Should Type Ia Supernova Distances Be Corrected for Their Local Environments?

We infer distances from the SNe Ia in the Pantheon and Foundation samples using the most recent version of the SALT2 light curve fitter (Guy et al. 2007) (SALT2.4; Betoule et al. 2014; Guy et al. 2010) and the Tripp estimator (Tripp 1998):

$$\begin{eqnarray}&&\mu ={m}_{B}-{ \mathcal M }+\alpha \times {x}_{1}-\beta \times c.\end{eqnarray} \tag{ 1 }$$

m_B is the log of the light curve amplitude, x₁ is the light curve shape parameter, and c is the light curve color parameter. α and β are nuisance parameters along with ${ \mathcal M }$, a parameter encompassing the SN Ia absolute magnitude at peak and the Hubble constant.

The Pantheon and Foundation analyses apply sample selection criteria using these SALT2 light curve parameters to ensure that the SNe Ia included can yield accurate distances. These include cuts on the shape and color to ensure that the SNe are within the parameter ranges for which the SALT2 model is valid (−3 < x₁ < 3, −0.3 < c < 0.3) and cuts to ensure that the shape and time of maximum light are well measured (x₁ uncertainty <1 day and time of maximum uncertainty <2 days). Here, we also require a Milky Way reddening of E(B − V) < 0.15 mag and z > 0.01 to remove SNe with large systematic peculiar velocity uncertainties.

The Foundation data have a few additional selection criteria, all of which were applied in Foley et al. (2018): the first light curve point must have a phase of <7 days, at least 11 total light curve points are required in gri_P1, and Chauvenet's criterion is applied to remove outliers. All samples remove spectroscopically peculiar SNe Ia (apart from 1991T-like SNe, which are included).

Finally, survey selection effects bias the SN distances, the light curve shapes, and the light curve colors. We apply bias corrections to the distances and light curve parameters using the BEAMS with Bias Corrections (BBC) method (Kessler & Scolnic 2017). The BBC method uses simulated SN samples to correct x₁, c, m_B, α, and β for observational biases and selection effects. Though the BBC method makes no corrections based on host galaxy information directly, the BBC corrections are important for this study because SN Ia light curve demographics depend on host properties (Childress et al. 2013) and SNe Ia with c < −0.2 and x₁ > 2 have mean Hubble residuals, before BBC correction, of up to 0.2–0.3 mag (Scolnic & Kessler 2016). These residuals are three to four times larger than the host mass step.

We use the simulations from Scolnic et al. (2018) and Foley et al. (2018) with the BBC method to generate these bias corrections (D. M. Scolnic et al. 2018, in preparation, will contain additional simulation details specific to the Foundation sample). The BBC method removes six additional SNe from the sample: three from Pantheon and three from Foundation. We cannot be certain that the bias corrections are valid for these six SNe as they lie in a region of shape, color, and redshift space that is not well sampled by the SN simulation. With the BBC method, we find α = 0.141 and β = 3.149 using the z < 0.1 SNe in this analysis.

After these additional cosmology cuts, 170 z < 0.1 SNe Ia are from the CSP or CfA surveys, 43 are from SDSS or PS1, and 170 are from the Foundation DR1 sample, for a total of 383 SNe Ia. We note that Foley et al. (2018) lists 180 SNe as passing all cosmology cuts. Of these, three are at z < 0.01, two are at z > 0.1, two do not pass cuts due to small changes in the SALT2 fitting parameters,¹³ and the remainder are lost due to BBC cuts. See Foley et al. (2018) and Scolnic et al. (2018; and references therein) for additional details on the sample selection.

We measure host galaxy properties with photometry from the PS1 first data release (Chambers et al. 2016) and the Sloan Digital Sky Survey Data Release 14 (SDSS DR14; Abolfathi et al. 2018). The PS1 DR1 has deep, grizy observations over 3π steradians of the sky and has observed at the locations of over 90% of SNe in the current low-z sample. PS1 y-band photometry in particular allows for a robust determination of host galaxy masses. SDSS has imaged ∼14,000 square degrees in the ugriz filters, including the locations of ∼65% of the SNe in the Pantheon+Foundation low-z sample. We measure SDSS u and PS1 grizy photometry within apertures of 3 kpc diameter at the location of each SN in this sample.

