FIRST-YEAR SLOAN DIGITAL SKY SURVEY-II SUPERNOVA RESULTS: HUBBLE DIAGRAM AND COSMOLOGICAL PARAMETERS

After the finish of the Fall 2005 season, all of the SN spectra were processed with IRAF (Tody 1993). Classification of the reduced SN spectra was aided by the IRAF package rvsao.xcsao (Tonry & Davis 1979), which cross-correlates the spectra with libraries of SN spectral templates and searches for significant peaks. Details of this analysis are described in Zheng et al. (2008). About half of the SN spectra had an excellent template match, while the other half required more human judgment for the SN typing. Based on this analysis, 130 candidates were classified as confirmed SNe Ia and 16 candidates were classified as probable SNe Ia.

For 29 of these 146 candidates, we have used the SDSS host-galaxy spectroscopic redshift as reported in the SDSS DR4 database; typical redshift uncertainties are 1–2 ×10⁻⁴. For SN 2005hj, a host-galaxy spectroscopic redshift and its uncertainty were obtained by Quimby et al. (2007). For 82 of the candidates that do not have a host spectroscopic redshift in the DR4 database, we use the redshift from host-galaxy spectral features obtained with our own spectroscopic observations. The redshift precision in those cases is estimated to be 0.0005, the rms difference between our host-galaxy redshifts and those measured by the SDSS spectroscopic survey (DR4) for a sample in which both redshifts are available. For the remaining 34 candidates, our redshift estimate is based on spectroscopic features of the SNe, with an estimated uncertainty of 0.005, the rms spread between the SN redshifts and host-galaxy redshifts. In summary, 77% of the spectroscopically confirmed and probable SNe Ia have spectroscopic redshifts determined from host-galaxy features, while the rest have redshifts based on SN spectral features. The redshifts are determined in the heliocentric frame and then transformed to the CMB frame as described in Section 8.

The redshift distribution for the 130 confirmed SNe Ia from the 2005 season is shown below in Figure 2(e). The relative deficit of confirmed SNe at redshifts between 0.15 and 0.25 is due to the finite spectroscopic resources that were available for the Fall 2005 campaign and to the relative priorities given to low- and high-redshift candidates for the different telescopes (Sako et al. 2008). Subsequently, host-galaxy redshifts have been obtained for most of the "missing" SN Ia candidates with SN Ia like light curves in this redshift range. These photometrically identified (but spectroscopically unconfirmed) candidates with host-galaxy redshifts are used in the determination of host-galaxy dust properties (Section 7), but we do not include them in the Hubble diagram for this analysis. Compared to the Fall 2005 season, spectroscopic observations during the 2006 and 2007 seasons were more complete around redshifts z ∼ 0.2.

To achieve precise and reliable SN photometry, we developed a new technique called "Scene Model Photometry" (SMP) that optimizes the determination of SN and host–galaxy fluxes. This method and the Fall 2005 SN photometry results are described in detail in Holtzman et al. (2008).

The basic approach of SMP is to simultaneously model the ensemble of survey images covering an SN location as a time-varying point source (the SN) and sky background plus time-independent galaxy background and nearby calibration stars, all convolved with a time-varying PSF. The calibration stars are taken from the SDSS catalog for stripe 82 produced by Ivezić et al. (2007). The fitted parameters are SN position, SN flux for each epoch and passband, and the host host-galaxy intensity distribution in each passband. The galaxy model for each passband is a 20 × 20 grid (with a grid scale set by the CCD pixel scale, 04 × 04) in sky coordinates, and each of the 400 × 5 = 2000 galaxy intensities is an independent fit parameter. As there is no pixel re-sampling or image convolution, the procedure yields correct statistical error estimates. Holtzman et al. (2008) describes the rigorous tests that were carried out to validate the accuracy of SMP photometry and of the error estimates.

Although we have obtained additional imaging on other telescopes for a subsample of the confirmed SNe Ia, only photometry from the SDSS 2.5 m telescope is used in this analysis. Figure 1 shows four representative SDSS-II SN Ia light curves processed through SMP and provides an indication of the typical sampling cadence and signal to noise as a function of redshift.

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The fluxes and magnitudes returned by SMP are in the native SDSS system (Ivezić et al. 2007). The SDSS photometric system is nominally on the AB system, but the native flux in each filter differs from that of a true AB system by a small amount. AB magnitudes are obtained by adding the AB offsets in Table 1 to the native magnitudes. The offsets are determined by comparing photometric measurements of the HST standard solar analogs P3330E, P177D, and P041C with synthetic magnitudes based on the published HST spectra (Bohlin 2007) and SDSS filter bandpasses. Since the standard stars are too bright to be measured directly with the SDSS 2.5 m telescope, the measurements are taken with the 0.5 m SDSS Photometric Telescope (the PT) and transformed to the native system of the SDSS telescope. The technique of transferring the PT magnitudes to the native SDSS system is identical to that used to obtain the SDSS photometric calibration (Tucker et al. 2006). The uncertainty in the AB offsets is estimated to be 0.003, 0.004, 0.004, 0.007, 0.010 mag (for u, g, r, i, z) based on the internal consistency of the three standard solar analogs. The uncertainties given in Table 1 are larger, since they also account for the ∼10 Å uncertainties in the central wavelengths (given in the same table) of the SDSS filters.

The Multicolor Light-Curve Shape method, known as mlcs2k2 in its current incarnation (JRK07), has been in use for more than a decade; the original MLCS version (Riess et al. 1998) was used by the High-z Supernova Team in the discovery of cosmic acceleration. For each SN, mlcs2k2 returns an estimated distance modulus and its uncertainty; the redshift and distance modulus for each SN are inputs to the cosmology fit discussed in Section 8.

mlcs2k2 describes the variation among SN Ia light curves with a single parameter (Δ). Excess color variations relative to the one-parameter model are assumed to be the result of extinction by dust in the host galaxy and in the Milky way. The mlcs2k2 model magnitude is given by

$$\begin{eqnarray} m_{\rm model}^{e,f} & = & {M}^{e,f^{\prime }} + p^{e,f^{\prime }}\Delta + q^{e,f^{\prime }}\Delta ^2 \nonumber \\ && + X_{\rm host}^{e,f^{\prime }} + K^e_{ff^{\prime }} + \mu + X_{\rm MW}^{e,f}, \end{eqnarray} \tag{ 1 }$$

where e is an epoch index that runs over the observations, f are observer-frame filter indices, f' = UBVRI are the rest-frame filters for which the model is defined, Δ is the mlcs2k2 shape-luminosity parameter that accounts for the correlation between peak luminosity and the shape/duration of the light curve, X_host is the host-galaxy extinction, X_MW is the Milky Way extinction, $K_{ff^{\prime }}$ is the K-correction between rest-frame and observer-frame filters, and μ is the distance modulus, which satisfies μ = 5log₁₀(d_L/10pc), where d_L is the luminosity distance. We use this model for SN epochs in the rest-frame time range −15 < T_rest < +60 days relative to rest-frame B–maximum. Observer-frame passbands are included that satisfy $3200 < \bar{\lambda }_f/(1+z) < 9500$ Å, where $\bar{\lambda }_f$ is the mean wavelength of the filter passband, and z is the redshift of the SN Ia. To account for larger model uncertainties in the rest-frame UV region, a K-correction uncertainty of $0.0006\times (3500-\bar{\lambda }_f)$ mag is added in quadrature to the model error for $\bar{\lambda }_f< 3500$ Å.

In the mlcs2k2 model, the shape-luminosity parameter Δ describes the intrinsic SN color dependence on brightness, and X_host describes SN color variations from reddening (extinction) by dust in the host galaxy, which is assumed to behave in a manner similar to dust in the Milky Way. In particular, the extinction is described by the parameterization of Cardelli et al. (1989) (hereafter CCM89), $X_{\rm host}^{e,f^{\prime }}=\zeta ^{e,f^{\prime }}(a^{f^{\prime }}+b^{f^{\prime }}/R_V)A_V$, where A_V is the extinction in magnitudes in the V band, a^V = 1, b^V = 0, and the relative extinction in other passbands is determined by the parameter R_V, the ratio of V-band extinction to color excess, R_V = A_V/E(B − V). For the Milky Way, the value of R_V averaged over a number of lines of sight is R_V = 3.1; this global value has been adopted in previous SN analyses using mlcs2k2. For the galaxies that host SNe Ia, we instead adopt R_V = 2.18 ± 0.50, as derived in Section 7.2 from the SDSS-II SN data.

The coefficients ${M}^{e,f^{\prime }}$, $p^{e,f^{\prime }}$, and $q^{e,f^{\prime }}$ are model vectors that have been evaluated using nearly 100 well-observed low-redshift SNe as a training set. $M^{e,f^{\prime }}$ is the absolute magnitude for an SN Ia with Δ = 0. Assuming a Hubble parameter h = H₀/100 km s⁻¹ Mpc⁻¹ = 0.65, the resulting absolute magnitudes at peak brightness are −20.00, −19.54, −19.46, −19.45, −19.18 mag for U, B, V, R, I, respectively. The p and q vectors translate the shape-luminosity parameter Δ into a change in the SN Ia absolute magnitude. The $p^{0,f^{\prime }}$ values (at peak brightness) vary among passbands from 0.6 to 0.8, and the $q^{0,f^{\prime }}$ vary from 0.1 to 0.9; therefore, intrinsically faint (bright) SNe have positive (negative) values of Δ.

We use model vectors based on the procedure outlined in JRK07, but with two notable differences. First, the vectors have been re-evaluated based on our determination of the dust parameter, R_V = 2.18. Since most of the nearby objects used in the training have low extinction, retraining with a different value of R_V has little effect on the vectors and therefore on the cosmological results. Note that the insensitivity of the mlcs2k2 training to the value of R_V does not imply that the estimated distances for high-redshift SNe are insensitive to the value of R_V, especially since the latter samples include highly extinguished SNe. The impact of R_V on the cosmology results is presented in Section 9.

The second change in the model vectors from JRK07 involves adjustments to the $M^{e,f^{\prime }}$ that were developed during the course of the WV07 analysis of the ESSENCE data. For the model training with the Nearby SN Ia sample, it was assumed that the observed A_V distribution has the functional form of an exponential distribution convolved with a Gaussian centered at A_V = 0. However, the mlcs2k2 training process resulted in a convolution Gaussian that is not centered at zero; adjustments were made in the vectors such that the Gaussian is centered at zero. The main caveat in this procedure is that the selection efficiency for the Nearby sample is small and unknown, and therefore it is not straightforward to model the observed A_V distribution in terms of an underlying population. The $M^{e,f^{\prime }}$ adjustments for UBVRI depend only on the passband and are independent of epoch. For the model vectors determined with R_V = 2.2, the magnitude adjustments relative to the values in JRK07 are

$$\begin{equation} \delta M^{\it {UBVRI}} = +0.050,+0.020, 0.0, -0.002, -0.033. \end{equation} \tag{ 2 }$$

K-corrections transform the mlcs2k2 SN rest-frame Landolt-system magnitudes to the magnitudes of a redshifted SN in an observed passband. K-corrections are computed following the prescription of Nugent et al. (2002), which requires an SN spectrum at each epoch, the spectrum of a reference star, and the reference star magnitude in each passband. As explained in Appendix A, we use a single template spectrum for each SN epoch and warp it to match the colors of the SN model. Since the Landolt photometry is not associated with a precisely defined set of filters, we use the standard UBVRI and B_X filters defined by Bessell (1990) and apply a color transformation to obtain photometry in the Landolt system. That procedure is detailed in Appendix B. In place of the traditional primary reference star Vega, we choose BD+17°4708 (Oke & Gunn 1983) as our primary reference because it has been measured by Landolt, it has a precise HST STIS spectrum (Bohlin 2007) and it is the primary reference for SDSS photometry. We have also carried out the analysis with Vega as the primary reference and include the difference as a systematic error. The primary magnitudes for each filter system are given in Table 22 of Appendix C for both BD+17 and Vega.

A light-curve fit determines the likelihood function $\cal L$ of the observed magnitudes or fluxes as a function of four model parameters for each SN Ia: (i) time of peak luminosity in rest-frame B band, t₀, (ii) shape-luminosity parameter, Δ, (iii) host-galaxy extinction at central wavelength of rest-frame V band, A_V, and (iv) the distance modulus, μ. The redshift (z) is accurately determined from the spectroscopic analysis, so it is not included as a fit parameter; the redshift uncertainty is included in the cosmology analysis (Section 8). For each SN, the log of the posterior probability P_post, or χ² statistic, is given by

$$\begin{eqnarray} \chi ^2 & = & -\!2\ln P_{\rm post}(t_0,\Delta,A_V,\mu | {\rm data}) \nonumber \\ & = & -\!2\ln {\cal L}({\rm data}|t_0,\Delta,A_V,\mu) -2\ln P_{\rm prior}(z,A_V,\Delta), \nonumber \\ \end{eqnarray} \tag{ 3 }$$

where P_prior is a Bayesian prior (see below), and the log-likelihood is given by

$$\begin{equation} -\!2\ln {\cal L} = \left\lbrace \sum _i \frac{\left[F_i^{\rm{data}} - F_i^{\rm{model}}(t_0,\Delta,A_V,\mu)\right]^2}{ \sigma _{i,\rm{stat}}^2 + \sigma _{i,\rm{model}}^2} \right\rbrace. \end{equation} \tag{ 4 }$$

Here the index i runs over all measured epochs and observer-frame passbands, and F^data_i is the observed flux for measurement i. The statistical measurement uncertainty, σ_stat, is estimated from the SMP as described in Section 3.2. For the model uncertainty, σ_model, we use the diagonal elements of the mlcs2k2 covariance matrix, which are estimated from the spread in the training sample of SNe. For example, at the epoch of peak brightness (t₀) these model errors are 0.11, 0.07, 0.08, 0.10, 0.11 mag for U, B, V, R, I, respectively; the uncertainties increase monotonically with time away from t₀. As explained below in the list of modifications, we do not use the off-diagonal mlcs2k2 correlations in this analysis.

Since A_V is a physical parameter that is always positive, and since it is not well constrained if peak S/N is low or if the observations do not span a large wavelength range, the mlcs2k2 fit includes a Bayesian prior on the extinction. The prior forbids negative values of A_V and encodes information about the distribution of extinction in SN host galaxies as well as the selection efficiency of the survey. Since there is degeneracy between the inferred values of A_V and μ, the prior leads to reduced scatter in the Hubble diagram. For the Nearby SN sample, which has high peak S/N for all objects, the prior has no impact on the Hubble scatter; for the other samples we employ, the prior reduces the Hubble scatter by a factor of 1.3–2. For this analysis, the prior is defined to be

$$\begin{eqnarray} P_{\rm prior}(z,A_V,\Delta) & = & P(A_V) P(\Delta) \nonumber \\ & & \times \epsilon _{\rm search}(z,A_V,\Delta) \epsilon _{\rm cuts}(z,A_V,\Delta), \end{eqnarray} \tag{ 5 }$$

where P(A_V) and P(Δ) are the underlying SN Ia population distributions of A_V and Δ, and we assume that these distributions are independent of redshift. We determine them from SDSS-II SN data in Section 7. For SNe passing the selection cuts, the parameter Δ is typically precisely determined by the light-curve fit, so the prior on Δ does not have a significant impact on the inferred parameters. The functions _search and _cuts are survey-dependent efficiency factors associated with the survey selection functions and with the sample selection cuts. The efficiencies are determined from Monte Carlo simulations in conjunction with the observed data distributions for each survey in Section 6. Tests with high-statistics simulations have verified that the prior in Equation (5) leads to unbiased results for cosmological parameters.

