Issue 
A&A
Volume 526, February 2011



Article Number  A81  
Number of page(s)  24  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201015792  
Published online  24 December 2010 
Do spectra improve distance measurements of Type Ia supernovae?^{⋆}
^{1}
Centre de Physique des Particules de Marseille (CPPM),
CNRS/IN2P3, 163 avenue de
Luminy, 13288
Marseille Cedex 9,
France
email: blondin@cppm.in2p3.fr
^{2}
HarvardSmithsonian Center for Astrophysics (CfA),
60 Garden Street, Cambridge, MA
02138,
USA
Received:
20
September
2010
Accepted:
24
November
2010
We investigate the use of a wide variety of spectroscopic measurements to determine distances to lowredshift Type Ia supernovae (SN Ia) in the Hubble flow observed through the CfA Supernova Program. We consider linear models for predicting distances to SN Ia using lightcurve width and color parameters (determined using the SALT2 lightcurve fitter) and a spectroscopic indicator, and evaluate the resulting Hubble diagram scatter using a crossvalidation procedure. We confirm the ability of spectral flux ratios alone at maximum light to reduce the scatter of Hubble residuals by ~10% [weighted rms, or WRMS = 0.189 ± 0.026 mag for the flux ratio ℛ(6630/4400)] with respect to the standard combination of lightcurve width and color, for which WRMS = 0.204 ± 0.029 mag. When used in combination with the SALT2 color parameter, the colorcorrected flux ratio ℛ^{c}(6420/5290) at maximum light leads to an even lower scatter (WRMS = 0.175 ± 0.025 mag), although the improvement has low statistical significance (<2σ) given the size of our sample (26 SN Ia). We highlight the importance of an accurate relative flux calibration and the failure of this method for highlyreddened objects. Comparison with synthetic spectra from 2D delayeddetonation explosion models shows that the correlation of ℛ(6630/4400) with SN Ia absolute magnitudes can be largely attributed to intrinsic color variations and not to reddening by dust in the host galaxy. We consider flux ratios at other ages, as well as the use of pairs of flux ratios, revealing the presence of smallscale intrinsic spectroscopic variations in the irongroupdominated absorption features around ~4300 Å and ~4800 Å. The best flux ratio overall is the colorcorrected ℛ^{c}(4610/4260) at t = −2.5 d from maximum light, which leads to ~30% lower scatter (WRMS = 0.143 ± 0.020 mag) with respect to the standard combination of lightcurve width and color, at ~2σ significance. We examine other spectroscopic indicators related to lineprofile morphology (absorption velocity, pseudoequivalent width etc.), but none appear to lead to a significant improvement over the standard lightcurve width and color parameters. We discuss the use of spectra in measuring more precise distances to SN Ia and the implications for future surveys which seek to determine the properties of dark energy.
Key words: supernovae: general / cosmology: observations
Appendices are only available in electronic form at http://www.aanda.org
© ESO, 2010
1. Introduction
Precise distances to Type Ia supernovae (SN Ia) formed the cornerstone of the discovery of cosmic acceleration (Riess et al. 1998; Perlmutter et al. 1999). These measurements use the shape of supernova light curves and their colors to tell which supernovae are bright and which are intrinsically dim (Phillips 1993; Riess et al. 1996; Prieto et al. 2006; Jha et al. 2007; Guy et al. 2007; Conley et al. 2008; Mandel et al. 2009). In this paper we explore the suggestion of Bailey et al. (2009) that spectra can contribute to improved distance measurements. We apply statistical tests to a subset of the ~250 SN Ia for which we have good light curves and spectra based on the ongoing program of supernova observations at the HarvardSmithsonian Center for Astrophysics (CfA; Matheson et al. 2008; Hicken et al. 2009a).
It is important to construct the best possible distance indicators to extract the maximum cosmological information from supernova surveys. The present stateoftheart gives distances to wellobserved individual objects with uncertainties of order 10%, so that samples of nearby (Hicken et al. 2009b) and distant SN Ia (ESSENCE, Miknaitis et al. 2007; SNLS, Astier et al. 2006) can be combined to constrain the equationofstate for dark energy, noted w. The first results show that for a flat universe with constant w, the dark energy is compatible with a cosmological constant (for which w = −1) within about 10% (Astier et al. 2006; WoodVasey et al. 2007). Constraints on the variation of w with redshift come from from highredshift observations with the Hubble Space Telescope (Riess et al. 2004, 2007). Presentday limits are weak, but future work with large, carefully calibrated samples from the ground (PanSTARRS, Dark Energy Survey, LSST) and from space (Euclid, WFIRST) will contribute to distinguishing the nature of dark energy (Albrecht et al. 2009). In designing the followup observations for these enterprises, it is worth knowing whether spectra will be useful only for classification and precise redshifts, or whether the spectra of the supernovae themselves can be used to improve the precision of the distances. The way we explore this is to analyze the CfA sample, using the difference between the distance derived from Hubble expansion with the distance predicted from our various models. This difference is the Hubble residual, which we use as a measure of the power of a particular model to predict the supernova distance. As described below, we explore models that combine quantitative information from the spectrum with information on light curve shape and color.
Spectroscopic information is fundamental to the success of employing SN Ia as distance indicators in large surveys. Cleanly separating Type Ia supernovae from corecollapse events like SN Ib and SN Ic improves the purity of the sample. More directly, Nugent et al. (1995) showed that some easilymeasured line ratios in SN Ia spectra are correlated with the luminosity. Measurements of line velocities (and gradients thereof), strengths, and widths and their relation to supernova luminosity have been explored recently by several authors (Benetti et al. 2005; Blondin et al. 2006; Bongard et al. 2006; Hachinger et al. 2006; Bronder et al. 2008). Likewise, Matheson et al. (2008) revealed spectroscopic variability amongst SN Ia of similar luminosity. But the first application of spectroscopic clues to improve distance estimates has come from Bailey et al. (2009). Using spectra of 58 SN Ia from the Nearby Supernova Factory, they showed that the ratio of fluxes in selected wavelength bins (flux ratios) could reduce the scatter of Hubble residuals by ~20% compared to the usual combination of lightcurve width and color parameters (σ = 0.128 ± 0.012 mag cf. 0.161 ± 0.015 mag). By using a flux ratio measured on a dereddened spectrum in combination with a color parameter they found a further ~5% improvement (σ = 0.119 ± 0.011 mag). We have sought first to see if we can reproduce their results using the CfA data set, and then to test additional ideas about ways to use spectra to improve the estimates of supernova distances.
In practice, the standardization of SN Ia magnitudes involves a term related to the width of the light curve and a correction due to color. While some methods attempt to separate intrinsic color variations from reddening by dust in the host galaxy (e.g., MLCS2k2; Jha et al. 2007) others use a single parameter for both effects (e.g. SALT2; Guy et al. 2007), exploiting the degeneracy between the two: underluminous SN Ia are also intrinsically redder than overluminous SN Ia (e.g. Tripp 1998). We adopt the latter approach in this paper, to match the method used by Bailey et al. (2009). An active area of research involves the use of SN Ia spectra to provide independent or complementary information on SN Ia luminosities that would help improve their use as distance indicators.
We consider models for predicting distances to SN Ia of the form: (1)where m_{B} is the apparent restframe Bband magnitude at peak, M is a reference absolute magnitude, “width” and “color” are the usual lightcurve parameters, and “spec” is some spectroscopic indicator; (α,β,γ) are fitting constants. We study the following five models:

1.
only a spectroscopic indicator is used [i.e. (α,β) = (0,0)],

2.
both a spectroscopic indicator and a lightcurve width parameter are used, but no color parameter (i.e. β = 0),

3.
both a spectroscopic indicator and a color parameter are used, but no lightcurve width parameter (i.e. α = 0),

4.
a spectroscopic indicator is used in addition to the lightcurve width and color parameters.

5.
both lightcurve width and color parameters are used, but no spectroscopic indicator (i.e. γ = 0). We refer to this as the “standard” model.
The paper is organized as follows: in Sect. 2 we present our lightcurve fitting and training method, as well as a crossvalidation procedure to evaluate the impact of each spectroscopic indicator. We present the CfA data set in Sect. 3. In Sect. 4 we study the flux ratios of Bailey et al. (2009), while in Sect. 5 we consider other spectroscopic indicators. We discuss the use of SN Ia spectra for distance measurements in Sect. 6 and conclude in Sect. 7.
2. Methodology
2.1. Lightcurve fitting
We use the SALT2 lightcurve fitter of Guy et al. (2007) to determine the width and color parameters for each SN Ia in our sample. A model relating distance, apparent magnitudes, and linear dependencies of the absolute magnitude is: (2)where (x_{1},c) are the SALT2 light curve width and color parameters, and is some spectroscopic indicator. The restframe peak apparent Bband magnitude is m_{B}, also obtained from the SALT2 fit to a supernova’s light curve. The distance modulus predicted from the light curve and spectral indicators is μ, and the constant M is a reference absolute magnitude. The distance modulus estimated from the redshift is μ(z) = 25 + 5log_{10} [D_{L}(z) Mpc^{1}] under a fixed cosmology, where D_{L} is the luminosity distance.
We use the exact same SALT2 options as Guy et al. (2007) to fit the SN Ia light curves in our sample, and only trust the result when the following conditions are met: reduced ; at least one Bband point before +5 d from Bband maximum, and one after +10 d; at least 5 B and Vband points in the age range −15 ≤ t ≤ +60 d; finally, we impose a cut on the SALT2 x_{1} parameter, namely −3 ≤ x_{1} ≤ 2. This last condition is equivalent to considering SN Ia in the range 0.8 ≲ Δm_{15}(B) ≲ 1.7 (i.e. subluminous 1991bglike SN Ia are excluded). We examined all the lightcurve fits by eye to ensure they were satisfactory given this set of conditions. Approximately 170 of the ~250 SN Ia with light curves from the CfA SN program pass these requirements.
2.2. Training
For estimating the coefficients of the model (training), we use a custom version of the luminosity distance fitter simple_cosfitter^{1} (A. Conley 2009, priv. comm.) based on the Minuit function minimization package (James & Roos 1975). This code minimizes the following expression with respect to the parameters (α,β,γ,ℳ): (3)where m_{B,s} is the restframe peak apparent Bband magnitude of the s^{th} SN Ia, and m_{pred,s} is the predicted peak apparent Bband magnitude, given by: (4)where D_{L} is the luminosity distance at redshift z_{s} for a given cosmological model described by the standard parameters (w,Ω_{m},Ω_{Λ}). Since our analysis only includes objects at low redshifts (z < 0.06), we do not solve for these parameters and simply assume a flat, cosmological constantdominated model with (w,Ω_{m},Ω_{Λ}) = (−1,0.27,0.73). The ℳ term is a collection of constants including the reference M.
The variance that appears in the denominator of Eq. (3) includes an error on the corrected magnitude (Eq. (2)), using the estimation error covariance of the lightcurve parameters and spectroscopic indicators, a variance due to peculiar velocities (σ_{pec,s} = (v_{pec}/cz_{s})(5/ln10), where we take the rms peculiar velocity v_{pec} = 300 km s^{1}), and an intrinsic dispersion of SN Ia magnitudes: (5)where σ_{int} is adjusted iteratively until (typically σ_{int} ≲ 0.2 mag for the standard (x_{1},c) model). For any particular model, the intrinsic variance accounts for deviations in magnitude in the Hubble diagram beyond that explained by measurement error or random peculiar velocities, and hence represents a floor to how accurately the model can predict distances. To limit the impact of the peculiar velocity error we restrict our analysis to SN Ia at redshifts z > 0.015 (σ_{pec} < 0.15 mag). Of the 170 SN Ia with satisfactory SALT2 fits, 114 are at redshifts greater than 0.015.
2.3. Crossvalidation
We consider several models described by Eq. (2) that use different subsets of the predictors ). If we train a model on the data of all the SN in the sample to estimate the coefficients , we can evaluate the fit of the model by computing the training error, e.g. the mean squared distance modulus residual, , over all SN s in the training set.
For finite samples, the average Hubble diagram residual of the training set SN is an optimistic estimate of the ability of the statistical model, Eq. (2), to make accurate predictions given the supernova observables. This is because it uses the supernova data twice: first for estimating the model parameters (training), and second in evaluating the residual error. Hence, the training set residuals underestimate the prediction error, which is the expected error in estimating the distance of a SN that was not originally in the finite training set. We refer to these data as “outofsample”. Furthermore, with a fixed, finite, and noisy training data set, it is always possible to reduce the residual, or training, error of the fit by introducing more predictors to the model. However, this may lead to overfitting, in which apparently significant predictors are found in noisy data, even though in reality there was no trend. These relationships are sensitive to the finite training set and would not generalize to outofsample cases. To evaluate predictive performance and guard against overfitting with a statistical model based on finite data, we should estimate the prediction error for outofsample cases. To do so, we use a crossvalidation (CV) procedure to evaluate the impact of using a spectroscopic indicator , alone and in conjunction with standard light curve parameters, on the accuracy of distance predictions in the Hubble diagram.
Crossvalidation seeks to estimate prediction error and to test the sensitivity of the trained statistical model to the data set by partitioning the full data set into smaller subsets. One subset is held out for testing predictions of the model, while its complement is used to train the model. This process is repeated over partitions of the full data set. This method avoids using the same data simultaneously for training the model and for estimating its prediction error. Crossvalidation was used before for statistical modeling of SN Ia by Mandel et al. (2009), who applied the .632 bootstrap method to evaluate distance predictions for SN Ia using near infrared light curves. A careful implementation of a crossvalidation method is particularly important for small samples, as is the case in this paper (e.g. 26 SN Ia at maximum light; see Sect. 4.3).
In this paper, the crossvalidation method we use is known as Kfold CV. The idea is to divide our SN Ia sample into K subsets, train a given model on K − 1 subsets, and validate it on the remaining subset. This procedure is repeated K times, at which point all SN Ia have been part of a validation set once. Typical choices of K are 5 or 10 (e.g., Hastie et al. 2009). The case K = N, where N is the number of SN Ia in our sample, is known as “leaveoneout” CV. In this case, each SN Ia in turn is used as a validation set, and the training is repeated N times on N − 1 SN Ia.
In practice, we run Kfold CV as follows:

1.
the sample of N SN Ia is randomly divided into K subsets of equal size (when N is not a multiple of K, the number of SN Ia between any two subsets differs by at most one).

2.
Looping over each K fold:

2a.
all the SN Ia in the Kth subsetare removed from the sample: they form the validation set.The remaining SN Ia define thetraining set.

2b.
the objects in the training set are then used to determine the bestfit values for the parameters in Eq. 3, as well as the intrinsic dispersion σ_{int} in Eq. (5).

2c.
using this set of parameters we predict the magnitudes of the SN Ia in the validation set (indexed j): (6)The Hubble residual, or error, of the predicted distance modulus is then (7)

2a.

3.
When the magnitude or distance of each SN Ia has been predicted once using the above scheme, we analyze the prediction errors (Sect. 2.4). When doing so, we check that the set of bestfit are consistent amongst all training sets.
2.4. Comparing model predictions
For each model, which we label by its predictors, e.g. , crossvalidation gives us a set of prediction errors {Δμ_{s}} for each SN s. To summarize the total dispersion of predictions, we computed the weighted mean squared error, (8)the square root of which is the weighted rms. We weight the contribution from each SN by the inverse of its expected total variance (the precision) . We prefer to use the rms of the prediction residuals rather than the sample standard deviation, since the former measures the average squared deviation of the distance prediction from the Hubble distance μ(z), whereas the latter measures the average squared deviation of prediction errors from the mean prediction error. Note that the mean squared error is equal to the sample variance plus the square of the mean error. Thus, the mean squared error will be larger than the sample variance if the mean error, or bias, is significant, but the two statistics will be the same if it is not. Since the mean prediction error is not guaranteed to be zero, we use the WRMS statistic to assess the total dispersion of distance prediction errors. We also estimate the sampling variance of this statistic (see Appendix A).
The WRMS measures the total dispersion in the Hubble diagram. However, we expect that some of that scatter is due to random peculiar velocities (influencing μ(z) with variance ), and some due to measurement error (). Using the crossvalidated distance errors, we also estimate how precisely we can expect a particular model to predict the distance to a SN Ia when these other sources of error are negligible. We call this variance estimate the rms intrinsic prediction error, a property of the model itself, and label it . Intuitively, this is the result of subtracting from the total dispersion the expected contributions of peculiar velocities and measurement uncertainties. It is similar to the intrinsic variance σ_{int} discussed in Sect. 2.2, in that it represents a floor to how accurately the model can predict distances. It is not strictly equivalent, however, since σ_{int} is adjusted during the training process so that , while σ_{pred} is estimated using the crossvalidated distance modulus prediction errors. In Appendix B, we describe a maximum likelihood estimate for σ_{pred} and its standard error from the set of distance predictions.
We are also interested in the intrinsic covariance of the distance prediction errors generated by two different models. Imagine that peculiar velocities and measurement error were negligible, and model P and model Q predict distances to the same set of SN Ia. We calculate the prediction errors, from each model. There is a positive intrinsic covariance if tends to be positive when is positive, and a negative intrinsic covariance if they tend to make errors in opposite directions. The intrinsic correlation is important because it suggests how useful it would be to combine the distance predictions of two models. If two models tend to make prediction errors in the same direction (positive correlation), then the combined model is not likely to do much better than the most accurate of the two original models. However, if two models tend to make prediction errors that are wrong in different ways (zero or negative correlation), then we expect to see a gain from averaging the two models.
Even if two models make prediction errors that are intrinsically uncorrelated, random peculiar velocities will tend to induce a positive correlation in the realized errors if the methods are used on the same set of SN. This is because the unknown peculiar velocity for a given SN is the same regardless of the model we use to generate its distance prediction. Hence, the expected contribution of random peculiar velocities to the sample covariance of predictions must be removed to estimate the intrinsic covariance between two models. In Appendix B, we describe a maximum likelihood estimator for the intrinsic covariance and its standard error using the set of distance predictions.
We use the maximum likelihood estimation method to estimate the intrinsic prediction error and intrinsic covariance of each model compared to the reference model (x_{1},c) that uses only light curve information.
3. Spectroscopic data
We have used a large spectroscopic data set obtained through the CfA Supernova Program. Since 1994, we have obtained ~2400 optical spectra of ~450 lowredshift (z ≲ 0.05) SN Ia with the 1.5 m Tillinghast telescope at FLWO using the FAST spectrograph (Fabricant et al. 1998). Several spectra were published in studies of specific supernovae (e.g., SN 1998bu; Jha et al. 1999), while 432 spectra of 32 SN Ia have recently been published by Matheson et al. (2008). We also have complementary multiband optical photometry for a subset of ~250 SN Ia (Riess et al. 1999; Jha et al. 2006; Hicken et al. 2009a), as well as NIR JHK_{s} photometry for the brighter ones (WoodVasey et al. 2008). All published data are available via the CfA Supernova Archive^{2}.
All the spectra were obtained with the same telescope and instrument, and reduced in a consistent manner (see Matheson et al. 2008 for details). The uniformity of this data set is unique and enables an accurate estimate of our measurement errors.
4. Spectral flux ratios
4.1. Measurements
Bailey et al. (2009) introduced a new spectroscopic indicator, calculated as the ratio of fluxes in two wavelength regions of a SN Ia spectrum binned on a logarithmic wavelength scale. This ratio, noted ℛ(λ_{X}/λ_{Y}) = F(λ_{X})/F(λ_{Y}) (λ_{X} and λ_{Y} being the restframe wavelength coordinates in Å of a given bin center), is measured on a deredshifted spectrum corrected for Galactic reddening using the Cardelli et al. (1989) extinction law with R_{V} = 3.1 in combination with the dust maps of Schlegel et al. (1998). A colorcorrected version of this flux ratio, noted ℛ^{c}(λ_{X}/λ_{Y}), is measured on a spectrum additionally corrected for the SALT2 color parameter using the color law of Guy et al. (2007). Figure 1 illustrates both measurements.
Fig. 1 Illustration of the flux ratio measurement. The upper panel shows the input spectrum (deredshifted and corrected for Galactic reddening; here SN 1999gd around maximum light), binned on a logwavelength scale. The gray vertical lines represent the fluxes in characteristic wavelength bins mentioned throughout the text. The lower panel shows the same spectrum corrected for SALT2 color, which is used to measure the colorcorrected flux ratios ℛ^{c}. 
We use the same binning as Bailey et al. (2009), namely 134 bins equally spaced in lnλ between 3500 Å and 8500 Å (rest frame), although most of the CfA spectra used here do not extend beyond ~7100 Å (see Sect. 4.3). The resulting ~2000 km s^{1} bin size is significantly less than the typical width of a SN Ia feature (~10 000 km s^{1}). The error on ℛ includes a flux error (from the corresponding variance spectrum), an error due to the relative flux calibration accuracy (see Sect. 4.2), and an error due to the SALT2 color precision. When there are several spectra of a given SN Ia within Δt = 2.5 d of the age we consider (see Sect. 4.3 for spectra at maximum light; Sect. 4.4 for spectra at other ages), we use the errorweighted mean and standard deviation of all flux ratios as our measurement and error, respectively. Bailey et al. (2009) also chose Δt = 2.5 d in their analysis, and we find that increasing Δt worsens the results while decreasing it leads to too small a sample.
Bailey et al. (2009) crosschecked the results for their best single flux ratio ℛ(6420/4430) using the the sample of SN Ia spectra published by Matheson et al. (2008) (and available through the CfA SN Archive). We checked the validity of our flux ratio measurements by comparing the values of ℛ(6420/4430) in the Matheson et al. (2008) sample with those reported in Table 2 of Bailey et al. (2009). In all cases, our measurements agree well within the 1σ errors. This also holds for SN 1998bu, accidentally removed from the Matheson et al. (2008) sample by Bailey et al. (2009) (H. Fakhouri 2010, priv. comm.). We note that we were unable to crosscheck the flux ratio measurements of Bailey et al. (2009) in a similar fashion, since none of their 58 SN Ia spectra are publicly available.
4.2. Impact of relative flux calibration and SALT2 color
When the λ_{X} and λ_{Y} wavelength bins have a large separation (≳ 1000 Å), ℛ(λ_{X}/λ_{Y}) is essentially a color measurement. We therefore expect flux ratios to be sensitive to the relative flux calibration accuracy of the spectra. Figure 2 shows the relation between uncorrected Hubble residuals [i.e. m_{B} − M − μ(z)] and our most highlyranked flux ratio ℛ(6630/4400) at maximum light (see Sect. 4.3). There is one data point per SN Ia, colorcoded according to the absolute difference in B − V color at maximum light derived from the spectrum and that derived from the photometry, noted Δ(B − V), which we use as a proxy for relative flux calibration accuracy. The bulk of the sample defines a highly correlated relation (dashed line), with several outliers all having Δ(B − V) ≥ 0.1 mag. We therefore restrict our analysis to SN Ia with spectra that have a relative flux calibration better than 0.1 mag.
Fig. 2 Uncorrected Hubble residual vs. flux ratio ℛ(6630/4400) at maximum light, colorcoded according to the absolute difference in B − V color derived from the spectrum and that derived from the photometry, noted Δ(B − V). The dashed line is a linear fit to the SN Ia with Δ(B − V) < 0.1 mag. The highlyreddened SN 2006br is not shown here. 
Bailey et al. (2009) noted that the highlyreddened SN 1999cl was a large outlier in their analysis, and attributed this to the nonstandard nature of the extinction towards this SN (R_{V} ≈ 1.5; Krisciunas et al. 2006). To explore the effects of reddening, in Fig. 3 (left), we show the relation between uncorrected Hubble residual and ℛ(6630/4400) at maximum light, for SN Ia at redshifts z > 0.005 that satisfy our requirement on the relative flux calibration accuracy. Using this lower redshift bound has the effect of including several highlyreddened SN Ia (including SN 1999cl; see Fig. 4), which are otherwise excluded based on the redshift cut we use elsewhere this paper (z > 0.015). For SN Ia with c < 0.5, ℛ(6630/4400) is highly correlated with uncorrected Hubble residuals, but those with red colors (c > 0.5) tend to deviate significantly from this relation (dashed line; this is not the case for SN 1995E, for which c ≈ 0.9), the two largest outliers corresponding to the reddest SN Ia (SN 1999cl and SN 2006X). Both are subject to high extinction by nonstandard dust in their respective host galaxies (A_{V} ≈ 2 mag for R_{V} ≈ 1.5; Krisciunas et al. 2006; Wang et al. 2008) and display timevariable Na i D absorption, whose circumstellar or interstellar origin is still debated (Patat et al. 2007; Blondin et al. 2009). The reddening curves in Fig. 3 (dotted lines) seem to corroborate the fact that the nonlinear increase of flux ratios at high values of the SALT2 color parameter is mainly due to reddening by dust with low R_{V}. Nonetheless, SN 1999cl still stands out in this respect as it would require a value of R_{V} ≲ 0.5 inconsistent with that found by Krisciunas et al. (2006). Moreover, while we obtain consistent R_{V} estimates for SN 2006X using other flux ratios, this is not the case for SN 2006br, for which some flux ratios are consistent with R_{V} = 3.1.
Fig. 3 Left: Uncorrected Hubble residual vs. flux ratio ℛ(6630/4400) at maximum light for SN Ia at z > 0.005 with Δ(B − V) < 0.1 mag, colorcoded according to the SALT2 color parameter, c. Points corresponding to SN Ia with c > 0.5 are labeled. The dashed line is a linear fit to the SN Ia with c < 0.5. The dotted lines are reddening curves for different values of R_{V}, normalized to the smallest ℛ(6630/4400) value. Right: Colorcorrected Hubble residual vs. colorcorrected flux ratio ℛ^{c}(6420/5290) at maximum light. 
Fig. 4 Histogram of the SALT2 color parameter (c) for SN Ia at z > 0.015 (open) and 0.005 < z < 0.015 (hatched). Bins that include SN Ia with c > 0.5 are labeled. 
The right panel of Fig. 3 shows the relation between colorcorrected Hubble residual [i.e. m_{B} − M − βc − μ(z)] and our most highlyranked colorcorrected flux ratio ℛ^{c}(6420/5290) at maximum light (see Sect. 4.3) for the same sample. SN Ia with a SALT2 color c > 0.5 are again outliers. As noted by Bailey et al. 2009, this shows that a single color parameter cannot encompass the variety of SN Ia intrinsic colors and extinction by nonstandard dust. We therefore impose a cut on SALT2 color in our analysis, only considering SN Ia with c < 0.5. Four of the five SNe with c > 0.5 in Fig. 3 are rejected anyway based on our redshift cut. The remaining one, SN 2006br, is then rejected based on our color cut.
4.3. Results on maximumlight spectra
4.3.1. Selecting the best flux ratios
After selecting SN Ia that satisfy both requirements on relative flux calibration accuracy and SALT2 color parameter, we are left with 26 SN Ia at z > 0.015 with spectra within Δt = 2.5 d from maximum light (see Table 1, where we also present selected flux ratio measurements). The spectra show no sign of significant contamination by hostgalaxy light, which can also bias the flux ratio measurements. We make no cut based on the signaltonoise ratio (S/N) of our spectra, as they are generally well in excess of 100 per logwavelength bin. We only consider flux ratios for wavelength bins represented in all the spectra. This leads to 98 bins between ~3690 Å and ~7060 Å, i.e. 9506 independent flux ratios.
SN Ia sample for flux ratio measurements at maximum light.
Top 5 flux ratios at maximum light from 10fold CV on 26 SN Ia.
We run the Kfold crossvalidation procedure outlined in Sect. 2.3, and consider the five models for estimating distances to SN Ia described in Sect. 1: When no color correction is involved (Eqs. (9)–(10)), we use the uncorrected flux ratio ℛ. When a color correction is involved (the −βc term in Eqs. (11)–(12)), we use the colorcorrected version of the flux ratio ℛ^{c}. Using ℛ in combination with color, or ℛ^{c} alone or in combination with x_{1}, severely degrades the predictive power of the model, so we do not report results using (c,ℛ); ℛ^{c} alone; or (x_{1},ℛ^{c}).
