A HUBBLE DIAGRAM FROM TYPE II SUPERNOVAE BASED SOLELY ON PHOTOMETRY: THE PHOTOMETRIC COLOR METHOD^*

The Carnegie Supernova Project¹⁵ (CSP; Hamuy et al. 2006) provided all the photometric and spectroscopic data for this project. The goal of the CSP was to establish a high-cadence data set of optical and NIR light curves in a well defined and well understood photometric system and obtain optical spectra for these same SNe. Between 2004 and 2009, the CSP observed many low-redshift SNe II (N_SNe ∼ 100 with z ≤ 0.04), 56 of which had both optical and NIR light curves with good temporal coverage. This was one of the largest NIR data samples. Two SN 1987A-like events were removed (SN 2006V and SN 2006au; see Taddia et al. 2012), leaving the sample listed in Table 1 with photometric parameters measured by Anderson et al. (2014b). Note that we do not include SNe IIb or SNe IIn.

2.2.1. Photometry

2.2.2. Spectroscopy

In the V band the determination of AvG can be applied using the extinction maps of Schlafly & Finkbeiner (2011). To convert AvG to extinction values in other bands, we need to adopt an extinction law and the effective wavelength for each filter.

SN II spectra evolve with time from a blue continuum at early times to a redder continuum with many absorption/emission features at later epochs. This implies that the effective wavelength of a broadband filter also changes with time (see the formula given Bessell & Murphy 2012, A.21). To calculate effective wavelengths at different epochs, we adopt a sequence of theoretical spectral models from Dessart et al. (2013) consisting of a SN progenitor with a main-sequence mass of 15 M_⊙, solar metallicity Z = 0.02, zero rotation, and a mixing length parameter of 3.¹⁷ The choice of this model is based on the fact that it provided a good match to a prototypical SN II such as SN 1999em. For each photometric epoch, we choose the closest theoretical spectrum in each epoch since the explosion, the extinction law from Cardelli et al. (1989), and in time R_V = 3.1 to obtain the MW extinction in the other filters.

Having corrected the observed magnitudes for Galactic extinction, we also need to apply a correction attributable to the expansion of the universe called the KC. A photon received in one broad photometric bandpass in the observed referential has not necessarily been emitted (rest-frame referential) in the same filter, which is why this correction is needed. For each epoch of each filter we use the same procedure to estimate the KC. Here we describe our method step by step for one epoch and a given filter X.

• 1.  
  We choose in our model spectral library (Dessart et al. 2013, model m15mlt3) the theoretical spectrum (rest frame) closest to the photometric epoch since the explosion time (corrected for time dilatation), with a rest-frame spectral energy distribution (SED), f^rest(λ_rest). Because our library covers a limited range of epochs from 12.2 to 133 days relative to explosion, observations outside these limits are ignored.
• 2.  
  We bring the rest-frame theoretical spectrum to the observer's frame using the (1 + z_hel) correction, where z_hel is the heliocentric redshift of the SN, f^obs(λ) = f^rest(λ_rest(1 + z_hel)) × 1/(1 + z_hel), where λ is the wavelength in the observer's frame.
• 3.  
  We match the theoretical spectrum to the observed photometric magnitudes of the SN (Hsiao et al. 2007). For this we calculate synthetic magnitudes (from the model in the observer's frame, f^obs(λ)) and compare them to the observed magnitudes corrected for MW extinction. We use all the filters available at this epoch. Then we obtain a warping function W(λ) (quadratic, cubic, depending on the number of filters used) and do a constant extrapolation for the wavelengths outside of the range of filters used. With our warping function, we correct our model spectrum and obtain ${f}_{\mathrm{warp}}^{\mathrm{obs}}(\lambda )=W(\lambda )\times {f}^{\mathrm{obs}}(\lambda )$. We compute the magnitude in the observer's frame:

  $$\begin{eqnarray*}&&{m}_{z}^{X}=-2.5{\mathrm{log}}_{10}\left(\displaystyle \frac{1}{\mathrm{hc}}\int {f}_{\mathrm{warp}}^{\mathrm{obs}}(\lambda ){S}_{\lambda }^{X}\lambda d\lambda \right)+{\mathrm{ZP}}_{X},\end{eqnarray*}$$

