Assessment of Systematic Uncertainties in the Cosmological Analysis of the SDSS Supernovae Photometric Sample

To match a SN to its most likely host galaxy, the d_DLR method is employed. This uses angular separation (${\rm{\Delta }}\theta$) and accounts for the galaxy spatial profile and orientation. The derivation for d_DLR is shown in the appendix. Gupta (2013) also provides a detailed derivation.

For each SN, all galaxies within 30'' are selected and sorted by ascending d_DLR values. The galaxy with the smallest d_DLR is considered to the host galaxy.

SN light-curve fits are done with the SALT2 light-curve model (Guy et al. 2010, hereafter G10) using the improved model from the Joint Light-curve Analysis (Betoule et al. 2014). The light-curve fitting and selection requirements are implemented with the SuperNova Analysis (SNANA) software package (Kessler et al. 2009b).

Selection requirements (cuts) are applied to reduce CC contamination and to define a sample that has distance biases that can be modeled with a Monte Carlo simulation. SNe with properties within the SALT2 training range of color (−0.3 < c < 0.3) and stretch (−3 < x₁ < 3 ) are selected. To ensure well-measured light-curve fit parameters, we apply cuts on the uncertainties for stretch (${\sigma }_{{x}_{1}}\lt 1$) and time of maximum brightness (${\sigma }_{{t}_{0}}\lt 2$ observer frame days). We also require that the SALT2 fit probability (based on χ² per degree of freedom) be >0.001. Next, we define a rest-frame age, ${T}_{\mathrm{rest}}=(\mathrm{MJD}-{\mathrm{MJD}}_{\mathrm{peak}})/(1+z)$, where MJD is the observation date, ${\mathrm{MJD}}_{\mathrm{peak}}$ is the date of peak brightness, and z is the redshift. We require that at least one observation satisfies ${T}_{\mathrm{rest}}\gt 10$ days, and that at least one observation satisfies ${T}_{\mathrm{rest}}\lt 0$ days. Finally, we require that at least two bands have an epoch in which the S/N is >5. A summary of these cuts on the data are is in Table 1.

Table 1.  Sequential Selection Requirements on the SDSS Transient Sample

Cut	Number	Comments
Total Candidates	10,258	Data Release (Sako et al. 2018)
Spec-z and dDLR	4356	Host galaxy redshift
${T}_{\mathrm{rest}}\lt 0$	3852	Light-curve sampling
${T}_{\mathrm{rest}}\gt 10$	3518	Light-curve sampling
S/N	2717	Signal-to-Noise Ratio > 5
SALT2 Fit Parameters	1219	c, x1, ${\sigma }_{{x}_{1}}$, ${\sigma }_{{t}_{0}}$
NN Classifier	700	Nearest Neighbour classifier

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We use a Nearest Neighbour (NN) classifier developed by Sako et al. (2018) and Kessler & Scolnic (2017). We simulate a large training sample of SN (Ia + CC) with the same selection requirements and light-curve fitting as for the data. The redshift, color, and stretch (z, c, and x₁) are used to define a three-dimensional space for the NN analysis. In this space, each data point (real or simulated) is the center of a sphere at $\{z,c,{x}_{1}\}$. Classification is done by counting the number of Ia and CC SN within the sphere, and the data point is classified to be the type that is most frequent inside the sphere. The size of the sphere is set with a metric determined by maximizing the product of the efficiency and purity (Kessler & Scolnic 2017). Data identified as SNe Ia with a probability of less than 0.5 are rejected.

Cosmological analysis within the BBC framework is done in three stages. The first stage classifies SNe as Ia or CC using the NN method, and assigns a probability (P_Ia) for each event to be a SNe Ia. The second stage separates the data into redshift bins and determines a mean distance modulus in each bin, after accounting for selection biases and CC contamination (Kessler & Scolnic 2017). Here we use 10 equal sized bins ranging from z = 0.02 to 0.5. The third stage performs a cosmological fit to the binned distances using BAO (Eisenstein et al. 2005) and CMB (Komatsu et al. 2009) priors, similar to Lasker et al. (2019) who found these priors to be sufficient for systematics studies.

To model the potential measurement biases of cosmological parameters based on selection of host galaxies, we first create a realistic library of host galaxies (HOSTLIB) with properties that match those of our data. We evaluate the quality of our HOSTLIB by comparing the distributions of the smallest and second smallest d_DLR (Section 3.1); these distributions are sensitive to galaxy spatial profile, survey depth, galaxy photo-z, and ${\rm{\Delta }}\theta$.