We rank the flux ratios in each case based on the intrinsic prediction error (σ_{pred}; see Sect. 2.4), but note that ranking based on the weighted rms of prediction Hubble residuals makes almost no difference. The results for the top five flux ratios are displayed in Table 2. We also report the bestfit γ, the weighted rms of prediction Hubble residuals (WRMS), the intrinsic correlation of residuals with those found using the standard (x_{1},c) predictors (noted ρ_{x1,c}; see Sect. 2.4), and the difference in intrinsic prediction error with respect to the standard (x_{1},c) model, noted Δ_{x1,c}. Since we compute the error on Δ_{x1,c} (see Appendix B), we also report the significance of this difference with respect to the standard (x_{1},c) predictors. This is a direct measure of whether a particular model predicts more accurate distances to SN Ia when compared to the standard approach, and if so how significant is the improvement. Figure 5 shows the resulting Hubble diagram residuals vs. redshift for the best flux ratio in each of the four models given by Eqs. (9)–(12), and using the standard (x_{1},c) predictors.
Fig. 5 Hubble diagram residuals for the highestranked flux ratios at maximum light. From top to bottom: prediction residuals using ℛ only; (ℛ,x_{1}); (ℛ^{c},c); (ℛ^{c},x_{1},c); and using the standard SALT2 fit parameters (x_{1},c). In each case we indicate the weighted rms of prediction Hubble residuals (gray highlighted region). 
Fig. 6 Correlation between the highestranked (ℛ,ℛ^{c}) at maximum light and the SALT2 fit parameters (x_{1},c). 
All the flux ratios listed in Table 2 lead to an improvement over the standard (x_{1},c) correction (i.e. Δ_{x1,c} < 0), as found by Bailey et al. (2009), but the significance is low: <1σ for ℛ only; ≪ 1σ for (x_{1},ℛ); ~1.5σ for (c,ℛ^{c}) and (x_{1},c,ℛ^{c}). This is in part due to the small number of SN Ia in our sample. Note that ρ_{x1,c} > 0.5 in all cases, i.e. the models that include a flux ratio tend to make prediction errors in the same direction as (x_{1},c), and we do not expect to gain much by combining these models.
Using best single flux ratio ℛ(6630/4400) by itself reduces the weighted rms of prediction residuals (as well as the intrinsic prediction error, σ_{pred}) by ≲10% when compared with (x_{1},c) (WRMS = 0.189 ± 0.026 mag cf. 0.204 ± 0.029 mag), although as noted above the significance of the difference in intrinsic prediction error is negligible (Δ_{x1,c} = −0.018 ± 0.025 mag, or 0.7σ).
Using ℛ in combination with x_{1} leads to no improvement over using ℛ alone (although this is not reported by Bailey et al. 2009, it is consistent with their findings; S. Bailey 2009, priv. comm.), and even leads to systematically worse results. Our best single flux ratio ℛ(6630/4400) yields a difference in intrinsic prediction error with respect to (x_{1},c) of Δ_{x1,c} = −0.018 ± 0.025 mag, when used on its own, while it yields Δ_{x1,c} = −0.006 ± 0.028 mag when combined with x_{1}. These differences are statistically indistinguishable from one another given the size of the error on Δ_{x1,c}, but they are systematic regardless of the flux ratio we consider.
This seems counterintuitive, as one might expect that including an additional predictor would result in more accurate distance predictions. However, this is not necessarily the case under crossvalidation. The reason is that x_{1} by itself is a poor predictor of Hubble residuals, and one does not gain anything by combining it with ℛ(6630/4400). This is not surprising, as the relation between lightcurve width and luminosity is only valid if the SN Ia are corrected for color or extinction by dust beforehand. In fact, ℛ(6630/4400) by itself accounts for most of the variation in Hubble residuals. When we crossvalidate, the extra coefficient α will tend to fit some noise in a given training set, and this relation will not generalize to the validation set. This results in an increase in prediction error because the added information is not useful. We see from Table 2 that adding x_{1} affects the bestfit value for γ [γ = −4.51 ± 0.15 cf. −4.37 ± 0.09 for ℛ(6630/4400) only]; moreover, we obtain α ≲ 0 when using [x_{1},ℛ(6630/4400)] where α ≈ 0.15 when using (x_{1},c), which again shows that α is fitting noise when ℛ(6630/4400) is combined with x_{1}. This illustrates the advantage of using crossvalidation in guarding against overfitting noise as more parameters and potential predictors are added.
Figure 6 (upper panel) shows why ℛ(6630/4400) alone is a good predictor of Hubble residuals. Its strong correlation with SALT2 color (Pearson correlation coefficient r = 0.92) shows that this ratio is essentially a color measurement. The correlation with x_{1} is less pronounced (r = −0.38), but this is largely due to a small number of outliers: removing the three largest outliers results in a Pearson correlation coefficient r = −0.65. The flux ratio ℛ(6630/4400) by itself is thus as useful a predictor as x_{1} and c combined.
The relation between ℛ(6630/4400) and x_{1} is not linear, but it is certainly true that SN Ia with higher x_{1} (i.e. broader light curves) tend to have lower ℛ(6630/4400) (the same is true for ℛ(6420/4430), the highestranked flux ratio by Bailey et al. 2009). Since the width of the lightcurve is a parameter intrinsic to each SN Ia (although its measurement can be subtly affected by hostgalaxy reddening; see Phillips et al. 1999), the correlation between x_{1} and ℛ(6630/4400) shows that the color variation measured by ℛ(6630/4400) is intrinsic in part. This is consistent with the socalled “brighterbluer” relation of Tripp (1998): overluminous SN Ia are intrinsically bluer than underluminous SN Ia (see also Riess et al. 1996).
Using a colorcorrected flux ratio ℛ^{c} in combination with color results in even lower Hubble residual scatter when compared with the single flux ratio case. Our best flux ratio in this case, ℛ^{c}(6420/5290), reduces the weighted rms of prediction residuals by ~15% with respect to (x_{1},c) [WRMS = 0.175 ± 0.025 mag cf. 0.204 ± 0.029 mag], and the intrinsic prediction error by ~20% [σ_{pred} = 0.148 ± 0.029 mag cf. 0.181 ± 0.032 mag]. Again, the significance of this difference is only 1.4σ (Δ_{x1,c} = −0.032 ± 0.023 mag). We see from Fig. 6 (middle panel) that ℛ^{c}(6420/5290) is strongly anticorrelated with x_{1} (r = −0.78), and that dereddening the spectra using the SALT2 color law is effective in removing any dependence of ℛ^{c}(6420/5290) on color, as expected.
One would naively think that combining our best colorcorrected ratio ℛ^{c}(6420/5290) with (x_{1},c) would lead to an even further improvement, but this is not the case. In fact, ℛ^{c}(6420/5290) ranks 298th when we consider the set of predictors (x_{1},c,ℛ^{c}). This is due to the strong anticorrelation of ℛ^{c}(6420/5290) with x_{1}. Adding x_{1} as an extra predictor when ℛ^{c}(6420/5290) already includes this information means α will tend to fit noise in a given training set, as was the case for the set of (x_{1},ℛ) predictors when compared with ℛonly. Indeed, the bestfit value for α for [x_{1},c,ℛ^{c}(6420/5290)] is again consistent with 0.
Nonetheless, several colorcorrected flux ratios do result in a further reduced scatter when combined with (x_{1},c), although the wavelength baseline for these ratios is much smaller (≲400 Å) and the wavelength bins forming the ratios are all concentrated in the region of the S ii λλ5454,5640 doublet. Our highestranked flux ratio in this case, ℛ^{c}(5690/5360), reduces the weighted rms of prediction residuals by ~20% with respect to (x_{1},c) [WRMS = 0.164 ± 0.023 mag cf. 0.204 ± 0.029 mag], and the intrinsic prediction error by ~25% [σ_{pred} = 0.134 ± 0.028 cf. 0.181 ± 0.032 mag]. Again, the significance of this difference is only 1.6σ (Δ_{x1,c} = −0.044 ± 0.028 mag). We see from Fig. 6 (lower panel) that this ratio is not correlated with x_{1} (r = −0.08) or c (r = 0.11), and thus constitutes a useful additional predictor of distances to SN Ia.
4.3.2. Twodimensional maps of all flux ratios
The results for all 9506 flux ratios are displayed in Fig. 7. The four rows correspond to the four models for estimating SN Ia distances that include a flux ratio (Eqs. (9)–(12)). The left column is colorcoded according to the weighted rms of prediction Hubble residuals (flux ratios that result in WRMS > 0.324 mag are given the color corresponding to WRMS = 0.324 mag), while the right column is colorcoded according to the absolute Pearson correlation coefficient of the correction terms with uncorrected Hubble residuals (e.g. for the set of predictors (c,ℛ^{c}), the correlation of (−βc + γℛ^{c}) with uncorrected residuals).
Fig. 7 Results from 10fold crossvalidation on maximumlight spectra. From top to bottom: ℛ only; (x_{1},ℛ); (c,ℛ^{c}); (x_{1},c,ℛ^{c}). The left column is colorcoded according to the weighted rms of prediction Hubble residuals, while the right column corresponds to the absolute Pearson crosscorrelation coefficient of the correction terms with uncorrected Hubble residuals. 
Only a very restricted number of wavelength bins lead to a low WRMS of prediction Hubble residuals when a flux ratio ℛ is used by itself (Fig. 7; upper left), namely λ_{X} ≳ 6300 Å and λ_{Y} ≈ 4400 Å (4 of the 5 best flux ratios in Table 2 for ℛonly have λ_{Y} ≈ 4400 Å). This is in stark contrast with the large number of flux ratios with absolute Pearson correlation coefficients r > 0.8 (Fig. 7; upper right). In general, a flux ratio with a higher correlation coefficient will result in a Hubble diagram with less scatter, but this is not systematically the case, and the relation between the two is certainly not linear. For Pearson correlation coefficients r > 0.8, the standard deviation of Hubble residuals can vary by up to 0.1 mag at any given r (Fig. 8, top panel). This is because the crosscorrelation coefficient does not take into account errors on ℛ or on the Hubble residual, and is biased by outliers and reddened SN Ia. The lower panel of Fig. 8 shows the impact of including the highlyreddened SN 2006br: at any given r, the resulting weighted rms of prediction Hubble residuals is 30–60% higher. Moreover, many flux ratios with high correlation coefficients (r > 0.8) result in Hubble diagrams with excessively large scatter (WRMS > 1 mag). This is counterintuitive, since the resulting scatter in these cases appears to be larger than when no predictors at all are used to determine distances to SN Ia (in which case WRMS ≈ 0.5 mag). The reason is that we consider the scatter under crossvalidation, as opposed to fitting all the SN Ia at the same time. In these aberrant cases, the trained model is sensitive to the inclusion or exclusion of some outlier in the training set, and this leads to large errors when the outlier is in the validation set. Last, including this SN leads to correlations with r > 0.95, where there are none otherwise. Figure 8 thus justifies our excluding SN 2006br from the sample (already excluded based on our cut on SALT2 color; see Sect. 4.2), and illustrates the advantage of selecting flux ratios based directly on the weighted rms of prediction Hubble diagram residuals, rather than on crosscorrelation coefficients. As already mentioned in Sect. 4.3.1, using crossvalidated prediction errors to select the best flux ratios guards us against overfitting a small sample: in the naive approach that consists in fitting the entire SN Ia sample at once, adding more predictors always leads to a lower scatter in Hubble residuals (this is known as “resubstitution”; see, e.g., Mandel et al. 2009).
When the SALT2 color parameter is used in combination with a colorcorrected flux ratio ℛ^{c}, there are again restricted wavelength regions that lead to a low weighted rms of prediction Hubble residuals (Fig. 7; third row left) (4 of the 5 best flux ratios in Table 2 for (c,ℛ^{c}) predictors involve wavelength bins at ~4900 Å and ~6500 Å). The SALT2 color parameter c does not attempt to distinguish between reddening by dust and intrinsic color variations. Dereddening the spectra using this parameter corrects for both effects regardless of their relative importance. However, since the SALT2 color law is very similar to the Cardelli et al. (1989) extinction law with R_{V} = 3.1 and E(B − V) = 0.1 mag (Guy et al. 2007, their Fig. 3), one generally assumes that the color correction removes the bulk of reddening by dust, and the remaining variations in the SED are primarily intrinsic to the supernova. If this is so, it is intriguing that the best flux ratios for the ℛonly and (c,ℛ^{c}) models share similar wavelength bins. The recent survey of 2D SN Ia models from Kasen et al. (2009) suggests that a significant part of the color variation measured by the ℛ(6630/4400) is indeed intrinsic (see Sect. 4.3.4).
The second row of Fig. 7 confirms that using the x_{1} parameter in combination with a flux ratio results in a slight degradation in the weighted rms of prediction residuals, while the correlations with uncorrected Hubble residuals are degraded with respect to cases where ℛ is used by itself. Last, the bottom row of Fig. 7 is a visual demonstration that (x_{1},c,ℛ^{c}) fares better than (c,ℛ^{c}) overall, although the best colorcorrected flux ratios do not perform significantly better. We see from the right panel that the correlations of (αx_{1} − βc + γℛ^{c}) with uncorrected residuals all have absolute Pearson correlation coefficients r ≲ 0.5. The two regions at λ_{X,Y} ≈ 5300 Å stand out in the 2D plot of WRMS residuals, and all the top colorcorrected ratios for (x_{1},c,ℛ^{c}) include a wavelength bin in that region (which corresponds to the absorption trough of the S ii λ5454 line).
Fig. 8 Weighted rms of prediction Hubble residuals vs. absolute Pearson crosscorrelation coefficient for all flux ratios at maximum light, excluding (top) and including (bottom) the highlyreddened SN 2006br. 
4.3.3. Comparison with Bailey et al. (2009)
We confirm the basic result of Bailey et al. (2009) using an independent sample and a different crossvalidation method: the use of a flux ratio alone or in combination with a color parameter results in a Hubble diagram with lower scatter when compared to the standard (x_{1},c) model. Using a flux ratio alone, Bailey et al. (2009) find ℛ(6420/4430) as their most highlyranked ratio, while we find ℛ(6630/4400) (see Table 2). The wavelength bins are almost identical, and in any case ℛ(6420/4430) is amongst our top 5 ratios. For this ratio we find γ = −3.40 ± 0.10, in agreement with γ = −3.5 ± 0.2 found by Bailey et al. (2009)^{3}.
The other four flux ratios given by Bailey et al. (2009) (their Table 1) are not part of our top5 ℛ. For two of these ratios the reason is trivial: they include wavelength bins redder than 7100 Å, not covered by most of our spectra. The other two flux ratios [ℛ(6420/4170) and ℛ(6420/5120)] lead to differences <5% on the Hubble diagram residual scatter with respect to the standard (x_{1},c) model according to Bailey et al. (2009) [σ = 0.166 ± 0.016 mag for ℛ(6420/4170) and σ = 0.154 ± 0.015 mag for ℛ(6420/5120), cf. 0.161 ± 0.015 mag for (x_{1},c)], and they rank 29th and 658th in our study, respectively. This discrepancy is in part due to the selection method: Bailey et al. (2009) select their best ratios based on crosscorrelation coefficients with uncorrected magnitudes, while we select them based on the intrinsic prediction error from crossvalidated Hubble diagram residuals. However, ranking our ratios using the same method as Bailey et al. (2009) does not resolve the discrepancy. It is possible that Bailey et al. (2009) are sensitive to their exact choice of training and validation samples, where we have randomized the approach. We note however that the impact on the weighted rms of prediction residuals is statistically indistinguishable for many flux ratios given our sample size (e.g. error on WRMS ~0.03 mag cf. differences of ≲ 0.01 mag in WRMS for the top 5 flux ratios; see Table 2), so that the exact ranking of flux ratios is not well determined and subject to revisions from small changes in the input data.
