Using Host Galaxy Photometric Redshifts to Improve Cosmological Constraints with Type Ia Supernovae in the LSST Era

The original PLAsTiCC simulation covers the first 3 yr of LSST with 18 models that include both extragalactic and galactic transients. For this analysis, we simulate only SNe Ia using the SALT2 model (Guy et al. 2007). This model includes measured populations of stretch and color from Scolnic & Kessler (2016) with stretch– and color–luminosity parameters (αあるふぁ = 0.14, βべーた = 3.1), an intrinsic scatter of the model spectral energy distribution (SED), and a near-infrared extension (Pierel et al. 2018) to include the i-, z-, and y-band wavelength range. Correlations between SNe and host galaxy mass are not included. Next, the model SED is modified to account for cosmic expansion (Ωおめが_M = 0.315, w = −1, flatness), redshift, and Galactic extinction from Schlafly & Finkbeiner (2011). Filter passbands are used to compute broadband fluxes at epochs determined by the DDF cadence from OpSim (Delgado et al. 2014; Reuter et al. 2016), and observing conditions (zero-point, PSF, and sky noise) are used to model flux uncertainties. The 5σしぐま limiting magnitudes for each of the ugrizy passbands are listed in Table 2 for both the low- and high-z samples. We adopt the detection efficiency versus signal-to-noise ratio (S/N) from the DC2 analysis as shown in Figure 9 of Sánchez et al. (2022). The simulated trigger selects events with two detections separated by at least 30 minutes.

Following PLAsTiCC, we define a "z_spec" sample consisting of two subsets of events with accurate spectroscopic redshifts (σしぐま_z ∼ 10⁻⁵). The first subset assumes an accurate redshift from spectroscopically confirmed events based on a forecast of the performance for the 4 m Multi-Object Spectroscopic Telescope spectrograph (4MOST; de Jong et al. 2019) ¹⁴ that is under construction by the European Southern Observatory. ¹⁵ The 4MOST is expected to begin operation in 2023 (similar to the LSST timeline) and is located at a latitude similar to that of the Rubin Observatory in Chile. The second subset includes photometrically identified events with an accurate host galaxy redshift using 4MOST. The second subset has ∼60% more events than the first subset, and each subset is treated identically in the analysis. The simulated efficiency versus redshift for each z_spec subset is shown in Figure 2.

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To estimate the host galaxy photometric redshifts, PLAsTiCC used a color-matched nearest-neighbor (CMNN) photometric redshift estimator (G18). The CMNN uses a five-dimensional color space grid to train a set of galaxies and defines a distance metric that is used on the test set to assign the redshift and associated uncertainty. Figure 3(a) shows the photo-z residuals, z_phot − z_true, as a function of z_true.

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To characterize the residuals, we follow Graham et al. (2018) and define metrics for an inner core resolution and outlier fraction using the quantity Δでるたz_(1+z) = ∣z_phot − z_true∣/(1 + z_phot). The resolution is the width of the interquartile distribution of Δでるたz_(1+z) divided by 1.349 and denoted by σしぐま_IQR. The outlier fraction (f_out) is the fraction of events satisfying

$$\begin{eqnarray}&&{\rm{\Delta }}{z}_{(1+z)}\gt 3{\sigma }_{\mathrm{IQR}}\ \mathrm{and}\ {\rm{\Delta }}{z}_{(1+z)}\gt 0.06.\end{eqnarray} \tag{ 1 }$$

For z_true < 0.4, nearly all of the events have a spectroscopic redshift, and for z_true > 1.4, the SNe are too faint for detection. For the relevant redshift range (0.4 < z_true < 1.4), σしぐま_IQR = 0.025 and f_out = 0.13.

