Should Type Ia Supernova Distances be Corrected for their Local Environments?

Recent analyses suggest that distance residuals measured from Type Ia supernovae (SNe Ia) are correlated with local host galaxy properties within a few kpc of the SN explosion. However, the well-established correlation with global host galaxy properties is nearly as significant, with a shift of 0.06 mag across a low to high mass boundary (the mass step). Here, with 273 SNe Ia at $z<0.1$, we investigate whether stellar masses and rest-frame $u-g$ colors of regions within 1.5 kpc of the SN Ia explosion site are significantly better correlated with SN distance measurements than global properties or properties measured at random locations in SN hosts. At $\lesssim2\sigma$ significance, local properties tend to correlate with distance residuals better than properties at random locations, though despite using the largest low-$z$ sample to date we cannot definitively prove that a local correlation is more significant than a random correlation. Our data hint that SNe observed by surveys that do not target a pre-selected set of galaxies may have a larger local mass step than SNe from surveys that do, an increase of $0.071\pm0.036$ mag (2.0$\sigma$). We find a $3\sigma$ local mass step after global mass correction, evidence that SNe Ia should be corrected for their local mass, but we note that this effect is insignificant in the targeted low-$z$ sample. Only the local mass step remains significant at $>2\sigma$ after global mass correction, and we conservatively estimate a systematic shift in H$_0$ measurements of -0.14 $\textrm{km}\,\textrm{s}^{-1}\textrm{Mpc}^{-1}$ with an additional uncertainty of 0.14 $\textrm{km}\,\textrm{s}^{-1}\textrm{Mpc}^{-1}$, $\sim$10\% of the present uncertainty.


INTRODUCTION
Type Ia supernovae (SNe Ia) have become increasingly precise cosmological distance indicators through improvements in how they are standardized. Beyond accounting for the light-curve shape and color of SNe Ia, the most recent and smallest effect to be routinely addressed in cosmological samples is a ∼0.06 mag correction derived from the empirical correlation of SN Ia distance residuals with host galaxy mass (the mass step; Kelly et al. 2010;Lampeitl et al. 2010;Sullivan et al. 2010). Cosmology analyses typically correct for the mass step (Sullivan et al. 2011;Betoule et al. 2014) despite a lack of understanding of the underlying cause. If mass serves only as a proxy for the underlying cause, for example, metallicity or progenitor age (Hayden et al. 2013;Chil-dress et al. 2014;Graur et al. 2015), a somewhat different correction may yield improved cosmological distance estimates from SNe Ia.
In cases where the progenitor has a short delay between formation and explosion (prompt SNe Ia), the environment near the site of the SN could be used as a better diagnostic of the properties of the progenitor than the global host environment. Up to 50% of SNe Ia could explode less than 500 Myr after the formation of their progenitor systems (Rodney et al. 2014;Maoz et al. 2014). Therefore, a correction based on the local environment may be a better method of standardizing SNe Ia than a correction based on the host galaxy as a whole. However, for SNe Ia with longer delay times such a correlation becomes less likely.
Evidence for a correlation between SN shape-and color-corrected magnitude (hereafter corrected magnitude) and local star formation rate within 1-2 kpc of the SN explosion site was reported by Rigault et al. (2013) using SN Factory data (Aldering et al. 2002) and Rigault et al. (2015) for a publicly available SN sample (Hicken et al. 2009a). Kelly et al. (2015) found evidence that the dispersion of SN corrected magnitudes was lower in highly star-forming local environments but had only a small sample of ∼20 SNe that were found in such environments. However, Jones, Riess, & Scolnic (2015) found that after applying updated light curve fitters and employing the same sample selection as used for cosmological analyses, the relationship between inferred SN Ia distance and local star formation was found to be insignificant in a sample of 179 z < 0.1 SNe Ia.
More recently, Roman et al. (2017) found a relationship between SN corrected magnitude and local, rest-frame U − V color. A "step" between blue and red colors was seen at 1.7σ significance at z < 0.1 and 6.9σ significance when using SNe Ia from 0.1 < z < 0.5 (and 7.0σ significance when all SNe are included). The z < 0.1 step measurement is 0.053 ± 0.032 mag, while the 0.1 < z < 0.5 step is 0.117 ± 0.017 mag. The reason for this difference is unclear, but factors could include statistical fluctuation, survey selection effects, different effective apertures due to blending at high-z, or a redshift-dependence local step. Similarly, Kim et al. (2018) used global properties to infer local properties for a subset of SNe Ia from 0.01 z 1, finding that the inferred local star formation correction was 0.081 ± 0.018 mag, 0.024 mag larger than the global mass step.
Here, we ask whether the evidence for a local step implies that host galaxy properties near the SN location contain additional information that could improve the standardization of SNe Ia. Alternatively, it may be that local regions merely trace global host galaxy properties. Roman et al. (2017), for example, found that the size of the local step decreases by just 0.022 mag when inferring local properties within an aperture of radius 16 kpc, an aperture that should contain nearly all the light from a galaxy. With the first data release of the Foundation low-z SN sample (Foley et al. 2018), we are now able to ask this question with up to 271 z < 0.1 SNe Ia, a low-z sample that is ∼40% larger than that used in previous cosmological analyses (Scolnic et al. 2017;Jones et al. 2018;Betoule et al. 2014).
We use measurements of the stellar mass and local, rest-frame u − g colors near the SN location. The local stellar mass is a natural first measurement to investigate, given the known correlation of SN distance residuals with global stellar mass. Measuring local stellar mass is also a convenient measurement; it only requires optical photometry, which is available for the entire low-z SN sample. Rest-frame u − g colors, on the other hand, are effectively the same as the local U −V colors used by Roman et al. (2017). u − g colors are sensitive to the host galaxy star formation without suffering from the resolution limitations of shorter-wavelength UV instruments such as GALEX (e.g. Jones et al. 2015).
We measure the Hubble residual "step" as a function of global properties, local properties, and the properties within 1.5 kpc apertures at random locations within each host galaxy. In §2 we present the SN sample and we measure host galaxy properties in §3. In §4 we measure the correlation of these data with host galaxy properties, and in §5 we examine the impact of our results on the Hubble constant. We conclude in §6.