In the mlcs2k2 fit, the estimated value and uncertainty for each model parameter, e.g., the distance modulus μ, are obtained by marginalizing the posterior (Equation (3)) over the three other parameters and taking the mean and rms of the resulting one-dimensional probability distribution. In the marginalization integrals, we use 11 bins in each of the parameters; this choice is dictated by the computational time required for the large number of systematics tests (see Section 9.1). We have compared results with 11 and 15 integration bins and find excellent agreement.

To implement the mlcs2k2 method, we have written a new version of the fitting package with several modifications from JRK07:

• 1.  
  We fit in calibrated flux instead of magnitudes. In previous analyses using mlcs2k2, the fits were carried out using magnitudes, and data with S/N<5 were typically excluded in order to avoid ill-defined magnitudes associated with negative flux measurements. The S/N cut results in a biased determination of the shape-luminosity parameter Δ and therefore of the distance modulus μ. Fitting in flux enables a proper treatment of errors for all measurements and results in a negligible bias in Δ and μ, as determined from a simulation. This change is crucial for our analysis, since ∼40% of the SDSS-II SN measurements (with −15 < T_rest < +60 days) have S/N <5.
• 2.  
  We have made two improvements to the treatment of K-corrections. First, we use the updated SN Ia spectral templates from Hsiao et al. (2007), which result in better consistency between the data and the best-fit mlcs2k2 model for observer-frame filters that map onto rest-frame R band. Second, we have improved the spectral warping used for K-corrections as explained in Appendix A.
• 3.  
  The mlcs2k2 model includes off-diagonal covariances in the model magnitudes to account for brightness correlations between different epochs and passbands; in this analysis, we ignore the off-diagonal covariances for two reasons. The primary reason is that the mlcs2k2 model covariances appear to display unphysical behavior. The correlation coefficient ρ_ij ≡ cov(i, j)/σ_iσ_j between epochs i and j decreases discontinuously from unity at t_i = t_j (Figure 4): the correlation between epochs separated by only one day is weak, 0.2 < ρ_{t,t + 1} < 0.8, and thus does not penalize (via χ²) random variations of ∼0.1 mag over one-day timescales. The observed smoothness of high-quality SN Ia light-curve data rules out such large intrinsic fluctuations, suggesting that random instrumental noise may have been included in the model covariance matrix. The impact of the off-diagonal covariances on determination of the cosmological parameters (w and Ω_M) from the SN data is much smaller than the statistical uncertainties. Second, there is a subtle limitation when measurements at the same epoch in two observer-frame passbands f₁, f₂ are matched onto the same rest-frame filter f' using λ_rest = λ_obs/(1 + z) for each passband. In the mlcs2k2 model, there is an artificial 100% correlation between the two rest-frame model magnitudes. This feature arises for the observed ugriz filters used by SDSS-II and SNLS, but does not appear for the Bessell filters used in the Nearby and ESSENCE samples.
• 4.  
  We have extensively modified the prior (Equation (5)). The mlcs2k2 prior in JRK07 is intended to reflect the true distribution of A_V. In analyzing the ESSENCE data, WV07 used a different A_V prior and multiplied it by a simulated efficiency that depends upon extinction, intrinsic luminosity, and redshift. In our analysis we use more detailed Monte Carlo simulations (Section 6) of each data sample to estimate the survey efficiencies that are incorporated into the priors, and we use the SDSS-II SN data sample to determine the underlying A_V distribution.
• 5.  
  In mlcs2k2, the reddening parameter R_V is treated as a fixed global parameter. In JRK07 and WV07, R_V was set to the average Milky Way value of 3.1. In our analysis, we use R_V = 2.18 ± 0.50 as empirically determined from the SDSS-II SN sample (Section 7).

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Some example fits for SDSS-II SN light curves using the modified version of mlcs2k2 are shown in Figure 1. Figure 5 shows the average fractional residuals between the mlcs2k2 model light curves and the data for the SNe in each survey and for each rest-frame UBVR passband. The overall data-model agreement is good, except for some late-time epochs and U band. The U-band residuals are discussed later in more detail (Section 10.1.3). Figure 6 shows the fit parameters A_V and Δ versus redshift for SNe in the different surveys. The impact of the prior requiring A_V > 0 is immediately evident in the top-left panel. If we split each SN sample at its median redshift, the average A_V for the lower-redshift SNe is larger than for the higher-redshift SNe; this A_V difference is 0.1 mag for the Nearby sample, and ∼0.05 mag for the other SN samples. The prior discussed above accounts for this redshift-dependent shift. The right panel in Figure 6 shows the fitted A_V versus redshift using a flat prior, P(A_V) = P(Δ) = 1 in Equation (5). Although there are many SNe with A_V < 0, in Section 7.3 we show that an underlying extinction distribution with A_V > 0, combined with measurement uncertainties, is consistent with the "negative-A_V" distribution obtained from fitting with a flat prior. Since Δ is well constrained by the light-curve fits, the Δ distribution with a flat prior is very similar to that using the nominal prior.

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We have checked the results of the modified mlcs2k2 fitter with the distance estimates derived by WV07 for the ESSENCE and Nearby SN samples. For this comparison, we use the WV07 extinction prior and efficiency, as described in their Equations (2) and (3). The WV07 efficiency function accounts for missing SNe at high redshift and for the bias arising from using only measurements with S/N >5; for comparison, we therefore use the same S/N cut. The modified fitter is run in a mode that replicates the original mlcs2k2 fitter, with two exceptions: first, as noted above, we fit in flux instead of magnitude. Second, the K-corrections use the average spectral template of Hsiao et al. (2007), while WV07 used a library of spectra and interpolated K-corrections to the desired epoch. To compare our "Nearby+ESSENCE" analysis with WV07, we fit light curves for the 45 Nearby SNe Ia (0.015 < z < 0.1) and 57 ESSENCE SNe Ia analyzed by WV07. The rms scatter between our fitted distance moduli (μ) and those from WV07 is 0.03 mag and 0.05 mag for the Nearby and ESSENCE samples, respectively. Our marginalized value for the dark energy equation of state parameter w, using the SDSS BAO prior (see Section 8), agrees to within 0.01 with the result of WV07.

We stress that this comparison with WV07 is a consistency check of our version of mlcs2k2 relative to previous versions. When we analyze the present SN samples with mlcs2k2 (Section 10), our different prior and mlcs2k2 model parameter values result in cosmological parameter estimates that differ significantly from those of WV07, as discussed in Section 10.1.4.

The salt-ii light-curve fitting method (Guy et al. 2007) has been developed by the SNLS collaboration. The salt-ii model employs a two-dimensional surface in time and wavelength that describes the temporal evolution of the rest-frame spectral energy distribution (SED) for SNe Ia. The temporal resolution of the model is 1 day, and the wavelength resolution is 10 Å, allowing accurate synthesis of model fluxes to compare with photometric data. The model is created from a combination of photometric light curves and hundreds of SN Ia spectra. When there are measurement gaps in the spectral surface, the unmeasured regions of the SED are determined from interpolations of the measured regions. The photometric data are mostly from the Nearby sample (JRK07) but also includes higher-redshift data (z > 0.1) to better constrain the rest-frame ultraviolet behavior of the model. For a complete list of SN light curves and spectra used for training, see Table 2 in Guy et al. (2007).

In salt-ii, the rest-frame flux at wavelength λ and time t (t = 0 at B-band maximum) is modeled by

$$\begin{eqnarray} {\frac{dF_{\rm rest}}{d\lambda }}(t,\lambda) & = & x_0 \times [M_0(t,\lambda) + x_1 \times M_1(t,\lambda)] \nonumber \\ && \times \exp [c \times CL(\lambda)]. \end{eqnarray} \tag{ 6 }$$

M₀(t, λ), M₁(t, λ), and CL(λ) are determined from the training process described in Guy et al. (2007). The M₀ surface represents the average spectral sequence, and is very similar to the sequence of average spectral templates (Hsiao et al. 2007) that we use for the mlcs2k2 K-corrections. M₁ is the first moment of variability about this average, accounting for the well-known correlation of both peak brightness and color with light-curve shape, and x₁ is the stretch parameter, the analog of the mlcs2k2 Δ parameter. CL(λ) is the mean color correction term, and c is a measure of SN Ia color. Although the color variation is not explicitly attributed to dust extinction, in the optical region CL(λ) is reasonably well approximated by the CCM89 extinction law with R_V ∼ 2. In the UV region, CL(λ) exceeds the CCM89 extinction by about 0.07 mag.

The spectral-time surfaces are defined for rest-frame times −20 < T_rest < +50 days relative to the time of maximum brightness, and for rest-frame wavelengths that span 2000 to 9200 Å. We use the salt-ii spectral surfaces obtained from retraining the model using Bessell-filter shifts based on HST standards, as discussed in Appendix B (Table 21), but otherwise using the same technique and data as described in Guy et al. (2007). The UBVRI magnitudes for the primary reference Vega are taken from Fukugita et al. (1996): these are 0.02, 0.03, 0.03, 0.03, 0.024, respectively, and are slightly different from those used to crosscheck the mlcs2k2 method. Although the wavelength coverage of the spectral surface is rather broad, the salt-ii model includes only those observer-frame passbands for which $2900< \bar{\lambda }_f/(1+z) < 7000$ Å, where $\bar{\lambda }_f$ is the mean wavelength of the filter and z is the SN Ia redshift.

To compare with photometric SN data, the observer-frame flux in passband f is calculated as

$$\begin{equation} F^f_{\rm obs}(t) = (1+z)\int d\lambda ^{\prime }\left[ \lambda ^{\prime }{dF_{\rm rest}\over d\lambda ^{\prime }}(t,\lambda ^{\prime }) T^f(\lambda ^{\prime }(1+z)) \right], \end{equation} \tag{ 7 }$$

where T^f(λ) defines the transmission curve of observer-frame passband f. For the SDSS-II, ESSENCE, SNLS, and HST samples, T^f(λ) is provided by each survey. For the Nearby sample, T^f(λ) is given by the Bessell (1990) UBVRI filter response curves, with wavelength shifts as described in Appendix B and listed in Table 21.

The model uncertainty accounts for the covariance between M₀(t, λ) and M₁(t, λ) at the same epoch and wavelength. Although spectral covariances between different epochs and wavelengths are not considered, the model does account for covariances between integrated fluxes at different epochs within the same filter. Each SN Ia light curve is fitted separately using Equations (6) and (7) to determine the parameters x₀, x₁, and c. However, the salt-ii light-curve fit does not yield an independent distance-modulus estimate for each SN. As discussed in Section 8.2, the distance moduli are determined as part of a global fit to an ensemble of SN light curves in which cosmological parameters and global SN properties are also determined. The salt-ii fits do not include informative priors on the fit parameters or the effects of selection efficiencies. We correct the salt-ii results for selection biases using a Monte Carlo simulation (see Sections 6 and 9.2).

In most cases, the salt-ii light-curve fits are qualitatively very similar to the mlcs2k2 fits on a per-object basis. The average rest-frame light-curve residuals for the salt-ii fits are shown in Figure 7 for each survey for filters UBVR; note that the U-band residuals for the Nearby SNe show some discrepancy, as will be discussed later. Figure 8 shows the fitted values for the color parameter c and stretch parameter x₁ versus redshift for SNe in the different surveys.

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As a crosscheck on our use of the public salt-ii code, we have compared our fits of the 71 SNe Ia from Astier et al. (2006) to fits done by the developer (J. Guy 2008, private communication) and find good agreement. The mean difference in the color (c) is 0.003 ± 0.003, with an rms dispersion of 0.02, and the mean difference in the shape-luminosity parameter (x₁) is −0.014 ± 0.021, with an rms dispersion of 0.18. The slight differences are attributable to the use of different versions of the code and of the error (dispersion) map.

We end this section by briefly comparing and contrasting the salt-ii and mlcs2k2 methods. The mlcs2k2 rest-frame model for the intrinsic SN brightness is defined in discrete UBVRI passbands corresponding to the Landolt system. For each SN light curve, a composite SN spectrum is warped based on the model fit to the observed SN colors at each epoch, and the warped spectrum is used to perform the K-corrections needed to transform the rest-frame model to the observer-frame fluxes. The salt-ii model uses a composite SN spectrum that depends on both the epoch and intrinsic luminosity as well as an epoch-independent color term. This spectrum is used to model the rest-frame fluxes. Both the salt-ii and mlcs2k2 models are trained using Nearby SN Ia data, but salt-ii training also includes higher-redshift data that reduces the dependence on the Nearby SN sample and provides better constraints on the rest-frame ultraviolet regions of the spectrum.

The salt-ii parameters x₁ and c are analogous to the mlcs2k2 parameters Δ and A_V. The parameters x₁ and Δ are essentially equivalent in describing the correlation between SN light-curve shape and brightness, but c and A_V have different meanings. mlcs2k2 assumes that all intrinsic SN color variations are captured in the model by the light-curve shape-luminosity correlation and that any additional observed color variation is due to reddening by host-galaxy dust. The color term (c) in salt-ii describes the excess color (red or blue) of an SN relative to that of a fiducial SN with fixed stretch parameter x₁. The excess color could be from host-galaxy extinction, from variations in SN color that are independent of x₁, or from other effects, and salt-ii does not attempt to separate these effects. salt-ii uses c to reduce the scatter in the Hubble diagram in a manner analogous to the use of x₁. The global salt-ii parameter β, defined below in Section 8.2, is the analog of the global mlcs2k2 dust parameter R_B = R_V + 1; one expects β ≃ R_B if excess color variation is purely due to host-galaxy extinction. The salt-ii β parameter is determined from the global fit to the Hubble diagram for the entire SN Ia sample under analysis; we determine the mlcs2k2 R_V parameter by modeling the observed colors of a specific subset of the SN data (Section 7.2).

Concerning correlations among model parameters, mlcs2k2 and salt-ii treat different aspects. The mlcs2k2 model includes covariances between different epochs and passbands, but we have excluded the off-diagonal covariances as explained in Section 5.1. The salt-ii model includes covariances between integrated fluxes at different epochs within the same passband, but covariances between passbands are not considered. salt-ii also includes the covariance between the spectral surfaces M₀(t, λ) and M₁(t, λ) at each epoch and wavelength bin (Equation (6)), but it does not include covariances between different epochs and passbands.

Within the mlcs2k2 framework, each light-curve fit yields an estimated distance modulus along with its estimated error, independent of cosmological assumptions. By contrast, in salt-ii the distance-modulus estimate for a given SN is based on a global fit to the ensemble of SNe within a parameterized cosmological model (see Section 8.2). A result of this global minimization in salt-ii is that a distance-modulus bias in a particular redshift range, such as could arise from including a poorly calibrated filter, will induce a bias over the entire redshift range of the sample. For the determination of cosmological parameters, this tends to reduce the sensitivity to systematic problems and hence can lead to smaller systematic uncertainties. However, this reduced sensitivity can also make biases more difficult to identify. An explicit example of this is described in Section 10.2.4.

Fitting with mlcs2k2 usually incorporates a Bayesian prior (Equations (3)–(5)) that reduces the scatter in the Hubble diagram by incorporating information about the underlying A_V distribution and the survey efficiencies. The prior and the resulting Hubble scatter do not depend on cosmological parameters. Because of the assumption that excess color variation is due to extinction by dust, the prior excludes values of A_V < 0. In mlcs2k2, SNe with very blue apparent colors (bluer than the template) are assigned A_V ≃ 0, and the data-model color discrepancy is attributed to fluctuations. In salt-ii, apparently blue SNe are assigned negative colors (c < 0) that result in larger luminosities and distance moduli compared to mlcs2k2.

