SUPERNOVA SIMULATIONS AND STRATEGIES FOR THE DARK ENERGY SURVEY

We employ the SNANA package (Kessler et al. 2009b) to simulate and fit Type Ia and Type Ibc/II SN light curves. We emphasize that while we are using SNANA to investigate the capabilities of the DES, it was originally developed and utilized for the analysis of observational SDSS SN data (Kessler et al. 2009a), was used by the Large Synoptic Survey Telescope Collaboration to forecast SN observations (Abell et al. 2009), and can be applied to any survey in general. Using the simulation requires a survey-specific library that includes the survey characteristics, e.g., filters, observing cadence, seeing conditions, zero points, and CCD characteristics.

For a rest-frame SN light curve model, such as MLCS2k2, the basic simulation steps are as follows:

• 1.  
  pick a sky position, redshift from observed SN rate distributions, and sequence of observer and rest-frame observation times;
• 2.  
  pick SN luminosity (Δ) and V-band host-extinction (A_V, the amount of dust extinction in magnitudes from Cardelli et al. 1989) parameters randomly drawn from their distributions;
• 3.  
  generate a rest-frame light curve from the SN light curve model: e.g., magnitudes in the U, B, V, R, and I filters (Bessell 1990) versus time;
• 4.  
  add host-galaxy extinction to each rest-frame magnitude using A_V (from Step 2 above) and the CCM dust model from Cardelli et al. (1989);
• 5.  
  add K-corrections (Nugent et al. 2002) to transform UBVRI to observer-frame filters¹⁵;
• 6.  
  add Galactic (Milky Way) extinction using data from Schlegel et al. (1998);
• 7.  
  use survey zero points to translate above-atmosphere magnitudes into observed flux in CCD counts;
• 8.  
  compute noise from the sky level, point-spread function (PSF), CCD readout noise (negligible for the DES), and signal Poisson statistics;
• 9.  
  in addition to Steps 4 and 5 above, apply an ad hoc Gaussian smearing model of intrinsic SN color variations to obtain Hubble residuals that match observations.

We make use of three light curve models that are integrated into SNANA to simulate and fit SN light curves: MLCS2k2, SALT2, and SNooPy. Note that SNANA uses MINUIT (James 1994) for minimization. The MLCS2k2 model is improved relative to the Jha et al. (2007) code (see Section 5.1 and Appendix B of Kessler et al. 2009a), e.g., it fits in flux instead of magnitudes and includes simulated efficiency in the prior. A key difference between MLCS2k2 and SALT2 is that the former fits for a distance for each SN while the latter does not. The SALT2 light curve model in SNANA is accompanied by a separate program called SALT2mu (Marriner et al. 2011) that is used to determine a distance for each SN so that the MLCS2k2 and SALT2 fit results can be treated in the same way (see Section 5.2 for additional information).

Construction of a survey-specific library as input to the SN simulation is crucial to obtaining realistic simulated light curves. For each DES SN observing field, this library includes information about the survey cadence, filters, CCD gain and noise, PSF, sky background level, and zero points and their fluctuations. The zero point encodes exposure time, atmospheric transmission, and telescope efficiency and aperture. These quantities vary with each exposure and so, for the DES study, we created a program that uses, among other things, the CTIO weather histories, ESSENCE zero point and PSF data, time gaps due to Blanco community use, and Moon brightness to estimate the parameters for the DES simulation library. Table 1 shows example entries in this library, and we now discuss the details of their creation.

The SN component of the DES is limited to about 10% of the total survey photometric time. In all cases, after a certain period of time (expected to be ∼8 days), if an SN field has not been observed it becomes the top observational priority of the survey even under photometric conditions. There are two main options being considered for the decision procedure to observe in shorter intervals than 8 days: (1) make maximal use of non-photometric time based on an infrared cloud camera (RASICAM; Lewis et al. 2010) or (2) decide based on the measured seeing, giving the non-SN DES components the best seeing for weak lensing and other science, and switch to the SN fields if the PSF is ≳ 1''. The final DES decision tree will probably be a combination of these two. In this analysis we have simulated option 1.

The separation of non-photometric and photometric time in the generation of the simulation library is accomplished by incorporating weather history maintained at CTIO for more than 20 years. This SN survey strategy leads to a two-component cadence: a peak in the number of observations at very short cadence due to several-day periods of non-photometric time when the SN fields dominate the observing time, and a broad secondary maximum around 8 days when photometric time is used (see Figure 4). Our SN observing also requires an air mass less than 2.0. This, combined with DES half-nights in January and February and long periods of photometric conditions, can lead to cadences longer than 8 days.

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The other critical components of the simulation input library are the PSF, sky background, and zero points. Usually, in this type of study, one takes averages of these quantities. In our case, we have used ESSENCE data to provide variations of the PSF and zero points at CTIO for each observation, as well as SDSS data for the dependence of sky background on relative moon position. The measured PSF variation of ESSENCE data is input directly into the simulation library after correcting for the wavelengths of the filter centroids and air masses for the mock DES observation. The resulting PSF distribution is shown in Figure 5. Recent improvements to the telescope and its environment, along with the optical design and mechanical (hexapod) control of DECam, are expected to deliver improved image quality compared to these data. The choice of PSF distribution is conservative for option 1 and consistent with option 2.

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Another key input to the simulation is the rate of SNIa explosions in the universe as a function of redshift (see Section 6.1 for a discussion of the input rate of core-collapse SNe). The total number of SNe that the DES will observe is clearly directly sensitive to that rate. The default SNIa rate we employ in SNANA is the power law from Dilday et al. (2008):

$$\begin{equation} {R_{{\rm SNIa}}\equiv {\rm SNIa\;rate}} = \alpha _{\rm Ia}\times (1+z)^{\beta _{\rm Ia}}, \end{equation} \tag{ 1 }$$

where α_Ia = (2.6 + 0.6 − 0.5) × 10⁻⁵ SNe h³₇₀ Mpc⁻³ yr⁻¹, h₇₀ = H₀/(70 km s⁻¹ Mpc⁻¹), where H₀ is the present value of the Hubble parameter,¹⁶ and β_Ia = 1.5 ± 0.6. In addition, Dilday et al. (2008) further found the correlation coefficient between α_Ia and β_Ia to be −0.80. Extrapolating this rate to redshifts greater than 1 is highly uncertain.