We evaluated three different HOSTLIBs to use in our simulations. The first two, the Advanced Camera for Surveys General Catalog (ACS-GC), and the Marenostrum Institut de Ciències de l'Espai Simulations Catalogue, were used in Gupta et al. (2016). For these two HOSTLIBs, the simulated ${\rm{\Delta }}\theta$ distribution does not match the SDSS data.

Therefore, we created a third library by compiling observed galaxies within Stripe 82 from the SDSS DR14 data release. To maximize completeness and exclude spectroscopic selection effects, we selected host galaxies with a photometric redshift. The HOSTLIB includes Sérsic profile information calculated from the Stokes values. These profiles are used in simulations to place SNe near a galaxy and to model Poisson noise from the host galaxy. The SNANA simulation only calculates the smallest d_DLR value for each SN, so the second smallest values were calculated separately from the HOSTLIB.

For data and simulation, Figure 3 compares distributions of ${\rm{\Delta }}\theta$, DLR, and d_DLR. Each distribution is shown separately for the smallest and second smallest d_DLR value. Also shown is the ratio of the smallest to second smallest d_DLR values (${r}_{\mathrm{DLR}}$). We find good agreement in all distributions. Note that a HOSTLIB with too-large separations between galaxies can result in good data/sim agreement for the smallest d_DLR in Figures 3(a)–(c), but would result in poor agreement for the second smallest d_DLR, and also underpredict misassociations. The good agreement for the second smallest d_DLR and ${r}_{\mathrm{DLR}}$  is therefore an important metric for reliably predicting the misassociation rate. We also show in Figure 2(e) that there is good agreement in the r-band magnitude of the host-galaxy distribution between data and simulations.

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Past studies have shown that there is a correlation between the stretch-and-color-corrected luminosity of SNe Ia and the host-galaxy stellar mass (${M}_{\mathrm{stellar}}$). This effect has been found in the SN Legacy Survey (Sullivan et al. 2010), the SDSS sample (Lampeitl et al. 2010; Gupta et al. 2011b; Hayden et al. 2013; Wolf et al. 2016), and the PS1 sample (S18). In this SDSS analysis, we simulate these correlations to estimate biases arising from our spectroscopic galaxy selection. For every galaxy in our HOSTLIB, we calculate ${M}_{\mathrm{stellar}}$ using the methodology from Taylor et al. (2011),

$$\begin{eqnarray}&&{M}_{\mathrm{stellar}}=1.15+0.7\times (g-i-0.4\times (i-{\mu }_{\mathrm{calc}})),\end{eqnarray} \tag{ 1 }$$

where g and i are the host-galaxy magnitudes in the respective band, and μ_calc is the calculated distance modulus using the galaxy redshift and the same ΛCDM cosmology parameters as in the simulations. Equation (1) is used to calculate masses for both the data and the HOSTLIB. A comparison of the host-mass distribution between data and simulations is shown in Figure 2(f); while the agreement is not as good as that for the host-galaxy magnitude comparison and could use further study, it is sufficient to assess systematics.

We introduce a −0.025 mag correction to the luminosity of SNe in galaxies with stellar mass M_⊙ > 10¹⁰, and a +0.025 mag correction to those with M_⊙ < 10¹⁰. We do not include additional correlations of SNe Ia properties (c and x₁) with host-galaxy properties; these correlations are discussed in Smith et al. (2014).

Using simulations generated with input from our HOSTLIB, we evaluate the selection efficiency of our host galaxy. In our Fiduciary analysis, we define the efficiency, ${\epsilon }_{\mathrm{host}}(r)$, to be a function of host-galaxy r-band magnitude as follows:

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{host}}(r)\equiv {N}_{\mathrm{data}}(r)/{N}_{\mathrm{sim}}(r),\end{eqnarray} \tag{ 2 }$$

where N_data(r) is the number of SNe in each host-galaxy r-band magnitude bin for data, and N_sim(r) is the number of SNe in each host-galaxy r-band magnitude bin for a simulation with ${\epsilon }_{\mathrm{host}}(r)=1$. We scale ${\epsilon }_{\mathrm{host}}(r)$ so that the maximum is 1 (Figure 4(a)).

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For a systematic test, we follow Jones et al. (2018b) and parameterize the selection function to depend on host spec-z (Figure 4(b)):

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{host}}(z)\equiv {N}_{\mathrm{data}}(z)/{N}_{\mathrm{sim}}(z),\end{eqnarray} \tag{ 3 }$$

where z is the host-galaxy spec-z, and N_data(z) and N_sim(z) are defined as in Equation (2), but using z bins.

