Issue 
A&A
Volume 529, May 2011



Article Number  L4  
Number of page(s)  6  
Section  Letters  
DOI  https://doi.org/10.1051/00046361/201116723  
Published online  08 April 2011 
Letters to the Editor
The reddening law of type Ia supernovae: separating intrinsic variability from dust using equivalent widths^{⋆}
^{1}
Université de Lyon, Université Lyon 1, CNRS/IN2P3, Institut de Physique Nucléaire de Lyon, 69622 Villeurbanne, France
email: nchotard@ipnl.in2p3.fr
^{2}
Physics Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
^{3}
LPNHE, Université Pierre et Marie Curie Paris 6, Université Paris Diderot Paris 7, CNRSIN2P3, 75252 Paris Cedex 05, France
^{4}
Department of Physics, Yale University, New Haven, CT 062508121, USA
^{5}
Physikalisches Institut Universitat Bonn, Nussallee 12 53115 Bonn, Germany
^{6}
Department of Physics, University of California Berkeley, 366 LeConte Hall MC 7300, Berkeley, CA, 947207300, USA
^{7}
Computational Cosmology Center, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
^{8}
Department of Astronomy, University of California, Berkeley, CA 947203411, USA
^{9}
Observatoire de Lyon, 69230 SaintGenis Laval, Université de Lyon, Université Lyon 1, 69003 Lyon, France
^{10}
Australian National University, Mt. Stromlo Observatory, The RSAA, Weston Creek, ACT 2611, Australia
^{11}
Centre de Physique des Particules de Marseille, 163 avenue de Luminy, Case 902, 13288 Marseille Cedex 09, France
^{12}
Tsinghua Center for Astrophysics, Tsinghua University, Beijing 100084, PR China
^{13}
New York University, Center for Cosmology and Particle Physics, 4 Washington Place, New York, NY 10003, USA
^{14}
National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, PR China
Received: 15 February 2011
Accepted: 28 March 2011
We employ 76 type Ia supernovae (SNe Ia) with optical spectrophotometry within 2.5 days of Bband maximum light obtained by the Nearby Supernova Factory to derive the impact of Si and Ca features on the supernovae intrinsic luminosity and determine a dust reddening law. We use the equivalent width of Si ii λ4131 in place of the light curve stretch to account for firstorder intrinsic luminosity variability. The resulting empirical spectral reddening law exhibits strong features that are associated with Ca ii and Si ii λ6355. After applying a correction based on the Ca ii H&K equivalent width we find a reddening law consistent with a Cardelli extinction law. Using the same input data, we compare this result to synthetic restframe UBVRIlike photometry to mimic literature observations. After corrections for signatures correlated with Si ii λ4131 and Ca ii H&K equivalent widths and introducing an empirical correlation between colors, we determine the dust component in each band. We find a value of the totaltoselective extinction ratio, R_{V} = 2.8 ± 0.3. This agrees with the Milky Way value, in contrast to the low R_{V} values found in most previous analyses. This result suggests that the longstanding controversy in interpreting SN Ia colors and their compatibility with a classical extinction law, which is critical to their use as cosmological probes, can be explained by the treatment of the dispersion in colors, and by the variability of features apparent in SN Ia spectra.
Key words: supernovae:general / dust, extinction / cosmology: observations
Table 1 is available in electronic form at http://www.aanda.org
© ESO, 2011
1. Introduction
Type Ia supernovae (SNe Ia) luminosity distances are measured via the standardization of their light curves using brightnesswidth (stretch, x_{1}, Δm_{15}) and color corrections (Phillips 1993; Tripp 1998; Guy et al. 2007; Jha et al. 2007). While the determination of the intrinsic dispersion related to the light curve shape is subject to small differences between fitters, the manner in which color is linked to dust is still controversial, because it may be affected by additional yet unidentified intrinsic dispersion. Whereas earlier work used the totaltoselective extinction ratio of the Milky Way, R_{V} = 3.1, direct estimates from the supernovae (SNe) Hubble diagram fits lead to lower values, from R_{V} = 1.7 to R_{V} = 2.5 (Hicken et al. 2009; Tripp 1998; Wang et al. 2009). While the derivation of this value is subject to assumptions about the natural color dispersion of SNe (Freedman et al. 2009; Guy et al. 2010), the reason for a difference between SNe data and the Milky Way average result have remained unknown.