The choice of the DES SN fields is driven by four primary considerations:

• 1.  
  visibility from CTIO;
• 2.  
  visibility from northern hemisphere, 8 m class telescopes for SN follow-up spectroscopy;
• 3.  
  past observation history as it pertains to the use of pre-existing galaxy catalogs and calibration;
• 4.  
  overlap with the survey area for the Visible and Infrared Survey Telescope for Astronomy (VISTA; Emerson et al. 2004; see Section 7.2).

Based on these criteria, we have tentatively chosen the five fields in Table 2 and Figure 1. In this paper, we consider five SN survey strategies (see Table 3). For the 10-field hybrid strategy, the fields are the two deep fields and three shallow fields from the 5-field hybrid plus five additional shallow fields clustered around the Chandra Deep Field-South field. In later sections we will compare in detail the results of these surveys, including projected constraints on cosmology.

Table 2. Likely Dark Energy Survey Supernova Fields

Field	Pointing R.A. and Decl.
(3 deg2 area)	(deg, J2000)
Chandra Deep Field-S	525, −275
XMM-LSS	345, −55
SDSS Stripe 82	550, 00
SNLS D1/Virmos VLT	3675, −45
ELAIS S1	05, −430

Notes. Not all of these fields satisfy all of the field choice optimization criteria discussed in the text; e.g., ELAIS S1 is not visible from northern hemisphere, 8 m class telescopes, but matches the other criteria well.

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We used SNANA to explore the choice of filters and exposure times, and the resulting effects on survey cadence. We evaluated the effect of the griz and grizY filter sets on DES SN observations. Figure 2 shows the chosen DES SN filters along with the DES CCD quantum efficiency. In this paper, we have selected five SN search strategies that span the range from ultra-deep and narrow to wide and shallow, including hybrid mixtures of the two. Table 4 shows the filter exposure times for the deep fields for the deep and 10-field hybrid strategies and the shallow fields for the 10-field hybrid strategy for the griz filter set (see the discussion about Y band at the end of this section). Table 5 shows the limiting magnitudes in each filter for the 10-field hybrid survey. For all the survey strategies considered, the deep fields have the exposure times listed in the second column of Table 4. For the shallow survey considered in this paper, as well as for the shallow fields in the 5-field hybrid strategy, each of the fields has one-third of the total exposure time per field of a deep field.

We define "epoch" to be an observation in a single filter on a given date (with no requirement on a source detection). In order to produce simulated sets of DES SN light curves that realistically represent the quality needed for the determination of cosmological parameters, we defined selection cuts that each simulated light curve must individually satisfy (see Table 6). The selection criteria ensure that a DES SN light curve used for analysis is well sampled, with measurements both when the light curve is rising and falling, and of sufficient quality to allow for a robust distance determination, which is essential for constraining cosmology. However, these cuts are relatively inefficient for SNIa retention at higher DES redshifts; studies of the use of looser cuts in conjunction with photometric SN typing methods are ongoing. The effects of different cuts on the maximum signal to noise in a given passband (SNRMAX) on simulated Type Ia and simulated core-collapse samples (described in more detail later) are shown in Figures 6 and 7, respectively. The tightest cuts shown, which are our defaults in this paper, produce the best sample purity at the expense of lower SNIa efficiency.

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Figure 8 shows example multi-band SNRMAX values for simulated DES SN light curves subject to the cuts described above assuming the 10-field hybrid strategy. Note how the g-band measurements have significantly reduced SNRMAX beyond a redshift of z ∼ 0.5 and are absent beyond z ∼ 0.8 due to the flux being redshifted out of the wavelength range of the light curve model.

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Our investigation of a grizY survey option showed that the Y-band SNRMAX barely reaches above 5 even when half of the deep z-band exposure time is devoted to it, and that the Y-band drops below SNRMAX of 5 at a redshift of ∼0.7. Thus, we elected not to use the Y filter for DES SN observations. Note, however, that the planned DES overlap with the VISTA Deep Extragalactic Observation (VIDEO) Survey will provide for Y-band and J-band light curves for a few percent of the DES SNe (see Section 7.2).

Figure 9 shows example DES light curves at redshifts of 0.25, 0.50, 0.74, and 1.07. Particularly noteworthy is that the flux errors projected for DES SN observations are very small at lower redshifts and remain reasonable even beyond a redshift of z = 1. The fact that the g band is absent for the z = 0.74 and z = 1.07 light curves highlights why high-redshift SNe only have three passbands for griz surveys.

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A key to planning a cosmological SN search is the tradeoff between survey area and depth. For the DES SN search, a motivation for deep observations is the advantage of the DECam red sensitivity, while a wide survey area is desirable because it returns a greater number of SNeIa at a given signal-to-noise ratio. In other words, the observing strategy should be both wide, to maximize SN statistics, and deep, to provide for a longer lever arm. Figure 10 shows the SNIa redshift distribution for the deep, shallow, and two hybrid survey strategies. We also considered an ultra-deep strategy (3 deg²) and found that it delivers only a marginal improvement in SNIa statistics beyond a redshift of z = 1 relative to the 10-field hybrid strategy, for example, while the latter results in a factor of 2.8 more SNeIa overall. In particular, we found that the 10-field hybrid has 42% more SNeIa in the redshift range of 0.6–1.0 relative to the ultra-deep strategy. In addition, the ultra-deep survey produces statistics inferior to the deep survey. Thus, the ultra-deep strategy is withdrawn from consideration, while the deep strategy is carried throughout this paper. Figure 10 also shows that the deep and shallow surveys exhibit a significant decrease in the number of SNe at low and high redshifts, respectively, relative to the two hybrid surveys. The hybrid surveys also retain a significant fraction of the low- and high-redshift SNe found in the shallow and deep surveys while avoiding a significant fraction of the selection bias of the shallow survey (see Section 5.1). The redshift distributions for the hybrid surveys including the deep and shallow components are shown in Figure 11. The 10-field hybrid strategy is preferred on the grounds of maximizing SN statistics in the intermediate redshift regime.

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In order to explore the sensitivity of the redshift distribution to the rate of SNeIa, we performed simulations including the α_Ia and β_Ia variations according to the uncertainties given by Equation (1). Since Dilday et al. (2008) found the correlation coefficient between α_Ia and β_Ia to be −0.80, we ran simulations assuming that the parameters are 100% anti-correlated. We found that the projected number of DES SNeIa would change by approximately 7% given such a rate variation.