DATA AND ANALYSIS
For this analysis, we combine 216 z < 0.1 SNe from the Pantheon compilation (Scolnic et al. 2017) with 178 SNe Ia from the Foundation first data release (DR1; Foley et al. 2018). This combined sample contains 394 SNe Ia and twice as many SNe Ia at z < 0.1 as recent cosmological analyses (e.g. Scolnic et al. 2017).
The Foundation survey uses the Pan-STARRS1 (PS1) telescope to follow nearby SNe Ia discovered by ASAS-SN (Holoien et al. 2017), ATLAS (Tonry et al. 2018), Gaia (Gaia Collaboration et al. 2016, and the Pan-STARRS Survey for Transients (PSST; Huber et al. 2015) among other surveys. SNe Ia from the Foundation DR1 are observed on the well-calibrated PS1 photometric system (Schlafly et al. 2012) and can therefore be used to measure distances with good control over systematic uncertainties. The Foundation DR1 includes 225 SNe Ia, 180 of which pass the cuts for inclusion in a cosmological analysis used in Foley et al. (2018) (2 Foundation SNe Ia are at z > 0.1 and are therefore excluded here).

Sample Selection Requirements using SALT2
We infer distances from the SNe Ia in the Pantheon and Foundation samples using the SALT2.4 light curve fitter (Betoule et al. 2014;Guy et al. 2010) and the Tripp estimator (Tripp 1998): (1) m B is the log of the light curve amplitude, x 1 is the light curve shape parameter, and c is the light curve color parameter. α and β are nuisance parameters along with M, a parameter encompassing the SN Ia absolute magnitude at peak and the Hubble constant. The Pantheon and Foundation analyses apply sample selection criteria based on these SALT2 light curve parameters to ensure a well-calibrated sample. 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 < x 1 < 3, −0.3 < c < 0.3), and cuts to ensure that the shape and time of maximum light are well-measured (x 1 uncertainty <1 day and time of maximum uncertainty <2 days). Here, we also require 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 P 1 , 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 . The BBC method uses simulated SN samples to correct x 1 , c, m B , α, and β for observational biases and selection effects. These corrections are important for this study because SN light curve parameters depend on host properties and SNe Ia with c < −0.2 and x 1 > 2 have mean Hubble residuals of 0.2-0.3 mag ). These residuals are 3-4 times larger than the host mass step.
We use the simulations from Scolnic et al. (2017) and Foley et al. (2018) with the BBC method to generate these bias corrections (Scolnic et al. 2018, in prep, will contain additional simulation details specific to the Foundation sample). The BBC method removes 6 additional SNe from the sample; 3 from Pantheon and 3 from Foundation. We cannot be certain that the bias corrections are valid for these 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, 3 are at z < 0.01, 2 are at z > 0.1, 2 do not pass cuts due to small changes in the SALT2 fitting parameters 12 and the remainder are lost due to BBC cuts. See Foley et al. (2018) and Scolnic et al. (2017) (and references therein) for additional details on the sample selection.