To observe only the regions within ∼3 kpc of the SN, we require the typical seeing of PS1 and SDSS to correspond to 3 kpc or less in physical size. PS1 images have a typical seeing of ∼1'', while SDSS images have a median seeing of approximately 138 in u. Blending of local and global effects may occur at higher redshifts. If we therefore restrict our sample to z < 0.1, where 3 kpc corresponds to an angle of ∼16, we can be assured that we are indeed probing local regions.

We also remove 29 SNe in galaxies with inclination angles >70° based on the Tully & Fisher (1977) axial ratio method, leaving 354. This cut increases the likelihood that local regions are truly local, as highly inclined galaxies could have non-local regions contained in the 3 kpc aperture due to projection effects. However, we note that projection effects will always be a concern in this type of study, particularly in early-type galaxies. Finally, we remove SNe for which the identification of the host galaxy is uncertain. SNe for which the host cannot be reliably identified should not be used in a sample that compares local to global measurements. We match SNe Ia to candidate host galaxies using the galaxy size- and orientation-weighted SN separation parameter, R (Sullivan et al. 2006):

$$\begin{eqnarray}\begin{array}{rcl}{R}^{2} & = & {C}_{{xx}}{x}_{r}^{2}+{C}_{{yy}}{y}_{r}^{2}+{C}_{{xy}}{x}_{r}{y}_{r}\\ {C}_{{xx}} & = & {\cos }^{2}(\theta )/{r}_{A}^{2}+{\sin }^{2}(\theta )/{r}_{B}^{2}\\ {C}_{{xy}} & = & 2\cos (\theta )\sin (\theta )(1/{r}_{A}^{2}+1/{r}_{B}^{2})\\ {C}_{{yy}} & = & {\sin }^{2}(\theta )/{r}_{A}^{2}+{\cos }^{2}(\theta )/{r}_{B}^{2},\end{array}\end{eqnarray} \tag{ 2 }$$

where x_r = x_SN − x_gal and y_r = y_SN − y_gal. r_A, r_B, and θ are galaxy ellipse parameters measured by SExtractor (Bertin & Arnouts 1996). Each R parameter corresponds to an elliptical radius about the host center. We consider the host ambiguous if the minimum R is greater than 5. This cut removes an additional 83 SNe, leaving a final sample of 273 SNe.

After all cuts, grizy images for measuring the local mass step are available for 273 SNe Ia. Of these SNe, 195 lie in the SDSS footprint and therefore have u measurements for measuring the rest-frame u − g color. We do not attempt to infer rest-frame u colors for host galaxies without u observations.

We treat the dependence of SN Ia shape- and color-corrected magnitude on host mass and u − g as a step function, as previous studies have found this to be well-motivated by the data (Betoule et al. 2014; Roman et al. 2018). There may be theoretical reasons to favor a step function as well; Childress et al. (2014) predict that the mean ages of SN Ia progenitors undergo a sharp transition between low-mass and high-mass galaxies. If Hubble residuals depend on the physics related to progenitor age, a step would naturally be produced in this model. The dust extinction law in passive versus star-forming galaxies could also change in a way that would produce a step.

To estimate the size of the mass and u − g color steps, we use the maximum likelihood approach from Jones et al. (2015). Our likelihood model treats SNe in low-mass and high-mass regions (or in regions with blue/red u − g colors) as belonging to two separate Gaussian distributions and simultaneously determines the maximum likelihood means and standard deviations of those two distributions. The four parameters of this model can be easily constrained with a standard minimization algorithm. The baseline approach considered here does not re-fit α and β on each side of the color or mass split, but we explore this approach in Section 4.2.