Using both a colorcorrected flux ratio ℛ^{c} and the SALT2 color parameter decreases the residual scatter further, as found by Bailey et al. (2009). Using the set of predictors [c,ℛ^{c}(6420/5290)] leads to ~15% lower WRMS with respect to (x_{1},c) [WRMS = 0.175 ± 0.025 mag cf. 0.204 ± 0.029 mag], and to ~20% lower σ_{pred} (σ_{pred} = 0.148 ± 0.029 mag cf. 0.181 ± 0.032 mag) at 1.4σ significance based on the difference in intrinsic prediction error, Δ_{x1,c}. None of the colorcorrected flux ratios listed by Bailey et al. (2009) (their Table 1) are part of our five highestranked ℛ^{c}, although our top ratios are formed with almost the same wavelength bins (ℛ^{c}(6420/5290) in this paper; ℛ^{c}(6420/5190) in Bailey et al. (2009)). The other colorcorrected ratios in Bailey et al. (2009) rank well below in our study, whether we select the best ℛ^{c} according to the resulting Hubble residual scatter or the crosscorrelation of (−βc + γℛ^{c}) with uncorrected residuals. The same caveats apply here as when selecting the best uncorrected flux ratios (see previous paragraph), although the ℛ^{c} measurement is probably even more sensitive to the relative flux calibration accuracy of the spectra.
We also crosschecked the results of Bailey et al. (2009) by simply validating their best flux ratios on our entire SN Ia sample. The results are displayed in Table 3, where we give the weighted rms of Hubble residuals from a simultaneous fit to the entire SN Ia sample (as done by Bailey et al. 2009), as opposed to prediction residuals under crossvalidation. For all flux ratios (both ℛ and ℛ^{c}) in Table 3, our own bestfit γ agrees within the 1σ errors with that found by Bailey et al. (2009) (noted γ(B09) in Table 3), although we have systematically larger errors. We note that most of the top ratios reported by Bailey et al. (2009) lead to no significant improvement over (x_{1},c), and even leads to slightly worse results for some ratios (e.g. ℛ(6420/5120) results in WRMS = 0.237 ± 0.032 mag cf. 0.194 ± 0.027 mag for (x_{1},c)). A closer look at Table 1 of Bailey et al. (2009) shows that this is also the case in their paper: for the ℛonly model, only one ratio out of five, namely ℛ(6420/4430), results in a lower Hubble diagram residual scatter. The other four are either consistent with no improvement (ℛ(7720/4370) and ℛ(6420/5120)), or yield slightly worse results (ℛ(6420/4170) and ℛ(7280/3980)). Again, this results from the way Bailey et al. (2009) selected their best ratios, based on the correlation with uncorrected Hubble residuals.
Validation of top 5 flux ratios at maximum light from Bailey et al. (2009) (noted B09).
Fig. 9 Absolute Pearson correlation coefficients of flux ratios at maximum light with uncorrected absolute magnitudes M_{B} in 2D delayeddetonation SN Ia models of Kasen et al. (2009) (left), and in data from the CfA SN Ia sample (right). 
We cannot directly compare the resulting scatter in Hubble diagram residuals with those reported in Table 1 of Bailey et al. (2009). First, they use the sample standard deviation (σ), whereas we use the weighted rms (see Sect. 2.4). Second, the scatter they find for the standard (x_{1},c) model is significantly lower than ours. We have refit the data presented in Table 1 of Bailey et al. (2009) to derive the weighted rms of Hubble residuals for the (x_{1},c) model from their sample, and find WRMS = 0.148 ± 0.014 mag, which is almost 0.05 mag smaller when compared to our sample (0.194 ± 0.027 mag). This difference in the Hubble residual scatter between the SNFactory and CfA samples is consistent with the difference found amongst other nearby SN Ia samples by Hicken et al. (2009b).
Interestingly, using the WRMS statistic as opposed to the sample standard deviation results in a smaller difference in residual scatter between the ℛonly and (x_{1},c) models. Using our own fits of the data presented in Table 1 of Bailey et al. (2009), we find WRMS = 0.131 ± 0.014 mag for ℛ(6420/4430), i.e. ~11% smaller scatter when compared to (x_{1},c), where the difference between the two models is ~20% when considering the sample standard deviation.
4.3.4. Comparison with 2D models
We use synthetic spectra based on a recent 2D survey of delayeddetonation SN Ia models by Kasen et al. (2009) to investigate the physical origin of the high correlation between several flux ratios and uncorrected SN Ia magnitudes. These models were found to reproduce the empirical relation between peak Bband magnitude and postmaximum decline rate. A more detailed comparison of SN Ia data with these models will be presented elsewhere.
Fig. 10 Absolute restframe Bband magnitude (M_{B}) vs. flux ratio ℛ(6630/4400) at maximum light in 2D SN Ia models of Kasen et al. (2009) (small dots), and in data from the CfA SN Ia sample (A_{V} < 0.45 mag: filled circles, A_{V} > 0.45 mag: open circles). The dotted and dashed lines are linear fits to the models and data, respectively, where models for which ℛ(6630/4400) > 0.25 and data for which A_{V} > 0.45 mag have been excluded from the fit. The slope (Γ) and Pearson correlation coefficient (r) are indicated for both cases. The data have been offset vertically for clarity. Including models for which ℛ(6630/4400) < 0.25 results in Γ = 5.71 ± 0.03 and r = 0.97, while including data with A_{V} > 0.45 mag results in Γ = 4.43 ± 0.47 and r = 0.92. The arrows indicate approximate reddening vectors for different values of R_{V}. 
We measured flux ratios in the same manner as we did for our data, and computed Pearson correlation coefficients with (uncorrected) absolute magnitudes synthesized directly from the spectra. The 2D correlation map is shown in Fig. 9 (left panel), alongside the same map derived from the CfA SN Ia sample (right panel). At first glance, the two maps appear similar, with two large ~1000 Åwide “bands” of flux ratios with strong correlations with uncorrected magnitudes, for λ_{X}(λ_{Y}) ≳ 6200 Å and λ_{Y}(λ_{X}) ≲ 6000 Å, although the correlations are even stronger in the models (several flux ratios have absolute Pearson correlation coefficients r > 0.95, where there are none in the data). A closer look reveals some important differences, the models having strong correlations for 6000 Å ≲ λ_{X}(λ_{Y}) ≲ 6200 Å and λ_{Y}(λ_{X}) ≳ 6200 Å that are not present in the data. The same applies to the regions with coordinates λ_{X}(λ_{Y}) ≈ 4200 Å. These differences are significant and illustrate the potential for such comparisons to impose strong constraints on SN Ia models.
In Fig. 10 we show the correlation of uncorrected absolute restframe Bband magnitudes (M_{B}) with our highestranked flux ratio ℛ(6630/4400), both from the 2D models and CfA data, where we have used the redshiftbased distance for the latter. The vertical offset is arbitrary and solely depends on the normalization adopted for the data, which we have chosen for sake of clarity. There are 1320 model points, each corresponding to one of the 44 2D delayeddetonation models of Kasen et al. (2009) viewed from one of 30 different viewing angles. The linear fits shown in Fig. 10 are done over the range 0.25 ≲ ℛ(6630/4400) ≲ 0.50, where the models and data overlap. For the data this is equivalent to excluding the three most highlyreddened SN Ia (open circles), for which the hostgalaxy visual extinction A_{V} was determined based on lightcurve fits with MLCS2k2 (Jha et al. 2007). This is justified since no reddening by dust is applied to the models. The slope of the relation between M_{B} and ℛ(6630/4400) is significantly steeper for the models (Γ = 7.38 ± 0.13) than for the data (Γ = 4.76 ± 1.04), and the correlation is much stronger (r = 0.89 cf. 0.69 for the data). This is not surprising since the data are subject to random measurement and peculiar velocity errors, which degrade the correlation. Including models for which ℛ(6630/4400) < 0.25 softens the slope to Γ = 5.71 ± 0.03 and results in a stronger correlation (r = 0.97), while including data with A_{V} > 0.45 mag results in Γ = 4.43 ± 0.47 and a much stronger correlation r = 0.92. This last value for Γ can be compared with the γ fitting parameter for this same flux ratio (γ = −4.42 ± 0.09; see Table 2), although the latter is based on a formal crossvalidation procedure and the opposite sign is a consequence of the convention when using the flux ratio to predict SN Ia distances. As noted in Sect. 4.3.2, the correlation of M_{B} with ℛ(6630/4400) is largely biased by the minority of highlyreddened SN Ia.
Fig. 11 Results from 10fold crossvalidation on spectra at t = −2.5, +0, +5, +7.5 d. (From top to bottom), colorcoded according to the weighted rms of prediction Hubble residuals. The left column is corresponds to ℛ only, while the right column corresponds to the (c,ℛ^{c}) model. 
The models yield values for ℛ(6630/4400) ranging between ~0.12 and ~0.44, all due to intrinsic color variations. Since these models reproduce the relation between M_{B} and postmaximum decline rate of Phillips (1993), they confirm the intrinsic nature of the correlation between ℛ(6630/4400) and { x_{1},c } shown in Fig. 6 (upper left panel).
The wavelength bins λ_{X} = 6630 Å and λ_{Y} = 4400 Å are close to the central wavelengths of the standard R and B broadband filters, hence ℛ(6630/4400) is a rough measure of the B − R color at Bband maximum. The 2D models of Fig. 10 indicate that a large part of the variation in ℛ(6630/4400) seen in the data is due to intrinsic variations in B − R color. Reddening in the host galaxy is then needed to explain values of ℛ(6630/4400) ≳ 0.4, while at lower values it is challenging at best to discriminate between the effects of intrinsic color variations and extinction by dust, since both affect ℛ(6630/4400) in the same manner, as illustrated by the reddening vectors in Fig. 10 (they are really reddening curves, cf. Fig. 3, but the behavior is almost linear over this small range in ℛ(6630/4400)).
The models also give a physical explanation for the correlation of ℛ(6630/4400) with absolute magnitude. Indeed, the variation of this ratio is largely caused by spectroscopic variations around 4400 Å, a region dominated by lines of Fe ii and Fe iii, with contributions from Mg ii (Ti ii provides an important source of opacity for the least luminous SN Ia), while the region around 6630 Å has little intrinsic variation (this was noted by Bailey et al. 2009). This translates to a standard deviation of peak Bband magnitudes (σ ≈ 0.40 mag) that is almost twice as large as the Rband magnitude (at B maximum; σ ≈ 0.26 mag) in the models. The relative contribution of Fe ii and Fe iii lines is related to the temperature of the lineforming regions in the SN Ia ejecta, itself a function of peak luminosity (dimmer SN Ia are generally cooler; see, e.g., Kasen & Woosley 2007). One thus expects a large luminositydependent spectroscopic variation in this wavelength region, although its exact shape and relation to temperature remains largely unknown.
While these models provide useful insights into the physical origin of these correlations, a direct comparison with the data reveals some of their shortcomings. In Fig. 10 we see that some models predict values of the flux ratio ℛ(6630/4400) ≲ 0.25 for the most luminous SN Ia, where the data are limited to values greater than this. Our sample includes several SN Ia at the high luminosity end that show no sign of extinction in their host galaxies (A_{V} < 0.05 mag based on lightcurve fits with MLCS2k2), so the differences are real and point to discrepancies between the data and the models, some of the latter having bluer B − R colors at Bband maximum. This is not surprising, as the models explore a larger range of parameter space than is realized in nature. Comparisons of this sort can then help constrain the range of model input parameters. A more detailed comparison of SN Ia data from the CfA SN program with these models will be presented elsewhere.
4.4. Results on spectra at other ages
Bailey et al. (2009) restricted their analysis to spectra within Δt = 2.5 d from Bband maximum. In this section we consider flux ratios measured on spectra at other ages. We impose the same cuts on relative flux calibration accuracy (Δ(B − V) < 0.1 mag), SALT2 color (c < 0.5), redshift (z > 0.015), and age range (Δt = 2.5 d) as those used for the maximumlight spectra in the previous section. We consider all ages between t = −2.5 d and t = +7.5 d, in steps of 2.5 d (for ages earlier than −2.5 d or later than +7.5 d the number of SN Ia with spectra that satisfy our cuts falls below 20, and we do not trust the results). We report the best ratio at each age in Table 4, for both the ℛonly and (c,ℛ^{c}) models.
Top flux ratio at ages −2.5 ≤ t ≤ +7.5 d from 10fold CV.
The 2D maps of Hubble diagram residual scatter for spectra at ages t = −2.5, +0, +5, and +7.5 d are shown in Fig. 11. As was the case for maximumlight spectra, adding the SALT2 x_{1} parameter leads to slightly degraded results when compared with ℛ alone, so we do not show plots for (x_{1},ℛ) in Fig. 11. Moreover, we do not show results for (x_{1},c,ℛ^{c}) since the best colorcorrected flux ratios in this case do not result in a significant improvement over (c,ℛ^{c}). At all the ages we consider here, the set of predictors (c,ℛ^{c}) results in lower weighted rms of prediction residuals than (x_{1},c), although the significance of the difference is ≲2σ and is lower for t ≥ + 5 d than for −2.5 ≤ t ≤ + 2.5 d. A flux ratio by itself only leads to an improvement over (x_{1},c) near maximum light (−2.5 ≤ t ≤ + 2.5 d). As was the case at maximum light, there is a positive intrinsic correlation in prediction error between all the distance prediction models that include a flux ratio and (x_{1},c) [0.4 ≲ ρ_{x1,c} ≲ 0.8].
The best set of predictors overall in the age range −2.5 ≤ t ≤ + 7.5 d is [c,ℛ^{c}(4610/4260)] at t = −2.5 d. The weighted rms of prediction residuals is reduced by ~30% with respect to (x_{1},c) [WRMS = 0.143 ± 0.020 mag], and the intrinsic prediction error by ~40% (σ_{pred} = 0.106 ± 0.028 mag), the significance of the difference being ~2σ (Δ_{x1,c} = −0.081 ± 0.037 mag). We show the correlation of ℛ^{c}(4610/4260) at t = −2.5 d and the SALT2 parameters (x_{1},c) in Fig. 12. This colorcorrected ratio is mildly correlated with x_{1} [r = 0.61; this drops slightly to r = 0.47 if we ignore the two points at ℛ^{c}(4610/4260) > 1.3] and uncorrelated with color (r = −0.40; this drops to r = 0.03 if we ignore the two points at c > 0.4). Interestingly, the wavelength bins that constitute this ratio are part of the two prominent spectral absorption features, predominantly due to irongroup elements, that were found to vary intrinsically between SN Ia based on the 2D models discussed in Sect. 4.3.4.
Fig. 12 Correlation between ℛ^{c}(4610/4260) at t = −2.5 d and the SALT2 fit parameters (x_{1},c). The Pearson coefficient of the correlation with color drops to r = 0.03 if we ignore the two points at c > 0.4. 
4.5. Results using two flux ratios