  where c is the light velocity in Å s⁻¹, h the Planck constant in erg s, λ is wavelength, ${S}_{\lambda }^{X}$ the transmission function of filter X, and ZP_X is the zero point of filter X (see Contreras et al. 2010; Stritzinger et al. 2011).
• 4.  
  We bring back the warping spectrum to the rest frame ${f}_{\mathrm{warp}}^{\mathrm{rest}}(\lambda )=(1+{z}_{\mathrm{hel}}){f}_{\mathrm{warp}}^{\mathrm{obs}}(\lambda \times 1/(1+{z}_{\mathrm{hel}}))$ and we obtain and calculate the magnitude:

  $$\begin{eqnarray*}&&{m}_{0}^{X}=-2.5{\mathrm{log}}_{10}\left(\displaystyle \frac{1}{\mathrm{hc}}\int {f}_{\mathrm{warp}}^{\mathrm{rest}}(\lambda ){S}_{\lambda }^{X}\lambda d\lambda \right)+{\mathrm{ZP}}_{X}.\end{eqnarray*}$$
• 5.  
  Finally, we obtain the KC for this epoch as the difference between the observed and the rest-frame magnitude, ${\mathrm{KC}}_{X}={m}_{z}^{X}-{m}_{0}^{X}$.
• 6.  
  To estimate the associated errors, we follow the same procedure but instead of using the observed magnitudes for the warping, we use the upper limit, i.e., observed magnitudes plus associated uncertainties.

As a complementary work on the KC and to validate our method, we compare the KC values found using the Dessart et al. (2013) model to those computed from our database of observed spectra. In both cases we use exactly the same procedure. First, the observed spectrum is corrected in flux using the observed photometry (corrected for AvG) in order to match the observed magnitudes. The photometry is interpolated to the spectral epoch. In Figure 1 we show a comparison between the KC obtained with the theoretical models and using our library of observed spectra at different redshifts. As we can see, the KC values calculated with both methods are very consistent. This exercise validates the choice of using the Dessart et al. (2013) models to calculate the KC. There are two advantages to use the theoretical models. First, we can obtain the KC for NIR filters (Y, J, H) for which we do not have observed spectra, and second, this method does not require observed spectra which are expensive to obtain in terms of telescope time, and virtually impossible to get at higher redshifts.

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To correct and standardize the apparent magnitude, we use two photometric parameters: s₂, which is the slope of the plateau measured in the V band (Anderson et al. 2014b), and a color term at a specific epoch. The color term is mainly used to take into account the dispersion caused by the host galaxy extinction. The magnitude is standardized using a weighted least-squares routine by minimizing the equation below:

$$\begin{eqnarray}&&{m}_{\lambda 1}+\alpha {s}_{2}-{\beta }_{\lambda 1}({m}_{\lambda 2}-{m}_{\lambda 3})=5\mathrm{log}({cz})+\mathrm{ZP},\end{eqnarray} \tag{ 1 }$$

where c is the speed of light, z the redshift, ${m}_{{\lambda }_{\mathrm{1,2,3}}}$ the observed magnitudes with different filters, and corrected for AvG and KC, while α, ${\beta }_{{\lambda }_{1}}$, and ZP are free-fitting parameters. The errors on these parameters are derived assuming a reduced chi square equal to one. In order to obtain the errors on the standardized magnitudes, an error propagation is performed in an iterative manner. Note that ${\beta }_{{\lambda }_{1}}$ is related to host galaxy R_V if we assume that the color–magnitude relation is due to extrinsic factors (the intrinsic color is degenerate with the ZP). We obtain:

$$\begin{eqnarray}&&{\beta }_{\lambda 1}=\displaystyle \frac{{A}_{\lambda 1}}{E({m}_{\lambda 2}-{m}_{\lambda 3})},\end{eqnarray} \tag{ 2 }$$

where A_{λ 1} is the host galaxy extinction in the λ1 filter and E the color excess. Assuming a Cardelli et al. (1989) law, there is a one-to-one relationship between R_V and ${\beta }_{{\lambda }_{1}}$. First we obtain the theoretical β for different R_V values using the Cardelli et al. (1989) coefficients (a and b):

$$\begin{eqnarray}&&\beta ({R}_{V})=\displaystyle \frac{{a}_{\lambda 1}+\displaystyle \frac{{b}_{\lambda 1}}{{R}_{V}}}{({a}_{\lambda 2}+\displaystyle \frac{{b}_{\lambda 2}}{{R}_{V}})-({a}_{\lambda 3}+\displaystyle \frac{{b}_{\lambda 3}}{{R}_{V}})}.\end{eqnarray} \tag{ 3 }$$

Then we derive R_V from the value of β_λ1 determined from the least-squares fit (Equation (1)). We will discuss the resulting R_V values in Section 6.5.

