THE EFFECT OF HOST GALAXIES ON TYPE Ia SUPERNOVAE IN THE SDSS-II SUPERNOVA SURVEY

In this analysis, we use the full data set from the SDSS-II Supernova Survey (Frieman et al. 2008), which provides one of the largest samples of SNe Ia currently available. The SDSS-II SN Survey was a dedicated search for transient objects using the SDSS 2.5 m telescope and imaging camera (York et al. 2000; Gunn et al. 2006) to perform repeat imaging of the "Stripe 82" region of the SDSS survey for three months a year from 2005 to 2007. The SDSS-II transient database contains many thousands of potential SN candidates, out to z ∼ 0.5, of which ≃500 were spectroscopically confirmed as SNe Ia during the survey period (Sako et al. 2008; Holtzman et al. 2008). The first year (2005) of the SDSS-II SN sample was recently used for detailed cosmological analyses (Kessler et al. 2009a; Sollerman et al. 2009; Lampeitl et al. 2010).

For the host galaxy analysis presented herein, we focus on the low-redshift part (z < 0.21) of the SDSS-II SN sample, where the SN light curves are measured to high accuracy, with multiple, high signal-to-noise ratio data points per light curve, and the k-corrections are empirically determined to be more reliable, i.e., minimizing the influence of the UV-part of the SN spectrum (see Foley et al. 2008; Kessler et al. 2009a). Furthermore, the efficiency of the SDSS-II SN Survey remains above 50% below this redshift limit as demonstrated in Dilday et al. (2010a).

To ensure that our SN sample is more complete, we also include photometric SNe Ia that have a light curve consistent with being a Type Ia, based on the Bayesian light-curve fitting of Sako et al. (2008), and a known host galaxy spectroscopic redshift. The likely non-Ia contamination within these additional photometrically classified Ia's is only ≃ 3% (Dilday et al. 2010a).

In total, this provides a sample of 361 supernovae (for z < 0.21), of which 258 are spectroscopically confirmed. We provide a breakdown of these SN numbers in Table 1 where "Spec Confirm" gives just the spectroscopically confirmed SNe. In the following section, we describe the light-curve fitting procedure and the host galaxy classification which lead to a further reduction in the available SN sample (see Table 1).

Several algorithms are available for fitting the light curves of SNe and determining cosmological distances and SN properties. The two most common, publicly available, fitting methods are SALT2 (Guy et al. 2007) and MLCS2k2 (Jha et al. 2007; Kessler et al. 2009a); we use both of these techniques to explore the dependences of our results on the details of the light-curve analysis. For the main results of this paper, we use the public SALT2 light-curve fitter and, in the Appendix, we provide a similar analysis using MLCS2k2. We find that the results and conclusions of this paper are consistent for both light-curve fitting algorithms and different SN selection criteria.

For our main SALT2 light-curve fitting analysis, we impose the following additional criteria to our SN sample (described in Section 2.1) to ensure robust measurements for the stretch and color of each SN (based on our experience with the first year SDSS-II SN cosmology analysis). First, we require at least five epochs in all the SDSS gri passbands, with at least one measurement before the light-curve maximum. We also require that the reduced χ² of the light-curve fit to the data in each filter is less than 3. These cuts reject 108 SNe from our sample and are shown in Table 1 labeled as "After LC cuts."

SALT2 reports for each individual SN an apparent brightness (m_B) in the B band, a stretch value (x₁), and a color (or c) term, which can then be used to calculate a distance modulus (μ) using

$$\begin{equation} \mu = (m_B - M)+\alpha x_1 - \beta c, \end{equation} \tag{ 1 }$$

where M is the "standardized" absolute SN Ia magnitude (at x₁ = c = 0), α describes the overall stretch law for the sample, and β is the color law for the whole sample.²⁶ We only report the uncalibrated values of M from SALT2 which are degenerate with our assumed value of H₀. We are only interested in relative differences in the absolute brightness of SNe, not the true brightness. In most SN cosmological analyses, it is assumed that these parameters are invariant to the type of host galaxy in the sample and do not evolve with redshift, but there is no a priori reason for such an assumption.