Within the salt-ii framework, scatter in the Hubble diagram is explicitly minimized by simultaneously adjusting global SN parameters along with the cosmological parameters; this minimization is described in Section 8.2. In contrast to mlcs2k2, the salt-ii Hubble scatter depends on the cosmological parameters, and there is no mechanism to account for the survey efficiency directly in the fits. To correct for biases related to the survey efficiencies (Section 8.2), we use the Monte Carlo simulations described in Section 6.

The mlcs2k2 and salt-ii light-curve fit residuals can be visually compared in Figures 5 and 7; the data and models are consistent for rest-frame passbands BVR, but there are discrepancies for U band in both cases. We address this issue in more detail in Sections 10.1.3 and 10.2.4, and we compare the mlcs2k2 and salt-ii results explicitly in Section 11.

Using the mlcs2k2-based simulation described above, the resulting scatter in the Hubble diagram for SDSS-II SNe at z < 0.15 is only 0.06 mag, well below the observed scatter of ∼0.15 mag. To make the model more realistic, we introduce intrinsic fluctuations in the simulated luminosity. The models for intrinsic fluctuations described below are empirically determined to match the observed Hubble scatter and are not based on a physical model.

We have implemented two models of intrinsic SN variations. The default method we use for mlcs2k2, called "color smearing," introduces an independent fluctuation in each passband, and the fluctuation is the same for all epochs within each passband. A random number $r_{f^{\prime }}$ from a unit-variance Gaussian distribution is chosen for each rest-frame passband f'. A magnitude fluctuation, $\delta m_{f^{\prime }} = r_{f^{\prime }} \sigma _{f^{\prime }}^0$, is added to the generated magnitude at all epochs, where $\sigma _{f^{\prime }}^0$ is the magnitude uncertainty at peak brightness given by the mlcs2k2 model in passband f'. In this method, the intrinsic model colors are randomly varied by typically ∼0.1 mag. Since the simulated color variations are the same at all epochs, this model does not respect the Lira law (Phillips et al. 1999), the empirical observation that intrinsic SN Ia colors have smaller variations at epochs later than about two months after explosion; this deficiency has a negligible impact on our analysis because our requirements of good light-curve coverage make our simulated efficiencies insensitive to the magnitudes at such late epochs.

The second model of intrinsic variation, which we use for the salt-ii method, and as a crosscheck for the mlcs2k2 method, is called "coherent luminosity smearing:" a coherent random magnitude shift, typically ∼0.15 mag, is added to all epochs and passbands. In the coherent smearing method, the intrinsic model colors are not varied.

A caveat in our implementation of intrinsic luminosity variations is that the mlcs2k2 simulation and fitter use different models of intrinsic fluctuations and covariances. Although the mlcs2k2 model includes a full covariance matrix, we argued in Section 5.1 that these covariances do not accurately reflect intrinsic correlations. Since we use only the diagonal elements of the mlcs2k2 covariance matrix in the fitter, a literal translation for the simulation would be to implement a random intrinsic fluctuation of ∼0.1 mag independently for each epoch and passband. Since the observed smoothness of high-quality SN Ia light-curve data rules out such large intrinsic epoch-to-epoch fluctuations, we have chosen the above methods to simulate intrinsically smooth light curves. In future training of SN Ia light curves, it will be desirable to extract a model of intrinsic fluctuations and covariances that can be used consistently in both the light-curve fitter and the simulation.

The final step in the simulation is to model losses related to the SN search. In particular, we need to account for SNe Ia that would have passed the light-curve selection cuts of Section 4 had they been identified, but that were missed due to inefficiencies in the SN search.

Search-related losses come from the following sources: (1) the image-differencing pipeline can fail to detect objects with very low signal to noise as well as objects with nearby artifacts such as a diffraction spike; (2) humans tasked with evaluating objects detected in subtracted images may not correctly identify them as possible SN candidates; (3) software used in spectroscopic targeting to fit light curves and photometrically classify SN candidates identified by humans may not correctly classify all SNe Ia; and (4) due to limited resources for spectroscopic observations, not all photometrically identified SN Ia candidates will be targeted spectroscopically or result in a spectrum with sufficient signal to noise to confirm the SN type and determine its redshift.

We define the overall survey efficiency, or survey selection function, as _survey = _search × _cuts, where the search efficiency is further decomposed as _search = _subtr × _spec. Here, _subtr describes the net search efficiency of the image-subtraction pipeline corresponding to step (1) above. The term _spec describes the combination of steps (2), (3), and (4), which depends in part on human judgment for each SN, and _cuts is the factor associated with the final selection cuts described in Section 4. These decompositions of efficiency components are convenient because, if sufficient information about the search is available, then _cuts and _subtr can be reliably simulated. By contrast, it is usually impossible to directly simulate _spec because it involves complex decision making under varying circumstances by many people involved in spectroscopic observations. However, if the spectroscopic efficiency is nearly 100% below some redshift for a given survey, then we can model _spec at higher redshifts by comparing observed distributions of SNe properties (including redshift) to simulated distributions expected for a spectroscopically complete survey. In summary, our philosophy is to make as detailed a model of the survey efficiency components as the data and survey information allow, and to use data-simulation comparisons to constrain the other components.

The different components of the efficiency are likely to be correlated, so that the overall efficiency is not really a simple product as defined above. As an extreme example, if both the search and selection cuts required only that the maximum S/N was greater than 5, then _search = _cuts, and the combined efficiency would be the same (not the product of the two). As discussed below for the SDSS-II survey, we can simulate the combined effect of the image-subtraction efficiency (_subtr) and selection cuts (_cuts). To simplify notation we write the combined efficiency as a product, _subtr × _cuts, but it should be understood that this product really refers to the combined efficiency taking into account all correlations. The _spec term is treated differently in that it is defined as an independent efficiency function that multiplies the other efficiency terms.

For the SDSS-II, ESSENCE, SNLS, and HST samples, we have obtained detailed observing conditions, and the simulation accurately describes the effects of sample selection criteria, _cuts. For the SDSS-II, we determine _subtr using information from the fake SNe Ia that were inserted into the images during the survey (see below). For the non-SDSS samples, we do not have access to the pixel-level data, so we set _subtr = 1 and absorb the image-subtraction pipeline efficiency into the spectroscopic efficiency, _spec = _search. For the Nearby sample, we do not even have the information needed to determine the impact of the selection cuts; we therefore absorb all sources of inefficiency into the spectroscopic efficiency, _spec = _survey.

As noted above, for the SDSS-II, fake SNe Ia are used to infer _subtr as a function of S/N, as shown in Figure 7 of Dilday et al. (2008). These efficiency curves are used by the light-curve simulation to probabilistically determine which measurements in which passbands result in detections; a single-epoch detection in any two of the three (gri) passbands is considered to be an object. An object found at two or more epochs results in an SN candidate and is considered to be discovered by the image-subtraction pipeline. For the SDSS-II, the image-subtraction pipeline efficiency (_subtr) is complete up to a redshift of z ∼ 0.2 and then drops gradually to about 60% at z ≃ 0.4.

For the HST simulation, we use the single-epoch search efficiency as a function of magnitude, as described in Strolger et al. (2005) and Strolger & Riess (2006). The search was done with the F850LP_ACS passband (mean wavelength: 9070 Å); it is fully efficient down to mag ∼23, and the efficiency drops to half at mag ∼26. Incorporating this single-epoch efficiency profile into our simulation and requiring any one detection to discover an SN, the simulated efficiency for finding SNe Ia is 100% up to z ∼ 1, and drops to about 65% at z = 1.6.

6.2.1. Modeling Spectroscopic Efficiency

Although selection effects associated with the photometry can be simulated based on available information, modeling the spectroscopic efficiency for each survey is more of a challenge. The SN redshift distributions N(z) for the non-HST data and simulations are shown in Figure 9. With the exception of the Nearby SN sample, the simulations include losses from selection requirements (Section 4); for SDSS-II, losses from the image-subtraction pipeline are also included. For the Nearby sample, the simulation is based on SDSS-II observing conditions and is chosen to be 100% efficient, since we do not have the information to model the effects of photometric selection requirements in that case; the simulated redshift distribution in narrow bins of redshift therefore scales as N(z) ∼ z² (ignoring evolution of the SN Ia rate at low redshift).

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Below some cutoff redshift, z_cut, the spectroscopic efficiency is assumed to be ∼100%; for z > z_cut the data-simulation discrepancy in N(z) is taken to be due to spectroscopic inefficiency. For the SDSS-II and SNLS samples, the cutoff redshifts are estimated to be 0.15 and 0.65, respectively, based on the levels of completeness given in their SN Ia rate measurements (Dilday et al. 2008; Neill et al. 2006). For the ESSENCE sample, we estimate z_cut ∼ 0.45 based on visual inspection of the redshift histogram in Figure 9. For the Nearby (z < 0.1) sample, there is no redshift below which the spectroscopic completeness is assumed to be 100% (see discussion below). All four of these samples show significant spectroscopic inefficiency at the upper ends of their redshift ranges.

The data-simulation redshift comparison for the HST sample is shown in Figure 10. Although the uncertainty in the SN Ia rate at these high redshifts is large, there is no evidence for significant spectroscopic inefficiency: the data-simulation redshift comparison is consistent with the claim that there are no significant losses for z < 1.4 (Riess et al. 2007). We therefore assume that _spec = 1 for the HST survey, and this sample is not included in the efficiency discussion below.

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The spectroscopic efficiency as a function of redshift is defined to be _spec ≡ _survey/(_subtr × _cuts), which corresponds to the ratios of the data and simulation histograms in Figure 9. For each sample, this efficiency can be fitted by an exponential function,

$$\begin{eqnarray} \epsilon _{\rm spec}(z) & = & \zeta _0\exp [-(z-z_{\rm cut})/\zeta _1]\; {\rm for }\; z > z_{\rm cut} \\ \epsilon _{\rm spec}(z) & = & 1\; {\rm for }\; z < z_{\rm cut}, \nonumber \end{eqnarray} \tag{ 8 }$$

where ζ₀ and ζ₁ are determined from the fit. The discontinuity at z_cut is motivated by the SDSS-II survey, for which our spectroscopic observation strategy targeted z < 0.15 candidates with very high priority. We have no evidence that the other surveys should have discontinuities in their redshift distributions, but the model above provides a reasonably accurate representation for the redshift distributions of the ESSENCE and SNLS samples as well. This functional form is adopted for computational convenience in generating large Monte Carlo samples.

Although the redshift dependence of the spectroscopic efficiency has now been estimated, we know that _spec(z) depends in reality on a variety of factors, not simply redshift, and we must model its dependence on those factors in order to properly model the selection function and associated biases. As noted above, since spectroscopic targeting is a complex process, the underlying mechanism determining _spec is not easily characterized. The simplest possibility would be that _spec depends purely on redshift: in this case, ^z_spec ≡ _spec(z) would be given by Equation (8), and there would be no impact of spectroscopic efficiency on the survey bias, since redshift is precisely measured for each SN in the samples we consider. However, this model is not a priori likely: it is more probable that _spec depends explicitly on both redshift and apparent SN Ia brightness, the well-known Malmquist bias.

To consider this alternative, we define a "magnitude dimming" parameter ${\mathcal M}_{\rm dim}$ as the difference between the simulated rest-frame magnitude and the magnitude of the brightest possible SN Ia at peak light in rest-frame V band (for mlcs2k2) or in rest-frame B band (for salt-ii). For the mlcs2k2 and salt-ii models, ${\mathcal M}_{\rm dim}$ is given by

$$\begin{eqnarray} {\mathcal M}_{\rm dim}({\rm MLCS2k2}) & = & A_V + \Theta \big[ p^{0,V}(\Delta - \Delta _{\rm ref})\nonumber\\ && + q^{0,V}\big(\Delta ^2 - \Delta ^2_{\rm ref}\big) \big] \end{eqnarray} \tag{ 9 }$$

$$\begin{equation} {\mathcal M}_{\rm dim}({\rm SALT\!-\!II}) = \beta (c - c_{\rm ref}) - \alpha (x_1 - x_{1,\rm ref}). \end{equation} \tag{ 10 }$$

For mlcs2k2, A_V is the host-galaxy extinction in the V band, p^0,V and q^0,V are model parameters at the epoch of peak brightness in the V band (see Equation (1)), and Δ_ref = −0.3 corresponds to nearly the most intrinsically luminous SN Ia in the training sample. For the SDSS-II, ESSENCE, and SNLS samples, Θ = 1; for the Nearby sample, we set Θ = 0, because the Δ distribution for the Nearby sample appears unbiased relative to that of the underlying SN Ia population (see Section 7.3) while its A_V distribution is clearly biased. For salt-ii, x₁ and c describe the light-curve shape and color for each SN Ia (see Section 5.2), $x_{1,\rm ref}=2.6$ and c_ref = −0.26 correspond to the brightest SN Ia, and α, β are global SN Ia parameters determined in the cosmology fit (see Section 8.2).

Using the magnitude dimming parameter, we model the spectroscopic inefficiency due to apparent magnitude as

$$\begin{equation} \epsilon _{\rm spec}^{\mathcal M}= \exp [-{\mathcal M}_{\rm dim}/m(z)], \end{equation} \tag{ 11 }$$

where m(z) is an exponential slope function determined by numerically solving

$$\begin{eqnarray} & & \int d{\mathcal M}_{\rm dim}N_{\rm SIM}(z,{\mathcal M}_{\rm dim}) \exp [-{\mathcal M}_{\rm dim}/m(z)] \nonumber \\ && \quad = N_{\rm DATA}(z). \end{eqnarray} \tag{ 12 }$$

Here, $N_{\rm SIM}(z,{\mathcal M}_{\rm dim})$ is the number of simulated events in a two-dimensional bin of redshift and simulated ${\mathcal M}_{\rm dim}$, and N_DATA(z) is the number of data events in the redshift bin. In practice, m(z) is evaluated in discrete redshift bins for z > z_cut and fit to an exponential function of redshift,

$$\begin{equation} m(z) = m_0\exp [-(z-z_{\rm cut})/m_1] + m_2, \end{equation} \tag{ 13 }$$

where m₀, m₁, and m₂ are determined in the fit, and m(z) = ∞ for z < z_cut. Since ${\mathcal M}_{\rm dim}$ is defined for V band in the mlcs2k2 model and for B band in the salt-ii model, the associated m₀, m₁ parameters are different. The parameter m₂ is non-zero only for the Nearby sample. Qualitatively, Equation (11) says that SNe that are intrinsically faint (large positive Δ) or extinguished (large A_V) will be under-represented at the high-redshift end of a sample.

In principle, we now have two models for the spectroscopic efficiency: one, denoted by ^z_spec, and explicitly given in Equation (8), depends solely on redshift; the other depends explicitly on apparent brightness and implicitly on redshift and is also constrained to match Equation (8). While the latter model corresponds to expectations for Malmquist bias, the "redshift-only" model ^z_spec may be better suited to modeling spectroscopic selection that assigns lower priority to SN candidates near the cores of host galaxies, since SN-host angular separation tends to decrease with redshift. Since both models satisfy Equation (8), these two efficiency models can be linearly combined into a more general model:

$$\begin{equation} \epsilon _{\rm spec}(z,{\mathcal M}_{\rm dim}) = (1-A_{\mathcal M})\epsilon _{\rm spec}^z+ A_{\mathcal M}\epsilon _{\rm spec}^{\mathcal M}. \end{equation} \tag{ 14 }$$

To break this model degeneracy and determine the coefficient $A_{\mathcal M}$, we compare the mean fitted extinction $\bar{A}_V(z)$ versus redshift for the data and the simulation after applying all efficiency factors, and we minimize the χ² for the data-simulation difference. For the SDSS-II, $A_{\mathcal M}= 0.8\pm 0.2$, indicating that most of the search inefficiency is related to SN Ia apparent brightness. For the ESSENCE survey, $A_{\mathcal M}= 0\pm 0.2$, indicating that most of the search inefficiency is simply a function of redshift. For the SNLS sample, the best-fit $A_{\mathcal M}= 1$; however, since the data-simulation χ² for $A_{\mathcal M}=0$ is only 0.3 larger than the minimum χ² at $A_{\mathcal M}=1$, there is essentially no information on $A_{\mathcal M}$. The large uncertainty on $A_{\mathcal M}$ is in part due to the relatively small spectroscopic inefficiency for the SNLS. For the Nearby sample, there is no redshift range for which the sample is 100% efficient, and we therefore need an additional constraint to determine the efficiency parameterization. Noting that the mean fitted extinction drops rapidly with redshift for the Nearby sample (see Figure 11), we assume that $A_{\mathcal M}=1$. Figure 11 shows that the simulated efficiency works well in reproducing the strong extinction gradient in the Nearby sample.