In this section we consider corrections using a linear combination of two flux ratios as follows: where the latter equation includes an additional correction due to color, hence the use of colorcorrected ratios ℛ^{c}. For both cases, we fix to the highestranked single flux ratio (e.g., at maximum light: ℛ_{1}(6630/4400) and ; see Table 2), but leave both (γ_{1},γ_{2}) as free parameters (i.e. we do not set γ_{1} equal to γ found in the single flux ratio case).
The results for the top five secondary flux ratios at maximum light are displayed in Table 5. In all cases, including a second flux ratio further reduces the weighted rms of prediction residuals by ≈15–20% [WRMS = 0.162 ± 0.022 mag for ℛ_{2}(5160/5290); WRMS = 0.151 ± 0.021 mag for ] with respect to the single flux ratio case (WRMS = 0.189 ± 0.026 mag for ℛ(6630/4400); WRMS = 0.175 ± 0.025 mag for ℛ^{c}(6420/5290)). Again the significance of the improvement (≲ 2σ) is difficult to gauge given our sample size, this despite the fact that γ_{2} is significantly different from zero in all cases. Our best secondary flux ratios, ℛ_{2}(5160/5290) and , are uncorrelated with the SALT2 fit parameters (x_{1},c) and with the highestranked primary ratios ℛ_{1}(6630/4400) and (see Fig. 13), and hence provide independent information on SN Ia luminosity.
Top 5 secondary flux ratios at maximum light from 10fold CV on 26 SN Ia.
Fig. 13 Correlation between the highestranked at maximum light and the SALT2 fit parameters (x_{1},c), and the highestranked . 
The secondary flux ratios ℛ_{2} listed in Table 5 have a much smaller wavelength baseline than the primary ratios ℛ_{1} and (see Table 2). The highestranked secondary ratios have a 130 Å and 140 Å baseline, respectively. These ratios do not measure SN Ia colors: they measure smallscale intrinsic spectroscopic variations. Interestingly, all the wavelength bins that form these secondary ratios are clustered around the S ii λλ5454,5640 doublet and the irongroupdominated absorption complex Fe ii λ4800 mentioned in Sect. 4.4 (see also Sect. 4.3.4). As was the case for a single flux ratio, the results using a secondary flux ratio are similar for −2.5 ≤ t ≤ +2.5 d, and tend to be worse at later ages. We choose not to discuss them further.
In a recent paper^{4}, Yu et al. (2009) have searched for flux ratio pairs that minimize Hubble diagram residuals (with no color correction), and find several such pairs which achieve a standard deviation σ ≲ 0.10 mag at ages between −3 d and +12 d from maximum light. We have validated the flux ratio pairs reported in their Table 4 and find that none of them lead to an improvement compared with the standard (x_{1},c) model. There could be several reasons for this disagreement: Yu et al. (2009) do not use a crossvalidation procedure and their sample size (anywhere between 17 and 24 SN Ia depending on the age and flux ratio pair considered) suggests they may be overfitting a small sample. Moreover, they use colorcorrected flux ratios (actually corrected for the hostgalaxy extinction, A_{V}, as opposed to the SALT2 color parameter) but do not include a color parameter in their equation to correct for the SN Ia magnitudes. When we use the SALT2 color parameter in addition to the same flux ratio pairs as Yu et al. (2009), several pairs indeed lead to an improvement over the standard (x_{1},c) model, but not over the correction using a single flux ratio (or a single flux ratio in combination with color) we find in this paper. Last, Yu et al. (2009) do not impose a redshift cut on their sample: only 14 out of 38 SN Ia are at redshifts z > 0.015, and 5 are at redshifts z < 0.005, where the magnitude error due to peculiar velocities is σ_{pec} > 0.4 mag. It is unclear why this should lead to a lower scatter in Hubble residuals (on the contrary one expects an increased scatter), but it certainly impacts their analysis.
5. Other spectroscopic indicators
In this section we consider other spectroscopic indicators, mostly related to spectral line profile morphology. Some of these indicators are also flux ratios, but the wavelengths correspond to precise locations of absorption troughs or emission peaks in the SN Ia spectrum, as opposed to the “blind” approach of computing flux ratios from all possible wavelength bins with no a priori physical motivation. We use the same approach as for the flux ratios, i.e. we consider models for predicting SN Ia distances which include a spectroscopic indicator, possibly in combination with a lightcurve parameter (cf. Eq. (1)), and we validate each model using Kfold crossvalidation (and present results for K = 10 in this section).
5.1. Measurements
We divide the SN Ia spectrum into several “features”, each labeled according to the strongest line in that wavelength range. Figure 14 shows the seven features we consider in this paper, from Ca ii λ3945 in the blue to Si ii λ6355 in the red. The wavelengths associated with each feature correspond to the gfweighted mean wavelength of the different atomic transitions for the specified ion (e.g. 3945 Å for the Ca ii H & K lines), except for the two large features dominated by Fe ii lines, where the wavelengths denote the approximate location of the deepest absorption (~4300 Å and ~4800 Å). The Fe ii λ4300 feature also includes contributions from Mg ii, and possibly Fe iii for the most luminous SN Ia. For the faintest, 1991bglike SN Ia, Ti ii constitutes a dominating source of opacity in this wavelength region. Since we do not include 1991bglike SN Ia in our analysis, however, we do not present measurements for Ti ii.
Fig. 14 Wavelength bounds of spectroscopic features for which we measured the various indicators shown in Fig. 15, illustrated using the maximumlight spectrum of SN 2006ax. 
The various spectroscopic indicators we consider are illustrated in Fig. 15, based on the Si ii λ6355 line profile in the spectrum of SN 2005ki at t = +1 d from maximum light. We first smooth the spectrum using the inversevariance Gaussian filter of Blondin et al. (2006) with a smoothing factor 0.001 < dλ/λ < 0.01 determined based on a χ^{2} test using flux errors from the variance spectra (Fig. 15; thick line). The smoothed spectrum makes it easier to define wavelength locations of local flux maxima on either side of the absorption component of the P Cygni profile (λ_{blue} and λ_{peak}), as well as the location of maximum absorption (λ_{abs}). The wavelengths λ_{abs} and λ_{peak} are then used to define the absorption and peak velocities, respectively (v_{abs} and v_{peak}), using the relativistic Doppler formula (see also Blondin et al. 2006). We also measure the heights of the local maximum (h_{blue} and h_{peak}), and define a pseudocontinuum between them. These latter quantities are measured on the original, unsmoothed spectrum. Division by this pseudocontinuum enables us to measure the relative absorption depth (d_{abs}) and fullwidth at halfmaximum (FWHM) of the absorption component, as well as its pseudoequivalent width (pEW; defined analogously to the equivalent width used by stellar spectroscopists for abundance determinations, but without the physical basis, hence “pseudo” EW; Fig. 15; right panel).
Fig. 15 Definition of the main spectroscopic indicators used in this paper, here illustrated using the Si iiλ6355 line profile in the spectrum of SN 2005ki at t = +1 d. The right panel shows the pseudocontinuum (dashed line), as well as the wavelength locations of the blue and red emission peaks (λ_{blue} and λ_{peak}) and their respective heights (h_{blue} and h_{peak}). The wavelength of maximum absorption (λ_{abs}) serves to define the absorption velocity, v_{abs}. The peak velocity v_{peak} is defined analogously. The left panel shows the same line profile normalized to the pseudocontinuum, and serves to define the (relative) absorption depth (d_{abs}), FWHM, and pseudoequivalent width (pEW; shaded gray region). In both panels, the thick line corresponds to the smoothed flux, where we have used the inversevariance weighted Gaussian filter of Blondin et al. (2006) with a smoothing factor dλ/λ = 0.005. 
We measure these quantities for all the features presented in Fig. 14, except for the complex Fe ii λ4300 and Fe ii λ4800 features for which we only consider the pseudoequivalent width. The error on each measured quantity includes errors due to redshift, relative flux calibration, hostgalaxy extinction and contamination, and of course the flux error. We only consider measurements for which the mean S/N over the entire feature is greater than 5 per Å, and require a minimum of 20 SN Ia with valid measurements. Note that we do not impose cuts on relative flux calibration accuracy or SALT2 color, as was the case for the flux ratios, since these quantities are mostly local measurements which are far less sensitive to the overall SED.
We also consider various spectroscopic ratios, which were found to correlate with absolute magnitude, defined below: The ratios ℛ(Ca) and ℛ(Si) were both defined by Nugent et al. (1995), and found to correlate well with the luminosity decline rate parameter Δm_{15}(B). To increase the S/N of the ℛ(Ca) measurement, Bongard et al. (2006) introduced the corresponding integral flux ratio ℛ(CaS), also found to correlate with absolute magnitude. Using a grid of LTE synthetic spectra to investigate the ℛ(Si) wavelength region, Bongard et al. (2006) also defined a ratio of the red local maximum of Si ii λ6355 to the red local maximum of S ii λ5640, noted ℛ(SiS). The corresponding integral flux ratio is ℛ(SiSS), again introduced by Bongard et al. (2006) to increase the S/N of the ℛ(SiS) measurement. Last, Hachinger et al. (2006) measured the absorption velocities and pseudoEW in 28 SN Ia spectra and found two additional pEW ratios, ℛ(S,Si) and ℛ(Si,Fe), that are good indicators of luminosity. Note that ℛ(Ca) and ℛ(SiS) are in fact flux ratios similar to those defined by Bailey et al. (2009).
5.2. Results
We present our results using the absorption velocity (v_{abs}; units of 10^{4} km s^{1}), the fullwidth at halfmaximum (FWHM; units of 10^{2} Å), the relative absorption depth (d_{abs}), the pseudoequivalent width (pEW; units of 10^{2} Å), and the various spectroscopic ratios ℛ(X) (Eqs. (16)–(22)) in Tables C.1–C.5 (Appendix C). We do not present results for the peak velocity (v_{peak}) as they are far worse than for the other indicators. There were not enough valid measurements for Ca ii λ3945, hence the absence of this line in Tables C.1–C.4. We only report results for the bluer absorption of the S ii doublet (S ii λ5454) in Tables C.1–C.3, but the pseudoequivalent width is that of the entire doublet (see Table C.4).
Based on the difference in intrinsic prediction error with respect to the standard model which uses the SALT2 fit parameters (x_{1},c), again noted Δ_{x1,c}, we see from Tables C.1–C.5 that none of these spectroscopic indicators alone leads to a lower weighted rms of prediction residuals (i.e. Δ_{x1,c} > 0). At best they are consistent with no improvement at all (e.g. pEW(Si ii λ4130), for which Δ_{x1,c} = 0.041 ± 0.033 mag)). The same is true at ages other than maximum light.
Nonetheless, several such indicators compete well with (x_{1},c), even leading to small improvements (albeit statistically insignificant), but only when combined with SALT2 color (pEW(Si ii λ4130) and ℛ(Si)) or in addition to (x_{1},c) (v_{abs}(Si ii λ6355 and d_{abs}(S ii λ5454)). We discuss these indicators in the two following sections.
5.2.1. Spectroscopic indicators in combination with SALT2 color
When used in combination with the SALT2 color parameter, both the pseudoequivalent width of Si ii λ4130 and the ℛ(Si) spectroscopic ratio compete well with the standard (x_{1},c) predictors (Δ_{x1,c} = 0.006 ± 0.014 mag and Δ_{x1,c} = −0.007 ± 0.030 mag). Both indicators are strongly anticorrelated with x_{1} and uncorrelated with SALT2 color (see Fig. 16, left and middle panels), while the correlation with colorcorrected Hubble residual is more pronounced for ℛ(Si) [r = 0.63] than for pEW(Si ii λ4130) [r = 0.35]. In a sense, both indicators act to replace the lightcurve width parameter x_{1}. The anticorrelation of ℛ(Si) with x_{1} has been recovered by several authors since its publication by Nugent et al. (1995), while the relation between pEW(Si ii λ4130) and lightcurve shape has more recently been mentioned by Arsenijevic et al. (2008) and Walker et al. (2010).
Fig. 16 Correlation between pEW(Si ii λ4130) and ℛ(Si) at maximum light and the SALT2 fit parameters (x_{1},c), and colorcorrected Hubble residual. The open circle in the lower panels corresponds to SN 2000dk. 
We show the Hubble residuals obtained when using pEW(Si ii λ4130) and ℛ(Si) in combination with SALT2 color in Fig. 17, where we also show the residuals from the standard (x_{1},c) model. The subluminous (but not 1991bglike) SN 2000dk stands out as a ≲2σ outlier for [c,ℛ(Si)], while this is not the case for (x_{1},c) (the point corresponding to SN 2000dk is highlighted in both Figs. 16 and 17). This single SN contributes a large fraction of the residual scatter (WRMS (incl.00dk) = 0.190 ± 0.025 mag cf. 0.196 ± 0.027 mag for (x_{1},c)), and excluding it from the sample leads to a ~10% decrease in the weighted rms of prediction Hubble residuals, resulting in a ~15% improvement over (x_{1},c) (WRMS (excl.00dk) = 0.171 ± 0.028 mag cf. 0.197 ± 0.028 mag for (x_{1},c)).
Fig. 17 Hubble diagram residuals for pEW(Si ii λ4130) (top) and ℛ(Si) (bottom) at maximum light. In each case we show the Hubble residuals obtained using SALT2 color and the spectroscopic indicator (upper panels), and using the standard SALT2 fit parameters (x_{1},c) (lower panels). We also indicate the weighted rms of Hubble residuals (gray highlighted region). For the ℛ(Si) spectroscopic indicator, we report the weighted rms both including and excluding SN 2000dk (open circle). 
5.2.2. Spectroscopic indicators in addition to the SALT2 fit parameters (x_{1}, c)
When used in addition to the standard SALT2 fit parameters (x_{1},c), both the absorption velocity of Si ii λ6355 and the relative absorption depth of S ii λ5454 result in a ≲ 10% decrease in the weighted rms of prediction residuals (Δ_{x1,c} = −0.020 ± 0.019 mag and Δ_{x1,c} = −0.022 ± 0.030 mag), although the apparent improvement for d_{abs}(S ii λ5454) is due to the fact that the weighted rms of prediction residuals for the (x_{1},c) model is somewhat larger for this particular sample (WRMS = 0.221 ± 0.031 mag). Both these spectroscopic indicators are uncorrelated with x_{1} or color (Fig. 18; the apparent correlation of v_{abs}(Si ii λ6355) with color (r = 0.59) is destroyed if we ignore the one point at c ≈ 0.2), and thus provide additional information independent of lightcurve shape or color. The correlation with (x_{1},c)corrected Hubble residuals (Fig. 18, right panels) is only modest ( r ≈ 0.40 for both indicators), and should be reviewed as more data become publicly available.
Fig. 18 Correlation between v_{abs}(Si ii λ6355) and d_{abs}(S ii λ5454) at maximum light and the SALT2 fit parameters (x_{1},c), and (x_{1},c)corrected Hubble residual. 
We show the Hubble residuals obtained when using v_{abs}(Si ii λ6355) and d_{abs}(S ii λ5454) in addition to the standard (x_{1},c) predictors in Fig. 19. One clearly sees from these diagrams that the impact of the additional spectroscopic indicator is fairly small, as the sign and magnitude of the residuals are almost the same for and (x_{1},c). This is further confirmed by looking up the value for the intrinsic correlation in prediction error for both indicators in Tables C.1 and C.3: ρ_{x1,c} = 0.83 ± 0.06 for v_{abs}(Si ii λ6355) and ρ_{x1,c} = 0.75 ± 0.10 for d_{abs}(S ii λ5454).
