SNLS3: CONSTRAINTS ON DARK ENERGY COMBINING THE SUPERNOVA LEGACY SURVEY THREE-YEAR DATA WITH OTHER PROBES

Our SN Ia samples are divided into two categories: those discovered and confirmed by the SNLS and those taken from the literature which sample different redshift ranges to SNLS. The SNLS uses data taken as part of the five-year Canada–France–Hawaii Telescope Legacy Survey (CFHT-LS). The CFHT-LS is an optical imaging survey, the deep component of which conducted repeat imaging of four fields every three to four nights in dark time with four filters, allowing the construction of high-quality multi-color SN light curves (G10). Spectroscopic follow-up is used to confirm SN types and measure redshifts, critical in obtaining clean samples of SNe Ia as reliable photometric identification techniques have yet to be developed, despite recent progress (Kessler et al. 2010; Bazin et al. 2011). Candidates were prioritized following the procedure outlined in Sullivan et al. (2006a). SNLS benefited from large time allocations on 8–10 m class telescopes—∼1500 hr over five years—including the Gemini North and South telescopes, the European Southern Observatory Very Large Telescopes, and the Keck telescopes. Nearly all our SN spectra are published²⁴ (Howell et al. 2005; Bronder et al. 2008; Ellis et al. 2008; Balland et al. 2009; Walker et al. 2011). All spectra are analyzed and uniformly typed according to the classification schemes of Howell et al. (2005) and Balland et al. (2009). Further information on all 242 SNLS SNe Ia that we use, including light-curve parameterizations, can be found in G10.

The SNLS data set is complemented with SNe Ia from the literature over redshift ranges that the SNLS sample does not cover. We use 123 SNe Ia at low redshift (z ≲ 0.08) from a variety of sources (primarily Hamuy et al. 1996; Riess et al. 1999; Jha et al. 2006; Hicken et al. 2009a; Contreras et al. 2010), 14 SNe Ia at z ≳ 0.8 from the HST-discovered sample of Riess et al. (2007), and 93 SNe Ia over 0.06 ≲ z ≲ 0.4 from the first year of the Sloan Digital Sky Survey-II SN search (Holtzman et al. 2008). Light-curve parameterizations and other data for these events, on the same relative system as that of G10, can be found in C11.

We also considered including other SN Ia samples that, at least in part, probe the same redshift range as SNLS. However, we do not do this for several reasons (see also C11 for a discussion of these points). SNLS is designed to control systematics as much as is possible—a single telescope survey with a well-understood photometric response and calibration, using deep exposures in filters that allow the same rest-frame colors to be measured for most of the redshift range that it probes. There is also significant published information on the host galaxies (Sullivan et al. 2010), essential for the cosmological analysis and which is not available for other higher-redshift samples. Finally, SNLS is by far the largest and best observed (i.e., highest signal-to-noise ratio for each event) SN Ia sample over 0.3 ≲ z ≲ 1.0. Adding other SNe to this might lead to marginally improved cosmological constraints from a purely statistical perspective, but would certainly lead to a much more complex and uncertain analysis of systematic uncertainties when combining data from many surveys conducted at many telescopes.

All SNe are corrected for Galactic extinction, Malmquist, and other selection biases (Perrett et al. 2010; Conley et al. 2011), and peculiar velocities (at low redshift).

We parameterize the SN Ia light curves for distance estimation using the combined results from updated versions of two independent light-curve fitters—SiFTO (Conley et al. 2008) and SALT2 (Guy et al. 2007). Both techniques provide an estimate of the SN peak rest-frame B-band apparent magnitude at the epoch of maximum light in that filter, a measure of the SN light-curve shape, and an estimate of the SN optical B − V color ($\mathcal {C}$). SiFTO parameterizes the light curve in terms of stretch (s), while SALT2 uses a related parameter x₁. The two light-curve fitters are compared in G10, which also provides details of the techniques used to average them into a single light-curve parameterization for subsequent distance estimation. The distance estimation technique used, based on the combined SiFTO/SALT2 light-curve parameters, is described in Section 3.

Other light-curve fitting and distance estimation techniques are available. In particular, the MLCS2k2 fitter (Jha et al. 2007) has been widely used in previous SN Ia analyses (e.g., Hicken et al. 2009b; Kessler et al. 2009), and the use of MLCS2k2 versus SALT2 has led to significantly different cosmological parameters in some cases. The two techniques are mathematically equivalent (to first order—see Section 4.2.3 in G10), and many of the apparent differences can instead be traced back to the training data and priors on SN color. We do not use MLCS2k2 in the SNLS3 analysis for several practical reasons, discussed at length in G10 and C11, and which we summarize here.

There are apparent calibration problems with observer U-band SN Ia data (which we do not use anywhere in our analysis) which MLCS2k2 is reliant upon for its distance-estimation training. These include the following.

• 1.  
  Observer-frame U-band SN Ia data show more scatter around individual SN Ia light-curve fits than can be accounted for by the published observational uncertainties. This large intrinsic scatter is not seen in SNLS and SDSS-SN SN observations transformed into rest-frame U-band.
• 2.  
  In U−B versus B−V color–color space, the SNe with observer-frame U-band data show a systematic offset compared with the other SN samples (see Figure 1).
• 3.  
  There is significant tension between SNe with observer-frame U-band data, and those without, in the cosmological fits. This tension disappears if this U-band data are removed from the light-curve fits for the low-z SNe.