Sample Selection Requirements using Host Galaxy
Properties 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. 2017). 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 in physical size or less. PS1 images have a typical seeing of ∼1 , while SDSS images have a median seeing of approximately 1.38 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 ∼1.6 , 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): 12 In order to match the Pantheon analysis, we reduce the wavelength range over which the SALT2 model is fit to the photometric data to a maximum of 7000Å.
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 271 SNe. After all cuts, grizy images for measuring the local mass step are available for 271 SNe Ia. 195 of these SNe 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.

HUBBLE RESIDUAL STEPS
The local photometry was measured within a circular aperture of radius 1.5 kpc, while the global host galaxy photometry was measured using elliptical aperture photometry. The size of the global host ellipse was set to be equal to the R = 4 ellipse measured by SExtractor on each PS1 r-band image. A uniform ellipse radius that extends just beyond the estimated isophotal radius of the galaxy ensures that all flux is captured and that a uniform aperture size is used for all photometric bands. An R = 4 ellipse is still small enough for contamination from neighboring stars or galaxies to be negligible. In addition, the difference between the PS1 and SDSS seeing is just 1.7% of the median R = 4 semimajor axis of the galaxies in this sample and therefore should not significantly bias the photometry, especially given that the elliptical aperture extends beyond each galaxy's isophotal radius.
We then fit the local and global ugrizy photometry to template SEDs following the method of Pan et al. (2014). We estimate galaxy masses and un-reddened, rest-frame u and g colors using the Z-PEG SED-fitting code (Le Borgne & Rocca-Volmerange 2002), which is based on spectral synthesis from PEGASE.2 (Fioc & Rocca-Volmerange 1997). Galaxy SED templates correspond to spectral types SB, Im, Sd, Sc, Sbc, Sa, S0 and E. We marginalize over E(B-V), which is allowed to vary from 0 to 0.2 mag, and the star formation rate. Uncertainties are estimated by generating Monte Carlo realizations of our photometric measurements. For each filter, we generate mock photometric points from a normal distribution with standard deviation equal to the photometric uncertainties, and use Z-PEG to fit SEDs to each realization of the photometric data. We then estimate the uncertainty in the host mass and rest-frame photometry from the spread in output values. The photometric uncertainties from this approach can occasionally be unrealistically small; for this reason we add 0.05 mag uncertainty in quadrature to the u−g rest-frame colors, approximately equal to the photometric errors for a 3 kpc region in a bright host galaxy, to account for systematic uncertainties in the SED fitting. -Local mass density and u − g maps from four representative galaxies in our sample. The local mass and colors used in this work are measured from the 3 kpc diameter regions indicated by the small circles. For illustration, the local mass density is computed per pixel and has a median value of log(M * /M ) -log(Area) ∼ 8 kpc −2 . To include regions of negative flux in the map, which have an undefined color measurement, the bottom row shows the probability that the true u − g color is < 1.6 mag. The approximate R = 3 isophotal radius of each galaxy is denoted by the ellipses. White colors in the map indicate regions on the border between locally high-mass and low-mass and blue u − g/red u − g (and may also indicate pixels with higher than average noise). For the purposes of this plot, we use observer-frame u − g colors.
Several studies have discussed whether local and global SED-fitting measurements are self-consistent. Sorba & Sawicki (2015) found a 0.1 dex bias in global host galaxy mass measurements of star-forming galaxies when fitting mass to the photometry of the entire galaxy instead of performing a pixel-by-pixel fit and summing the individual measurements. This level of bias will not affect our results, as we look at global and local mass independently (defining the step location separately for global and local measurements). The location of the step is also not known to within 0.1 dex (Scolnic et al. 2017). Other studies have found that summing the results of pixel-bypixel SED fitting give the same parameters as a SED fit to the photometry of the whole galaxy (Salim et al. 2016;San Roman et al. 2018).