Based on the observed values of our SN stretch and color distributions, we remove a further 19 light curves by imposing the limits of −4.5 < x₁ < 2 and −0.3 < c < 0.6, which are labeled in Table 1 as "After LC fitter limits." These limits were empirically determined to remove SN events where SALT2 reports extreme values either for x₁ or c not resembling the majority of SN in our sample (see Figure 2), leaving 234 SNe Ia for our main SALT2 analysis (as shown in Table 1).

Following the usual SALT2 prescription, we determine M, α, and β in Equation (1) by minimizing the scatter about a fiducial redshift–distance relation. We adopt as the reference cosmological model a flat universe with an energy density of matter of Ω_M = 0.272, taken from Komatsu et al. (2009), and H₀ = 65 km s^-1 Mpc⁻¹. We have confirmed that the main results of this paper do not depend on the details of this assumed cosmological model.

A key part of our analysis is the host galaxy properties for each SN, which were determined using the techniques outlined in detail in Smith (2008). We begin by matching SN positions, within a 0.25 arcminute search radius, with SDSS galaxies detected in the deep optical stacked images of "Stripe 82" constructed from the SDSS-I/II photometry (Abazajian et al. 2009) and choose the closest match as the host galaxy. We also require that all SN host galaxies have a measured SDSS model magnitude of r ⩽ 23. These two constraints remove 7% of our SNe with either a missing or too faint host. We then visually confirm each host, via inspection of images with and without the SN present, to ensure that the correct host has been associated with each SN. In six cases, where the host is extended or de-blended into multiple objects by the SDSS automated software, we adjust the host position to the center of the underlying galaxy.

Each of the SNe Ia in our sample has a spectroscopic redshift, either from the SN itself or its host galaxy. This redshift is combined with the five SDSS photometric measurements (model magnitudes in ugriz passbands corrected for Galactic extinction) for the host galaxy to determine the star formation rate (SFR) and stellar mass of each system, using the PÉGASE2 spectral energy distributions (Fioc & Rocca-Volmerange 1997, 1999) and the Z-PEG software package (Le Borgne & Rocca-Volmerange 2002). In detail, we used the eight star-forming scenarios, as listed in Table 1 of Le Borgne & Rocca-Volmerange (2002), and assume a Kroupa (2001) initial mass function. In these scenarios, the SFR is defined for most galaxies via SFR = ν × M_gas, where ν (in units of Gyr⁻¹) ranges from 0.07 to 3.33, while for irregular galaxies, the SFR is defined as SFR = 0.065M^1.5_gas (M_gas is the density of gas in solar masses). We use the default modeling of internal dust as discussed in Le Borgne & Rocca-Volmerange (2002) where a King profile is used for the elliptical template, whilst a plane-parallel slab distribution is used for spiral and irregular galaxies. Each scenario is then evaluated at 69 different time-steps in their evolution, thus resulting in 552 possible galaxy template spectra covering a wide range of possible evolutionary scenarios.

These templates were fit to the galaxy fluxes (converted from their model magnitudes after correcting them to the AB-system), with the redshift fixed to the redshift of the SN or host galaxy, to determine the best template for each host galaxy. The normalization is related to the total stellar mass of these templates and is a free parameter which is determined as part of the fitting procedure to the host. We also use the best-fit template to estimate the recent star formation rate of the host galaxy by integrating the best-fit template over the last half a gigayear of its evolution, e.g., if the best-fit template had an age of 8 Gyr, then the recent SFR of the host was calculated over the period of 8–7.5 Gyr. Error bars on these estimates were determined by propagating the observed galaxy photometric errors. Our technique is similar to that used by Sullivan et al. (2006), and the Z-PEG software and the spectral energy distributions have been used substantially in the literature (Glazebrook et al. 2004; Grazian et al. 2006; M. Smith et al. 2010, in preparation).