Following PLAsTiCC, we use the volumetric rate model R(z) based on Dilday et al. (2008) for z < 1 and Hounsell et al. (2018) for z > 1. The rate R(z) we adopt is given by

$$\begin{eqnarray}&&R(z)=2.5\times {10}^{-5}{\left(1+z\right)}^{1.5}\,{\mathrm{yr}}^{-1}{\mathrm{Mpc}}^{-3}\,\,(z\lt 1),\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&R(z)=9.7\times {10}^{-5}{\left(1+z\right)}^{-0.5}\,{\mathrm{yr}}^{-1}{\mathrm{Mpc}}^{-3}\,\,(z\gt 1)\,.\end{eqnarray} \tag{ 3 }$$

We simulate a spectroscopically confirmed low-z sample based on the WFD cadence from DC2. We assume accurate spectroscopic redshifts and 100% efficiency up to redshift z < 0.08. The simulation code and SN Ia model are the same as for the high-z sample. Compared to DDFs, the WFD cadence has 30% fewer observations, on average, and a 1 mag shallower depth (Table 2).

To standardize the SN Ia brightness, we fit each light curve to the wavelength-extended SALT2 light-curve model from Pierel et al. (2018), the same model used in the simulations. The SALT2 light-curve fit determines the time of peak brightness (t₀), amplitude (x₀), stretch (x₁), and color (c). Previous cosmology analyses have all used SNe with accurate z_spec; thus, redshift had always been a fixed parameter in the SALT2 fit. In our analysis, the SALT2 fit uses the methodology in Kessler et al. (2010) in which the redshift is floated as a fifth parameter, which we call "z_phot."

Following Equation (1) in Kessler et al. (2010), the five-parameter SALT2 fit uses minuit (James & Roos 1975) to minimize the following χかい²:

$$\begin{eqnarray}\begin{array}{rcl}{\chi }^{2} & = & \displaystyle \sum _{i}\left\{\displaystyle \frac{{\left[{F}_{i}^{\mathrm{data}}-{F}_{i}^{\mathrm{model}}({\vec{x}}_{5})\right]}^{2}}{{\sigma }_{F,i}^{2}}+2\mathrm{ln}({\sigma }_{F,i}/{\tilde{\sigma }}_{F,i})\right\}\\ & & +\left\{\displaystyle \frac{{\left({z}_{\mathrm{phot}}-{z}_{\mathrm{host}}\right)}^{2}}{{\sigma }_{z,\mathrm{host}}^{2}}\right\},\end{array}\end{eqnarray} \tag{ 4 }$$

where ${F}_{i}^{\mathrm{data}}$ is the SN flux for the ith observation, ${F}_{i}^{\mathrm{model}}({\vec{x}}_{5})$ is the SALT2 model flux computed from the five SN-dependent parameters ${\vec{x}}_{5}=\{{t}_{0},{x}_{0},{x}_{1},c,{z}_{\mathrm{phot}}\}$, and σしぐま_F,i is the quadrature sum of statistical and SALT2 model uncertainties. The second term in Equation (4) accounts for the fact that the SALT2 model uncertainties depend on one of the fitted parameters (z_phot) because of the dependence on rest-frame wavelength and epoch. A reference uncertainty (${\tilde{\sigma }}_{F,i}$) is computed after the first fit iteration so that the $2\mathrm{ln}({\sigma }_{F,i}/{\tilde{\sigma }}_{F,i})$ term is close to zero in the second and third fit iterations. The second row in Equation (4) is the host galaxy photo-z prior, where z_host is the mean of the photo-z probability density function (PDF), and σしぐま_z,host is the rms.

To estimate the initial SALT2 parameters prior to the fit, z_phot = z_host. The remaining initial parameter estimates are obtained by minimizing the χかい² in a very coarse grid search. After each minuit fit iteration, the wavelength range for each LSST passband is transformed to the rest frame using a fitted z_phot. If the central rest-frame wavelength is outside of the SALT2 model range (2600–11000 Å), the passband is dropped in the next fit iteration; a previously dropped passband can be added if it is within the model range. If any passband is dropped or added, the fit iteration is repeated to ensure that a consistent set of passbands are included in the fit.