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As an illustration, the inferred efficiencies for the SDSS-II SN sample are shown in Figure 12 as a function of A_V, for different values of the redshift and shape-luminosity parameter Δ. The high quality of the simulation for the SDSS-II, ESSENCE, SNLS, and HST samples is illustrated in Figures 13 and 14, where we compare the observed redshift and flux distributions to simulations that include the efficiency functions derived above. The redshift comparisons have excellent χ²/dof as a result of how the spectroscopic efficiency is determined. For the flux comparisons, the χ²/dof vary from 1 to a few; the larger χ² values are due to a few notable discrepancies in some of the flux bins.

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We have also compared the distributions of the number of epochs, earliest and latest epochs, noise, and peak colors, and find good data-simulation agreement in these distributions as well.

The results of Section 6.2.1 indicate that current SN samples suffer from significant spectroscopic selection effects (see Figure 9). Since spectroscopic selection is likely to be biased against highly extinguished SNe, use of purely spectroscopic SN samples may lead to biased estimates of the distribution of host-galaxy dust properties and thereby to potentially biased distance estimates when a dust-distribution prior is applied. To address this issue, we use a nearly complete set of SDSS-II SNe Ia to measure dust properties. For SDSS-II events with SN Ia like light curves that were not spectroscopically confirmed as SNe Ia, we have embarked upon a program to obtain host-galaxy spectra and measure spectroscopic redshifts; we call these photometric SN Ia candidates. Based on distributions of the mlcs2k2 fit parameters as well as visual inspection of the light-curve fits, we find that the requirement of a good SN Ia light-curve fit (see cut 5 in Section 4) to a well-sampled light curve is a good substitute in identifying SNe Ia when a confirming SN spectrum is lacking.

To identify photometric SN Ia candidates, we start with all 4100 candidate events that were detected by the on-mountain frame-subtraction pipeline on two or more epochs in the 2005 observing season (see Section 2). We process these candidate light curves through SMP (Section 3.2), fit them with the mlcs2k2 method, and prioritize them for host-galaxy spectroscopy based on the quality of the light curve and the fit. We have obtained host-galaxy redshifts for the majority of candidates for which the host-galaxy r-band magnitude satisfies r ≲ 20 and for a large subsample of fainter hosts as well. Adding this "host-z" photometric SN Ia sample to the spectroscopically confirmed and probable SN Ia sample, the combined sample appears to be nearly spectroscopically complete to z ≃ 0.3, when compared with the simulated sample. For z < 0.3, after the selection cuts of Section 4 are applied, the combined sample comprises 81 confirmed SNe Ia, 6 probable SNe Ia, and 73 host-z SNe Ia. We refer to these 160 SNe Ia as the SDSS-II dust sample. To illustrate the importance of including the host-z subset, we note that the average fitted extinction (A_V) is about 0.2 for the spectroscopically confirmed sample and almost 0.4 for the host-z sample: ignoring the host-z subset would clearly lead to biased results for the distribution of host-galaxy dust properties. On the other hand, giving highest priority to bright host galaxies for the host-z follow-up program may preferentially select hosts of more extinguished SNe.

To illustrate our understanding of the efficiency, Figure 15 shows the redshift distribution for the dust sample, compared with the simulation that includes known losses from the image-subtraction pipeline and selection cuts but does not include losses related to spectroscopic observations. This comparison shows that the dust sample is indeed nearly complete for redshifts z < 0.3. We can obtain an independent estimate of the dust-sample completeness by counting the number of photometric SN Ia candidates that pass our selection cuts, that have a photometric redshift (based on the host galaxy or the SN light curve) less than z_phot = 0.3, and that have not yet been targeted for host-galaxy spectroscopy. There are nearly 40 such events, indicating a completeness, after selection cuts, of about 80% for the dust sample at z < 0.3.

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Although R_V varies along sight lines through the Milky Way and likely varies with local environment within galaxies, we follow standard practice in treating it as a global parameter, because most current SN data are not adequate to determine it on an object-by-object basis. After briefly reviewing previous determinations of R_V, we describe the method we have developed to determine R_V and its results.

Measurements of dust properties in the Milky Way have favored an average value of R_V ∼ 3.1 (Fitzpatrick & Massa 2007) along sight lines with moderate to substantial extinction, and this has been used as a canonical value in the literature. However, studies of stellar colors in the direction of several thin cirrus clouds in the Milky Way indicate a value of R_V ∼ 2 for those relatively low-extinction environments (Szomoru & Guhathakurta 1999). For galaxies that host SNe Ia, two methods have been commonly used to determine R_V. The first infers an average or global R_V value based on statistical averages of optical light-curve colors. This method has been applied to the Nearby SN sample (Phillips et al. 1999; Altavilla et al. 2004; Reindl et al. 2005; Riess et al. 1996; Nobili & Goobar 2008), resulting in R_V values in the range of 2–3, somewhat lower than the Milky Way average value. The second method uses SNe Ia that have densely sampled, high signal-to-noise optical and NIR photometry, for which R_V can be estimated for individual events: this method has resulted in R_V ∼ 1–2 (Krisciunas et al. 2007, 2006; Elias-Rosa et al. 2006; Wang et al. 2008), significantly below the canonical Milky Way value. Calculations and simulations have shown that multiple scattering by circumstellar dust can lead to lower values for the reddening parameter inferred from SNe Ia, R_V ∼ 1.5–2.5 (Wang 2005; Goobar 2008).

Within the framework of the mlcs2k2 light-curve model, we have used the SDSS-II dust sample and a variant of the average color method to make a new determination of R_V. Previous measurements with this method were based on samples with large and unknown selection effects; our determination of R_V is based upon an SN Ia sample with a well-understood selection efficiency. We assume that reddening is due to dust extinction with a wavelength dependence described by the CCM89 model and that the mlcs2k2 model parameters (M, p, q in Equation (1)) accurately describe the SN Ia brightness and colors. The measurement of R_V is based upon comparing the average colors as a function of SN epoch of the SDSS-II data to those of the simulation. By using the nearly complete z < 0.3 dust sample, we minimize potential selection bias against extinguished SNe Ia; this sample also has a selection efficiency that is well described by the simulation.

For the dust sample, we compute three mean observed SN Ia colors, g − r, r − i, and g − i, in one-day bins in rest-frame epoch, as shown in Figure 16. Although the third color (g − i) is redundant in most cases, it provides information in the few cases in which the r-band measurement is not available. The rest-frame time is relative to the time of peak brightness in the B band as determined by the mlcs2k2 light-curve fit (see Section 5.1). While the light-curve fits include all observations regardless of signal to noise, the estimates of the mean observed colors include only those observations for which the S/N is greater than 4. Since the g, r, i measurements are taken simultaneously in SDSS-II, the colors are determined directly from the data without the need to interpolate in time.

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We compare these color-versus-epoch measurements with those of a grid of simulated samples, where each sample is generated with a different value of R_V and $\overline{A}_V$, and where $\overline{A}_V$ describes the generated A_V distribution, with $P(A_V) = \exp (-A_V/\overline{A}_V)$. The R_V and $\overline{A}_V$ grid sizes are 0.2 and 0.05, respectively. The simulated and data samples are subject to the same selection cuts and fit with mlcs2k2 in the same way. We determine the best-fit values of R_V and $\overline{A}_V$ by minimizing the following χ² statistic between the data and the grid of simulations,

$$\begin{eqnarray} \chi ^2 & = & \sum _e \left[\frac{ [ {\langle g-r\rangle _e}^{\rm data} - {\langle g-r\rangle _e}^{\rm sim}) ]^2 }{ [{\sigma _{\langle g-r\rangle _e}^{\rm data}}]^2 + [{\sigma _{\langle g-r\rangle _e}^{\rm sim}}]^2} \right] \left[\frac{\sigma _{\langle g-r\rangle _0}^{\rm model}}{\sigma _{\langle g-r\rangle _e}^{\rm model}}\right]^2 \nonumber \\ && + \sum _e (g,r \rightarrow r,i) + \sum _e (g,r \rightarrow g,i), \end{eqnarray} \tag{ 15 }$$

where the epoch-index e runs over 1-day bins with −5 < Tⁱ_rest < 30 days, the data averages 〈〉^data_e are taken over all dust-sample SNe and epochs e surviving the cuts above, and the measured colors in the simulated samples depend upon the input values of R_V and $\overline{A}_V$. Each color uncertainty, e.g., $\sigma _{(g-r)_e}$, is estimated as ${\rm rms}/\sqrt{N_e}$, where N_e is the number of color measurements (summed over all SNe) at epoch e. The second term in brackets is a weighting to account for the mlcs2k2 model uncertainty; $\sigma _{\langle g-r\rangle _0}^{\rm model}$ is the minimum model uncertainty at the epoch of peak brightness, and $\sigma _{\langle g-r\rangle _e}^{\rm model}$ is the model uncertainty at epoch e. The model-uncertainty ratio is therefore unity at T_rest = 0 and decreases as the model uncertainty increases for epochs away from peak brightness, so that epochs with large errors are downweighted. We have tested this method with 100 simulated mock data samples as described in Section 6; the input values of R_V and $\overline{A}_V$ are recovered, and the statistical uncertainties, although they are smaller than the grid sizes, match the spread in recovered values.

Using the SDSS-II data and simulations, we find

$$\begin{equation} {R_V = 2.18\pm 0.14_{\rm stat} \pm 0.48_{\rm syst}} \end{equation} \tag{ 16 }$$

$$\begin{equation} {\overline{A}_V= 0.358\pm 0.026_{\rm stat} \pm 0.068_{\rm syst}} \end{equation} \tag{ 17 }$$

and a correlation coefficient of 0.17. The relatively small correlation coefficient confirms that the method is independently sensitive to R_V and $\overline{A}_V$. Figure 16 compares the average observed colors with those of the simulation using the best-fit values in Equation (16). The χ² values are somewhat larger than expected, particularly for g − r. There is also a notable data-simulation discrepancy for epochs past about 10 days for the g − r and g − i colors, although these late-time epochs carry less weight in the χ² due to the increasing model errors.

To estimate the systematic uncertainties in this measurement, we have varied aspects of the procedure and determined their effects on the recovered R_V and $\overline{A}_V$. In particular, we lowered the redshift cutoff for the dust sample to z < 0.25 (nominal is 0.3), varied the simulated redshift distribution of the underlying SN Ia population, dN/dz = α(1 + z)^β, within the correlated 1σ errors on α and β from Dilday et al. (2008), varied the minimum epoch from −10 to 0 days (nominal is −5 days), varied the maximum epoch from +25 to +35 days (nominal is +30 days), varied the minimum signal to noise between 2 and 8 (nominal is 4), excluded each of the three colors g − r, g − i, and r − i individually from the χ² minimization, and ignored the mlcs2k2 model-uncertainty terms in the χ² statistic. We also ran the simulation without intrinsic color fluctuations (see Section 6.1). The changes due to each of these variations to the nominal results for R_V and $\overline{A}_V$ were added in quadrature to obtain an estimate of the total systematic uncertainty. Table 3 summarizes the contributions to the systematic uncertainties. The largest source of uncertainty comes from reducing the redshift range from 0.3 to 0.25, which changes R_V by 0.36 ± 0.17, and the error reflects the uncorrelated uncertainty. Although this R_V shift is marginally consistent with a random fluctuation, we have included the shift as a systematic error.

Table 3. Summary of Uncertainties for the Determination of R_V and $\overline{A}_V$

Source	σ(RV)	$\sigma (\overline{A}_V)$
Statistical	0.14	0.026
Redshift range 0.25 − 0.3	0.36	0.031
Assumed dN/dz: β = 0.9 − 2.1	0.11	0.004
Epoch range	0.25	0.045
S/N >2, 8	0.11	0.011
Exclude one color	0.08	0.030
Ignore mlcs2k2 model uncertainty	0.05	0.018
Remove color-smearing model	0.14	0.017
Total systematic	0.48	0.068
Total uncertainty	0.50	0.072

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As a crosscheck of how the inferred R_V depends on the assumed exponential form of the extinction distribution, P(A_V), we have repeated the procedure using different distributions for A_V in the simulation. In particular, we use a flat, truncated A_V distribution, P(A_V) = 1 for $A_V \le 2\overline{A}_V$, and P(A_V) = 0 when $A_V > 2\overline{A}_V$; this results in a negligible change in the inferred values of R_V and $\overline{A}_V$. Thus, the inferred value of R_V appears to be insensitive to the assumed A_V distribution.

7.2.1. The salt-ii Approach to Measuring R_V

The mlcs2k2 simulation and the prior used in the mlcs2k2 fitting method require knowledge of the underlying distribution for the V-band extinction due to host-galaxy dust, P(A_V). The distribution is also needed for the shape-luminosity parameter Δ, although the latter has less impact on the results, because Δ is better-determined by the light-curve data for each SN. We determine these distributions using the Bayesian unfolding method of D'Agostini (1995), which essentially uses underlying trial distributions in the simulation to make predictions for the observed distributions of fitted A_V and Δ. To limit selection biases in this procedure, we use the SDSS-II dust sample of Section 7.1.

We fit the SDSS-II light curves with mlcs2k2 using a flat prior on A_V, i.e., the fitted A_V value is allowed to be negative. In the fit, the extinction as a function of wavelength is given by the CCM89 model with R_V = 2.18, as derived in Section 7.2. The underlying distributions for A_V and Δ are determined such that when they are input into the simulation, the fitted distributions from the simulated light curves match the distributions from the mlcs2k2 fits to the SDSS-II data. Technical descriptions of this procedure and of the assumed underlying distributions are given in Appendix D.

The results for the data and Monte Carlo simulation are shown in Figure 17. Although the generated A_V distribution includes only non-negative values of A_V, there is a tail of fitted negative values that is well described by the simulation and arises from photometric errors and intrinsic SN color variations. Our procedure determines the underlying A_V distribution without assuming a functional form for P(A_V), but it turns out that this distribution is well described by an exponential, P(A_V) = exp(−A_V/τ_V), with

$$\begin{equation} \tau _{\rm V} = [ 0.334\pm 0.072] + 0.10\times (R_V- 2.18) \end{equation} \tag{ 18 }$$

$$\begin{equation} \hspace{-7pc} = 0.334\pm 0.088. \end{equation} \tag{ 19 }$$

The uncertainty in τ_V includes statistical and systematic uncertainties as described in Appendix D. The R_V dependence is shown explicitly in Equation (18), along with the internal measurement error. Equation (19) shows the total uncertainty including the uncertainty on R_V. The agreement between τ_V and $\overline{A}_V$ (from Section 7.2) is well within the expected dispersion based on simulated tests.