To test the validity of our PÉGASE2-based methodology, we have compared the stellar masses determined using the photometric data on over 350,000 SDSS Main galaxies (taken from DR4; Adelman-McCarthy et al. 2006) to masses derived by Kauffmann et al. (2004) using a technique based on SDSS spectral features. We find no mean difference between the two mass estimates, with a variance of only 3%.

In Figure 1, we show the separation of host galaxies according to their stellar mass and SFR obtained from the PÉGASE2 analysis above. We classify each host as either passive (i.e., shows no sign of recent star formation activity) or star-forming (i.e., having evidence for recent star formation). To ensure a clean separation between these two galaxy classes, we require that the measured 1σ error on the estimated SFR for each galaxy is smaller than the separation in SFR for the two classes of galaxies, i.e., it is then unlikely that statistical errors on an individual SFR measurement can scatter a galaxy from one host galaxy type to the other. We also exclude star-forming host galaxies that have a specific SFR (i.e., sSFR; defined as the SFR per stellar mass) in the range log(sSFR) < −10.5 as illustrated in Figure 1 as a dashed line. This limit excludes the locus of star-forming galaxies at the bottom of the blue cloud of points in Figure 1, which are predominantly fit by the PÉGASE2 lenticular galaxy scenario, thus leading to an unclear interpretation of their star formation activity. These cuts and limits ensure that we have two, well-separated, samples of host galaxies. We show in Table 1 the numbers of SNe available in these two host galaxy classes and note that many SNe have been excluded from further analysis because of the ambiguity of their host galaxy type. In Table 2, we provide the SN designation, host galaxy coordinates, host galaxy stellar mass, and star formation rate (derived from our best-fit PÉGASE2 model). We also provide whether the SN was spectroscopically confirmed (sp) or classified by just its light curve (lc), and if we classify the host galaxy as passive (p) or star-forming (sf). More sophisticated stellar population models could have been used to determine the star formation histories of our host galaxies (Maraston et al. 2009) but we find that our classification of galaxies into two broad classes of star formation activity is relatively unaffected by the choice of templates (see Smith 2008 for more details). Also, our host galaxy analysis is in good agreement with a simple cut on the color of the galaxies, e.g., dividing the galaxies at u − r = 2.22 (Strateva et al. 2001).

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Table 2. Host Properties Used in the Main SALT2 Analysis

Designation		Host Position		Stellar Mass	SFR
SN ID	IAU	α(J2000)	δ(J2000)	(log M☉)	(log M☉ yr−1)	SNa	Hostb	Sample
1032	2005ez	03h07m11016	+01°07'1196	10.47+0.09−0.07	N/A	sp	p
1580	2005fb	03h01m17544	−00°38'3863	7.72+1.00−0.32	−0.98+1.02−0.36	sp	sf	r
15421	2006kw	02h14m57912	+00°36'0980	10.17+0.12−0.10	0.80+0.18−0.22	sp	sf	r
11172	N/A	21h29m39120	−00°12'0788	10.18+0.14−0.03	1.00+0.02−0.33	lc	sf	r
				...

Notes. Hosts with negligible star formation rates are indicated with N/A and members of the restricted sample with r. ^aSN classification as an Ia based on spectra (sp) or light-curve shape (lc). ^bOur host classification either as passive (p) or star-forming (sf).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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As discussed above, there is a clear trend for the x₁ distribution with host type, but no obvious trend for the color distribution. We explored if the constants in Equation (1) (M, α, β) are dependent on host type by using a Markov chain Monte Carlo (MCMC) simulation where we minimize the χ² for the fit to the distance modulus versus redshift, as a function of (M, α, β) separately for passive and star-forming galaxies. Fits were obtained by running the MCMC chains with 50,000 accepted steps and adjusting the step size empirically to achieve a typical frequency for accepted steps of ≈20%. One sigma errors on each parameter are provided by marginalizing over the remaining other parameters from the MCMC chains. We perform this analysis assuming an intrinsic dispersion of σ_int = 0.14 mag, which is added in quadrature to the errors on the distance modulus to achieve a reduced χ² close to 1 (i.e., χ²/ndf ≈ 1; see Lampeitl et al. 2010 for further discussion of this intrinsic dispersion).