For the subset with accurate z_spec, the redshift prior is so precise (0.0001) that such fits are essentially equivalent to fixing the redshift in a four-parameter fit. Note that z_phot refers to the fitted redshift for all events, including the z_spec subset.

We apply the following selection requirements (cuts) based on analyses using real data:

• 1.  
  at least three bands with maximum S/N >4,
• 2.  
  successful light-curve fit,
• 3.  
  ∣x₁∣ < 3.0,
• 4.  
  ∣c∣ < 0.3,
• 5.  
  stretch uncertainty σしぐま_x1 < 1.0,
• 6.  
  time of peak brightness uncertainty σしぐま_t0 < 2.0 days,
• 7.  
  P_fit > 0.05, ¹⁶
• 8.  
  0.01 < z_phot < 1.4, and
• 9.  
  valid bias correction (see Section 4.4).

The SALT2 light-curve fits for several events are shown by the smooth curves in Figure 1. After the selection requirements, the redshift distribution is shown in Figure 5(a) for the subset with and without z_spec.

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The z_phot residual versus z_true is shown in Figure 3(a) for all galaxies in the catalog and Figure 3(b) for host galaxies after the SN Ia trigger and selection cuts. After the selection cuts, SNe associated with the host galaxy photo-z outliers tend to be excluded by the SALT2 fit and P_fit cut; the core resolution is reduced by 10%, and the outlier fraction is reduced by 20%.

To compare the photo-z precision between the host and SN, we performed SALT2 light-curve fits without a host galaxy photo-z prior to determine the SN-only z_phot residuals (Figure 3(c)); the SN-only z_phot core resolution is slightly (∼1.1) better than for the galaxies in Figure 3(a), although the outlier fractions are the same. For the combined SN+host SALT2 fits, Figure 3(d) shows z_phot residuals versus z_true; compared to fitting SNe only, the SN+host z_phot resolution is 30% smaller and has ∼15% fewer outliers.

To evaluate systematic uncertainties, the SALT2 light-curve fits and BBC fit are repeated 17 times, ¹⁷ each with a separate variation shown in Table 3. Each variation results in a distance modulus variation, and we compute a systematic covariance matrix (COV_syst) using Equation (6) in Conley et al. (2011).

We include variations in Galactic extinction, calibration, z_spec, and host galaxy z_phot. We do not include SALT2 modeling and training uncertainties, nor do we include uncertainties on the stretch and color populations.

The galactic extinction uncertainty (row 2 in Table 3) is σしぐま_MWEBV = 0.05 · MWEBV and taken from the Pantheon analysis (Scolnic et al. 2018). The Hubble Space Telescope (HST) calibration uncertainty (row 3) is from the DES SN Ia cosmology analysis (Table 4 in Brout et al. 2019) and based on Bohlin et al. (2014). The zero-point uncertainty (row 4) is from the LSST science road map (Section 3.3 in Ivezić & The LSST Science Collaboration 2018) and consistent with the Pan-STARRS 3πぱい internal calibration accuracy (Schlafly et al. 2012; Magnier et al. 2013). The wavelength calibration uncertainty (row 5) is from the Pantheon analysis in Scolnic et al. (2018).

The spectroscopic redshift uncertainty (row 6) is from Table 4 in Brout et al. (2019), which is based on low-redshift constraints on local density fluctuations (Calcino & Davis 2017). For the host galaxy photo-z bias uncertainty (row 7), the statistical bias in our PLAsTiCC simulation is well below 0.01, as shown in the lower panel of Figure 2 in G18. This statistical bias is valid for the galaxy training set, but the bias for the subset of SN Ia host galaxies is likely to be larger. Without a photo-z bias estimate for SN Ia host galaxies, we make an ad hoc estimate from the DES weak-lensing cosmology analysis in which Myles et al. (2021) found a statistical z_phot bias of ∼0.001, while their weighted z_phot bias is 0.01, an order of magnitude larger. We use their weighted z_phot bias of 0.01 as the systematic uncertainty. The uncertainty in the host z_phot uncertainty (row 8) is from the variation in robust standard deviations in the upper panel of Figure 2 in G18.