Equivalent widths are good spectral indicators for addressing this question because they probe the intrinsic variability of SNe Ia and by construction little depend on extinction owing to their narrow wavelength baseline. Arsenijevic et al. (2008) showed a strong correlation between the equivalent width of the Si ii λ4131 feature and the SALT2 x_{1} width parameter (Guy et al. 2007), and Bronder et al. (2008) showed its correlation with M_{B}. Walker et al. (2011) used it to standardize the Hubble diagram, but were hindered from drawing firm conclusions by the quality of the low redshift data.
In this work, we take advantage of the Nearby Supernova Factory (SNfactory) spectrophotometric sample to revisit these conclusions, using both spectral data and derived UBVRIlike synthetic photometry. We present in Sect. 2 the SNe Ia sample and the definition of the Si ii λ4131 and Ca ii H&K equivalent widths, which will be used in Sect. 3.1 to correct the Hubble residuals. These corrected magnitudes are used to derive the relative absorption in each wavelength band, δA_{λ}. The correlations between the δA_{λ} from SN to SN across different bands provide the reddening law, as described in Sect. 3.2, as well as the dispersion matrix between bands. In Sect. 4 we show that the resulting reddening law agrees with a Cardelli extinction law (CCM, Cardelli et al. 1989; O’Donnell 1994). Our reddening law has a value of R_{V} which agrees with the MilkyWay value of 3.1, when the proper dispersion matrix is used. We then discuss these results in Sect. 5 and conclude in Sect. 6.
2. Data set and derived quantities
This analysis uses flux calibrated spectra of 76 SNe Ia obtained by the SNfactory collaboration with its SNIFS instrument (Aldering et al. 2002) on the University of Hawaii 2.2m telescope on Mauna Kea. This subset is selected in the same way as in Bailey et al. (2009), using only SNe with a measured spectrum within 2.5 days of Bband maximum, but with an enlarged data set with a redshift range of 0.007 < z < 0.09. The SALT2 x_{1} and c parameters (Fig. 1) and the spectra phases with respect to the Bband maximum light are derived using fits of the full light curves in three observerframe top hat bandpasses corresponding approximately to BVR.
Fig. 1 Correlations of ew^{Si}, SALT2 x_{1} (top left) and c (top right) parameters and measured peak absolute magnitude up to a constant term, δM_{B} (bottom). The open circles in the bottom panel are the data points excluded from the fit, which is displayed as a solid line. is equivalent for the Bband to the red curve shown in Fig. 3a. 
After flux calibration, host galaxy subtraction, and correction for Milky Way extinction, the 76 spectra are transformed into rest frame. For the spectral analysis, they are rebinned with a resolution of 1500 km s^{1}. In addition, synthetic magnitudes are derived from the spectra after photon integration in a set of five UBVRIlike tophat bandpasses with a constant resolution over the whole spectral range (3276 − 8635 Å, see Fig. 3 and caption). The uncorrected Hubble residuals δM(λ) = M(λ) − ⟨ M(λ) ⟩ are independently computed for each band of mean wavelength λ, relative to a flat ΛCDM universe with Ω_{M} = 0.28 and H_{0} = 70 km s^{1} Mpc^{1}. For the spectral analysis, the Hubble residuals are corrected for phase dependence using linear interpolation. For broad bands, we instead interpolate the magnitudes to the date of the Bmaximum, using the light curve shape of each SN Ia as defined by the SALT2 x_{1} parameter, but with each band’s peak magnitude fitted separately. The data are presented in Table 1. The errors on Hubble residuals include statistical, calibration (~2% for peak luminosity), and redshift uncertainties. They are correlated between bands with a correlation coefficient varying from 0.64 to 0.98.