Fig. 19 Hubble diagram residuals for v_{abs}(Si ii λ6355) (top), and d_{abs}(S ii λ5454) (bottom) at maximum light. In each case we show the Hubble residuals obtained using the spectroscopic indicator in addition to the SALT2 fit parameters (x_{1},c) (upper panels), and using (x_{1},c) only (lower panels). We also indicate the weighted rms of Hubble residuals (gray highlighted region). 
5.2.3. Results using multiple indicators
We have also considered models involving a linear combination of two spectroscopic indicators (i.e. ) or a ratio of two indicators (i.e. ), also for cases including the SALT2 fit parameters (x_{1},c). No combination of two of these spectroscopic indicators leads to an improvement over the single indicator case, regardless of the age considered.
6. Discussion: do SN Ia spectra really help?
The central question this paper addresses is whether spectra yield useful information to predict distances to SN Ia better than lightcurve width and color alone. The answer to this question can have a significant impact on the way future SN Ia surveys are planned, namely whether or not they should include spectroscopic (or spectrophotometric) capabilities. This has been (and remains!) an active area of discussion for proposals for spaceborne missions within the framework of the Dark Energy Task Force (Albrecht et al. 2009) or the US Astronomy & Astrophysics Decadal Survey^{5}.
Of all the spectroscopic indicators considered in this paper, the concept of flux ratio introduced by Bailey et al. (2009) appears to be the most promising, yielding up to ~30% lower Hubble residual scatter than when using the standard lightcurve parameters. However, given the limited sizes of the SNFactory (58 SN Ia) and CfA (26 SN Ia) samples on which the method has been applied, the results are at best statistically significant at the ≲ 2σ level, and the method should be validated on much larger samples. It should be noted that the measurement of flux ratios requires accurate relative flux calibration, as well as minimal contamination by hostgalaxy light. Both requirements impose strong conditions on future SN Ia surveys that plan to use this method.
The other spectroscopic indicators we consider in this paper are intimately linked to lineprofile shapes of specific SN Ia spectral features. One would have hoped that such a physicallymotivated approach would yield interesting results, but this is not the case. At best, these indicators yield ≲1σ lower residual scatter compared with the standard lightcurve parameters. This is rather disappointing, but also points to potential problems with the measurement method we use. It is largely automated, but requires some human interaction to ensure the correct local maxima used to define the wavelength bounds of each feature are selected. Moreover, while some indicators (such as the absorption velocity v_{abs}) are largely insensitive to hostgalaxy reddening, others (such as the pseudoEW) are strongly affected. Recent unbiased techniques based on wavelet transforms have been proposed that are largely insensitive to these measurement issues (Wagers et al. 2010), and the present analysis could be repeated with such techniques.
Last, given our spectroscopic data we have focused exclusively on the optical region, but there appears to be spectroscopic indicators that correlate with luminosity in other wavelength regions (UV: Foley et al. 2008; NIR: Marion et al., in prep.). An increased spectroscopic sample at these wavelengths might reveal spectroscopic quantities that lead to even more precise distances to SN Ia than optical flux ratios.
7. Conclusions
We have investigated the use of spectroscopic indicators which, when used alone or in conjunction with lightcurve parameters (width and color), predict distances to SN Ia better than when using the standard combination of lightcurve width and color. We have carried our a Kfold crossvalidation analysis on a large spectroscopic data set obtained through the CfA Supernova Program. We constructed and implemented maximum likelihood estimators for the rms intrinsic prediction error of a given method, and the intrinsic covariance of prediction errors of different methods. We used these estimates to compare predictive models for SN Ia distances in a quantitative manner.
We first considered the spectroscopic flux ratios of Bailey et al. (2009), highlighting the importance of an accurate relative flux calibration and the failure of this method for highlyreddened objects (SALT2 color c > 0.5). At maximum light, our best single flux ratio ℛ(6630/4400) from 26 SN Ia at z > 0.015 leads to a ~10% lower weighted rms of crossvalidated prediction Hubble residuals (WRMS = 0.189 ± 0.026 mag) than when using the standard SALT2 lightcurve width (x_{1}) and color (c) parameters (WRMS = 0.204 ± 0.029 mag), at 0.7σ significance. When used in combination with the SALT2 color parameter, our best colorcorrected flux ratio ℛ^{c}(6420/5290) leads to ~15% lower weighted rms (WRMS = 0.175 ± 0.025 mag), at 1.4σ significance. We thus confirm the use of flux ratios in improving distance measurements of SN Ia magnitudes, although the significance of the difference with respect to the standard purely photometric approach is difficult to gauge given our sample size. We also point to differences between the best ratios found in this paper and those reported by Bailey et al. (2009), in part due to the way these ratios are selected: Bailey et al. (2009) select their best ratios based on crosscorrelation coefficients with uncorrected magnitudes, while we directly select them using the rms intrinsic error of crossvalidated distance predictions in the Hubble diagram.
Comparison of our results with synthetic spectra from a 2D survey of delayeddetonation explosion models of Kasen et al. (2009) shows that a large part of the variation in our best single flux ratio ℛ(6630/4400) is intrinsic and not due to reddening by dust. The correlation of this ratio with SN Ia magnitudes is due to the luminositydependent spectroscopic variation in the irongroup dominated absorption features around ~4300 Å. While the models confirm the presence of many flux ratios that correlate strongly with absolute magnitude, significant deviations exist with respect to the data. Such deviations can in principle be exploited to impose strong constraints on SN Ia models.
We extended the analysis of flux ratios to SN Ia spectra at other ages (− 2.5 ≤ t ≤ +7.5 d from maximum light (see Table 4). The best set of predictors overall in this age range is the colorcorrected ℛ^{c}(4610/4260) at t = −2.5 d combined with SALT2 color, which leads to ~30% lower weighted rms of prediction Hubble residuals with respect to (x_{1},c) (WRMS = 0.143 ± 0.020 mag), and to ~40% lower intrinsic prediction error (σ_{pred} = 0.106 ± 0.028 mag), at ~2σ significance. The wavelength bins that constitute this ratio are part of the two prominent spectral absorption features predominantly due to irongroup elements, labeled Fe ii λ4300 and Fe ii λ4800, and which were found to vary intrinsically between SN Ia based on 2D models. Flux ratios at t ≥ +5 d fare worse than at maximum light.
We also considered distance predictions based on two flux ratios. We find that the improvement over the standard (x_{1},c) model is at the ≲2σ level at best, and tends to be worse for ages t ≥ +5 d. At maximum light, our best secondary ratios are ℛ_{2}(5160/5290) and , whose wavelength bins are clustered around the S ii λλ5454,5640 doublet and the irongroupdominated absorption complex Fe ii λ4800. Both ratios measure intrinsic smallscale differences between SN Ia that are uncorrelated with lightcurve shape or color, and thus provide independent information on their luminosity.
We also considered spectroscopic indicators associated with spectral lineprofile morphology: the absorption (and peak) velocity, the fullwidth at half maximum, the relative absorption depth, the pseudoequivalent width, as well as other spectroscopic ratios. None of these spectroscopic indicators alone leads to a lower weighted rms of prediction Hubble residuals. Only when they are combined with SALT2 color do several indicators compete well with the standard predictors. Such is the case of the Si ii λ4130 pseudoEW and spectroscopic ratio ℛ(Si). Both indicators are correlated with x_{1} and act as a replacement to lightcurve shape in the distance prediction. When used in addition to (x_{1},c), the Si ii λ6355 absorption velocity and S ii λ5454 relative absorption depth lead to a small improvement, albeit statistically insignificant. Using a linear combination of two such spectroscopic indicators and ratios thereof leads to no further improvement, whether at maximum light or at other ages.
Do spectra improve distance measurements of SN Ia? Yes, but not as much as we had hoped. The statistical framework developed here should be applied to an independent and larger sample to find out whether the effort of obtaining spectra for a cosmological sample will be repaid with better knowledge of dark energy.
While this paper was in the final stages of the refereeing process, Foley & Kasen (2010) posted a preprint on the arXiv server, in which they reanalyze the results of Wang et al. (2009) to show that SN Ia with different Si ii λ6355 absorption velocities at maximum light have different intrinsic colors. Accounting for these intrinsic color differences reduces the scatter of Hubble residuals by ~30%, while using SN Ia from a “normal” subsample reduces the scatter by ~40%. Although Foley & Kasen (2010) do not crossvalidate their results or comment on their statistical
significance, their analysis suggests that spectroscopy could be used to select a subsample of “wellbehaved” SN Ia for more precise distance measurements.
Online material
Appendix A: Sampling variance of weighted mean square error
If we assume the prediction errors are distributed normally with total variance, Eq. (5): , then the sampling variance of the weighted mean square error is (A.1)The standard error on WRMS is then .
Appendix B: Maximum likelihood estimators for intrinsic prediction error and covariance
Let the intrinsic variance of predictions () from method P,Q be . Denote the intrinsic covariance between distance prediction errors from P and Q as c_{PQ}. The intrinsic correlation is ρ_{PQ} = c_{PQ}/(σ_{P}σ_{Q}). Let θ = (σ_{P},σ_{Q},ρ_{PQ}) be the vector of this triplet of quantities. These parameters can be arranged in an intrinsic covariance matrix: (B.1)We wish to estimate them from the crossvalidated predictions. We derive estimators for this intrinsic covariance using maximum likelihood, and also derive its uncertainty.
We assume that the pair of prediction errors are jointly distributed normally around zero with total covariance (B.2)The measurement error covariance matrix, , contains the measurement variances, , for model P and model Q, on the diagonal, and any covariance due to observational error in the offdiagonal. Since supernova s is subject to the same random peculiar velocity under both models P and Q, the covariance from peculiar velocity dispersion is (B.3)The negative log likelihood for the unknown (σ_{P},σ_{Q},ρ_{PQ}), given the set of distance predictions is (B.4)We numerically maximize the likelihood with the constraints σ_{P},σ_{Q} > 0 and ρ_{PQ} < 1. Once we have found the maximum likelihood estimate (MLE) , we can compute its error by
numerically evaluating the Hessian of the negative log likelihood, ). The sampling covariance (error) of the MLE is estimated from the Fisher information: . The standard errors in each of (σ_{P},σ_{Q},ρ_{PQ}) are the square roots of the diagonal elements of . The offdiagonal elements contain the estimation covariance between the three parameters. If the difference in intrinsic prediction error between the two models is Δ = σ_{P} − σ_{Q}, the sampling variance of Δ is (B.5)where is the (i,j) element of the error covariance matrix of the MLE. This error estimate accounts for covariance from random peculiar velocities and the intrinsic correlation between two models. Notably, a large ρ_{PQ} will affect the significance of the difference, Δ.
From the prediction errors of a single method, {Δμ_{s}} , we can estimate the rms intrinsic prediction error σ_{pred}. The negative log likelihood simplifies to (B.6)The maximum likelihood estimate is found by minimizing this or finding the zero of the score function . If N is large enough, the standard error on can be estimated using the Fisher information at the MLE: (B.7)An estimate of the sampling variance of the maximum likelihood estimate of the intrinsic variance is the inverse of the Fisher information . The standard error of itself is the square root of . This estimate of the intrinsic dispersion “subtracts” out the contribution of random peculiar velocities and measurement error to the total dispersion.
Appendix C: Results for other spectroscopic indicators at maximum light
We present our results using the absorption velocity (v_{abs}), the fullwidth at halfmaximum (FWHM), the relative absorption depth (d_{abs}), the pseudoequivalent width (pEW), and the various spectroscopic ratios ℛ(X) [Eqs. (16)–(22)] in Tables C.1–C.5.
v_{abs} (units of 10^{4} km s^{1}) at maximum light from 10fold CV.
FWHM (units of 10^{2} Å) at maximum light from 10fold CV.
d_{abs} at maximum light from 10fold CV.
pEW (units of 10^{2} Å) at maximum light from 10fold CV.
ℛ(Ca) and ℛ(Si) at maximum light from 10fold CV.
In fact Bailey et al. (2009) find γ = +3.5 ± 0.2, but this is due to a typo in their equation for the distance modulus: γℛ really appears as a negative term in their paper (S. Bailey 2010, priv. comm.).
At the time of writing, the paper by Yu et al. (2009) has not been accepted for publication. Here we refer to the 2nd version of their paper, dated 30th June 2010.
Acknowledgments
We acknowledge many useful conversations with members of the RENOIR group at the CPPM, in particular Florent Marmol and André Tilquin. We thank Stephen Bailey for his patience in explaining the details of his flux ratio measurements and validation procedure. Alex Conley shared a nonpublic custom version of his simple_cosfitter code and provided invaluable help with cosmology fits. We further thank Julien Guy and Gautham Narayan for advice on using the SALT2 lightcurve fitter, and Dan Kasen for sending us the output of his 2D radiative transfer calculations. Support for supernova research at Harvard University, including the CfA Supernova Archive, is provided in part by NSF grant AST 0907903.
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All Tables
Validation of top 5 flux ratios at maximum light from Bailey et al. (2009) (noted B09).
All Figures
Fig. 1 Illustration of the flux ratio measurement. The upper panel shows the input spectrum (deredshifted and corrected for Galactic reddening; here SN 1999gd around maximum light), binned on a logwavelength scale. The gray vertical lines represent the fluxes in characteristic wavelength bins mentioned throughout the text. The lower panel shows the same spectrum corrected for SALT2 color, which is used to measure the colorcorrected flux ratios ℛ^{c}. 