To implement distance bias corrections in BBC (Section 4.4), we generate a large sample of 3.1 × 10⁶ events (after cuts; Section 3.1), which consists of 2.6 × 10⁶ high-z events and 4.4 × 10⁵ low-z events. The bias correction is applied independently for high- and low-z, and thus the relative number of events in each subsample need not match the data. The simulation procedure is identical to that used for the simulated data, except for αあるふぁ and βべーた. While fixed values are used for the data sample, a 2 × 2 αあるふぁ, βべーた grid is used for the "biasCor" simulation to enable interpolation in BBC.

The BBC reads the SALT2-fitted parameters (high- and low-z) from the data and biasCor simulation and produces a bias-corrected Hubble diagram, both unbinned and in redshift bins. For each event, the measured distance modulus is based on Tripp (1998),

$$\begin{eqnarray}&&\mu ={m}_{B}+\alpha {x}_{1}-\beta c+{M}_{0}+{\rm{\Delta }}{\mu }_{\mathrm{bias}}\,,\end{eqnarray} \tag{ 5 }$$

where ${m}_{B}\equiv -2.5{\mathrm{log}}_{10}({x}_{0})$, {αあるふぁ, βべーた, M₀} are global nuisance parameters, and Δでるたμみゅー_bias = μみゅー − μみゅー_true is determined from the biasCor simulation in a five-dimensional space of {z, x₁, c, αあるふぁ, βべーた}. A valid bias correction is required for each event, resulting in a few percent loss. The distance uncertainty (σしぐま_μみゅー ) is computed from Equation (3) of KS17. Since there is no contamination from non–SNe Ia, all SN Ia classification probabilities are set to 1, and we do not use the BEAMS formalism.

There are two subtle issues concerning the use of z_phot and its uncertainty σしぐま_z . First, the calculated distance error from σしぐま_z (${\sigma }_{\mu }^{z}$ in Equation (3) of KS17) is an overestimate because it does not account for the correlated color error that reduces the distance error. By floating z_phot in the SALT2 fit, redshift correlations propagate to the other SALT2 parameter uncertainties; therefore, we set ${\sigma }_{\mu }^{z}=0$. The second issue concerns the μみゅー_bias computation, where μみゅー_true is computed at a SALT2-fitted z_phot rather than the true redshift.

To avoid a dependence on cosmological parameters, the BBC fit is performed in 14 logarithmically spaced redshift bins. The fitted parameters include the global nuisance parameters (αあるふぁ, βべーた, M₀) and bias-corrected distances in 14 redshift bins. The unbinned Hubble diagram is obtained from Equation (5) using the fitted parameters.

If the same selection requirements are applied to each systematic variation for computing COV_syst, small fluctuations in the fitted SALT2 parameters and redshift result in slightly different samples, and these differences introduce statistical noise in COV_syst. We avoid this covariance noise by defining a baseline sample for events passing cuts without systematic variations, and we use this same baseline sample for all systematic variations. For example, if an event has a fitted SALT2 color parameter c = 0.299 and migrates to c = 0.3001 for a calibration systematic, this event is preserved without applying cuts that require ∣c∣ < 0.3.

To avoid sample differences from the valid bias-correction requirement, the BBC fit is run twice, and the second fit only includes events that have a valid bias correction in all systematic variations. Finally, for redshift systematics that result in migration to another redshift bin, the original (no syst) redshift bin is preserved for the BBC fit.