In previous SNIa Hubble diagram analyses, cosmological constraints have been obtained using mostly spectroscopic confirmation of the SN, which not only afforded an extremely precise determination of the redshift, but also the additional advantage of accurate SN typing. For the DES, it is impractical to obtain spectra for every SN at high-z. The DES will use photometric typing for most of the SNe observed (see Section 6). This technique works very well, and will be further validated by obtaining a spectrum for a significant fraction of low-redshift SNe. In addition, a sample of 10%–20% of SNe at higher redshifts, with a spectrum taken with 6–10 m class telescopes, will be used to study SN evolution, photo-z's, and sample purity. Note that SNe with host galaxies too dim to obtain a host spectrum are another sample that could trigger taking of a follow-up SN spectrum.

Obtaining spectroscopic redshifts of host galaxies, assuming correct host identification, yields precise SN redshifts. In addition, large numbers of the host galaxies can be measured simultaneously with an MOS. We will target every visible SN host, but we expect that the efficiency of obtaining a valid redshift will decrease significantly for galaxies dimmer than apparent i-band magnitude m_i = 24, as indicated by the follow-up of SNLS galaxies (D. Hardin et al. 2012, in preparation). For the purposes of our study, we have approximated the efficiency of obtaining a galaxy redshift as 100% for m_i < 24 and 0% for m_i > 24. For forecasting SN analyses, as well as planning follow-up telescope resources, it is important to estimate the fraction of SN hosts with m_i < 24. Measurements of SNIa host magnitudes from SNLS (D. Hardin et al. 2012, in preparation) have large statistical uncertainties at the highest SNLS redshifts. Therefore, we have constructed a model described in Appendix A. This is a non-trivial task, however, given uncertainties in the SNIa rate dependencies on galaxy mass, luminosity, and type and of redshift evolution. Appendix A describes, in detail, our estimates of SNIa host-galaxy brightnesses in redshift bins, and the sources of significant uncertainty at large redshift. A model estimate is shown in Table 7, where we present the fractions of SNIa host galaxies satisfying the apparent-magnitude limit m_i < 24 for z-bin values from 0.1 to 1.2. Within the uncertainties, the data and model agree. In this study, we choose to use the model (since it lacks the statistical fluctuations of the data) to remove from our cosmology analysis SNe without a host spectrum by applying the stated fractions (for the 10-field hybrid strategy, this cuts out 429 SNe, mostly at high redshift). We will present the impact of this choice on cosmological constraints in Section 8.

Photo-z's, both of the host galaxy and the SN, will play four roles in the DES: they will provide (1) interim SN redshifts before host-galaxy spectra are available, (2) an opportunity to supplement the SN sample with redshifts if the host galaxy is dimmer than m_i = 24 or if the host spectrum cannot be obtained for other reasons, (3) a check on host galaxy identification when redshift comparisons are possible, and (4) help in classifying SNe during the search and in prioritizing them for spectroscopic follow-up. Two key elements of our photometric redshifts are: (1) a deep, ∼35 measurement co-add, per season, of the host galaxy; and (2) a combined SN+host photo-z fit using the host photo-z as a prior. In the next two sections, we show that the combination of these two elements give photo-z's the precision needed to play the roles in the DES SN analysis mentioned above.

The DES photo-z's will come from a combination of host-galaxy photo-z and SN photo-z measurements. The host-galaxy photo-z is expected to be relatively accurate since each SN field will be sampled more than 100 times over the 5 year survey. The limiting magnitudes of the SN host galaxy, one-season co-add will be ∼26th mag, compared to ∼24th mag for the standard DES field. The limiting magnitude of a five-season co-add will be ∼27th mag. DES expects to have at least 60K host-galaxy spectroscopic redshifts for training photo-z's (H. Lin 2011, private communication). In our simulations, the host-galaxy photo-z is determined by a neural-net algorithm described in Oyaizu et al. (2008). Also from Oyaizu et al. (2008), the photo-z error is estimated by the Nearest Neighbor Error algorithm. Figures 12 and 13 show scatter plots of photometric versus true redshifts for galaxies with a magnitude less than 24th and the histograms for the difference of host/SN photometric redshifts and true redshifts, respectively. The host-galaxy photo-z's have a Gaussian sigma of ∼0.027 and a non-Gaussian tail. The SN photo-z is fit with SNANA, using the host-galaxy photo-z as a prior (Kessler et al. 2010b), and is seen to have a Gaussian sigma of ∼0.022 and much-reduced tails. When added to the spectroscopic redshifts provided by SN follow-up, these redshifts are precise enough to begin an interim analysis of DES SNe before host spectra are available.

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The second role for photo-z's is to supplement redshifts from host spectra at high-z, assuming that the host spectra are only available for m_i < 24. We have prepared a simulated sample (detailed in Appendix A) of galaxy photo-z's that has been trained on a sample of m_i < 24 galaxies, and applied to galaxies with 24 < m_i < 26. Figure 14 shows histograms of the host photo-z residuals from this sample, and the combined SNe+host photo-z. We will investigate the impact of using these photo-z's on a cosmology analysis in Section 8.

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At this time, we are assuming that SNe with hosts dimmer than m_i = 26 will not be used in a cosmology analysis, although with a five-season co-add it is likely that many of those hosts will be observed and may provide an interesting sample to study.

The use of a prior on the MLCS2k2 extinction parameter A_V improves the determination of the distance modulus when the measurement error on A_V becomes wider than the width of the A_V distribution. The improvement is noticeable in the simulated DES data at high redshifts where the SN colors are determined by measurements in only three bands: r, i, and z. However, the use of a prior is susceptible to the introduction of biases if implemented with incorrect information. While measurement errors, in principle, average to zero when the measurements of many SNe are combined, a bias in the prior will not average to zero. Inaccuracies in the prior, which is essentially the distribution of SNe in A_V, can arise from purely experimental errors. However, unknown astrophysics, including the evolution of the host-galaxy population or the SN colors with redshift, pose serious challenges to the use of a prior in a high-precision survey like the DES. While we do provide some estimate of potential systematic errors resulting from the use of a prior based on the SDSS analysis, our estimates must currently be considered preliminary.