The equivalent widths ew^{Si} and ew^{Ca} corresponding to the Si ii λ4131 and the Ca ii H&K features are computed in the same way as in Bronder et al. (2008, Eq. (1)). The errors are derived using a Monte Carlo procedure that takes into account photon noise and the impact of the method used to select the feature boundaries. ew^{Si} and ew^{Ca} are insensitive to extinction by dust, changing by less than 1% when adding an artificial reddening with a CCM law with R_{V} = 3.1 and E(B − V) = 0.5.
3. Method
3.1. Intrinsic corrections and absorption measurements
The Hubble residuals exhibit a dependency on observables such as ew^{Si} and ew^{Ca}, which are uncorrelated with dust extinction. As shown in Fig. 1 as an example, the δM_{B} dependence on ew^{Si} exhibits a linear behavior, with an asymmetrical magnitude dispersion attributed to extinction and remaining intrinsic variability. A similar dependence with equivalent widths is found for other bands. We may thus model the Hubble residuals for a given SN, i, as a sum of intrinsic and dust components, making various assumptions about the number of spectral energy distribution (SED) correction vectors, s_{λ}, from none (Eq. (1a)) to two (Eq. (1c)): After finding and the by a leastsquares minimization over all SNe from Eq. (1b), with an asymmetrical 2σ clipping, is kept constant and and the are then computed in the same manner. An example of the fit using ew^{Si} for the Bband is given is Fig. 1 (solid line), being the correction applied to find the .
3.2. The empirical reddening law
Fig. 2 Relation between δA_{U} and δA_{V} after ew^{Si} correction (triangles up) and after ew^{Si} and ew^{Ca} corrections (triangles down). The δA_{U} are displayed with an added arbitrary constant. The ellipses represent the full measured covariance matrix between the two bands. They are enlarged by the additional subtraction error after ew^{Ca} correction. The result of the fits are also displayed. The and respectively correspond to the ones in Figs. 3b and c for the Uband. 
Because we expect the relation between the δA_{λ} to be linear for dust extinction, as shown in Fig. 2 for the U and Vbands, we model the empirical reddening law as (2)where δA_{λ,i} are the measured values, the slopes γ_{λ} ( = ) the reddening law coefficients, the fitted relative extinction for the SN i, and η_{λ} a free zero point. , γ_{λ} and η_{λ} are obtained by a χ^{2} fit using the full wavelength covariance matrix C_{i} of the δA_{λ,i} measurements. However, the data are more dispersed than their measurement error (Fig. 2), and another source of dispersion must be introduced. We adopt an iterative approach by using the fit residuals, , to determine the covariance remaining after accounting for the measurement error covariance, C_{λ1λ2,i}. This empirical covariance matrix, D, is given by (3)where N is the number of SNe. In the next iteration, the total covariance matrix is given by C_{i} + D. We have checked that the converged matrix does not depend on initial conditions.
4. Results
Fig. 3 Black: reddening law presented as as a function of wavelength. Filled circles correspond to the results obtained using the UBVRIlike bands, curves are for the spectral analysis. Red: linear slope s_{λ} (mag/Å) of equivalent widths versus δM_{λ}. Dotted lines: CCM law fit corresponding to the broad bands analysis. Panel a): δM_{λ} corrected only for the phase dependence (Eq. (1a)). Panel b): δM_{λ} corrected for phase and ew^{Si} (Eq. (1b)). Panel c): δM_{λ} corrected for phase, ew^{Si} and ew^{Ca} (Eq. (1c)). The vertical dotted lines represent the UBVRIlike bands boundaries. The shaded vertical bands represent the Si and Ca domain. The shaded band around the curves are the statistical errors. 