In the text 
Fig. 2 Uncorrected Hubble residual vs. flux ratio ℛ(6630/4400) at maximum light, colorcoded according to the absolute difference in B − V color derived from the spectrum and that derived from the photometry, noted Δ(B − V). The dashed line is a linear fit to the SN Ia with Δ(B − V) < 0.1 mag. The highlyreddened SN 2006br is not shown here. 

In the text 
Fig. 3 Left: Uncorrected Hubble residual vs. flux ratio ℛ(6630/4400) at maximum light for SN Ia at z > 0.005 with Δ(B − V) < 0.1 mag, colorcoded according to the SALT2 color parameter, c. Points corresponding to SN Ia with c > 0.5 are labeled. The dashed line is a linear fit to the SN Ia with c < 0.5. The dotted lines are reddening curves for different values of R_{V}, normalized to the smallest ℛ(6630/4400) value. Right: Colorcorrected Hubble residual vs. colorcorrected flux ratio ℛ^{c}(6420/5290) at maximum light. 

In the text 
Fig. 4 Histogram of the SALT2 color parameter (c) for SN Ia at z > 0.015 (open) and 0.005 < z < 0.015 (hatched). Bins that include SN Ia with c > 0.5 are labeled. 

In the text 
Fig. 5 Hubble diagram residuals for the highestranked flux ratios at maximum light. From top to bottom: prediction residuals using ℛ only; (ℛ,x_{1}); (ℛ^{c},c); (ℛ^{c},x_{1},c); and using the standard SALT2 fit parameters (x_{1},c). In each case we indicate the weighted rms of prediction Hubble residuals (gray highlighted region). 