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A sample of low-z SNe from the Carnegie Supernova Project (Contreras et al. 2010) with observer-frame u' data shows none of these three problems above. That is, the properties of the low-z SN Ia U-band data are inconsistent with SNLS at z ∼ 0.5, with SDSS-SN at z ∼ 0.25, and with the CSP sample at z ∼ 0.02.

Various possibilities for the origin of this U-band anomaly are outlined in Section 2.6 of C11 (see also Kessler et al. 2009). For this to be an evolutionary or astrophysical effect, any evolution must be astonishingly sudden (i.e., turning on at z ∼ 0.25 and then not evolving any further out to z = 1)—yet somehow only effect one of the two sources of low-z data (those with U, instead of u', data). It must also somehow not manifest itself in maximum-light spectral comparisons between low-z and high-z (e.g., Ellis et al. 2008; Foley et al. 2008; Cooke et al. 2011). Note that the small, and not very significant, evolution that is seen in Cooke et al. (2011) is in the opposite sense to that implied by the U-band photometry problem.

The overwhelming likelihood is that this is a problem with the notoriously difficult calibration of the observer-frame U-band data—there is no evidence that it is an astrophysical effect, and significant evidence that it is not. Note that an MLCS2k2 trained without the U-band data severely impacts the science that can be done—as a distance estimator, MLCS2k2 requires z ≲ 0.06 SNe Ia for the training, and therefore cannot be supplemented with SNLS/SDSS SN data sampling the rest-frame U-band as is the case with SALT2 and SiFTO.

Even with a version of MLCS2k2 not trained using the current U-band data, additional problems remain. These are discussed at length by G10 in their Section 4.2. In the analysis, the authors noted that MLSC2k2 requires the use of priors that color variation in SNe Ia is caused by dust extinction, which can introduce additional biases into the estimated distances—there is no evidence that color variation in SNe Ia is caused purely by dust extinction, and significant evidence that intrinsic SN properties make the story more complex (e.g., Conley et al. 2007; Kasen et al. 2009; Foley & Kasen 2011; Chotard et al. 2011; Nordin et al. 2011; Maeda et al. 2011). Additionally, a Cardelli et al. (1989)-like color-variation law for the SNe is assumed, which again is not supported by the data (Guy et al. 2007, 2010).

Disentangling the effect of intrinsic color variation from color variation due to dust would require additional parameters in MLCS2k2, SALT2, or SiFTO, which currently do not exist—at present no light-curve fitter correctly disentangles the two effects. In particular, a weakness of the approach used in this paper (see Section 3) is to conflate intrinsic and dust effects into a single parameter, β, during the distance estimation. However, the tests that we are able to perform—for example, examining the evolution in this β parameter with redshift (Figure 14 in C10)—show no evidence of any significant systematic effects in the SALT2/SiFTO distance estimation method. The case of maximal evolution in β supported by our data is included in our error budget (see Section 2.5).

In conclusion, the differences reported in Kessler et al. (2009) are now understood and can be traced back to a combination of the U-band calibration and color priors—and should not, therefore, be considered as systematics. These above issues were a significant motivation for us to develop two independent light-curve fitters within SNLS that differ conceptually in the way that color is handled (Guy et al. 2007, 2010; Conley et al. 2008). Differences in the light-curves fits from these two codes are carried through as an uncertainty in our analysis—so our final quoted errors on the cosmological parameters fully include this effect.

We apply various selection cuts to the SN Ia samples designed to ensure an adequate wavelength and phase coverage in the light-curve fits. These are described in detail in G10 and C11, and essentially ensure that an accurate light-curve width, rest-frame color, and peak rest-frame B-magnitude can be measured. Additionally, each SN must be spectroscopically confirmed (see Howell et al. 2005 for a discussion of our spectroscopic classification criteria), have a minimum CMB frame redshift (z_cmb) of 0.010 (after peculiar velocity correction), be spectroscopically normal, have a Galactic extinction of E^mw_{B − V} < 0.2, and be of normal stretch (0.7 < s < 1.3) and color ($-0.25<\mathcal {C}<0.25$). We also remove six outliers on the Hubble diagram; see C11 for details.

Recent analyses have found correlations between SN Ia luminosity and their host galaxies, even after the well-known fainter–faster and fainter–redder corrections have been made (Kelly et al. 2010; Sullivan et al. 2010; Lampeitl et al. 2010). At present, it is not clear from an astrophysical perspective which host galaxy parameter is the optimal choice to use (as the third parameter) in the analysis. The observed effects could be due to progenitor metallicity (e.g., Kasen et al. 2009), progenitor age (e.g., Krueger et al. 2010), or possibly some other parameter. Empirically, trends in SN Ia luminosity are seen with a variety of derived host galaxy parameters, including stellar mass (M_stellar), star formation rate (SFR), and inferred stellar age. However, many of these parameters are strongly correlated when derived from broadband photometry available for the host galaxies.

We use the host galaxy stellar mass as the third variable in our cosmological fitting. This has the advantage of being easiest to determine from sometimes limited host data, and shows significant trends with the SN Ia luminosities. We derive the host information by following the method in Sullivan et al. (2010), fitting the broadband spectral energy distribution (SED) of the host galaxies using simple galaxy population synthesis models (Le Borgne & Rocca-Volmerange 2002). The host galaxy information can be found in G10 and C11 for all the SNe used in our analysis.