An important systematic in cosmological analyses of photometric samples is the collection of CC models used to simulate training samples for classifiers. The most recent study on this systematic was done by Jones et al. (2018b), which used a compendium of publicly available CC templates and adjusted the luminosity functions of the library to match the Hubble residual tail region after selection cuts. Light-curve templates of SNII were adjusted by 1.1 mag to be more luminous and SNIb/c were similarly adjusted by 1.2 mag.

Since Jones et al. (2018b), the PLAsTiCC library (Kessler et al. 2019) was released, which has enhanced previous CC template libraries. Compared with Jones et al. (2018b), here we include MOSFiT (Modular Open-Source Fitter for Transients) for SNIbc (Kessler et al. 2010a; Villar et al. 2017; Guillochon et al. 2018; Pierel et al. 2018), Nonnegative Matrix Factorization (NMF) for SNII (Kessler et al. 2010a; Villar et al. 2017; Guillochon et al. 2018; Pierel et al. 2018), SNIax (Jha 2017), and SNe Ia-91bg. We included an additional 0.9 mag smear for SNII-NMF as discussed in Kessler et al. (2019). Our Fiduciary analysis includes SNII-NMF, SNIbc-MOSFiT, and SNe Ia-91bg. As we will discuss in the next section, SNIax were excluded from both the simulated training and data samples.

The PLAsTiCC models include SNIax, which typically have lower luminosity, lower ejecta velocity, and greater variation in photometric parameters than their SNe Ia counterparts. The SED model used in PLAsTiCC is based on the real SNIax, 2005hk. The SED model was augmented with other spectra and the luminosity function was inferred from the sample studied in Jha (2017). Light curves are generated to match the absolute magnitude (M_V), rise time (t_rise), and decline rate in the B and R bands (Δ m₁₅(B) and Δm₁₅(R)) detailed in Stritzinger et al. (2015) and Magee et al. (2016c).

We apply the analysis (Section 3) to our simulated data sample, and fit for nuisance parameters α, β, γ, and cosmological parameter w. The recovered values for these parameters in our Fiduciary analysis are consistent with their input values of 0.14, 3.2, 0.05, and −1, respectively. More precise validation tests with simulations are described in Section 6 of Brout et al. (2018).

Figure 3 shows the properties used to validate the simulations from which the misassociation rate was determined. Comparing each true host galaxy in our simulation to the d_DLR-selected galaxy, we determine the host-galaxy misassociation to be 0.6%.

Figure 6 shows the effect of misassociated hosts on redshifts (Figure 6(a)), as well as the distributions of Δθ (Figure 6(b)) and d_DLR (Figure 6(c)) for those misassociated hosts; Figure 6(d) is a histogram of ${r}_{\mathrm{DLR}}$ for misassociated SN. In each panel, the distribution for misassociated hosts is much broader compared to correctly identified hosts.

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We find that the recovery of w is biased by Δw = 0.0007 by this misassociation, which is consistent with 0.

We evaluate the bias on w due to our host-galaxy selection by comparing our recovered distances using ${\epsilon }_{\mathrm{host}}(r)$ to those using ${\epsilon }_{\mathrm{host}}(z)$. The impact on the binned distances is shown in Figure 5. For most of the redshift range, the impact is less than 2 milli-mags and the only significant impact is at the higher end of our redshift range. This produces a w-bias of Δw = −0.0072 ± 0.0037, significantly smaller than the statistical uncertainty. The rms around this value is 0.05σ.

Figure 7 shows the Hubble residual distributions using five different CC models in the analysis, and each model is indicated in the panel. Figure 7 contains our Fiduciary case, where CC templates used in the simulations are from the PLAsTiCC models with an adjusted luminosity function and without SNIax; the PLAsTiCC model with neither adjusted luminosity function nor SNIax; the PLAsTiCC model with adjusted luminosity function and SNIax; the CC templates from Kessler et al. (2010b) without luminosity adjustments (K10); and the CC templates from Jones et al. (2018a) with adjustments. The smallest contamination (P_CC) is 1% for K10, and the highest is 5.9% for the Fiduciary with Iax analysis. Still, we see that the contamination in the data for the positive tail of the Hubble residual distribution is better predicted in some cases than others. We find that the PLAsTiCC and K10 models do not match the data well in this region, confirming the need for luminosity function corrections to match the high Hubble residual tail suggested in Jones et al. (2018a).