In the text 
Fig. 6 Correlation between the highestranked (ℛ,ℛ^{c}) at maximum light and the SALT2 fit parameters (x_{1},c). 

In the text 
Fig. 7 Results from 10fold crossvalidation on maximumlight spectra. From top to bottom: ℛ only; (x_{1},ℛ); (c,ℛ^{c}); (x_{1},c,ℛ^{c}). The left column is colorcoded according to the weighted rms of prediction Hubble residuals, while the right column corresponds to the absolute Pearson crosscorrelation coefficient of the correction terms with uncorrected Hubble residuals. 

In the text 
Fig. 8 Weighted rms of prediction Hubble residuals vs. absolute Pearson crosscorrelation coefficient for all flux ratios at maximum light, excluding (top) and including (bottom) the highlyreddened SN 2006br. 

In the text 
Fig. 9 Absolute Pearson correlation coefficients of flux ratios at maximum light with uncorrected absolute magnitudes M_{B} in 2D delayeddetonation SN Ia models of Kasen et al. (2009) (left), and in data from the CfA SN Ia sample (right). 

In the text 
Fig. 10 Absolute restframe Bband magnitude (M_{B}) vs. flux ratio ℛ(6630/4400) at maximum light in 2D SN Ia models of Kasen et al. (2009) (small dots), and in data from the CfA SN Ia sample (A_{V} < 0.45 mag: filled circles, A_{V} > 0.45 mag: open circles). The dotted and dashed lines are linear fits to the models and data, respectively, where models for which ℛ(6630/4400) > 0.25 and data for which A_{V} > 0.45 mag have been excluded from the fit. The slope (Γ) and Pearson correlation coefficient (r) are indicated for both cases. The data have been offset vertically for clarity. Including models for which ℛ(6630/4400) < 0.25 results in Γ = 5.71 ± 0.03 and r = 0.97, while including data with A_{V} > 0.45 mag results in Γ = 4.43 ± 0.47 and r = 0.92. The arrows indicate approximate reddening vectors for different values of R_{V}. 

In the text 
Fig. 11 Results from 10fold crossvalidation on spectra at t = −2.5, +0, +5, +7.5 d. (From top to bottom), colorcoded according to the weighted rms of prediction Hubble residuals. The left column is corresponds to ℛ only, while the right column corresponds to the (c,ℛ^{c}) model. 

In the text 
Fig. 12 Correlation between ℛ^{c}(4610/4260) at t = −2.5 d and the SALT2 fit parameters (x_{1},c). The Pearson coefficient of the correlation with color drops to r = 0.03 if we ignore the two points at c > 0.4. 

In the text 
Fig. 13 Correlation between the highestranked at maximum light and the SALT2 fit parameters (x_{1},c), and the highestranked . 

In the text 
Fig. 14 Wavelength bounds of spectroscopic features for which we measured the various indicators shown in Fig. 15, illustrated using the maximumlight spectrum of SN 2006ax. 

In the text 
Fig. 15 Definition of the main spectroscopic indicators used in this paper, here illustrated using the Si iiλ6355 line profile in the spectrum of SN 2005ki at t = +1 d. The right panel shows the pseudocontinuum (dashed line), as well as the wavelength locations of the blue and red emission peaks (λ_{blue} and λ_{peak}) and their respective heights (h_{blue} and h_{peak}). The wavelength of maximum absorption (λ_{abs}) serves to define the absorption velocity, v_{abs}. The peak velocity v_{peak} is defined analogously. The left panel shows the same line profile normalized to the pseudocontinuum, and serves to define the (relative) absorption depth (d_{abs}), FWHM, and pseudoequivalent width (pEW; shaded gray region). In both panels, the thick line corresponds to the smoothed flux, where we have used the inversevariance weighted Gaussian filter of Blondin et al. (2006) with a smoothing factor dλ/λ = 0.005. 

In the text 
Fig. 16 Correlation between pEW(Si ii λ4130) and ℛ(Si) at maximum light and the SALT2 fit parameters (x_{1},c), and colorcorrected Hubble residual. The open circle in the lower panels corresponds to SN 2000dk. 

In the text 
Fig. 17 Hubble diagram residuals for pEW(Si ii λ4130) (top) and ℛ(Si) (bottom) at maximum light. In each case we show the Hubble residuals obtained using SALT2 color and the spectroscopic indicator (upper panels), and using the standard SALT2 fit parameters (x_{1},c) (lower panels). We also indicate the weighted rms of Hubble residuals (gray highlighted region). For the ℛ(Si) spectroscopic indicator, we report the weighted rms both including and excluding SN 2000dk (open circle). 

In the text 
Fig. 18 Correlation between v_{abs}(Si ii λ6355) and d_{abs}(S ii λ5454) at maximum light and the SALT2 fit parameters (x_{1},c), and (x_{1},c)corrected Hubble residual. 

In the text 
Fig. 19 Hubble diagram residuals for v_{abs}(Si ii λ6355) (top), and d_{abs}(S ii λ5454) (bottom) at maximum light. In each case we show the Hubble residuals obtained using the spectroscopic indicator in addition to the SALT2 fit parameters (x_{1},c) (upper panels), and using (x_{1},c) only (lower panels). We also indicate the weighted rms of Hubble residuals (gray highlighted region). 

In the text 
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