The step between low-mass/high-mass and bluer/redder u − g may correspond roughly to the boundary between passive and star-forming galaxies. The median rest-frame, host galaxy dust-corrected u − g color of this sample is 1.27, and we adopt this value as an agnostic choice for the location of the step following Roman et al. (2018). For the local mass step, we again choose the divide between the locally "low" and "high" mass galaxies to be the median local mass of our sample, log(M_*/M_⊙) = 8.83. The local mass, as defined here, is the stellar mass in the cylinder within a circular aperture of diameter 3 kpc. Unlike the local mass or color steps, the location of the global host mass step has been well measured by multiple independent data sets and analyses. For this reason, we adopt the standard global host mass step location of log(M_*/M_⊙) = 10 (Sullivan et al. 2010; Betoule et al. 2014; Scolnic et al. 2018). Figure 1 shows the local mass per pixel and u − g maps for four representative galaxies in our sample.

Zoom In Zoom Out Reset image size

We also consider whether the global step is driven by the local step and if so, how "local" the local measurement needs to be (Rigault et al. 2015; Roman et al. 2018). To address this question, we place 150 random apertures of 3 kpc diameter in each galaxy and measure the local mass and u − g color within those apertures. We use the SED template-fitting approach discussed above to fit the photometry in each aperture individually. We use these random measurements to ask whether the region near the SN is better correlated with SN luminosity than the regions far from the SN.

We again use the galaxy size- and orientation-weighted SN separation parameter R to choose where to place the apertures. We first use SExtractor to measure the ellipse that best approximates the shape of a given host galaxy. Each region with a given R parameter lies at the same elliptical radius about the host center. Regions with R = 0 are at the host center, while regions with R = 3 are approximately at the isophotal limit of the galaxy (shown in Figure 1). Regions with R = 5 are outside the isophotal limit of the galaxy and lie far enough away from the host center that identifying the true host galaxy begins to become ambiguous. To include as many apertures near the galaxy center as far from it, we place random apertures so that 25 have 0 < R < 1, 25 have 1 < R < 2, and so on out to R = 5, which is the Sullivan et al. (2006) criteria for matching an SN to its likely host galaxy.

We use these random measurements to explore how the local mass and color steps change if host properties are inferred from regions far from the SN location. For random apertures with a given distance from the SN location or a given R, we measure the physical properties associated with each SN from the random location instead of the SN location. We use these random measurements to find the maximum likelihood mass and color steps, and compare to the mass and u − g steps using the properties of the host galaxy at the SN location. For each set of random measurements, we choose the median of those measurements for the step location. This prevents a situation where the vast majority of the sample is on one side of the step location, which can occur as apertures move preferentially toward or away from the host galaxy center.

The spacing of these random apertures will be less than the seeing of the images in most cases, meaning that many random measurements will be partially correlated. However, we can avoid statistical complications by using just one random measurement per SN at a given time and avoiding regions within 3 kpc of the true SN location.

Roman et al. (2018) measured a step from z < 0.1 SNe Ia that was 0.038 ± 0.034 mag smaller than the step they measured from the full sample, though the difference was not statistically significant. If confirmed, this difference could either be due to a redshift evolution of the local step or differences in low-z versus high-z survey methodology. Specifically, much of the low-z data are from surveys that target a pre-selected set of (usually NGC) galaxies. None of the high-z surveys target pre-selected galaxies. Targeted surveys also collect SNe that are more like the sample of SNe Ia within ∼40 Mpc that are calibrated by Cepheids and used as a rung on the distance ladder for measuring H₀. On the other hand, all z > 0.1 data used for measuring the dark energy equation of state come from surveys that do not target specific galaxies. It may also be relevant that the CfA and CSP low-z SNe were observed on the Johnson filter system, while Foundation and the z > 0.1 data were primarily observed on the Sloan filter system. Because SNe observed on the Sloan filter system have g as the bluest band, there could be differences if host galaxy biases affect SN Ia luminosity in a wavelength-dependent manner.