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For the case of Fiduciary with SNIax, the SNIax make up 40% of the CC contamination. While the SNIax luminosity distribution is fainter than that of SNe Ia, we also find that the x₁ and c values satisfy the selection requirements (see Section 2), and therefore the NN classification poorly separates SNe Ia from SNIax. Jones et al. (2018b) do not include SNIax in the contamination library due to the expectation that they are too red (c > 0.3), and because the SNIax model was not available. As discussed in Section 4.3, only a single SN was used to generate the SNIax model. However, if we perform SALT2 light-curve fits on the four known SNIax in SDSS (including 2005hk), we find that all the SNIax have fitted color values c > 0.3 and thus fail our selection requirements. Therefore, these light-curve fits suggest that the SNIax contamination is overestimated and further study is needed.

From simulations, the impact on w due to the systematics related to CC libraries is given in Table 2. We find the mean bias due to the CC systematics is ${\rm{\Delta }}w\lt | 0.01|$, with a statistical uncertainty of ∼0.001. The rms in w from the simulations due to the systematics is 0.01–0.02.

In the last two columns of Table 2, we show the impact on w of the systematics studied in this analysis for the real data sample. This is shown for all the systematics except for the misassociated host, as there we cannot apply the same technique on the data as we did for the simulations. To assess whether the changes seen for the data sample are consistent with predictions from the simulations, we define the number of standard deviations (N_σ) for each systematic as

$$\begin{eqnarray}&&{N}_{\sigma }=({\rm{\Delta }}{w}_{\mathrm{sim}}-{\rm{\Delta }}{w}_{\mathrm{data}})/{w}_{\mathrm{rms}},\end{eqnarray} \tag{ 6 }$$

where ${\rm{\Delta }}{w}_{\mathrm{sim}}$ is the Δw recovered in simulations, Δw_data is the Δw recovered from the data, and w_rms is the rms of the Δw recovered from simulations. We find that the highest deviation compared to the simulations is seen for the "No LF adjustment" systematic at 2.1σ. All other deviations near or below <1σ. Therefore, we conclude that the impacts of the systematics seen in the simulations are consistent with those seen for the data.

We start with the radial equation of an ellipse as measured from the center, with semimajor and semiminor axes a and b and orientation angle θ:

$$\begin{eqnarray}&&r(\theta )=\displaystyle \frac{{ab}}{\sqrt{{\left(a\sin \theta \right)}^{2}+{\left(b\cos \theta \right)}^{2}}}.\end{eqnarray} \tag{ 7 }$$

We define the SN angle as the angular difference between a line that goes through the SN position and galaxy center and a line that passes through north and the galaxy center. Combined with the orientation of the galaxy as given by the galaxy position angle we define θ by subtracting the SN angle from the position angle of the galaxy. With a, b, and θ, the DLR is defined from Equation (7) as the effective radius of the galaxy at angle θ.

The position angle of the galaxy can be found using the Stokes parameters Q and U given in the SDSS DR14 data release. DR14 uses a slightly unconventional notation for U; this is corrected with a factor of 2 in the position angle.

Since these parameters are not fits to a model, but rather based on pixel data, they are more robust for fainter galaxies. The position angle ϕ can be expressed as

$$\begin{eqnarray}&&\phi =\displaystyle \frac{1}{2}\arctan \left(\displaystyle \frac{2U}{Q}\right).\end{eqnarray} \tag{ 8 }$$

The ratio of the semimajor and minor axes can also be computed with the Stokes parameters. Defining $\kappa \equiv {Q}^{2}+{U}^{2}$, the ratio a/b is then expressed as

$$\begin{eqnarray}&&\displaystyle \frac{a}{b}=\displaystyle \frac{1+\kappa +2\sqrt{\kappa }}{1-\kappa }.\end{eqnarray} \tag{ 9 }$$

Following Sako et al. (2018), we set a equal to the Petrosian half-light radius within the r-band and b is determined using Equation (9). Finally, we define a distance-weighted d_DLR as

$$\begin{eqnarray}&&{d}_{\mathrm{DLR}}=\displaystyle \frac{{\rm{Angular}}\,{\rm{separation}}}{r(\theta )}.\end{eqnarray} \tag{ 10 }$$

It is important to note that DLR and d_DLR are survey-dependent quantities and are not easily comparable across surveys. Of particular note are magnitude cutoffs. Establishing a magnitude limit does allow for fine-tuned control of ${\rm{\Delta }}\theta$ density, but does not account for apparent ellipticity. At higher magnitudes, the apparent ellipticity as measured by the Stokes parameters begins to increase. At fainter brightness, noise begins to dominate the signal and leads to unrealistic ellipticity measurements. But differences in magnitude limits for different surveys, combined with those in image processing, can alter the apparent size of a galaxy. Self-consistent DLR measurements within the survey are more accurate than solely ${\rm{\Delta }}\theta$ determinations, but cross comparison would not be effective.