For the analysis presented here, the prior has the following definition:

$$\begin{equation} P_{\rm prior} = P(A_{\rm V})\times P(\Delta)\times \epsilon _{\rm cuts}(z,A_{\rm V},\Delta)\rm, \end{equation} \tag{ 2 }$$

where P(A_V) and P(Δ) are the underlying physical A_V and Δ (luminosity parameter) distributions and _cuts is the fraction of SNe that pass the selection cuts for a given redshift, A_V, and Δ. For this work, following Kessler et al. (2009a), P(A_V) is given by dN/dA_V = $\exp (-A_{{\rm V}}/\tau _{A_{\rm V}}$) with $\tau _{A_{\rm V}}$ = 0.334, and P(Δ) is an asymmetric Gaussian with peak position, Δ₀, and positive and negative side widths, σ₊ and σ₋, respectively, given by Δ₀ = −0.24, σ₊ = +0.48, σ₋ = +0.23. In addition, we set dN/dA_V = 0 for A_V < 0.

For a given survey, e.g., the 5-field DES hybrid scenario, _cuts is calculated using SNANA by cyclically simulating SN light curves and checking which light curves pass the defined selection cuts until the desired efficiency accuracy is reached. Figure 15 shows the selection efficiencies for various classes of SNeIa. Both the deep and shallow observation fields within the 5-field hybrid survey exhibit statistical completeness for nearby and/or bright SNe. However, Figure 15 shows the vastly higher efficiency of the deep relative to the shallow fields for distant and faint and/or heavily extincted SNe. Figure 16(a) shows our application of efficiencies to the hybrid survey simulation in order to avoid the bias in the fitted distance modulus that would arise from MLCS2k2 light curve fitting with an incorrect prior, e.g., one with the assumption of a flat prior on efficiency.

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As discussed above, the introduction of priors can easily lead to biases if the effects of the survey selection efficiency are poorly understood. Of particular concern is the bias manifested as a difference between observed (i.e., "fitted") and true (i.e., "simulated") distance modulus ("μ_fit" and "μ_sim" hereafter) that can arise. Figure 16(f) shows such a departure of μ_fit–μ_sim from zero beyond a redshift of ∼0.7. The bias illustrates the size of the μ-correction that the DES SN data would need if the efficiency prior were incorrectly assumed to be flat. Note, one does not expect the selection bias to have a significant effect at low redshift because there the SN sample is essentially complete. The fact that A_V is driven toward zero, while the trend in Δ is negative, as redshift increases beyond ∼0.5 (see Figure 17), implies that only less extincted and/or brighter SNe pass the selection cuts, and strongly supports our identification of the bias in μ as a selection bias. In addition, this selection effect explains the small drop in rms beyond a redshift of z = 1.0 exhibited in Figure 16(a). Figure 16(d) also shows, when compared with Figure 16(f), one of the key motivations for the hybrid survey. A systematic check is enabled by the ability to compare the less biased distance moduli from the deep part of the data set at higher redshifts with the more biased shallow part. If that cross-check is validated, then confidence is increased in the highest region of the deep component of the survey (i.e., redshifts greater than 1.0) where the deep component suffers a similar bias to that experienced by the shallow component at intermediate redshifts. Figure 16(e), showing the case of the deep-only strategy, is included for completeness.

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In this section, we discuss MLCS2k2 flat prior and SALT2 model fitting. Such fits avoid the issue of selection efficiency bias discussed above. The tradeoff is an increase in the rms spread in the distance modulus, as is clearly evident in the comparison of Figure 16(a) with Figure 16(b). In addition, Figure 16(b) shows a high-redshift μ bias evident in MLCS2k2 fits with flat priors. This is due to the fact that such fits allow negative values of A_V, for which the fitter compensates by pulling the distance modulus to higher values.

The SALT2 light curve fitter in SNANA is accompanied by a separate program called SALT2mu (Marriner et al. 2011) that fits the SALT2 parameters α and β that are used to determine the standard SNIa magnitudes. The parameters that correlate distance modulus with x₁ (a stretch-like parameter) and c (the color) are α and β, respectively. We have chosen to fit for the α and β parameters independent of the cosmology using SALT2mu, which allows us to apply the same cosmological fitting procedure to the outputs of the MLCS2k2 and SALT2 light curve fits.

The resulting distance modulus residuals are shown in Figure 16(c). The trend in the rms spread of the distance modulus is rather similar to that obtained with MLCS2k2 with the use of a flat prior. While it would be possible to apply a prior on the color in the SALT2 fit, we have followed normal practice in not doing so here. For the remainder of this paper, we will use MLCS2k2 fits with correct priors (corresponding to Figure 16(a)) in our analysis, with the exception that we use SNooPy in Section 7.2 and include SALT2 in the discussion of the DES SN cosmology fits in Section 8.

In order to simulate SNcc, we use the input SN rate parameterization of Dilday et al. (2008), which found the SNIa rate from SDSS to be of the form α(1 + z)^β with α_Ia = 2.6 × 10⁻⁵ SNe h³₇₀ Mpc⁻³ yr⁻¹, and β_Ia = 1.5. For SNcc, we take β_cc = 3.6 to match the star formation rate. Various studies, the most recent being SNLS (Bazin et al. 2009), have shown this assumption to be valid, albeit with low statistics and limited redshift range. This leaves the determination of α_cc. Taking the ratio of SNcc/SNeIa to be the SNLS value of 4.5 for redshifts of <0.4 (Bazin et al. 2009), we calculate the value α_cc must have in order to obtain the ratio of 4.5: α_cc = 6.8 × 10⁻⁵ SNe h³₇₀ Mpc⁻³ yr⁻¹. Note that with this value of α_cc, the SNcc/SNeIa ratio increases to ∼10 out to a redshift of 1.2. A caveat in this estimate is that one of the largest uncertainties is the actual population near the detection threshold. Direct measurements of the SNcc rate beyond a redshift of z = 0.4 would be very helpful in the determination of SNIa sample purity.

In this section, we discuss the relative fraction of the SNcc subtypes. The most important fraction is that of Type Ib/c, since they most commonly pass the combination of cuts on SNRMAX and MLCS2k2 fit probability that the SN is an SNIa (f_p = $P_{\chi ^2}$, the probability from fit χ² and the number of the degrees of freedom). The literature contains several estimates of the ratio of Type Ib/c to Type Ib/c plus II SNe (see Table 8 for examples). The most complete references, in terms of fractions being given for each type of SNcc, are Li et al. (2011b) and Smartt et al. (2009), and the Type Ib/c fractions are in good agreement. We have used the Smartt et al. (2009) values (see Table 9) as the default set of fractions in this analysis, as they give a more conservative amount of SNcc misidentification relative to Li et al. (2011b).