To ensure that our results are not driven by a few outliers, we also perform our analyses on a restricted subset of SNe with tighter allowed ranges of c and x₁ values. This restricted sample is illustrated in Figure 2 as the inner dot-dashed box and reduces the sample from 162 SNe (see Table 1) to 116 SNe.

In Table 3, we summarize our results for fitting M, α, β for the full sample and the restricted subsample discussed above. We see a correlation between the host galaxy type and the absolute magnitude (M) of the supernovae, i.e., after the SNe have been standardized using the SALT2 light-curve fitting algorithm, there is still a difference of ≃ 0.1 mag, with SNe Ia in passive galaxies being brighter (more negative absolute magnitudes). To illustrate this effect, we present in Figure 3 the residuals to the Hubble diagram (after removing the fiducial redshift-distance relation) for the best-fit SALT2 parameters of (M, α, β) = (−30.11, 0.12, 2.86), which were determined from fitting the whole SN sample regardless of host galaxy type. As can be seen, there is a visible offset between SNe in passive and star-forming galaxies.

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The interpretation for the other SALT2 parameters (α and β) is less clear. First, we see no clear evidence for differences in α with host galaxy type given the statistical errors. Next, the fitted values of β (the color law) for star-forming galaxies do appear to be larger than those found for passive galaxies, i.e., β values for star-forming galaxies are above 3, while for passive galaxies we find values below 3. The significance of this difference in β varies between the full and restricted samples, which is not too surprising, as excluding the outliers in the color range will clearly increase the statistical error on the slope of the color law seen in Table 3. The mean slope (β) is similar for both the full and restricted sample.

In Figure 4, we show the corrected absolute magnitude for SNe in passive galaxies as a function of their fitted color and stretch values. The left-hand panels show the color-corrected absolute magnitude as a function of x₁, i.e., only the color part of Equation (1) (βc) has been applied to m_B. The right-hand panels show the stretch-corrected absolute magnitude, as a function of c, when only the stretch component of Equation (1) (αx₁) has been applied to the distance modulus. In the upper panels, we show the best-fitting law (Equation (1)) assuming the best-fit values of M, α, β for passive galaxies in Table 3, i.e., (M, α, β)_P = (−30.19, 0.16, 2.42), respectively. In the lower row of panels, we show the best-fit law again but now assuming the best-fit parameters for star-forming galaxies, namely (M, α, β)_SF = (−30.10, 0.12, 3.09).

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Comparing the top and bottom left-hand panels in Figure 4, it is clear we see that the amplitude (M) of the best-fitted relationship is different between the two and clearly wrong in the bottom panels (i.e., using the star-forming best-fit SALT2 parameters for SNe in passive galaxies). In Figure 5, we show the same analysis as in Figure 4, but this time the data plotted is for the star-forming SN sample. Again, we see that the amplitude of the fitted law (Equation (1)) is different and inappropriate if used to describe the wrong type of galaxy.

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Before we interpret these results, it is important to understand potential systematic uncertainties in our analysis. First, we have tested if our result depends on the inclusion of a subset of photometrically confirmed SNe. We see negligible changes in the central values of M₀, α, and β (for SALT2) which are significantly smaller than the errors on these parameters. We also verify the robustness of our results to reasonable changes in the fiducial cosmological model and find no significant effect as expected. Likewise, we have increased the redshift range of the sample used in our analysis, e.g., increasing the limit to z < 0.45, which more than doubles the size of the sample but makes the sample more incomplete. We find that the observed differences with host galaxy type are still present, but the significance is decreased. This decrease in significance is likely caused by the decrease in signal-to-noise ratio for both the SN light curves and the galaxy photometry, as well as increases in the sample incompleteness (both spectral confirmations and Malmquist bias effects). The uncertainties in the k-corrections are also increased as the UV part of the SN spectrum becomes more important. We note that Sullivan et al. (2010) see similar results but for the higher redshift SNLS sample, thus suggesting that any decrease in significance we witness is probably caused by observational issues rather than evolution in the SN population. For these reasons, we have chosen to focus on the cleanest, most efficient sample of SDSS-II SN at z < 0.21.