Measuring the Mass and Color Steps
We treat the dependence of SN Ia shape-and colorcorrected magnitude on host mass and u − g as a step function, as previous studies have found this to be wellmotivated by the data (Betoule et al. 2014;Roman et al. 2017). 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 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, Riess, & Scolnic (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 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 u − g color of this sample is 1.6, and we adopt this value as an agnostic choice for the location of the step following Roman et al. (2017). 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 ) = 8.9. 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 datasets and analyses. For this reason, we adopt the standard global host mass step location of log(M * /M ) = 10 Betoule et al. 2014;Scolnic et al. 2017). Figure  1 shows local mass per pixel and u − g maps for four representative galaxies in our sample.

Measuring the Mass and Color in Random
Apertures 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. 2017). To address this question, we place 150 random apertures of diameter 3 kpc 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 The dependence of SN luminosities on the mass and u−g color within 1.5 kpc of the SN location. Colors indicate the probability that a SN is in a low-mass host galaxy (left) or a galaxy with blue rest-frame color (right). We see 2σ correlations with both quantities. The gap in rest-frame u − g colors at ∼1.4 mag may be due to a gap in the colors of the PEGASE.2 SED templates.
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 a 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 towards 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.

RESULTS
Using the methods described above, we measure a local mass step of 0.067 ± 0.017 mag (3.9σ significance) and a local color step of 0.040 ± 0.020 mag (2.0σ). These steps are shown in Figure 2. If we use global properties instead of local to measure the size of the step, we find the global mass step to be 0.049 ± 0.018 mag and the global color step to be 0.046 ± 0.020 mag. The local mass step is larger than the global mass step, while the local u − g step is smaller than the corresponding global step. Table 1 summarizes each global and local step measured from these data, both before and after correcting for the maximum likelihood global mass step of 0.049 ± 0.018 mag. Most significantly, we find a local mass step of 0.059 ± 0.017 mag after correcting for the global mass (3.5σ). If we instead correct for the local mass step before measuring the global step, we find a global mass step of 0.046 ± 0.019 (2.5σ). Table 1 also divides the sample into SNe from surveys that target a pre-selected set of galaxies and those that do not ( §4.1 below).
Estimating the statistical significance of the difference between the global and local steps is somewhat difficult due to the fact that global and local measurements are partially correlated. 56% of the SNe in this sample are either globally and locally high-mass or globally and locally low-mass (72% for local color). To estimate the 1σ uncertainty on the difference between the global and local step with correlated measurements, we simulate 1,000 SN samples using our real local and global measurements but with Hubble residuals drawn from a Gaussian centered on 0 and with dispersion equal to the real dispersion of our sample. We find that 68% of the Monte Carlo samples have a local/global difference < 0.020 mag for the mass measurements, and <0.018 mag for the color measurements. These correspond to the 1σ uncertainties on the local/global difference, and are slightly smaller than the uncertainties that would be obtained just by adding the local and global mass uncertainties in quadrature. With this approach, we find that sizes of the global and local measurements for both mass and color are consistent at the 1σ level. Therefore, the data do not indicate that the local steps are intrinsically more significant than the global steps. Measuring the local mass step from just the 195 SNe Ia with SDSS u data gives a local mass step of 0.062±0.019, just 5 mmag less than the step measured from the full sample. The slight gap in the dust-corrected, rest-frame u−g colors ( Figure 2) is likely due to a gap in the colors of the PEGASE.2 SED templates, and should not adversely impact our results. The local and global measurements used in this work are available online 13 .

Targeted Versus Untargeted Surveys
Roman et al. (2017) 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 either 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. All of the high-z surveys do not 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 H 0 . 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. In addition, it may 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 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). We see a 2.1σ increase in the local mass step, a 1.9σ increase in the global color step, and statistically insignificant differences in the global mass and local color steps when untargeted surveys are used instead of targeted surveys. These differences are not highly significant but could indicate that 13 http://pha.jhu.edu/~djones/localcorr.html the correlation of SN distance with host galaxy properties is sensitive to survey selection effects.
We also check the significance of a local step vs 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 0.090 ± 0.026 mag (3.5σ) and a local color step of 0.067 ± 0.033 mag (2.0σ). We find a global mass step of 0.049 ± 0.027 mag and a global color step of 0.072 ± 0.035 mag, both consistent with the local steps. We find a 1.7σ difference between the Foundation and non-Foundation steps.