The absolute brightness of SNcc is a critical parameter in the number of SNcc misidentified as SNeIa, since most are too dim to pass typical SNRMAX cuts (e.g., those shown in Table 6). Two references for absolute SNcc brightnesses, Richardson et al. (2002) and Li et al. (2011b), are compared in Tables 10 and 11. The numbers in Table 10 have been corrected for the significant Malmquist bias evident in that data. The correction assumed a threshold of 16 mag in apparent brightness, and took into account the larger volume sampled by intrinsically brighter SNe than for fainter SNe. The volume-limited analysis in Li et al. (2011b) is already corrected for Malmquist bias, but Conley et al. (2011) used Richardson et al. (2002) in their analysis, noting that Li et al. (2011b) perhaps missed a bright Type Ib/c component by avoiding low-luminosity galaxies. To take the conservative approach, we used the single-Gaussian-approximation brightnesses from Richardson et al. (2002) as our default.

Table 10. The Absolute B-band Magnitudes and Widths for the Single-Gaussian Fits from Richardson et al. (2002), Corrected for Malmquist Bias

Richardson et al. (2002)
SN Type	MB	$\sigma _{M_{B}}$
IIP	−14.40 ± 0.42	0.81
Ib/c	−16.72 ± 0.23	0.62
IIL	−17.19 ± 0.15	0.47
IIn	−17.78 ± 0.41	0.74

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Table 11. The Absolute R-band magnitudes and widths from Li et al. (2011b)

Li et al. (2011b)
SN Type	MR	$\sigma _{M_{R}}$
IIP	−15.66 ± 0.16	1.23
Ib/c	−16.09 ± 0.23	1.24
IIL	−17.44 ± 0.22	0.64
IIn	−16.86 ± 0.59	1.61

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SNcc are observed to be a much more heterogeneous class than SNeIa and, in contrast to SNeIa, there is no parameterization available that describes the diversity of SNcc light curves. Therefore, we take a template approach to modeling SNcc. Three sets of templates are compared, with each being a spectral sequence as a function of time. The first set is of 40 templates from the Supernova Photometric Classification Challenge (Kessler et al. 2010a), the second set is of the composite spectral templates constructed by Nugent,¹⁷ and the third set is of Type Ib/c and IIP templates from Sako et al. (2011), augmented by the Nugent templates for Types IIL and IIn. All templates were converted to SDSS filter magnitudes, and SNANA performs the K-corrections into the DES filters. Note that there is no template for Type IIb, which in Li et al. (2011b) is more numerous than Types IIL or IIn. The Type IIL template is expected to be the closest to Type IIb SNe, and, therefore, we used it for the Type IIb sub-sample.

In the SNANA simulation, the templates are corrected to the absolute brightnesses discussed in the previous section. In addition, the Nugent templates are composite spectra and do not include absolute brightness fluctuations, therefore they are also smeared by the Gaussian-fitted widths tabulated in the previous section in order to better reflect the observations. The Kessler et al. (2010a) and Sako et al. (2011) templates already have sufficient variation in brightnesses and require no additional smearing. The templates from the Supernova Photometric Classification Challenge are the most complete set and are used as the default in the rest of this paper. In particular, note that these templates contain a "1 + z_temp" bug in that each SNcc template is too dim by a factor of 1 + z_temp (see Table 4 and Section 2.6 of Kessler et al. 2010a), where z_temp is the redshift of the template. In the next section, we include a discussion of the effect of this bug on our simulations.

Using the inputs discussed above, and spectroscopic host redshifts, we simulated the DES SN sample including SN Types Ia, Ib/c, IIL, IIn, and IIP subject to the selection criteria listed in Table 6. The SNe in this combined sample are fit to the SNIa MLCS2k2 model, giving a fit probability f_p variable cut that can be customized for each analysis, and for the amount of SNcc observed in a sub-sample with spectroscopic follow-up. Figure 18 shows the distribution of f_p for the SNIa and SNcc samples, after all other selection cuts have been applied. The number of SNe of each type with no f_p cut, and with f_p > 0.1, is shown in Table 12. For those results, the effect of the 1 + z_temp SNcc template bug discussed at the end of the previous section is an increase in the SNIa purity by ∼2%, which has no impact on our conclusions. Figures 19 and 20 show redshift distributions of these samples subject to a fit probability cut f_p > 0.1. Table 13 shows comparisons in the total SNcc number with variations in the simulation inputs discussed above, with a range of × 3 in total number of SNcc. Note that the sample purity is better than that obtained with the same analysis performed in Kessler et al. (2010a); this is due to the correction for Malmquist bias applied to the SNcc simulation sample, which reduces their expected absolute brightness and therefore the number passing SNRMAX cuts.

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We perform an analysis of the color variations in the SN colors g − i, g − z, r − i, and r − z for a grid of values of R_V and $\tau _{A_{\rm V}}$, where $\tau _{A_{\rm V}}$ is the parameter that controls the width of the simulated A_V distribution, as described in Section 5.1. As $\tau _{A_{\rm V}}$ increases, the A_V distribution extends to larger extinctions and, thus, produces SNe with redder colors. Our reference color sample is a simulation with the values of R_V = 2.18 and $\tau _{A_{\rm V}}$ = 0.334, which are the best-fit values from Kessler et al. (2009a). The results presented here are for the redshift range 0.4 < z < 0.7. This range has the highest SN statistics for the DES. For the redshift range z < 0.4, the SN statistics are much less, but SNRMAX is substantially better, so that the precision of the color measurements is comparable to those presented here.

We constructed a suite of simulations with a grid of R_V and $\tau _{A_{\rm V}}$ values in order to assess the effects of changes in R_V and $\tau _{A_{\rm V}}$ on SNIa colors. An example of the effects on the g − z color is shown in Figure 21. The differences in color between simulations with R_V = 2.69 and $\tau _{A_{\rm V}}$ = 0.25 and our reference sample parameters are shown as a function of phase. A signal-to-noise cut of 0.5 is applied at every phase. Figure 21 has two noteworthy features: the fitted average color level for phase <+11 days and the significant drop in color for later phases. The average color level of a given SN color for phase <+11 days has, in general, a complicated dependence on R_V, $\tau _{A_{\rm V}}$, and the redshift range of the data sample. In special cases, for simulations with fixed R_V, A_V, and certain values of fixed redshift, this dependence can be predicted from the CCM dust model (Cardelli et al. 1989) and the parameterization from Jha et al. (2007). We have verified that our simulations agree well with the predictions in these cases.

