantly the scatter in the diagram as compared with the fits to the individual bands.

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We measured correlation coefficients, r, among residuals in Bugri to be 0.8–0.9, and 0.3–0.6 between any band and YJHK. Most likely the lower correlation in the NIR is attributable to the difficulties of measuring precise peak magnitudes in these bands as discussed earlier. We compared the observed correlation coefficients with those estimated for each SN based on the measurement uncertainties and the form of the model in Equation (7). We found the expected correlation coefficients to be r < 0.05 for all cases, except between B and i bands where we estimate r ≈ 0.6, and between B and Y where we expect r ≈ 0.3. For these two cases, the high correlation may in part be due to the use of E(B − V) in the color-excess term. For all the rest of the cases, the contribution to the correlations due to the fit model is negligible. We thus interpret the dispersion in Figure 19 to be mostly due to peculiar velocities. If this is correct, then the true precision of the SN distances is the dispersion about the relations in Figure 20, which for the ugri bands amounts to 0.06–0.09 mag in distance modulus, or 3%–4% in distance.

In computing the denominator of the χ² function, care must be taken in estimating not only the variances of the data, but also the covariances between the different observables. We outline below how these variances and covariances are estimated. For each SN, the total variance is (we omit the subscript i for clarity)

$$\begin{eqnarray} \sigma ^2 & = & \sigma ^2(\mu)\;+\;\sigma ^2(m_X)\;+\;b_X^2\,\sigma ^2(\Delta m_{15}(B))\;+\;\beta _X^2\,\sigma ^2(c)\nonumber \\ &&-\,2\,b_X\,\mathrm{cov}[m_X,\Delta m_{15}(B)]\;-\;2\,\beta _X\,\mathrm{cov}[m_X,c]\nonumber \\ &&+\,2\,b_X\,\beta _X\,\mathrm{cov}[\Delta m_{15}(B),c], \end{eqnarray} \tag{ A5 }$$

where σ²(q) denotes the variance in quantity q, cov[p, q] denotes the covariance between quantity p and q, and we have assumed cov[μ, q] = 0. Furthermore, we assume the peak magnitudes in two different bands are uncorrelated (cov[m_X, m_Y] = δ_XYσ²(m_X)). The three covariance terms in Equation (A5) are computed as follows.

• 1.  
  For cov[m_X, Δm₁₅(B)], we recall that Δm₁₅(B) ≡ m_B(t = 15) − m_B(t = 0) and so

  $$\begin{eqnarray} \mathrm{cov}[m_X,\Delta m_{15}(B)] & = & \mathrm{cov}[m_X,m_B(t=15)]\nonumber \\ -\,\mathrm{cov}[m_X,m_B(t=0)]& = & -\delta _{XB}\,\sigma ^2(m_X), \end{eqnarray} \tag{ A6 }$$

  where we have assumed cov[m_X, m_B(t = 15)] = 0. Note that we have neglected any possible correlated error introduced by the subtraction of the host-galaxy image in the B band. As discussed by Contreras et al. (2010), such errors are less than the quoted statistical errors of the photometry.
• 2.  
  For cov[m_X, c] we have, from Equation (A2),

  $$\begin{eqnarray} \mathrm{cov}[m_X,c] & = & \mathrm{cov}[m_X,(m_Y-m_Z)]\nonumber\\ &&-\,B_0\,\mathrm{cov}[m_X,\Delta m_{15}(B)]\nonumber \\ & = & (\delta _{XY}\;-\;\delta _{XZ}\;+\;B_0\,\delta _{XB})\,{\sigma ^2(m_X)}.\quad \end{eqnarray} \tag{ A7 }$$
• 3.  
  For cov[Δm₁₅(B), c], we also have from Equation (A2)

  $$\begin{eqnarray} \sigma _{(\Delta m_{15}(B),c)} & = & \mathrm{cov}[\Delta m_{15}(B),(m_Y-m_Z)]\nonumber\\ &&-\,\mathrm{cov}[\Delta m_{15}(B),B_0\,\Delta m_{15}(B)]\nonumber \\ & = & (\delta _{ZB} - \delta _{YB})\,\sigma ^2{m_B} - B_0\,\sigma ^2\nonumber\\ &&\times (\Delta m_{15}(B)). \end{eqnarray} \tag{ A8 }$$

Substituting Equations (A6), (A7), and (A8) into Equation (A5) for σ² we obtain the final expression for the measurement variances,

$$\begin{eqnarray} \sigma ^2 &=& \sigma ^2(\mu)+[1+2\,\delta _{XB}\,b_X \nonumber\\ &&-\,2\,\beta _X(\delta _{XY}\,-\,\delta _{XZ}\,+\,B_0\,\delta _{XB})]\,\sigma ^2(m_X)\nonumber \\ &&+\,\left[b_X^2\;-\;2\,b_X\,\beta _X\,B_0\right]\,\sigma ^2(\Delta m_{15}(B))\nonumber\\ &&+\,\beta _X^2\,\sigma ^2_{c} + 2b_X\beta _X(\delta _{ZB} - \delta _{YB})\sigma ^2(m_B), \end{eqnarray} \tag{ A9 }$$

where we recall that B₀ = 0 for the case when c represents a pseudocolor.

The χ² minimization is performed by analytically marginalizing Equation (A4) over M_X(0) (see Goliath et al. 2001) at each point in a grid of values of {b_X; β_X}, followed by a search for the minimum value of χ². This process is repeated for various values of σ_SN—usually between 0 and 0.3 mag—in order to find the value ${\hat{\sigma }}_{\rm SN}$ that produces a minimum reduced χ² of χ²_min/(N − 3) = 1.

Once ${\hat{\sigma }}_{\rm SN}$ is found, it is used for computing the probability density ρ(χ²_marg) at each point in the grid of {b_X; β_X} values. This probability is given by

$$\begin{equation} \rho (b_X,\beta _X)\,=\,\exp \left\lbrace \frac{-\chi ^2_{\mathrm{marg}}}{2}\right\rbrace. \end{equation} \tag{ A10 }$$

Finally, the expectation value of each parameter, $\hat{\theta }$, in the grid of ${\vec{\theta }} \equiv \, \lbrace M_X(0);b_X;\beta _X\rbrace$ is computed as the moment

$$\begin{equation} \hat{\theta }\,=\,\frac{\int \theta \,\rho ({\vec{\theta }})\,d{\vec{\theta }}}{\int\!\! \rho ({\vec{\theta }})\,d{\vec{\theta }}}, \end{equation} \tag{ A11 }$$

and the variances in the fit parameters, $\sigma _{\hat{\theta }}$, are estimated as the second moment

$$\begin{equation} \sigma _{\hat{\theta }}^2\,=\,\frac{\int (\theta -\hat{\theta })^2\,\rho ({\vec{\theta }})\,d{\vec{\theta }}}{\int\! \rho ({\vec{\theta }})\,d{\vec{\theta }}}. \end{equation} \tag{ A12 }$$