We consider a variety of systematic uncertainties in our analysis, discussed in detail in C11. Details of the construction of the covariance matrices that encode this information can be found in that paper. For each systematic, we estimate its size and adjust that variable in the light-curve fits. All the light curves are refit, including a re-training of the light-curve model where required, and the derived light-curve parameters (m_B, s, $\mathcal {C}$) compared for each SN with and without the inclusion of the systematic. These differences are converted into a covariance matrix following C11.

We use three approaches²⁵ to perform our cosmological fits. For relatively simple cosmological fits involving a small number of parameters, we use a grid technique that computes χ² of Equation (3) at every point converting into a probability via $P\propto \exp \left(-\frac{1}{2}\chi ^2\right)$, with the proportionality set by normalizing over the grid. The "nuisance parameters" α, β, and ${\mathcal M_{B}}$ are marginalized over when generating confidence contours in the parameters of interest, and we report the expectation value of the marginalized parameters. Due to the (relatively) fast run time, and the contour visualization, this fitting technique is particularly well suited to analyzing the magnitude of the individual sources of systematic uncertainty in our analysis, which would be impractical with more complex and slower fitting approaches.

The second approach is a χ² minimization routine which simply reports the best fit. The results of this technique should be close to the reported values from the grid marginalization, but should not be expected to agree exactly, and we provide both. Note that σ_int calculated by C11 is performed for the marginalization approach fits—when these σ_int are used in the χ² minimization fits, a χ²/dof < 1 should be expected as the best-fit parameters from the marginalization fits will not lie at a minimum in χ².

The third approach is the CosmoMC program (Lewis & Bridle 2002), which uses a Markov–Chain Monte Carlo technique to explore cosmological parameter space. We use this approach for our main cosmological results. We made the following modifications to the 2010 May version of the CosmoMC package to handle SNLS3 data: first, we properly marginalize over the SN nuisance parameters α and β rather than holding them fixed; second, we keep track of the difference between heliocentric and CMB frame redshifts, which enter into the luminosity distance differently, important for some of the lowest-z SNe; third, we have added the ability to fit for the host-dependence of SN Ia absolute luminosities as described in Sullivan et al. (2010).

The first item above is handled most efficiently by explicitly fitting for α and β along with the cosmological parameters, as internally marginalizing over their values is computationally more expensive for the SNLS3 sample. The consequences of incorrectly holding the nuisance variables fixed, or of simply substituting the values that minimize the χ², both true of the default CosmoMC SN Ia implementation, are discussed in Section 4.6 and Appendix C of C11, as well as Section 4 of this paper. In brief, this simplified approach leads to both underestimated uncertainties and biased parameter estimates, due to small correlations between α, β, and the cosmological parameters.

For CosmoMC fits, where we allow for a time-varying dark energy equation of state (w(a) = w₀ + w_a(1 − a), where a is the scale factor), we follow the prescription of Fang et al. (2008). Furthermore, in the CosmoMC fits we do not consider massive neutrinos and assume a simple power-law primordial power spectrum (i.e., we neglect tensor modes and any running of the scalar spectral index).

We include several external non-SN data sets in our fits. For the grid marginalization and χ² minimization fits, we use two external constraints. The first is the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7) Baryon Acoustic Oscillations (BAO) measurements of Percival et al. (2010). This is a Gaussian prior on the distance ratios r_s(z_d)/D_V(z) at z = 0.2 and z = 0.35, where r_s(z_d) is the comoving sound horizon at the baryon drag epoch, and D_V(z) is a spherically averaged effective distance measure given by D_V(z) = [(1 + z)²D²_A(z)cz/H(z)]^1/3 (Eisenstein et al. 2005), where D_A(z) is the proper angular diameter distance. The second is a prior based on the Wilkinson Microwave Anisotropy Probe 7-year (WMAP7) "shift" parameter R (Bond et al. 1997), the "acoustic scale" l_a, and the decoupling redshift z_*, as defined in Komatsu et al. (2011), following the prescription of Komatsu et al. (2009). This prior includes most of the power of the CMB data for measuring dark energy (e.g., Wang et al. 2007).

For our main cosmological fits, with the CosmoMC program, we use different external constraints: the power spectrum of luminous red galaxies (LRGs) in the SDSS DR7 (Reid et al. 2010) in place of the BAO constraints, the full WMAP7 CMB power spectrum (Larson et al. 2011) in place of the shift parameters, and a prior on H₀ from the SHOES (Supernovae and H₀ for the equation of state) program (Riess et al. 2009, 2011). This Gaussian H₀ prior, H₀ = 73.8 ± 2.4 km s⁻¹ Mpc⁻¹, makes use in its construction of many of the low-redshift z < 0.1 SNe Ia used in this paper. Their absolute magnitudes are calibrated directly using Cepheid variables in eight local SN Ia host galaxies (Riess et al. 2011), the Cepheids themselves calibrated using different techniques: the geometric maser distance to the galaxy NGC 4258, trigonometric parallax distances for Milky Way Cepheids, and eclipsing binary distances for Cepheids in the Large Magellanic Cloud. The H₀ prior was derived with SN Ia parameters from the MLCS2k2 distance estimator (Jha et al. 2007) given in Hicken et al. (2009b). In principle, for complete consistency with this work, we would use the combined SALT2/SiFTO fits for the same SN events to re-estimate H₀ in a consistent way. However, the maser distance used in Riess et al. (2011) is not published, so we defer this exercise to a future analysis. However, we note that the uncertainty in H₀ given by Riess et al. (2011) does include an allowance for the systematic of using SALT2 in place of MLCS2k2 (they quote an increase in H₀ of 1.0 km s⁻¹ Mpc⁻¹ with SALT2), so most of the systematic difference is likely already included in our error budget.