4.1. EW^{Si} and EW^{Ca} impacts on the derived extinction law
Results for the SED correction vector, s_{λ}, and the reddening law, γ_{λ}, are presented in Fig. 3 for different assumptions about the number of intrinsic components. If SNe were perfect standard candles affected only by dust as assumed by Eq. (1a) and Fig. 3a, the empirical reddening law would be a CCMlike law with an average R_{V} for our galaxy sample. However, clearly exhibits smallscale SNlike features. These features correlate strongly with some of the features in the ew^{Si} correction spectrum, , derived via Eq. (1b) and illustrated in red in Fig. 3a.
The reddening law obtained after ew^{Si} correction (Fig. 3b) is closer to a CCM law, except in the Ca ii H&K and IR triplet, and the Si ii λ6355 region. This indicates the presence of a second source of intrinsic variability. We select ew^{Ca} to trace a second spectral correction vector, (Eq. (1c)), since Ca is clearly a major contributor to the observed variability and ew^{Ca} and ew^{Si} also happen to be uncorrelated (ρ = 0.06 ± 0.12). As shown in Fig. 3b, does a good job of reproducing the shape of the deviation of relative to the CCM law.
The mean reddening law, , obtained after the additional correction by ew^{Ca} (Fig. 3c) is a much smoother curve with small residual features and agrees well with a CCM extinction law. Thus it appears that these two components can account for SN Ia spectral variations at optical wavelengths. Any intrinsic component that might remain would have to be largely uncorrelated with SN spectral features fixed in wavelength, as well as being coincidentally compatible with a CCM law.
4.2. R_{V} determination
We apply the same treatment as above to our UBVRIlike synthetic photometric bands, and find agreement with the spectral analysis, as shown by the black points in the three panels in Fig. 3. After the ew^{Si} correction (Eq. (1b), Fig. 3b), the U and Iband values deviate significantly from a CCM law, which is recovered after the full ew^{Si} and ew^{Ca} correction (Eq. (1c), Fig. 3c). The empirical fit presented in Sect. 3.2 can be forced to follow a CCM extinction law by substituting , where a_{λ} and b_{λ} are the wavelengthdependent parameters given in Cardelli et al. (1989) and O’Donnell (1994), and a single R_{V} is fit over all bands. This fit applied to leads to an average R_{V} = 2.78 ± 0.34 for the SN host galaxies in our sample. This value is compatible with the Milky Way average of R_{V} = 3.1 The quoted uncertainty is statistical and derived with a jackknife procedure, removing one supernova at a time.
We tested the robustness of our R_{V} determination in several ways. Since the s_{λ} are measured in sequence, each one using the corrected magnitudes from the previous step, we swap the order in which the correction is applied, and find R_{V} = 2.70. Using the whole sample to compute the s_{λ} instead of applying a 2σ clipping cut leads to R_{V} = 2.79, which shows the small influence of clipping at this stage. Host galaxy subtraction is performed using the full spatiospectral information from the host obtained after the SN Ia has faded, and we do not see evidence for residual host galaxy features in our spectra. The measurement covariance matrix C_{i} depends on the assumed calibration accuracy: for the present result, this is estimated from repeated measurements of standard stars. Another estimate can be obtained from the residuals to a SALT2 lightcurve fit. Using this in C_{i} values leads to R_{V} = 2.76. The stability of the result with respect to our choice of bandpasses is checked by removing one band at a time, and we obtain a rms of ± 0.23 around the mean result. Finally, a Monte Carlo simulation was performed using the converged γ_{λ}, , η_{λ} and dispersion matrix D as an input, and setting R_{V} = 2.78. The δA_{λ,i} are then randomly generated adding Gaussian noise with covariance C_{i} + D. Over 100 generations, the D matrix is recovered with a maximal bias of 5% on the diagonal elements and the mean fitted value is R_{V} = 2.68 ± 0.03, indicating that, if anything, our method slightly underestimates R_{V}.