We select only SNe located in the Hubble flow, i.e., with cz_CMB ≥ 3000 km s⁻¹ in order to minimize the effect of peculiar galaxy motions. Our available sample is composed of the entire sample in the Hubble flow except for 3 SNe. We eliminate two SNe due to the fact that the warping function cannot be computed, thus the KC (SN 2004ej and SN 2008K). We also take out the outlier SN 2007X and found for this object particular characteristics like clear signs of interaction with the circumstellar medium (flat H alpha P-Cygni profile; see C. P. Gutiérrez et al. 2016, in preparation).

SNe II are supposedly characterized by similar physical conditions (e.g., temperature) when they arrive toward the end of the plateau (Hamuy & Pinto 2002), which is why we use the end of the optically thick phase measured in the V band (as defined by Anderson et al. 2014b) as the time origin in order to bring all SNe to the same timescale. When the end of the plateau is not available, we choose 80 days post-explosion, which is the average for our sample.

Given that SNe II show a significant dispersion in the plateau duration driven by different evolution speeds, we decide to take a fraction of the plateau duration and not an absolute time to ensure that we compare SNe II at the same evolutionary phase. Thus, in the following analysis, we adopt OPTd*X% as the time variable where OPTd is the optically thick phase duration and X is percentage ranging between 1% and 100%.

In Figure 2 we present the variation with evolutionary phase of the dispersion in the Hubble diagram using the filters available and the (V − i) color. The lowest rms values in the optical is found for the r band, and at NIR wavelength using the Y/J band. Note that the coverage in the Y band is better than in the J band, hence hereafter we use the Y band. For these two bands we can obtain the median rms over all the epochs (from 0.2*OPTd to 1.0*OPTd) and the standard deviation. We find for the r band 0.47 ± 0.04 mag and for the Y band 0.48 ± 0.04 mag. In Figure 3 we do the same as above but we change colors. Fixing the r band and using different colors, we show the variation of the rms. This figure shows that the color that minimizes the rms is (V − i) ((r − J) yields a lower dispersion but the time coverage is significantly less). We find a median rms over all the epochs of 0.47 ± 0.05. For this reason we decide to combine the r band and the (V − i) color for the Hubble diagram. Note also that the best epoch for the r band is close to the middle of the plateau, 55% of the time from the explosion to the end of the plateau, whereas the best epoch in the Y band is later in phase post-explosion, around 65%. In general, the best epoch to standardize the magnitude is between 60% and 70% of the OPTd for NIR filters and for optical filters between 50% and 60% of OPTd. Physically these epochs correspond in both cases more or less to the middle of the plateau. Note that we tried other time origins such as the epoch of maximum magnitude instead of the end of the plateau, but changing the reference does not lower the rms.

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In Figure 4 we present a Hubble diagram based entirely on photometric data using s₂ and color term for two filters, the r band and the Y band. In the r band the rms is 0.44 mag (with 38 SNe) which allows us to measure distances with an accuracy of ∼20%. We find the same precision using the Y' band with a rms of 0.43 mag (30 SNe). Note that the color term is more important for the optical filter than for the NIR filter. Indeed, for the r band the rms decreases from 0.50 to 0.44 mag when the color term is added, whereas for the NIR filter the improvement is only of 0.004 mag. Using all available epochs, we find a mean improvement of 0.025 ± 0.011 in the r band and 0.014 ± 0.013 in the Y band. This shows that the improvement is significant in the optical but less in the NIR. The drop using the optical filter is not surprising because this term is probably at least partly related to host galaxy extinction which is more prevalent in optical wavelengths than in the NIR, so adding a color term for NIR filters does not significantly influence the dispersion. Note that if we use the weighted root mean square (WRMS) as defined by Blondin et al. (2011) we find 0.40 mag and 0.36 mag for the r band and Y band, respectively, after s₂ and color corrections.