The Ω_m–w flat universe fits (Table 2) represent a 5.2% statistical measurement of w and a 7.6% measure with systematics (i.e., ≃ 5.5% with systematics only). The total systematic uncertainty is therefore comparable to, but slightly larger than, the statistical uncertainty. The full systematic uncertainty error budget can be found in Table 3. Systematic uncertainties generate about a ∼70% increase in the size of the area of the SNLS3+BAO+WMAP7 Ω_m–w 68.3% confidence contour relative to a fit considering statistical errors only (compared to an 85% increase in the SN-only contour; see C11).

The dominant systematic uncertainty is calibration (as in C11), and in particular how well known the colors and SED of the flux standard (BD 17° 4708) are—each of these two terms provides about a 20% increase in the contour area size over the statistical-only fit. The SNLS instrumental zero points and filter responses also have a large effect, generating a ∼15% increase in each case. In part, this is because the SNLS data are calibrated to the Landolt (1992) system (for comparison to the low-redshift literature SNe), for which the color terms from the SNLS filters are large (Regnault et al. 2009). This situation should improve in the near future as new low-redshift SN Ia samples observed in a filter system similar to SNLS become available, dramatically reducing these calibration uncertainties.

By contrast, systematics caused by potential evolution in SN Ia properties (the parameters α and β) are considerably smaller. As discussed in C11, we find no evidence that α varies with redshift, and only marginal evidence of redshift variation in β: explicitly dα/dz = 0.021 ± 0.07 and dβ/dz = 0.588 ± 0.40 (C11). It is unlikely that this β evolution is real (G10); however, we conservatively adopt dα/dz = 0.07 (the uncertainty in the slope) and dβ/dz = 1.0 in our systematics analysis. Even this amount of redshift evolution in α and β contributes an almost negligible effect (Table 3). The largest identified systematic uncertainty related to the astrophysics of SNe Ia is the implementation of the host-galaxy-dependent term in Equation (2).

The lower left panel of Figure 2 shows the Ω_m–w contours with all systematics included, except those related to calibration. These "no-calibration-systematics" contours are very similar to the statistical-only contours (only a factor 1.07 larger), with w = −1.048 ± ^+0.057_{− 0.058}. This represents a total error in w of ≃ 5.5%, and a systematic contribution of ≃ 1.8%, significantly smaller than when the calibration systematics are included. With our current knowledge and fitting techniques for SNe Ia, this represents the systematic floor given a negligible photometric calibration uncertainty.

We now present our main cosmological results. We consider various combinations of the SNLS3, WMAP7, SDSS DR7 LRGs, and H₀ data sets: WMAP7+SNLS3+DR7 is the most similar to the constraints used in the grid marginalization approach (Table 2), but still differs as it uses the full matter power spectrum of LRGs rather than the BAO constraint, and the full WMAP7 power spectrum rather than the shift parameters. The best-fitting value of w (Table 4) is therefore slightly different due to these differing external constraints even though the SN Ia constraints are identical, but the percentage error in w is the same at 7.6%.

4.2.1. Constant w Fits

All the results are consistent with a spatially flat, w = −1 universe. Our results for a flat universe with a constant dark energy equation of state are

$$\begin{eqnarray*} \nonumber \Omega _{m}&=&0.269\pm {0.015,}\\ \nonumber w&=&{-}1.061^{+0.069}_{-0.068}, \end{eqnarray*}$$

and, relaxing the assumption of spatial flatness,

$$\begin{eqnarray*} \nonumber \Omega _{m}&=&0.271\pm {0.015,}\\ \nonumber \Omega _{k}&=&{-}0.002\pm {0.006,}\\ \nonumber w&=&{-}1.069 ^{+0.091}_{-0.092}, \end{eqnarray*}$$

including external constraints from WMAP7 and SDSS DR7 and a prior on H₀ (all quoted uncertainties in this section include both the SN statistical and systematic components). The confidence contours are in Figures 4 and 5, and the corresponding best-fit cosmological parameters for various combinations of external constraints can be found in Table 4.

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In Table 5, we give a full list of all the best-fit parameters from the constant w CosmoMC fits with the WMAP7, SDSS DR7, and H₀ external data sets.²⁶ This includes some parameters which the SN Ia data do not directly constrain. In particular, Ω_b and Ω_c are the fractional energy densities of baryons and dark matter, τ is the reionization optical depth, n_s is the scalar spectral index, A₀₅ is the amplitude of curvature perturbations at k = 0.05 Mpc⁻¹, and σ₈ is the normalization of the matter power spectrum at 8 h⁻¹ Mpc.