For cosmology fitting, we use a fast minimization program that approximates a cosmic microwave background (CMB) prior using the R-shift parameter (e.g., see Equation (69) in Komatsu et al. 2009) computed from the same cosmological parameters that were used to generate the SNe Ia. The R uncertainty is σしぐま_R = 0.006, tuned to have the same constraining power as Planck Collaboration et al. (2021). We fit with wCDM and w₀ w_a CDM models, where w = [w₀ + w_a (1 − a)]. The statistical+systematics covariance matrix is used. We fit both binned and unbinned Hubble diagrams.

For the w₀ w_a CDM model, the figure of merit (FoM) is computed based on the Dark Energy Task Force definition in Albrecht et al. (2006),

$$\begin{eqnarray}&&\mathrm{FoM}\simeq \displaystyle \frac{1}{\sigma ({w}_{a})\sigma ({w}_{0})\sqrt{1-{\rho }^{2}}}\,,\end{eqnarray} \tag{ 6 }$$

where ρろー is the reduced covariance between w₀ and w_a .

For the wCDM cosmology fits, Table 5 shows the average w bias and uncertainty among the 25 samples. The average w bias is consistent with zero for both the z_spec and z_phot samples and with and without systematic uncertainties. The w-bias precision is ∼0.002. The average w uncertainty (〈σしぐま_w 〉) for the z_phot sample is 0.023 with systematics and is only slightly improved compared to 〈σしぐま_w 〉 = 0.025 for the z_spec sample. The additional sensitivity from the host galaxy z_phot sample is small because the increased statistics are at higher redshifts, where the dark energy density fraction is much smaller compared to lower redshifts, where the sample is dominated by spectroscopic redshifts.

Table 5. Summary of wCDM Cosmology Fits

Redshift
Source	Systematics 〈w Bias〉 a	〈σしぐまw 〉 b
zspec	Stat. only	−0.0008 ± 0.0020	0.020
	Stat.+syst.	−0.0027 ± 0.0025	0.025
zphot	Stat. only	−0.0003 ± 0.0017	0.020
	Stat.+syst.	−0.0009 ± 0.0018	0.023

Notes.

^a Average bias among 25 samples with uncertainty of std/$\sqrt{25}$. ^b Average fitted uncertainty among 25 samples.

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For the w₀ w_a CDM model, the average bias, uncertainty, and FoM are shown in Table 6. While there was little improvement using the z_phot sample with the wCDM model, the w₀ w_a CDM improvement is much more significant because higher-redshift events, which are enhanced by the z_phot sample, are more sensitive to evolving dark energy (w_a ). With systematics, 〈FoM〉 = 95 for the z_spec sample and 145 for the z_phot sample. The w₀–w_a constraining power is shown in Figure 7 for a single simulated data sample.

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Table 6. Summary of w₀ w_a CDM Cosmology Fits

zsource	Syst.	〈w0 Bias〉 a	〈wa Bias〉	$\langle {\sigma }_{{w}_{0}}\rangle$ b	$\langle {\sigma }_{{w}_{a}}\rangle$	〈FoM〉
zspec	Stat. only	0.0083 ± 0.0143	−0.0658 ± 0.0674	0.076	0.353	136
	Stat.+syst.	0.0067 ± 0.0140	−0.0683 ± 0.0662	0.092	0.418	95
zphot	Stat. only	0.0029 ± 0.0082	−0.0228 ± 0.0342	0.048	0.211	237
	Stat.+syst.	0.0011 ± 0.0091	−0.0202 ± 0.0363	0.071	0.294	145

Notes.

^a Average bias among 25 samples with uncertainty of std/$\sqrt{25}$. ^b Average fitted uncertainty among 25 samples.

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The average bias is consistent with zero for both w₀ and w_a . For the z_spec sample, the bias precision is ∼0.015 and ∼0.07 for w₀ and w_a , respectively. For the z_phot sample, the bias precision is improved to ∼0.009 and ∼0.04. The w₀–w_a average bias is shown in Figure 8 and compared to the the w₀–w_a contours (statistical+systematic) for a single sample.