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The underlying A_V distribution, including the uncertainties, is shown as the hatched region in Figure 18(a); we use this P(A_V) as part of the mlcs2k2 prior in Equation (5). Our inferred A_V distribution is marginally consistent with that of JRK07, who derived a prior of exponential form from the Nearby sample, with τ^JRK_V = 0.46 (the dashed curve in Figure 18(a)). Our A_V distribution is also somewhat consistent with the "galactic line-of-sight" (glos) prior, based on theoretical considerations (Hatano et al. 1998; Commins 2004; Riello & Patat 2005), that was used by WV07 for the ESSENCE analysis (the solid curve in Figure 18(a)).

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The underlying distribution of Δ is described by an asymmetric Gaussian with mean Δ₀ = −0.24, and Gaussian widths of σ₋ = 0.24 for Δ < 0 and σ₊ = 0.48 for Δ>0 (see Appendix D for uncertainties). As shown in the right panel of Figure 17, when input into the simulation this distribution leads to a distribution of fitted Δ that is in good agreement with the observed distribution of fitted Δ from the SDSS-II dust sample. The underlying Δ distribution, P(Δ), is shown in Figure 18(b). In addition to the asymmetric Gaussian, we have added a tail to the Δ distribution at positive Δ, allowing for underluminous SNe where the underlying distribution is poorly measured due to small-number statistics; when used in the mlcs2k2 prior of Equation (5), this tail ensures that the fitter is not heavily biased against underluminous SNe. Since Δ is well determined from the light-curve shape, the fitted value of Δ has little sensitivity to the functional form of P(Δ). To check the effect of the arbitrary tail in P(Δ), we have run the fits with amplitude of the tail region multiplied by half and by two; in both cases, the rms variation in fitted Δ is 0.01.

To check that the derived extinction distribution reflects that of the global SN population rather than that of a (possibly biased) SDSS-II sub-sample, we compare the fitted A_V distributions for the data and for simulations generated with the underlying A_V and Δ distributions derived above for each SN sample in Figure 19. Here, the fits are carried out with a flat A_V prior, allowing A_V < 0. The overall agreement is good for all samples. This test illustrates that the inferred negative extinction values (when an uninformative prior is used) are consistent with being artifacts of the combination of low signal to noise and intrinsic color fluctuations, since the simulation is generated with A_V ⩾ 0. Note that the procedure used to determine the underlying A_V distribution ensures that the data and simulation agree well for positive A_V for the SDSS-II sample, but parameters have not been adjusted to match the distribution for the other samples, or for negative A_V.

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As described in Section 5.1, mlcs2k2 provides an estimate of the distance modulus, μ, for each SN. Cosmological parameter estimates are derived by minimizing the following χ² statistic (= − 2ln of the posterior probability) for the SN Ia sample over a grid of model parameter values,

$$\begin{eqnarray} \chi ^2_{\mu }& = & \left\lbrace \sum _i \frac{ [\mu _i - \mu (z_i;w,\Omega _{\rm M},\Omega _{\rm DE},H_0) ]^2}{{\sigma _{\mu } }^2} \right\rbrace \nonumber \\ && + \chi ^2_{\rm BAO}+ \chi ^2_{\rm CMB}, \end{eqnarray} \tag{ 26 }$$

where μ_i is the distance modulus estimated from the mlcs2k2 fit for the ith SN, z_i is its spectroscopically determined redshift, and μ(z_i; w, Ω_M, Ω_DE, H₀) = 5log(d_L/10pc) is computed from Equation (20). The χ²_BAO and χ²_CMB terms in Equation (26) incorporate the prior information from the SDSS LRG BAO measurement (Equation (23)) and the WMAP CMB measurement (Equation (25)). When reporting values and errors on individual cosmological parameters, we use the values at the χ²_μ-minimum, marginalized only over H₀ (due to the degeneracy between the Hubble parameter and the fiducial peak rest-frame model magnitude). In determining w, marginalizing over Ω_M shifts the value from the χ²_μ-minimum by 0.025 for the SDSS-only sample and by ∼0.01 for the other sample combinations.

In Equation (26), the distance-modulus uncertainty is given by

$$\begin{equation} {\sigma _{\mu } }^2 = \big(\sigma _{\mu }^{\rm fit}\big)^2 + \big(\sigma _{\mu }^{\rm int}\big)^2 + \big(\sigma _{\mu }^{z}\big)^2, \end{equation} \tag{ 27 }$$

where σ^fit_μ is the statistical uncertainty reported by mlcs2k2, σ^int_μ = 0.16 is the additional (intrinsic) error added so that the χ² per degree of freedom is unity for the Hubble diagram constructed from the Nearby SN Ia sample (Section 10.1) and σ^z_μ is calculated from the redshift uncertainty as described below.

The redshift uncertainty contains two components: from spectroscopic measurements and from peculiar motions of the host galaxy. The first source of uncertainty, σ_zspec, is from the uncertain redshift determination, either from the spectrum of the SN or from its host galaxy. For SDSS-II SNe we use σ_zspec = 0.0005 for host-galaxy-based redshifts, σ_zspec = 0.005 for SN-based redshifts, and the reported SDSS redshift errors for host redshifts from the SDSS spectroscopic galaxy survey (Section 3.1). For the ESSENCE and SNLS data, we use the estimated redshift errors reported in their public data tables. For the Nearby sample, redshift measurement errors were usually not reported; we take 50 km s⁻¹ as a conservative estimate (M. Hamuy 2008, private communication). The second source of redshift uncertainty, σ_zspec, arises from peculiar velocities of and within host galaxies. We take σ_zspec = 0.0012, the quadrature sum of typical galaxy peculiar velocities of 300 km s⁻¹ and typical internal motions of 200 km s⁻¹. For most of the SDSS-II SNe, for which the redshift is obtained from the host galaxy, the contribution from internal motions is overestimated because these spectra are averaged over the internal galaxy motions. The final redshift uncertainty is defined to be σ²_z = σ²_zspec + σ²_zspec. To simplify the treatment of redshift errors, we project these uncertainties onto distance modulus using the expression for the distance–redshift relation for an empty universe,

$$\begin{equation} \sigma _{\mu }^{z} = \sigma _z \left(\frac{5}{\ln 10}\right) \frac{1+z}{z(1+z/2)}. \end{equation} \tag{ 28 }$$

Using a different cosmological model to compute σ^z_μ from σ_z leads to negligible changes in the Hubble diagram fits.

Galaxy peculiar velocities are not random, as the above treatment assumes, but are spatially correlated, since they are induced by the gravitational effects of large-scale structure. SN observations are affected by the peculiar velocities of both the host galaxies and of the Milky Way. We have accounted for the Milky Way peculiar velocity by correcting for the CMB dipole: all SN redshifts have been transformed into the comoving frame of the CMB. This is particularly important for the SDSS-II, because the equatorial stripe comes within 7⁰ of the CMB dipole direction, and the redshift correction from the heliocentric frame is negative along the entire stripe. We use $(1+z) = (1+z_{\odot })/(1-\vec{v}_0\cdot \hat{n})$, where z_☉ is the heliocentric redshift as described in Section 3.1, $\hat{n}$ is the unit vector pointing from the Earth to the SN, and $\vec{v}_0$ is the CMB velocity vector, with a magnitude of 371 km s⁻¹ and direction given by Galactic coordinates l = 264.14⁰, b = +48.26⁰ (Fixen et al. 1996). The CMB-frame redshifts for the non-SDSS-II samples are taken from the literature.

In transforming to the CMB frame and making no other corrections for velocity correlations, we are implicitly assuming that the peculiar velocities of the host galaxies are uncorrelated with that of the Milky Way and approximately uncorrelated with each other. That assumption is a good approximation for the SDSS-II and higher-redshift SN samples, which cover large spatial volumes and are distant from the Milky Way. It is not a good approximation for the Nearby SN sample on both theoretical (Hui & Greene 2006; Cooray & Caldwell 2006) and observational (Neill et al. 2007) grounds. Not including velocity correlations means that the low-redshift SNe are overweighted in the χ²_μ statistic and that SN-derived cosmological parameter errors in the literature have been underestimated. However, inclusion of the SDSS-II SN sample significantly reduces the impact of the uncertainties due to low-redshift peculiar velocities.

In the first stage of the salt-ii analysis framework, the photometry for each SN light curve is fitted to an empirical model (Section 5.2) to determine a shape-luminosity parameter (x₁), a color parameter (c), and an overall flux normalization (x₀). For the salt-ii Hubble diagram analysis, the reference B-band magnitude is m*_B = −2.5log [x₀∫dλ'M₀(t, λ')T^B(λ')]. The flux integral is the same as Equation (7) with z = c = x₁ = 0 and f = B. The fitted parameter x₀ depends on the luminosity distance and the SN brightness.

To estimate cosmological parameters, the above parameters are related to the distance modulus for each SN by the expression

$$\begin{equation} \mu _i = {m_B^*}_i - M + {\alpha } \cdot x_{1,i} - {\beta } \cdot c_i. \end{equation} \tag{ 29 }$$

The global parameters M, α, and β describe the SN Ia population and are estimated simultaneously with the cosmological parameters by carrying out a χ² minimization using an expression analogous to Equation (26). The minimization and error estimation are performed using the program minuit.⁴² The expression for the distance-modulus uncertainty, σ_μ, is similar to that in Equation (27) for mlcs2k2, and the redshift uncertainty is treated identically. The intrinsic dispersion (σ^int_μ) is determined separately for each sample combination by setting the global best-fit χ² for that sample combination to unity, in contrast to the mlcs2k2 method for which we determine σ^int_μ solely from the Nearby sample. In addition, the salt-ii expression for χ²_μ includes a covariance matrix to account for correlations between the parameters x₁, c, and x₀ (Section 9.2). In determining cosmological parameters, the Hubble parameter is marginalized over, but the parameters α and β are not.

If the color corrections were due only to extinction by host-galaxy dust, the salt-ii parameter β would be equal to the Cardelli et al. (1989) extinction parameter, R_B (Conley et al. 2007). Further, if host-galaxy extinction were similar to the mean extinction in the Milky Way, one would expect β ≃ R_B = R_V + 1 ≃ 4. However, in salt-ii, β is left as a free parameter that is determined in the global fit to the Hubble diagram.

To account for selection bias, we have applied the salt-ii fitting method to simulated sample combinations (a)–(f). The Monte Carlo simulations are generated from the salt-ii model using parameters based on our analysis of the data (Section 10.2). In particular, α = 0.13 and β = 2.56, and the simulated distributions of c and x₁ are described by Gaussians with $\bar{c} = 0.04$, σ_c = 0.13, $\bar{x_1}=-0.13$, $\sigma _{x_1} = 1.24$. These distributions were estimated directly from the light-curve fits rather than from simulating underlying c and x₁ distributions such that the observed and simulated distributions match. The intrinsic luminosity is randomly varied using a coherent luminosity smearing factor of 0.12 mag, and the spectroscopic efficiency is based on the salt-ii parameters described in Section 6.2.1.

To isolate a potential salt-ii bias from a shift due to the CMB and BAO priors, the simulated SN sample is generated with the cosmological parameter values obtained from fitting without any SNe, w⁰ = −0.80 and Ω⁰_M = 0.30. The bias in the cosmological parameters is estimated to be w − w⁰ and Ω_M − Ω⁰_M, where w and Ω_M are obtained from the analysis of the simulated SN sample combined with the BAO+CMB priors. To increase the significance of the measured bias, 500 data-sized samples were simulated and analyzed; the resulting statistical uncertainty on the w-bias is typically 0.004. The average bias for each sample combination and cosmological model is shown in Table 5. For the results shown below in Tables 16 and 17, these bias corrections have been added to the cosmological parameters.

Rest-frame U band. As discussed in detail in Section 10.1.3, there are systematic discrepancies between the mlcs2k2 rest-frame U-band model and the light-curve data for all but the Nearby SN Ia sample. We have therefore carried out a test in which the observer-frame filter corresponding to rest-frame U band is excluded from the light-curve fits and the resulting distance estimates. Figure 30 shows that for the SDSS-only sample, the exclusion of the U band causes a mean systematic shift in estimated distance modulus of 0.12 ± 0.02 mag for z > 0.21. The resulting tilt in the Hubble diagram leads to a systematic change of −0.310 in the equation of state parameter w for both the SDSS-only (a) and Nearby+SDSS (c) sample combinations; we include this shift as an asymmetric systematic uncertainty, as indicated by the minus signs in the first row of Tables 6 and 7.

For the other sample combinations ((b), (d), (e), (f)), which include the ESSENCE and SNLS samples, the exclusion of the U band results in w-shifts of 0.04–0.08. Based on tests with simulations, we cannot distinguish these shifts from random fluctuations; we therefore include the largest shift, δw = 0.08, as a symmetric systematic uncertainty for these four sample combinations.

Minimum redshift for Nearby SN sample. As discussed in Section 4, the choice of minimum redshift z_min for the Nearby sample is complicated by the so-called Hubble bubble, a jump in estimated SN Ia distance modulus around z ∼ 0.025 within the Nearby sample (JRK07). This shift is shown in the residual Hubble diagram in Figure 20, which compares the fitted SN distance modulus in redshift bins, μ_fit(z), with the calculated distance modulus (μ_calc(z)) for a cosmological model with w = −1 and Ω_M = 0.3. Since μ_calc(z) is a smooth function of redshift, the jump between the two leftmost points in the plot reflects a ≃0.12 mag discontinuity in SN distance modulus (μ_fit(z)) at z ≃ 0.025. The significance of the shift in μ between SNe at z < 0.025 and z > 0.025 is 2.4σ. JRK07 found a comparable shift, corresponding to 0.14 mag. To further explore this issue, we also show the same residuals for the lower-redshift portion of the SDSS-II sample. The fitted SDSS-II SN distances are consistent with those of the z > 0.025 subset of the Nearby SNe Ia and less consistent with the z < 0.025 subset (∼2.5σ discrepancy).

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The Hubble bubble may be a real feature due to a local, large-scale void, an artifact of selection biases in the Nearby SN sample, or an artifact of the light-curve fitter assumptions about host-galaxy dust and color variations. Regardless of its interpretation, this feature has been used as an argument to discard SN Ia measurements at z < 0.025 from cosmological fits (e.g., Riess et al. 2007). We have decided to make a more agnostic choice for z_min. We carry out cosmological fits using two sample combinations, Nearby+SDSS ((c)) and Nearby+SDSS+ESSENCE+SNLS (d), and vary z_min from 0.015 to 0.03; for mlcs2k2, the resulting variation in w is shown in the first two panels of Figure 21. For both sample combinations, the approximate midpoint of the w-variation occurs at z_min = 0.02, which we therefore take as the nominal choice. As z_min is varied around this value, the w-variations are approximately ±0.060 for sample combination (c) and ±0.040 for (d); we include these variations as systematic uncertainties in Table 6, with associated results for Ω_M in Table 7.

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mlcs2k2 SN Ia model parameters. The mlcs2k2 model vectors M, p, and q in Equation (1) were determined by training the light-curve fitter on a Nearby SN Ia sample, as described in JRK07. As discussed in Section 5.1, our analysis uses a set of model vectors that includes the adjustments to the $M^{e,f^{\prime }}$ values given in Equation (2). To estimate the sensitivity of the results to these adjustments, we have also carried out the cosmology analysis using the mlcs2k2 model vectors without those adjustments. These two sets of model vectors result in values for w that differ by δw = 0.01–0.04, depending on the sample combination.

Galactic extinction. The wavelength dependence of the Milky Way Galactic extinction is expressed as A(λ) = A_V × (a(λ) + b(λ)/R_V), where R_V = 3.1, A_V = R_V × E(B − V), E(B − V) is the color excess, and the functions a(λ) and b(λ) are defined by Cardelli et al. (1989). To estimate the systematic uncertainty for each SN, we coherently decrease the color excess by 0.01 + 0.16 × E(B − V) mag (but requiring non-negative color excess) relative to the nominal value in Schlegel et al. (1998). The color-excess uncertainty of 0.01 mag is based on optical-versus-IR discrepancies (Burstein 2003). The resulting w-uncertainties are 0.01–0.02. Note that we have not varied the functions a(λ) and b(λ), and therefore this uncertainty may be underestimated. The mean Galactic extinctions in the r band are 0.20 (Nearby), 0.14 (SDSS), 0.07 (ESSENCE), 0.05 (SNLS) and 0.01 mag (HST).