Table 5. Full Set of Cosmological Parameters Obtained with the CosmoMC Fitter

Class	Parameter	Constant w	Constant w
		Flat	Non-flat
Primary	100Ωbh2	2.258+0.054− 0.054	2.265+0.056− 0.056
	Ωc	0.1149+0.0041− 0.0041	0.1145+0.0047− 0.0047
	θ	1.0398+0.0026− 0.0027	1.0401+0.0026− 0.0026
	τ	0.087+0.006− 0.007	0.088+0.007− 0.007
	Ωk	...	−0.002+0.006− 0.006
	w0	−1.061+0.069− 0.068	−1.069+0.091− 0.092
	ns	0.969+0.013− 0.013	0.970+0.014− 0.013
	log [1010A05]	3.095+0.033− 0.033	3.094+0.033− 0.033
	α	1.451+0.109− 0.109	1.454+0.112− 0.111
	β	3.265+0.111− 0.111	3.259+0.111− 0.109
Derived	ΩDE	0.731+0.015− 0.015	0.731+0.015− 0.015
	Agea	13.71+0.11− 0.11 Gyr	13.78+0.31− 0.31 Gyr
	Ωm	0.269+0.015− 0.015	0.271+0.015− 0.015
	σ8	0.850+0.038− 0.038	0.847+0.038− 0.038
	zreb	10.55+1.20− 1.19	10.55+1.20− 1.18
	H0	71.57+1.65− 1.65 km s−1 Mpc−1	71.18+1.92− 1.87 km s−1 Mpc−1

Notes. ^aThe current age of the universe. ^bThe redshift at which the reionization fraction is a half.

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Of particular note is the high importance of the SN Ia data set in placing meaningful constraints on w. Assuming a flat universe, WMAP7+DR7 alone only measure w to ∼20%, and adding the H₀ prior (i.e., WMAP7+DR7+H₀) only decreases this uncertainty to ∼11%. Including the SNLS3 data set with WMAP7+DR7, by contrast, reduces the uncertainty to 7.7%. WMAP7+SNLS3 together also provide a 7.7% measurement. With all external constraints, including the H₀ prior, the measurement of w is 6.5%, comparable to WMAP7+SNLS3 alone. Note that the DR7 constraint has almost no effect on the uncertainty in the measurement of w—WMAP7+H₀+SNLS3 has essentially the same uncertainty as WMAP7+DR7+H₀+SNLS3.

The situation is slightly different when making no assumption about spatial flatness, but the basic result of the high importance of the SN Ia data remains. In this case, the WMAP7+DR7+SNLS3 data provide an 8.8% measurement, compared to 11.3% without DR7 (the DR7 data make important contributions toward constraining Ω_m). WMAP7+DR7+H₀ alone can only make a ∼20% measurement; adding SNLS3 improves this dramatically to ∼8.5%.

4.2.2. Variable w Fits

The final set of fits allows the equation-of-state parameter w to vary simply as a function of the scale factor, a, as w(a) = w₀ + w_a(1 − a), with a cosmological constant equivalent to w₀ = −1, w_a = 0. We use a hard prior of w₀ + w_a ⩽ 0, from the constraint of matter domination in the early universe. The confidence contours are shown in Figure 6 assuming a flat universe. The best-fit parameters are listed in Table 6. Again, we find no evidence of deviations from the cosmological constant. Assuming a flat universe, we find

$$\begin{eqnarray*} \nonumber \Omega _{m}&=&0.271^{+0.015}_{-0.015},\\ \nonumber w_0&=&{-}0.905^{+0.196}_{-0.196},\\ \nonumber w_a&=&{-}0.984^{+1.094}_{-1.097}. \end{eqnarray*}$$

The H₀ prior has only a small effect on the w(a) fits (Figure 7). As a comparison, with no H₀ prior, we find w₀ = −0.949^+0.198_{− 0.201} and w_a = −0.535^+1.109_{− 1.111}. The SN Ia data are critical for a constraining measurement—without the SN data, fits for w₀ and w_a did not converge.

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Table 6. Cosmological Results Obtained with CosmoMC Assuming a Variable Dark Energy Equation of State and a Flat Universe

Parameter	WMAP7+DR7	WMAP7+DR7	WMAP7+DR7
	+SNLS3	+SNLS3+H0	+SNLS3+H0
		(Stat. Only)	(Stat+Sys)
Ωm	0.282+0.019− 0.019	0.274+0.014− 0.014	0.271+0.015− 0.015
w0	−0.949+0.198− 0.201	−0.870+0.139− 0.139	−0.905+0.196− 0.196
wa	−0.535+1.109− 1.111	−0.938+0.821− 0.827	−0.984+1.094− 1.097
H0	70.26+2.40− 2.43	71.38+1.40− 1.38	71.99+1.68− 1.69
FoMa	10.6	21.5	11.1

Note. ^aThe DETF (Albrecht et al. 2006) figure of merit (FoM), implemented here as $1/(\sigma _{w_p}\sigma _{w_a})$; see Section 4.3.

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We compare our results to previous constraints on dark energy using SN Ia data. Komatsu et al. (2011), with a combination of WMAP7, BAO, the Riess et al. (2009) H₀ measurement, plus the Hicken et al. (2009b) SN Ia data set, found w₀ = −0.93 ± 0.13 and w_a = −0.41^+0.72_{− 0.71}. Although this may appear to be better than our constraints, it did not include a proper handling of SN Ia systematics due to the lack of a consistent, published data set at that time, and the uncertainties will therefore be underestimated. Using a slightly larger SN Ia data set ("Union2"), the same WMAP7/BAO constraints, and the Riess et al. (2009) H₀ measurement, Amanullah et al. (2010) find Ω_m = 0.274^+0.016_{− 0.015}, Ω_k = −0.002 ± 0.007, and w = −1.052^+0.092_{− 0.096}, comparable to our results (Section 4.2). However, those authors left α and β fixed when computing systematic uncertainties, which may underestimate the size of the final uncertainties (C11; see also Figure 2 and Table 3), and likely underestimated the magnitude of the photometric calibration uncertainties (see C11 for discussion).