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For the z_phot sample, the FoM averaged over 25 samples is 〈FoM〉 = 237 with only statistical uncertainties and drops to 〈FoM〉 = 145 when systematic uncertainties are included. Since there are many systematics contributing to the decrease in 〈FoM〉, we quantify the impact of each systematic i by recomputing the covariance matrix separately for each systematic (COV_syst,i ) and repeating the cosmology fit for each COV_syst,i . We finally compute the FoM ratios,

$$\begin{eqnarray}&&{{ \mathcal R }}_{\mathrm{FoM},i}={\mathrm{FoM}}_{\mathrm{syst},i}/{\mathrm{FoM}}_{\mathrm{stat}},\end{eqnarray} \tag{ 7 }$$

where FoM_syst,i is the FoM including only systematic i, and FoM_stat is the FoM without systematic uncertainties. Note that ${{ \mathcal R }}_{\mathrm{FoM},i}\leqslant 1$. Table 7 shows the ${{ \mathcal R }}_{\mathrm{FoM},i}$, and the FoM degradation is dominated by the calibration systematics.

Table 7. FoM Ratio ${{ \mathcal R }}_{\mathrm{FoM},i}$ for Each Systematic (w₀ w_a CDM Model)

	${{ \mathcal R }}_{\mathrm{FoM},i}$
Systematic(s)	zphot zspec
None (stat. only)	1.00 1.00
MWEBV	0.99 1.00
ZERRSCALE	0.99 1.00
zSHIFT	0.98 0.99
Photo-z shift	0.98 0.99
CAL_WAVE	0.90 0.94
CAL_Zp	0.71 0.75
CAL	0.64 0.71
Stat. + all syst.	0.61 0.70

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The 0.01 photo-z shift systematic has a small (2%) effect on FoM for three reasons. First, the combined SN+host light-curve fit results in an average fitted redshift error of ∼0.004, or about half the host photo-z error. Second, this photo-z systematic does not affect z_spec events that dominate the lower-redshift region below about 0.5 (Figure 5(a)), and this z_spec region is most sensitive to redshift errors. The final reason is that the fitted z_phot and SALT2 color are anticorrelated; thus, a larger (smaller) z_phot results in bluer (redder) color, and this change in color self-corrects the distance error as illustrated in Figure 9.

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To describe this distance self-correction, we first define Δでるたz_syst−z as the difference between a SALT2-fitted z_phot with 0.01 host galaxy photo-z shift and nominal photo-z and similarly define Δでるたμみゅー_syst−z as the distance difference from Equation (5). Figure 9(a) shows Δでるたμみゅー_syst−z versus Δでるたz_syst−z and a linear fit for the slope, d μみゅー/dz, in one of five z_true bins. Figure 9(b) shows the measured d μみゅー/dz slope in five z_true bins (black circles) along with the ΛらむだCDM theory curve in red. In the ideal limit, where the measured d μみゅー/dz exactly equals theory d μみゅー/dz, the distance self-correction is perfect and results in no systematic uncertainty. Here the measured d μみゅー/dz are close to the theory curve; thus, the distance error is mostly corrected.

To gain further insight into the photo-z sensitivity, we first consider a naive systematic of shifting the fitted z_phot by 0.01 after the light-curve fit for the subset without a z_spec. In this test, the compensating d μみゅー/dz points in Figure 9 are forced to be zero, and there is no systematic reduction from a combined SN+host fit. Fitting the w₀ w_a CDM model without COV_syst, the w₀ and w_a biases are 0.03 and 0.15, respectively. Next, we consider the realistic case of shifting the host photo-z before the SALT2 light-curve fit; the corresponding w₀ and w_a biases are 0.001 and 0.003, more than an order of magnitude smaller than the naive systematic. While we have included an explicit host galaxy photo-z systematic, there is no explicit analog for the SN. The SN photo-z systematic is accounted for by the calibration and Galactic extinction contributions to the systematic uncertainty budget in Table 3, but it is difficult to untangle the impact of these systematics on distance and photo-z.