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In the literature, a majority of the studies used SNe IIP for their sample. To check if we can include all the SNe II (fast and slow decliners), we did some analysis of the SNe and investigate if any of the higher residuals arise from intrinsic SN properties. The overall conclusion is that, at least to first order, we did not find any correlation between SNe II intrinsic differences (s₂, OPTd,...) and the Hubble residuals. This suggests that SNe within the full range of s₂ values (i.e., all SNe II) should be included in Hubble diagram.

Following the work of Folatelli et al. (2010) for SNe Ia, we investigated the combined Hubble diagram using all the filters available (by averaging the distance moduli derived in each filter) but the dispersion obtained is not much better. We found the same correlation between the distance modulus residuals in one band versus those in another band, as found by Folatelli et al. (2010), suggesting that the inclusion of multiple bands does not improve the distance estimate.

If we include SNe in the Hubble flow (cz ≥ 3000 km s⁻¹) and very nearby SNe (cz ≤ 3000 km s⁻¹) for the r band, the dispersion increases from 0.44 to 0.48 mag (46 SNe), whereas in the Y band the rms increase from 0.43 mag to 0.45 (41 SNe).

We also try to use two different epochs, one for the magnitude and the other for the color, but again this does not improve the rms. Finally, we try also to use the total decline rate (between maximum to the end of the plateau) instead of the plateau slope. Using the total decline rate does not lower the rms (dispersion around 0.47 mag for 45 SNe in the r band) but could be useful for high redshift SNe.

To apply the SCM, we need to measure the velocity of the SN ejecta. One of the best features is Fe iiλ 5018 because other iron lines such as Fe iiλ 5169 can be blended by other elements. Expansion velocities are measured through the minimum flux of the absorption component of P-Cygni line profile after correcting the spectra for the heliocentric redshifts of the host galaxies. Errors were obtained by measuring many times the minimum of the absorption changing the trace of the continuum. The range of velocities is 1800–8000 km s⁻¹ for all the SNe. Because we need the velocities for different epochs in order to find the best epoch (as done for the PCM), i.e., with less dispersion, we do an interpolation/extrapolation using a power law (Hamuy 2001) of the form:

$$\begin{eqnarray}&&V(t)=A\times {t}^{\gamma },\end{eqnarray} \tag{ 4 }$$

where A and γ are two free parameters obtained by least-squares minimization for each individual SN and t the epoch since explosion. In order to obtain the velocity error, we perform a Monte Carlo simulation, varying randomly each velocity measurement according to the observed velocity uncertainties over more than 2000 simulations. From this, for each epoch (from 1 to 120 days after explosion) we choose the velocity as the average value and the incertainty to the standard deviation of the simulations. The median value of γ is −0.55 ± 0.25. This value is comparable with the value found by other authors (−0.5 for Olivares et al. 2010, −0.464 by Nugent et al. 2006, and −0.546 by Takáts & Vinkó 2012). Note that, as found by Faran et al. (2014a), the iron velocity for the fast decliners (SNe IIL) also follow a power law but with more scatter. Indeed, for the slow decliners (s₂ ≤ 1.5), we find a median value γ = −0.55 ± 0.18, whereas for the fast decliners (s₂ ≥ 1.5) we obtain γ = −0.56 ± 0.35. More details will be published in an upcoming paper (Gutiérrez et al.).

To standardize the apparent magnitude, we perform a least-squares minimization on:

$$\begin{eqnarray}&&{m}_{\lambda 1}+\alpha \mathrm{log}(\displaystyle \frac{{v}_{\mathrm{Fe}\;{\rm{II}}}}{5000\;\mathrm{km}\;{{\rm{s}}}^{-1}})-{\beta }_{{\lambda }_{1}}({m}_{\lambda 2}-{m}_{\lambda 3})\\ &&\qquad =5\mathrm{log}({cz})+\mathrm{ZP},\end{eqnarray} \tag{ 5 }$$

where c, z, and m_λ1,2,3 are defined in Section 4.1 and α, ${\beta }_{{\lambda }_{1}}$, and ZP are free-fitting parameters. The errors on the magnitude are obtained in the same way as for the PCM but the epoch is different. For the SCM, the photospheric expansion velocity is very dependent on the explosion date. This is why, after trying different epochs and references, we found that the best reference is the explosion time as used in Nugent et al. (2006), Poznanski et al. (2009), and Rodríguez et al. (2014). The same epoch for the magnitude, color, and iron velocity is employed. Just like for the PCM, we use the same color, (V − i) and the same filters (r, Y band). For some SNe, we are not able to measure an iron velocity due to the lack of spectra (only one epoch) and our sample is thus composed of 26 SNe.