We also compare our results with those predicted by the Dark Energy Task Force (DETF; Albrecht et al. 2006) for an experiment of this type by calculating the figure of merit (FoM) for our combination of data sets. The exact definition of the FoM has some ambiguity: in the DETF report it is defined as proportional to the reciprocal of the area of the error ellipse in the w₀–w_a plane that encloses 95% of the total probability, but the constant of proportionality is never stated. The values given in Albrecht et al. (2006) are based on performing a transform of variables from w₀ to w_p, the value of w at the so-called pivot redshift z_p, where w_p and w_a are uncorrelated. The FoM is then simply taken to be $1/(\sigma _{w_p}\sigma _{w_a})$. This prescription is the most commonly used in the literature (e.g., Eisenstein et al. 2011), although some authors have been more literal in taking the area of the ellipse. FoM calculations should not assume a flat universe; however, current data are poorly constraining without this constraint, so we follow the practice in the literature and assume flatness in our FoM numbers.

For our WMAP7+DR7+SNLS3+H₀ fit in a flat universe, we find z_p ≃ 0.19 and w_p = −1.063 ± 0.082. Combined with our measurement of w_a, we find an FoM of 11.1 (see also Table 6). Excluding the SHOES H₀ prior gives an FoM of 10.6. Directly taking the reciprocal of the area of the 95% confidence intervals gives 0.56 and 0.44, respectively.

In the DETF terminology, the SNLS3 sample represents a stage II SN experiment, and the final combination of this and other stage II experiments is predicted to give an FoM of ≃ 50. However, these figures are difficult to compare with our results. The DETF calculations assume a far larger SN Ia sample of 1200 events (including, significantly, 500 at low redshift), together with a (superior) CMB prior from the Planck satellite (Planck Collaboration et al. 2011) rather than WMAP, and also include cluster and weak lensing experiments. They do not include BAO or H₀ information, although the latter only has a small effect on the FoM.

We first test for any variations in w as a function of the field in which the SNLS SN occurred. SNLS observes in four fields distributed in right ascension (see Sullivan et al. 2006a, for the field coordinates), so we can test for a combination of photometric calibration systematics, as well as a physically varying w in different directions, by comparing the cosmological parameters we derive from each field. To do this, we adjust our χ² minimization approach to fit for global nuisance parameters (α, β, and $\mathcal M_{B}$) for all SNe, but with a different w and Ω_m in each of the four different SNLS fields. The average w and Ω_m for the four fields is applied to the external SNe and for combination with WMAP7 and the BAO constraints.

This approach effectively adds six new terms to the fit. The results can be found in Table 7. The χ² only drops from 418.1 (for 466 dof) to 414.9 (for 460 dof), indicating that the fields are consistent. Comparing the individual field Ω_m and w values to the average values of the four fields, and allowing for the covariances between the individual values, gives a χ² of 3.38 for 6 dof, consistent with the χ² distribution and indicating no significant variability among the different fields.

The SN Ia light-curve shape is well known to vary systematically as a function of the SN environment. Fainter SNe Ia with faster light curves are preferentially located in older stellar populations, while the brighter examples with broad light curves tend to explode in late-type spiral or star-forming systems (e.g., Hamuy et al. 1995, 2000; Sullivan et al. 2006b). When coupled with the evidence for both a young and old component to the SN Ia progenitor population (Mannucci et al. 2005, 2006; Sullivan et al. 2006b; Brandt et al. 2010), or at least a wide range in the SN delay times (Pritchet et al. 2008; Totani et al. 2008), a natural prediction is a subtle change in the mix of SN light-curve shapes with redshift (Howell et al. 2007).

If the stretch–luminosity relation is universal across SN stretch and progenitor age, this predicted drift will not impact the determination of the cosmological parameters—a low-stretch SN should correct equally well as a high-stretch SN. To test this, we split our SN sample into two groups (s  <   1 and s  ⩾  1) and perform independent cosmological fits to each subsample. This obviously restricts the lever arm in stretch and so the α coefficient is less well determined; nonetheless, this is a useful test of the utility of SNe Ia across different environments.

When considering subsamples of SNe classified by SN properties (for example, stretch or color), the Malmquist corrections that are applied globally to the sample will not be appropriate. For example, low-stretch SNe Ia are intrinsically fainter and will suffer from a larger selection effect at high-redshift than high stretch events. Rather than applying different Malmquist corrections for these different subsamples (which could in principle be derived from simulations such as those in Perrett et al. 2010), we instead restrict the SNe to the redshift ranges over which selection effects are reduced. For the SNLS sample, we restrict to z < 0.75, and for the SDSS sample, we restrict to z < 0.3. These data also tend to be better observed, being brighter, with smaller error bars on the SN parameters. We also discard the small HST sample. We perform both statistical and statistical+systematic uncertainty fits—the comparison is useful as we are interested in the differences between SN Ia subsamples, and many of systematics affect subsamples in a similar way.

The results are given in Table 8 and Figure 8, the latter generated from the covariance matrices of the fits. The derived cosmological parameters are consistent between low- and high-s SNe Ia, although the nuisance parameters show some differences: the α values are consistent between low-s and high-s, but much larger than the full sample. The M_B values show large differences between low-s and high-s. However, there are significant (and expected) covariances between α and M_B in these fits: for the low-s group, M¹_B increases as α decreases, while for the high-s group M¹_B decreases for decreasing α (and a similar trend is seen for M²_B). As M_B is defined at s = 1, which the SNe in neither group sample, the two parameters become quite interdependent. When considering the joint confidence contours between α, M¹_B, and M²_B (Figure 8) only mild tensions are seen. A slightly larger tension is seen in the value of β for the two subgroups: the values 2.90 ± 0.16 and 3.25 ± 0.14 differ at ∼2.2σ (Table 8).