Because because Foundation data come predominantly from untargeted surveys (Gaia, ASAS-SN, PSST), our data can be used to determine whether SNe from targeted surveys have a different local or global step than SNe from untargeted surveys. Foundation includes some data from targeted surveys only because untargeted surveys would likely discover these SNe if the targeted surveys did not exist (Foley et al. 2018). We therefore treat Foundation as an untargeted survey in this analysis.

In Table 1, we compare the local and global steps measured from z < 0.1 SNe in targeted surveys (CfA and CSP) and z < 0.1 SNe from surveys that are not targeted (Foundation, PS1, and SDSS). After global mass correction, we see a 2.0σ increase in the local mass step, a 1.8σ increase in the local color step, and a 2.1σ increase in the global color step when untargeted surveys are used instead of targeted surveys. Only the difference in the global mass step is statistically insignificant. These differences are not highly significant but could indicate that the correlation of SN distance with host galaxy properties is sensitive to survey selection effects. In the Appendix, we examine the differences in intrinsic dispersion on either side of the mass and color divide, finding 1σ–2σ evidence that SNe in locally low-mass or locally blue regions may have lower dispersions.

One might expect the difference in significance to be affected by the fact that SNe Ia in targeted surveys could be biased toward regions with higher stellar mass. Though this is the case for the global mass, it is not the case for the local mass as many SNe Ia in the targeted sample are far from the centers of their host galaxies. Of the SNe Ia in targeted surveys used in this study, 83% (91 of 110) have global masses >10 dex, the global mass step divide. However, 51% (56 of 110) occurred in locally massive regions locally massive (local mass >8.83 dex).

We also check the significance of a local step versus a global step using the Foundation sample alone. Our sample includes 127 Foundation SNe with grizy data that can be used to measure the local mass step and 80 Foundation SNe with SDSS u observations that can be used to measure the local color step. We find a local mass step of 0.091 ± 0.024 mag (3.8σ) and a local color step of 0.120 ± 0.030 mag (4.0σ). We find a global mass step of 0.055 ± 0.027 mag and a global color step of 0.104 ± 0.033 mag, both consistent with the local steps. We find a 2.1σ difference between the Foundation and non-Foundation local mass step and a 2.9σ difference between the Foundation and non-Foundation local color step.

The correlation of SN shape and color may also change as a function of host galaxy properties; β, in particular, could be subject to change due to the change of dust properties as a function of host mass or color. For this reason, we tested the effect of adding separate α and β parameters to the likelihood model for each side of the mass or color step.

The results of α and β variation are shown in Figure 4. We find that α is universally higher in locally red regions and locally massive regions (∼2σ significance). We find a significant difference in β only for SNe in locally red regions of their host galaxies, which most likely implies that the effect is driven by dust obscuring the SN.

When α and β are allowed to vary, the local mass step increases by 0.005 mag, while the local color step increases by 0.038 mag. Both increases are only marginally significant, but similarly to Rigault et al. (2018), we find that allowing α and β to be fit simultaneously with the local or global step tends to increase the size of the step and reduce the dispersion. We find a dispersion of just 0.047 mag for SNe in locally red regions but with a high uncertainty, such that the difference between locally red versus blue regions is not statistically significant. Previous results, e.g., Rigault et al. (2015), Kelly et al. (2015) have found lower dispersion in locally blue regions, but did not allow α and β to vary (our sample also gives this result at 1σ significance).

Table 1 shows that after global mass correction, only the local mass step remains significant at >3σ (0.056 ± 0.017 mag). Previous studies (e.g., Roman et al. 2018) have seen a similar effect, which they interpret as evidence that local regions encode information about the SN progenitor that is not captured by a global correction.