Simultaneously Fitting a Global and Local
Step Table 1 shows that after global mass correction, only the local mass step remains significant at >2σ (0.057 ± 0.017 mag). Previous studies (e.g. Roman et al. 2017), 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 3 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 mass 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 ∼2-3σ evidence (mass) and ∼1σ evidence (color) that the Hubble residuals are different from 0 only in the two quadrants where local and global agree.
In the previous sections, we have measured only a single step at a time. Beginning with the standard likelihood approach presented in §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 3.6σ local mass step and a 2.0σ global mass step. The intrinsic dispersion about the Hub-  ble diagram (the dispersion after photometric uncertainties are taken into account) is 7% 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 only marginally significant and the Bayesian Information Criterion does not support the addition of an extra step to the likelihood model. However, making either a global step or local step alone leaves an additional step with between 2.5σ and 3.5σ significance. Therefore, the possibility that local and global may reinforce each other remains intriguing.

Random Apertures
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 §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 0.028±0.016 mag. This step is smaller than the true local step by 0.025±0.016 mag, a difference with 1.6σ significance. As discussed at the beginning of §4, these uncertainties incorporate the correlation between the local and random measurements. We measure a u−g step of 0.029±0.020 mag, 0.012±0.023 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.
The last five rows of Table 3 show false local steps using a set of representative R parameters and distances from the SN. Seven of these measurements yield steps smaller than the local step, including all local mass measurements, while three measurements yield steps larger than the local step. However, only one measurement is smaller than the baseline steps by >1σ (after taking the local/random correlation into account). 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 4 expands the results in Table 3 to show change in the local mass and color steps as a function of both ∆R, the difference in R between the SN and the aperture (left), and of the aperture's physical distance from the SN (right). Negative ∆R indicates that physical properties are inferred from regions closer to the galactic center than the SN location, while positive ∆R means that the physical properties are inferred from regions farther from the galactic center than the SN location.
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 in play for different values of ∆R; for a SN at the center of its host galaxy, having a random aperture with ∆R < 0 is impossible. Similarly, a SN near the edge of its host could not have a large ∆R. Small hosts in particular will have a restricted range of ∆R and physical

Local
Step 0.061 ± 0.016 0.053 ± 0.016 0.041 ± 0.019 0.023 ± 0.020 Random Step b 0.045 ± 0.017 0.028 ± 0.017 0.029 ± 0.020 0.011 ± 0.020 5 kpc from SNe 0.039 ± 0.017 0.015 ± 0.017 0.052 ± 0.019 0.029 ± 0.020 10 kpc from SNe 0.040 ± 0.018 0.023 ± 0.018 0.062 ± 0.021 0.048 ± 0.020 R < 1 0.020 ± 0.019 0.002 ± 0.019 0.034 ± 0.022 0.013 ± 0.022 1 < R < 2 0.050 ± 0.018 0.031 ± 0.018 0.031 ± 0.020 0.017 ± 0.021 2 < R < 3 0.048 ± 0.017 0.030 ± 0.017 0.067 ± 0.020 0.048 ± 0.020 Note. -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 5 rows exclude regions within 3 kpc of the SN location. Also in the last 5 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". a The size of each step after applying the maximum likelihood global mass correction of 0.049 ± 0.018 mag. b Regions >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. Fig. 4.-After global mass correction, the "false local step" (black): the correlation of SN distance measurements with the masses and u − g colors of different regions in the host galaxy. ∆R is the difference in R between the SN location and the random location after excluding all regions within 3 kpc of the true SN location. The local step after global mass correction and its uncertainty are indicated by the shaded region. For each false local step, the true local step at the SN location (red line) is plotted using the same set of SNe used to measure the random step. Note. -We show the effect of applying a local step after correcting for a 0.06 mag mass step following Riess et al. (2016). We note that the H0 correction appears to be stronger in untargeted surveys of SNe Ia than it does in targeted surveys such as the  sample. Note that the "global mass" correction increases H0, as we measure a slightly smaller mass step of 0.05 mag in this work. However, the steps applied are nearly identical to those listed in the "Global Mass Corr." columns of Table 1. a Significance of the step after 0.06 mag correction based on global mass. 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 ∆R parameters and distances. For this reason, in Figure 4 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 ∆R steps are not completely statistically independent. However, the observed differences between random and local are consistent with the observed 0.057 ± 0.017 mag local mass step after global mass correction. We see no significant difference between the local and random color step.
We find a 0.012 mag decrease in the random color step compared to the local color step, which is consistent with Roman et al. (2017). Roman et al. (2017) 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. (2017), and a difference of 0.022 is comparable to the 1σ local color uncertainties. However, this test constrains the effect of a non-local measurement to 0.04 mag.