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The sensitivity to parameters R_V and $\tau _{A_{\rm V}}$ of the small-phase average g − z difference is shown in Figure 22. The error bars show the statistical uncertainty for each parameter choice. Overall, the trend is to increase the value of the g − z difference by approximately 0.3 mag as R_V increases from 1.1 to 3.1, which is a plausible range for R_V, and $\tau _{A_{\rm V}}$ increases from 0.16 to 0.52. From this figure, simulated values of R_V within ∼0.38 of the reference value, and of $\tau _{A_{\rm V}}$ within ∼0.06 of the reference value, can be distinguished at the 1σ level. Similar plots for other SN colors and other redshift ranges show slightly different dependencies on the parameters, and hence can be used to further lower the above uncertainties in R_V and $\tau _{A_{\rm V}}$. Figure 22 also shows several degenerate combinations of R_V and $\tau _{A_{\rm V}}$ that lead to the same level of g − z difference. This occurs because SN colors are largely dependent only on the ratio of A_V to R_V, and so a given color difference can only determine the ratio of A_V to R_V to some uncertainty. This degeneracy is reduced by considering the behaviors of other color differences and their redshift dependence. In addition, the second feature of Figure 21, namely, the drop-off in the g − z difference at late phases,¹⁸ can also be used to resolve this degeneracy. In this analysis, we assume that the degeneracy can be broken by an SDSS-like analysis (Kessler et al. 2009a), which took all such effects into consideration. Therefore, in our analysis in Section 8, we take the uncertainty in R_V and $\tau _{A_{\rm V}}$ to be 0.38 and 0.06, respectively.

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7.2.1. The DES+VIDEO Overlap

7.2.2. The DES+VIDEO Supernova Sample

Based on VIDEO Survey SN data from the first season, we constructed an SNANA simulation library (see Section 2.2) with the following characteristics: typical Y- and J-band PSF is of order 1 arcsec, sky noise on the order of 200–400 photoelectrons, and zero points ranging from 31.5 to 32.0 mag. Two seasons of VIDEO/DES overlap are expected, and the simulation library assumes that the observing conditions will be similar during both seasons. In addition, there are 10 observations in Y band and 13 in J band, each with 32 minutes of exposure time. Based on this simulation library, we estimate that the DES+VIDEO combined SNIa sample from years 2013 and 2014 could consist of approximately 108 SNe with z < 0.5 in the common Chandra Deep Field-South, XMM-LSS, and ELAIS S1 fields (see Table 2). Figures 23 and 24 show VIDEO SNRMAX for the 108 overlapping SNe, and an example simulated combined light curve, respectively. As shown in Figure 23, some of the SNe have SNRMAX that are relatively low (e.g., <5), and may not be useful for all SNe analyses. As a follow-up to this analysis, a study is planned to utilize SNANA simulated DES+VIDEO SNIa light curves to evaluate the benefit to SN color determinations to be gained by adding VIDEO infrared SNIa data to the DES SN analysis.

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Constraints on cosmological parameters are obtained by comparing the theoretical values of distance moduli, μ(z, θ_c), to the values inferred from the light curve fits of the SN simulations, μ_fit(z), where θ_c ≡ {Ω_DE, w₀, w_a, Ω_k} is the set of cosmological parameters. The likelihood for an individual SN at redshift z_i, L(μ_fit|z_i, θ_c), is taken to be Gaussian with a mean given by the μ(z_i, θ_c) at redshift z_i, for the cosmological parameters θ_c, with a standard deviation σ^μ_i given by the MLCS2k2 light curve fit errors and an intrinsic dispersion σ_int = 0.13 added in quadrature. In the case of SNe with photometrically determined redshifts, we add an error of |∂μ(z, θ_c)/∂zδz| in quadrature. The simulated SN observations are independent, and the likelihood is analytically marginalized over the nuisance parameter combination of the Hubble constant, H₀, and the absolute magnitude, M, with a flat prior. This results in a likelihood for the μ_obs(z) for all SNe that is Gaussian and has a covariance matrix Cov, which can be calculated from the above errors σ^μ_i on each SN discussed above.

Following the DETF Report (Albrecht et al. 2006), we evaluated the performance of survey options in terms of the DETF Figure of Merit (FoM), albeit with the following modification. The DETF FoM is defined to be the inverse of the area of the 95% confidence-level (CL) error ellipse in the w₀, w_a plane when all other parameters have been marginalized over. However, the FoM is often calculated as [$\sigma _{w_p}\sigma _{w_a}$]⁻¹, or equivalently $[{\rm det}({\rm Cov}_{w_0w_a})]^{-0.5}$. We follow the latter convention, as also used by SNLS (Sullivan et al. 2011). Our FoMs may be converted to the DETF FoM defined by Albrecht et al. (2006) upon dividing by factor of 18.8. Henceforth, we refer to our modified FoM as the DETF FoM. Along with DES SNe, and the low-redshift samples discussed above, we used prior constraints from the DETF Stage II experiments plus Planck in the form of a Fisher matrix for these experiments obtained from the DETF (W. Hu 2009, private communication). Henceforth, we refer to this prior as "Stage II." It should be noted that this prior constrains cosmological parameters much more strongly than current data. Thus, priors used in parameter estimates based on current data releases, such the SNLS results (Sullivan et al. 2011), are much weaker, and consequently the FoMs computed from such surveys are much lower due to the weaker priors. The error covariance matrix C on all the parameters is estimated as the inverse of the Fisher matrix F evaluated at a fiducial set of parameters Θ_p (which is chosen to be the set suggested by DETF, i.e., Ω_DE = 0.73, Ω_K = 0, w₀ = −1, w_a = 0):