In Figure 5 we present the Hubble diagram and the residual for two different filters. The dispersion is 0.29 mag (or 0.30–0.28 mag in WRMS for the Y band and r band, respectively) for 24 SNe (some SNe do not have color at this epoch). These values are somewhat better than previous studies (Hamuy & Pinto 2002; Nugent et al. 2006; Poznanski et al. 2009; D'Andrea et al. 2010; Olivares et al. 2010) where the authors found dispersions around 0.30 mag with 30 SNe (more details in Section 6.3). Note the major differences between our study and theirs is that they included very nearby SNe (cz ≤ 3000 km s⁻¹), only slow declining SNe II (SNe II with low s₂, historically referred to as SNe IIP), did not calculate a power law for each SN as we do, and used a different epoch. Note also the work done by Maguire et al. (2010), where they applyed the SCM to NIR filters (J band and (V − J) color) using nearby SNe (92% of their sample with cz ≤ 3000 km s⁻¹), finding a dispersion of 0.39 mag with 12 SNe (see Section 6.3). To finish, we tried a combination of the PCM and SCM, i.e., adding a s₂ term to the SCM but this does not improve the dispersion.

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A significant fraction of values from Anderson et al. (2014b) do not correspond to the slope of the plateau but sometimes to a combination of s₁ (initial decline) and s₂. Indeed, for some SNe, it was impossible to distinguish two slopes and the best fit was only one slope. For this reason, we decide to define a new sample composed only by 12 SNe with values of s₁ and s₂ and with cz_CMB ≥ 3000 km s⁻¹. From this sample and using the r/(V − i) combination, we obtain a dispersion of 0.39 mag with 12 SNe, which compares to 0.48 mag from the entire sample. From the Y band, the dispersion drops considerably from 0.44 to 0.18 mag with only 8 SNe. However, this low value should be taken with caution due to possible statistical effects which are discussed later (see Section 6.4).

In Figures 6 and 7 we compare the Hubble diagram obtained using the SCM and the PCM. For both methods we use the same SNe (Hubble flow sample) and the same set of magnitude–color. The dispersion using the r band and the Y band is 0.43 mag for the PCM, whereas for the SCM is 0.29.

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In general, the SCM is more precise than the PCM but the dispersion found with the PCM is consistent with the results found by the theoretical studies done by Kasen & Woosley (2009; distances accurate to ∼20%) but the authors used other photometric correlations (plateau duration). Unfortunately, as suggested by Anderson et al. (2014b), using this parameter the prediction is not seen in the observations. We tried to use the OPTd values as an input instead of the s₂ and we did not see any improvement on the dispersion. Note also the recent work of Faran et al. (2014b), in which the authors found a correlation between the iron velocity and the I-band total decline rate. Although in this paper we do not use the total decline rate but another quantity related to the plateau slope, our work confirms the possibility of using photometric parameters instead of spectroscopic.

In this section we compare our SCM with other studies. First, we use only optical filters to compare with Poznanski et al. (2009) and Olivares et al. (2010). Both studies used the (V − I) color and also the I band. Note that Olivares et al. (2010) also used the B and V bands but here we consider only the I band for consistency. Poznanski et al. (2009) found a dispersion of 0.38 mag using 40 slow decliners. In our sample, instead of using the I band we used the sloan filter, i band, and (V − i) color. Using our entire sample, i.e., SNe (37 SNe in total for all the redshift range), we derive a dispersion similar to Poznanski et al. (2009) of 0.32 mag (epoch: 35 days after explosion). We can also compare the parameter α derived from the fit. Again we obtain a consistent value, α = 4.40 ± 0.52, whereas Poznanski et al. (2009) found α = 4.6 ± 0.70. The other parameters are not directly comparable due to the fact that the authors assumed an intrinsic color which is not the case in the current work. Using a Hubble constant (H₀) equal to 70 km s⁻¹ Mpc⁻¹, we can translate our ZP to an absolute magnitude (ZP = M_corr − 5log(H₀) + 25) M_i = −17.12 ± 0.10 mag that it is lower than the results obtained by Poznanski et al. (2009; M_I = −17.43 ± 0.10 mag). This difference is probably due to the fact that the corrected magnitude has not been corrected for the intrinsic color in our work.