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Table 8. Fits for Ω_m and w Using SN Ia Subsamples

Sample	N	α	β	M1B	M2B	σdiffa	Ωm	w	rms
Statistical
Allb	368	1.354 ± 0.081	3.097 ± 0.094	−19.119 ± 0.020	−19.197 ± 0.016	5.1	0.281 ± 0.015	−1.013 ± 0.055	0.139
s ⩽ 1.0	174	1.736 ± 0.180	2.837 ± 0.141	−19.147 ± 0.031	−19.253 ± 0.029	4.1	0.280 ± 0.016	−1.020 ± 0.068	0.135
s > 1.0	194	1.998 ± 0.251	3.275 ± 0.136	−19.054 ± 0.034	−19.126 ± 0.029	3.3	0.284 ± 0.016	−0.994 ± 0.070	0.143
$\mathcal {C}\le 0.0$	178	1.462 ± 0.125	3.792 ± 0.370	−19.069 ± 0.036	−19.126 ± 0.028	2.0	0.284 ± 0.017	−0.990 ± 0.072	0.132
$\mathcal {C}>0.0$	190	1.267 ± 0.120	3.797 ± 0.197	−19.173 ± 0.031	−19.286 ± 0.026	4.4	0.281 ± 0.017	−1.012 ± 0.072	0.148
Mstellar ⩽ 10	134	1.337 ± 0.199	3.575 ± 0.169	−19.142 ± 0.032	...	...	0.275 ± 0.017	−1.064 ± 0.090	0.141
Mstellar > 10	234	1.319 ± 0.087	2.802 ± 0.114	...	−19.185 ± 0.016	...	0.283 ± 0.016	−0.990 ± 0.057	0.134
Statistical+Systematic
All	368	1.341 ± 0.082	3.084 ± 0.099	−19.128 ± 0.028	−19.193 ± 0.024	3.0	0.282 ± 0.018	−1.004 ± 0.084	0.138
s ⩽ 1.0	174	1.753 ± 0.185	2.895 ± 0.161	−19.142 ± 0.040	−19.260 ± 0.036	3.4	0.280 ± 0.018	−1.022 ± 0.093	0.137
s > 1.0	194	1.981 ± 0.254	3.246 ± 0.141	−19.063 ± 0.040	−19.133 ± 0.037	2.5	0.284 ± 0.019	−0.993 ± 0.098	0.143
$\mathcal {C}\le 0.0$	178	1.478 ± 0.135	3.981 ± 0.416	−19.077 ± 0.046	−19.118 ± 0.037	1.2	0.281 ± 0.020	−1.011 ± 0.109	0.135
$\mathcal {C}>0.0$	190	1.246 ± 0.123	3.826 ± 0.199	−19.191 ± 0.040	−19.297 ± 0.038	3.0	0.282 ± 0.020	−1.003 ± 0.107	0.149
Mstellar ⩽ 10	134	1.320 ± 0.202	3.481 ± 0.178	−19.158 ± 0.043	...	...	0.272 ± 0.020	−1.085 ± 0.125	0.139
Mstellar > 10	234	1.308 ± 0.089	2.854 ± 0.124	...	−19.177 ± 0.024	...	0.282 ± 0.018	−0.999 ± 0.084	0.134

Notes. ^aThe significance of the difference between M¹_B and M²_B, including covariances. ^b"All" means all SNe in the redshift-restricted sample described in Section 5.2.

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In a similar vein, we also split the sample by SN color at $\mathcal {C}=0$. Unlike stretch, SN color (examined independently of SN luminosity) shows no clear evidence of variation with environment. No robust trends have been found despite rigorous examinations (e.g., Sullivan et al. 2010; Lampeitl et al. 2010), and we also find no significant trends of the nuisance parameters varying as a function of SN color.

We also segregate the SNLS3 sample according to the environment in which the SN exploded. This is a fundamentally different test, dividing the sample according to the environment of the SN rather than its direct properties (although significant correlations exist between, for example, SN stretch and host galaxy M_stellar or star-formation rate). For these tests, we use M_stellar to segregate the sample, as used in the cosmological fits, following Sullivan et al. (2010). As we already use a different M_B for SNe Ia in low- and high-M_stellar galaxies in our standard fits, we simply adapt this approach to additionally fit for a different α and β in the two host classes. This has the advantage that we are no longer comparing the results of different fits as we fit for all the nuisance parameters simultaneously, and we can use the full systematics covariance matrix, as well as the full SN Ia sample. (We also provide the fit results when we physically divide the sample into two, for comparison with Section 5.2, in Table 8.) Note that in these fits we do not include the host galaxy systematic term, as we are trying to examine the effect of any host galaxy dependence.

The "multi nuisance parameter" fit results are shown in Table 9, which gives the values of the nuisance parameters themselves, and Table 10, which gives the effect on w and χ². The joint confidence contours for some of the nuisance parameters are shown in Figure 9. For completeness, we also give results when only one M_B is used.