5. Discussion
In this work, we selected ew^{Si} as a first variable for correction because it provides a good proxy for x_{1}, and it because is a modelindependent variable. In our dataset the Pearson correlation coefficient of ew^{Si} with x_{1} is − 0.83 ± 0.04 (Fig. 1), which confirms the result found in Arsenijevic et al. (2008). Computing the Hubble residuals in the Bband using (x_{1},c) or (ew^{Si},c) to standardize SNe both lead to a dispersion of residuals of 0.16 mag, also confirming this result. We find that ew^{Si} is uncorrelated with c (ρ = −0.04 ± 0.12, Fig. 1), contrary to the recent claim by Nordin et al. (2010). Because ew^{Si} and x_{1} are highly correlated, it is possible to redo the analysis with x_{1} in place of ew^{Si}. The resulting SED correction vector is similar to , and the conclusions identical, yielding R_{V} = 2.69.
Computing the Hubble residuals with (ew^{Si},ew^{Ca},c) to standardize supernovae leads to a dispersion of 0.15 mag, which is a small improvement. Indeed, we do observe a correlation of ew^{Ca} and c, with ρ = 0.34 ± 0.10. This correlation is even increased to ρ = 0.50 when c is computed with a Uband in addition to BVR in the lightcurve fit. This shows that c contains an intrinsic component, and that the SALT2 analysis is already accounting for some of this Ca effect. This is illustrated in Fig. 3b: the SALT2 color model is derived taking into account a variability measured by one intrinsic parameter only, and thus can be compared to . The observed Uband contribution in broad filters explains the UV rise in the empirical color models. The presence of a second intrinsic parameter induces an increased variability in the UV, as was already noticed by Ellis et al. (2008), but without a clear attribution to the Ca ii H&K line.
Much lower effective R_{V} values have been found previously: R_{V} = 1.1 in Tripp (1998), R_{V} = 2.2 in Kessler et al. (2009); Guy et al. (2010) and R_{V} ≈ 1 − 2 in Folatelli et al. (2010). These values were derived accounting only for one intrinsic parameter beyond color. If we derive an effective value after the sole ew^{Si} correction to mimic these analyses, we obtain R_{V} = 3.1, so the explanation for the difference has little to do with the number of intrinsic parameters entering the correction. This difference is explained by the assumption on the dispersion matrix. Indeed, if we instead use the one of Guy et al. (2010), corresponding to an rms between 0.09 and 0.11 mag on the diagonal, and all offdiagonal terms with an identical rms of 0.09 mag, we obtain R_{V} = 1.86. This value is representative of previous analyses, but the matrix used is incompatible with our bestfit matrix. D has diagonal values of 0.07, 0.04, 0.05, 0.05 and 0.10 for UBVRI respectively, lower than the values quoted by Guy et al. (2010). However, while adjacent bands are correlated in our matrix, an anticorrelation growing to − 1 when the wavelength difference increases is observed for other bands, which implies a large color dispersion. As shown by Freedman et al. (2009) and Kessler et al. (2009), increasing the color dispersion leads to a higher R_{V} value. This longrange anticorrelation implies that uncorrected variability in the SN spectra and/or the reddening law remains.
6. Conclusions
Equivalent widths are essentially independent of host reddening and provide a handle on the intrinsic properties of the SNe Ia.
In particular, ew^{Si} measured at maximum light is confirmed to be as powerful as stretch. After correction for this intrinsic contribution to the magnitude, we derived a reddening law and proposed a natural method to assess the dispersion caused by additional fluctuations. We found that this empirical reddening law is affected by features linked to the SN physics, and showed that the Ca ii H&K line provides a second intrinsic variable that is uncorrelated with ew^{Si} or x_{1}. Correcting Hubble residuals with ew^{Si} and ew^{Ca} leads to a reddening law consistent with a canonical CCM extinction law, while adding a dispersion in colors during the fit leads to a value of R_{V} close to the Milky Way value of 3.1. Owing to the coupling of R_{V} with the residual dispersion matrix, the derived value of R_{V} may be affected as the remaining variability becomes better understood and corrected. Our plan to study the same SNe Ia over a range of phases should help address this issue. The evolution of SNe Ia will be one of the main limitations to their use in precision cosmology surveys at high redshift. Our findings show that the accurate measurements of the Ca ii H&K line provides an additional tool to improve the separation of dust and intrinsic variability. The presence of remaining scatter offers the possibility of improvements in the future.