In Figure 5, we show the relationship between the local and global measurements in this work to understand which SNe are being corrected by the global versus the local steps. We show the global and local mass densities instead of the global and local masses used elsewhere in this analysis, in order for the local and global units to be the same in this figure. In particular, there are a number of SNe far from the centers of their host galaxies that have high global mass densities but low local mass densities. We label the weighted average of the Hubble residuals in each quadrant. If the local step were driving the global step, we would expect to see a change in Hubble residual only along the x-axis (the local measurement axis). Similarly, if the global measurement were driving the local correction, we would expect the average Hubble residual to change only along the y-axis. Instead, we see ∼4σ evidence (mass) and ∼2.6σ evidence (color) for a step when considering only the two quadrants where local and global agree.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In the previous sections, we have measured only a single step at a time. Beginning with the standard likelihood approach presented in Section 3, we now expand the method to simultaneously measure a combined local and global step for mass and color. The results are summarized in Table 2. By measuring global and local mass steps together, we find a 3.1σ local mass step and a 2.2σ global mass step. The intrinsic dispersion about the Hubble diagram (the dispersion after photometric uncertainties are taken into account) is 3%–5% lower than the dispersion after correcting for a single step. The combined local and global u − g step is less significant than the mass step.

The evidence for a combined local/global mass step is marginally significant, with a Bayesian Information Criterion that is slightly lower (ΔBIC = 3.4) when an extra step is included in the likelihood model. Making either a global step or local step alone leaves an additional step with ≳3σ significance. Therefore, the possibility that local and global may reinforce each other is intriguing.

Having seen evidence for a local mass step after global mass correction, the question remains how "local" the local measurement would need to be to correct SN distances. To answer this question, we use the measurements of mass and color within random apertures discussed in Section 3.2. We summarize the results of these random tests in Table 3. By inferring local properties from random regions >5 kpc from the SN location after first correcting for the global mass step, we measure a "false local" mass step of 0.029 ± 0.017 mag. This step is smaller than the true local step by 0.027 ± 0.017 mag, a difference with 1.6σ significance. As discussed at the beginning of Section 4, these uncertainties incorporate the correlation between the local and random measurements. We measure a u − g step of 0.027 ± 0.015 mag, 0.011 ± 0.027 mag smaller than the local step. We therefore see only marginal evidence that measurements of host galaxy properties within 5 kpc of the SN location are important for SN distance corrections. In the Appendix, we examine the differences in intrinsic dispersion between local and random regions, finding no significant difference in dispersion when local mass and color are inferred from random locations instead of locations near the SN.

Table 3.  Comparing Local to Random Measurements

	ΔM		${{\rm{\Delta }}}_{u-g}$
	No Global Mass Corr.	Global Mass Corr.a	No Global Mass Corr.	Global Mass Corr.a
Local Step	0.067 ± 0.017	0.056 ± 0.017	0.060 ± 0.019	0.038 ± 0.019
Random Step	0.047 ± 0.018	0.029 ± 0.017	0.040 ± 0.020	0.027 ± 0.020
5 kpc from SNe	0.025 ± 0.019	0.000 ± 0.018	0.031 ± 0.021	0.012 ± 0.021
10 kpc from SNe	0.032 ± 0.021	0.015 ± 0.020	0.051 ± 0.021	0.036 ± 0.021
R < 1	0.021 ± 0.019	0.002 ± 0.019	0.039 ± 0.023	0.011 ± 0.023
1 < R < 2	0.041 ± 0.019	0.018 ± 0.019	0.046 ± 0.021	0.031 ± 0.021
2 < R < 3	0.044 ± 0.018	0.021 ± 0.019	0.053 ± 0.020	0.037 ± 0.020

Notes. R is the distance from the center of the galaxy in units of the normalized elliptical radius of the galaxy (Sullivan et al. 2006). The last five rows exclude regions within 3 kpc of the SN location. Also in the last five rows, the step location is taken to be the median of every sample to avoid a situation in which 90% or more of the sample is considered "high mass" or "low mass."

^aThe size of each step after applying the maximum likelihood global mass correction of 0.058 ± 0.018 mag. ^bRegions >5 kpc from SN are randomly sampled. One random region is chosen per SN, the step is measured, and this process is repeated 100 times. The steps listed here are the mean of 100 samples.