Dust parameter R_V. As described in Section 7.1, the SDSS dust sample was used to determine the dust reddening parameter, R_V = 2.18 ± 0.50 (Section 7.2), and the extinction distribution exponential slope, τ_V = 0.334 ± 0.088 (Section 7.1). Propagating the correlated uncertainties on R_V and τ_V, the uncertainty on w is δw = 0.036 for the SDSS-only sample. For the combined data samples, δw ≲ 0.03. This color-related uncertainty appears rather small, considering that the issue of color variations has been a major source of uncertainty in previous studies such as WV07.

Simulated spectroscopic efficiency. The simulated efficiency is part of the mlcs2k2 fitting prior, as indicated by the terms _search and _cuts in Equation (5). The main effect of an error in determining the spectroscopic efficiency is to introduce a bias in the mlcs2k2 prior (Equation (5)). As discussed in Section 6, the largest uncertainty in simulating the efficiency is related to modeling of the incompleteness of the spectroscopic observations, _spec.

For the Nearby, SDSS, and SNLS samples, _spec depends mainly on the ${\mathcal M}_{\rm dim}$ component, i.e., $A_{\mathcal M}\simeq 1$ in Section 6.2.1. We explore the systematic error due to spectroscopic efficiency modeling for these samples by repeating the analysis with _spec set to unity, an extreme variation from the fiducial efficiency model of Section 6.2.1. Since the ESSENCE sample favors a purely redshift-dependent _spec ($A_{\mathcal M}= 0$ in Equation (14)), setting _spec = 1 has no impact in that case; we instead evaluate the systematic error for that sample by replacing the purely z-dependent _spec with a purely ${\mathcal M}_{\rm dim}$-dependent _spec, i.e., by setting $A_{\mathcal M}= 1$. In these tests, the efficiency due to photometric selection cuts (_cuts) is included in the fitting prior as well as the image-subtraction efficiency (_subtr) for the SDSS sample.

For the above systematic changes in _spec, the corresponding changes in distance modulus versus redshift are shown in Figure 22 for both the data and the Monte Carlo simulations. One clearly sees that incorporating a non-trivial spectroscopic efficiency model has a significant systematic impact on distance estimates for the more distant SNe in each sample. Moreover, the data-simulation agreement is good, adding confidence in our implementation of the spectroscopic efficiency model in the fitting prior.

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From Figure 9, there is no doubt that _spec is significantly less than one and that our model for _spec is closer to reality than the extreme assumption of setting _spec = 1. To be conservative, we use half of the absolute-value difference in w between the fiducial and extreme efficiency models as our estimate of the systematic error associated with uncertainty in the efficiency model. The largest uncertainty related to the simulated efficiency occurs for sample combinations (a) and (c); the w-uncertainties (half the shifts) are δw = 0.062 for the SDSS-only sample and δw = 0.072 for the Nearby+SDSS sample combination (c). We also note that including versus not including the simulated efficiency in the Nearby SN sample changes w by a few hundredths for the other sample combinations that include the Nearby sample, half of which is included as a systematic uncertainty.

Calibration of the primary reference star, BD+17. As discussed in Appendix B, the UBVRI magnitudes of the primary spectrophotometric reference star, BD+17, are used in the K-corrections to relate the flux calibration of the Nearby SN sample to that of the higher-redshift samples. To evaluate the uncertainty in the BD+17 magnitudes, we compare the consistency of the synthetic magnitudes of the HST spectra with the SDSS photometric magnitudes. We first compute u, g, r, i, z zero-point offsets as the difference between synthetic BD+17 magnitudes computed from the HST-measured spectrum (Bohlin 2007) and the magnitudes measured by Landolt (2007). These offsets are then compared to those based on three solar analogs (P3330E, P177D, P041C). The differences between the zero-points offsets are −0.004, −0.013, 0.005, 0.010, 0.012 mag for u, g, r, i, z, respectively. We therefore assume a 1% uncertainty in each of the U, B, V, R, I magnitudes for BD+17. To propagate this error, we change each BD+17 magnitude by 1% independently in each passband; for each change, all of the SN light curves are re-fitted with the mlcs2k2 model and cosmological parameters are extracted. The resulting five independent w-shifts are then added in quadrature and included as a systematic uncertainty. The resulting w-uncertainties are 0.02–0.03.

As a crosscheck, we have replaced BD+17 with Vega as the primary reference star. The resulting changes in w are consistent with the changes from applying 1% shifts in the U, B, V, R, I magnitudes of BD+17. The differences between BD+17 and Vega are therefore not included as a systematic uncertainty.

Landolt–Bessell color transformations. As discussed in Appendix B, we use color transformations to transform between the Landolt network and the filter system defined by Bessell (1990). Table 20 shows the color terms (k_i) between Landolt magnitudes and synthetic magnitudes using the Bessell (1990) filters. We vary each color term independently by one standard deviation, and the corresponding changes in w are added in quadrature. The resulting change is δw < 0.01 for all sample combinations.

Shifted Bessell filters instead of color transformation. As an alternative to the Landolt–Bessell color transformation, we consider the approach of Astier et al. (2006), using a modified set of UBVRI filter responses in which the shapes of the Bessell (1990) response curves are held fixed, but the central wavelength of each filter passband is shifted so that the color terms (Equation (B2)) are zero. The corresponding wavelength shifts are given in Table 21 under the column "HST standards." Using these wavelength shifts and no color transformations in place of the color transformation method of Appendix B, w shifts by δw = 0.007 for the SDSS-only sample. For the other sample combinations, δw ∼ 0.01.

SDSS AB offsets. As discussed in Section 3.2, the uncertainties in the SDSS AB offsets are 0.009, 0.009, 0.009 mag for g, r, i, respectively (see Table 1). These uncertainties include zero-point uncertainties of 0.004, 0.004, 0.007 mag based solely on the consistency of the solar analogs and also account for uncertainties in the central wavelengths of the SDSS filters: δλ ≃ 7, 16, 25 Å for g, r, i, respectively. We independently vary each AB offset by one standard deviation, and the resulting changes in w are added in quadrature. The resulting change in w is a few hundredths for the sample combinations that include SDSS-II SNe. Note that the w-uncertainty for SDSS-only (δw = 0.004) is much smaller than that for the Nearby+SDSS combination (δw = 0.030). For SDSS-only, the effect of changing the AB offsets is mainly to shift all of the distance moduli by the same amount; the corresponding change in cosmological parameters is small. For Nearby+SDSS, the SDSS-II distances are shifted relative to those of the Nearby sample, and the abrupt feature in the Hubble diagram results in a larger change in the cosmological parameters.

ESSENCE R −I color zero point. WV07 report an uncertainty of 0.02 mag in their R − I color zero point, resulting in a systematic uncertainty of δw = 0.04 for their analysis of the Nearby+ESSENCE data sets. As a crosscheck, we propagate their R − I uncertainty into our analysis and find δw = 0.05 for the same Nearby+ESSENCE combination, in reasonable agreement with WV07. We apply their R − I uncertainty to our sample combinations and find that the uncertainty in w is much smaller, δw ∼ 0.01 for the combinations ((b), (d), (e), (f)) that include the ESSENCE sample. Note that these sample combinations also include the SNLS sample, suggesting that the impact of the ESSENCE R − I uncertainty is reduced by the presence of another high-redshift data sample.

SNLS g, r, i, z zero points. Astier et al. (2006) reported zero-point uncertainties of 0.01, 0.01, 0.01, 0.03 for their g, r, i, z passbands, respectively. In their analysis of the Nearby+SNLS combination, they vary each zero point independently and add the corresponding w-shifts in quadrature: the resulting systematic is δw = 0.05. As a crosscheck, we propagate their zero-point uncertainties into our analysis and also find δw = 0.05 for the same Nearby+SNLS combination. We apply their zero-point uncertainties to our sample combinations and find slightly smaller uncertainties for the combinations ((b), (d), (e), (f)) that include the SNLS sample.

HST zero points. The HST zero-point uncertainties are 0.02 mag for the F110W and F160W filters⁴³ (mean wavelengths are 1.14 μm and 1.61 μm, respectively), and 0.01 mag for the optical filters.⁴⁴ For the analysis of the five-sample combination that includes HST (e), the resulting w-uncertainty is δw = 0.012.

The cosmological parameter shifts due to all of these sources of systematic error are added in quadrature to derive total systematic-error estimates δw and δΩ_M for the mlcs2k2 analysis of the FwCDM model; these are given in Tables 6 and 7.

To examine systematic uncertainties in the context of the salt-ii model, we undertake an analysis similar to that carried out for mlcs2k2. We also determine systematic uncertainties for the salt-ii parameters α and β that enter Equation (29). Since the salt-ii training software is not available for public use, with one exception we make approximations in cases where re-training of the spectral surfaces is needed. In such cases we either use the nominal salt-ii surfaces and propagate changes only in the light-curve fits, or we use the uncertainties based on the mlcs2k2 analysis. The systematic uncertainties in w and Ω_M for the different sample combinations are given in Tables 8 and 9.

Rest-frame U band. As discussed in detail in Section 10.2.4, there are systematic discrepancies between the salt-ii rest-frame U-band model and the observer-frame U-band light-curve data for the Nearby SN Ia sample. As was noted in Section 9.1, a related issue is seen for mlcs2k2. We carry out a test of the salt-ii fits in which the observer-frame filter corresponding to the rest-frame U band is excluded from the fits. Figure 40 shows that for the SDSS-only sample, the exclusion of the U band results in a redshift-dependent change in the distance modulus. This results in a w-shift of −0.100 for the SDSS-only sample and −0.133 for the Nearby+SDSS combination; these shifts are included as asymmetric systematic uncertainties.

For the other sample combinations, which include the higher-redshift ESSENCE and SNLS samples ((b),(d),(e),(f)), the exclusion of the rest-frame U band results in w-shifts of 0.04–0.09. Based on tests with simulations, we cannot distinguish these shifts from random fluctuations; we therefore include the largest shift, δw = 0.09, as a symmetric systematic uncertainty for these four sample combinations. If the observer-frame U band is excluded from the Nearby sample, while the rest-frame U band is included in the higher-redshift samples, the change in w is no more than 0.01.

Minimum redshift for Nearby SN sample. The dependence of the salt-ii results for w on the z_min cut is shown in the third and fourth panels of Figure 21. For z_min ≳ 0.018, the inferred value of w is fairly insensitive to the value of z_min. However, when z_min is reduced to 0.015, w changes by almost 0.09 for the Nearby+SDSS combination, and by 0.04–0.05 for the combinations that include the higher-redshift (ESSENCE and SNLS) samples. To account for these variations, we have assigned a systematic uncertainty of δw = 0.05 for the Nearby+SDSS combination and δw = 0.03 for the other sample combinations that include the Nearby SN Ia sample.

Galactic extinction. The systematic uncertainty from Galactic extinction is described in Section 9.1, resulting in δw ∼ 0.02.

salt-ii training with SDSS-II data. Here we examine the salt-ii training process that produces the spectral surfaces M₀(t, λ) and M₁(t, λ) in Equation (6). Because SDSS-II SN probes a relatively unexplored range of SN redshifts, the rest-frame behavior of the SDSS-II light curves may not be as well described by the salt-ii model as that of other SN samples that were used in the training of the model. To quantify this issue, J. Guy has retrained the salt-ii spectral surfaces twice, first including the light curves of the SDSS spectroscopic sample and second including those of the SDSS dust sample (Section 7.1). Evaluating cosmological parameters obtained with each retrained set of spectral surfaces and comparing the results with those from the standard salt-ii training, we include the larger of the two w-shifts as a systematic uncertainty. For the SDSS-only and Nearby+SDSS ((a) and (c)) sample combinations, the uncertainty is δw ∼ 0.02. For the other combinations, δw ∼ 0.01.

salt-ii dispersions. Recall from Section 5.2 that the spectral surfaces, M₀(t, λ) and M₁(t, λ), were retrained using the Bessell filter shifts based on HST standards (Table 21). The model dispersions around these surfaces, however, were not determined in the retraining, and we therefore use the model dispersions from Guy et al. (2007). To allow for the resulting uncertainty in the dispersions, we assign a systematic uncertainty of half the difference between using and ignoring the dispersions. The resulting uncertainties are δw ∼ 0.01–0.02 for combinations that include the higher-redshift ESSENCE and SNLS samples. For the SDSS-only and Nearby+SDSS sample combinations, the w-uncertainty is negligible.

β-variation with redshift. If the salt-ii SN parameters (α, β, M) are allowed to vary independently in redshift bins, while the cosmological parameters are fixed, we find a strong redshift dependence of β for z > 0.6 (see Section 10.2.3). To estimate the corresponding systematic uncertainty, the Hubble diagram fits have been redone allowing α, β, and M to vary with redshift using a simple model in which each SN parameter is constant for z < 0.6, and is then allowed to vary linearly with redshift for z > 0.6. Compared to the nominal salt-ii model with redshift-independent parameters, the largest change, δw = +0.073, occurs for SDSS+ESSENCE+SNLS (b) in which the Nearby sample is excluded. For sample combinations (d) and (f), δw ∼ 0.04, and for the full SN set (e) that includes the HST, δw ∼ 0.01. These w-shifts are included as asymmetric systematic uncertainties.

Simulated bias correction. For salt-ii, we have determined bias corrections from simulations, as described in Section 8.2 (see Table 5). We include half the w-shift as a systematic uncertainty. The largest uncertainty is δw = 0.020 for SDSS-only.

Calibration of primary reference star, Vega. We assume uncertainties of 0.01 mag in the calibration of U, B, V, R, I for the primary reference, Vega. Since a full accounting of this effect would require another retraining of the salt-ii surfaces, we instead adopt the uncertainties derived from the mlcs2k2 analysis (Table 6). In Astier et al. (2006), the corresponding uncertainty is δw = 0.024 for the Nearby+SNLS combination; as a crosscheck, we have evaluated this uncertainty for the same sample combination and find good agreement, δw ≃ 0.021. For the sample combinations analyzed here, the resulting w-uncertainties are 0.02–0.03.

Calibration: shifted Bessell filters for Nearby data. As discussed in Section 5.2, we use the Bessell (1990) filter responses with wavelength shifts given in Table 21 of Appendix B. Since these shifts differ from those in Astier et al. (2006), we use the difference in cosmological results derived from both sets of wavelength shifts to define an additional systematic uncertainty on w. This uncertainty is δw ∼ 0.02 for sample combinations that include the Nearby sample.

Calibration: zero-point offsets for SDSS, ESSENCE, SNLS, HST. Zero-point uncertainties for the SDSS, ESSENCE, SNLS, and HST bandpasses are propagated in the same manner as for the mlcs2k2 method (Section 9.1). The SNLS zero points are varied in the fit, but not in the training of the spectral surface, and therefore these w-uncertainties might be slightly overestimated. The other survey samples were not used in the training, and therefore varying the zero points in the fit is sufficient to estimate the systematic error. Note that the w-uncertainty from the SDSS AB offsets is smaller for SDSS-only than for Nearby+SDSS, as explained in Section 9.1.

The parameter shifts due to all of these systematic errors are added in quadrature to derive the total systematic-error estimates in Tables 8 and 9.