$$\begin{eqnarray} F_{ij} (\Theta _p) &\equiv & F_{ij}^{{\rm DES}} +F_{{\rm Stage\;II}}\nonumber \\ F_{ij}^{{\rm DES}}(\Theta _p) &\equiv & \left\langle -\partial _i \partial _j \ln (L^{{\rm DES}}(\mu _{{\rm obs}}\vert \theta _c)\bigg \vert _{\Theta _p} \right\rangle \nonumber \\ & = & \frac{\partial \mu ^{a}}{\partial \Theta _c^{i}}{\rm Cov}^{-1}_{{\rm ab}}\frac{\partial \mu ^{b}}{\partial \Theta _c^{j}}\bigg \vert _{\Theta _p} + \frac{\partial ^2\ln ({\rm det} ({\rm Cov}))}{2\partial \Theta _i\partial \Theta _j}\bigg \vert _{\Theta _p}, \quad\quad \end{eqnarray} \tag{ 3 }$$

where a, b index each SN and i, j index the four cosmological parameters. The calculated DETF FoM, assuming spectroscopic host-galaxy redshifts and statistical uncertainties only,¹⁹ ranges from 214 to 228 (see Table 15). The Stage II experiments plus Planck, without any additional data, yield an FoM of 58. Hence, with statistical uncertainties only, the relative improvement is by a factor of 3.69–3.93. In our FoM estimates, we include a reduction in the SN sample size due to incompleteness in the sample of host galaxies based on the fractions in Table 7. The FoM before trimming is typically a factor of 1.07 larger. Augmenting the sample with photometric redshifts, described in Section 4.3, results in an increase in relative FoM by a factor of 1.03. The hybrid 5-field is presented for both MLCS2k2 and SALT2 simulations and analyses. The difference in FoM between the two is due to the smaller Hubble residuals for the MLCS2k2 model with a dust extinction prior (as shown in Figure 16). In the next section, we augment our FoM calculations via the inclusion of systematic uncertainties, both with and without the effect of the dust prior.

The DES SN search precision will depend strongly on our ability to control systematic uncertainties. In this section, we discuss the inclusion of such uncertainties in our FoM forecasts (the details of the calculation are given in Appendix B). Unless specified, the numbers in this section assume an accurate redshift derived from a host-galaxy spectrum. We consider three other fundamental sources of systematic uncertainties and one tied to an analysis option in this paper:

• 1.  
  filter zero points (fundamental);
• 2.  
  filter-centroid wavelength shifts (fundamental);
• 3.  
  SNcc in the SNIa sample (fundamental);
• 4.  
  the use of a dust prior for R_V and A_V (derived from an analysis choice).

In addition, all of the calculations in this section include the inter-calibration of the low-redshift anchor data sets and DES. We acknowledge the existence of astrophysical systematic effects that are not included here. Such effects are community-wide concerns and are beyond the scope of this study.

The filter zero point uncertainties are taken as independent and to have the value of 0.01 mag, which is an estimate of the final survey precision. The shift in the distance modulus is computed for each filter zero point change. The effect of a change in the i-band filter and the corresponding change in μ are displayed in Figure 27. Two sets of data points are shown, one for a 0.01 mag shift in i band, and the other a 0.1 mag shift but with the μ change divided by 10. This demonstrates the linearity in the μ change for a shift in zero point. A Markov chain Monte Carlo cosmology calculation,²⁰ with four independent cosmology parameters, was used to evaluate the shift in the maximum likelihood value of w₀ due to a shift in filter zero point (see Figure 28). These cosmology shifts are meant as an example and are not used in the FoM calculation described earlier and in Appendix B. The FoM including the μ changes caused by filter zero point shifts, as illustrated in Figure 27, is shown in Table 16 for the hybrid 10-field survey. This is the most important of the fundamental systematic uncertainties listed above.

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Table 16. DETF Figure of Merit (Modified as Described in Section 8.1) for the MLCS2k2 Model Including Various Systematic Changes in the DES SNIa Hybrid 10-field Survey (Including a Low-redshift Anchor and a Simulated SDSS Sample)

Systematic	FoM
Change	with
Included	Systematic
None	228
Filter zero point shift	157
Inter-calibration	188
Filter λ shift	179
Core-collapse misid.	226
RV and $\tau _{A_{\rm V}}$	128
Total without RV and $\tau _{A_{\rm V}}$	124
Total with RV and $\tau _{A_{\rm V}}$	101

Note. The 5-field hybrid total values without and with R_V and $\tau _{A_{\rm V}}$ are 120 and 94, respectively.

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The systematic effects have also been evaluated for the hybrid 5-field survey, and the impact on the FoM was found to be very similar to that for the 10-field survey. Thus, from the point of view of constraining the CPL parameters, after combining with prior data, the two strategies are essentially equivalent. The strategy choice is then motivated by the potential for other kinds of studies that can, e.g., test and verify the accuracy of SNIa light curve models and of the redshift independence of SNIa standardized luminosities. Such studies typically require a large sample size, so that one can study correlations with other observables, e.g., SNIa host properties. The 10-field survey provides a much larger number of well-measured SNeIa at redshift ranges where the potential for observing host properties, or obtaining SNIa spectra, is high. Thus, the 10-field hybrid strategy is more suitable for such studies.

The systematic uncertainties in the filter centroids are derived in a similar fashion to the zero points, using 10 Å as the expected wavelength precision for the DES. The resulting FoMs are also presented in Table 16.

The systematic uncertainty due to SNcc misidentification is caused by the fitted-μ difference between SNeIa and SNcc (see Figure 26). SNcc are generally dimmer than SNeIa, and, in a fit for SNIa parameters, this causes a shift in μ to larger values. In this analysis, the fraction of SNcc in the SNIa samples is small, typically <5%. The resulting small average μ shift, and the fact that the SNcc that pass selection cuts are all at low redshift where the low-redshift anchor suppresses their effect, causes a relatively small decrease in FoM (see Table 16).

The final systematic uncertainty considered is the use of an incorrect dust extinction prior in the SNIa fitting procedure. Figure 16 showed that the use of the prior in the MLCS2k2 fit improved the rms scatter of the Hubble diagram, compared to the SALT2 fit, which had no prior. However, the tradeoff is an additional systematic uncertainty since an incorrect prior in the fit can bias the distance modulus. The uncertainty in R_V and $\tau _{A_{\rm V}}$ is derived from the analysis of one SNIa color presented in Section 7.1 and in Figure 22. Values of R_V within ∼0.38 of the SDSS reference value, and $\tau _{A_{\rm V}}$ within ∼0.06 of the SDSS reference value, were used to derive the FoM. The resulting effect on the FoM is actually more significant than that of the fundamental systematics (see Table 16). These uncertainties can be improved by using all the color information available; on the other hand, they ignore possible redshift dependence. Our analysis indicates that the effect of the current dust prior systematic is much larger than the effect of increased Hubble residuals in the SALT2 analysis FoM (see Table 15).