IMPACT ON THE HUBBLE CONSTANT
A leading approach for measuring the Hubble Constant, H 0 , calibrates the luminosity of SNe Ia in nearby galaxies using Cepheid variables and compares them to SNe Ia in the Hubble flow (typically z 0.01 − 0.02). A potential bias may enter if there are differences in the mean host properties of the two SN samples for some of the host properties considered here.
The determination of H 0 in Riess et al. (2016) corrects the two SN samples for the global mass step using a value of 0.06 mag (Betoule et al. 2014), consistent with the 0.049 ± 0.018 mag global step we measure in this work. After the 0.06 mag global mass step is applied to our sample, instead of the 0.049 mag global mass step determined in §4, we measure residual, local step sizes of 0.057 ± 0.017 mag (mass) and 0.019 ± 0.020 mag (color). Of these, only the local mass step may be considered significant and may indicate a bias. Here we calculate the size of a possible bias in H 0 . We also note that for a local step to resolve the discrepancy between the local measurement and the CMB-inferred value (Planck Collaboration et al. 2015), the effect would also have to be present in SN Ia J-band luminosity (Dhawan et al. 2018).
The bias to the Hubble constant due to a local mass step is given by: local is the size of the local step after removing the global step. ψ HF and ψ C are the fractions of SNe Ia in the Hubble flow and in galaxies with Cepheid observations, respectively, that occurred in locally massive regions of their hosts. We use the recent measurement of H 0 = 73.48 ± 1.66 from Riess et al. (2018) as our baseline. ψ HF is computed using only the SNe in this analysis that are also included in Riess et al. (2016). 16 of 19 total Cepheid calibrators have PS1 imaging, as 3 (SN 2001el, SN 2012fr, and SN 2015F) are too far south for PS1. An additional 2 SNe lack SDSS u imaging (SN 2005cf andSN 2007sr). For these 5 SNe, we use SkyMapper photometry (Wolf et al. 2018) instead of PS1 and SDSS photometry to determine the local masses, global colors, and local colors.
Because the fraction of SNe Ia with local masses above or below the step is fairly well balanced across the Cepheid calibrator and Hubble flow samples, with a fractional sample difference of just under 0.15, the effect on H 0 is a small fraction of the step, reducing it by 0.26 km s −1 Mpc −1 . This shift is 16% of the present uncer-tainty in H 0 . A slightly larger sample difference is seen for local u − g colors. We find that 94.7% of Cepheid calibrators are in u − g < 1.6 galaxies. In contrast, ∼47.7% of the Hubble flow sample are in u − g < 1.6 galaxies. However, because the significance of the local color step (after global mass correction) is ∼ 1σ, no correction is warranted.
For the local mass, global mass, local u − g and global u − g steps, Table 4 gives the estimated bias to H 0 using the measurements in this work after a global mass correction. These range from 0.08 to -0.31 km s −1 Mpc −1 . However, only the local mass step is significant and thus could be considered meaningful.
A caveat to applying even the local mass step correction may be drawn from the differences in steps suggested in the previous section for targeted and non-targeted surveys. Both the Cepheid calibrated and Hubble flow samples used in Riess et al. (2016) came exclusively from targeted surveys in which all local steps with or without the global mass correction applied are smaller and not significant with only ∼ 1σ confidence. If the present hint of a difference in step sizes between these survey types is established with larger surveys, we would conclude that no additional correction to H 0 would be warranted for these local steps. At present a conservative approach would be to apply half the shift to H 0 and consider half the shift as part of the systematic uncertainty.
An alternative approach to accounting for differences in the host properties of SN samples could be to ensure both samples are homogeneous. For the determination of H 0 using Cepheids to calibrate SNe Ia, it is necessary to select calibrators from late-type galaxies. Placing this same selection criterion on the Hubble flow sample, as done in Riess et al. (2016), has a negligible impact on the uncertainty in H 0 because the number of SNe Ia in late-type hosts in the Hubble flow is much larger than the number of calibrators.