Using 30 slow declining SNe in the Hubble flow and very nearby SNe (z between 0.00016 and 0.05140), (V − I) color, and the I band, Olivares et al. (2010) derived a dispersion of 0.32 mag which is the same as what we obtained. However, the parameters derived by Olivares et al. (2010) are different. Indeed, using the same Equation (5) and the entire sample they obtained α = 2.62 ± 0.21, β = 0.60 ± 0.09, and ZP = −2.23 ± 0.07 instead of α = 4.40 ± 0.52, β = 0.98 ± 0.31, and ZP = −1.34 ± 0.10 for us. From their ZP (H₀ = 70 km s⁻¹ Mpc⁻¹) we derive M_I = −18.00 ± 0.07 mag (M_i = −17.12 ± 0.15 mag for us). When the authors restrict the sample to objects in the Hubble flow, they end up with 20 SNe and a dispersion of 0.30 mag. If we perform the same cut, we find a dispersion of 0.29 for 24 SNe. We obtain consistent dispersion for both samples using similar filters. Note that reducing our sample to slow decliners alone (s₂ ≤ 1.5, the classical SNe IIP in other studies) in the Hubble flow does not improve the dispersion. As mentioned in Section 5.3, the difference in dispersion between Olivares et al. (2010) and our study can be due, among other things, to the difference in epoch used, or that we calculate a power law for each SN for the velocity.

With respect to the NIR filters, Maguire et al. (2010) suggested that it may be possible to reduce the scatter in the Hubble diagram to 0.1–0.15 mag and this should then be confirmed with a larger sample and more SNe in the Hubble flow. The authors used 12 slow decliners but only 1 SN in the Hubble flow. Using the J band and the color (V − J), they found a dispersion of 0.39 mag against 0.50 mag using the I band. From this drop in the NIR, the authors suggested that using this filter and more SNe in the Hubble flow could reduce the scatter from 0.25 to 0.3 mag (optical studies) to 0.1–0.15 mag. With the same filters used by Maguire et al. (2010), and using the Hubble flow sample, we find a dispersion of 0.28 mag with 24 SNe. This dispersion is 0.1 mag higher than that predicted by Maguire et al. (2010; 0.1–0.15 mag). To derive the fit parameters, the authors assumed an intrinsic color (V − J)₀ = 1 mag. They obtained α = 6.33 ± 1.20 and an absolute magnitude M_J = −18.06 ± 0.25 mag (H₀ = 70 km s⁻¹ Mpc⁻¹). If we use only the SNe with cz_CMB ≥ 3000 km s⁻¹ (24 SNe), we find α = 4.64 ± 0.64 and ZP = −2.44 ± 0.18, which corresponds to M_J = −18.21 ± 0.18 mag assuming H₀ = 70 km s⁻¹ Mpc⁻¹. If we include all SNe at any redshift, the sample goes up to 34 SNe and the dispersion is 0.31 mag. From all SNe we derive α = 4.87 ± 0.52 and ZP = −2.44 ± 0.20 which corresponds to M_J = −18.21 ± 0.20. To conclude, the Hubble diagram derived from the CSP sample using the SCM is consistent and somewhat better with those found in the literature.

More recently, Rodríguez et al. (2014) proposed another method to derive a Hubble diagram from SNe II. The PMM corresponds to the generalization of the SCM, i.e., the distances are obtained using the SCM at different epochs and then averaged. Using the (V − I) color and the filter V, the authors found an intrinsic scatter of 0.19 mag. Given that the intrinsic dispersion used by Rodríguez et al. (2014) is a different metric than that used by us (the rms dispersion), we computed the latter from their data, obtaining 0.24 mag for 24 SNe in the Hubble flow. Using the V band and the (V − i) color and doing an average over several epochs,, we found a dispersion of 0.28 mag which is similar to the value found from the SCM and comparable with the value derived by Rodríguez et al. (2014). From the Y band and (V − i) color we find an identical dispersion of 0.29 mag.