The total FoM, including our current estimates of systematic uncertainties, is shown in Table 16, for the cases both with and without the dust prior. The 95% CL limits on w₀ and w_a are displayed in Figure 29, for statistical uncertainties and including all systematic uncertainties. Figure 30 displays the total 95% CL limits on w as a function of redshift, and includes curves for the DETF Stage II prior alone.

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Overall, the DES SNIa sample, augmented by a low-redshift anchor set, is expected to constrain a time-dependent parameterization of w, and improve the DETF FoM by at least a factor of 1.75 over the Stage II value of 58.

The main differences in our method, compared with that presented in the DETF report, are as follows.

• 1.  
  We have an improved estimate of the effect of photo-z uncertainties, coming from the full simulation of the DES survey.
• 2.  
  The uncertainty in absolute magnitudes referred to in the DETF report is analytically marginalized in our likelihood function. Therefore, our "statistics-only" Fisher matrix accounts for this in the correlated covariance matrix.
• 3.  
  In our SN analysis, we include two low-redshift samples as anchors: the 3 year SDSS sample with ∼350 SNe and a sample of 300 SNe taken to be at redshift of 0.055 (Li et al. 2011b), each with a Gaussian error of 0.13 mag. Neither of these samples were included in the calculation of the Stage II prior Fisher matrix. In each of these low-redshift samples, the dominant systematic uncertainty is expected to be a step μ offset of the kind described in the DETF report. Therefore, we include a step offset for each of the low-redshift samples. We also assume a Gaussian prior on each of these step offsets of width δM_lowz = δM_SDSS = 0.01. This is consistent with the suggestion of the DETF report.
• 4.  
  In the DETF report, all other systematic effects were treated as an effective linear and quadratic shift in μ(z). The relevant nuisance parameters were taken to be the first and second-order redshift coefficients. We have used a more realistic estimate of the effects of systematics on μ(z). In our simulations, we vary the systematic parameters θ_n, which model the instrumental setup and SN properties, from their assumed values and study how the μ^obs(z) changes. Assuming that we are in the linear regime, we write

  $$\begin{equation} \mu ^{{\rm obs}}(z,\Theta) = \mu ^{{\rm obs}}(z,\Theta _p) + \sum _{j}\frac{\partial \mu ^{{\rm obs}}(z,\Theta)}{\partial \theta _n^j} \bigg \vert _{\Theta _p}\Delta \theta _n^j, \end{equation} \tag{ B2 }$$

  where the sum runs over j, which indexes the list of systematic parameters θ_n.

By varying each systematic parameter under consideration, one at a time, in our simulations, we estimate the average values of the partial derivatives (∂μ^obs(z, Θ))/(∂θ^j_n) in redshift bins of 0.1 in terms of fitting functions that involve 3–6 parameters. Using these, we evaluate the Fisher matrix in the parameter space Θ, which includes both the cosmological parameters θ_c and the systematic parameters θ_n. The parameters θ_n will be set or measured to a fiducial value with an estimated uncertainty either in the process of calibration (parameters related to survey conditions) or by other experiments (SNIa light curve model parameters or SNcc fractions). These measurements allow us to use appropriate priors on the deviation of these parameters from their fiducial values. Therefore we will marginalize over all the systematic parameters with such priors on each of them. We now proceed to discuss the relevant set of systematic parameters.

We first identify the relevant set of systematic effects and parameterize them. These are the zero points in each filter band, the wavelength of the centroid of the filters, the fraction of SNe that are SNcc but misidentified as SNeIa, and the values of $\tau _{A_{\rm V}}$ and R_V. First, we describe these parameters, and the priors on them due to independent measurements.

Zero points. We include the deviation in zero points in each band {g₀, r₀, i₀, z₀} from the fiducial values as parameters. Further, we expect to be able to calibrate these independently to an error of 0.01 mag. Hence we assume a Gaussian prior on each of these parameters with a width σ^g_zpt = σ_zpt^r = σⁱ_zpt = σ_zpt^z = 0.01.

Wavelength of filter centroid. We include the deviation of the wavelength {λ_g, λ_r, λ_i, λ_z} of the centroid of each filter from the fiducial values. Further, we expect to be able to calibrate these independently to an error of 10 Å. Hence we assume a Gaussian prior on each of these parameters with a width σ^g_λ = σ_λ^r = σⁱ_λ = σ_λ^z = 10 Å.

Core-collapse fraction. The sample purity for the hybrid 10-field survey is 98%. We are taking the uncertainty in this value to be 2%, and have shown that the effect on the FoM is still smaller than the other uncertainties. To be specific, we use f_cc = 0 as a fiducial value and $\sigma _{f_{{\rm cc}}}=0.02$ as the Gaussian uncertainty in f_cc.

CCM dust model parameters. μ_fit depends on the true values of the parameters $\tau _{A_{\rm V}}$ and R_V values. Our simulations are based on SDSS results (Kessler et al. 2009a) and, hence, we assume fiducial values of $\tau _{A_{\rm V}}$ = 0.334 and R_V = 2.18. Assuming that these variables are correlated in the way determined by the SDSS SN survey (Kessler et al. 2009a), we expect to have independent measurements determining these parameters as a correlated Gaussian with a covariance matrix C with $C_{R_{\rm V}R_{\rm V}} = 0.1444, C_{\tau _{A_{\rm V}},\tau _{A_{\rm V}}} =0.003136, C_{\tau _{A_{\rm V}}R_{\rm V}}= 0.0036176$.

As described in Section 2, these parameters are inputs to our simulations. Therefore, we can study changes in the fitted values of μ by changing these input parameters in SNANA. For each of the parameters that we shall describe, we compute $({\partial {\mu _{{\rm eff}}(z,\Theta)}}/{\partial \Theta _i})\vert _{\Theta _p}$ by numerically evaluating the partial derivative of μ_eff(z, Θ) as a function of redshift. For doing so, μ_eff(z, Θ) is estimated as the sample mean of obtained values of the fitted distance modulus μ_fit in redshift bins of 0.1. Having obtained these estimated values, along their dispersions, we fit these values at discrete redshifts bin centroids to simple functions of the redshift. This allows us to evaluate the partial derivative at any redshift in the range of observation (0, 1.2).