CONCLUSIONS
We used up to 271 SNe from the Pantheon and Foundation samples to determine whether the physical properties of the regions near the location of SNe Ia are as correlated with SN light curve parameters and inferred SN distances as global host properties or random regions within those same host galaxies. We see a significant correlation between local stellar mass and SN distance residuals. However, even with the largest sample of z < 0.1 SNe Ia to date, we were unable to definitively prove that local information is better-correlated with SN Ia distance measurements than global or random information. This sample is ∼40% larger than the low-z sample used in recent measurements of cosmological parameters. Our measurements of local masses and local, rest-frame u − g colors for the full sample are available online 14 .
We find ∼2σ evidence for a correlation between Hubble residuals of SNe for which local and global measurements agree. The difference between the inferred distances of SNe in both locally high-mass regions and globally highmass galaxies versus those in locally/globally low-mass regions is 0.099 ± 0.026 mag. The evidence that such an effect exists is not definitive, but is plausible given that a 3.5σ local mass step remains after correcting for global mass, and a 2.5σ global mass step remains after correcting for local mass. We see no evidence for a local or global step, as a function of either mass or color, in a sample of SNe Ia for which global and local indicators disagree. Figure 3 summarized the Hubble residuals in each local versus global quadrant.
We find that the local mass step is more significant than a local color step. We find 1.2σ evidence for a local u − g step after correcting for a global host mass step.
Though the results here do not prove that SNe Ia are more correlated with their local host environments than their global environments, we use these results and their uncertainties to put limits on the estimated bias to cosmological parameters due to local effects. The only step detected at >2σ significance, the local mass step, would give an estimated systematic shift in H 0 of -0.13 km s −1 Mpc −1 with an additional uncertainty of 0.13 km s −1 Mpc −1 , ∼10% of the current uncertainty on H 0 .
Lastly, we find 2.1σ evidence for tension between measurements of the local mass step from surveys that target a pre-selected set of galaxies (the previous low-z sample) and surveys that do not. Previous work has also shown that different samples may have different step sizes and it is not clear why (e.g. Rest et al. 2014;Scolnic et al. 2014). Roman et al. (2017) found that the targeted lowz sample has marginal evidence for a local color step of 0.049 ± 0.046 mag (1.1σ significance), but they found a local step that was nearly twice as large when including data with 87% of SNe from untargeted surveys (7.0σ significance). The fact that the untargeted surveys here were observed on the Sloan filter system, while the targeted surveys used Johnson filters may also perhaps play a role. Though the samples included in Roman et al. (2017) cannot determine whether this result is due to redshift evolution of the step or survey-specific effects, our data − and future Foundation data releases − can break this degeneracy.
We remain agnostic about the reasons for sampleto-sample differences, but it is clear that pre-selecting galaxies will alter the demographics of the SN sample and therefore may change the measured relationships of SNe Ia with their hosts. As most SNe used in the Riess et al. (2016) H 0 measurement are from targeted searches, it is unclear whether it is appropriate to apply a correction to the current H 0 analysis if that correction is measured from untargeted samples. This question is unlikely to be resolved without a better understanding of the relationships between SNe Ia and their environments.
The existing low-z sample is also subject to significant calibration uncertainties and selection biases. A local mass step in particular could be biased by difference imaging residuals in SN Ia photometry. In Foundation, we have multiple epochs of PS1 3π with no SN light that can be used to test and correct for the possibility of small difference imaging biases in future work. When SNfactory (Aldering et al. 2002) and the Foundation second data release are publicly available, these data may reveal correlations that our data are unable to probe.
As the connection between SN environments and their progenitors remains unclear, the SN-host relation will remain a possible source of systematic uncertainty in cosmological analyses for the foreseeable future. If future studies find evidence for a relationship between SN Ia corrected magnitudes and their local environments, we propose that these studies adopt the methodology presented here to determine the "locality" of the correlation. If global host properties will be sufficient to correct SN Ia magnitudes for host galaxy biases, space-based imaging will not be needed for precision cosmology. If, on the other hand, convincing evidence is shown that regions 5 kpc from the SN location are not as well correlated with the SN Ia corrected magnitude as regions 2 kpc from the SN location, this would have enormous consequences for future cosmological analyses and the resources such analyses would require.