Download table as:  ASCIITypeset image

The last five rows of Table 3 show false local steps using a set of representative R parameters and distances from the SN. All of these measurements yield steps smaller than the local step, typically by ∼2σ significance for mass and ∼1σ for color. The R measurements in Table 3 do not include regions within 3 kpc of the SN, so that no measurements include the true local fluxes at the SN location. We also restrict distance measurements to R < 5.

Figure 6 expands the results in Table 3 to show the change in the local mass and color steps as a function of both ΔR, the difference in R between the SN and the aperture (left), and of the aperture's physical distance from the SN (right). A negative ΔR indicates that physical properties are inferred from regions closer to the galactic center than the SN location, while a positive ΔR means that the physical properties are inferred from regions farther from the galactic center than the SN location.

Zoom In Zoom Out Reset image size

As distances from the SNe increase, the sampling of random apertures becomes slightly more sparse, and therefore the mass and color steps are not always computed using the full SN sample. There is a similar effect at play for different values of Δ R; for an SN at the center of its host galaxy, having a random aperture with Δ R < 0 is impossible. Similarly, an SN near the edge of its host could not have a large ΔR. Small hosts in particular will have a restricted range of ΔR and physical distances >10 kpc from the SN location may be outside the R = 5 ellipse. Therefore, there are significant biases in the global host demographics for different ΔR parameters and distances. For this reason, in Figure 6 we always compare the false local steps to the true local steps measured using the exact same set of SNe.

There are hints that the SN distance measurement becomes less correlated with the localized host galaxy mass at ≳5 kpc from the SN. We also find that a number of mass step measurements are smaller than the local step by ≳0.03 mag (∼2σ). The statistical significance of these differences is limited, and different ΔR steps are not completely statistically independent. However, the observed differences between random and local are consistent with the observed 0.056 ± 0.017 mag local mass step after global mass correction. However, we see no statistically significant difference between the local and random color step.

We find a 0.011 mag decrease in the random color step compared to the local color step, which is consistent with Roman et al. (2018). Roman et al. (2018) find a decrease in the size of a local color step of 0.022 mag when changing from their nominal local radius of 3 kpc to a radius of 16 kpc, approximately the maximum distance from the SN location considered here. Because we use only a low-z sample to examine local regions, our uncertainties are larger than those of Roman et al. (2018), and a difference of 0.022 is comparable to the 1σ local color uncertainties. However, the 1σ uncertainties on this test constrain the effect of a non-local measurement to ≲0.04 mag.

Recent work from Rigault et al. (2018) has suggested that the local specific star formation rate (LsSFR) has a strong correlation with SN Ia residuals in SNfactory data. Though we lack the local Hα measurements used by Rigault et al. (2018), ugriz photometry should enable us to investigate whether such a correlation is present in our data, though we caution that Hα may be a more robust diagnostic of sSFR.

As Z-PEG has difficulty measuring sSFR in passive galaxies, we use LePHARE (Arnouts & Ilbert 2011) with Bruzual & Charlot (2003) spectral templates as an alternative SED-fitting method to infer the LsSFR from our sample. We also use LePHARE for consistency checks on our data in Section 4.6 below. The SED-fitting parameters from LePHARE are broadly consistent with those from Z-PEG; the median u − g color from LePHARE is within 0.04 mag of the Z-PEG color and the median host galaxy mass is just 0.09 dex lower.

Using the median LsSFR of our sample, −10.6 dex, as the divide between SNe Ia in low- and high-sSFR regions, we measure a step size of 0.051 ± 0.020 mag (2.5σ). The global sSFR step size is nearly identical, 0.054 ± 0.020 mag (2.7σ). The significance of both steps becomes <2σ after global mass step correction (the local step becomes 0.035 ± 0.021 mag and the global step becomes 0.029 ± 0.020 mag). Rigault et al. (2018) find a step of 0.125 ± 0.023 mag (0.163 ± 0.029 mag after allowing α and β to be fit simultaneously with the step). Their result is statistically inconsistent with ours at the 2.4σ level, though removing the BBC corrections would reduce that discrepancy to 2.1σ. If this discrepancy is not due to statistical fluctuation or unforeseen systematic effects due to differences in sample selection, calibration, or sSFR measurement methods, it may be additional evidence that targeted versus untargeted surveys affect the measured step sizes. Interestingly, the sSFR step sizes we measure do not significantly change if we use only SNe from targeted or untargeted samples.