In analyzing the Hubble diagram, the figure of merit is the rms and the holy grail is to obtain very low dispersion in the Hubble diagram (i.e., low distance errors). In our work, we show that in the Y band we can achieve a rms around 0.43–0.48 mag using the Hubble flow sample (30 SNe) and the entire sample (41 SNe), whereas using the golden sample (8 SNe) we obtain a dispersion of 0.18 mag. It is important to know if this decrease in rms is due to the fact that we used well studied SNe within the golden sample or if it is due to the low number of SNe. For this purpose we do a test using the Monte Carlo bootstrapping method.

From our Hubble flow sample, we remove randomly one SN and compute the dispersion. We do that for 30,000 simulations and the final rms corresponds to the median, and the errors to the standard deviation. Then after removing 1 SN, we remove randomly 2 SNe and again estimate the rms and the dispersion over 30,000 simulations. We repeat this process until we have only four SNe, i.e., we remove from one SN to (size available sample—four SNe). For each simulation, we compute a new model, i.e., new fit parameters (α, β, and ZP).

From this test we conclude that when the number of SNe is lower than 10–12 SNe the rms is very uncertain because the parameters (i.e., α, β, and ZP) start diverging (see the Appendix). This implies that the rms is driven by the reduced number of objects so it is difficult to conclude if the model for the golden sample is better because the rms is smaller or because it is due to a statistical effect.

As stated in Section 4.1, the ${\beta }_{{\lambda }_{1}}$ color term is related to the total-to-selective extinction ratio if the color–magnitude relation is due to extrinsic factors (dust). In the literature, for the MW, R_V is known to vary from one line of sight to another, from values as low as 2.1 (Welty & Fowler 1992) to values as large as 5.6–5.8 (Cardelli et al. 1989; Fitzpatrick 1999; Draine 2003). In general for the MW, a value of 3.1 is used which corresponds to an average of the Galactic extinction curve for diffuse interstellar medium. Using the minimization of the Hubble diagram with a color term, in the past decade the SNe Ia community has derived lower R_V for host galaxy dust than for the MW. Indeed, they found R_V between 1.5 and 2.5 (Krisciunas et al. 2007; Elias-Rosa et al. 2008; Goobar 2008; Folatelli et al. 2010; Phillips et al. 2013; Burns et al. 2014). This trend was also seen more recently using SNe II (Poznanski et al. 2009; Olivares et al. 2010; Rodríguez et al. 2014). This could be due to unmodeled effects such as a dispersion in the intrinsic colors (e.g., Scolnic et al. 2014).

We follow previous work in using the minimization of the Hubble diagram to obtain constraints on R_V for host galaxy dust. Using the PCM, the Hubble flow sample, and the r band, we find β_r close to 0.98. Using a Cardelli et al. (1989) law, we can transform this value in the total-to-selective extinction ratio, and we obtain ${R}_{V}={1.01}_{-0.41}^{+0.53}$. Following the same procedure but using the SCM, we also derive low R_V values, but consistent with those derived using the PCM.

At first sight, our analysis would suggest a significantly different nature of dust in our Galaxy and other spiral galaxies, as previously seen in the analysis of SNe Ia and SNe II. However, we caution that the low R_V values could reflect instead intrinsic magnitude–color for SNe II not properly modeled. To derive the R_V (or pseudo R_V) values, we assume that all the SNe II have the same intrinsic colors and same intrinsic color–luminosity relation; however, theoretical models with different masses and metallicity show different intrinsic colors (Dessart et al. 2013). Disentangling both effects would require to know the intrinsic colors of our SN sample. Indeed, with intrinsic color–luminosity corrections the ${\beta }_{{\lambda }_{1}}$ color term could change and thus we will be able to derive an accurate R_V. In a forthcoming paper, we will address this issue through different dereddening techniques (T. de Jaeger 2016, in preparation) that we are currently investigating.