Online material
Supernovae used in this work, including synthetic magnitudes after phase correction δM_{λ}, SALT2 fit parameters x_{1} and c, equivalent widths ew^{Si} and ew^{Ca} in Å and phases in days.
Acknowledgments
We are grateful to the technical and scientific staff of the University of Hawaii 2.2m telescope, Palomar Observatories, and the High Performance Research and Education Network (HPWREN) for their assistance in obtaining these data. We also thank the people of Hawaii for access to Mauna Kea. This work was based in part on observations with UCO facilities (Keck and Lick 3 m) and NOAO facilities (GeminiS (program GS2008BQ26), SOAR, and CTIO 4 m). We thank UCO and NOAO for their generous allocations of telescope time. We thank Julien Guy for fruitful discussions on color law derivation as well as the anonymous referee and Alex Kim for constructive comments on the text. This work was supported in France by CNRS/IN2P3, CNRS/INSU, CNRS/PNC, and used the resources of the IN2P3 computer center. This work was supported by the DFG through TRR33 “The Dark Universe”, and by National Natural Science Foundation of China (grant 10903010). This work was also supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics and the Office of Advanced Scientific Computing Research, of the US Department of Energy (DOE) under Contract Nos. DEFG0292ER40704, DEAC0205CH11231, DEFG0206ER0604, and DEAC0205CH11231; by a grant from the Gordon & Betty Moore Foundation; by National Science Foundation Grant Nos. AST0407297 (QUEST), and 0087344 & 0426879 (HPWREN); by a Henri Chretien International Research Grant administrated by the American Astronomical Society; the FranceBerkeley Fund; by an Explora Doc Grant by the Region Rhone Alpes; and the Aspen Center for Physics.
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All Tables
Supernovae used in this work, including synthetic magnitudes after phase correction δM_{λ}, SALT2 fit parameters x_{1} and c, equivalent widths ew^{Si} and ew^{Ca} in Å and phases in days.
All Figures
Fig. 1 Correlations of ew^{Si}, SALT2 x_{1} (top left) and c (top right) parameters and measured peak absolute magnitude up to a constant term, δM_{B} (bottom). The open circles in the bottom panel are the data points excluded from the fit, which is displayed as a solid line. is equivalent for the Bband to the red curve shown in Fig. 3a. 

In the text 
Fig. 2 Relation between δA_{U} and δA_{V} after ew^{Si} correction (triangles up) and after ew^{Si} and ew^{Ca} corrections (triangles down). The δA_{U} are displayed with an added arbitrary constant. The ellipses represent the full measured covariance matrix between the two bands. They are enlarged by the additional subtraction error after ew^{Ca} correction. The result of the fits are also displayed. The and respectively correspond to the ones in Figs. 3b and c for the Uband. 

In the text 
Fig. 3 Black: reddening law presented as as a function of wavelength. Filled circles correspond to the results obtained using the UBVRIlike bands, curves are for the spectral analysis. Red: linear slope s_{λ} (mag/Å) of equivalent widths versus δM_{λ}. Dotted lines: CCM law fit corresponding to the broad bands analysis. Panel a): δM_{λ} corrected only for the phase dependence (Eq. (1a)). Panel b): δM_{λ} corrected for phase and ew^{Si} (Eq. (1b)). Panel c): δM_{λ} corrected for phase, ew^{Si} and ew^{Ca} (Eq. (1c)). The vertical dotted lines represent the UBVRIlike bands boundaries. The shaded vertical bands represent the Si and Ca domain. The shaded band around the curves are the statistical errors. 

In the text 
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