In the Appendix, we provide a parallel analysis of our data using the MLCS2k2 light-curve fitting technique and find similar results to those seen in the SALT2 analysis in Section 3, namely, differences in the absolute magnitude of SNe Ia between passive and star-forming host galaxies, as well as differences in the best-fit color laws (as represented by R_V). This confirms that our results are not sensitive to the details of how the light curves were analyzed and suggests that the trends we see are either inherent to the supernovae, especially as we still see a correlation for the restricted sample of SN. However, further analysis is required to conclusively determine the fundamental origin of the observed correlations including potential improvements in the light-curve fitting methodologies, e.g., a better representation of local SNe in passive hosts within the MLCS2k2 training sample or a more sophisticated parameterization of the x₁ and c distributions in SALT2 to better accommodate the fast declining SNe Ia in passive hosts.

Throughout this analysis, we have assumed an intrinsic dispersion for our SN sample of σ_int = 0.14 mag, which is consistent with the value obtained for the first year SDSS-II data analyzed in Lampeitl et al. (2010). In Table 3 (and Table 5), we present the separate χ² values for SNe in both passive and star-forming galaxies (for both SALT2 and MLCS2k2). The star-forming subsample has a reduced χ² above unity, but for passive hosts, the reduced χ² value is now less than 1. This observation suggests that SNe in passive galaxies would favor a smaller intrinsic dispersion and are thus a more homogeneous population, or the observed errors are more representative of the scatter about the Hubble diagram.

To investigate this matter further, we have re-fit both the passive and star-forming SN samples (with no restrictions in c and x₁) adjusting the intrinsic dispersion σ_int to a value that gives a reduced χ² close to 1. In the case where we fit the passive sample with the parameters derived from just the passive sample we find σ_int = 0.13, but these values give χ² of 321 (for 122 SNe) for the star-forming sample. In the reverse, we fit the star-forming sample with the best-fit star-forming parameters and find σ_int = 0.17, where now the passive sample yields χ² = 55 for 40 SNe. This result suggests that by using the larger σ_int for SNe in passive hosts, we are effectively down-weighting these SNe (by increasing their errors) because of the offset in M between the SNe Ia in the two types of galaxies.

As discussed in Section 2.3, we have classified our SN host galaxies into two well-separated classes, namely passive and star-forming. We initially separated the galaxies in this way because previous studies of the properties, and rates, of SNe Ia have shown clear correlations with the star formation activity of the host (e.g., Hamuy et al. 2000; Sullivan et al. 2006; Mannucci et al. 2005; Dilday et al. 2010b). Gallagher et al. (2008) showed that the measured metallicity for local SN host galaxies was correlated to the residuals around the best-fit distance–redshift relation, which has prompted some authors to look for correlation with the host galaxy stellar mass as a proxy for the metallicity (using the known mass–metallicity relationship; Tremonti et al. 2004). Both Kelly et al. (2010) and Sullivan et al. (2010) find such a correlation with stellar mass and Sullivan et al. (2010) exploit this correlation to improve the cosmological fits to the three year SNLS data set.

We present here a first analysis of the SDSS-II Hubble residuals as a function of the host galaxy stellar mass. The advantage of such an analysis is that the host stellar mass is a continuous parameter thus potentially avoiding uncertainties associated with binning galaxies into two distinct samples. Unfortunately, estimating the stellar mass from broadband photometry is challenging and therefore the measurements of stellar mass can be noisy and potentially biased. For the analysis below, we therefore restrict ourselves to the clean sample defined in Section 2.3, i.e., we still do not include galaxies with an ambiguous classification between star-forming and passive.