Using the mlcs2k2 method, we present cosmological results for the six sample combinations (a)–(f) of Table 4. Table 10 shows the spectroscopically determined redshift and marginalized mlcs2k2 fit parameters for SNe that pass the selection cuts described in Section 4. We use the ensemble of redshifts z_i and estimated distance moduli μ_i for each sample combination to fit cosmological model parameters, as explained in Section 8.1.

10.1.1. Goodness of Fit and Hubble Scatter

10.1.2. mlcs2k2 Hubble Diagrams and Cosmological Parameters

Figure 24 shows the differences between the estimated SN distance moduli μ_i and those for an open CDM model with no dark energy (Ω_M = 0.3, Ω_DE = 0) as a function of redshift, for each of the six SN sample combinations; the large square (pink) points show weighted averages of these residuals in redshift bins. Also shown is the Hubble distance-residual curve between the best-fit FwCDM model for each sample combination (including the BAO+CMB prior) and that for the open model (solid curves). In each panel, the Hubble parameter for the open CDM model has been adjusted to agree with that for the best-fit FwCDM model, so that the FwCDM versus open CDM residuals vanish at z = 0. Since the Hubble parameter is not determined in this analysis, a constant vertical offset in Figure 24 is irrelevant: what is significant are the slope and curvature of the points and the best-fit (solid) curves versus redshift. Figure 25 shows the distributions of normalized residuals, (μ_i − μ_FwCDM)/σ_μ, where μ_FwCDM is the distance modulus from the best-fit FwCDM model for sample combination (e) including the BAO+CMB prior, and σ_μ is the total uncertainty defined in Equation (27). The bulk of the distribution of all 288 normalized residuals (the upper left panel of Figure 25) is well fitted by a Gaussian with σ = 0.77; outliers increase the rms to 0.90. For the Nearby sample, the rms is one as expected, because σ^int_μ = 0.16 mag is determined such that χ²_μ/N_dof = 1.

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Figures 26 and 27 show the mlcs2k2 statistical-uncertainty contours for the FwCDM and ΛCDM models: for each of the six sample combinations, the SN Ia, BAO, and CMB contours are shown along with the combined constraints. For the combined SN+BAO+CMB results, we include systematic uncertainties to derive total (statistical plus systematic) uncertainty contours, as explained in Appendix F, with results shown in Figures 28 and 29 for the FwCDM and ΛCDM models (note the zoomed axis scales compared to the previous figures). The best-fit cosmological parameter values and uncertainties, including the BAO and CMB priors, are given in Tables 12 and 13 for the FwCDM and ΛCDM models. The distance modulus versus redshift curve for the best-fit FwCDM cosmological parameter values (relative to that of an open CDM model) are shown as the solid curves in the Hubble residual plots in Figure 24.

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Table 12. For the FwCDM Model, Constraints on w and Ω_M from mlcs2k2 SN Distances Combined with SDSS BAO and WMAP-5 CMB Results

Parameter	Result for Sample Combination
	(a)	(b)	(c)	(d)	(e)	(f)
χ2μ	170.9	406.7	279.5	517.8	568.1	341.9
Ndof	102	220	135	253	287	150
rmsμ	0.15	0.20	0.16	0.20	0.21	0.23
w	−0.84	−0.71	−0.92	−0.76	−0.76	−0.78
σw(stat)	0.15	0.09	0.13	0.08	0.07	0.08
σw(syst)	+0.08−0.32	0.11	+0.10−0.33	0.11	0.11	0.12
σw(tot)	+0.17−0.35	0.14	+0.16−0.35	0.14	0.13	0.14
ΩM	0.289	0.319	0.273	0.306	0.307	0.302
$\sigma _{\Omega _{\rm M}}({\rm stat})$	0.033	0.025	0.028	0.021	0.019	0.022
$\sigma _{\Omega _{\rm M}}({\rm syst})$	+0.019−0.054	0.027	+0.023−0.056	0.026	0.023	0.026
$\sigma _{\Omega _{\rm M}}({\rm tot})$	+0.038−0.064	0.036	+0.036−0.062	0.033	0.030	0.034

Notes. ^aSDSS-only. ^bSDSS+ESSENCE+SNLS. ^cNearby+SDSS. ^dNearby+SDSS+ESSENCE+SNLS. ^eNearby+SDSS+ESSENCE+SNLS+HST. ^fNearby+ESSENCE+SNLS.

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Table 13. For the ΛCDM Model, Constraints on Ω_M and Ω_Λ from mlcs2k2 SN Distances Combined with SDSS BAO and WMAP-5 CMB Results

Parameter	Result for Sample Combination
	(a)	(b)	(c)	(d)	(e)	(f)
χ2μ	55.2	171.7	87.1	203.9	237.9	145.8
Ndof	102	220	135	253	287	150
rmsμ	0.15	0.20	0.16	0.20	0.21	0.23
ΩΛ	0.735	0.714	0.735	0.713	0.705	0.718
$\sigma _{\Omega _{\Lambda }}(\mbox{stat})$	0.019	0.019	0.019	0.019	0.018	0.019
$\sigma _{\Omega _{\Lambda }}(\mbox{syst})$	0.000	0.000	0.006	0.004	0.004	0.004
$\sigma _{\Omega _{\Lambda }}(\mbox{tot})$	0.019	0.019	0.019	0.019	0.018	0.019
ΩM	0.274	0.300	0.274	0.302	0.312	0.294
$\sigma _{\Omega _{\rm M}}({\rm stat})$	0.023	0.023	0.023	0.023	0.022	0.023
$\sigma _{\Omega _{\rm M}}({\rm syst})$	0.000	0.000	0.001	0.001	0.001	0.001
$\sigma _{\Omega _{\rm M}}({\rm tot})$	0.023	0.023	0.023	0.023	0.022	0.024

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Among the six SN sample combinations, the best-fit values of the dark energy equation of state parameter w fall roughly into two groups. In the first group, the highest-redshift sample is from SDSS-II: for the SDSS-only sample (a) and Nearby+SDSS sample combination (c), w = −0.84 and −0.92. The agreement between these values is consistent with the expected rms spread of 0.07 based on simulations. The second group comprises the other four sample combinations, which include the higher-redshift ESSENCE and SNLS samples: w = −0.71, − 0.76, − 0.76, − 0.78 for sample combinations (b), (d), (e), (f), respectively. Simulations predict an rms spread in w of ∼0.1 between the results from these two groups; the observed difference is therefore not statistically significant but may nevertheless be an indicator of systematic effects.

Table 12 also shows that the statistical and systematic errors in w and Ω_M for sample combinations (b) and (f) are very similar. Since these two sample combinations differ only in the substitution of the Nearby SN Ia sample with the SDSS-II sample, this indicates that the first-season SDSS-II SN sample anchors the Hubble diagram with comparable constraining power to the Nearby sample.

Using the Nearby+SDSS+ESSENCE+SNLS+HST sample combination (e), which covers the widest redshift range, we obtain w = −0.76 ± 0.07(stat) ± 0.11(syst) and Ω_M = 0.307 ± 0.019(stat) ± 0.023(syst). Although this value for w is higher than that obtained from other recent SN measurements, we stress that the difference is not due to inclusion of the SDSS-II SN data: as Table 12 shows, we infer nearly identical parameter values for sample combination (f), which excludes the SDSS-II data. By contrast, WV07 inferred w = −1.07 ± 0.09(stat) ± 0.13(syst), Ω_M = 0.267^+0.028_−0.018(stat) using mlcs2k2 for a sample combination nearly identical to (f) and including the BAO (but not CMB) constraints. In Section 10.1.4, we trace the differences between the WV07 result and ours to changes in mlcs2k2 model parameters and assumptions.

For the ΛCDM model, in comparison with the FwCDM model, the SN results carry less weight relative to the combined BAO and CMB results in constraining the parameters. In particular, Figure 27 and Table 13 show that the SDSS-only (a) and Nearby+SDSS (c) SN samples have almost no impact on the maximum likelihood parameter values and uncertainties. For the other four SN sample combinations, there is some tension between the SN and BAO results: the SN results pull the maximum likelihood parameter values along the CMB contour, away from the BAO contours. Since the BAO contours are in all cases narrower than those for the SNe, however, this shift is small, corresponding to δΩ_DE ≃ −0.03, δΩ_M ≃ 0.04 or less.

10.1.3. U-Band Anomaly with mlcs2k2

As noted in Section 9.1, the largest single contribution to the systematic-error budget comes from consideration of the rest-frame U band. Within the mlcs2k2 framework, the issue is manifest as a difference between the Nearby sample (JRK07) and the other SN Ia samples. We have carried out a series of tests in which the observer-frame passband corresponding to rest-frame U band is excluded from the mlcs2k2 light-curve fits. We compare the resulting distance-modulus estimates, μ_noU, with those for which rest-frame U band is included, μ, and define Δμ_noU, ≡ μ_noU − μ. If the mlcs2k2 model is a good description of the data, we would expect Δμ_noU, to scatter about zero.

The left panel of Figure 30 shows the resulting change in the SDSS-II SN Hubble diagram, Δμ_noU, versus redshift. For z > 0.21, observer-frame g corresponds to rest-frame U; excluding g band in this redshift range results in a shift in the distance modulus of Δμ_noU, = (0.12 ± 0.02) mag. In the redshift interval 0.21 < z < 0.285, the remaining observer-frame SDSS passbands r and i correspond to rest-frame V and R; for z > 0.285, they correspond to B and V. In both of those redshift intervals, we see a distance-modulus shift of Δμ_noU, ≃ 0.1–0.15 mag. For z < 0.21, the gri filters do not map onto rest-frame U band, therefore Δμ_noU, ≡ 0 in this redshift range. Since Δμ_noU, is determined from strongly correlated samples, i.e., the r- and i-band data are the same in fits with and without U band, its uncertainty is estimated to be ${\rm rms}/\sqrt{N}$, where rms is the root mean square and N is the number of SNe Ia. Applying the same U-exclusion test to simulations shows that the observed shift has a significance of ∼6σ, consistent with our estimate of the uncertainty. The large tilt in the Hubble diagram that results from excluding g band at z > 0.21 results in a shift of δw ∼ 0.3 for the SDSS-only and Nearby+SDSS sample combinations, by far the dominant contribution to the systematic-error budget for those samples.

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We further investigate the U-band anomaly by comparing the fitted distance moduli with and without rest-frame U for three subsamples: (i) observer-frame UBV versus BV for 17 Nearby SNe Ia; (ii) observer-frame ugr versus gr for nine low-redshift SDSS-II SNe for which the u-band signal to noise is sufficient (z < 0.1), and (iii) observer-frame gri versus ri for 13 SNLS SNe with 0.21 < z < 0.5 and that have at least one g-band measurement within ±10 days of maximum brightness. In each case, the subsamples are chosen such that the light curves pass the selection criteria of Section 4 and include three observer passbands, one of which maps onto rest-frame U band. The differences in distance modulus (Δμ_noU,) between the two- and three-band fits (without and with rest-frame U) are shown in the right panel of Figure 30. The SDSS-II and SNLS subsamples show a consistent shift of about 0.1 mag. For comparison, the right panel of Figure 30 also shows the average shift for the points in the left-panel test at z > 0.21, again showing consistency. By contrast, the Nearby subsample is consistent with no shift or a slightly negative shift. Since the mlcs2k2 model is trained on a superset of the Nearby SN Ia data, we would expect no significant shift for the Nearby subsample. The redshift range z < 0.1 in the left panel of Figure 30 is not the same as the SDSS-ugr (z < 0.1) test in the right panel: the former is based on observer-frame gri and does not map onto rest-frame U band, while the latter is based on ugr versus gr in order to test excluding rest-frame U band.

Since the U band is particularly sensitive to host-galaxy extinction, it is worth exploring whether this anomaly might be an artifact of the assumed extinction law. We have repeated the test above, replacing the CCM89 color law nominally used in mlcs2k2 with the empirically determined salt-ii color law, CL(λ), in Equation (6). The salt-ii color law results in a U-band extinction that is 0.07 mag larger than that from using CCM89 with R_V = 2.18 and the mean extinction value, A_V = 0.35. The differences in the extinction for the other passbands are much smaller: 0.014, 0, 0.007, −0.002 for B, V, R, I. Repeating the U-band exclusion tests with the salt-ii color law results in distance-modulus offsets for the subsamples in the right panel of Figure 30 that are ∼20% smaller than for the nominal test using the CCM89 color law.

Although excluding rest-frame U band reveals a problem, it does not definitively indicate that rest-frame U, as opposed to one of the other passbands, is the source of the problem. To study this in more detail, we carry out a similar test in which observations in passbands corresponding to rest-frame B band are excluded. For SDSS-II SNe with redshifts z < 0.21, this B-exclusion test corresponds to comparing distance moduli from observer-frame gri (rest-frame BVR) with those from just r and i (rest-frame VR). The difference in the average distance modulus, μ_VR − μ_BVR, is −0.01 ± 0.02 mag, consistent with no shift. This test suggests that rest-frame B is not the source of the anomaly and strengthens the circumstantial evidence that rest-frame U is the source of the anomaly.

To further diagnose the U-band anomaly, we compare the light-curve data and the mlcs2k2 model light-curve fits as a function of epoch for the different rest-frame passbands and the five different SN samples. Figure 5 shows the data-model residuals for the nominal fits. Figure 31 shows the residuals when the filter corresponding to rest-frame U band is excluded from the fit; to see the U-band residuals, we must use the nominal mlcs2k2 model parameters that include the U band in the training. When the U band is excluded for the Nearby SNe Ia, there is a negligible change in the U-band residuals; this is expected because mlcs2k2 is trained on the nearby data. For the SDSS-II, excluding U band from the fits results in a ∼0.05 mag shift in the U-band residuals near the time of peak brightness (T_rest = 0). The ESSENCE survey used only two passbands, R and I, and only a few of their SNe Ia probe rest-frame U band, so we cannot use this sample to probe the problem. For the SNLS sample, the residuals have been plotted for those SNe Ia that have observer-frame g-band measurements mapping into rest-frame U band (most of the SNe with z < 0.48), i.e., the same subset used in Figure 30. When the rest-frame U band is excluded, the corresponding shift in the U-band residuals is consistent with the shift seen in the SDSS-II. We have also examined the higher-redshift SNLS subset for which the observer-frame r band maps onto the rest-frame U band; the results of those tests are consistent with the tests based on the g band, but with much larger uncertainties. Note that the SNLS subset with an excluded g band has a spectroscopic efficiency _spec = 1 (see Section 6.2), while the SDSS-II spectroscopic efficiency is significantly smaller than 1. Although the corresponding mlcs2k2 priors are quite different, the U-band anomaly is consistent between them, indicating that it is not caused by errors in _spec.

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Since the mlcs2k2 model is trained with the Nearby sample, the U-band anomaly in the other SN Ia samples results in significant systematic uncertainties that limit the precision of cosmological parameters obtained with the current implementation of the mlcs2k2 method. There are a number of possible causes for the U-band discrepancy: (1) the SN Ia U-band flux could be redshift dependent; (2) selection effects for the Nearby sample could result in a U-band flux distribution that is not representative of the true SN Ia population; (3) there is a problem with the mlcs2k2 model; (4) the observer-frame U-band flux for the Nearby sample is not properly translated into the Landolt system; (5) the SN SED in the UV region is not adequately constrained, leading to errors in the K-corrections.