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The data do not support the addition of a different α parameter in low- and high-mass galaxies—when this is added to the fits, the α values are generally consistent and the quality of the fit, as indicated by χ², is unchanged. This is true even when only one M_B is used in the fits, and suggests that α is fairly insensitive to the details of the environment and characteristics of the SN Ia progenitor stellar population.

However, there is evidence of different M_B (as already fit for) and different β. The value of β is ∼3.7 in low-mass galaxies, versus ∼2.8 in high-mass galaxies, regardless of whether two M_B are used. A similar trend is seen when physically dividing the sample into two and performing separate independent fits as in the previous section (Table 8), and is consistent with the β difference seen between low- and high-stretch SNe in Section 5.2, as low-stretch SNe are preferentially found in massive host galaxies. Generally, the two β values show only a very small positive covariance and differ at the ∼4.3σ level. There is also a substantial reduction in χ² of the fit when including two β terms. For example, fitting for two βs and two M_B reduces χ² to 405.4 from 423.1 obtained if only one β is used (for 465 and 466 dof). An F-test indicates that this additional term is required at ≃ 4.5σ.

The variation of β with host properties has been observed at lower significance by Sullivan et al. (2010) in the SNLS sample, and by Lampeitl et al. (2010) in the SDSS SN Ia sample (a larger SDSS SN Ia sample than the one used in this paper). Lampeitl et al. (2010) find β ∼ 2.5 in passive host galaxies and β ∼ 3.1 in star-forming host galaxies. Using their full sample, the significance is ∼3.5σ, although this drops to <2σ when considering only cosmologically useful events with normal stretches and colors using cuts similar to those used in this paper.

The most significant result from our analysis of the SN subsamples is the additional variation of β, as well as M_B, between low- and high-M_stellar host galaxies. This effect appears real in our data and so should be accounted for appropriately in our cosmological results. We therefore examine the systematic effect of not including this term, and compare to our existing systematic uncertainty error budget.

Compared to a statistical uncertainty-only fit, the addition of two αs and two βs, gives a ΔΩ_m = 0.00 and Δw = 0.005 (Table 10). The mean statistical-only errors on Ω_m and w are 0.0148 and 0.0545 (Table 3). Adding in quadrature the shifts measured when including the two αs and βs increases the w uncertainty to 0.0547. This total uncertainty on w is smaller than the uncertainty obtained when including host galaxy systematic term (0.0559) listed in Table 3, i.e., the effect on w of introducing two βs is smaller than our current host galaxy systematic term.

Note that this would not be the case if we had neglected all nuisance parameter variation, i.e., had only used one M_B in our cosmological fits. In this case, ΔΩ_m = 0.005 and Δw = 0.055 (the shift from considering one M_B to considering two M_B); in the case of w, this is a shift larger than our statistical uncertainty, and is comparable to our total systematic uncertainty, becoming the dominant term in the error budget. Thus, while the use of different M_B is essential for an SN Ia cosmological analysis, the use of two βs and two M_Bs is not. Note that no previous SN Ia cosmological analysis has performed this host galaxy correction, indicating that systematic uncertainties will be significantly underestimated in these studies.

A similar argument can be made using the fits including systematic uncertainties. Here, we compare fits that do include the host galaxy systematic term (unlike the numbers in Table 10), as we wish to examine whether the size of any shift in the cosmological parameters with the addition of two βs is accounted for by our existing systematic uncertainty error budget. In this case, Δw = 0.015. Our total error in w is 0.0810, compared to 0.0800, excluding the host systematic term (see Table 3). Adding 0.015 in quadrature to this 0.0800 gives a w uncertainty of 0.0813, a total uncertainty on w almost the same as that obtained when using the host systematic term. Therefore, our conclusion is that while the two β effect appears real in our data, it is adequately accounted for by our systematic uncertainty error budget.

Although the variation of β with host parameters is not a concern for this cosmological analysis, it does have implications for the physical origin of color variation in SNe Ia, which may impact future surveys. A long-standing observation is that the slope of the relation between M_B and B − V (i.e., β) is ≪4.1 (Tripp 1998; Astier et al. 2006), the value expected based on Milky Way like dust if β is interpreted as the ratio of total-to-selective extinction R_B (where R_B ≡ R_V + 1, and R_V ≃ 3.1 for the Milky Way). The effective β for SNe Ia is likely a conflation of different physical effects, including extinction by dust (which may vary with host type; e.g., Lampeitl et al. 2010; Sullivan et al. 2010, and this paper) and intrinsic variation in SN color that does not correlate with the SN light-curve shape (e.g., Folatelli et al. 2010), and which may depend on variables such as explosion asymmetry or observational viewing angle (Kasen et al. 2009; Maeda et al. 2011). Recent work has shown that SN color is also correlated with SN Ia spectral features, with SNe possessing faster ejecta velocities having redder colors at fixed M_B (or equivalently brighter M_B at fixed color) in samples with very red SNe excluded (Foley & Kasen 2011).

Under the assumption that any intrinsic SN Ia color–luminosity relation has a smaller effective β than that from dust (as seems likely given we observe β < 4.1), our results are qualitatively consistent with a scenario in which dust extinction modifies this intrinsic color–luminosity relation. The lowest-M_stellar host galaxies are those with the largest specific SFRs, and therefore the largest dust content. We would therefore expect to find SNe Ia with a larger effective β (i.e., closer to the true dust value) in lower M_stellar hosts, which is consistent with our observations. In more massive, passive host galaxies, we are likely observing a β closer to the intrinsic value.