In Figure 6, we show the Hubble residuals as a function of stellar mass. As expected, the passive galaxies have preferentially higher stellar masses than the star-forming subsample. A linear fit to the Hubble residuals is shown as the inclined line (triple-dot-dashed) and has a slope of 0.069 ± 0.014 mag per log(M_☉). We see the same result, at the same statistical significance, with our MLCS2k2 analysis in the Appendix. Therefore, these results imply that adding an additional parameter, dependent on the host stellar mass, to Equation (1) would give a better fit to the SDSS-II SN Hubble diagram.

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To quantify this statement, we have repeated our MCMC analysis in Section 3.1, but now minimizing over four parameters (M, α, β, γ) where calculating the distance modulus using

$$\begin{equation} \mu = (m_B - M)+\alpha x_1 - \beta c+\gamma m_{st}, \end{equation} \tag{ 2 }$$

where m_st is defined as

$$\begin{equation} m_{st} = \log (m_{\rm host}) - 9.5, \end{equation} \tag{ 3 }$$

where m_host is the stellar mass of the host in units of solar mass derived from our PÉGASE2, fits to the host galaxy colors (Section 2.3).

The results of this new analysis are presented in Table 4. First, we see that adding an additional parameter has not significantly changed the fitted values of M, α, and β; they are close to the values in Table 3 for star-forming hosts (which dominate the whole sample). Secondly, we see a significant non-zero value for γ at >4σ. Finally, we can compare the three- and four-parameter fits (Equations (1) and (2), respectively) to the full SN sample, regardless of host galaxy type, using the Bayesian Information Criteria (BIC; Liddle 2004). BIC is a penalized likelihood statistic that accounts for models with different numbers of parameters, and we find that the BIC score for Equation (2) is 117 compared to 134 for Equation (1). The smaller BIC score demonstrates that the additional parameter is justified.

In addition to the SALT2 light-curve fitter in Section 2.2, we also provide a parallel analysis based on the MLCS2k2 SNANA light-curve technique (Jha et al. 2007; Kessler et al. 2009a, 2009b). In this way, we can determine if the correlations seen in Section 3.1 are just an artifact of the light-curve analysis. As in Section 2.2, we assume a flat cosmology of Ω_M = 0.272 from Komatsu et al. (2009) as well as assume that the true absolute magnitude of SNe Ia is M = −19.44 (Kessler et al. 2009a) and an intrinsic dispersion of σ_int = 0.14 mag.

In SALT2, the parameters α and β describe the global stretch and color laws of SNe Ia and are determined through minimizing the scatter around a cosmological model. MLCS2k2 takes a different approach in that the distance is both linearly and quadratically dependent on the light-curve decline rate parameter (Δ) compared to the purely linear dependence of x₁ in SALT2. The global correction (analogous to α in SALT2) is determined from a well-measured, low-redshift training set, prior to fitting. Any excess color variation is assumed to be due to extinction by dust in the host galaxy, and is parameterized using the Cardelli et al. (1989) law, where the excess is E(B − V) = A_V/R_V. For our Galaxy, R_V = 3.1, whilst previous SN Ia studies favor values of R_V ∼ 2.1 (see references in Section 1).

In this analysis, we attempt to constrain R_V as a function of host galaxy type. MLCS2k2 does not minimize the scatter on the Hubble diagram but assumes a value for R_V in the fitting process. Thus, we allow R_V to vary between 0.1 < R_V < 4.0, in increments of 0.1, and minimize the χ² of the fit to the Hubble diagram to determine the best-fitting R_V value. For A_V, we assume a "standard" prior distribution in the fitting process as outlined in Kessler et al. (2009a), who used P(A_V) = exp(−A_V/τ) with τ = 0.33, based on the first year SDSS-II data. To quantify the effect of this assumption on our results, we also consider the case where no prior on the A_V distribution is assumed, thus allowing it to take any value and mimicking the SALT2 approach.

As with SALT2, and following the methodology of Kessler et al. (2009a, 2009b), we require that each light curve in our MLCS2k2 analysis has at least five photometric observations in the SDSS gri passbands located between −20 and +60 days relative to maximum light in the SN rest frame. Of these epochs, at least one must be 2 or more days prior to maximum light, and at least one must be 10 or more days past maximum light²⁷, with at least one epoch with an S/N>5 in each of the passbands (although not necessarily the same epoch each time). These criteria ensure that we have well-measured light curves for 256 SNe at z < 0.21 (see Table 1 for details).