The first possibility, a redshift-dependent flux in the U band, is unlikely, given our test based on the ugr passbands for nine SDSS-II SN Ia with 0.04 < z < 0.09 (the right panel in Figure 30). Although this redshift range is still slightly higher than that for the Nearby sample $(\bar{z} \sim 0.03)$, a very rapid redshift evolution of SN Ia properties would be needed to account for the discrepancy. The second possibility is motivated by the very low selection efficiency for the Nearby sample, as indicated by the data-simulation comparison of the redshift distribution in the upper-left panel of Figure 9. However, the Nearby sample shows good agreement with the other SN Ia samples in the B, V, R passbands, so one would have to postulate a selection effect that biases the U band more than the other bands. The third possibility, of a problem with the light-curve model, is difficult to exclude, but the residuals in the Nearby sample ("Nearby" column of the left panel of Figure 5) look reasonable; this rules out obvious problems, thus narrowing potential problems to the extrapolation to higher redshifts and to different passbands.

The fourth possibility seems at first sight unlikely, since calibration errors are typically quoted at the level of 0.01–0.02 mag, but it could be that these errors have been underestimated. A mis-calibration of more than 0.1 mag in the U band would be needed to account for the observed anomaly. JRK07 reported that the light-curve residuals are 40% larger in the U band than in the other bands, but that does not necessarily point to a calibration problem. The U-band residuals for the Nearby sample (upper-left panel in Figure 5) vary with epoch as the SN becomes redder, suggesting a problem with the definition of the U-band filter.

The last possibility, a difference in the UV region of the SN SED, has been suggested in previous works (Foley et al. 2008; Sullivan et al. 2009). Both mlcs2k2 and salt-ii models assign larger uncertainties in the UV region compared to the redder bands. Ellis et al. (2008) found that maximum-light SN Ia SEDs varied significantly more at wavelengths λ < 4000 Å than for redder wavelengths and concluded that the additional dispersion is not due to extinction from host-galaxy dust.

Since the U-band anomaly appears in multiple samples (SDSS-II and SNLS) as well as in multiple redshift ranges for the SDSS-II sample (z < 0.1 and z > 0.21 in the right panel of Figure 30) we believe that the problem lies within the Nearby SN sample, i.e., with observations in the observer-frame U band. The most likely source of the problem is either the training procedure or the translation into the Landolt system. We will address this issue in the future by retraining light-curve models with SDSS-II SNe; the modeling of the rest-frame U band will make use of u-band measurements at z < 0.1 and g-band measurements at z > 0.21.

10.1.4. Comparison with WV07 Result

The Nearby+ESSENCE+SNLS sample combination (f) corresponds to one of the combinations analyzed by the ESSENCE collaboration in WV07 (with the exceptions of a different minimum redshift cut and different light-curve selection criteria). To compare with those results, we repeat the sample (f) analysis using the same BAO prior as in WV07, i.e., without the CMB prior. We find w = −0.75 ± 0.11(stat), which differs from the WV07 value by 0.32, or about 3σ_stat. Since the two analyses are based on the same data, the statistical significance of the discrepancy is much larger than 3σ_stat. If we add the systematic uncertainties in quadrature (δw_syst = 0.18), which is clearly an overestimate, the discrepancy is still fairly significant (1.8σ). Both analyses are based on the mlcs2k2 method, but our changes in mlcs2k2 parameters and priors result in systematic differences that are explained below.

We run the mlcs2k2 fitter in the "WV07 mode" (Section 5.1), using the WV07 parameter choices and reproducing their result for w, and then make cumulative changes in the analysis that evolve it toward the parameter inputs and light-curve fitting code used in our fiducial analysis. The resulting shifts in w relative to the WV07 value are shown in Figure 32. The changes from WV07 that we implement sequentially are change the minimum redshift for the Nearby sample from 0.015 to 0.02 (z > 0.02); set off-diagonal model covariances to zero (off-diag); implement K-correction improvements in item 2 of Section 5.1 (Kcor); replace the Bessell filter shifts introduced by Astier et al. (2006) and adopted by WV07 to the color transformation method used in our analysis and simultaneously change the primary standard from Vega to BD+17 (Calibration); use mlcs2k2 model parameters M, p, q (Equation (1)) corresponding to R_V = 2.18 (Section 7.2), but still use R_V = 3.1 and WV07 priors in the light-curve fits; use R_V = 2.18 with WV07 priors in the light-curve fits; use the A_V prior and efficiency from this analysis and remove the WV07 requirement that each observation has S/N >5 (prior).

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The largest source of change in w (∼0.25) results from our different assumptions about host-galaxy dust and the mlcs2k2 fitting prior. Our fitting prior is based on a measurement of R_V and of the A_V distribution using the SDSS dust sample (Section 7) along with a comprehensive model of survey efficiencies using Monte Carlo simulations. In contrast to our analysis, the fitting prior used in WV07 is based on the assumption that R_V = 3.1, that the A_V distribution is represented by the glos distribution, and that the spectroscopic targeting efficiency is unity.

Using the salt-ii method (Section 8.2), we present cosmological results for the six sample combinations (a)–(f) of Table 4. Table 14 gives the spectroscopic redshift and derived salt-ii fit parameters for each SN that passes the selection cuts of Section 4. Recall that the fit parameters x₁ and c are estimated from the light-curve fits for each object. For each sample combination, these fit parameters are used to estimate the cosmological parameters, the global parameters α and β in Equation (29) (see Tables 16 and 17), and the distance moduli, by minimizing the scatter in the Hubble diagram. Since the distance-modulus estimate for each SN depends upon the sample combination in which the SN is included, upon the cosmological model parameterization, and upon the BAO and CMB priors, we provide tables for each of the six sample combinations and for both the FwCDM and ΛCDM models. Although the cosmological parameters have been corrected for the selection bias using simulations (Table 5), the distance moduli in Table 14 do not include any bias corrections. The entries in this Table should therefore not be used to derive cosmological constraints.

10.2.1. salt-ii Hubble Dispersion

In the salt-ii method, an intrinsic dispersion (σ^int_μ) is added in quadrature to the distance-modulus uncertainty (Equation (27)) such that the resulting Hubble diagram χ²_μ/N_dof is equal to 1. Using the FwCDM model parameterization and the BAO+CMB prior, Table 15 gives the σ^int_μ values obtained from fitting each SN sample independently and setting χ²_μ = N_dof. For the Nearby, SDSS-II, and ESSENCE samples, the salt-ii values are similar to those from mlcs2k2 in the fourth line of Table 11. For the SNLS sample, the salt-ii dispersion is significantly smaller than that from mlcs2k2, while for the HST sample the salt-ii dispersion is larger. The smaller dispersion for SNLS may derive in part from the fact that the salt-ii model was partially trained on SNLS data. For sample combinations (a)–(f), the σ^int_μ values are in Tables 16 and 17.

Table 15. Intrinsic Dispersion (σ^int_μ) Required for χ²_μ = N_dof for Each SN Ia Sample Fit Separately with the salt-ii Method

	σintμ for Sample
	Nearby	SDSS-II	ESSENCE	SNLS	HST
Independent fits	0.15	0.08	0.17	0.11	0.23

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Table 16. For the FwCDM Model, Constraints on w and Ω_M from salt-ii SN Distances Combined with SDSS LRG BAO and WMAP-5 CMB Results

Parameter	Result for Sample
	(a)	(b)	(c)	(d)	(e)	(f)
σintμ	0.084	0.124	0.105	0.128	0.140	0.160
rmsμ	0.178	0.219	0.170	0.210	0.232	0.231
w	−0.87	−0.98	−0.92	−0.98	−0.96	−0.95
σw(stat)	0.12	0.08	0.11	0.07	0.06	0.08
σw(syst)	+0.06−0.12	0.15	+0.07−0.15	0.13	0.12	0.13
σw(tot)	+0.14−0.17	0.17	+0.13−0.18	0.15	0.14	0.15
ΩM	0.281	0.256	0.271	0.264	0.265	0.267
$\sigma _{\Omega _{\rm M}}$ (stat)	0.030	0.019	0.025	0.017	0.016	0.019
$\sigma _{\Omega _{\rm M}}({\rm syst})$	+0.015−0.025	0.033	+0.015−0.029	0.028	0.025	0.027
$\sigma _{\Omega _{\rm M}}({\rm tot})$	+0.034−0.039	0.038	+0.029−0.038	0.033	0.030	0.033
α	0.127	0.123	0.113	0.107	0.124	0.106
σα(stat)	0.017	0.015	0.014	0.013	0.014	0.019
σα(syst)	0.020	0.021	0.016	0.020	0.023	0.023
σα(tot)	0.026	0.026	0.021	0.024	0.027	0.030
β	2.52	2.62	2.50	2.66	2.64	2.56
σβ(stat)	0.16	0.13	0.15	0.12	0.12	0.17
σβ(syst)	0.11	0.19	0.11	0.19	0.18	0.25
σβ(tot)	0.19	0.23	0.19	0.22	0.22	0.30

Notes. ^(a)SDSS-only ^(b)SDSS+ESSENCE+SNLS ^(c)Nearby+SDSS ^(d)Nearby+SDSS+ESSENCE+SNLS ^(e)Nearby+SDSS+ESSENCE+SNLS+HST ^(f)Nearby+ESSENCE+SNLS

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Table 17. For the ΛCDM Model, Constraints on Ω_M and Ω_Λ from salt-ii SN Distances Combined with BAO and CMB Results

Parameter	Result for Sample
	(a)	(b)	(c)	(d)	(e)	(f)
σintμ	0.085	0.123	0.105	0.128	0.140	0.160
rmsμ	0.177	0.220	0.170	0.210	0.232	0.231
ΩΛ	0.734	0.735	0.734	0.734	0.727	0.734
$\sigma _{\Omega _{\Lambda }}(\mbox{stat})$	0.019	0.017	0.019	0.017	0.016	0.017
$\sigma _{\Omega _{\Lambda }}(\mbox{syst})$	0.019	0.026	0.018	0.020	0.019	0.018
$\sigma _{\Omega _{\Lambda }}(\mbox{tot})$	0.027	0.031	0.026	0.026	0.025	0.025
ΩM	0.275	0.274	0.275	0.275	0.279	0.275
$\sigma _{\Omega _{\rm M}}({\rm stat})$	0.023	0.021	0.023	0.020	0.019	0.021
$\sigma _{\Omega _{\rm M}}({\rm syst})$	0.014	0.021	0.013	0.020	0.017	0.016
$\sigma _{\Omega _{\rm M}}({\rm tot})$	0.027	0.030	0.027	0.029	0.026	0.027
α	0.126	0.123	0.113	0.113	0.116	0.105
σα(stat)	0.017	0.015	0.014	0.013	0.014	0.019
σα(syst)	0.020	0.021	0.016	0.019	0.023	0.023
σα(tot)	0.027	0.026	0.022	0.023	0.027	0.030
β	2.58	2.64	2.50	2.57	2.65	2.46
σβ(stat)	0.15	0.12	0.15	0.12	0.12	0.17
σβ(syst)	0.13	0.18	0.09	0.17	0.18	0.19
σβ(tot)	0.20	0.22	0.17	0.20	0.22	0.25

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10.2.2. salt-ii Hubble Diagrams and Cosmological Parameters

Figure 33 shows the differences between the salt-ii-estimated SN distance moduli μ_i and those for an open CDM model with no dark energy (Ω_M = 0.3, Ω_DE = 0) as a function of redshift. Figure 34 shows the distribution of normalized residuals, (μ_i − μ_FwCDM)/σ_μ, where μ_FwCDM is the distance modulus from the best-fit FwCDM model for sample combination (e) and σ_μ is the total uncertainty defined in Equation (27). The bulk of the distribution of all 288 normalized residuals (the upper-left panel of Figure 34) is well fitted by a Gaussian with σ = 0.75; outliers increase the rms to 0.92. The rms for combination (e) is slightly smaller than 1 because covariances from the fit are not included in the calculation of σ_μ.

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Figures 35 and 36 show the salt-ii statistical-uncertainty contours for the FwCDM and ΛCDM models; for the latter, the salt-ii SN contours are more consistent with the BAO+CMB constraints than the mlcs2k2 contours were. For the combined SN+BAO+CMB results, the total uncertainty contours, including systematic errors, are shown in Figures 37 and 38. The best-fit cosmological parameter values and uncertainties, marginalizing over H₀ and incorporating the bias corrections of Table 5, are given in Tables 16 and 17, respectively. The salt-ii statistical errors on cosmological parameters are consistent with those from mlcs2k2. The systematic errors for the two methods are similar for sample combinations (b) and (d)–(f), but the salt-ii systematic uncertainty is significantly smaller for the sample combinations in which SDSS-II is the high-redshift sample (a) and (c). The large difference in the systematic uncertainty is driven by the U-band anomaly.

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Among the six SN sample combinations, the best-fit values of w for the FwCDM model again fall into two groups, as for the mlcs2k2 results, but with the opposite trend in w compared to mlcs2k2. For sample combinations (a) and (c), in which SDSS-II is the high-redshift sample, the salt-ii method results in w = −0.87 and −0.92, comparable to the mlcs2k2 results of −0.84 and −0.92 for the same sample combinations. However, for the other four sample combinations, which include higher-redshift SNe, salt-ii yields w = −0.98 to −0.95, compared with the mlcs2k2 result of w = −0.71 to −0.76 for the same sample combinations. Based on studies with simulated sample combinations, the observed difference between these two groups of salt-ii results is not statistically significant. However, the difference between the mlcs2k2 and salt-ii results for the higher-redshift samples appears to be significant. We discuss these differences further in Section 11.

10.2.3. Redshift Evolution of salt-ii Parameters

In the salt-ii model fits, α, β, and M (see Equation (29)) are global parameters that are assumed to be independent of redshift. To test the consistency of this assumption, we have carried out salt-ii fits separately in five redshift bins for sample combinations (d) and (e). For the fit in each redshift bin, the cosmological parameters w and Ω_M in the FwCDM model are fixed to the values from the sample (e) fit, and the BAO+CMB prior is applied. The redshift dependence of the best-fit results for α, β, and M are shown in the left panels of Figure 39. There is evidence for redshift evolution of the parameters, particularly for the color parameter β, which falls with increasing redshift above z ∼ 0.6. This trend is more evident without the HST sample, suggesting that the β-variation is driven primarily by the SNLS sample. We have performed this test with high-statistics simulations (right panels of Figure 39) and find that the fitted salt-ii parameters are consistent across all redshift bins. Since the simulation accounts for the Malmquist bias, selection effects, and measurement errors, the redshift dependence favored by the data is likely due to some other effect.

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10.2.4. U-Band Anomaly with salt-ii

Here, we study how the U-band anomaly is manifest in the salt-ii method. Figure 40 shows the change in the binned SDSS-II SN Hubble diagram when the salt-ii fit excludes data from the observer-frame passband corresponding to the rest-frame U band, i.e., excluding g-band data for z > 0.21. Although this change only affects the light-curve fits for z > 0.21, it can alter the estimated distance moduli at all redshifts since they are derived from a global fit to the Hubble diagram. For z < 0.21, the average distance-modulus shift is (0.012 ± 0.003) mag; for z > 0.21, the mean shift is (0.079 ± 0.028) mag relative to the z < 0.21 shift. This relative shift is about half of that for mlcs2k2, but it is still significant: for the SDSS-only sample, the exclusion of the U band in the salt-ii fits results in a shift in w of −0.100 that is included as a systematic uncertainty.

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The data-model residuals for the salt-ii light-curve fits are shown in Figure 7 for all of the SN Ia samples in all of the rest-frame passbands. Figure 41 shows the residuals when the filter corresponding to the rest-frame U band is excluded from the fit. In both cases, there are systematic discrepancies between the model and the data in the rest-frame U band for the Nearby SN Ia sample at all epochs. Since the salt-ii rest-frame U band model was trained primarily on higher-redshift SNe, i.e., downweighting observer-frame U-band data from Nearby SNe, this points to a systematic offset between the nearby, observer-frame U-band data and the higher-redshift, rest-frame U-band data, which was also qualitatively seen in the mlcs2k2 fits.

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