We show the local and global LsSFR steps in Figure 7 and include the LsSFR measurements from LePHARE in our online data.

Zoom In Zoom Out Reset image size

In this section, we present several consistency checks to validate the SED-fitting procedures in this work. The Z-PEG SED-fitting method is significantly different from that of Roman et al. (2018), for example, who also base their results on the PEGASE.2 templates but warp those templates to match the observed photometry of galaxies in the SuperNova Legacy Survey fields.

However, 54 SNe Ia in this sample are also included in Roman et al. (2018). We compare our rest-frame U – V colors and observed u − g colors to those measured by Roman et al. (2018) using their online data. Though we measure u − g colors within a 1.5 kpc radius while Roman et al. (2018) use a 3 kpc radius, we observe a median color of 1.52 mag, just 0.06 mag bluer than that of Roman et al. (2018). Though we use rest-frame u − g colors in this work, after fitting with Z-PEG, we verified that the rest-frame U − V colors were consistent with those of Roman et al. (2018): we find a median rest-frame U − V color of 0.83 mag compared to 0.77 mag for Roman et al. (2018). If we measure a U − V step instead of a u − g step, we find that the step size increases by just 5 mmag.

Z-PEG returns a set of "pseudo-observed" model magnitudes, which have been reddened and redshifted to match the observed data. These model magnitudes should be close to the observed data if our SED-fitting procedure is reliable. For 141 of the 194 SNe Ia with u observations, the pseudo-observed magnitudes are within the 2σ uncertainties on the local u − g color observations. In reality, there is some additional uncertainty on the model, which would increase the statistical agreement between model and data. If we restrict our sample to just these 141 SNe Ia, we measure a local color step of 0.044 ± 0.022 mag, but consistent with the 0.060 ± 0.019 mag step measured from the full sample.

For the local mass step, a simple consistency check may be performed using the relationship between gi photometry and the host galaxy stellar mass given by Taylor et al. (2011):

$$\begin{eqnarray}&&\mathrm{log}({M}_{* }/{M}_{\odot })=1.15+0.70(g-i)-0.4{M}_{i},\end{eqnarray} \tag{ 3 }$$

where the absolute i magnitude M_i is estimated using Ω_M = 0.3, Ω_Λ = 0.7, and H₀ = 70 km s⁻¹ Mpc⁻¹. The uncertainty of the relation is 0.1 dex, which we add in quadrature to the propagated photometric uncertainty.

Although this equation does not k-correct the photometry, it is still a reasonable approximation for these low-redshift data. Using this approximation instead of Z-PEG, we measure a local mass step of 0.077 ± 0.017 mag, consistent with the measurement of 0.067 ± 0.017 mag measured from the full sample.

Lastly, we use the LePHARE SED-fitting software (Arnouts & Ilbert 2011) with Bruzual & Charlot (2003) templates (the version used for the COSMOS mass function; Ilbert et al. 2009) to independently check the mass and color measurements from Z-PEG. We find a median color just 0.04 mag redder than the Z-PEG measurements and a median host mass 0.09 dex smaller than the Z-PEG measurements. LePHARE yields less model-dependent colors than Z-PEG, as it uses the SED templates for k-corrections but interpolates using those k-corrections from the observed magnitudes themselves. We measure local mass and color steps that are consistent with, though slightly smaller than, the Z-PEG measurements: with LePHARE, we measure a nearly identical local mass step of 0.066 ± 0.017 mag and a local color step of 0.047 ± 0.019 mag.