In addition, we impose further constraints on the determined light-curve shape and fit probability. We follow the procedure of Kessler et al. (2009a) and remove any object with a low probability of being a SN Ia (i.e., those with P(I_a) < 10⁻³) or which has an unphysical value of Δ < −0.4. These cuts remove a further 42 events (Table 1). As the value of A_V is dependent on R_V, we require that each SN satisfies the criteria above for all values of R_V used. Using the standard A_V prior above, we are left with 214 SNe Ia (labeled as "After LC fitter limits" in Table 1). To test the robustness of our results to outliers, we also restrict the range of A_V values allowed (to 0.0 < A_V < 1.0), which would reduce the sample to 198 SNe. Finally, we also apply the same cuts on the host galaxy classifications as described in Section 2.3 and presented as "Valid host galaxy type" in Table 1.

In Table 5, we present the best-fitting values of R_V and H₀ for our MLCS2k2 analysis as well as their one sigma errors, which we calculated from the χ²_minimum + 1 range holding the other parameters constant at their best-fit values. In the case where the error on R_V cannot be determined because it is smaller than the discrete binned values of the R_V parameter, we then conservatively assume an error of 0.1 (the spacing between the R_V grid points). The top of Table 5 presents results assuming the standard A_V prior distribution and we see a clear preference for higher values of Hubble constant for SNe in passive galaxies compared to star-forming galaxies (and the whole sample together). This preference is seen regardless of the A_V range allowed.

This observed difference in Hubble constant can be due to differences in the assumed absolute magnitude (ΔM) of SNe in different galaxy types, and we can convert between the two parameters using

$$\begin{equation} \Delta M = 5 \log _{10} (\Delta {H}_{0}), \end{equation} \tag{ A1 }$$

where ΔH₀ is the difference in Hubble constants between the two samples. Therefore, in Table 5, we see ΔH₀ = 3.1 km s⁻¹ Mpc⁻¹ (assuming the standard A_V prior distribution and the full SN sample), which translates to a ≃ 0.12 mag difference (using Equation (A1)), with SNe in passive galaxies being brighter. This result is consistent with our SALT2 analysis in Section 3.1 and demonstrates that this result is independent of the details of the analysis.

Table 5 also contains the best-fitting R_V values, and we see a difference between the R_V laws for SNe in the passive and star-forming galaxy samples: supernovae in passive galaxies appear to favor R_V ≃ 1 compared to R_V ∼ 2 in star-forming galaxies. This difference in R_V, as a function of host galaxy type, could be worrying for SN cosmology analyses which typically use a fixed value of R_V for all SNe and often constrain the allowed range to 1.7 < R_V < 2.5. This range is consistent with our star-forming SN samples (and the whole sample) but inconsistent with our passive galaxy SN sample.

In Figure 7, we show how the reduced χ² of our fits to the Hubble diagram varies as a function of R_V for our sample. The solid lines indicate the R_V values when all values of A_V are allowed, while the dashed lines are only using SNe Ia with 0 < A_V < 1. This figure re-enforces the results in Table 5 in that there is a difference in the minima of these curves for the different galaxy types. As with the SALT2 analysis in Section 3.1, we see that constraining the color range allowed decreases the difference as we are removing outliers to the main color laws.

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The bottom row of Table 5 shows the results assuming no prior distribution on A_V during the MLCS2k2 light-curve fitting, i.e., any A_V value is allowed. In theory, this should be the closest match to the SALT2 analysis presented in Section 3.1. Again, we see differences in the fitted H₀ values between the passive and star-forming host galaxy samples. Intriguingly, the evidence for a difference in R_V between the two host galaxy types is now less, which is consistent with the SALT2 results in Section 3.1 where we only witnessed a slight dependence on β for the galaxy type. Clearly the choice of A_V prior, as well as the allowed range of A_V values, can have a significant effect on the best-fitting color parameters.